摘要:

Farday D~SCUSS.,1996, 103, 1-18 Overview Lecture Hydration processes in biological and macromolecular systems John L. Finney Department of Physics and Astronomy, University College London, Gower Street, London, UK WCIE 6BT I discuss several themes arising from the papers presented at this Faraday Discussion that are, in my view, particularly interesting and/or important. With respect to model systems, recent progress in understanding structural aspects of the hydrophobic interaction, and of through-solvent interactions in general, is highlighted, together with the need to continue to develop more sophisticated (theoretical and computational) interpretational tech- niques if we are to exploit to the full the power of present-day experimental techniques. The current state of our knowledge of hydration effects on the structure and dynamics of biomolecules is discussed, and the importance of being able to see how molecular-level structural effects control behaviour at the important mesoscopic level is underlined.Issues relating to recent progress in characterising solvent effects in more complex systems and pro- cesses, including those of industrial interest, are raised, and the necessity of using a range of appropriate experimental techniques when tackling such complex problems is stressed. Progress since the 1975 Royal Society Dis- cussion is highlighted, and interesting issues ripe for fruitful discussion at this meeting are raised. A strong case can now be made that our under- standing of both the structure and dynamics of water as a function of its local environent is now sufficiently good to enable us to use water as a probe of complex system behaviour, rather than, as heretofore, an object of study in itself. Introduction Many of the papers to be discussed at this meeting begin by making similar points.‘Water has been recognised . . . as one of the major structuring factors of biomolecules’.’ ‘As the natural solvent of biological macromolecules, water influences many aspects of biological function’.’ ‘The water-protein interaction has long been recognised as a major determinant of chain folding, conformational stability, internal dynamics, and binding specificity of globular protein^'.^ Although the majority of the papers to be discussed focus on biological macromolecules, similar comments hold for non-biological ones: that we need a better understanding of the effect of hydration on structural and dynamical properties in order to control industrially relevant processes on such systems better is evident from the papers that examine the effects of water on the dynamical properties of cellulose acetate4 and on the gelatinisation of ~tarch.~ Yet it is also clear from these papers that our understanding of solvent effects at the molecular level remains imperfect.With respect to protein stability, for example, we need a much better understanding if we are to input into, for example, protein engineer- ing: as the free energy of stability of a typical protein is only some 10 to 20 kcal mol-’, equivalent to a handful of hydrogen bonds, it takes only a small shift in the free energy 1 2 Overview Lecture balance to make our engineered protein unstable (or too stable) compared with a closely similar molecule under the same solvent conditions.We do not even have an adequate understanding of solubility of small molecules, let alone biological or non-biological macromolecules. This poor knowledge relates to our imperfect understanding of the solvent interactions involved. Statements like these have been made for decades, and many meetings have taken place to assess the state of contemporary knowledge, and to try to move it forward. A Faraday Discussion is an excellent forum in which to explore ideas on topics of such intrinsic importance and fascinating complexity.I hope we can contribute positively in both respects at this discussion. 2 A rational strategy? The overall approach to trying to understand hydration processes over the past few decades might be summarised by asking the following three questions: First, in a given system, where is the water? What is the structure of the hydration water, and of that further away from the macromolecule interface? Can we characterise the detailed hydration geometry at the interface? Secondly, how does the water move? What is its dynamics? Over what range of timescales can we characterise the dynamics, and what timescales (picoseconds to hours?) are relevant to understanding a particular process or function? How does the water dynamics couple to that of the macro-molecule? Does one control the other, and if so, which? Thirdly, once we have some understanding of these structural and dynamic ques- tions, how can this understanding explain the behaviour of a particular system, not only at the molecular (e.g.protein-substrate binding) and macroscopic (e.g. flow properties) levels, but also on the increasingly important mesoscopic scale (e.g. the operation of membranes)? And, increasingly importantly, how can we exploit this understanding to modulate interactions and modify operation in a controlled manner? In trying to follow this kind of strategy, we need to recognise the complexity of macromolecule-water systems, a complexity which relates to the multicomponent nature of typical systems of interest, and to the wide ranges of both spatial and temporal scales involved.We need also to devise tactics to tackle the problems raised by this complexity. Here, I would pick out three strands. First, understanding the behaviour of sensibly chosen model systems can inform on important aspects of real systems. There are good examples at this Discussion in the papers on interactions in solution, in particular of the non-polar molecules that are used to try to improve our understanding of the hydropho- bic interaction. Secondly, it is no longer realistic to tackle a complex system using one experimental technique, and then to build a model to explain only that single set of results. It is this kind of single-technique approach which has often led us astray in the past.It is no use for us to construct a model to explain our state-of-the-art NMR data if this model is inconsistent with what we have learned from e.g. neutron scattering. We need to use all the different experimental techniques we can usefully find, in order to try to obtain complementary information (on consistent temporal and spatial scales), an approach typified by ref. 4. Thirdly, as we employ more powerful and more sophisti- cated experimental techniques on our samples, we need increasingly sophisticated inter- pretational techniques in order to try to understand the results of our experiments. It is here that computer simulation has made a substantial mark over the past twenty years, enabling us to move away from the oversimplistic models that we were previously more- or-less enforced to employ because of our inability to deal with this degree of complexity in a 'straight' theoretical manner.With these points in mind, the task I have set myself in this lecture is to focus on some themes from the papers to be discussed that I think are particularly interesting and/or important. I want to set some of these in context, and pick out some questions, J. L. Finney 3 many of which are raised in the papers themselves, that we might usefully address at this discussion. In Section 3, I start by considering some of the work presented on model systems, emphasising (a) our understanding, or lack of understanding, of the hydropho- bic interaction, and (b) the need for increasingly sophisticated interpretational tech- niques to enable us to understand the results we are now able for the first time to get as a result of advances in experimental techniques.Sections 4 and 5 look at biomolecules specifically, in terms of structure and dynamics, respectively, underlining among other things examples where we can now begin to see how molecular-level structural effects can be seen to control behaviour at the mesoscopic level, We might also ask if our understanding of the structure and/or dynamics of water in macromolecular systems is good enough for us to be able to use water as an experimental probe, rather than as something to be studied to improve our understanding of the behaviour of water itself in a particular environment.In Section 6, I make some brief comments on more complex systems and processes. Finally, in Section 7, I summarise where I believe we have made particular progress in the past 20 years or so, and draw together some of the issues that I think, on the basis of the papers that are to be presented here, we can usefully address at this discussion. 3 Model systems 3.1 Aqueous solutions Interactions in solution are central not only to biomolecule stability and interactions, but also to much of chemistry and chemical engineering. We can thus probe the proper- ties of appropriate model solution systems to help us understand the through-solvent interactions of important chemical groups, and the relation of the effective potential of mean force to hydration of the chemical groups themselves.To understand the struc- tures of solutions, we ideally would like to know three things. First, how are the solute molecules, or important chemical groups on those molecules, hydrated (the solute- solvent structural correlations), and how does this hydration change as conditions such as temperature, pressure, pH, ionic strength etc. of the solvent are varied. Secondly, how is the structure of the hydration water perturbed from that in the bulk solvent (the solvent-solvent correlations). Thirdly, what is the structure from the solute molecule’s viewpoint :what are the solute-solute correlations. Building on the pioneering work of Enderby and his group in exploiting neutron isotope substitution difference methods,6 and exploiting recent advances in neutron sources and instrumentation, we can now begin to access this information directly.By using selective H-D substitution, for example on methyl groups, we can go even further, and obtain crucially important information on the orientational pair correlation func- tions. We have never been able to get this information before. Thus, recent work on the tetramethylammonium ion shows us that this group hydrates as a non-polar molecule, with the hydration structure relating to a disordered clathrate-like cage. Furthermore, the disorder in the hydration structure can be quantified, as shown in Fig. 1. In the same system, the solvent structure in the hydration shell can be measured uia the water hydrogen-hydrogen correlation function, and the solute-solute correlation function also determined. Thus, we can access all three structurally-important pair distribution func- tions.Its essentially spherical symmetry makes the TMA ion a relatively simple case to deal with, by using recent spherical harmonic expansion developments pioneered for this kind of work by Soper.8 For the tert-butyl alcohol system, the molecular symmetry is sufficiently high that we can still obtain useful information by careful analysis in this way. For example, we have been able to determine for the first time the dominant rela- tive orientations of two contacting tert-butyl alcohol molecule^,^ confirming that the Overview Lecture Fig.1 Probability density of the water molecule around a TMA ion in a 2.0 M aqueous solution of TMAC1.' The regions of maximum intensity (shaded dark) occur at orientations of the water molecule that correspond to the HOH plane and the HOH bisector lying tangential to the surface of the TMA ion. intermolecular approaches are predominantly through methyl head-group contacts, as would be expected from the classical ideas of the hydrophobic interaction. We are thus currently in the position of being able to measure relatively complex solution structures at a level of detail that has been previously out of reach. However, there are increasing difficulties as we move to solute molecules of lower symmetry, and our ability to interpret the data needs further development.The early neutron work on aqueous methanol, which was the first measurement to confirm the disordered cage nature of the hydration of non-polar groups," resulted from an experiment performed in early 1989. It was several years before Soper' developed the analysis techniques that enabled us to interpret the results in a useful way. As we are now moving to work with more complex molecules, these methods themselves become less useful, and other inter- pretive approaches become necessary. One particularly promising way forward is dis- cussed in Soper's paper at this meeting," in which he builds on the Reverse Monte Carlo method12 to generate structures consistent with the experimental data via his Empirical Potential Monte Carlo simulation technique.Such developments in analysis procedures, allied to increasingly imaginative experiments on well chosen model solu- tion systems, will have a major effect on our understanding of solution structure. The influence of these procedures on understanding liquid structures could be, in its way, as fundamental as that of Rietveld methods for refining structures from crystalline powder data.13 We now have the experimental and data analysis techniques with which to begin to open up the field of 'liquid state crystallography'. J. L. Finney 5 3.2 Hydrophobic interaction This is the name given to the entropic driving force that tends to bring together two non-polar groups in aqueous solution.It is significantly temperature sensitive, and can be modulated by changing the properties of the solvent, for example by adding ions. Studies over many years have shown how this modulation depends on the type of ion, the relative effectiveness in enhancing the interaction being ordered in the lyotropic, or Hofmeister series.14 The conventional wisdom of the hydrophobic interaction states that the entropic driving force derives from some kind of ‘ordering’ of the water molecules in close contact with a non-polar group. When two such groups are brought together, part of this ‘ordered’ hydration shell is expelled to the bulk, with a consequent loss in entropy to the systern.l5 Since the seminal paper of Kauzmann in 1959,16 there has been a general acceptance that the hydrophobic interaction is the dominant driving force in processes such as protein folding, enzyme-substrate and protein-protein interactions.We might note, however, that this generally accepted view has never really been established experimen- tally. Over the past 15 years or so, queries concerning its validity have been raised, drawing on both good thermodynamic evidence17 and calculations based on our know- ledge of protein surface structure.18 These doubts have intensified over the past five years or so, with several worker^'^*^* subjecting this conventional wisdom to critical examination. For the present, however, this concept is so almost universally invoked in structural molecular biology that the adjective ‘hydrophobic’ has been transferred in general usage from the interaction itself to the non-polar groups.Thus there is talk of ‘hydrophobic groups’, implying that the groups themselves avoid water. This is not true, as Haymet’s paper stresses.2’ Perhaps this meeting could take the opportunity of emphasising the fact that ‘hydrophobic’ is a property of a two-(or more) component system, not a pro- perty of a single kind of molecule. To call a molecule or chemical group ‘hydrophobic’ is terminology which is not only incorrect (as Haymet’s paper makes clear, there is an attraction between a non-polar group and water) but, also misleading, signifying incor- rect physics which does not help our understanding of this subtle, but crucially impor- tant, effect.It would be far preferable to call the moiety concerned non-polar or apolar. Returning to the molecular-level origin of the hydrophobic interaction itself, this is still unclear. As recent neutron scattering measurements have shown (Fig. 2), there is no evidence that the structure of the hydration water close to a non-polar group is more ordered than that of water in the bulk. Again, there is increasing evidence from other directions that this conventional explanation of the entropic driving force is wanting, as pointed out in a recent excellent review.22 The nature of the entropic driving force is addressed in the papers of Haymet21 and Pa~laitis.~~ I hope this meeting will be able to tackle this question, and help us move on from the conventional wisdom that appears to be wanting.Although hydrophobic interaction studies tend to concentrate on the hydration of the non-polar moiety, it is the end result, the solute-solute interaction, that is the impor- tant consequence. Its strong temperature dependence is a signature of the effect, and the work reported by Skipper24 successfully reproduces the expected increase in solute- solute interaction with rising temperature, followed by a subsequent decrease with a further rise in temperature. These simulations contain a wealth of data that I hope will be mined further for molecular-level information that could well throw light on the solvent-related processes that are controlling the interaction. We might also note that these simulations are in at least qualitative agreement with very recent small angle neutron scattering measurements on tetrahydrofuran solutions, in which increased self- association of the molecules is found with increasing temperature up to about 60T, further temperature rise resulting in a decrease (see Fig.3).25 Ouerview Lecture 2 1.5 h & vI' cn 0.5 0 0 2 4 6 8 r/A Fig. 2 The hydrogen-hydrogen pair correlation function for a 1 :9 mole ratio methanol-water mixture (line) compared with the same function for pure water (circles)." The second and third peaks relate to intermolecular distances, and, within the errors of the data, show no significant structural differences. 0.23 mole fraction tetrahydrofuran in D20 30 25 n 20 .-u)CI c32-15> c_-v)ca, c .-t lo 5 0 0 0.05 0.1 0.15 0.2 0.25 Q /Ad' Fig.3 Small angle neutron scattering data taken on 0.23 mol fraction solutions of tetra-hydrofuran in D20,as a function of temperature. The increase in the small angle scattering signal as the temperature is increased to 60 "C indicates increasing aggregation of THF molecules. As the temperature is raised further to 70 "C, the process reverses.25 J. L. Finney 4 Water in biomolecules: structure 4.1 Water structure at the biomolecule interface I noted with interest that there is no crystallographic paper at this meeting to tell us the present state of knowledge of the structure of water at, for example, a protein surface.We can contrast this with the situation 21 years ago, when a Royal Society Discussion on Water Structure and Transport in Biology was organised by Rex Richards and Felix Franks.26 At that time, we were just beginning to have enough information from reason- able resolution protein crystallographic studies to begin to characterise the structure of the hydrating water. Since then, techniques have improved, and a large number of systems have been studied at higher resolution; consequently, our structural knowledge of the water-biomolecule interface has improved very significantly. One of the major advances in our structural knowledge of biomolecule hydration has come from neutron crystallographic studies. Unlike X-rays, neutrons can see hydro- gen (or preferably, for technical reasons, deuterium) atoms clearly.Whereas assignment of hydrogen bonding from X-ray studies required us to make assumptions concerning hydrogen positions, (sometimes, in hindsight, found to be incorrect 27), the specific loca- tion of hydrogens enabled us to at last understand the important orientational structure of the water molecules at the interface.28 Furthermore, data of sufficiently high quality has become available on a limited number of systems to enable us to unravel the struc- tural disorder that is usually present at such interfaces. An example is given in Fig. 4, which shows the neutron-determined structure of the water in vitamin B,, coenzyme,29 the data being taken at 15 K to reduce the thermal disorder. The dashed contours indicate hydrogen atoms, thus clearly showing the ordered water molecules in the ‘pocket’ region.In the ‘channel’ region, the data are good enough to allow us to pick out two separate water networks, as indicated by the solid and dashed lines linking neighbouring water molecules in this region. We might note that the conclusions from infra-red measurements in Marechal’s paper3’ are consistent with the structural information we have built up by this kind of study. The crystallography, however, does not require any interpretive models, and can give quantitative data on intermolecular distances and angles. Using these geometrical data, we have been able to see for the first time the structural regularities that appear to control the structures of hydrogen bonding networks.28 These regularities, formulated in terms of anisotropic repulsions, have given us a new understanding of hydration, and new tools with which to examine hydration structure^.^^ It can be argued that we now understand water interaction geometries sufficiently well to use this structural know- ledge to advance further our understanding of hydrated systems.We can now see why simple distance criteria for hydrogen bonding can be misleading if we do not know the hydrogen positions, and we can set tighter, and more justifiable, hydrogen bond distance and angle constraints for use in protein structure refinement. Water tetrahedrality can be explained without recourse to the strong, orientation-dependent attractions that are often enshrined in potential function models used in simulations, and the dominance of trigonal geometry for short hydrogen bonds that is observed in high-resolution hydrate structures can be explained.Finally, these structural regularities explain rather beauti- fully the complex structure of the ice phase diagram.31 Thus, from studies of the hydration of complicated molecules, we have learned new and interesting things about water-water, and hydrogen bonding interactions in general, that we can now feed back into more basic hydration studies. These results bear on some of the water potential function issues raised in Robinson’s paper.32 I would like to set some of these in context. For example, some of us have been dissatisfied with pair additive models for many years.To my knowledge, the earliest attempts were made independently in the early 1970s by the teams at Birkbeck and Groningen,j3-js both of whom made significant progress, which some more recent similar approaches take Overview Lecture Fig. 4 Water networks in coenzyme B,, crystals at 15 K, determined from single crystal neutron scattering measurements. Large open circles: 0;small open circles: H. Dashed contour lines are negative neutron-density levels (as expected for the negatively scattering proton), while solid contour lines are positive neutron-density levels. Solid and dashed lines between the solvent mol-ecules represent a1 ternative water networks. f~rther.~~-~~However, I felt in the early 1980s that, although polarisability would need to be taken into account eventually, there were other more basic problems with the standard potentials being used.38 As the work deriving from the B,, studies just dis-cussed showed, one of these problems turned out to be the isotropic repulsions generally used, despite the fact that high quality quantum mechanical calculations had shown an anisotropy consistent with the experimental results.39 It is perhaps no accident that those models that depart from spherical symmetry by placing a repulsive centre on the hydrogen (e.g.MCY4') tend to perform better. Whatever potential function we use for water, however, I believe it should be based on reasonable physics, rather than one which uses a non-physical device designed to fit a particular characteristic of one set of structural data, as I believe the approach set out in ref.32 does. The approach set out there is also strongly reminiscent of the old mixture models that were popular before the 1970s. These could fit the experimental data that they were designed to fit, but offered little in the way of further physical insight. 4.2 C-H 'hydrogen bonds' The idea of C-H hydrogen bonds has been around a long time, crystallographic evi- dence for them having been published by Sutor in 1963,41 with a whole book being J. L. Finney published in 1974.42 When protein structures began to be examined in detail in the 1970s, in order to try to begin to understand general structural principles of proteins, it was noted that for ribonuclease-S, lysozyme, and bovine pancreatic trypsin inhibitor, internal interactions within the proteins were dominated, not by apolar-apolar group contacts as might be expected from the conventional wisdom of the dominance of the hydrophobic driving force (see above discussion in Section 3.2), nor even by polar-polar contacts, but by polar-apolar ones.18 As these conclusions were based on the assign- ment of interaction areas between the various groups, and noting that it would be an oversimplification to relate contact area to energy, we should tread carefully when trying to draw strong inferences from these data, but the high relative frequency of such con- tacts is suggestive, and should be noted.Looking in more detail at the distances between non-polar carbon atoms and water oxygen (and other polar) atoms, however, clear evidence was found for contact distances significantly less than the van der Waals radius sum that we would expect to find for a normal non-hydrogen bonding interaction.18 At that time, one could perhaps query if the accuracy of the coordinate data was sufficient to allow such conclusions to be drawn. However, more recent work, using neutron scattering at essentially atomic resolution, has given further strong evidence for short polar-non-polar distances. Some examples are shown for the coenzyme B,, structure in Fig. 4. Here, the dis- tance between the oxygen of water 418 to the carbon of the nearby methyl group is about 3.2 8, very much less than the van der Waals sum. The C-H...O angle is, however, far from straight at ca.120”. So we should perhaps beware of calling this interaction a hydrogen bond. However, other C-H groups do point more directly to the lone pair regions of water molecules (for example waters 213, 223, and 222), and these also show short C..-O contacts. There is thus ample, quantitative evidence for polar-non-polar interactions that are stronger than a simple van der Waals interaction. The simulation work reported in Westhof’s paper’ tries to quantify this kind of interaction. Although I believe the criteria they use for identifying C-H hydrogen bonds is too broad, and will include even those pairs of groups whose contact distances are the same as the van der Waals sum, there are clearly a number of short contacts that do imply strengthened interactions.Three points might be made. First, the strongest evidence for short contacts is to those hydrogen atoms that have been assigned relatively high partial charges, the charges of >0.2 electrons assigned to the H(5) hydrogens not surprisingly results in shortened distances. These charges may well be realistic, in that they are assigned to hydrogens that are attached to conjugated ring carbons, for which significant electron withdrawal from the proton is expected. Secondly, if we are to call these interactions hydrogen bonds, then the water oxygen atoms must be significantly closer to the ‘non-polar’ group than the water hydrogen.Looking at Fig. 3 of ref. 1, we see this is only clearly the case for the H(5) proton already mentioned. It might therefore be misleading to claim the other cases as hydrogen bonds. Thirdly, as the examples given in the stereo pairs of Fig. 5 to 7 of ref. 1 show, short polar- non-polar contacts to water molecules tend to occur in situations where other good hydrogen bonds are made. It is as if the water molecule’s .environment is such that groups to which good hydrogen bonds might be made by the remaining ‘hydrogen-bonding sites’ of the water molecule are not available. In that circumstance, in order to minimise the energetic penalty of a ‘missing’ hydrogen bond, the polar group optimises its local geometry to regain as much as it can of the lost energy by a closer approach to the non-polar group.These configurations obtained in this simulation do indeed fit the patterns observed experimentally. Attempts made some 15 years ago to estimate the possible energetic contribution of short polar- non-polar interactions concluded that they might ‘recover’ some 5 to 10% of the energy of the ‘lost’ hydrogen bond. In a large molecule, although these contributions individually may be small (of the order of room temperature kT),the total energetic contribution over a large number of such interactions may not be insig- 10 Overview Lecture nificant with respect to the free energy of folding or interaction of a macromolecule. I believe that it would, however, be a mistake to exaggerate their importance. There is a good case for considering all these interactions as ‘opportunistic’, with the system com- pensating as best it can for the imperfect hydrogen bonding forced on it by an insufi- cient number of polar groups available for hydrogen bonding.4.3 How good must water predictions be? Both the papers by Hummer2 and Pa~laitis~~ make the very valid point that enormous amounts of computer time are needed to perform adequate simulation studies of bio- molecule hydration. There are also those who would regard the results of such simula- tions with caution, bearing in mind the simplifications that must be made in potential functions if we are to obtain results in a finite time. Hummer and Paulaitis thus present a theoretical approach, based on the potential of mean force, that aims to give informa- tion on the water density as a function of position, and show results on small molecules that relate (ref.23) to the hydrophobic interaction (see section 3.2 above) and (ref. 2) to the hydration of a range of protein and nucleic acid systems. While not feeling competent to pronounce on technical aspects of this approach, the work is interesting and seems to me to be potentially promising. As in the early days of computer simulations, it does raise an issue which I think is worth discussing, namely, how good need a prediction of hydration be to be useful? This raises perhaps two associated points. First, how do we perform a quantitative comparison between the predictions and experiment, especially when the system is large and complicated? In the early days of computer simulation calculations of water posi- tions, rms deviations of simulated water positions from experimentally-determined loca- tions were made, but how small a value of this quantity should we aim for? For example, an rms deviation of 1.5 8, may sound impressive, but when we consider a typical hydrogen-bond length of 3 A, is this any better than random? Moreover, is it sensible to only use a statistical measure over all water molecules, when the water- involved process in which we are interested may depend on only one or a handful of molecules which may not be well predicted? Secondly, how well do we need to repro- duce coordination numbers? The comment is made in ref.2 that the method is best able to identify local regions of high water density, but that the actual density values are less reliable. Yet, when we recall that it is a delicate balance of forces that holds together an enzyme in its native conformation, is this degree of prediction good enough? These are questions to which I see no easy answer, but they are ones which we could usefully raise at this discussion. 4.4 Membranes: From the molecular to the mesoscopic scale I would like to conclude this discussion of issues related to biomolecule structure by focussing on some of the results presented in be rend sen'^^^ simulation of several membrane-water systems. In Section 2, I tried to stress that one major purpose of our understanding of the detailed molecular level nature of the water-biomolecule interface was to use this to understand processes at the meso- and macroscopic scales.I think it is fair to say that we have so far very few examples we can quote where our molecular- level understanding has illuminated what happens at a higher level. Ref. 43 is one of these, in which simulations at the molecular level have enabled us to see the connection to mesoscopic-scale behaviour that is important in the operation of these systems. Four points in particular emerge. First, two ordering principles are elucidated that control the water organisation, namely compensation of local charge density, and opti- misation of water geometry. These are principles which we have suspected for some time J.L. Finney 11 are important in protein structure, and it is comforting to have their relevance confirmed in this case. The authors conclude that the sign of the electrostatic potential derives from the delicate balance of these two effects. Thus, we have a reasonable explanation of how the electrostatic potential results from subtle structural effects at the molecular level. Secondly, the relatively complex non-polar lipid-water interface is found to be similar to a free water interface. Similar situations in which the properties of a relatively complex water interface are found to be similar to those of a simpler system, or to behave in a way that is consistent with general models or theories, crop up in several of the papers in different contexts.For example, the water dynamics close to a protein is argued44 to be consistent with a general model that is more usually applied to glass transitions in supercooled liquids. We should therefore keep our minds open to the possibility that the water-related effect we are trying to understand may be explained in terms of apparently simpler ideas than those we might originally have felt we needed to invoke to under- stand a particular situation. Thirdly, the membrane simulations show how performing a dynamical simulation smears out the oscillatory behaviour that has been seen in earlier static simulations, and on which conclusions had been drawn about the nature of the interface. We use static models at our peril.Fourthly, the simulations suggest that the interpretation of the hydration force is complex, deriving from the interplay between the water geometry and surface protrusions. Thus, there seems not to be a simple explana- tion, and the deviation from molecular-level ‘flatness’ of the membrane surface appears to be an important factor. We might note in passing the early neutron reflectometry results that were interpreted in terms of a (molecular-level) rough surface that involved protrusions of the polar head-group into the aqueous phase, and significant penetration of water into the lipid region.45 To ignore these ‘imperfections’ when trying to under- stand the important physical properties and behaviour of membrane systems, e.g. in constructing theories or in performing static simulations, would appear to be a serious mistake, neglecting aspects of the system that are crucial to the physical behaviour we are trying to understand.5 Water in biomolecules: Dynamics There have been major advances since the 1975 Royal Society Discussion in our under- standing of the dynamics of water close to a macromolecule surface. At that time, the model which the majority of the community accepted specified two fractions of water in the hydration region. A first fraction was considered to be anisotropically tumbling, with a correlation time z x 1 ns, while a second fraction with a much longer z w 1-10 ps, was considered to be a ‘dynamically-oriented bound state’.46 As we now know, this picture was wrong.Much controversy has followed in the past 20 years, and the picture has been clarified, in significant part through the NMR disper-sion measurements of the kind discussed here by Halle,43 and the multidimensional nuclear Overhauser enhancement techniques developed by Wuthrich’s school that allow individual water molecules to be observed directly uia their double couplings with protein proton^.^' As we began to exploit deuterium and “0 relaxation measurements, it became clear that the interpretive models that had been used to derive the accepted picture led to strong, and irreconcilable, incon~istencies.~*-~ As Halle states in ref. 3, the ‘field was plagued by interpretational controversies’. Hence, new interpretive models had to be developed, resulting, with more recent high quality data as instrumentation has improved, in the present model as set out in ref.3. I would here stress the problems of using interpretive models to explain the results from a particular experimental technique. I have already tried to underline the need to use as many relevant techniques as possible in trying to unravel hydration problems. Where an interpretive model is used, it must be consistent with results obtained from Overview Lecture complementary techniques; if it is at clear variance with such, then the model has prob- lems. If it is not consistent with known results from other experiments, conclusions we draw by using the model to try to understand other data must be suspect.We should be aware of the dependence of our conclusions from an experiment on any interpretive models used, and try to improve these to make our conclusions less model dependent. A case in point here is the neutron inelastic scattering results discussed by Doster in ref. 52, where he specifically tries to move away from model-dependent interpretations of his data. The less model-dependent our conclusions are, the more faith we can have in them. If we are forced to use such models, we must be clear what their limitations are, and that our conclusions, being model-dependent, may well not be unique, or correct. Where the complexity of a problem forces us to use interpretive models, we have to use them, at least until we can find a better way. We should choose a good model, consistent with what we know from other considerations.But we should not fall in love with it. In Halle’s NMR relaxation paper,3 he shows that the dynamics of water close to large proteins behaves as expected from work on smaller proteins, and argues convinc- ingly that cavities that appear empty by crystallography are likely to contain water molecules. He further makes the intetesting comment that the hydrogen bonding struc- ture is not the prime determinant of the water ‘residence time’, and that it is necessary to go beyond simple hydrogen bond counting if we are to understand the mechanistic basis of water dynamics. He also comments that the key to the fast dynamics in bulk water lies in a co-operative mechanism.This is in interesting contrast with the conclu- sions of Luzar (ref. 53)on water dynamics close to small molecules, where a co-operative mechanism is not invoked. The other NMR paper from Wuthri~h~~ exploits the high resolution techniques developed by the ETH group to focus on the rate processes gov- erning water exchange between hydration sites on the macromolecule and the bulk solvent. A novel NMR experiment is described, together with the complementary use of long-time molecular dynamics simulations to refine the analysis of NMR data on hydra- tion water. This seems to me to be an excellent use of the simulation technique to dig a little deeper into what may be happening. In the discussion of water structure in Section 4.1 above, it was suggested that our knowledge of water interaction geometries might now be suficiently good to enable us to use this knowledge in advancing further our understanding of hydrated interfaces, for example in refining crystallographic data on protein structures.Halle makes a related point with respect to water dynamics. In the past, much work on hydration has concen- trated on trying to understand the structure and dynamics of water in such complex systems as macromolecules. A case can now be made that our understanding on both the structural and dynamical fronts is sufficiently good that we can now use the water mol- ecule as a probe through which to improve our understanding of the macromolecular system. This is a strong claim. I think it has considerable validity.Water dynamics is also examined in two inelastic neutron scattering This is a technique which has shown considerable growth in the past ten years, exploit- ing improvements in neutron sources and instrumentation. The two papers are an inter- esting complementary pair, in that they both aim to extract the overall water dynamics of (different) hydrated proteins by removing the contribution from the protein, but the two approaches remove the protein in different ways. Belli~sent-Funel~~ takes advantage of our increasing ability to make fully deuteriated proteins: as the inelastic spectrum as measured by neutrons is dominated by hydrogen, the measured signal from a deuteriat- ed protein is largely that of the (hydrogenated) water, together with that of the hydrogen atoms that have exchanged with the protein.In contrast, Doster5’ performs two experi- ments at the same hydration level, one with H,O, the second with D,O. Subtracting the two results should again give the dynamics of the hydration water plus that of the exchangeable hydrogens on the protein. It is a tribute to the experimental work in both papers that similar quantitative J. L. Finney results are obtained. Interpretation, however, takes different directions, with Doster trying to avoid problems of drawing model-dependent conclusions by working up a model-independent approach. One interesting common conclusion of both papers is that the water dynamics can be described by reference to mode-coupling theory.In Bellissent-Funel’s it is argued that the water dynamics in the protein powder has similarities to the water dynamics at a porous Vycor surface. This implies that the detailed nature of the local constraints on water molecules may not be important in determining the dynamics, another example of similarity between a property of a complex hydrated interface and that of a simpler interface. A related point is made by Simon~on,’~the results of their polar fluctuation simulation suggesting that deviations from continuum theory are not large when ionisable side-chains are not charged. They do, however, comment that continuum theory may be inadequate when charges are considered, suggesting that we may need to go beyond a continuum approach when we consider the activity of proteins under realistic solvent conditions. The final paper on dynamicss5 uses dielectric relaxation to try to understand the motion of both ions and water close to DNA, and draws some interesting conclusions especially with respect to the ion motion.This work also shows a strong dependence of the measured dielectric loss as a function of hydration (Fig. 1 of ref. 55) which raises the question of how valid are studies of hydrated powders in the study of water dynamics in proteins, and indeed of the dynamics of the protein itself. We should also note here the impressive simulations of the polar fluctuations of cytochrome c presented in ref. 54 by Simonson. This latter paper points out that the dynamics of dry powders bears little resemblance to proteins in solution.It also shows the necessity to consider the charged side-chains at the surface if we are to obtain a reasonable representation of the dynamics of the system. We might recall here the work done in the 1980s which showed that enzymes became active at relatively low hydration levels. For example, the work of Rupley et alS6on lysozyme showed that enzyme activity commenced in a hydrated powder at ca. 0.2 wt.% water. Other works7-s9 had suggested that by this hydration level, enough water was available for the ionisable groups to reach their equilibrium charged states, and also that a dynamic transition had occurred which had increased the flexibility of the protein to enable its activity to commence. Simonson’s polar fluctuation simulations may throw more light on our understanding of these earlier results.On the basis of results such as these, it has become generally accepted that we can usefully study protein and solvent dynamics at relatively low hydration levels, and so gain information that is relevant to understanding the role of dynamics in activity and other processes. I suspect we are beginning to uncover evidence that the situation may not be so simple. First, there are comments in Bellissent-Funel’s paper44 implying the water dynamics may be, up to a point, controlling the dynamics of the protein itself; if this is the case, then adding more water in progressing to solution might be expected to ‘loosen up’ the dynamics very much more. Secondly, very recent neutron measurements on both hydrated protein powders and solutions6* showed a strong dependence of the measured rms displacement of the protein on the solvent conditions, in experiments probing timescales of about tens of picoseconds.Results on hydrated powders confirmed the earlier results obtained by Doster, Cusack and Petry6’ that there is a transition from essentially harmonic to anharmonic behaviour at ca. 200 K. However, similar measure- ments on solutions resulted in this transition temperature moving upwards to the melting temperature of the solvent. Although these were preliminary measurements, and were restricted to probing dynamics over a relatively short timescale (ca. 10-“-10-lo s)which may be shorter than those relevant to e.g.enzyme activity (turnover numbers for which go up to only lo7 s-’), one implication is that the protein is strongly coupled to the solvent, and that perhaps the solvent may largely control the dynamics of the protein at these timescales. We might note that Bellissent-Funel’s paper44 suggests that 14 Overuiew Lecture the slow dynamics of the hydration water in her powder sample could be the cause of the retardation of the protein itself, and possibly the source of the observed anharmoni- city. The implications of the solution measurements mentioned above may be consistent with this. Furthermore, Simonson’s paper54 finds a strong electrostatic coupling between the protein and solvent which results in much of the microscopic detail of the protein dynamics being smoothed by solvent screening.6 Complex systems and processes Several papers at this discussion look at aspects of macromolecular hydration that can be regarded as of a higher degree of complexity than those discussed above. The two papers dealing with processes in proteins have an interesting common factor, namely that they both use a water-perturbational approach: Nicholls62 and Go~dfellow~~ use the addition and/or removal of water to probe (experimentally and computationally, respectively), the role of water in processes involving proteins. Perturbational approaches can be very powerful, as we know from the development of physics in general. The application of such methods, carefully planned and controlled, might simi- larly have high promise in assisting our understanding of the role of water in complex macromolecular processes, including giving us useful information on the molecular level interactions involved.Using cytochrome c oxidase, the work presented by Nicholl~~~ investigated the role of hydration as a control mechanism in membrane proteins. Using deviations from osmotic ideality, he identifies several water-involved processes, such as water uptake on binding, a reduction in activity of water-stressed states, and the involvement of water in modulating transitions between different states of the enzyme. In each of these processes, water appears to play a distinct mechanistic role. With the involvement of water in these processes, we have a possible tool for modifying behaviour by modifying the hydration, using the kinds of osmotic control mechanisms used here.Effective control does, however, require understanding of the precise molecular-level mechanisms of the solvent involvement. I suspect that here, carefully planned computer simulations may be of significant help. Although Goodfellow’s paper63 uses a pertur- bational approach to probe possible protein unfolding paths, and no doubt we can discuss later the information we might thus obtain to help understanding possible unfolding pathways, and perhaps throw light on the popular concept of the molten globule as an intermediate state in the folding process, the kind of approach exampled here can be used to help our understanding of water involvement in the kinds of pro- cesses discussed by Nicholls.There would also seem to be a need for such simulations to be carried out in parallel with appropriate experimental measurements. Here we can perhaps see a call for further development of measurements of dynamics as a function of water perturbation by several techniques (in particular NMR and neutrons?), and perhaps also of small angle scattering to give spatial information. The computational water-perturbational approach described in this paper can also give us potentially inter- esting information on the details of water interactions in protein systems, and perhaps even throw some light on how the marginal stability that is important for enzyme activ- ity is maintained, and perturbed by denaturing and protective agents.Nicholls work also gives evidence that water requirements are linked to proton involvement in electron and proton transport, and here there is a possible link with Zundel’s paper64 which argues for a strong role for water in proton transport processes. This hypothesis has been discussed for at least two decades, and some early protein structures managed to locate water molecules in appropriate positions that would be consistent with such a role for solvent. This possibility was raised during the 1975 Royal Society Discussion. In my view, however, evidence is still largely circumstantial. J. L. Finney 15 Although we are dealing with complicated systems, and devising appropriate experi- ments may be difficult, it would seem worthwhile to try using methods that allow more direct interpretation of the results than does IR spectroscopy.In terms of the interpreta- tion of the IR continuum in terms of chains having high proton polarisability, it would seem that an appropriate neutron inelastic scattering experiment could be devised as a good test. Although most of the papers at this discussion relate specifically to biological macro- molecules, there are two papers which are brave enough to tackle hydration problems relating to macromolecules that are, in some respects, much less-ordered, and hence perhaps more difficult to tackle. It is these kinds of systems that are of intense interest industrially, and for which we need to develop an understanding on the role of hydra- tion on a variety of physical properties if we are to be able to improve our ability to usefully exploit and modify their properties.The paper to be presented by Waigh5 reports work that is relevant to sorting out ways in which we might best take advantage of genetically engineered strains so as to be able to optimise starch production so that gelatinisation and the staling properties are optimised, while still retaining nutritional value (and hopefully, from my perspective, taste!). To do this effectively, we need an understanding of how the water interactions at the molecular level affect mesoscopic level processes, and this requires continued development of techniques such as the small angle methods at the centre of this paper.The impressive work presented by McBrierty4 similarly tackles the problem of rela- ting microscopic-level water-involved processes to macroscopic properties, in particular the mechanical relaxation behaviour of cellulose acetate. This work is a beautiful example of the use of several techniques (here differential scanning calorimetry, dielectric relaxation, dynamic mechanical thermal analysis, and NMR) to try to devise a plausible model of the mechanical behaviour of this complex system. The complexity of this system, and the need to be imaginative in tackling it, is underlined by the assignment of at least seven relaxation processes. Fig. 11 of this paper underlines the complexity of this system and its dynamical behaviour.The fact that the ordinate axis covers eleven orders of magnitude in frequency (from 10-' to 10" Hz) tells its own story. In tackling indus- trially relevant problems such as this, we need to devise an appropriate range of experi- ments, using what we have learned from work on simpler systems, and, again using what we have learned about hydration in simpler systems, try to build up an explanation of the system's behaviour that is consistent with all the experimental data, and with what we know from relevant work on hydration of macromolecular interfaces in general. 7 Summary: Some interesting issues The papers to be discussed at this meeting demonstrate much of the progress that has been made in understanding the water-macromolecule interface since the 1975 Royal Society Discussion.26 Increasingly sophisticated experimental techniques can now give us information that we have wanted to access for a long time, and there are several examples at this discussion.We are becoming more adept at interpreting results on these inherently complex systems in a less model-dependent way. Hence, we are less liable to serious pitfalls when constructing interpretive models to help us understand what our data mean. And we are learning enough about the molecular-level behaviour to begin to be able to move from knowledge of the molecular-level structure and/or dynamics to understanding how these give rise to properties and behaviour at the mesoscopic, and macroscopic, levels.From the papers to be discussed, a number of points emerge that I think are of particular interest and/or importance. Many of these should provide stimulating dis- cussion material over the next two days, and hopefully we will be able to resolve some of Overview Lecture the outstanding issues that are ripe for resolution. Of particular interest to me are the following points. (a) The power of neutron techniques, both elastic and inelastic, is beginning to be exploited effectively to improve our understanding of both model and ‘real’ systems. To fully exploit the potential of the neutron, we need further more sophisticated interpretive techniques, both theoretical and computational, and some ways forward are contained in papers presented at this discussion. (b) Our understanding of the physical basis of the hydrophobic interaction, of central importance when trying to understand biomolecular interactions in particular, is begin- ning to progress, and perhaps moving away from the conventional interpretation of the origin of the entropic contribution to the driving force.We should clean up the termino- logy, recognise that a single molecule is not of itself hydrophobic, as current loose usage implies. We must have a two- (or more) component system before we can observe hydrophobic behaviour. I believe we should reserve the term for such systems, and refer to the single moieties as non-polar or apolar. The origin of the entropic contribution is still unclear, and I hope the papers presented here will be able to clarify this.(c) Are CH ‘hydrogen bonds’ energetically important? Although the ideas involved here are over 30 years old, there seems to be a major renewal of interest in the inter- action between polar and non-polar groups. We should be able, on the basis of current knowledge, to assess whether or not such interactions are energetically important in e.g. protein and nucleic acid stability, or whether their contribution is incidental to other stronger interactions. (d) How well need a theory, or a simulation, of hydration perform to be ‘good enough’, or useful? The delicate energy balance that maintains the structural stability of the native structure of a protein implies quite stringent requirements on such predictive methods.Similarly stringent requirements are implied by the fact that only small pertur- bations of a structure (e.g. a single point mutation) may result in large changes in stabil- ity. What means do we have to even begin to answer this question? (e) Both structural and dynamical studies on a variety of macromolecule-water inter-faces seem increasingly to fit ideas obtained from simpler systems, and those consistent with relatively general theories. This suggests we should query under what circum- stances the molecular-level details of the constraints on water molecules are really imp0 r tant. (f) Dynamical smoothing can give results that differ from those of static predictions. The membrane and polar fluctuation simulations reported at this meeting suggest we draw conclusions from static simulations at our peril.(g) What are the limits to studies of hydrated protein powders, as in the neutron inelastic scattering papers presented at this meeting? When trying to draw conclusions from such measurements on the role of hydration in e.g. enzyme activity, what time- scales are most appropriate, and are motions on these timescales significantly affected by working on hydrated powders rather than in solution (which is more difficult)? (h) What are the most effective computational routes to follow to probe the role of solvent effects in protein folding? Do the structures obtained by water insertion tech- niques tell us useful things about folding intermediates? (i) Do we now understand both the structure and dynamics of water well enough, as a function of its local environment, to use water as a probe, rather than, as has usually been the case until recently, an object of other experimental probes? Rather than contin- uing to try to find out more about water in these systems, do we now understand water sufficiently well to be able to use its measured properties at a given interface to tell us something about the macromolecular system itself? (j) In a similar vein, are we beginning to learn enough about the involvement of water in macromolecular processes to be able to use water activity modification to control processes? J.L. Finney 17 I think these are interesting issues. I hope we can discuss them in an interesting, stimulating, meeting over the next two days.References 1 P. Auffnger, S. Louise-May and E. Westhof, Faraday Discuss., 1996,103,151. 2 G. Hummer, A. E. Garcia and D. M. Soumpasis, Faraday Discuss., 1996,103, 175. 3 V. P. Denisov and B. Halle, Faraday Discuss., 1996,103,227. 4 V. J. McBrierty, C. M. Keely, F. M. Coyle, H. Xu and J. K. Vij, Faraday Discuss., 1996,103,255. 5 T.A. Waigh, P. J. Jenkins and A. M. Donald, Faraday Discuss., 1996,103,325. 6 J. E. Enderby & G. W. Neilson, in Water: A Comprehensive Treatise, ed. F. Franks, 1979,vol. 6,p. 1. 7 J. Z. Turner, A. K. Soper and J. L. Finney, J. Chem. Phys., 1995,102,5438. 8 A. K. Soper, J. Chem. Phys., 1994,101,6888. 9 D. T.Bowron, J. L. Finney and A. K. Soper, in preparation.10 A. K. Soper and J. L. Finney, Phys. Rev. Letts., 1993,71,4346. 11 A. K. Soper, Faraday Discuss., 1996,103,41. 12 R. L. McGreevy and L. Pusztai, Mol. Sim., 1988,1,359. 13 R. A. Young, The Rietveld Method, Oxford University Press, Oxford, 1993. 14 F. Hofmeister, Arch. Exp. Path. Pharmakol. ,1888,24,247. 15 H.S.Frank and M. W. Evans, J. Chem. Phys., 1945,13,507. 16 W. Kauzmann, Ado. Prot. Chem.,1959,14,1. 17 P. D. Ross and S. Subramanian, Biochemistry,l981,30,3096. 18 J. L. Finney, B. J. Gellatly, I. C. Golton and J. Goodfellow, Biophys. J., 1980,32, 17. 19 K.P. Murphy, P. L. Privalov and S. J. Gill, Science, 1990, 247, 559. 20 B. Lee, Proc. Natl. Acad. Sci. USA, 1986,83,8069. 21 A. D. J. Haymet, K. A. Silverstein and K. A. Dill, Faraday Discuss., 1996, 103, 117.22 W. Blokzijl and J. B. F. N. Engberts, Angew. Chem. Int. Ed. Engl., 1993,32,1545. 23 S. Garde, G. Hummer and M. E. Paulaitis, Faraday Discuss., 1996,103, 125. 24 N. T.Skipper, C. H. Bridgeman, A. D. Buckingham and R.L. Mancera, Faraday Discuss., 1996, 103, 141. 25 D. T.Bowron, J. L. Finney and P. A. Timmins, unpublished work. 26 Phil. Trans Roy. SOC.Lond. B., ed. R. E. Richards and F. Franks, 1977,vol. 278. 27 H. F. J. Savage, Biophys. J., 1986,50,947. 28 H. F. J. Savage and J. L. Finney, Nature (London), 1986,322,717. 29 J. P. Bouquiere, J. L. Finney and H. F. J. Savage, Acta Crystallogr, 1994,BSO, 566. 30 Y.Marechal, Faraday Discuss., 1996,103, 349. 31 H. F. J. Savage, in Water Science Reviews, ed. F. Franks, 1986,vol.2,p. 67. 32 C. H. Cho, S. Singh and G. W. Robinson, Faraday Discuss., 1996,103, 19. 33 P. Barnes, in Rep. of CECAM Workshop on Molecular Dynamics and Monte Carlo Calculations on Water,ed. H. J. C. Berendsen, CECAM, Orsay, France, 1972,p. 77. 34 H.J. C. Berendsen and G. A. van der Velde, in Rep. of CECAM Workshop on Molecular Dynamics and Monte Carlo Calculations on Water, ed. H. J. C. Berendsen, CECAM, Orsay, France, 1972, p.63. 35 P. Barnes, D. V. Bliss, J. L. Finney and J. E. Quinn, Nature (London), 1979,282,459. 36 P. Ahlstrom, A. Wallqvist, S. Engstrom and B. Jonsson, Mol. Phys., 1989,68, 563. 37 J. Brodholt, M. Sampoli and R. Vallauri, Mol. Phys., 1995,85, 81. 38 J. L. Finney, J. 0.Baum and J. E. Quinn, in Water Science Reviews, ed.F. Franks, 1985,vol. 1, p. 93. 39 K. Hermansson, PhD. Thesis, Uppsala, 1984. 40 0.Matsuoka, E. Clementi and M. Yoshimine, J. Chem. Phys. 1976,64,1351. 41 J. D.Sutor, J. Chem. SOC.,1963, 1105. 42 R. D.Green, Hydrogen Bonding by CH Groups, Macmillan, London, 1974. 43 S-J. Marrink, D. P. Tieleman, A. R. van Buuren and H. J. C. Berendsen, Faruday Discuss., 1996, 103, 191. 44 M-C. Bellissent-Funel, J-M. Zanotti and S. H. Chen, Faruday Discuss., 1996, 103,281. 45 E. M. Lee, R. K. Thomas, J. Penfold and R. C. Ward, J. Phys. Chem., 1989,93,381. 46 K. J. Packer, Phil. Trans Roy. SOC.Lond. B., 1977,278,59. 47 K. Wuthrich, M.Billeter, P. Guntert, P. Luginbiihl, R. Riek and G. Wider, Faraday Discuss., 1996, 103,245. 48 S. H. Koenig and W. S. Schillinger,J.Biol. Chem., 1969,244,3282. 49 S. H. Koenig, K. Hallenga and M. Shporer, Proc. Natl. Acad. Sci. USA, 1975,72,2667. 50 K. Hallenga and S. H. Koenig, Biochemistry, 1976, 15,4255. 51 J. L. Finney, J. M. Goodfellow and P. L. Poole, in Structural Molecular Biology, ed. D. B. Davies, W. Saenger and S. S. Danyluk, Plenum Press, New York, 1982,p. 387. 52 M.Settles and W. Doster, Faraday Discuss., 1996, 103, 269. 18 Overview Lecture 53 A. Luzar, Faraday Discuss., 1996,103,29. 54 T. Simonson and D. Perahia, Faraday Discuss., 1996,103,71. 55 R. S. Lee and S. Bone, Faraday Discuss., 1996,103, 59. 56 G. Careri, E. Gratton, P-H. Yang and J. A. Rupley, Nature (London), 1980,284,572. 57 P. L. Poole and J. L. Finney, Int. J. Biol. Macromol., 1983,5,308. 58 J. C. Smith, S. Cusack, P. L. Poole and J. L. Finney, J. Biomol. Struct. and Dynamics, 1987,4, 583. 59 P. L. Poole and J. L. Finney, Comments in Mol and Cell Biophys., 1984,2, 129. 60 R. M. Daniel, J. L. Finney, J. C. Smith and M. Ferrand, unpublished results. 61 W. Doster, S. Cusack and W. Petry, Nature (London), 1989,337,754. 62 P. Nicholls, Y. Sternin, J. Loewen, T. Jennings and B. Tattrie, Faraday Discuss., 1996,103,313. 63 J. M. Goodfellow, M. Knaggs, M. A. Williams and J. M. Thornton, Faraday Discuss., 1996,103,339. 64 B. Brzezinski and G. Zundel, Faraday Discuss., 1996, 103, 363. Paper 6103459D; Received 17th May, 1996

ISSN:1359-6640    

DOI:10.1039/FD9960300001

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年代:1996

数据来源: RSC

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General Discussions of the Faraday Society/Faraday Discussions of the Chemical Society Date 1907 1907 1910 191 1 1912 1913 1913 1913 1914 1914 1915 1916 1916 1917 1917 1917 1918 1918 1918 1918 1919 1919 1920 1920 1920 1920 1921 1921 1921 1921 1922 1922 1923 1923 1923 1923 1923 1924 1924 1924 1924 1924 1925 1925 1926 1926 1927 1927 1927 1928 1929 1929 1929 1930 1931 1932 1932 1933 Subject Volume Trans. 3 3 6 7 8 9 9 9 10 10 11 Osmotic Pressure Hydrates in Solution The Constitution of Water High Temperature Work Magnetic Properties of AlloysColloids and their Viscosity The Corrosion of Iron and Steel The Passivity of Metals Optical Rotary Power The Hardening of Metals The Transformation of Pure Iron Methods and Appliances for the Attainment of High Temperatures in a Laboratory 12 Refractory Materials 12 Training and Work of the Chemical Engineer 13 Osmotic Pressure 13 Pyrometers and Pyrornetry 13 The Setting of Cements and Plasters 14 Electric Furnaces 14 Co-ordination of Scientific Publication 14 The Occlusion of Gases by Metals 14 The Present Position of the Theory of Ionization 15 The Examination of Materials by X-Rays 15 The Microscope : Its Design, Construction and Applications 16 Basic Slags :Their Production and Utilization in Agriculture 16 Physics and Chemistry of Colloids 16 Electrodeposition and Electroplating 16 Capillarity 17 The Failure of Metals under Internal and Prolonged Stress 17 Physico-Chemical Problems Relating to the Soil 17 Catalysis with special reference to Newer Theories of Chemical Action 17 Some Properties of Powders with special reference to Grading by Elutriation 18 The Generation and Utilization of Cold 18 Alloys Resistant to Corrosion 19 The Physical Chemistry of the Photographic Process 19 The Electronic Theory of Valency 19 Electrode Reactions and Equilibria 19 Atmospheric Corrosion.First Report 19 Investigation on Oppau Ammonium Sulphate-Nitrate 20 Fluxes and Slags in Metal Melting and Working 20 Physical and Physico-Chemical Problems relating to Textile Fibres 20 The Physical Chemistry of Igneous Rock Formation 20 Base Exchange in Soils 20 The Physical Chemistry of Steel-Making Processes 21 Photochemical Reactions of Liquids and Gases 21 Explosive Reactions in Gaseous Media 22 Physical Phenomena at Interfaces, with special reference to Molecular Orientation 22 Atmospheric Corrosion, Second Report 23 The Theory of Strong Electrolytes 23 Cohesion and Related Problems 24 Homogeneous Catalysis 24 Crystal Structure and Chemical Constitution 25 Atmospheric Corrosion of Metals, Third Report 25 Molecular Spectra and Molecular Structure 26 Colloid Science Applied to Biology 26 Photochemical Processes 27 The Adsorption of Gases by Solids 28 The Colloid Aspect of Textile Materials 29 Liquid Crystals and Anisotropic Melts 29 1933 Free Radicals 30 Date Subject 1934 Dipole M-oments 1934 Colloidal Electrolytes 1935 The Structure of Metallic Coatings, Films and Surfaces 1935 The Phenomena of Polymerization and Condensation 1936 Disperse Systems in Gases: Dust, Smoke and Fog Volume 30 31 31 32 32 1936 Structure and Molecular Forces in (a)Pure Liquids, and (b)Solutions 33 1937 The Properties and Function of Membranes, Natural and Artificial 33 1937 Reaction Kinetics 34 1938 Chemical Reactions Involving Solids 34 1938 Luminescence 35 1939 Hydrocarbon Chemistry 35 1939 The Electrical Double Layer (owing to the outbreak of the war the meeting was abandoned, but the papers were printed in the Transactions) 35 1940 The Hydrogen Bond 36 1941 The Oil-Water Interface 37 1941 The Mechanism and Chemical Kinetics of Organic Reactions in Liquid 37 Systems 1942 The Structure and Reactions of Rubber 38 1943 Modes of Drug Action 39 1944 Molecular Weight and Molecular Weight Distribution in High Polymers (Joint Meeting with the Plastics Group, Society of Chemical Industry) 40 1945 The Application of Infra-red Spectra to Chemical Problems 41 1945 Oxidation 42 1946 Dielectrics 42 A 1946 Swelling and Shrinking 42 B 1947 Electrode Processes Disc.1 1947 The Labile Molecule 2 1947 Surface Chemistry (Jointly with the Socieitei de Chimie Physique at Bordeaux Published by Butterworths Scientific Publications Ltd 1947 Colloidal Electrolytes and Solutions Trans.43 1948 The Interaction of Water and Porous Materials Disc. 3 1948 The Physical Chemistry of Process Metallurgy 4 1949 Crystal Growth 5 1949 Lipo-proteins 6 1949 Chromatographic Analysis 7 1950 Heterogeneous Catalysis 8 1950 Physico-chemical Properties and Behaviour of Nuclear Acids Trans. 46 1950 Spectroscopy and Molecular Structure and Optical Methods of InvestigatingCell Structure Disc. 9 1950 Electrical Double Layer 1951 Hydrocarbons1951 The Size and Shape Factor in Colloidal Systems 1952 Radiation Chemistry 1952 The Physical Chemistry of Proteins 1952 The Reactivity of Free Radicals 1953 The Equilibrium Properties of Solutions on Non-electrolytes 1953 The Physical Chemistry of Dyeing and Tanning 1954 The Study of Fast Reactions 1954 Coagulation and Flocculation 1955 Microwave and Radio-frequency Spectroscopy 1955 Physical Chemistry of Enzymes1956 Membrane Phenomena 1956 Physical Chemistry of Processes at High Pressures 1957 Molecular Mechanism of Rate Processes in Solids 1957 Interactions in Ionic Solutions Trans.47 Disc. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1958 Configurations and Interactions of Macromolecules and Liquid Crystals 25 1958 Ions of the Transition Elements 26 1959 Energy Transfer with special reference to Biological Systems 27 1959 Crystal Imperfections and the Chemical Reactivity of Solids 28 1960 Oxidation-Reduction Reactions in Ionizing Solvents 29 1960 The Physical Chemistry of Aerosols 30 1961 Radiation Effects in Inorganic Solids 31 1961 The Structure and Properties of Ionic Melts 32 1962 Inelastic Collisions of Atoms and Simple Molecules 33 1962 High Resolution Nuclear Magnetic Resonance 34 1963 The Structure of Electronically Excited Species in the Gas Phase 35 1963 Fundamental Processes in Radiation Chemistry 36 1964 Chemical Reactions in the Atmosphere 37 1964 Dislocations in Solids 38 1965 The Kinetics of Proton Transfer Processes 39 1965 Intermolecular Forces 40 Date Subject 1966 The Role of the Absorbed State in Heterogeneous Catalysis 1966 Colloid Stability in Aqueous and Non-aqueous Media 1967 The Structure and Properties of Liquids 1967 Molecular Dynamics of the Chemical Reactions of Gases 1968 Electrode Reactions of Organic Compounds 1968 Volume 41 42 43 44 45 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 1980 1980 1981 1981 1982 1982 1983 1983 1984 1984 1985 1985 1986 1986 1987 1987 1988 1988 1989 1989 1990 1990 1991 1991 1992 1992 1993 1993 1994 1994 3 994 1995 1995 1995 Homogeneous Catalysis with Special Reference to Hydrogenation and 46Oxidation 47Bonding in Metallo-organic Compounds 48Motions in Molecular Crystals 49Polymer Solutions 50The Vitreous State 51Electrical Conduction in Organic Solids 52Surface Chemistry of Oxides 53Reactions of Small Molecules in Excited States The Photoelectron Spectroscopy of Molecules 54 Molecular Beam Scattering 55 Intermediates in Electrochemical Reactions 56 Gels and Gelling Processes 57 Photo-effects in Adsorbed Species 58 Physical Adsorption in Condensed Phases 59 Electron Spectroscopy of Solids and Surfaces 60 Precipitation 61 Potential Energy Surfaces 62 Radiation Effects in Liquids and Solids 63 Ion-Ion and Ion-Solvent Interactions 64 Colloid Stability 65 Structures and Motion in Molecular Liquids 66 Kinetics of State Selected Species 67 Organization of Macromolecules in the Condensed Phase 68 Phase Transitions in Molecular Solids 69* Photoelectrochemistry 70* High Resolution Spectroscopy 71* Selectivity in Heterogeneous Catalysis 72* Van der Waals Molecules 73* Electron and Proton Transfer 74* Intramolecular Kinetics 75* Concentrated Colloidal Dispersions 76* Interfacial Kinetics in Solution 77* Radicals in Condensed Phases 78* Polymer Liquid Crystals 79* Physical Interactions and Energy Exchange at the Gas-Solid Interface 80* Lipid Vesicles and Membranes 81* Dynamics of Molecular Photofragmentation 82* Brownian Motion 83* Dynamics of Elementary Gas-phase Reactions 84* Sdvation 85* Spectroscopy at Low Temperatures 86* Catalysis by Well Characterised Materials 87* Charge Transfer in Polymeric Systems 88 Structure of Surfaces and Interfaces as studied using Synchrotron Radiation 89* Colloidal Dispersions 90* Structure and Dynamics of Reactive Transition States 91* The Chemistry and Physics of Small Metallic Particles 92* Structure and Activity of Enzymes 93* The Liquid/Solid Interface at High Resolution 94* Crystal Growth 95* Dynamics at the Gas/Solid Interface 96* Structure and Dynamics of Van der Waals Complexes 97* Polymers at Surfaces and Interfaces 98* Vibrational Optical Activity: From Fundamentals to Biological Applications 99* Atmospheric Chemistry : Measurements, Mechanisms and Models loo* Gels 101* Unimolecular Reaction Dynamics 102* * Available for purchase, for current information on prices etc.please contact the Sales and Promotion Depart- ment, The Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge, UK CB4 4WF.

ISSN:1359-6640    

DOI:10.1039/FD996030X003

出版商:RSC    

年代:1996

数据来源: RSC

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FU~UY 1996,103, 19-27 D~SCUSS., Liquid water and biological systems: The most important problem in science that hardly anyone wants to see solved Chul Hee Cho, Surjit Singh and G. Wilse Robinson* SubPicosecond and Quantum Radiation Laboratory, Department of Chemistry and Physics, Texas Tech University, Lubbock, Texas 79409, USA The main emphasis of this paper is the design of a water model that gives the correct temperature-dependent density. Water interaction models cur- rently used have been highly oversimplified and are presently incapable of producing, over even a modest range of temperature and pressure, the properties of real water. The new feature is a modified second-neighbour non-hydrogen-bonded interaction to match those known to exist in the moderately dense ice polymorphs or the high density amorphous solid. Combined dynamically in the liquid with ordinary tetrahedral bonding, the more dense metastable structure tends to grow in with increasing tem- perature because of the entropic driving force, creating the density maximum.With this new model, more realistic structural and dynamic properties of liquid water near surfaces, particularly biologically important macromolecules and membranes, can be expected in future work. One keeps hearing that the most abundant substance on Earth is one of the most mysterious materials known. It is ‘a still poorly known liquid’,’ and an ‘anomalous liquid and solvent’.2 ‘It is the most well known and the least understood compo~nd’.~ ‘We know how to work out the properties of ice and steam, but we have no clear idea why there is such a thing as ordinary liquid water!’4 One of the most familiar mysteries is the density anomaly: at sufficiently low tem- peratures, warming the liquid causes it to shrink.There is an increasing awareness that understanding this one property will reveal the origins of all the other anomalous properties of water, and thus will provide a complete description of this most important liquid in its bulk phase, as well as in the interfacial states of this substance. However, current popular computational models of water are simply unable to reproduce this anomaly.596 It would seem that, if a model does not possess this basic property of the real liquid, it cannot provide any sort of accurate picture in biological, chemical or geophysical applications, on which an ever increasing amount of computational effort is being spent. In spite of these feelings, motivations and precepts for developing a much better molecular-level picture.of liquid water, from the experimental, theoretical and computa- tional points of view, are clear. First of all, one would like to make predictions with regard to properties that either have never been considered experimentally or that have been dificult to measure in the laboratory. This requires that the theoretical model be capable of appropriately reproducing all the known phenomena, particularly the density maximum and other key anomalies. Furthermore, any model claiming to represent real water must correctly reflect variations in its most important properties over wide ranges of temperature, pressure, surface perturbations, etc.Unfortunately, none of the existing models has yet come close to reaching this though a few at temp^"-'^ to develop a model that reproduces some properties of water from the gas to condensed phases have been made. 19 Liquid water and biological systems Recent experimental investigations have suggested important avenues of approach towards the ultimate resolution of this problem. For example, vibrational Raman data14 for the liquid over a range of temperatures from the supercooled region indicate a mixture of resonances, one exhibiting strong hydrogen bonding as in normal ice, the other having a lower degree of such bonding.This bonding picture has been carried farther by Vedamuthu et a/., where it was found to be accurately consistent with a number of experimental properties of water: the temperature-dependent density' of H,O from about -30 to +40 "C within the five-decimal-place precision of the experi- mental data;16 the temperature-dependent density' of D20 to its four-decimal-place precision;" the densitylg of T20;20 and the isothermal compressibility of H20, includ-ing its minimum,21 to the three- to four- decimal-place precision of its measurement.16 Recently, based on the same type of considerations, a thermal lag effect has been dis- covered,20 which normalizes the structural properties of H,O, D20 and T20.Only in this way can isotope effects for a variety of dynamic properties, including the viscosity and self-diffusion coeficients, be understood. Surprisingly, perhaps, until this discovery, isotope effects in water have not fit any sensible pattern.22 Still more revealing are isochoric differential X-ray scattering data for liquid D20,23 again spanning both supercooled and normal temperature ranges. These data show the presence of at least two second-neighbour peaks in the radial distribution function, one near 3.4 A, which grows with increasing temperature differential, the other near 4.5 A, which decreases with temperature. Apparently, as the temperature increases, more densely packed second neighbours near 3.4 A are created in the liquid at the expense of the ordinary second neighbours of the open tetrahedral network.These new neighbours have been described as 'fifth neighbours' in the work of Sciortino et al.24Recognition of the presence of these neighbours, as well as the identification of their origin, is currently providing the 'key to the strong-box' that holds all the secrets of this most mysterious material. The purpose of the present paper is twofold. The first is to report some analytical results on an exactly soluble one-dimensional fluid model of liquid water. This model very simply illustrates the origin of the density maximum and its pressure dependence. The second purpose of the paper is to report on work in progress on molecular dynamics (MD) simulations using intermolecular potentials that have been augmented by an additional second-neighbour potential well.Both of these results act as first steps in the development of a water-water potential model that correctly reproduces the temperature-pressure dependence of the properties of the liquid, including the known anomalies. The second-neighbour shell Current water-water interaction models are mainly concerned with the formation and disruption of the tetrahedral structure at the nearest-neighbour level, with the hope that the longer range features will fall out naturally. This hope is weakened when one realizes that the bonding characteristics in the ice forms of moderately high density are inconsis- tent with the random hydrogen-bond network so often discussed and employed in descriptions of bulk or perturbed liquid water.For example, the 0. -0Lennard-Jones contribution in current water-water interaction potentials is chosen so as not to disturb this tetrahedral picture too much. Rowlinson' was satisfied with the Lennard-Jones diameter 0 = 2.725 A because it is much closer to the intermolecular distance in ice. The same value was used in later Monte Carlo calculations by Barker and Watts.25 A slight-ly larger 0, equal to that in neon, was introduced in the Ben-Naim-Stillinger p~tential~~.~~for water, then revised to o = 3.10 A in the later work of Stillinger and Rahman.28 However, the continuing focus on tetrahedral hydrogen bonding has obscured recognition of other possible bonding characteristics in condensed water C.H. Cho et al. systems. Precisely for this reason, it has not been possible to reproduce the moderately dense ice forms from any of these models. In fact Morse and Riceg created a ‘proving ground’ for some of the early water MD models, including ST2 and MCY, and it was found that none of these models was consistent with experimental ice polymorph data. Part of this problem was correctly ascribed to the absence of many-body effects in the intramolecularly rigid non-polarizable MD models then available, a problem that is currently being addressed through the introduction of polarizability and bond fle~ibility.~ Perhaps now is the time to go back to the insightful ideas of Morse and Riceg before more effort is consumed on MD studies of the liquid or of the interfacially perturbed liquid.Surely, the density maximum in the liquid and the structures of the most stable variable forms of ice, which are, after all, nearly equienergetic with that of normal ice Ih, must be correctly repro- duced by an interaction model before any such model can be taken seriously for other types of bulk or interfacial water problems. The presence of dynamically variable open and compact second-neighbour bonding in the liquid can certainly explain the density anomaly, as already shown through the empirical, yet highly precise, correlations reported by Vedamuthu et a1.15*17The basic concept rests on the disappearance on the average, with increasing temperat~re~~ or pre~sure,~’of the open low density intermolecular tetrahedral bonding in the liquid, in favour of regions having the compact 3.4 A second-neighbour structure of the denser ices3’ or the high density amorphous solid.31 This structure depends on two minima, a close-in shallow one near 3.4 A and a farther-out deeper one at ca.4.5 A. The shallow minimum could actually arise from collective many-body interactions in the condensed phase and thus may not be evident from quantum calculations on small water clusters, particularly the dimer. In any case, such interactions could be difficult to detect by current quantum mechanical investigations, since fairly large clusters may be required before they show up. A good question might be, ‘How large does a water cluster have to be in order to produce a second-neighbour potential minimum near the 3.4 A minimum that is known to occur in the ice forms of moderate density?’; or, one may ask, ‘By increasing the pressure on the cluster, is modern quantum theory capable of reproducing this structure at all?’ As mentioned earlier, this type of second-neighbour bonding in the liquid has been dramatically confirmed by differential X-ray scattering experiment^.^^ These latter data, shown in Fig.1, strikingly reveal the pronounced peak in liquid water near 3.4 A,which grows with an increasing temperature differential at constant density. The presence of this peak, just beyond the ordinary minimum in the radial distribution function of liquid water, is probably the reason that integration of this function out to the minimum always gives more than four nearest neighbours.The distance 3.4 A does not occur in the low density open tetrahedral structures, ice Ih and Ic,~~ so this distance serves as a fingerprint for the absence of the open tetrahedral bonding, or the presence of the compact bonding, whichever way one wants to look at it. This contracted bonding in dense ice accounts for the remarkably large ca. 30%, increase of density in these crys- talline forms. In addition to the 3.4 A second-neighbour feature in liquid water, the differential X-ray scattering results23 also show that the second-neighbour O--.Opeak near 4.5 A decreases markedly in intensity with increasing temperature. The decreased importance of this peak with increasing temperature, in concert with the increased intensity of the 3.4 A peak, provides direct experimental evidence, not only for the presence of both the open and the more compact bonding in the liquid, but also for the transformation of the open form to the more dense form with rising temperture.It is intuitively obvious that this produces the density maximum in liquid water. Any real test of a water-water potential will need to consider pressure effects as well as temperature effects. This has seldom been done. In this respect, we quote from a Liquid water and biological systems 0.2 c U c 0.1 bU 0 -0.1 2 4 6 8IiA Fig. 1 Differential O---Opair correlation function of liquid water at different temperatures but constant density, showing the growth and regression, respectively, of the 3.4 8,and 4.5 8, second nearest-neighbour peaks in liquid water with increasing temperature.Reproduced from ref. 23 with the kind permission of S-H. Chen. recent paper29 on neutron scattering experiments of liquid D,O under pressure and for -65 < T/"C < +53: 'The effect of pressure is subtle in the sense that it does not change substantially the number of hydrogen bonds, but the O...O.-.Oangles are modified; in fact the tetrahedral symmetry is partly lost.' This is exactly what happens when the open ice Ih type O...O bondin at 4.5 A in the second-neighbour shell changes over to the compact structure at 3.4 x. The hydrogen bonds bend.30 All these structural considerations are in perfect concordance with the presence in the liquid of two types of second-neighbour O...O configurations, a more stable one at sca. 4.5 A and a metastable one at ca.3.4 A.The various s ~ c c ~ sof this ~ s ~ simple idea, which provides a visual picture of the liquid and gives better than a semi- quantitative description of all its properties, including the anomalous ones, suggest that future realistic MD modelling of liquid water should include this type of bonding.34 In these respects, it is most curious why computational models of the past have used ice Ih and ice VII or VIII as benchmark^'^ for obtaining a realistic potential for the liquid. Actually, both 0.--0hydrogen-bonded and non-hydrogen-bonded distances lie at 2.8 A in ice VII and VIII, and thus these ice forms, though 'not disturbing the fully tetrahedral picture too much', would seem to provide a poor representation of anything anomalous taking place in the liquid.Takahashi fluid model The feasibility of the above ideas can be tested with an exactly solvable analytical model. In this model, we consider only the second neighbours in liquid water, ignoring the first neighbours altogether. The justification for this simplification rests on two experimental properties of liquid water: (1) the number density and structure making up the first shell are not strongly affected by temperat~re~~ or pressure29; and (2) the oxygen-oxygen nearest-neighbour distances in all forms of water and ice are the same, ca.2.8 81.23930-32 For these reasons, nearest neighbours cannot play a significant role in determining ther- mally induced or pressure-induced density variations in liquid water, except, of course, through normal thermal expansion and compressibility effects. A central molecule and its nearest neighbours can, in fact, be considered to be a core, dynamically exchanging water molecules to be sure, but on the average invariant. One might then expect that variations in the second-neighbour structure determine the density of the liquid. C. H.Cho et al. The foregoing considerations allow an exactly solvable one-dimensional model to be employed for the investigation of the density maximum in water. We ignore nearest neighbours and treat the second neighbours in the real liquid as first neighbours in this theoretical model.Within this framework, the potential in the one-dimensional model can be simply treated as two square wells between each particle, a shallow metastable inner well and a deeper outer well. The main purpose of introducing the model into the water problem is to study the effects on the temperature- and pressure-dependent density from the presence of the two second-neighbour potential wells. The difference between the real three-dimensional structure comes only from the placement of the two second-neighbour wells in a line rather than at some angle. This type of problem was first solved for a hard sphere potential by T~nks,~’ and was later generalized by Taka~hashi~~ The Hamiltonian in the Takahashi and model is where N particles each of mass rn and momenta pi have the potential energy, N-1 tv(qi) = C ~( qi+l -4iI)i= 1 Since the system is to be studied for a range of temperatures and pressures, it is convenient to work in the isobaric-isothermal (NPT) ensemble in which the partition function38 is where L is the volume in the one-dimensional system, P is the pressure, which is actually just the tension at the two ends of the fluid, and the other symbols have their usual meanings.The single integral sign collectively stands for N momentum integrals, with limits -GO to + co,and N position integrals, 0 to L. The Gibbs energy is G(P, T)= -kB T In Q,,(P, T) (4) It can be shownj4 that, for the Takahashi model, the Gibbs energy in the thermody- namic limit in which L -+ GO, N -+ a,while the volume per particle I = L/N is held constant, is given by36937 where K(P,T) is the Laplace transform of exp( -Pu); K(P,T) = dx exp(-PPx) x exp[-Pu(x)] The total volume of the system in this limit is3’ N &(P, T) L=(E),= -E( ap ), and the volume per particle 1 and the density p are given by Liquid water and biological systems Therefore, to obtain the density, one must first choose a potential, then perform the IC integral in eqn.(6), and differentiate the result with respect to P. The chosen potential contains the two second-neighbour minima described above, a close-in shallow one and a farther-out deeper minimum.Intuitively, using this picture, one would expect that at zero pressure and T = 0 K, the system exists in a crystalline state with all the particles occupying the outer, deeper potential well. As is normal for any substance, the application of pressure would force the particles into greater confinement, in this case into the inner well, creating a higher overall density of the system, as in the higher density ice polymorphs. More inter- estingly, keeping the pressure fixed at not too high a value and raising the temperature from T = 0 K, particles in the outer well tend to ‘boil’ into the inner well, increasing the overall density. This initial density increase through the low temperature region, fol- lowed by the inevitable expansion at high temperatures, would then produce a density maximum.In the original Takahashi there is a hard core for 0 < x < a and the potential is zero for x > 2a. Keeping this picture, but extending the model to two wells, we divide the region between a and 2a into three equal parts, an inner well, a plateau and an outer well. The potential can then be written as, a; O<x<a; hard core -6; a < x < 4a/3 inner well u(x) = I0; 4a/3 < x < 5a/3 plateau (9)-3s; 5a/3 < x < 2a outer well 0; x > 2a The quantities 3, and E are positive. The integral for K yields 1 IC = -e”4 + O(l -4) -P(1 -8)(l -491 BP where 8 = exp( -PPa/3) and 4 = exp(P&). Using eqn. (S), the equation of state becomes Pa 6[1 -4 -8(2 -38)(1 -4’)]P(1-a) = kBT + -3 4 + O(1 -4) -Q2(1 -8)(l -4A) It is easy to show that this equation reduces to the expected result in all appropriate limits; hard sphere, single well, ideal gas, etc.Studying eqn. (1 l), at low temperatures it is easy to see that the zero temperature limit of volume per particle is a discontinuous function of pressure, P > P, 1 = p3; P = P, 5a/3; P < P, where P, is a critical pressure equal to 343, -1)/2a. Physically, the three possibilities correspond to all the particles being in the inner well, in both wells with equal probabil- ity, or in the outer well. Of course, as soon as the temperature is not zero, the discontin- uity in volume disappears and, as would be expected, the volume is a continuous monotonically decreasing function of the pressure.More interestingly, for pressures lower than critical, a density maximum as a func- tion of temperature arises from eqn. (1 1). To see this by a specific example, we choose A = 10 and P = 10&/a.Since this pressure is less than the critical pressure, which equals 13.5~/a,a density maximum is indeed obtained. This is seen in Fig. 2, where palm is plotted as a function of k, T/e. C. H. Cho et al. 0.64 I 1 0.52 I I I I I 1 3 5 7 9 reduced temperature Fig. 2 The reduced density palm as a function of reduced temperature icB TIEfor A = 10 and for P = 10~/afor the double well model. For explanation of symbols, see text. Though it is, of course, unrealistic to compare p(T)for the one-dimensional modd with that of real water, it is of interest to note that by appropriate parameter scaling of the one-dimensional result, one can match exactly the p(T)curve of the real liquid, both at normal and elevated pressures.There is thus no doubt that this can also be done for a three-dimensional second-neighbour double well computational model for water. In fact, increasing the pressure in the one-dimensional model shows that the density maximum becomes broader and shifts to lower temperatures, finally disappearing at sufficiently high pressures, i.e. when P = P,. This is exactly the type of behaviour observed in the real liquid.40 Three-dimensional double-well potential The revised potential must not only depend on the distance, but also on the angles.Here, we start with an angular independent potential in order to become familiar with the problem from a computational point of view. Once the proper type of potential is developed, it will be possible to solve the molecular-level problem of liquid water under a variety of perturbations and thermodynamic conditions. In these preliminary simula- tions, we first choose a very elementary liquid argon-type potential with a double well, where the quantities A, rG,and B are adjustable parameters. The parameter A corre-sponds to the second well depth, and B defines the width of the second well. The param- eter rG is chosen depending on the distance between the first and second neighbours. We can adjust these parameters to obtain results that best show the density maximum. Clearly, an angular dependence is required for a better representation of the water- water intermolecular potential.Though relevant information can be obtained using the simpler NVE or NVT molecular dynamics algorithms, a direct determination of the density at constant pres- sure and temperature requires use of NPT ensemble methods in MD simulations devel- oped by Evans and M~rris.~~,~~ The NPT MD simulations have been carried out on the system of 256 atoms interacting with the double well pair potential of eqn. (13), whose interactions were truncated at a fixed value of r equal to 2.50. Since we are mainly interested in the density here, and not the dielectric properties, the long-range Liquid water and biological systems corrections (r > 2.50) for energies and pressure were calculated by integration over the potential cut-off, assuming that the pair radial distribution function has become unity.Periodic boundary conditions (PBC) and the minimum image convention were utilized in the usual way.43 The starting configuration for each temperature is an fcc lattice structure with a density chosen to be approximately the density of the liquid at the given T and P. Initial atomic velocities were assigned according to the Maxwell-Boltzmann distribution. An equilibration phase of 5000 time steps was required to ‘melt’ the lattice. An NVE run with intermittent temperature scaling to a desired temperature was used for this purpose. The time-step size in the NVE simulations was At* = 0.0046, corresponding to At = 10 fs for the present problem.The results of such simulations were found to be independent of the starting configurations. The NPT simulations started from the con- figuration prepared by the NVE runs and were carried out over a duration of 10000 integration steps with intermittent momentum and pressure scaling. Another 10000 time steps were needed to get average thermodynamic quantities. From these calculations, a density maximum has been obtained, but longer run times are required to ensure that equilibrium has been established in these computationally intensive runs. Also, angular dependent potentials, but still with a one-site ‘water’ model, are being tested. Preliminary work on this type of potential will be carried out using the much less intensive NVE ensemble method, realizing of course that, although the pressure changes contain similar information as the volume changes in NPT, the pressure is extremely sensitive to the input parameters. All these preliminary calculations in this rather unknown computational territory will provide the required experience for reaching the final goal of developing a water model showing the correct temperature- pressure dependent density.Conclusions What is most amazing about this approach is that almost everyone in molecular-level water research seems to be ‘looking the other way’ rather than dealing directly with the intuitively simple experimental truths concerning liquid water that are now available.When new bulk water models, containing rapidly responsive polarizability, flexible intramolecular bonding and realistic second-neighbour characteristics, have been opti- mized so that the various liquid-ice phase transitions and thermodynamic properties of the liquid over an extended range of temperature and pressure are realistically produced, it will be time to ‘start over again’ applying such models to various interfacial states of water, including drug design, protein folding problems, etc., of biological interest. The intermolecular bonding characteristics of water that determine the structure and dynamics of the various liquid-ice and ice-ice phase transitions and the anomalous properties of the bulk liquid must be particularly important when the liquid is subjected to an interfacial perturbation.For example, it has been claimed by many workers, using currently available MD models, that perturbations on the structure and dynamics of water extend no more than ca. 10 A from a surface. With the new features built into the water model, one may well ask if this 10 A value is still a valid assessment. References 1 Y. Marechal, in Hydrogen Bond Networks, ed. M. C. Bellisent-Funel and J. Dore, NATO AS1 Series, Kluwer, Dordrecht, 1994. 2 J. Jonas and A. Jonas, Annu. Rev. Biophys. Biomol. Structure, 1994,23,287. 3 H. J. C. Berendsen, Natuurkd. Voordr,, 1981,59,85. 4 M. C. Gutzwiller, Phys. Today, 1995,48, 15. 5 S. R. Billeter, P. M. King and W. F.van Gunsteren, J. Chem. Phys., 1994, 100,6692. C. H.Cho et al. 27 6 A. Wallqvist and P-0. Astrand, J. Chem. Phys., 1995, 102, 6559. 7 S-B. Zhu, S. Singh and G. W. Robinson, Adv. Chem. Phys., 1994,85,627. 8 A. G. Kalinichev and K. Heinzinger, in Thermodyn. Data, ed. S. Saxena, Springer, New York, 1992, ch. 1. 9 M. D. Morse and S.A. Rice, J. Chem. Phys., 1982,76,650. 10 J. R. Reimers, R. 0.Watts and M. L. Klein, Chem. Phys., 1982,64,95. 11 S.-B. Zhu and G. W. Robinson, Proc. Znt. Supercomputing Znst., 1988, I, 300. 12 J. S. Rowlinson, Trans. Faraday SOC., 1951,47, 120. 13 P. Cieplak, P. Kollman and T. P. Lybrand, J. Chem. Phys., 1990,92,6755. 14 G. DArrigo, G. Maisano, F. Mallamace, P. Migliardo and F. Wanderlingh, J. Chem. Phys., 1981, 75, 4264.15 M. Vedamuthu, S. Singh and G. W. Robinson, J. Phys. Chem., 1994,98,2222. 16 G. S. Kell, J. Chem. Eng. Data, 1975,20,97. 17 M. Vedamuthu, S. Singh and G. W. Robinson, J. Phys. Chem., 1994,98,8591. 18 G. S. Kell, J. Chem. Eng. Data, 1967, 12, 66. 19 M. Goldblatt, J. Phys. Chem., 1964,68, 147. 20 M. Vedamuthu, S. Singh and G. W. Robinson, J. Phys. Chem., 1996,100,3825. 21 M. Vedamuthu, S. Singh and G. W. Robinson, J. Phys. Chem., 1995,99,9263. 22 T. DeFries and J. Jonas, J. Chem. Phys., 1977,66,5393. 23 L. Bosio, S.-H. Chen and J. Teixeira, Phys. Rev., 1983, A27, 1468. 24 F. Sciortino, A. Geiger and H. E. Stanley, Phys. Rev. Lett., 1990,65,3452. 25 J. A. Barker and R. 0.Watts, Chem. Phys. Lett., 1969,3, 144. 26 A. Rahman and F.H. Stillinger,J. Chem. Phys., 1971,553336. 27 A. Ben-Naim and F. H. Stillinger, in Water and Aqueous Solutions, ed. R. A. Horne, Wiley Interscience, New York, 1972. 28 F. H. Stillinger and A. Rahman, J. Chem. Phys., 1974,60, 1545. 29 M-C. Bellissent-Funel and L. Bosio, J. Chem. Phys., 1995, 102, 3727. 30 B. Kamb, in Structural Chemistry and Molecular Biology, ed. A. Rich and N. Davidson, W. H. Freeman and Co., San Francisco, 1968, pp. 507-542. 31 0.Mishima, L. D. Calvert and E. Whalley, Nature (London), 1984,310,393. 32 D. Eisenberg and W. Kauzmann, The Structure and Properties of Water, Oxford University Press, New York, 1969. 33 M-P. Bassez, J. Lee and G. W. Robinson, J. Phys. Chem., 1987,91,5818. 34 C. H. Cho, S. Singh and G. W. Robinson, Phys. Rev. Lett., 1996,76, 1651. 35 L. Tonks, Phys. Rev., 1936,50,955. 36 H. Takahashi, Proc. Phys.-Math. SOC.Jpn. 1942,24,60. 37 See,for example, E. H. Lieb and D. C. Mattis, Mathematical Physics in One Dimension, Academic, New York, 1966. 38 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, Oxford, 1990. 39 K. Huang, Statistical Mechanics, Wiley, New York, 1987. 40 H. Kanno and C. A. Angell, J. Chem. Phys., 1980,73,1940. 41 D. J. Evans and G. P. Morriss, Chem. Phys., 1983,77,63. 42 D. J. Evans and G. P. Morriss, Phys. Lett., 1983,98A, 433. 43 W. W. Wood, in Physics of Simple Liquids, ed. H. N. V. Temperley, J. S. Rowlinson and G. S. Rush-brooke, North-Holland, Amsterdam, 1968, ch. 5. PaDer 6/00067C: Received 3rd Januarv. 1996

ISSN:1359-6640    

DOI:10.1039/FD9960300019

出版商:RSC    

年代:1996

数据来源: RSC

摘要:

Faraday Discuss., 1994,103,2940 Water hydrogen-bond dynamics close to hydrophobic and hydrophilic groups Alenka Lumr Department of Chemistry, University of California, Berkeley, California 94720-1460, USA We discuss the analysis of molecular dynamics calculations where we simu- late the kinetics of the breaking and forming of hydrogen bonds in distinctly different environments: liquid water and concentrated aqueous solution of DMSO. In our analysis, we consider reactive flux correlation functions com- puted for a variety of specific conditions and identify rate constants for bond making and breaking in terms of the liquid’s molecular dynamics. According to the proposed mechanism of hydrogen-bond kinetics, hydrogen bonds most frequently break during a process of switching allegiance with a newly formed bond replacing the broken one.Simulations reveal bond dynamics in the mixture to be significantly slower than in pure water. This is inter- preted in terms of reduced likelihood of fluctuations in the hydrogen-bond network, related to the presence of free hydrogen-bonding sites, which par- ticipate in the process of switching allegiances. Implications of our analysis for future experimental work are briefly discussed. 1 Introduction The dynamics of hydration in aqueous solutions involves the making and breaking of hydrogen bonds. We have studied the statistical evolution of these processes. The ele- mental process of the breaking and forming of hydrogen bonds in liquids has been probed indirectly through a variety of experimental techniques.’ Although typically limited to classical models, the method of molecular dynamics is a powerful tool to obtain a more direct microscopic picture of hydrogen-bond dynamics in molecular liquids.2 Factors controlling the dynamics can be determined from trajectory calcu- lations of correlation function^.^ Analysis of computer generated Newtonian trajectories in liquid water reveal long time non-exponential kinetics of hydrogen bond^.^-^ The relaxation of hydrogen bonds in water therefore appears very complicated, with no simple characterization in terms of a few relaxation times or rate constants.Most of the speculations about this apparently complicated kinetics pointed to cooperativity between neighbouring hydrogen bonds as the source of the ~omplexity.~ Our recent work5 demonstrates that the mechanism lies elsewhere, with virtually self evident coup- ling that exists between translational diffusion and hydrogen-bond dynamics. Diffusion governs whether a specific pair of water molecules are near-neighbours. Hydrogen bonds between such adjacent pairs form and persist at random, with average lifetimes determined by rate constants for bond making and breaking.In this paper, our model and analysis’ is applied to study the hydrogen-bond dynamics in the hydration process of dimethyl sulphoxide (DMSO). DMSO is a solute that is miscible in all proportions and forms strong hydrogen bonds with water, but also has methyl groups which do not form associative bonds.The water-DMSO mixture therefore provides a simple molecular model for studying competing effects of hydro- 29 30 Water hydrogen-bond dynamics phobicity and hydrophilicity on hydrogen-bond kinetics. An analysis of computed and experimental distribution functions revealed an enhancement of water-water hydrogen-bond correlations in the presence of hydrophobic and hydrophilic groups on DMS0.'v9 The timescales associated with these correlations are examined in Section 3. Recently performed computer simulations, presented here for the first time, show that on long timescales the relaxation behaviour of hydrogen bonds in the concentrated aqueous solutions of DMSO exhibit the analogous non-exponential kinetics, as was observed in pure water. Adopting the same analysis as for pure water, we determine rate constants for breaking and making of water-water and water-DMSO hydrogen bonds in the mixture in the same section.First, however, the methods of the analysis, and the applica- tion to the hydrogen-bond kinetics in pure liquid water are outlined in Section 2. 2 Hydrogen-bond kinetics in liquid water We studied the time dependence of hydrogen-bond dynamics in liquid water using com- puter simulations, 596 In particular, we performed molecular dynamics computations of hydrogen-bond population correlation functions and a variety of configuration specific reactive flux correlation functions. In the following, we describe basic methods and definitions introduced in this analysis.2.1 Reactive flux correlation functions and rate constants Let us first briefly sketch the correlation function formalism for the calculation of the rate constants of chemical reactions in solution (in this case hydrogen-bond making and breaking), treated in the framework of classical statistical mechanics. According to a configurational criterion for whether a particular pair of molecules is hydrogen bonded, we can define a hydrogen-bond population operator, h. It is unity if the particular tagged pair of molecules is hydrogen bonded, according to the adopted and is zero otherwise. The average number of hydrogen bonds in an equilibrium of N water molecules is $N(N -l)(h), where (h) denotes the time average of h. In the dynamical equilibrium, h fluctuates in time.These fluctuations are characterized by the hydrogen-bond corre-lation function, c(t): c(t)= <h(O)h(t))/(h) (1) where the angular brackets indicate the ensemble average over initial times. c(t)denotes the probability that the hydrogen bond is intact at time t, given it was intact at time zero. At equilibrium, the probability that a specific pair of molecules is bonded in a large system is negligibly small. c(t) therefore relaxes to zero. According to the fluctuation- dissipation theorem (or equivalently, Onsager's regression hypothesis)," the time evolu- tion of c(t)is the same as that for an initially prepared non-equilibrium concentration of hydrogen bonds. The rate of relaxation to equilibrium is characterized by the reactive flux correlation function, k(t): k(t) = -dc/dt = (j(O)[l-h(t)])/(h) where j(0) = -dh/dt It=* is the integrated flux departing the hydrogen-bond config- uration space at time zero.The function k(t) is the average of this integrated flux for those trajectories where the bond is broken at a time t later, hence the terminology 'reactive flux '. Its zero time value is the statistical transition-state theory (TST) approx- imation to the rate constant, kTsT .l2-I3 Dynamical corrections to the TST estimate of the rate constant are conveniently expressed in terms of the transmission coefficient K, which relates the classical TST A. Luzar 31 exp~ession’~to the exact forward-rate constant :12 k = Kkm (3) Roughly speaking, K is the fraction of successful or undeterred activated trajectories.For the simplest dynamical process, functions like k(t) will settle to a plateau value after an initial transient relaxation period. This plateau value would then determine the rate constant, k, for the process. The existence of a well defined plateau in time would imply a linear rate That is, if k(t)exhibited a plateau value k, then at times long compared with the initial decay, k(t)e k exp(-kt), with l/k corresponding to the average hydrogen-bond lifetime. It is evident, however, from our calculations (Fig. 1) that k(t)does not relax exponen- tially. Beyond the transient period of ca. 0.2 ps, which corresponds to librations on a timescale of less than 0.1 ps, and inter-oxygen vibrations on a timescale of 0.1-0.2 ps, the slope of log k(t) increases monotonically with time.’ Note that the behaviour of k(t) beyond the short time transient period is invariant to physically reasonable changes in hydrogen-bond definition.5*6 We,18 as many others,’ have speculated that the source of non-exponential kinetics of tagged hydrogen bonds in water is due to correlations between different hydrogen bonds.We explored this possibility in detail by partitioning the trajectories that contrib- ute to the reactive flux according to the particular environment of the hydrogen bond? These calculations unambiguously show that what happens to the dynamics at the tran- sition state and beyond is to a good approximation uncorrelated with the initial bonding state.Lack of correlations between the dynamics of a tagged hydrogen bond 0 4 8 12 tips Fig. 1 Reactive flux correlation functions, k(t), for water at room temperature, and density 1 g cm-3, sampled over 240 ps of physical time. SPC16 potential model was employed, using periodic boundary conditions and Ewald sums.” The inset panel shows the portion of the trajectory, from which k and k’ were evaluated according to eqn. (7). Water hydrogen-bond dynamics and that of its neighbours should lead to a single relaxation time and therefore these dynamics should be characterized by well defined rate constants. And indeed, rate con- stants do exist, although the rate function k(t) clearly exhibits non-exponential kinetics.The reason for this apparent inconsistency has to do with diffusion. Because of diffusion, recrossings in and out of the hydrogen-bond configuration space never stop and these recrossings affect k(t)at all times. The source of significant non-exponential relaxation for water at ambient conditions is therefore the coupling between hydrogen-bond population and diffusion.’ In particu- lar, two molecules can diffuse apart only after the hydrogen bond between them breaks, and a broken bond can reform if a molecule reverses its direction and diffuses back to its partner. This aspect of hydrogen- bond dynamics introduces a continuum of relaxation times. Quantitative consequences of diffusion were examined by partitioning the contrib- utors to k(t) according to whether or not a pair has moved apart after its bond has broken.From this partitioning we have computed a restricted reactive flux function, kin(t) k(t) = (XO)C1 -h(t)lH(t))/(h) (4) where H(t) is unity if the oxygen-oxygen distance of the tagged pair is not larger than Rgb (Table 1) and it is zero otherwise. The probability at time t that a pair of initially bonded molecules are now unbonded but remain separated by less than Rgb, is evalu- ated by n(t): n(t)= dt’ki,(t’) (5)L The time dependence of this population shows (Fig. 2) that there is quite a significant amount of bonds that break, but the molecules that previously had a common bond remain nearest neighbours. The probabilities c(t)and n(t)correspond to local populations that can interconvert.The simplest possible kinetics of this interconversion in the first coordination shell of water molecules is : dc/dt = -kc(t) + k’n(t) (6) where k and k’ are the respective rate constants for the breaking and making of hydro- gen bonds between a near-neighbour pair of molecules. As the interconversions described by eqn. (6) exclude significant translations, they therefore involve molecular rotations. At very short times translational diffusion does not play a significant role. Therefore we can extract the rate constants from the small time limit of eqn. (6). The only instantaneous contribution to n(t)is the breaking of the bond. Diffusion occurs one step later in time. Hence, the rate of change of dn/dt is equal to the rate of change of hydrogen-bond breaking, k.Determining the slope of k(t) at time zero and its value at time zero by taking into account the boundary conditions, c(0) = 1 and n(0)= 0, we Table 1 Cut-off distances between oxygen atoms, Rgb and between oxygen and hydrogen atoms, Rg)H,and 0-H-* -0angle @“), used in our geometrical definition of a hydrogen bond for pure water, water-water pair and water-DMSO pair in lDMSO : 2H,O (Their determination is explained in ref. 8 and 18) pure water 3.5 2.45 30 water-water pair in the mixture 3.5 2.45 30 water-DMSO pair in the mixture 3.2 2.40 30 A. Luzar 33 0.15 s 0.10 he cv E 0.05 1------0.00 " 0 5 10 15 20 ups Fig. 2 Populations n(t),eqn.(9,for pure water (solid line), n,,(t) for 11 pair in lDMSO :2H20 mixture (dotted line), and n,,(t) for 12 pair in the mixture (dashedline) derive the small time limit of k(t), immediately following the transient time: k(t) z k exp[-(k + k')t] (7) Finding the longest straight line we can draw through the data on Fig. l(inset panel), we estimate (within the statistical uncertainties) the initial slope and extract k and k' (Table 1). It is important to note that the same values for k and k', which we obtain from the short time analysis [eqn. (7)] also optimize the agreement between eqn. (6) and simula- tion results in the range of 1-20 PS.~ Bond making rate constant k' is only 30-40% bigger than bond breaking rate con- stant, k. That means that in the simulation, the free energy of the bonded state is not much lower than that of unbonded nearest neighbour.In the unbonded state, the bond between the tagged pair of molecules no longer exists. Instead, these molecules are now bonded to other nearby partners. In the bond broken state, bonded water molecules have simply switched allegiances. The average frequency of switching allegiance is k + k', which is 1.7 ps-' in the simulation m0de1.~ We have thus confirmed and quantified what Stillinger wrote more than 15 years ago: '. .. the hydrogen-bond network has a local preference for tetrahedral geometry, but it contains a large proportion of strained and broken bonds. These strained bonds appear to play a fundamental role in kinetic properties, because their presence enhances prospects for molecules to switch allegiances, trading a bond here for one there and thus altering the network topology'.'' By drawing only three water molecules for the sake of simplicity, and monitoring the bond between the tagged pair of molecules 1 and 2, the picture that emerges from our calculations5 is sketched schematically below (Fig.3). According to Fig. 3, molecule 1 was not a nearest neighbour to molecule 3 at time zero (I).All three molecules were in the preferred tetrahedral arrangement. Every once in a while molecule 3, which carries an empty hydrogen-bond site, moves closer to mol- ecule l (11). Transition state (TS), a state that must be visited during the passage from one stable state to another, occurs because fluctuations in the environment from per- fectly ordered state make an empty hydrogen bonding site accessible.The network of hydrogen bonds is bent and strained. A correlated fluctuation for bond breaking to happen is not required.6 There is a sufficient concentration of this necessary disorder present on average, providing the opportunity for bond breaking by a switch of alle-giance (111).After the switch, the strained network relaxes and pulls molecules 1 and 2 Water hydrogen-bond dynamics R<Rg I I1 TS I11 IV Fig. 3 Mechanism and kinetics of water pair dissociation. Circles represent three hydrogen- bonded (solid lines) water molecules. The dashed line is used in representing the situation where bonds between molecules 1 and 2, and between 1 and 3, are equally likely.Through computer graphics visualization, we have noted that the formation of a new hydrogen bond tends to occur between two water molecules that first have a third water as a common hydrogen-bonded neigh- bour. The equation below the schematic picture denotes the corresponding populations and relax- ation times that are measured in molecular dynamics. apart, so that they are no longer nearest neighbours. Again, the three molecules find themselves in a preferred tetrahedral arrangement, but this time with molecules 1 and 3 as nearest neighbours (IV). To the extent that n(t) is non-zero, the two tagged molecules 1 and 2 are no longer bonded to each other but remain nearest neighbours.The population n(t)is therefore a measure of local strain in the hydrogen-bond network. n(t) relaxes not only by conver- sion back to the bonded state, but also by diffusion. If this diffusion were to occur slowly enough, the sum of c(t) + n(t), which represents the probability that initially bonded pairs of molecules stay nearest neighbours at time t, would appear to be a constant. The separate populations, c(t) and n(t),would each change by interconversion only, relaxing as single exponentials. We can estimate the relevant timescale of diffusion in terms of the elementary diffusion time, z, i.e. the average time it takes for a molecule to diffuse from the domain (of width a) in which a bond with another tagged molecule can exist. In the parlance of ref.20, z represents the mean first passage time in an escaping process. It can be approximated by the average diffusion time (averaged over initial positions) from a sphere of radius a, z x a 15D.The self diffusion constant D is larger than cm2, and a is less than ca. 1.5 1. These numbers suggest that z is less than 1 ps for water at standard conditions, a value that is comparable to l/k and l/k’. And indeed, the diffu- sion model for hydrogen-bond kinetics, developed in ref. 5, qualitatively describes our simulation results for two independent correlation functions, i.e. k(t) and ki,(t) for liquid water, with the value of z = 0.4[1 & 0.25)~~. While the kinetics in the first coordination shell has been established to be very simple [eqn.(6)],relaxation of n(t) by diffusion occurs through a continuum of time- scales. We can estimate the degree to which diffusion is significant and we have done so elsewhere. Most importantly, however, our analysis establishes the existence of well defined rate constants, k and k’, and their connection to microscopic correlation func- tions [eqn. (6)].This connection can be used in general to analyse the dynamics of other hydrogen-bonded liquids. In the next section we use it to analyse the kinetics of hydro- gen bonds in concentrated aqueous solutions of DMSO. In this case, we are dealing with two distinct types of hydrogen bonds corresponding to water-water (ll), and water-DMSO (12)pairs. Here, the quantities that appear in eqn. (1)-(7)refer to a speci- fied type of bond which will be denoted by subscripts (ij)= (11) or (12).A. Luzar 35 3 Hydrogen-bond kinetics in water-DMSO mixtures In our earlier work, we studied the water-DMSO system by neutron diffraction with H/D isotope substit~tion,~*~l and by theoretical model- by computer simulati~n,~~~ ling.22*23In most of our investigations, the concentration of lDMSO :2H20 was used, since this concentration corresponds to extrema in several thermodynamic24925 and dielectric26 properties, and also, at this high concentration, all water molecules are in close proximity to a DMSO molecule and any significant effect on water structure should be readily observable. As a result of these investigations it is now much clearer what happens when DMSO and water mix.In spite of big differences between molecular geometries of DMSO and water, even at high concentration, DMSO fits rather well into the structure of water, with only minor changes in the local coordination of water mol- ecules. Water readily forms hydrogen bonds with an oxygen on DMSO, but can also rather easily form hydrogen-bonded cages around the hydrophobic methyl groups of DMSO.’ 3D histogram extracted from neutron diffraction data of the distribution of water molecules around DMSO molecule in solution points to accumulation of water around both the oxygen atom of DMSO and around the methyl groups of DMS0.27 Let us now consider dynamical consequences of these structural effects. Water near the hydrophobic groups has less opportunity to make hydrogen bonds.Water molecules that are hydrogen bonded to an oxygen atom on DMSO tend to stay in that bond, because water-DMSO hydrogen bond is stronger than water-water hydrogen bond.22*25-28Clearly, the environment of an average water molecule in the mixture con- tains more confined regions and therefore fewer accessible hydrogen-bond sites. There- fore, the likelihood of fluctuations in the hydrogen-bond network that can lead to hydrogen-bond breaking and to switching allegiance is lower in the mixture than it is in pure water. Thus, we should expect slowing down of water bond dynamics in the pres- ence of DMSO. Functions kiJ(t) and niJ(t)computed for the water-water pair in the mixture (from now on denoted as 11 pair), and water-DMSO pair in the mixture (12 pair), presented in Fig.2 and 4, confirm this expectation. 3.1 Potentials and method The interactions between water and DMSO molecules were assumed to be composed of pairwise additive potential functions between atomic sites. For water-water interactions we used the rigid SPC model.I6 For DMSO-DMSO interactions, the P2 potential developed in our earlier w~rk,~~~’ was used. For water-DMSO interactions, we used Lorentz-Berthelot combining rules.8 No parameter optimization was carried out for the mixture, once the parameters for pure components have been determined. Classical molecular dynamics simulation was performed on 162 H20 + 88 DMSO molecules at 300 IS, and at the experimental density for the mixture.25 The analysis of dynamical results was carried out on 20 consecutive trajectories, performed in the microcanonical ensemble, each of 10 ps in length, after 200 ps were completed in the canonical ensemble.The Nosk-Hoover thermostat was used to control the temperature for the 200 ps portion. Equations of motion were integrated using a velocity predictor-corrector method with a time step of 1 fs. Time derivatives of the hydrogen-bond correlation functions were calculated each time step. Periodic boundary conditions were used together with the minimum image convention for non-Coulombic interactions. Ewald summation was applied to evaluate the long-range Coulombic forces.’ 4 Results According to time evolution of niJ(t)for the 11 and 12 pairs in the mixture in compari- son with n(t)for pure water (Fig.2), breaking of a hydrogen bond involves less temporal strain in the mixture than it does in pure water, because the mixture is, on average, more Water hydrogen-bond dynamics strained than pure water. The temporal strain, however, relaxes more slowly than it does in water (in terms of absolute times) since the mixture is more constrained and tighter. Reactive flux correlation functions computed for the 11 and 12 pairs in the mixture kilt) show (Fig. 4) a rapid transient decay to values more than an order of magnitude below its initial value. The inset panel shows that the transient period in the mixture is over within 0.5 ps. Reactive flux functions do not show a typical plateau that would be found if we would be dealing with a single elementary process.Slopes of logarithms of kiit) for both the 11 and 12 pairs increase monotonically with time. The conclusions that we draw from these calculations are significantly different than those we obtained from 20 times shorter preliminary runs.* The reason for the ever changing slope of In ki,(t)is diffusion, as we discussed in Section 2. The self-diffusion constant for water in 1DMSO :2H20 mixture is ca. three times less than the corresponding value in pure water.30 The elementary diffusion times zij, which are expected to scale as 1/D, should therefore be ca. 1.5 ps. Note that by using the diffusion model for hydrogen-bond kinetics, developed in ref. 5, the best fit (not shown in this work) to molecular dynamics results for two independent correlation functions, kiJt) and niJt), at longer times is obtained with zij z 1-1.5 ps.These results will be presented elsewhere. Here we concen- trate on determining rate constants for the 11 and 12 pairs in the mixture. In the previous section, we discussed the formalism where we eliminate the effect of diffusion to extract rate constants for the breaking and making of hydrogen bonds, k and k', eqn. (7). We use this generally applicable procedure to determine the correspond- 0 2 4 6 tlps Fig. 4 Reactive flux correlation functions for 11 pair in lDMSO : 2H,O mixture (thick line), and 12 pair in the mixture (thin line), sampled over 200 ps of physical time.The inset panel shows portions of trajectories, from which k, and kfj were evaluated according to eqn. (7). In the main figure, In k12(t)has been shifted down by 3. A. Luzar 37 Table 2 Rate constants for breaking and making hydrogen bonds, k and k', deter-mined from eqn. (7), the corresponding TST estimates, k,, and transmission coeficients, ~c[eqn.(3)] k/ps -' k'lps -' kmlps-Ic water 0.7 (1 f0.10) 1.0 (1 & 0.20) 5.1 (1 +_ 0.01) 0.14 (1 f0.11) 11 pair 0.2 (1 & 0.30) 1.0 (1 +_ 0.35) 3.2 (1 f0.15) 0.06 (1 f0.45) 12 pair 0.1 (1 0.30) 0.8 (1 k0.30) 2.1 (1 f0.15) 0.05 (1 f0.45) ing rate constants, k, and kij in the mixture. Their values are listed in Table 2. The values for hydrogen-bond breaking constants in the mixture are much smaller than in pure water, while the values for hydrogen-bond making constants remain approximately the same.Switching allegiance (Fig. 3) takes less time if many available sites exist in the neighbourhood. In the mixture, there are fewer neighbours with which a new bond can be formed. It therefore takes longer until an appropriate partner for the formation of a new bond is encountered. The rate of returning to the previous tagged bond (measured in terms of k') appears to be less affected by these differences. One can see significant deviations from the statistical transition-state approximation for the hydrogen-bond breaking rate constant (Table 2). The values of transmission coefficients, estimated from the ratio of hydrogen-bond breaking constant k, and its transition-state theory approximation kTsT, indicate that the dynamical corrections to the statistical TST depend slightly (within a factor of two) upon the concentration and the type of hydrogen bond. The observed differences in K might however also be due to the fact that we do not identify the optimal transition state, as revealed by low values of K.And it may well be that all the difference in K for pure water compared with the mixture case stems from having even less optimal transition states in the mixture, than in pure water. This point deserves further investigation. Note that it is a very difficult task to locate the transition state(s) in complex systems with many degrees of freedom. An efficient stochastic sampling method proposed by Pratt31 is currently being applied in our laboratory in order to determine transition states.From determined rate constants we can estimate the difference in the free energies going from state I to state I11 (Fig. 3), AF,, i.e. free energies for switching allegiance. According to the detailed balance condition exp(-PAF,) = k/k' (8) where denotes the inverse of Boltzmann constant times the absolute temperature. In the mixture, the ratio k/k' is replaced by the ratio of corresponding rate constants for the pair under consideration. The values of AF, for pure water, the 11 and the 12 pair in the mixture, are presented in Table 3. From molecular dynamics data for pure water we also estimate the activation free energy at the apparent transition state, AF* exp(-PAF*) = P,dPs, (9) where PTSis the probability for being at the transition state (TS), which is the ratio between the time the TS is visited and the total time of the simulation, and Pss is the probability for being at a stable state, i.e.hydrogen bonded. Note, however, that we do not know the optimal choice of the transition state, TS. This leads to an uncertainty in our estimate for P, which is somewhat bigger than at the true TS. The optimization of transition states should therefore lead to a value of AF* higher than the value listed in Table 3. Planned studies of temperature dependence for exp( -PAF*), eqn. (9), will deter- mine the activation energy AE* for water from simulation. This will allow an interesting comparison with experimental values for AE*, estimated from the Rayleigh scattering32 and quasi-elastic incoherent neutron scattering3 experiments.Water hydrogen-bond dynamics Table 3 Free energies for switching allegiance, estimated from eqn. (8), AF,, and activation free energies at the transition state, estimated from eqn. (9) and (1l), AF*, for water, 11 and 12 pairs in the mixture AFJkJ mol-' AF*/kJ mol-' water 0.9 (1 f0.50) 13.4 (1 & 0.02) 11 pair 4 (1 i-0.30) 16.5 (1 & 0.07) 12 pair 5 (1 k0.25) 18.5 (1 & 0.07) Errors in AF,, AFf,, and AFT,, were esti- mated from errors in rate constants, k and k'. Error in AF* for pure water was estimated from differences between the results for various length of trajectories. The formulation for the rate constantI4 partially includes the influence of the con- densed phase environment through the free energy of activation, AF*, as well as the effect of dynamics at or near the transition state: k = IC exp(-PAF*) (10) If we, in the first approximation, assume that the dynamical features which are mani- fested in the value of the transmission coefficient K do not change from one system to the other, we can attribute all the changes to a statistical effect through AF*.Therefore, we can estimate the free energies of activation for the 11 and 12 pairs from the ratio of the values of bond breaking constants of the 11 and 12 pairs us. the corresponding value in pure water : k, ,/k = exp[ -fl(AF7, -AF*)] k,,/k = exp[ -P(AFT2 -AF*)] (1 1) The transition state occurs due to the fluctuations in the hydrogen-bond network that make an empty hydrogen-bond site accessible.The probability for finding such an empty site for the 11 pair in the mixture is lower compared with pure water, as the 11 pair in the mixture is partly surrounded by bulky hydrophobic methyl groups on DMSO. The activation free energy for water in the mixture is therefore higher than in pure water. The increase in AFT, of ca. 3 kJ mol-' is due to entropic effects. The same consideration applies to the 12 pair, however, an additional energetic effect is also revealed. A larger increase of ca. 5 kJ mol-' in AFT, is consistent with earlier observ- ations that the energy of the hydrogen bond for the 12 pair is stronger than for the 11 pair.2 2.2 5.28 It would, of course, be interesting to consider separate estimations for acti- vation energies, AE*, and activation entropies, AS*.Studying the temperature depen- dence of k(t), needed for these calculations, is left for future investigation. Also, up to this point, our analysis of the effect of hydrophobicity on hydrogen-bond dynamics has been based on the net effect of adding DMSO to water. This, of course, is what is done in real experiments. With simulation, more can be done, such as decomposing reactive flux correlation functions according to hydrogen-bond propinquity to methyl groups. We conclude the discussion of our results by emphasizing that in the case of the 11 and 12 pair in the mixture, the hydrogen-bond lifetimes, l/kij, are much larger than the hydrogen bond deadtimes, l/kij, and the elementary diffusion times, zij.These results could imply that we have a rate-limiting step in the dynamics of hydrogen bonds in the mixture case. Coupling between hydrogen-bond dynamics and diffusion, however, pre- A. Luzar 39 vents the separation in timescales and concomitant monoexponential kinetics. As already pointed out, the experimentally determined diffusion coefficient for water in lDMSO :2H20mixture is ca. 3 times smaller than the corresponding value in water.30 Computed bond-breaking constants are between ca. 4 and ca. 7 times smaller than in pure water. Therefore, in the case of the mixture, the time allowed in a practical com- puter experiment covers only a part of the time interval in which relevant dynamics occurs.k, and niJ(t) in the mixture are both essentially smaller than in pure water, therefore k, ci,-(t) remains comparable to kijniJ-(t)and we cannot neglect the second term in eqn. (6). Finally, we would like to spur some discussion about how our model might be experi- mentally verified, and what its implications are for measurements and interpretations. We believe that our phenomenological model can be applied to interpret incoherent neutron scattering3 and perhaps transient vibrational spectro~copy.~~ Experimental verification of the model will, however, require coordinated application of two different techniques, because c(t) and n(t) are independent functions that vary, at least in the case of pure water, on the same timescale.The random jump diffusion model has been used by experimentalists to determine the residence time, i.e. the characteristic time between diffusive jumps of a water molecule, from quasi-elastic neutron scattering (QENS).34936 It is estimated to be ca. 2 ps at room temperature. This time might be compared with the characteristic residence time from our model, that is the average time a molecule spends in a domain in which a bond with another molecule can exist: l/k + z x 1.8 ps. As yet no rigorous way of deducing hydrogen-bond lifetimes from experimental mea- surements has been developed. Problems come from the fact that the experimental tech- niques are sensitive to residence time.In a liquid where hydrogen bonds lead to the formation of dimers, like formic acid, the residence time and the hydrogen-bond lifetime are not very different. In situations where several hydrogen bonds are involved in the coordination shell of a molecule, e.g. liquid water, the residence time is larger than the hydrogen-bond lifetime. For water, some attempts are made to obtain an indirect deter-mination of the hydrogen-bond lifetime by associating the hydrogen-bond lifetime with a molecular motion related to the breaking of the bond. The same assumption, namely that the mechanism for hydrogen-bond breaking is due to large amplitude librational motions, is made in the interpretation of both depolarized light scattering (DLS)32 and QENS.34 The very low frequency component in the DLS spectrum of liquid water was attributed to the hydrogen-bond making/breaking process.The hydrogen-bond lifetime as estimated from DLS for room temperature water is 0.6 This time might corre- spond to the time needed to switch allegiance in our model, e.g. 1/(k + k') = 0.6 ps. QENS, on the other hand, measures translational diffusion, rotational movements and hydrogen jumps corresponding to the breakage of a bond. Separation of these different contributions is, however, ambiguous. According to our model, which agrees with molecular dynamics calculations, there is a strong coupling between translational and rotational motions for water at room temperature. We hope that the extension of the model will lead to a more reasonable expression to interpret QENS data.For the mixture water-DMSO, no QENS measurements have yet been performed. We believe that there are several advantages in the interpretation of the data in the mixture case. First, our model predicts l/kll+ T~~ x 6 ps, and l/ki2 + r12x 11 ps for the characteristic residence times for the 11 and 12 pairs in the mixtures. In this system the residence times almost coincide with hydrogen-bond lifetimes estimated from the inverse values of the corresponding hydrogen-bond breaking constants, i.e. ca. 5 ps and ca. 10 ps. 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Luzar and D. Chandler, Nature (London), 1996,379,55. 6 A. Luzar and D. Chandler, Phys. Rev. Lett., 1996,76,928. 7 A. Geiger, P. Mausbach, J. Schnitker, R. L. Blumberg and H. E. Stanley, J. Phys. (Paris), 1984, 45, C7; F. Sciortino and S. Fornili, J. Chem. Phys., 1989,90,2786;I. Ohmine, J. Phys. Chem., 1995,99,6767;R. Lamanna, M. Delmelle and S. Cannistraro, Phys. Rev. E, 1994,49, 2841. 8 A. Luzar and D. Chandler, J. Chem. Phys., 1993,98,8160. 9 A. K. Soper and A. Luzar, J. Phys. Chem., 1996,100,1357. 10 M. Ferrario, M. Haughely, I. R. McDonald and M. L. Klein, J. Chem. Phys., 1990,93, 5156. 11 D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University, Press, New York, 1987.12 D. Chandler, J. Chem. Phys., 1978,68,2959; D. Chandler, J. Stat. Phys., 1986,42,49. 13 B. J. Berne, in Multiple Time Scales, ed. J. U. Brackbill and B. I. Cohen, Academic Press, New York, 1985, p. 419; P. Hanggi, P. Talkner and M. Borkovec, Rev. Mod. Phys., 1990, 62, 251, and references therein. 14 P. Pechukas, Annu. Rev. Phys. Chem., 1981,32, 159. 15 R. Zwanzig, Annu. Rev. Phys. Chem., 1965,16,67. 16 H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren and J. Hermans, in Intermolecular Forces, ed. B. Pullman, Riedel, Dordrecht, 1981, p. 331. 17 M. P. Alan and D. J. Tildesley, Computer Simulation ofliquids, Clarendon, Oxford, 1981. 18 A. Luzar and D. Chandler, in Hydrogen Bond Networks, ed.M. C. Bellisent-Funel and J. C. Dore, Kluwer Academic, Dordrecht, 1994, p. 239. 19 F. H. Stillinger, Science, 1980, 209, 451. 20 C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Science, Springer, Berlin, 1985. 21 A. K. Soper and A. Luzar, J. Chem. Phys., 1992,97, 1320. 22 A. Luzar, J. Chem. Phys., 1989,91,451. 23 A. Luzar, J. Mol. Liquids, 1990,46, 221. 24 M. F. Fox and K. P. Whittingham, J. Chem. Soc., Faraday Trans., 1974, 75, 1407; J. Kenttammaa and J. Lindberg, Suom. Kemistil, 1960, B33, 32; P. Westh, J. Phys. Chem., 1994,98, 3222; J. T. W. Lai, F. W. Lau, D. Robb, P. Westh, G. Nielsen, C. Tradum, A. Hvidt and Y. Koga, J. Solution Chem., 1995,24, 89. 25 J. M. G. Cowie and P. M. Toporowski, Can. J. Chem., 1964,39,2240. 26 E. Tommila and A. Pajunen, Suom. Kemistil, 1969, B41, 172. 27 A. K. Soper, Faraday Discuss. 1996, 103,41. 28 R. Fuchs, G. E. Mcray and J. J. Bloomfield, J. Am. Chem. Soc., 1961,83,4281. 29 A. K. Soper, A. Luzar and D. Chandler, J. Chem. Phys., 1993,99,6836. 30 K. J. Packer and D. J. Tomlinson, Trans. Faraday Soc., 1971,67, 1302. 31 L. R. Pratt, J. Chem. Phys., 1986,85, 5045. 32 0.Conde and J. Teixeira, J. Phys. (Paris), 1983,44, 525. 33 S. H. Chen, J. Teixeira and R. Niclow, Phys. Rev. A, 1982,26,3477. 34 J. Teixeira, M. C. Bellisent-Funel, S. H. Chen and A. Dianoux, J. Phys. Rev. A, 1985, 31, 1913. 35 S. Bratos and J. C. Leickman, J. Chem. Phys., 1995,103,4887. 36 S. H. Chen and J. Teixeira, Adv. Chem. Phys., 1986,64, 1; S. H. Chen in ref. 1, p. 289. Paper 61017725; Received 13th March, 1996

ISSN:1359-6640    

DOI:10.1039/FD9960300029

出版商:RSC    

年代:1996

数据来源: RSC

摘要:

Faraday Discuss., 1996,103,41-58 Neutron scattering studies of solvent structure in systems of chemical and biological importance A. K. Sopert ISIS Facility, Rutherford Appleton, Laboratory, Chilton, Didcot, Oxon, UK OX11OQX Empirical Potential Monte Carlo simulation (EPMC) (driven by a set of interatomic potential-energy functions which are perturbed continuously throughout the simulation in a manner such that the calculated distribution functions reproduce the measured distribution functions) is applied to two outstanding problems :the structure of liquids with ffexible molecules, and the hydration of organic molecules (DMSO in this case) in aqueous solu- tion. Providing the relevant neutron data are available, the system tackled by this procedure can be arbitrarily complex, at least in principle.1 Introduction Neutron diffraction is a powerful tool for investigating aqueous solutions and organic molecular liquids, because neutrons are scattered strongly by all the common constitu- ents of these liquids, C, 0, N, C1, H, etc. By performing separate measurements on stoichiometrically identical solutions but with different isotopic compositions, it is pos- sible to isolate from a complex diffraction pattern specific correlations associated with the isotopically substituted sites. Hydrogen isotope substitution is a particularly power- ful technique in this regard because of the large change in neutron scattering length with isotope (-3.74 fm for H, and +6.67 fm for D), because deuterium-labelled compounds are often readily available, and because hydrogen is a major component of many of the systems of interest.Moreover since hydrogen atoms tend to lie on the outside of mol- ecules and in specific directions relative to other atoms of the molecules, such as C, N and 0,the H-H and H-other correlation functions often contain vital information on both the internal conformation of the molecules of the liquid, and the relative orienta- tion of neighbouring molecules. The present paper will not dwell overly long on the mechanical process of extracting these correlation functions from a set of diffraction data: this process is not trivial, since various approximations are invoked,' but it is now a relatively routine process to get correlation functions with an acceptable degree of accuracy.Perhaps the largest uncer- tainty in the experiment arises from the nuclear recoil distortion which can only be estimated approximately. Instead, the emphasis here is on a more topical question: what useful structural information is contained in these distribution functions for complex liquids and how can it be extracted? Recent progress in this regard has been demonstrated by the develop- ment of the Empirical Potential Monte Carlo (EPMC) simulation method:2 this approach, which is analogous to Reverse Monte Carlo (RMC) ~imulation,~ models the 7 Also affiliated with Dept of Physics and Astronomy, University College London, Gower Street, London, UK WClE 6BT. 41 42 Neutron scattering studies at solvent structure material in question by generating distributions of molecules which reproduce the mea- sured site-site correlation functions.In the EPMC technique the simulation is per-formed exactly as a conventional Monte Carlo computer simulation of a fluid, i.e. acceptable configurations of molecules are determined by the potential energy of inter- action. The potential-energy function is the sum of three terms: a harmonic term corre- sponding to intramolecular interactions ; a Lennard-Jones term, which prevents molecules approaching one another too closely, and, in the case of flexible molecules, prevents parts of the same molecule overlapping one another; and an empirical term which is updated continuously throughout the simulation and which ensures that the calculated site-site correlation functions approach the measured functions as close as is practically possible.The distributions of molecules generated in this way can then be used to address a variety of questions about the way the molecules are organised in the liquid. The paper is structured as follows: Section 2 reviews the types of site-site correlation functions that can be extracted from the experiment, particularly with reference to hydrogen isotope substitution. Section 3 describes the EPMC method. Section 4 describes new analyses of existing neutron diffraction data from the pure liquids water, ethylene glycol, n-octane and from a concentrated solution of DMSO in water. Finally some concluding remarks are made in Section 5. 2 Neutron diffraction and hydrogen isotope substitution For a fluid composed of several atomic species, a, the total interference differential scat- tering cross-section per atom, Itot(Q)as a function of wave vector transfer, Q, obtained by neutron diffraction is given in terms of the atomic fractions, ca, of each species, the neutron scattering length of each species, (b,), averaged over the spin and isotope states of that atom, and the set of site-site partial structure factors, HUB(Q),which define the structure of the system: where p is the total atomic number density of the system, and g,&) is the corresponding site-site pair correlation function.For a fluid composed of molecules, we assume that the total mole fraction of mol- ecules of type ‘T’is mT with the normalization xTmT = l, and that there are N, atoms on molecule T.In this case the total interference function can be split up into two components, intramolecular interference terms for atoms on the same molecule, and intermolecular terms for atoms on distinct molecules : where Iintra(Q)= b, b, (4) and A. K. Soper 43 while linte,.(Q)is defined by an equation similar to (l), but with Hap(Q) replaced by H$'ter)(Q),which represents only the correlations between atoms on distinct molecules. In eqn. (S), r!;) represents the separation of atoms i and j on molecule T, and the sum- mation over ij is carried out only for pairs of a, /3 atoms, respectively, on the same molecule.If isotope exchange occurs between atoms on different molecules (hydroxy hydrogen is the obvious example here) then the scattering lengths that appear in eqn. (4) are the same as those in eqn. (1). However, if such exchange does not occur, (e.g. methyl hydrogens) then the usual isotopic averaging does not take place for atoms on the same molecule, so there will be separate intramolecular form factors for molecules with differ- ent isotopic compositions. In the examples that follow both types of situation arise. Because hydroxy hydrogens do not exchange with methyl hydrogens the two types of hydrogen can be labelled separately, as H and M, respectively. All other atoms such as 0,C, N, C1, etc., and any hydrogens that are not labelled, are treated as a composite atom, X, with atomic fraction c, = za#H,ML+H,M c, and average scattering length b, = l/cx c, (ba).This gives rise to a minimum of six partial structure factors that can be determined by neutron diffraction : where +-ci 1 bia.PfH. M (ba)(bP)H$tra)(Q)-HEF)(Q) xx and bl, b2 are the scattering lengths of hydrogen and deuterium, respectively. Note that the M-M intramolecular contribution appears in the X-M and X-X structure factors because methyl hydrogens do not exchange? Thus, H,, in eqn. (5) will only contain intermolecular contributions, while HMHand HHH will contain intramolecular contribu- tions as well. Eqn. (6)-(8) have been derived for the situation where the experiment is conducted in a particular way, that is diffraction data are taken from three samples: the first has the relevant sites, H and/or M, deuteriated, the second has the same sites protonated and the third sample is a simple, typically equimolar, mixture of the first two.The reason for this sequence is that hydrogen has a markedly larger incoherent scattering cross-section than deuterium. Thus the incoherent scattering level from the mixture sample should be in exact relationship to the atomic percentage of hydrogen in this sample. This provides an internal check that all aspects of the neutron diffractometer have performed reliably in the course of the experiment, and that the mixture sample has indeed been made up as intended. More importantly, the method used to subtract this incoherent scattering assumes the third sample is a mixture of the first two.' 44 Neutron scattering studies at solvent structure To measure each of the functions HH, MM and MH, it is strictly necessary to perform three separate diffraction experiments for each function, making a total of nine experiments, but the deuteriated sample is common to all experiments and so is only needed once, making the real total only seven.The XH, XM and XX structure functions can then be deduced from the residual data, after removing the HH, MM and MH functions. Obviously, for hydrogen sites which do not exchange there is an almost endless number of possibilities for using isotope substitution to investigate the corre- lation functions associated with those sites. By the same token, other substitutions such N and C1 an be invoked as well.3 Empirical potential Monte Carlo simulation2 It is assumed at the outset that a set of ‘data’, the site-site pair correlation functions for the material of interest, g:p(r), exist. In the present case these will have been derived by Fourier transforming the partial structure factors extracted from the neutron diffraction experiment. Thus there may be up to six of them, namely H-H, M-H, M-M, X-H, X-M and X-X. The starting point for the development is the potential of mean force, $ap(r), between sites a and p in the fluid,5 where and gus(r)is the site-site pair correlation function between a and p. For the data there is an equivalent potential of mean force: $:p(r) = -kT ln[g:’(r)].These potentials are by definition pairwise potentials, but cannot be used in a computer simulation because they already contain the many body cooperative effects arising from the packing of particles in the liquid. However, if a computer simulation is performed with an assumed initial potential, then the potential of mean force can be used to indicate where that potential needs to be modified if it is to reproduce the measured site-site pair correlation func- tions accurately. The procedure therefore is to set up a model fluid with the correct density and temperature of the material in question, using an assumed potential, the reference poten- tial, U$(r) between sites a and p. For molecules the reference potential incorporates harmonic contributions for pairs of atoms known to be held at particular separations within the molecule, and a Lennard-Jones term for most sites, to prevent neighbouring molecules approaching one another too closely, and to prevent different parts of a flex- ible molecule overlapping one another.The parameters for the Lennard-Jones potentials can be taken from table^,^ or else chosen to ensure that the corresponding pair corre- lation function is zero over a sensible region at low r. The simulation with the reference potential gives rise to an initial estimate of the site-site pair correlation functions. Setting the initial potential, Uyp(r),and the site-site correlation functions, gas(r), equal to their initial (reference) values, UZf(r) and gzf(r), respectively, the potential of mean force [eqn.(9)] is used to generate a new potential- energy function, U$(r), as a perturbation to the initial potential: The new potential-energy function U,“,(r)now replaces U:p(r) in the simulation, which proceeds as before but with the revised potential. Subsequently, after a sufficient number of particle moves to bring the simulation into equilibrium with the new potential, the simulation is stopped again and a further pertur- bation to the empirical potential is estimated, based on the current site-site pair corre- lation functions. This process is repeated until such a point that, assuming the process described here is convergent, A. K.Soper for all r and all atom pairs. At that point a set of empirical potential-energy functions has been derived which is able to reproduce the observed site-site pair correlation func- tions of the material in question.It will probably already be apparent that in the case of neutron data from molecules, a full separation of the diffraction data into all the site-site correlation functions is not possible. In particular the terms X-H, X-M and X-X refer to a weighted sum over several distinct atomic sites. In this situation the best that can be done is to treat all the atoms that contribute to X as if they were a single composite atom (even though the reference potential itself does distinguish between those sites). Thus the potential energy associated with, say, the X-M correlation is given by UxM(r)which can be determined from the X-M pair correlation function data using eqn.(10) as for the other potential functions. Clearly, therefore, the greatest accuracy in the EPMC simulation will be achieved if the maximum possible number of isotope substitutions are made. It is possible to show that the algorithm described here must be convergent within the pairwise approximation.2 In practice, some refinements are necessary.2 Perhaps the most important of these is that because the pair correlation functions estimated by com- puter simulation are noisy by virtue of the finite number of particles involved, direct application of eqn. (10) produces a noisy potential which cannot be used in the simula- tion. Therefore the perturbation is smoothed at each iteration.In principle this smooth- ing might put a limitation on how well the simulation can reproduce a given set of diffraction data, but in practice this does not appear to present a difficulty, providing the perturbation is not smoothed too heavily. The perturbation is applied at each iteration of the algorithm: if a particular simulated correlation function starts to deviate too far from the data, then the perturbation to the potential grows in the opposite direction to bring it back. The net result is a distribution of molecules whose calculated site-site correlation functions, after a large number of computer steps, coincide with the mea- sured correlation functions. A great benefit of this kind of analysis is that the statistical error on the calculated distribution functions becomes smaller and smaller, the longer the simulation is run, in the same way that the statistical error on the diffraction data is reduced by long counting times.This is strong evidence that the simulation will readily proceed on a true random walk, and therefore sample a wide region of phase space. Note that the same technique is used to refine intramolecular correlations as well as intermolecular correlations, so that details of molecular structure and conformation are available from the same simulation. Having generated physical distributions of molecules which fit the diffraction data, a number of other quantities can be estimated. Perhaps the most obvious of these is the local bond angle distribution around a particular headgroup.Fig. 1 defines the geometry 7lr Fig. 1 Geometry used to define the O.--O-H bond angle distribution. 8 is the angle between the 0-H bond and the 0 -* * 0 positive radial direction. Thus 8 = 180"(cos 0 = -1) corresponds to one of the 0-H bonds on the water molecule pointing directly towards a neighbouring oxygen. 46 Neutron scattering studies at solvent structure for the case of a water molecule around the oxygen atom of another molecule. Further information about the orientational structure of the system can be obtained by calcu- lating the spherical harmonic coefficients of the orientational pair correlation function.' These can be used to reconstruct a specified orientational distribution in the liquid.8 Finally, in the case of flexible and chain molecules, it may be interesting to estimate the angle of rotation, 4, about various internal bonds,6 and the degree to which these are correlated: such information leads to a picture of the typical molecular conformation in a situation where many variants may be present.Of course it has to be realised that any reconstruction technique such as that described here may not necessarily lead to the correct structure. What it will do, however, is to lead to the most likely structure that is consistent with both the known boundary constraints and the measured data. 4 Results for pure liquids and solutions 4.1 Liquid water Because of its central importance to much solution chemistry and biological inter- actions, the structure of water has been studied in phenomenal detail compared with most other liquids.Remarkably, the controversy about the true nature of hydrogen bonding is still hotly pursued.' Neutron diffraction, by providing a somewhat inconsis- tent view of water over the years from different experimenters, has fuelled this debate. The truth is unfortunately that too much emphasis has been placed on determining whether this or that peak in the measured correlation functions is sharper or weaker than the same peak in the correlation functions obtained by other experimenters or in computer simulations of water. Now for the first time there may be a tool at our dis- posal for interrogating the diffraction experiment as to what it is really telling us about water.Several recent diffraction experiments on water"*' ' have explored the structure of water away from ambient conditions. These results caused sufficient controversy12 that they have subsequently been repeated' under different experimental conditions, and the old data analysed in a different way.14 There is no essential qualitative change to the earlier conclusions but some quantitative differences so that the measured site-site correlation functions for superheated water can be realized by physical distributions of water molecules, a feature lacking in the original data.12 These new data have been subjected to EPMC simulation analysis and compared with similar analyses of ambient water data. Fig. 2 shows the computed bond angle distribution 0-H about 0 as esti- mated by this method from the ambient water data, and compares it with the same distribution for water at 400°C and at a density of 0.022 molecules per i.e.roughly double the critical point density. Clearly the distribution of bond angles in supercritical water is greatly distorted compared with ambient water. Nonetheless, if we allow hydro- gen bonds to be bent by up to say +30", then the degree of hydrogen bonding in supercritical water may be as much as 40% of its value in ambient water. Therefore the sharpness and shape of the first peak in the OH pair correlation function, whose decline was previously used to argue that hydrogen bonding had been largely eradicated in the supercritical liquid," cannot be directly interpreted in terms of the degree of hydrogen bonding: the level of hydrogen bonding can only be estimated once a definition of the hydrogen is established.' The neutron evidence is therefore that water structure is greatly distorted in the supercritical state, but that a residual degree of hydrogen bonding still exists.Fig. 3(a) shows Kusalik's spatial distribution function (SDF),16 go&, Q), for ambient water as determined from EPMC simulation of the ambient water neutron diffraction data, and this can be compared with the same distribution, extracted by the same method, in the supercritical liquid, Fig. 3(b). The characteristic hydrogen-bonded lobe of room tem- A. K. Soper 47 0--0-H bond angle distributionh 200 It i 53 100 a Lw rn a 50 0 -1 -0.5 0 0.5 1 cos 8 Fig.2 O...O-H bond angle distribution for water at two temperatures as determined by EPMC simulation of neutron diffraction data (line = 25 "C, circles = 400 "C) perature water is greatly diminished and is broader in the supercritical state, but it is still present at a level of about one quarter of its density in the ambient liquid (after taking account of the fact that the density at 400°C was about two-thirds of the ambient density). Therefore by either measure, i.e. Fig. 2 or 3, it would be incorrect to say that hydrogen bonding had completely disappeared in the supercritical liquid. 4.2 Liquid ethylene glycol Compared with water, rather little is known about the structure of ethylene glycol in the liquid state, even though it is widely used in the chemical industry, and is well known for its cryoprotective properties. Fundamentally, the molecule can be regarded as a pair of methanol molecules joined at the methyl groups, with the OH groups free to form hydrogen bonds either with themselves or with neighbouring molecules.The whole question of the degree and nature of hydrogen bonding at the microscopic structure level in organic molecular liquids is largely uncharted territory, and unlike the case of water, there is very little previous diffraction work. However, as long ago as 1949, 0. Bastiansen" determined by electron diffraction that the internal torsion angle of the ethylene glycol molecule about the central C-C axis was ca.74", corresponding to the so-called gauche conformation, raising the possibility of an internal hydrogen bond for this molecule. This observation was quite remarkable for its time, given that there were no computers available for performing Fourier transforms. The other conformations usually considered are cis, in which the two C-C-0 planes of the molecule are coin- cident, and trans, in which the same two planes are at 180" to one another. The cis configuration is generally believed to be excluded by virtue of steric hindrance between the two oxygen atoms. Ab initio studies18 of the isolated molecule support the gaucheconformation with an internal hydrogen bond as being the most favourable energeti- cally, but some trans conformations are not ruled out.In the liquid state it is not clear whether the possibilities for intermolecular hydrogen bonding would not offset any ten- dency to form intramolecular hydrogen bonds. There are apparently no recent diffrac- tion data on the pure liquid. The neutron diffraction data in this case were measured on the Special Environment Diffractometer at the IPNS pulsed neutron source at Argonne.lg Three samples were Neutron scattering studies at solvent structure I 50 75 25 50 Fig. 3 The 00 spatial distribution functions for water(a) under ambient conditions, and (b) at 400°C and a density of 0.022 molecules per The definition of the molecular geometry is identical to that of Svishchev and Kusalik." made up, DO-CD,-CD,-OD, HO-CD,-CD,-OH, and an equimolar mixture of these two liquids.The diffraction data were used to extract the H-H, X-H and X-X partial structure factors are described in Section 2, and then subsequently Fourier trans- formed to r space. No attempt to subtract the intramolecular structure was made at this stage because for ethylene glycol the intramolecular structure overlaps the intermolecu- lar structure and the molecule is probably flexible in the liquid. Since the internal con- formation of the molecule was not known a priori for the liquid it was imperative that A. K. Soper 49 the intra- and inter-molecular structures were refined simultaneously. Also no substitut- ions were made on the methyl hydrogen sites which meant that the X-X correlation function includes all the M-M and M-other correlations.In fact, as will be seen below, it appears that ethylene glycol assumes quite a well defined conformation in the liquid state. The diffraction data were transformed to P space, and the resulting H-H, X-H and X-X pair correlation functions are shown in Fig. 4. They were then subjected to the EPMC simulation technique described in Section 3. Near-neighbour intramolecular dis- tances were constrained by a harmonic potential to the following values: c-C to 1.54 A, C-0 to 1.43 A, C-M to 1.08 A and 0-H to 0.98 A. In addition, all second- neighbour bond angles (such as C-C-0, M-C-M, C-0-H, etc.) were constrained to the tetrahedral angle, 109.47",again with a harmonic potential.This potential serves to preserve the underlying molecular geometry while allowing significant deviations for individual molecules from the ideal molecular structure. Free rotations of radicals about the C-C bond and both C-0 bonds were allowed in the simulation. Intermolecular interactions and intramolecular interactions not affected by the harmonic potential (intramolecular 0-0 interaction is an example of this) were controlled by a Lennard- Jones potential, with parameters determined from a simulation of methanol :20 ELJ(OO)= 0.7307 kJ mol-', aLJ(O0)= 3.083 A, ELJ(CC)= 0.9151 kJ mo1-', oLJ(CC)= I I I I I II I I 2 4 6 8 10 rfA Fig. 4 Measured site-site pair correlation functions for liquid ethylene glycol at 20 "C (circles).The line shows the same correlation functions calculated from one configuration the EPMC simu- lation of these data. Notice the pronounced peak at ca. 2.4 8, in the HH correlation function, indicating the presence of hydrogen bonding in the liquid. It is not clear from these data alone whether this is due to intramolecular or intermolecular hydrogen bonding. Neutron scattering studies at solvent structure 3.861 A. No charges were placed on these sites however, and no Lennard-Jones param- eters for methyl and hydroxy hydrogens were defined. Apart from pairs of atoms defined to lie in the harmonic potential, there was an overall restriction that no pair of atoms, whether intermolecular or intramolecular, could approach one another closer than 1.5 A for the H-H correlation, 1.5 A for the X-H correlation and 1.8 A for the X-X corre-lation.Once the system of molecules had been equilibrated with this reference potential, the empirical potential was refined: as shown in Fig. 4 an excellent representation of the diffraction data was obtained in all three cases. Subsequently a number of ancilliary quantities were calculated. The first quantity, Fig. 5, is the distribution of torsion angles, 4, about the C-C axis. The definition of torsion angle here is identical to that used by Allen and Tildes- ley.6 It will be observed immediately that these distributions peak at 4 = +69", which is remarkably close to the value discovered by Bastiansen,17 but there is fair6 broad range of values about the peak value, so it is already clear that there is probably no unique value of this torsion angle for ethylene glycol in the liquid state.Note that in this simu- lation there was no assumed torsional potential, so this observation is purely a conse- quence of the neutron diffraction data and the imposed constraints on atomic overlap. Fig. 6 shows the same O..-O-H bond angle distribution, this time for ethylene glycol, as was estimated for water in Fig. 2. It can be seen immediately that the shape of this distribution is quite different from that in water: for ethylene glycol there is no sharp peak at cos I3 = -1. Instead there is a broad distribution, with a marginal prefer- ence for 0 values greater than 90". From this result it might be concluded that hydrogen bonding is weak or non-existent in ethylene glycol, but the truth is different.In Fig. 7 is shown the spatial distribution function around the H-0-C unit, with the oxygen atom at the origin, and the positive z axis defined by the line joining 0 and the point midway between H and C [see Fig. 7(a)].An intense, isolated, lobe in this distribution is seen in the I3 range 0-20". The calculation was repeated with intramolecular correlations excluded, whereupon the lobe disappeared: this intensity must therefore be caused by intramolecular hydrogen bonding. Such a hydrogen bond would be highly non-linear, which explains the absence of any strong peak in Fig. 6. Further analysis of the ethylene glycol data is deferred to a more detailed acc~unt,'~ but it is already clear that a wealth of information has been extracted by EPMC simula-tion in this case.C-C torsion angle distribution m c.-U 0 -1 00 0 100 4 / degrees Fig. 5 Distribution of C-C torsion angles for ethylene glycol in the liquid, as determined by EPMC simulation of the diffraction data A. K.Soper 15 0"""""""'~""'-1 -0.5 0 0.5 1 cos e Fig. 6 Bond angle distribution 0 + -~0-H for ethylene glycol. Compared with Fig. 2 for water, linear hydrogen bonds appear to occur infrequently in liquid ethylene glycol (but see Fig. 7). 4.3 Liquid n-octane The structure of long chain molecules in the liquid state has received little attention by experimentalists so far,21322 because the large number of possible molecular conforma- tions makes direct interpretation of the neutron data difficult.However, several com- puter simulations of long chain hydrocarbon liquids have been performed, and appear to give a reasonable account of the microscopic liquid proper tie^.^^ Comparison of the available diffraction data with these simulations is limited but in general is satisfactory. What has not been established, however, is whether useful information can be extracted from the neutron diffraction experiment on long chain molecules: there are many over- lapping intra- and inter-molecular distances. The example chosen here is n-octane because of its relatively simple molecular structure, but the results have implications for many other systems of interest, including the structure of flexible .molecules dissolved in water.The data described here were measured on the SANDALS neutron diffractometer at the ISIS pulsed neutron source.24 Hydrogen isotope substitution of the methyl hydro- gens was performed, leading to the three composite partial structure factors, M-M, X-M and X-X, where the X-X structure factor consists of C-C correlations (inter- and intra-molecular) and M-M correlations (intramolecular only), because these hydrogens do not exchange. The set of correlation functions derived from the diffraction data are shown in Fig. 8. The great similarity between the M-M correlation function obtained here and that for n-decane and n-eicosane22 is to be noted. The data were subject to the EPMC simulation procedure already described.Internal C-C bonds were constrained to 1.54 A, C-H bonds to 1.08 A, and all internal bond angles were constrained to 109.47'. However, rotations about the C-C bonds were unrestricted, apart from the requirement that atoms not bonded by the harmonic potential could not approach one another closer than 1.5 A for the M-M correlation, and 2.0 A for each of the X-M and X-X correlations, irrespective of whether they were on the same or distinct molecules. As a result the molecules had in principle a consider- able degree of molecular flexibility, which would only be restricted by the neutron data. The reference potential was initially defined by Lennard-Jones parameters oCc= 3.872A,E~~ = 0.396 kJ mol-1,23 but once the distribution had been equilibrated, the value of was set to zero, and the molecular distributions were controlled by the empirical potential alone, in order to ensure the simulation would sample the largest possible range of structures. Attempted Monte Carlo moves included translations of individual Neutron scattering studies at solvent structure (a) Fig.7 (a) Geometry used to define the spatial distribution function for ethylene glycol. The C-OH plane of the central oxygen defines the x-z plane of the coordinate system, with the z axis running from the central oxygen to the midpoint of the line joining the carbon and hydroxyl hydrogen atoms. (h)The spatial distribution of oxygen atoms around a central oxygen in ethylene glycol.This distribution includes oxygens on the same molecule as the oxygen at the origin. The pronounced lobe of intensity near the positive z axis indicates an intramolecular hydrogen bond is present, and explains the absence of linear hydrogen bonds in the bond angle distribution. atoms, rotations about all internal C-C bonds, whole molecule rotations and whole molecule translations. As can be seen in Fig. 8, the fits obtained to these data were reasonable, but not as good as for ethylene glycol. Better fits could be obtained, but they required the local internal molecular geometry, especially the tetrahedral bond angles, to A. K. Soper t 1 10 5 I I II 1 0 2 4 6 8 10 r/A Fig. 8 Composite site-site pair correlation functions for liquid n-octane, as obtained by neutron diffraction with hydrogen isotope substitution on the methyl hydrogens (circles).The lines show the fits to these data using Empirical Potential Monte Carlo simulation of the data. be significantly distorted. These structures were therefore discounted as being unphysical. An important issue for such flexible molecules is the degree to which neighbouring C-C bonds along the molecule are in the trans (torsion angle 4 = +180") and gaucheconformations (torsion angle $ = &-60"). Simulations of longer chain hydrocarbons predict a roughly 2 : 1 split between the two conformations, re~pectively.~~ Fig. 9(a) shows the distribution of C-C torsion angles, $, for n-octane as obtained from the EPMC simulation.(This torsion angle is identical to that defined by Allen and Tildesley)? Pronounced peaks show up for 5, x & 180", but no distinct bands appear at 4 = f60". However, there is a broad, low band in the region -120 < 4/< 120 degrees, indicating that, as expected, roughly one third of the C-C bonds are not in the trans conformation. The average internal length of the molecule (carbon to carbon distance) was 6.0 & 1.2A, which is significantly shorter than the theoretical length (8.8 A) of a linear octane molecule. However, it was found that the fits shown in Fig. 8 could generally be improved by forcing a greater degree of linearity into the molecules. Thus according to the neutron data the molecules in this liquid appear to be rather linear in character (Fig.10).To emphasise this point, Fig. 9(b) shows the distribution function n(Gk),where Gk= x;=2e4n+j-l(n refers to any one of the C-C bonds), for k = 1-5. This distribution function, which measures the extent to which neighbouring $ values for k adjacent C-C bonds are correlated, is averaged over all the n possible starting bonds for this Neutron scattering studies at solvent structure 6 4 6h i' 2 n -(b) C-C torsion angle distribution -500 0 500 @k Fig. 9 (a) Distribution of torsion angles, 4, about the C-C bond in n-octane, as determined by EPMC simulation of the data of Fig. 8. (b)The ensemble averaged distributions, n(Qk),for k = 1 to 5, which represent the degree of correlation between the torsion angles on adjacent C-C bonds (see text).n(@Jpeaks strongly at cP2 = 0, which suggests that the n-octane molecule tends towards being linear in the liquid state. sum. For n-octane, with 7 C-C intramolecular bonds, there is a total of five such distribution functions, since the torsion angles of the first and last C-C bonds are not defined. Thus Q1 is defined between 4 = f180°, @j2 is defined between 4 = +360", etc. Obviously, n(Gl) is identical to the distribution shown in Fig. 9(a). A. K. Soper Fig. 10 View of a typical, nearly linear, octane molecule found in the EPMC simulation of the diffraction data. Considerable variations about this structure were observed. Fig. 9(b) shows that n(a2)and n(Q4) have strong peaks at @k = 0, or multiples of Gk= & 360°,while n(Q1), n(@& and n(@J have peaks at @k = f180, @k = 540", etc.This indicates that the torsion angle on adjacent C-C bonds are highly correlated, with rotations about adjacent C-C bonds being opposite and equal along the chain. Hence the molecules will on average tend to be nearly linear in the liquid. If there were a significant degree of helicity along the chain, the peaks in these torsional correlation functions would not occur in multiples of 180 or 360". 4.4 Dimethyl sulfoxide in water Dimethyl sulfoxide, (CH,),SO (DMSO), provides an excellent opportunity for studying water around organic molecules in solution by neutron diffraction. It has a large dipole moment, which means it mixes with water in all proportions, and water readily forms 4J = 0" Fig. 11 Spatial distribution function of water around water in a concentrated solution of DMSO and water (2 moles of water to 1 mole of DMSO).The similarity with Fig. 3(a)is clear, but note that the correlations around the hydrogen bond are actually enhanced compared with pure water. 56 Neutron scattering studies at solvent structure hydrogen bonds with the oxygen atom on DMSO. Moreover, it is possible to perform isotope substitution on the methyl hydrogens to establish the nature of water structure around the hydrophobic groups. A description of the experiment and subsequent data analysis using the EPMC technique have already appeared,25 so here the discussion concentrates on the results of that analysis.A total of six composite partial structure factors were obtained in this case, namely H-H, M-H, M-M, X-H, X-M and X-X, where X represents a weighted sum of 0 (water), 0 (DMSO), S (DMSO) and C (DMSO). As in the previous sections, to interpret fully the diffraction data the EPMC simulation must model all atoms of the system. In this case the reference potential parameters were taken from those used in previous simulations of water and DMS0.26 Excellent fits to the diffraction data were obtained. Fig. 11 shows the spatial distribution function for water around water in DMSO aqueous solutions, and is to be compared with the case of pure water in Fig. 3. It will be seen that the hydrogen bonding between water molecules is, if anything, enhanced com- pared with pure water: the number of hydrogen bonds on a water molecule does not fall off as rapidly as the average number density of water in the solution.Around the DMSO, Fig. 12, the signature of hydrogen bonding to the oxygen on DMSO is given by the pronounced peak in the spatial distribution at (0 z 115", 4 = (Fig. 12(b)), but it OO), is not as sharp as the equivalent peak in water-water correlations, probably because the geometry of water around DMSO is not as well defined as for water around water. OO),Other lobes can be seen at (0 = 35", 4 = and (0 z 35", 4 = 180"),with a broad low band of intensity for (0 z 0-llOo, 4 = &90"), corresponding to the hydration of the methyl groups.Water molecules in this region are found with their dipole moments lying roughly tangential to the methyl-water axis2 5 Discussion and Conclusion The results of the previous section have hopefully shown the great wealth of detail about local molecular conformation and structure in liquids and solutions that it is possible to extract by neutron diffraction. Prerequisites for this analysis to succeed are access to an intense and reliable neutron source, the ability to label specific sites on molecules and determine the local correlations with those sites, and a technique for developing physical models of the structure under investigation. All three elements are crucial: high fluxes allow investigation of dilute species, the labelling of sites with isotopes yields distinct information which is not accessible by a single diffraction experiment alone, and the ability to model the structure establishes the limits on what the structure can and cannot assume.Although there are no sharp Bragg diffraction peaks as in the case of a crys- talline sample, the shapes of the various pair correlation functions that can be deter- mined by diffraction play an important role in refining the structure. As a result, the diffraction data are found to be very sensitive to both the degree of hydrogen bonding between water molecules, whether in the pure liquid or in solution, and to the internal conformation of molecules. All this begs one obvious question, however. If the detailed structure determined by simulation is sensitive to the shape of the pair correlation function, how accurate are the data? In the case of supercritical water, it was argued12 that a systematic error in the diffraction pattern from light water caused the 0-0 correlation function determined by diffraction to be too high at small r, and the 0-H correlation function to be too low at small r.Something similar happens in the case of the n-octane data described here: the X-M and X-X correlations from the data tend to be too low in the region 2-5 A compared with the simulation, which causes the simulation to try to stretch the mol- ecules out as much as possible, making them linear. However, the M-M correlation A. K. Soper 9 Fig. 12 Spatial distribution functions for water around DMSO (r,8plots): (a)Definition of angles (O,$); (b)spatial distribution function for # = 0";(c) for # = +90" and (d) for 4 = 180".tends to be too high compared with the simulation, which has the opposite effect of trying to produce more density at short distances. This can only be achieved by curling the molecules back on themselves. In that situation the best that can be done is to fit all the available correlation functions as well as is practicable, while maintaining the con- straints on atomic overlap and molecular bond lengths and angles. That will then lead to a structure most consistent with the data, even if it does not completely reproduce the extracted correlation functions. In a sense the table has turned full circle: having demon- strated how structural information is to be extracted from diffraction data on liquids there is once again a great need to improve the accuracy of diffraction experiments on liquids.That development will almost certainly hinge on whether an accurate method of coping with the distortion to the diffraction pattern from nuclear recoil effects can be found. 58 Neutron scattering studies at solvent structure Much of the work in this paper would not have happened without the considerable assistance of numerous colleagues, especially M. A. Ricci, A. Luzar, D. Montague and C. Shelton. Part of the work was supported under EEC Twinning Agreement # SC1-CT91-0714. References 1 A. K. Soper and A. Luzar, J. Chem. Phys., 1992,97, 1320.2 A. K. Soper, Chem. Phys., 1996,202,295. 3 R. L. McGreevy and L. Pusztai, Mul. Sim., 1988, 1,359. 4 A. Luzar, A. K. Soper and D. Chandler, J. Chem. Phys., 1993,99,6836. 5 J. P. Hansen and I. R. McDonald, Theory ufSimple Liquids, Academic Press, London, 1986. 6 M. P. Allen and D. I. Tildesley, Computer Simulation of Liquids, OUP, Oxford, 1987. 7 C. G. Gray and K. E. Gubbins, Theory ofhfolecular Liquids, OUP, Oxford, 1984. 8 A. K. Soper, J. Chem. Phys., 1994,101,6888. 9 J. Li and D. K. Ross, Nature (London), 1993,365,327. 10 P. Postorino, R. H. Tromp, M. A. Ricci, A. K. Soper and G. W. Neilson, Nature (London), 1993, 366, 668. 11 P. Postorino, M. A. Ricci and A. K. Soper, J. Chem. Phys., 1994,101,4123. 12 G. Loffler, H. Schreiber and 0.Steinhauser, Ber.Bunsen-Ges. Phys. Chem., 1994,98, 1575. 13 F. Bruni, M. A. Ricci and A. K. Soper, 1996, submitted. 14 A. K. Soper, F. Bruni and M. A. Ricci, 1996, submitted. 15 A. A. Chialvo and P. T. Cummings, J. Chem. Phys., 1994,101,4426. 16 I. M. Svishchev and P. G. Kusalik, J. Chem. Phys., 1993,99, 3049. 17 0.Bastiansen, Acta Chem. Scand., 1949,3, 415. 18 J. Almlof and J. Stymne, Chem. Phys. Lett., 1975,33, 118. 19 A. K. Soper and D. Montague, in preparation. 20 M. Haughney, M. Ferrario and I. R. McDonald, J. Phys. Chem., 1987,91,4934. 21 A. H. Narten, A. Habenschauss, K. G. Connell, J. D. McCoy, J. G. Curro and K. S. Schweitzer, J. Chem. SOC.,Faraday Trans., 1992,88, 1791. 22 J. D. Londono, B. K. Annis, J. Z. Turner and A. K. Soper, J. Chem. Phys., 1994,101,7868. 23 G. D. Smith and D. Y. Yoon, J. Chem. Phys., 1994,100,649. 24 A. K. Soper and C. G. Shelton, 1994, unpublished report. 25 A. K. Soper and A. Luzar, J. Phys. Chem., 1996,100, 1357. 26 A. Luzar and D. Chandler, J. Chem. Phys., 1993,98,8160. Paper 6/01 188H; Received 19th February, 1996

ISSN:1359-6640    

DOI:10.1039/FD9960300041

出版商:RSC    

年代:1996

数据来源: RSC

摘要:

Faraday Discuss., 1996,103,59-69 Time domain reflectometry techniques used as an analytical tool to investigate water molecules bound to solid-state deoxyribonucleic acid (DNA) Richard S. Lee and Stephen Bone Institute of Molecular and Biomolecular Electronics, University of Wales, Dean Street, Bangor, Gwynedd, UK LL57 I U T In this study, the dielectric properties of ion-reduced Na- and Mg-DNA have been investigated over a frequency range 105-10'0 Hz as a function of hydration and temperature, using time domain reflectometry techniques. Three distinct dispersions have been identified, centred at frequencies of ca. lo', 10' and 10'' Hz, respectively. The lowest frequency dispersion has been attributed to localised relaxation of counter-ions associated with the phos- phate groups of the DNA double helix (G.P. Archer, S. Bone and R. Pethig, J. Mol. Electron., 1990, 6, 199). The highest frequency dispersion is thought to be the result of the rotational relaxation of DNA-bound water molecules. The origin of the dispersion centred in the lo8 Hz region is discussed in terms of a number of possible relaxation processes. X-Ray diffraction studies performed over 40 years ago found that different X-ray diffrac- tion patterns could be obtained for the same DNA fibre exposed to different relative humidities.' Later work showed that native DNA is able to adopt a number of different conformations depending on the hydration and ionic nature of the The most commonly observed conformations have been designated the A, B and C In addition, it has been found that double-stranded synthetic DNA polymers can exist in novel conformations such as D and S (or Z).' These conformations, even if present in only limited regions of DNA, could have an important role in forming distinc- tive molecular recognition sites. Water molecules have a primary influence on the conformation stablity of the double-stranded DNA structure.A dominant factor is associated with the fact that water molecules are highly polarisable and can provide dielectric screening between elec- trically charged entities. The presence of water in the DNA structure therefore moder- ates the electrostatic forces which exist between negatively charged phosphate groups and also between the phosphate groups and positively charged counter-ions.Since the energy of interaction between a cation and water is very specific for each type of cation, there will be a complex interplay between the short range ion-water interactions and the relatively long-range electrostatic interactions. Dielectric spectroscopy carried out over a wide frequency range is ideally suited to probe the nature of the interactions between DNA, water and counter-ions. Experimen- tal investigation and discussion of the dispersion centred around the lo7 Hz range has been previously performed by Archer et aL8 In this paper we present the experimental findings of a high-frequency dielectric study on native DNA over a range of hydrationsand ion types. 59 Water molecules bound to deoxyribonucleic acid Experimental Na-DNA (type XIV: derived from herring testes) was obtained from Sigma Chemicals.Excess sodium ions were removed by dialysing against ultra-pure water (Millipore Mill- R060 water purification system), for a week at a constant temperature of 4 "C.The water was refreshed at regular intervals. The solution containing the ion-reduced DNA was then freeze dried and stored in a desiccated container at 4 "Cuntil required. Samples of ion-reduced Mg-DNA were produced using a similar method, except that before freeze drying the ion-reduced DNA was dialysed against a 0.2 moll-' solution of MgCl, -6H,O for 48 h. The solution was then dialysed against ultra-pure water, regu- larly refreshed, for a week, to remove excess magnesium ions.The solution was then freeze dried and stored in the same fashion as the Na-DNA. Dielectric measurements Time domain reflectometry (TDR) was employed to obtain dielectric data in the fre- quency range 105-3 x lo9 Hz. This difference method used is described in detail else- where.' In essence, the technique involves applying a train of fast rising voltage pulses to a 50 a, low loss, coaxial transmission line. The transmission line is terminated by an open-ended sample cell. A sampling oscilloscope is used to record reflected waveforms from the cell containing the sample, a reference material with known dielectric proper- ties and air, respectively. The frequency range of dielectric data that can be obtained from experimental data is determined by the period (time window) that is used to capture the waveform data and the diameter and length of the inner conductor in the sample cell.For the measure- ments reported here, the sample cell took the form of a 7 mm diameter outer electrode with an inner electrode of length 1.5 mm and diameter 5 mm. After time referencing and mathematical manipulation, the waveforms were transformed from the time domain to the frequency domain using dedicated software. Mathematical analysis of the data was based upon the theory of Nakamura et al." The DNA powder packed into the sample cell (density 700 kg m-3), was hydrated by evacuating the cell to a pressure of 4.8 x low3bar and allowing the atmosphere to equilibrate with saturated salt solutions of known water partial pressure.The samples were left until equilibrium with the saturated salt solution was achieved, the process taking between 24 and 48 h. Gravimetric measurements Isotherms for each DNA sample were measured at 298 K using a vacuum micro-balance (Sartorius 4433MP8). The partial pressure measurements were made by manometer (0-15% relative humidity, rh) and a calibrated Humilab HL2D vacuum humidity probe (10-75% rh). For partial pressures corresponding to 70-85% rh, saturated salt solutions were used to provide the required humidity. Results The complex permittivity E* (E* = E' -id') of each of the DNA samples was determined using the TDR technique discussed above. Three dielectric dispersions were identified with relaxation frequencies centred around lo7, 10' and above 3 GHz, respectively.All three dispersions increased in magnitude with increasing hydration but their character- istic relaxation frequencies were found to be relatively independent of hydration. The dispersion centred around lo7 Hz was discussed by Archer et al.,' who suggested that the relaxation process is due to activated localised hopping of counter-ions in double minima potential-energy wells. R. S. Lee and S. Bone Dielectric dispersion centred around los Hz Fig. 1 shows the dielectric loss component of the complex permittivity for the relaxation centred on lo8 MHz using ion-reduced Na-DNA as a function of hydration. A disper-sion centred at the same frequency was also exhibited by samples of ion-reduced Mg-DNA.The magnitude of a dielectric dispersion can be represented by the dielectric increment, Ad,which is given by: AE’= E, -E, where and E, are the low- and high-frequency relative permittivities describing the relaxation process, respectively. The hydration dependence of the dielectric increment for this dispersion is shown in Fig. 2 for both ion-reduced Na- and Mg-DNA. Information pertaining to the amount of water bound to DNA was obtained using gravimetric measurements. The hydration isotherms for Na- and Mg-DNA are shown in Fig. 3. Two methods of analysis were carried out on the hydration isotherms, namely following the Brunauer-Emmett-Teller model (BET)l and the three-parameter model developed by Gascoyne and Pethig.” The BET model assumes all water bound in excess of the monolayer water has properties identical to that of normal bulk water.In the three-parameter model it is assumed that the water absorbed in the first layer is different to that in subsequent layers, which itself is different from that of bulk con- densed vapour. Using this model, the total amount of water absorbed ~(x)is given by: v, abxv(x) = (1 -bx)[1 + b(a -1)x) where x is the water partial pressure, t), is the monolayer hydration value corresponding to the hydration of the primary binding sites, ab is the activity of the water absorbed in the first layer and b is the relative activity of all subsequent layers. Eqn. (1) can be 2.5 T 2 +lQ%+TSW 1.5 -- .w - 1 --0.5-0 1o6 10’ 1o8 1oQ frequency/Hz Fig. 1 Variation of the dielectric loss for ion-reduced Na-DNA as a function of frequency and hydration Water molecules bound to deoxyribonucleic acid 0 0 0 2 4 6 8 10 12 14 16 number of water molecules per DNA base pair Fig. 2 Dielectric increment, Ad, for the dispersion centred in the 10' Hz region plotted as a function of hydration +Na-DNA Data uMg-DNA Data 0 0.2 0.4 0.6 0.8 partial pressure of water Fig. 3 Hydration isotherm for ion-reduced Na- and Mg-DNA R. S. Lee and S. Bone expressed in the form : X -= A(1 + Bx -CX2) where A = (t’,ab)-’, B = b(a -2) and C = b2(a-1). Using the isotherm data in this form, least-squares-fit methods were used to obtain values for the hydration parameters.These are shown in Table 1 together with the corresponding values for the BET analysis. The results indicate that on average 4.2 water molecules per base pair in ion- reduced Na-DNA and 4.8 water molecules per base pair in ion-reduced Mg-DNA are bound in primary hydration sites. For dielectric materials at low hydration levels, the Onsager formulation for the local electric field can be assumed to be relevant. It can be shown13 that the dipole moment p, is related to experimental measurements by: where N is the number density of relaxing dipoles, e0 is the permittivity of free space and kT is the Boltzmann energy. The variation of the left-hand side of eqn. (2), denoted by F(E),as a function of hydration, is shown in Fig.4. Thermodynamic information can be obtained from the temperature variation of the dielectric relaxation time. The enthalpy of activation, AH, for the underlying relaxation process, assuming an activated Arrhenius-type behaviour, is related to the characteristic relaxation time, z, by : z cc exp( g) (3) A plot of the natural logarithm of the characteristic relaxation time against reciprocal temperature can be used to calculate values of the activation enthalpy for the relaxation process. For both Na- and Mg-DNA, AH was found to be less than 0.3 kcal mol-’ and was indistinguishable from variations caused by experimental error. Dielectricdispersion above lo9Hz The hydration dependence of the high-frequency permittivity of the dispersion centred at lo8 Hzand the relatively large values of this parameter measured at high hydrations (E, = 5.7 at 13.2 water molecules per base pair for Na-DNA) indicated the existence of a Table 1 Derived hydration values obtained for ion-reduced Na- and Mg-DNA Na-DNA Mg-DNA urn a b urn U b BET analysisthree-parameter analysis 11.5 (4.2)” 11.6 (4.3)” 21.57 23.38 -0.91 13.2 (4.7)” 13.5 (4.8)” 16.38 20.21 -0.79 om primary binding site hydration capacity (% water uptake); (ub), water activity in the mono-layer ;b, relative activity of water in subsequent layers. Number of water molecules per base pair DNA.Water molecules bound to deoxyribonucleic acid 4 Na-DNA MQ-DNA 0 5 10 15 20 number or water molecules per DNA base pair Fig.4 Dielectric parameter f(~)plotted as a function of hydration for the dispersion centred around lo8 Hz.The straight line indicates the behaviour which bound water would be expected to show assuming that the water dipoles were rotationaly unhindered. 4 0 2 4 6 8 10 I2 14 16 number of water molecules per DNA base pair Fig. 5 Dielectric increment, ALE',for the dispersion centred above 1 GHz plotted as a function of hydration R.S. Lee and S. Bone -1.2 1 0.8 Na-DNAdata n-'c. 0.6 0 Mg-DNAdata-water fit 0.4 0 5 10 15 20 number of water molecules per DNA base pair Fig. 6 Dielectric parameterf(s) plotted as a function of hydration for the higher frequency disper- sion.The straight line without symbols indicates the behaviour which bound water would be expected to show assuming that the water dipoles were rotationally unhindered. further dispersion at higher frequencies. Since it was not possible to measure accurately the dielectric parameters above 5 GHz in this low-loss system, the hydration dependence of the magnitude of the high-frequency dispersion was inferred from the increase in permittivity in the low GHz range with increasing DNA water content and can be seen in Fig. 5. By applying the Onsager formulation for the local electric field to this dispersion data, it was possible to obtain values for the dielectric parameter,f(E), as a function of DNA water content. This is plotted in Fig. 6.The way in which the dielectric parameter, f(~),would vary with hydration, assuming a dielectric loss associated with DNA-bound water where each molecule exhibited a dipole moment equal to that of unhindered vapour phase water (6.14 x C m), is shown by the straight line plot. Discussion This study of hydrated DNA has revealed the presence of a dielectric dispersion centred around lo8 Hz. The existence of a further, higher frequency, dispersion has also been indicated by the study. The observed dielectric dispersion may arise from one of the following three possible mechanisms :(i) rotational relaxation of dipoles associated with the DNA molecule; (ii) the relaxation of Maxwell-Wagner/interfacial polarisations ;or (iii) the field-induced redistribution of ionic charges associated with the DNA.It is apparent from Fig. 2 that the dispersion is both hydration and ion-type depen- dent. For both Na- and Mg-DNA, there exists a population of bound water which does not contribute to the polarisability of the DNA molecules. With sodium as the counter- ion, the dispersion first appears at a hydration greater than 4 water molecules per base pair. When magnesium is the counter-ion the dispersion is not detectable until eight water molecules per base pair. This observation indicates that the dispersion is likely to originate from a process involving the phosphate-counter-ion-bound water system Water molecules bound to deoxyribonucleic acid rather than the rotational relaxation of polar bases in the DNA core.For non-intercalating ions such as sodium and magnesium, such relaxation processes associated with base pairs would be unlikely to be significantly affected by the type of counter-ion. The possibility that the process involves the relaxation of bound water molecules can also be discounted. It can be seen that the gradient of the straight line plot in Fig. 4, which indicates the characteristic expected for bound water, is considerably larger than that observed for the 100 MHz dispersion. In addition, the temperature dependence of the relaxation time for this dispersion indicated a very small activation energy. It would be expected14 that the relaxation of bound water would produce a larger activation energy, 4.5 kcal mol-', associated with the breaking of a hydrogen bond. Other polar species associated with the DNA molecule include those which the back- bone of the DNA strands contribute, i.e.the sugar and phosphate groups. These groups are rotationally hindered through being covalently bonded and it is unlikely that in the solid state the double helix possesses sufficient flexibility to produce the observed relax- ation. However, more detailed analysis of the dipoles associated with these groups is necessary before this can be completely discounted. DNA, like many biological systems, can be expected to exhibit a number of Maxwell-Wagner or interfacial polarisations as a result of charge build up at bound- aries between phases with different electrical properties.Although the DNA fibres are packed into the sample cell to form a compact configuration, it is possible that inter- fibre polarisations may occur between the fibre boundaries. However, a dispersion resulting from this interfacial phenomenon would be expected to be broad with a wide distribution of relaxation times, owing to the heterogeneous nature of the boundary phases. The experimentally observed dispersion, on the other hand, is close to a single Debye relaxation, and for this reason this type of interfacial effect is not considered responsible. A further possible interfacial polarisation may result from a build-up of charge at the DNA fibre/water + ion interface. Although simple, the two-layer capacitance model can be applied to estimate the dielectric parameters of a relaxation process originating from this phenomenon (see Diagram 1).It can be shown that the magnitude of the dispersion as defined by E: is,'4 where ELis the magnitude of the maximum dielectric loss per cycle of the Maxwell- Wagner dispersion. The corresponding relaxation time is given by : Medium 1 can be assigned to be the DNA molecule (E' = 3.0m o1 = lo-'' S m-', d = 25 x lo-'' m) and medium 2 the water and ionic phase (E~= 80, o1= lop2S m-', d = 2.5 x lo-'' m). Applying these values to the model gives ck = 0.006 and fm = 1/(2m)= 1.634 x lo7. Both the magnitude of the calculated dispersion and the fre- quency of the modelled data are considerably smaller than the experimentally observed parameters. However the processes cannot be completely dismissed, owing to the approximate nature of the values selected and the simplicity of the model employed.The third category of possible relaxation processes is ion or proton redistribution as a result of the application of the electric field. The dependence of the dispersion strength on the counter-ion type would seem to discount the possibility of proton translocation uia hydrated bases in the core of the DNA molecule. However, the motion of protons R. S. Lee and S. Bone Diagram 1 The system consists of two parallel slabs of differing dielectric material of thickness dl and d2 placed together between two electrodes (d = d, + d2). The dielectric phases possess permit- tivities of E, and e2 and conductivities of c1and 02,respectively.associated with the H-bond structure of the hydration shell of the phosphate-counter- ion pair must also be considered. For a relaxation process involving proton trans- location via an interconnected water-based H-bond network, it might be expected that the resulting dielectric dispersion should be observable from the lowest hydrations where the water molecules are likely to be structured by multiple H-bonding. This is not consistent with experimental findings, which indicate that the dispersion is only appar- ent after the completion of this strongly bound population. The counter-ions associated with DNA in both solid and solution states have been found to be and this property appears to be relevant to our dielectric data.IR of the dimethylphosphate anion in solvents such as dimethyl sulfoxide, methanol and water reveal a number of different counter-ion configurations. Localised hopping of counter-ions between differeht sites close to the hydrated phosphate group has been proposed as the cause of a dielectric dispersion centred at ca. 8 MHz for hydrated DNA.' It is apparent that there are striking similarities between the characteristics of the 100 MHz dispersion and the lower frequency dispersion centred at 8 MHz also observed for hydrated DNA. Both exhibit relaxation times which are relatively independent of hydration and ion type, with dielectric increments showing the same sort of hydration dependence and ion sensitivity. From this and the previous discussion it therefore seems most likely that the 100 MHz relaxation is related to motion of the counter-ion.It has been shown that the activation energy associated with the relaxation process is very small, which would seem to indicate that the counter-ions are only very loosely bound to the phosphate groups of the DNA molecule. This observation is consistent with the model of counter-ion condensation proposed by Manning,' 9-2 in which the electro- static considerations of DNA as a polyelectrolyte with closely spaced negative charges along the molecular chain result in a merging of the associated potentials and the attrac- tion of a condensed phase of oppositely charged counter-ions. Counter-ions in this phase are not associated with any individual phosphate group and are freely mobile Water molecules bound to deoxyribonucleic acid within the condensed phase.The dielectric increment and relaxation time for ion motion within this phase are given by: Ccr -Cz2q2nL,2A&=--3Eo 36~0kT and where C is the number density of subunits along which the ions migrate, z is the valence of the counter-ion, q is the electronic charge, n is the number of condensed counter-ions and rn is the counter-ion mobility. In this case, the increase in A€ with increasing hydration can result from either an increase in the number of ions involved in the relaxation process or an increase in the average distance, L,, travelled by the ions. Since the relaxation time is independent of hydration level, for the latter process to be valid, any increase in L, must be balanced by a rise in the value of the mobility of the ion with increasing hydration, i.e.L;/p must be approximately constant. Significant hydration-dependent variation of p on this time- scale, however, would appear to be unlikely since the activation energies measured were very small. Using eqn. (7) and a value of ,u = 5 x lo-' m2 Vs-', corresponding to the mobility of Na' ions in aqueous solution, it is possible to calculate a value for L, of ca. 3.2 nm. This distance is close to the diameter of the DNA helix (2.5 nm). A possiblemechanism which is consistent with the calculated value of L, would be the polarisation of the ionic phase in a direction perpendicular to the long axis of the DNA molecule. However, the exact physical meaning of L, in this model is unclear.A relaxation process involving counter-ion motion appears to be consistent with the hydration and ion-type sensitivity of the magnitude of the 100 MHz dispersion. Magne- sium ions are known to bind water molecules very strongly" and this means that the approximately five water molecules which compose the hydration sheath around the Mg2+ are of restricted mobility. These water molecules will therefore exhibit a lower effective permittivity and will provide a poor electrostatic screen between the Mg2 i.ions and the phosphate charges. On the other hand, for Na' ions there is little structure in the hydration shell and this results in these counter-ions being significantly screened from the electrostatic forces which might otherwise hinder their mobility.As a result of these differences in the water-binding properties of the two ions, it is to be expected that Mg2+ ions bind much more strongly to DNA compared with Na' ions. This property has been verified experimentally.22 Applying this ion hydration model to the DNA system, for low hydrations where the counter-ions are in a dehydrated state, it is likely that the ions are localised and reside in positions close to the negatively charged phosphate groups. When the ion becomes fully hydrated and the bound water molecules are able to screen the electrostatic charges dielectrically, the ions become free to migrate along the DNA chain in the condensed counter-ion phase.The appearance of the 100 MHz dispersion and the initial rise in the dielectric increments (Fig. 2) for both ion-types would then mark the transition between localised counter-ions restricted to the phosphate groups and fully mobile counter-ions. Dielectric data pertaining to the dispersion above 1 GHz indicates that this dielectric dispersion is only observable for hydrations greater than three water molecules per base pair for Na-DNA and four water molecules per base pair for Mg-DNA (Fig. 5). This is consistent with the experimentally obtained isotherm data which showed that 4.3 and 4.8 water molecules per base pair were tightly bound to primary hydration sites in Na- and Mg-DNA, respectively. This population of bound water can be expected to be rota- tionally hindered and therefore unable to contribute significantly to the relaxation process. R.S. Lee and S. Bone The magnitude of the polarisation, in this frequency region, for Na-DNA is similar to that expected for bound water (see Fig. 6). However, this is not the case for Mg-DNA, which exhibits much lower polarisation values. For Mg-DNA, water molecules bound subsequently to the monolayer hydration possess dipole moments which, on average, lie between irrotationally bound water and free vapour phase water. A possible explanation might be that this secondary population of water molecules contains both multiply H-bonded water molecules bound to other polar groups such as the sugars and purine and pyrimidine bases.Conclusions This work indicates the importance of water in the establishment and stability of a mobile counter-ion phase along the length of the DNA chain. In the absence of water the ions are localised and restricted to the immediate vicinity of the phosphate groups. Hydration above a critical transition appears to produce a population of ions which exhibit unrestricted motion, a property which is consistent with the existence of a con- densed counter-ion phase. The interactions between water, counter-ions and the DNA molecule are of crucial importance to the role of DNA in living systems. For example, electrostatic interactions and counter-ion release have been implicated in the stabilisation of repressor proteins to DNA and in the ability of duplex DNA to be condensed into compact structures in chromatin and phage heads.We would like to thank Mr. George Stevens for his technical assistance, the University of Wales for the award of a postgraduate studentship to R. S.L. and also the Company Research Laboratory of British Nuclear Fuels for providing additional funding to R. S. L. References 1 R.E. Franklin and R. G. Gosling, Acta Crystallogr. Sect. A, 1953,34,673.2 D. A. Marvin, M. Spencer, M. H. F. Wilkins and L. D. Hamilton, J. Mol. Biol.,1961,3,547. 3 W. Fuller, M. H. F. Wilkins, H. R. Wilson and L. D. Hamilton, J. Mol. Biol., 1965,12,60.4 P. J. Cooper and L. D. Hamilton, J. Mol. Biol., 1966,16, 562. 5 R. Langridge, D. A. Marvin, W. E.Seeds, H. R. Wilson, C.W. Hooper, M. H. F. Wilkins and L. D. Hamilton, J. Mol. Biol., 1960, 2, 38. 6 S.Arnott and D. W. L. Hukins, J. Mol. Biol., 1973,81,937 A. H. J. Wang, G. J. Quigley, F. J. Kolpak, J. L. Crawford, J. H. van Boon, G. van der Mare1 and A. Rich, Nature (London), 1979,282,680. 8 G. P. Archer, S.Bone and R. Pethig, J. Mol. Electron., 1990,6, 199. 9 S.Bone, Biochim.Biophys. Acta, 1987,916, 128. 10 H. Nakamura, S. Mashimo and A. Wada, Jpn. J. Appl. Phys., 1982,21,1022. 11 S. Brunauer, P. H. Emmett and E. Teller, J. Am. Chem. Soc., 1938,60,309. 12 P. R. C. Gascoyne and R. Pethig, J. Chem. SOC.,Faraday Trans., 1977,73,171. 13 M. Davies, Some Electrical and Optical Aspects of Molecular Behauiour, Pergamon, Oxford, 1965,p. 73. 14 R. Pethig, Dielectric and Electronic Properties of Biological Materials, John Wiley & Sons, Chichester, 1979. 15 M. Neubert, R. Bakule and J. Nedbal, Pruc. 5th Zntl. Symp. Electrets, ed. N. J. Piscataway G. M. Sessler and R. Gerhard-Multhaupt, IEEE Publ., 1985,pp. 825-830. 16 A. Bonincontro, C. Cametti, A. Di Biasio and F. Pedone, Biophys. J., 1984,45,495. 17 G. P. Archer, PhD Thesis, University of Leicester, 1989. 18 K. Patel, PhD Thesis, University of Leicester, 1985. 19 G. S.Manning, J. Chem. Phys., 1969,51,924.20 G. S.Manning, Q. Rev. Biophys., 1978,11, 2, 179. 21 G. S.Manning, Biophys. Chem., 1978,9,71. 22 I. Sissoeff,J. Grisvard and E.Guillk, Prog. Biophys. Mol. Biol., 1976,31, 165. Paper 6/00219F;Received 10th January, 1996

ISSN:1359-6640    

DOI:10.1039/FD9960300059

出版商:RSC    

年代:1996

数据来源: RSC

摘要:

Faraday Discuss., 1996,103,71-90 Polar fluctuations in proteins: Moleculardynamic studies of cytochrome c in aqueous solution Thomas Simonson' and David Perahiab a I.G.B.M.C., BP 163, C.U. de Strasbourg, 67404Illkirch, France Laboratoire d'Enzymologie Physico-Chimique et Molkculaire, Universitk de Paris-Sud, 91405 Orsay, France The equilibrium fluctuations (the polar fluctuations), of yeast cytochrome c are studied using nanosecond molecular-dynamic simulations in a spherical droplet of water, with a particular emphasis on the fluctuations of the total dipole moment, which determine the average relative permittivity. These fluctuations follow a simple probability distribution, .predicted by contin- uum electrostatics, and already observed in simulations of several polar liquids.An important component consists of diffusive, mutually indepen- dent, motions of the charged side chains at the protein surface. A quasi-harmonic normal mode analysis of the trajectory shows that while motions covering a large range of frequencies contribute to the polar fluctuations, the four lowest frequency modes account for 50% of the overall static relative permittivity of ca. 25. The fluctuations of the protein bulk, i.e. parts other than the charged side chains, are distributed over a larger number of modes. Modes up to at least 60 cm-' contribute to the average relative permittivity of the protein interior of cu. 4. The water surrounding the protein, despite the structural perturbation represented by the protein, has fluctuations similar to pure water, consistent with the idea of a linear solvent response to the protein charges.The relationship between the microscopic fluctuations seen in the simulations and simple continuum models is discussed. An important aspect of protein hydration is the dielectric screening of charged and polar protein groups by water molecules and by other protein groups. Dielectric screening influences most aspects of protein structure, dynamics, and activity.' v2 Charged groups are extruded from the protein interior, for example, towards the higher dielectric solvent. In enzyme reactions, reorganization associated with charge-transfer and charge-separation steps is reduced by segregating the active site (or redox group) from bulk water.Electrostatic interactions between protein and solvent are very complex; the protein surface is often highly charged, and the dielectric properties of this interfacial region are quite different from the protein bulk. This article describes the molecular dynamics of cytochrome c in water, focusing on the fluctuations of polar groups, which determine the protein's dielectric properties. The dielectric properties of proteins have been studied recently using a number of techniques. Experimental techniques include measurements of pK, shifts3 and redox potential shifts4*' upon mutating charged or polar groups; energy transfer,6 fluorescence,' and Stark effect measurement^;**^ and measurements of dielectric dispersion by dry protein powders.''*' 'Computer simulations have also been used increasingly in recent years to probe charge screening, including several studies of electron and proton transfer in pro- teins,12-' s and direct calculations of the relative permittivity or microscopic dielectric susceptibility of several proteins.' 6--25 These studies show that the protein interior tends 71 Polar JEuctuations in proteins to be much less polarizable than bulk water, leading to a low relative permittivity despite the presence of rather rigid polar groups in the interior of many proteins.This low polarizability makes it costly to ionize buried groups, and, together with relatively fixed polar groups in enzyme active sites, plays a role in reducing the activation barriers for proton and electron transfer in enzymes.26 Computer simulations can give detailed microscopic information on the fluctuations of polar protein groups which determine the dielectric response.However, dipolar fluc- tuations relax slowly, so that fairly long simulations are required, and a realistic descrip- tion of the surrounding solvent is needed, usually including a large number of explicit water molecules. Nanosecond simulations in bulk water are necessary to calculate reli- ably the average protein relative perrnitti~ity.l~*~~ Such simulations have only recently become possible owing to improved computer performance. Thus many basic questions remain open. Proteins are heterogeneous, with both polar and non-polar regions, and mobile and rigid segments.The exact contribution of these to the average relative permittivity, and to local dielectric properties are still unclear. Tightly bound and bulk water both contribute to charge screening, and affect the protein relaxation. Protein dynamics are complex, with vibrational and diffusional motions covering a broad range of timescales, all of which contribute to dielectric relxa- tion. The spatial variation of charge screening throughout a protein’s interior has only begun to be studied. To answer some of these questions, the present article analyses the polar fluctuations of the electron transfer protein cytochrome c in solution, using long molecular-dynamic simulations. We focus on three questions: what are the relevant timescales for relaxation of polar groups; what is the nature of the motions contributing to the dielectric proper- ties; and to what extent are the average static dielectric properties consistent with simple continuum electrostatics.This will give a molecular basis for understanding the relative permittivity of the protein, as well as more local aspects of the dielectric response. We pursue the analysis begun in two previous which examined the protein’s average relative permittivity, and its response to perturbing test charges. We analyse the fluctuations of the total dipole moment, which directly determined the relative permit- tivity, as well as the contributions of different structural groups, and the contributions of collective motions of different frequencies.Comparisons can be made with other polar media, including polar liquids, whose dielectric properties are currently an area of inten- sive in~estigation.~ 7-3 The next section describes computational details of the simulations and the analysis. The third section describes numerical results, based on 1 nanosecond molecular-dynamic simulations of yeast ferri- and ferro-cytochrome c in a spherical droplet of 1400 water molecules. Experimental System and Simulation Ferro- and ferri-cytochrome c are abbreviated by their Protein Data Bank codes, lYCC (ferro-) and 2YCC (ferri-). The simulation conditions were described earlier, but are repeated here for completeness. lYCC and 2YCC were each simulated in a spherical droplet of 1400 TIP3P water molecules,34 starting from recent high-resolution crystal structure^.^^ Electrostatic interactions were approximated by a Coulomb term shifted to zero beyond 12 A.36 The Charmm/Param19 empirical force field was used.36 Arg, Lys, Asp, Glu, and His side chains were fully charged, except for His 18, bound to the haem iron, and His 26, though to be ne~tral.~’ The crystallographic sulfate ion was included; otherwise no counterions were used.The total net charge in the system is +6 (1YCC) or +7 (2YCC). Bond lengths were constrained to ideal values,38 and a soft spherical T. Simonson and D. Perahia 73 boundary potential3’ of radius 24 A was used to confine the system. The system was weakly coupled to a heat bath at 293 K40 One nanosecond of simulation was run for each protein, and the last 900 ps used for analysis.Simulations were done with the program X-PLOR.~’ Relative permittivity and atomic group contributions With the spherical geometry of our simulations, the protein relative permittivity cs is related to the mean square dipole moment through a Frohlich-Kirkwood fluctuation formula :23 where E, is the relative permittivity of water, r,, is the mean radius of the protein, f is the function 9% (2)fbl, E2) = (81 + 282x82 + 2) -2(rp/rw)3(1-&2)(&1 -E2) and rw is the radius of the surrounding solvent region (24 A). The term on the left-hand side of eqn. (1) is known as the G-factor, by analogy to the Kirkwood 8-factor of a homogeneous liquid?2 The variance (AM2) of the total protein dipole moment is simply related to the covariance matrix of atomic displacements : where qi is the partial charge of atom i, ui is its instantaneous displacement from its mean position, the brackets represent an ensemble (or time) average, and the sum is over all pairs of protein atoms.We can define the contribution of a given group of atoms to (AM2) by simply limiting the sum to the pairs of atoms within the group. We can also define the contribution of correlations between two groups of atoms by letting the index i run over the first group, andj run over the second group. Below we will discuss the contributions of individual residues, and of correlations between groups of residues, to AM^). Principal component analysis of the dipole fluctuations It is well known that the eigenvectors of the mass-weighted displacement covariance matrix of the protein correspond to preferential directions of collective motion, some- times referred to as quasimodes of ~ibration.4~’~~The reciprocal of the eigenvalues 1, of this matrix represent effective frequencies of motion o,along these quasimodes: = 2kT/;1, (4) where k is Boltzmann’s constant and T the absolute temperature.In the limit where the energy surface is harmonic, the quasimodes coincide with the normal modes of the protein. For a harmonic system, the variance of the protein dipole moment can be written as a sum over the modes of vibration: where @‘) is the normalized displacement vector of atom i in the mode p and miis its mass.Polar Juctuations in proteins In the general, anharmonic, case, the variance of the protein dipole moment is related to the unweighted displacement covariance matrix, as indicated in eqn. (3). It can be written as a sum over the eigenvectors Fk of this matrix: where e)is the normalized displacement vector of atom i in the eigenvector p, and pupis the eigenvalue. This decomposition is exact, and does not depend on any physical inter- pretation of the eigenvectors. The eigenvectors are referred to as principal fa~tors.~~~~~ In proteins, the atomic masses are all similar except for the hydrogen atoms, and simulations are sometimes carried out with uniform, artificial, atomic masses, for pur- poses of efticiency. Thus in practice, the low frequency quasimodes of vibration, and the first principal factors, are expected to be similar.The quasimodes have the advantage of a clearer physical meaning, while the principal factors have the advantage of an exact connection to the variance of the protein dipole moment. We have analysed the polar fluctuations with both approaches. The quasiharmonic modes also provide a normal coordinate system with which to analyse sampling and convergence of the collective protein motions. At a time t, the instantaneous protein displacement can be projected onto a given mode p, giving a normal coordinate Qp(t).The principal factors provide another generalized coordinate system. The projection of the instantaneous protein displacement onto a principal factor is known as a principal component.In the next sections we analyse the variations and mutual correlations of the first few normal coordinates and principal components over the course of the trajectory. It is of interest to compare the behaviour of the normal coordinates and principal components with the behaviour of a set of Langevin os ill at or^.^^ The Langevin is a one-dimensional oscillator that is stimulated by a random force and damped by friction. If the protein motions were nearly harmonic, the Langevin oscil- lators should give a good approximate picture of the dynamics of each normal coordi- nate, with the friction and random force crudely representative of solvent effects. The autocorrelation function of a Langevin oscillator is where r is the friction coefficient, a = (r2-4~$)”~,and coo is the frequency of the undamped oscillator.Results Overall protein fluctuations and relative permittivity With the spherical geometry of our simulations, the protein relative permittivity E, is related to the mean square protein dipole moment through the Frohlich-Kirkwood fluctuation formula, eqn. (1). The left-hand side of eqn. (1) is referred to as the G-factor. Values for G and E, from these simulations were reported earlier.23 The protein radius rp is not uniquely defined; our definition is such that a homogeneous sphere of radius rp has the same radius of gyration as the protein. It is of interest to calculate G with and without the contribution of the charged protein side chains.Indeed, by excluding the charged portions of the charged side chains from the analysis, we effectively analyse the dielectric properties of the protein bulk, which turn out to be quite different from those of the charged portions of the protein surface. The G-factor was therefore calculated for both 1YCC and 2YCC, both including all protein charges, and excluding the charged portions of the charged side chains. The protein radius r is 16.5 A with all atoms T. Simonson and D.Perahia included, and 15.9 A with the charged groups excluded. When charged side chains are excluded from the calculation, we find (AM2)= 23.9 (eA)2 for lYCC and 15.8 (e.$)2 for 2YCC; the G-factors are 3.6 for 1YCC and 2.3 for 2YCC.This gives E~ = 4 f 1. The dipole fluctuations are roughly isotropic. When charged side chains are included, we find (AM2)= 127 (e%i)2 for 1 YCC and 132 (eA)2 for 2YCC; the G-factors are 15.9 and 16.6. This gives E~ = 25 f10. The dipole fluctuations are very anisotropic in this case. The uncertainty in the relative permittivity was estimated by varying the model parameters rp, r,, E, over a reasonable range, and taking into account statistical uncertainty in (AM2).23 The convergence of G is shown in Fig. 1 for both 1YCC and 2YCC. The cumulative time averages converge approximately over the 1 nanosecond simulation. The three Cartesian components of (AM2) also converge approximately (not shown). Another indication of the degree of convergence is the approximate agreement between the G-factors taken from our two simulations.Since the two protein structures are very similar e~perimentally,~ the two simulations are expected to give similar results. Taking the average over the two simulations, we effectively have a total of two nanoseconds of simulation data. The large variation of the G-factor near the end of the 2YCC simula- tion is due to a large, short-lived, fluctuation occurring in the interval t = 920-960 ps. Eight charged residues, and only eight undergo conformational rearrangements during this interval, and account for the variation of the total dipole moment (Glu -4, Lys 5, Lys 11, Arg 13, Glu 21, Glu 61, Glu 66 and Asp 90). The mean square dipole moment can be written as a sum over all pairs of atoms in the protein [eqn.(3)]. Using this decomposition, we can distinguish the contribution of different interaction ranges to G. By summing over pairs of atoms within a certain cut-off distance, we can plot G as a function of interaction range. Notice that this inter- action range is not to be confused with the electrostatic cut-off distance used in the simulations. The result is shown for lYCC in Fig. 2. Results for 2YCC are similar. The G-factor does not converge until distances of at least 25 A. Comparing the curves obtained with different simulation lengths, we see that the short-range interactions con- verge more quickly with increasing simulation length, reaching stable values after only 550 ps. Interactions over distances less than the 12 A cut-off actually used in the simula- tion converge the fastest.The long-range interactions do not converge until after 850 or lo00 ps. Kusalik has rep~rted~~.~~ that for several polar liquids, including water, the fluctua- tions of the total dipole moment follow a simple probability distribution, P(AM2)= A I AM I exp(-KAM~), (8) "I\ . x10 I 200 400 600 800 lo00 simulation lengthips Fig. 1 Convergence of G-factor during the simulation. The cumulative average of G is shown for 1YCC (solid) and 2YCC (dashed), with (bold) and without (light) the contribution of the charged portions of the charged side chains. Results without the charged side chains are scaled by 10 for clarity. 76 Polar ,fluctuations in proteins 20 19 4 17 3 18 16e 15 14 5 10 15 20 25 30 35 interactionrange/A Fig.2 Convergence of G-factor as a function of the interaction range included in the calculation, and the simulation length; lYCC, all charges. Each curve is labelled by the simulation length (in ps). where A is a normalization constant, and K is simply related to the mean square fluctua- tion : 3 K= 2(AM2) ‘ (9) This probability distribution can be derived from continuum electrostatic^.^^*^^ Devi-ations from it will arise if a system does not behave as a macroscopic continuum; for example, if it is too small. Deviations can also be due to insufficient sampling, especially in the region of the largest, least probable, fluctuations.Sampling the entire probability distribution of M is obviously more demanding than simply estimated its expectation value. Fig. 3 and 4 show the probability distribution of the protein (AM2) for lYCC and 2YCC, respectively. Differences between the lYCC and 2YCC results can reasonably be attributed to sampling limitations, since in other respects the two are similar (e.g. in size). The probability distributions are also shown for the fluctuations along the three Cartesian axes: AM:, AM; and AM,”(small grey dots in the lower panels of Fig. 3 and 4). Continuum electrostatics predicts that these three distributions should be equivalent (if we assume the protein is spherical). We see that when the charged portions of the charged side chains are excluded from the calculation, the fluctuations follow the dis- tribution, eqn.(8), rather well. Anisotropy is moderate. When all protein charges are included in the calculation, some of the sampling limitations become apparent, and possibly also deviations from the continuum theory. The 1YCC dipole moment follows the theoretical distribution quite well, despite a strong anisotropy of the fluctuations. The anisotropy leads to three distinct probability distributions for each Cartesian axis, clearly visible in the figure, above, below, and on top of the total probability distribu- tion. The 2YCC dipole moment deviates strongly from the continuum prediction. Differ- ences between lYCC and 2YCC are of the same magnitude as the differences of either from the continuum prediction.As suggested by Kusalik, a fit of the simulation data to the line in the figure provides a numerical route to the mean G-factor, and the relative permittivity which may be less sensitive to noise than a simple average over the AM2. In our case the two approaches give similar results. The fluctuations in these simulations can be compared to vacuum and quasivacuum simulations. Earlier simulations of tuna cytochrome c in vacuum led to an overall rela- tive permittivity of 4, owing to the more restricted polar fluctuations of charged groups in the absence of solvent.” Recent simulations of yeast ferricytochrome c in a quasi- vacuum environment (protein solvated by 195 water molecules) led to an overall relative permittivity of 8-10.47 T.Simonson and D. Perahia charged groups excluded charged groups included 0.2: 0.04a 0.15 s 0.038 z 0.11 0.02' 0.05! 0.01' n.- 01 0 5 10 15 20 0 20 40 60 80 100 120 Of' -21: 1 -10. 0 5 10 15 20 0 20 40 60 80 100 120 g=AM2/kT r3 Fig. 3 Probability distribution of 1 YCC total dipole moment. Right-hand panels include the contributions of the charged portions of the charged protein side chains; left-hand panels do not. The theoretical form of the probability distribution^^^ are superimposed (solid lines). The small grey dots in the lower panels correspond to the results for the three Cartesian components of g: g,., gu and gz (see text). charged groups excluded charged groups included 0 2 4 6 81012 0 20 40 60 80 100 120 3 -4- -8 ' a -12 4' -1OL.-~ 1 -14~ $ 0 2 4 6 81012 0 20 40 60 80 100 120 g= AM2/kT r3 Fig. 4 Same as Fig. 3 for the 2YCC total dipole moment Polar fluctuations in proteins Comparisons to polar liquids Goldman and Joslin3' have recently classified several polar liquids into two distinct families: hydrogen-bonded liquids and dipolar aprotic liquids, based on a comparison of their relative permittivities, and their dipolar strength functions. The dipolar strength function is given by y = 47-cNp2/9VkT,where p is the molecular dipole moment, and N the number of molecules in volume V.The hydrogen-bonded liquids have larger relative permittivities for comparable values of y, and the two families of liquids occupy distinct regions in the (y, E) plane.We reported earlier that, from the dipole fluctuations observed in our simulations, the relative permittivity of cytochrome c is about 4 f1, excluding charged side chains.23 We can define the dipolar strength function of the protein by analogy to a simple liquid, by summing the square dipole moments of the backbone peptide groups and the polar side chain groups. In the dipolar strength func- tion Np2 is then replaced by xi(piack+ pzide),where pba&is the dipole moment of the peptide group, psi& is the dipole moment of the side chain, and the sum is over all the residues. The result is 2300 Debyes2 = 102 (eA)2, of which 80% comes from the back- bone peptide groups.The protein y value is then 4.7. A relative permittivity of 4 f1 puts the protein clearly in the region of the (y,E) plane occupied by the dipolar aprotic liquids, which include N,N-dimet hyl formamide and N,N-dimethyl acetamide. These liquids, while quite polar, are presumably not as tightly organized by hydrogen bonds as the hydrogen-bonded class, which includes water, formamide, N-methyl formamide, and N-methyl acetamide. If charged side chains are included in the analysis, the protein relative permittivity is ca. 25 10. The contribution of the charged side chains to y can be defined simply as xRQ$(Gut), where R is a charged residue, of charge QR= 1, and duRis the displacement of the side chain centre-of-mass from its mean position.This sum is 60 (eA)2) for 1YCC. Adding this into y gives y = 6.7. With relative permittivity of 25, the protein falls even further into the dipolar aprotic region. Recent simulations of trypsin yielded a relative permittivity of 10 in the active site region;" assuming the dipolar strength function is similar to, or larger than, that of cytochrome c, trypsin also falls well in the dipolar aprotic region. Data for trypsin inhibitor and lysozyme also appear to fall in this region.19 Collective motions:sampling and convergence To understand further the molecular dynamics underlying the calculated relative permit- tivities, we performed a quasiharmonic analysis of the trajectories. Analysis of the normal coordinates of proteins is known to give information about the shape of the phase space region on which the system evolves (its attractor), and the correlations that arise between collective degrees of freedom as the system moves over that region.45v48,49 Such an analysis does not assume that the protein dynamics are actually harmonic.Rather, the quasimodes of vibration provide a convenient generalized coordinate system, which can help to reveal correlations between collective motions, and the degree of sampling of those motions, even if they are not harmonic. We projected the 2YCC trajectory onto the first 300 quasimodes, to obtain the first 300 normal coordinates Qp(t). The covariance matrix was calculated only for the protein atoms. Correlations with solvent atoms were not included explicitly.The eigenvalues are ordered by decreasing magnitude. Probability distributions of the first ten normal coor- dinates are shown in Fig. 5. On the timescale studied here, displacements along the first few normal coordinates are very anharmonic, clustering in a few discrete regions, rather than being normally distributed. Displacements along the normal coordinates 10-300 have more Gaussian probability distributions, with a single peak and a roughly Gauss- ian shape. A Kolmogorov-Smirnov white noise test (Fig. 5) gives the probability that the observed deviations from a Gaussian shape would be seen by chance, given an T. Simonson and D.Perahia underlying Gaussian distrib~tion.~' Of modes 10-300, just 12 have Kolmogorov- Smirnov probabilities below 67%, and 6 below 50%.Overall results for the principal components are very similar (not shown). Similar behaviour was seen in a principal component analysis of lysozyme dynamics in solution, where only the first few principal components deviated strongly from a Gaussian probability di~tribution.~' It may be that the probability distributions of the first few normal coordinates would also con- verge to a Gaussian shape given a longer simulation time. Despite the roughly Gaussian distribution of normal coordinates 10-300, the dynamics along the 300 normal mode directions are very anharmonic. This is illustrated by the autocarrelation functions of the first ten normal coordinates in Fig. 6. The normal modes oscillate at frequencies much lower than the ones predicted by the eigen- values of the covariance matrix.The autocorrelation functions shown correspond roughly to underdamped Langevin oscillators [eqn. (7)] with frequencies of 0.04-0.27 cm-l, periods of 800-115 ps, and friction coefficients of 0.004-0.03 cm? The autocor- relation functions of the corresponding Langevin oscillators are shown as dashed lines in the figure. In contrast, the quasiharmonic frequencies, calculated from the covariance matrix, range from 2 to 7 an-'. Thus the quasiharmonic frequencies are 25-50 times 2:::iZI0.01 ** -100-50 0 50 100 3 0.04 4 15% I 0.02On03( 0.0i 83811 -75-50-25 0 25 50 75 5 I -8-0.01 .* -40 -20 0 20 40 8 0.02 0.02 0.01 0.0 1 Fig.5 Probablity distributions for the first ten normal coordinates from the 2YCC trajectory (dots). Panels are labelled by mode number. The Gaussian with the same standard deviation is shown as a solid line. The Kolmogorov-Smirnov confidence level Pks(marked on each diagram in %) is the probability that the observed deviations from the Gaussian would occur by chance, if the underlying probability distribution is indeed Gaussian. 4.75-Polar fluctuations in proteins mode 1 mode 2 0 t2.0 0 = 2.4 0.25 0.25 -0.25 -0.5 -0.75 50 100 150 200 250 300 350 50 100 150 200 250 300 350 mode 3 ‘R mode 4 ;0.7 o =3.1i%=.07 0.8 \, W =3.9 0.5 q=.13 0.25 -0.5 ”. ./*-0.75 50 100 150 200 250 300 350 50 100 150 200 250 300 350 mode 5 w -4.6 1 0 =4.9 clp.11 q#= .12A,*-----0.25 -0.25 -0.25“i-1 -0.5 -0.5 50 100 I50 200 250 300 350 50 100 150 200 250 300 350 mode 7 mode 8 1, O.*A\ 0 =5.4 Ei\\ y=20 1 0.25 -0.25 -0.5 I 50 100 150 200 250 300 350 50 100 1SO 200 250 300 350 mode 10 vc mode 9 -w =5.8 o = 6.4 0.2 50 100 150 200 250 300 350 50 100 150 200 250 300 350 timelps Fig.6 Autocorrelation functions for the first ten normal coordinates from the 2YCC trajectory. Panels are labelled by the mode number. Fits to a Langevin oscillator are shown as a dashed line. The Langevin oscillator frequency wL and the quasiharmonic frequency w are given in cm-’. Friction coefficients vary from 0.04 to 0.3 ps-’. too fast. For modes 11-300, the quasiharmonic frequencies slowly approach the actual oscillation frequencies seen in the autocorrelation functions; for modes 200-300, the quasiharmonic frequencies are still about two times too fast.Scatter plots of pairs of normal coordinates show that the first three normal coordi- nates are strongly correlated with each other (Fig. 7). In the subspace corresponding to these three coordinates, the system wanders between a few discrete regions by infrequent jumps. In our nanosecond simulation, the system has only visited each region once or twice. Therefore a full exploration of this subspace would require much longer times, and our sampling of these degrees of freedom must be viewed as incomplete. In contrast, the other normal cordinates (4-300) are more weakly correlated, and appear to be rea- sonably well sampled. For the first few modes, the wandering motions seen in Fig.7 are much more suggestive of a stochastic diffusive walk than of quasiharmonic vibrations. T. Simonson and D. Perahia p =1 p 12 p =3 I --40 0 40 -40 0 40 P=4 p =5 P=6 -40 0 40 40 0 40 p r7 p =9 4 -4 I I I I -40 0 40 -40 0 40 -40 0 40 QP Fig. 7 Scatter plots giving the joint distributions of pairs of normal coordinates (Q,, Q,+ from the 2YCC trajectory [in 8, (g mol-') 1/2]. Each plot is labelled by p. Quasiharmonic analysis of the dipole fluctuations Collective motions in different frequency ranges contribute to the dielectric properties, with low-frequency modes expected to contribute most.Indeed, when a perturbingcharge is introduced, structural deformation will tend to follow the directions corre- sponding to soft, polar, modes of oscillation of the unperturbed system. This is expressed mathematically in eqn. (5) and (6),where the G-factor is decomposed into a sum over either the quasiharmonic modes of vibration, or the principal factors. We have calcu- lated the contribution of the quasiharmonic modes of vibration to G. The modes are calculated from the complete protein covariance matrix, including charged portions of the charged side chains. The G-factor can then be calculated with or without the contri- bution of these charged portions. Fig. 8 shows the contribution of individual modes to G for 2YCC as a function of frequency. Despite its limitations, the quasiharmonic model gives a good estimate of the G-factor with and without charged side chains.When all charges are included, the G-factor converges very rapidly as a function of frequency. The first four quasimodes account for 50% of G. The convergence is almost complete after the first 90 modes. This rapid convergence probably occurs because most of G is accounted for by a few charged side chains, which can be described by just a few degrees of freedom. When charged groups are excluded, the convergence is more gradual, with approximate convergence after 300 modes. This frequency range includes many types of protein deformations, such as motion of secondary structure elements and backbone deformations.Fluctuations of individual side chains The dipolar fluctuations of the protein are dominated by the charged side chains, as noted earlier.23 Table 1 gives the individual contributions of the charged side chains to Polar Juctuations in proteins v v) I0 !,number of modes35 . " 0 10 20 30 40 50 60 frequencyicm-1 Fig. 8 Contribu ion of quasimodes to the G-factor for 2YCC. The cumulative G is shown as a function of frequency. All charges (black dots); charged groups excluded (small dots, result scaled by 10 for clarity). Solid line: number of modes per cm-' (scaled by 5). Dashed line: number of modes up to the current frequency (scaled by 1/20). (AM2) for both lYCC and 2YCC, along with the average rms displacement of the side chain atoms during the simulation.The correlation between these two quantities is over 80%. Some of the charged side chains contribute almost as much to (AM2) as all the uncharged side chains together. Fig. 9 shows snapshots of the charged side chains at regular intervals along the trajectory, illustrating the broad range of motion of some of the side chains. Correlations between the charged side chains are weak, and contribute only 2-10% of the G-factor. Much of the dipolar fluctuations come from the 16 positively charged lysine residues. To characterize the local motions of these residues we show the probability distributions Table 1 Dipole fluctuations of charged side chains lYCC 2YCC 1YCC 2YCC residue MRR rmsf MRR rmsf residue mRR rmsf w~RR rmsf Glu -4 7.67 2.5 1.68 1.3 Lys -2 6.02 1.6 3.7 1 1.3 Lys 4 1.63 1.o 1.09 0.9 Lys 5 5.46 1.3 2.97 1.2 Lys 11 2.9 1.1 5.58 1.4 Arg 13 2 1.o 6.42 1.7 Glu 21 4.81 1.8 4.29 1.8 Lys 22 1 1.45 2.3 9.38 2.2 Lys 27 5.69 1.7 7.11 2.1 His 33 0.70 0.9 0.86 1.o Arg 38 0.22 0.5 0.26 0.5 His 29 2.46 1.4 0.49 0.8 Glu 44 2.19 1.5 1.72 1.2 Asp 50 0.9 1.1 0.59 0.8 Lys 54 1.85 1.2 3.12 1.4 Lys 55 2.63 1.o 3.92 1.1 Asp 60 0.73 1.o 2.06 1.4 Glu 61 0.7 0.9 3.07 1.5 Glu 66 2.62 1.4 2.16 1.2 Lys 72 4.78 1.5 1.85 0.9 Lys 73 7.74 1.6 4.77 1.2 Lys 79 4.91 1.5 3.25 1.2 Lys 86 7.4 1.8 6.83 1.8 Lys 87 6.78 1.5 5.83 1.4 Glu 88 4.49 1.7 1.23 1.o Lys 89 9.16 1.7 2.29 1.o asp 90 1.85 1.2 0.86 1.1 arg 91 0.59 0.6 0.31 0.5 Asp 93 0.3 0.6 0.23 0.5 Lys 99 5.14 1.3 2.2 1.2 Lys 100 7.96 1.6 7.88 1.7 Glu 103 2.23 0.9 3.78 1.3 CRR mRR 126 101 CR + R,mRR* -2 13 <AM2> 127 132 rmsf) 0.82 0.84COV(MRR, Charged residue contributions ~~zRRto total (AM2) in (eA)' and rms fluctuations (rmsf) around mean positions in A. T.Simonson and D. Perahia Lysll6 Lysl00 4 Fig. 9 Snapshots of 2YCC taken at regular intervals along the trajectory. The protein backbone, the haem group, and the charged side chains are shown. Several of the charged side chains are labelled with their name and number, and their contribution to the variance of the protein dipole moment [in (eA121. of their side chain dihedral angles, from the 2YCC trajectory, in Fig. 10. Note that the probability distributions are very similar for all 16 lysines, i.e.they are only weakly dependent on the local environment of each lysine. Therefore the wide range of contri-butions of the individual lysines to G must come from different motions of the protein backbone. The free energy barriers for transitions between side chain rotamers (not shown) vary from about 1 kcal mol-' (g+ +g-) to 3 kcal mol-' (t +g+). The (xl, x2)scatter plot can be compared with a recent survey of 191 high resolution protein X-ray structures.'l The main qualitative difference is the higher population of the (g', g-) and (t, g-) regions in the simulation; these regions are only weakly populated in the X-ray structures. Notice that each X-ray structure normally only contains the dominant con- former occupied by each residue, with conformational disorder modelled simply as a Gaussian dispersion around the dominant conformation. On the other hand the simula- Polar fluctuations in proteins ... 30G 60 60 180 300 60 180 300 0.04 0.02 0.02 60 180 300 60 180 300 60 180 300 x' x2 x3 Fig.10 Cumulative statistics for the 16 lysines' side chain torsion statistics. Upper panels: (xi,xi+ scatter plots; lower panels : probability distributions p(xi). tion (and the real system) undoubtedly samples minor conformers for some fraction of the time. Thus the X-ray survey is expected to underrepresent minor conformers. Solvent fluctuations The fluctuations of the solvent layer surrounding the protein were analysed and com- pared with fluctuations occurring in a pure water sphere of the same dimensions (radius 24 A).The mean square fluctuations of concentric shells are reported in Table 2. Shells 2 A thick are considered, with inner radii of 18, 20 and 22 A. The results are normalized by dividing (AM2) by the average number of water molecules in each shell. Results are quite similar in the two systems, protein + water and pure water. The mean square fluctuations are about 10% larger in the pure water system, in each of the three shells. This is comparable with the statistical uncertainty. In the outermost shell the water density is higher in the pure water sphere than in the protein + water sphere (357 us. 298 molecules). This may be due partly to electrostriction around the charged protein side chains, and partly to the ad hoc, commonly used, method to overlay water around the protein when preparing the simulation.The relative permittivity of the pure water sphere was calculated earlier,23 and found to be rather high (ca. 110). It was recently show that when the same water sphere is simulated with no cut-off for the electrostatic interactions, a much more accurate rela- tive permittivity of 80 is obtained.52 Table 2 Dipole fluctuations of concentric water layers (N) (AMZ)l(N)l(e~)2 system 18-20 8, 20-22 8, >22 8, 18-20 8, 20-22 8, >22 8, (N) 2YCC + water 320 369 298 0.32 0.34 0.45 (N) water sphere 351 360 357 0.34 0.40 0.51 (N) is the average number of waters in each concentric region.T.Sirnonson and D. Perahia Discussion Limitations of the simulation The main limitations of the simulation were discussed earlier;24 they include the use of a 12 A electrostatic cut-off, the modest size of the water droplet surrounding the protein (1400 molecules, 24 A radius), and the limited sampling accomplished in one nanosecond of molecular dynamics. It is important to emphasize that we are analysing a finite system, with no long- range interactions, so that an electrostatic cut-off is not as dramatic an approximation as in a bulk medium. Unfortunately it is very expensive to run simulations at different cut-offs, particularly since the quantities we are interested in converge slowly. Simula- tions of the pure water sphere with no electrostatic cut-off gave a more accurate relative permittivity (80 compared with 110 with the 12 A ~ut-off).~~~~~suggest that a 12 A cutoff is at least qualitatively reasonable.'' Studies of other proteins The size of the water sphere was dictated by available computer resources.Dielectric theory indicates that the overall protein dipole fluctuations should be only weakly affected by the finite size of the water sphere, compared with an infinite sphere.24 Never- theless, the surrounding vacuum is expected to induce local distortions in both the struc- ture and dynamics of charged protein groups at the protein surface. This artefact can only be reduced by increasing significantly the simulation cost. Increasing the size of the sphere to 28 A radius, for example, would double the simulation cost.The quality of conformational sampling in this work was analysed in some detail above. The quasiharmonic decomposition of the 2YCC trajectory showed that except for the first three or four modes, collective motions along the normal mode directions are well sampled. Scatter plots of pairs of normal coordinates give a single dense cluster which is exhaustively covered a large number of times in the course of the simulation. In contrast, motions along the first three normal mode directions are only partly sampled. Along these directions, the system makes infrequent jumps between a few discrete regions, each of which is explored only once during the simulation.For the system to visit each region a large number of times, a much longer simulation would be necessary. As a result, individual correlation coefficients between several pairs of distant residues (Lys86-LyslOO, Lys22-Lys 100, Lys27-Lys73) are unrealistically large (on the order of &0.4), the residues in these pairs having large displacements in the first three normal modes. A similar slow convergence of several interatomic correlation coefficients was noted in simulations of my~globin.~' Nevertheless, the G-factor is seen to converge rea- sonably well for both lYCC and 2YCC, as the vast majority of the protein degrees of freedom are well sampled. In particular, the lYCC and 2YCC sulations independently give quite similar G-factors and relative permittivities, based on a total of two nanose- conds of data.Timescale and nature of polar fluctuations The time scale of polar fluctuations and relaxation has important consequences for the protein's interactions with mobile charges and changing electric fields in its biological environment. It also has technical consequences for molecular simulations of electro- static processes, such as charge transfer, since the quality of sampling will be influenced by the timescale of polar fluctuations. Experimentally, dry protein powders present very flat dielectric dispersion curves, indicating a broad range of relaxation times.'0q" We see here that for the overall dipole moment, and therefore the average dielectric properties, relaxation times can be quite long.The protein G-factor relaxes slowly, converging over several hundred picoseconds, with short-range interactions converging more rapidly than long-range ones. The groups that contribute the most to G are the charged protein side chains, which undergo large, Polar fluctuations in proteins diffusive, mutually uncorrelated motions. While the local motions of the 16 lysine resi- dues around the torsion angles xl,xz, x3 are all similar, different motions of the protein backbone lead to a range of contributions of the lysines to G. The quasiharmonic analysis shows that the largest contribution to G comes from collective motions in the frequency range 0-25 cm-', or 0-60 cm-' when charged side chains are excluded. Only a few of the low frequency collective degrees of freedom are strongly coupled.These few degrees of freedom make a significant contribution to G, however. The dynamics of these degrees of freedom correspond to infrequent jumps between a few discrete regions, each region being sampled only once or twice in a nano-second. Except for the first few modes, the probability distribution of motions along each mode is roughly Gaussian, in agreement with the quasiharmonic model. The auto- correlations of these modes can be fit roughly by low frequency, weakly damped Lange- vin oscillators. However, the frequencies from this fit are much lower than the frequencies predicted by the quasiharmonic model. This means that the time actually required to explore the full range of motions along each normal mode is much longer than the simple harmonic prediction (which is l/co = <A2)/2kT,where A is the ampli- tude of motion of the mode and co its frequency).Thus the dynamics of the low fre- quency modes analysed here are primarily diffusive, rather than vibrational. Despite this limitation, the quasiharmonic model gives a good estimate of the G-factor (which is a static property, and does not depend on the timescale). Dielectric properties of cytochrome c While a great deal of experimental and theoretical work has been performed on electro- static interactions and charge screening in proteins, several basic questions remain unan- swered. Poisson-Boltzmann models routinely treated proteins as homogeneous low-dielectric media, with charged and polar groups modelled as fixed embedded charges, surrounded by a high-dielectric solvent.This model has been surprisingly suc- cessful in numerical calculations, and has influenced our way of thinking about protein electrostatics. However, many different variants of the Poisson-Boltzmann model have been used, sometimes giving very different results.54 The premises of a homogeneous, isotropic polarizability within a protein, of a sharp protein/solvent boundary, and a linear response to perturbing fields, clearly need further microscopic investigation. Dielectric dispersion measurements in dry protein powders have also been used to probe charge screening in cytochrome c and othe proteins.'0*" While these powders do have a low relative permittivity of 2-5, the relation to proteins in solution is unclear.Ionizable protein side chains are probably in their neutral form in the dry powder, and their mobility is very reduced compared with bulk solution. The powder relative permittivity increases rapidly with even very low solvent contents, and this could be due either to ionization of some of the side chains, increased mobility (as suggested by Bone and Pethig), the direct contribution of added solvent to the overaIl polarity, or all three effects. The present simulation gives a relative permittivity of 3-5 for the protein bulk, in close agreement with the dry powder measurements, and a relative permittivity of ca. 25 for the entire molecule, in good agreement with simulations of six different proteins despite different simulation condition^.^^^^ Simulations of a pure water sphere in the same conditions gave a water relative permittivity in fair agreement with e~periment,~ though a no-cut-off simulation is required to achieve very good agreement.52 The polar fluctuations of cytochrome c in our simulations, to a first approximation, are the ones that are expected for a homogeneous dielectric medium, as long as the charged portions of the charged side chains are excluded from the analysis. This amounts to viewing the charged portions either as part of a distinct, intermediate, medium, between protein and solvent, or simply as a part of the outer, solvent, medium.The calculated protein relative permittivity is then 4 1, in agreement with powder T.Simonson and D.Perahia measurements, as well as amide crystal measurements. Notice that this low relative per- mittivity is an average over the entire protein interior. A higher relative permittivity (ca. 10) was recently computed for the active site of trypsin.I8 It is unclear how many charged residues were included in this calculation. We emphasize that the low relative permittivity found here does not imply that the protein interior has no polar groups.56 It simply means that these groups have limited mobility, and therefore are weakly pol- arizable. The fluctuations of the protein dipole moment are roughly isotropic, and the system is roughly homogeneous in the sense that contributions to the relative permit- tivity come from the entire protein bulk, rather than only a local region such as the surface.The probability distribution of the polar fluctuations agrees with the prediction of simple continuum electrostatic^.^^^^^ In addition, the fluctuations of the solvent around the protein are close to the ones seen in a pure water sphere of the same dimen- sions: while the protein groups distort the solvent structure, they do not change the equilibrium fluctuations of the solvent very much. This is consistent with a continuum view of the solvent. In a continuum view the perturbing protein groups distort the structure of the solvent, and this distortion takes the form of induced charge at the protein/solvent boundary; but they do not affect the relative permittivity of the solvent, and therefore do not affect its equilibrium fluctuations. We also showed earlier that the probability distribution of the electrostatic potential at any given location within the protein is nearly Gaussian.24 This is a necessary condition for the protein to respond to perturbing charges as a linear medium, as it should in a continuum picture.The normal coordinates of the protein also follow roughly Gaussian distributions, except for the first few modes. At a more detailed level, of course, the continuum picture is unrealistic. If we attempt to view the entire protein, charged side chains included, as a single homogeneous, iso-tropic medium, a serious inconsistency arises as discussed earlier.2 The calculated rela- tive permittivity of 25 & 10 arises mainly from large, mutually uncorrelated, anisotropic, fluctuations of the charged side chains at the protein surface. This is contrary to the simple homogeneous continuum view.Even when charged side chains are excluded from the analysis, the polar fluctuations of the remainder of the protein are not strictly iso- tropic or uniform, but rather increase going from the centre of the protein towards its surface. Thus the calculated relative permittivity is ca. 2 in the innermost part of the protein, increases to 4 in the outer part, and rises to 25 if the charged groups at the protein surface are included. In a similar way, the local susceptibility in response to a perturbing point charge increases fairly smoothly as the point charge is moved from the protein centre towards its surface24 (although there are more complicated local varia- tions superimposed on this simple trend).An improved Poisson-Boltzmann model could incorporate this variation of the dielectric properties in a simple way. An early attempt to do this was made by States, Karplus and co-w~rkers.~~ Dielectric saturation is also found in our model, in contrast to a linear dielectric continuum picture. While the protein polar fluctuations are approximately Gaussian, deviations from a linear response are found to occur for perturbing charges of about e/4 in the protein interior, and this is confirmed by a more detailed investigation of oxida- tion of the haem.58 In simulations of a few other proteins, however, a linear resonse has been seen for larger perturbing charges, such as a redox electron, at specific sites.13 Why are continuum models as successful as they are, given the tremendous complex- ity of protein electrostatics? Two features are of interest.First, while continuum models are sensitive to a whole series of empirical parameters, such as charge radii, they do have a certain robustness with respect to the exact position of the protein/solvent boundary. Whether two charged protein groups are modelled as just inside, or just outside, the boundary does not change their self-energies (interaction with solvent) or their inter- action free energy too strongly. This can be shown very schematically for the simple case of a planar boundary, with numerical parameters suitable for typical groups at the Polar fluctuations in proteins protein surface.For two charges q = e, the interaction free energies from the two models (charges inside or outside the boundary) depend on the exact distance of the charges from the boundary. With typical parameters, the interaction free energies in the two models differ by 8 kcal mol-' when the charges are 4 A apart, and 2 kcal mol-' when they are 8 A apart. The charge self-energies, on the other hand, depend sensitively on the charge radii, and the two models can be made to agree if different radii are used in each. While these energy differences between the two models are not negligible by any means, they must be compared with the much larger total free energy and with the considerable uncertainty surrounding the optimal choice of model parameters.A second feature, noted in our earlier arises from the electrostatic coupling between protein charges and solvent. When protein charges move, the solvent adjusts on a picosecond timescale, so as to screen the protein charges, and reduce the electrostatic potential within the protein. In this way, when we look at the electrostatic potential, much of the microscopic detail of the protein dynamics is smoothed away by solvent screening. This is illustrated by the gradual variation of the local susceptibility in response to a perturbing test charge, which increases slowly by a factor of two as the test charge is moved from the protein centre towards the protein surface.24 Protein and solvent make large, almost equal, contributions to this susceptibility, and these contribu- tions vary wildly when the test charge is moved from place to place.However, the electrostatic coupling between protein and solvent makes a third, almost compensating contribution, so that the total susceptiblity varies smoothly. In effect, while the contin- uum model does not capture the rapidly varying, separate protein and solvent contribu- tions, it gives a reasonable estimate of their sum. Conclusion This article has given a detailed microscopic picture of fluctuations of polar groups in cytochrome c. Fluctuations of the total protein dipole moment, which determine the overall relative permittivity, were broken down into contributions from different struc- tural groups, and motions in different frequency ranges. This provides a basis for a detailed understanding of both the average relative permittivity and more local dielectric properties, including the screening of a perturbing point charge, such as a proton or a redox electron.While a few of the lowest frequency motions are not fully sampled in the nanosecond simulations, the vast majority of the collective degrees of freedom appear well sampled. Although the protein motions are by no means harmonic, motions along the normal mode directions are approximately independent of each other, and follow a Gaussian distribution, as noted also in the case of lysozyme and bovine pancreatic trypsin inhibit~r.~~,~' This suggests that protein dynamics in solution are in a sense simpler than previously thought.The fluctuations seen in the simulations are consistent, to a first approximation, with a continuum model of the protein, as long as the charged portions of the charged side chains are viewed as part of the outer, solvent, region, and not as part of the protein bulk. The low relative permittivity calculated in this case agrees with measurements on protein powders. It contrasts with the somewhat higher local relative permittivity (ca. 10) found in the active site of trypsin." If the entire protein is included in the calcu- lation, a large overall relative permittivity of ca. 25 is found. Qualitatively very similar results have since been observed in four additional proteins.55 The low relative permit- tivity of the protein bulk does not mean that the protein interior is a homogeneous non-polar medium.It simply means that polar groups there have limited mobility, and thus limited ablity to rearrange in response to a perturbing electric field. At a more detailed level, the relative permittivity is lower in the innermost part of the protein (ca. 2), increases to 4 in the outer part, and rises to 25 if the charged groups at the protein surface are included. Thus an improved Poisson-Boltzmann model could be obtained in T. Simonson and D.Perahia principle by incorporating a spatial variation of the dielectric proper tie^.^' With these limitations in mind, continuum electrostatics can serve as a useful framework for a qual- itative understanding of protein electrostatics.Simulations were performed on the Cray C98 of the IDRIS supercomputing centre of the Centre National de la Recherche Scientifique. References 1 R. Pethig, Dielectric and electronic properties of biological materials, John Wiley, New York, 1979. 2 A. Warshel and S. Russell, Q. Reu. Biophys., 1984 17,283. 3 A. Russell and A. Fersht, Nature (London), 1987 328,496. 4 N.Rogers, G. Moore and M. Sternberg, J. Mol. Biol., 1985,182, 613. 5 R. Varadarajan, T. Zewart, H. Gray and S. Boxer, Science, 1989,243,69. 6 S. Northrup, T. Wensel, C. Meares, J. Wendolowski and J. Matthew, Proc. Natl. Acad. Sci. USA, 1990 87,9503 7 D.Pierce and S. Boxer, J. Phys. Chem., 1992,96,5560. 8 D. Lockhardt and P. Kim, Science, 1992,257,947. 9 M. Steffen, K. Lao and S. 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Perahia, Proc. Natl. Acad. Sci. USA, 1995,92, 1082. 24 T. Simonson and D. Perahia, J. Am. Chem. Soc., 1995,117,7987. 25 T. Simonson and D. Perahia, Comp. Phys. Comm., 1995,91,291. 26 A. Warshel, Proc. Natl. Acad. Sci. USA, 1978,85, 5250. 27 M. Neumann, J. Chem. Phys., 1986,85,1567. 28 H. Alper and R. Levy, J. Chem. Phys., 1989,91,1242. 29 G. King and A. Warshel, J.Chem. Phys., 1990,93,8682. 30 S. Goldman and C. Josh, J.Phys. Chem., 1993,97, 12349. 31 2. Kurtovic, M. Marchi and D. Chandler, Mol. Phys., 1993,78, 1155. 32 P. Kusalik, Mol. Phys., 1993,80,225. 33 P. Kusalik, M. Mandy and I. Svischev, J. Chem. Phys., 1994,100,7654. 34 W. Jorgensen, J. Chandrasekar, J. Madura, R. Impey and M. Klein, J. Chem. Phys., 1983,79,926. 35 A. M. Berghuis and G. D. Brayer, J. Mol. Biol., 1992,223,959. 36 B. Brooks, R. Bruccoleri, B. Olafson, D. States, S. Waminathan and M. Karplus, J. Comput. Chem., 1983,4, 187. 37 G. Hartshorn and G. Moore J. Biochem, 1988,258,595, 38 J. Ryckaert, G. Ciccotti and H. 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Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1986. 51 R. Abagyan and M. Totrov., J. Mol. Bid, 1994,235,983. 52 T. Simonson, Chem. Phys. Lett., 1996,250,45. 53 R. Loncharich and B. Brooks, Proteins 1989,6, 32. 54 J. Antosiewicz, J. McCammon and M. Gilson, J. Mol. Biol., 1994,238,415. 55 T. Simonson and C. L. Brooks, 111, J. Am. Chem. Soc., in press. 56 A. Warshel, Nature (London), 1987,330, 15. 57 M. Delepierre, C. Dobson, M. Karplus, F. Poulsen, D. States and R. Wedin, J. Mol. Biol., 1987, 197, 111. 58 T. Simonson, unpublished work. Paper 5f08274I; Received 20th December, 1995

ISSN:1359-6640    

DOI:10.1039/FD9960300071

出版商:RSC    

年代:1996

数据来源: RSC

摘要:

Faraday Discuss., 19!96,103,91- 1 16 General Discussion Prof. Lynden-Bell opened the discussion of Prof. Robinson’s paper: This was a stimulating paper, but are you saying that water potentials should include a next-nearest neighbour interaction explicitly or will this interaction come out of a potential with the correct physics? Prof. Robinson replied: Absolutely, correct physics will of course give the correct many-body potential surface for water-water interactions. Right at the moment, however, this is just too much to ask. A good representation of these forces may require perhaps 100 water molecules, maybe many more, to be examined in quantum mechani- cal detail, including van der Waals contributions, which are always difficult quantum mechanically, and with regard to a huge number of important configurations.For various very complicated reasons, no one has yet been able to perform absolutely correct quantum mechanics on even the dimers of water and ammonia! See our papers on this topic.lV2 Clementi and co-workers3 only touched the front surface of the problem in an attempt to create an ab initio potential, using three- and four-body corrections, then examining by Monte Carlo 27 different water polymers, 4 < n < 8. Experimentally, the problem is at least as obtuse, with up to only a pentamer ring having been examined in any detail.4 Thus, my feeling is that, as in the past, the most useful liquid water interaction models for the foreseeable future, particularly those attempting to provide accurate non-nearest-neighbour structure, will be empirical, being only roughly guided by the results from ‘correct physics’.Empirical incorporation of instantaneously responsive ‘electronic ’ polarization and intramolecular bond flexibility, including disso- ciation, have already been described in many models,’ but are as yet too computa- tionally costly for extensive use in biological systems. As described in our paper, experiments have shown that neither temperature nor pressure has much effect on the inner tetrahedral structure, so the seemingly bizarre properties of the liquid cannot depend very much on variations within this inner struc- ture. Our explicit incorporation of non-nearest-neighbour or outer structural effects is simply a new step in an empirical direction.We are currently experimenting with the idea of confining the angular dependence of the H-bond interaction to a much narrower range than heretofore so it does not intrude on the more lobal van der Waals inter- action with a known O.--Opotential minimum near 3.4 1. Optimisation of this new model will be made to match the density maximum and the isothermal compressibility minimum. Near surfaces, such as biological surfaces, the added variability from the outer water structure could very well provide a qualitatively different picture than obtained with only the open tetrahedral structure that is built into current models. In fact, it may be possible to add this outer structural feature into current models without making other drastic modifications.This could improve water-water potentials without increasing the computational cost very much. 1 F. F. Muguet and G. W. Robinson, J. Chem. Phys., 1995,102,3648. 2 F. F. Muguet, G. W. Robinson and M. P. Bassez-Muguet, J. Chem. Phys. 1995,102,3655. 3 K. S. Kim,M. Dupuis, G. C. Lie and E. Clementi, Chem. Phys. Lett. 1986,131,451. 4 K. Liu, M. G. Brown, J. D. Cruzan and R.J. Saykally, Science, 1996,271,62; in a personal communica- tion, Dr. Saykally has informed me that a hexamer will be described soon in Nature (London). 5 G. W.Robinson, S-B. Zhu, S. Singh and M. W. Evans, Water in Biology, Chemistry and Physics: Experi- mental Overviews and Computational Methodologies, World Scientific, Singapore, 1996.Prof. Luzar commented: Prof. Robinson seems to believe that the mysterious anomaly of water has been successfully modelled for the first time in his papers.’V2 I 91 General Discussion have to point out that in a series of papers3-' nearly 30 years ago, Bell et al. worked on one-dimensional and three-dimensional models for water-like fluid, including the one- dimensional Takahashi model? Maxima on the curves of density against temperature at constant pressure and minima in isothermal compressibility curves were found at certain parameter values. Much more recently, an extension of the van der Waals equation has been pro- p~sed,~which incorporates, in an approximate fashion, the effects of the network of hydrogen bonds. The resulting model qualitatively predicts the unique thermodynamic properties of water, including the density maximum.' I also have to make a comment on the statement given in the present paper,' namely that current computer simulations of liquid water do not reproduce the density maximum.There were computer simulations that failed to reproduce the density maximum,'0*'' and those that successfully reproduced With those that failed,'0*'' the simulation runs were usually not long enough. One has to realize that at constant pressure the volume fluctuations of a relatively small model system (usually up to 300 particles) are large and the proper sampling of these density fluctuations requires simulation times of the order of nanosecond^.'^ The existence of the density maximum was sought from constant volume simulations of ST2, TIP4P and SPC/E, where pres- sure was seen to go through a minimum at a certain temperat~re.'~ In the isobaric ~irnulation,'~the SPC/E 1 bar isobar is characterized by a temperature of maximum density (TMD) of 235 K and a corresponding density 1.026 g ~m-~.The -40 MPa isobar is instead characterized by a TMD of ca. 250 K and the corresponding density of 1.000 g cm-3,14 in fair agreement with the experimental pressure dependence of the TMD line. The pressure-dependent density maximum is obviously captured by the exist- ing pair potentials for water. 1 C. H. Cho, S. Singh and G. W. Robinson, Phys. Rev. Lett., 1996, 76, 1651. 2 C, H. Cho,S. Singh and G. W. Robinson, Faraday Discuss.,1996,103, 19.3 G. M. Bell and H.Sallouta, Mol. Phys., 1975, 29, 1621. 4 G. M. Bell and D. W. Salt, Mol. Phys., 1973,26,887. 5 G. M. Bell, J. Phys. C, 1972, 5, 889. 6 G. M. Bell and D. A. Lavis, J. Phys. A, 1970,3,568. 7 G. M. Bell, J. Math. Phys., 1969, 10, 1753. 8 G. M. Bell, J. Phys. A, 1980, 13, 659. 9 P. H. Poole, F. Sciortino, T. Grande, H. E. Stanley and C. A. Angell, Phys. Rev. Lett., 1994,73, 1994. 10 S. L. Billeter, P. M. King and W. F. van Gunsteren, J.Chem. Phys., 1994,100,6692. 11 A. Wallqvist and P. 0.Astrand, J. Chem. Phys., 1994,85627. 12 P. H. Poole, F. Sciortino, U. Essmann and H. E. Stanley, Nature (London), 1992, 360, 324. 13 L. A. Baez and P. Clancy, J. Chem. Phys., 1994, 101,9837. 14 P.H. Poole, F. Sciortino, U.Essmann and H. E. Stanley, Phys. Rev. E, 1993,48, 3799. Prof. Robinson responded: Since one-dimensional models have been known for a long time, it is not surprising that someone discussed their application to water long before we did. We thank Dr. Luzar for providing us with the references. It turns out that Bell and co-workers did work on one-dimensional lattice gas and fluid models. He did find a density maximum, but because his potential wells were unphysical, the pressure dependence of the density maximum in his models often went the wrong way: the temperature of maximum density increased rather than decreased with increasing pressure. In Dr. Luzar's ref. 3, Bell solves, both exactly and approx- imately, two- and three-dimensional decorated lattice gas models, equivalent to deco- rated Ising models, with two kinds of nearest-neighbour interactions.He finds a maximum in the density and a minimum in the isothermal compressibility. In these models, the temperature of maximum density does decrease with increasing pressure, but the decrease is much faster than in real water. In the next reference, Bell solves exactly a one-dimensional lattice gas model, with two types of nearest-neighbour bonding. General Discussion However, in this model, the temperature at which the density maximum occurs increases instead of decreases with pressure. In Dr. Luzar's ref. 5, Bell solves approximately a three-dimensional lattice gas model with two types of nearest-neighbour bonding. He finds a density maximum, but also a density minimum at a temperature lower than that of his density maximum.In ref. 6,Bell approximately solves two-dimensional lattice gas models (he calls them lattice fluids even though nothing is flowing) with two types of nearest-neighbour bonding, but the pressure dependence of his density maximum is opposite to that of real water. Dr. Luzar's ref. 7 is the first of Bell's papers in which the one-dimensional lattice gas and Takahashi fluid models, with two types of nearest-neighbour bonds, are solved exactly. In the fluid model, there is a displaced single well, shifted from the hard core. The pressure dependence of the density maximum is wrong because of the unphysical nature of this potential. In the last of Bell's references cited by Dr.Luzar, the one- dimensional lattice gas and Takahashi fluid models are compared. The pressure depen- dence is not studied. Bell employs what he calls a double-well potential, but, in reality, it is a single well with two different depths, a well with a 'step', unlike our genuine double- well model. The extended van der Waals model recently proposed by Poole et al. (Dr. Luzar's ref. 9) is very promising. By making the probability of strong H-bond formation depend on the volume of the system, they have tried to explain the anomalous properties of water. Contrary to their opinion, however, we believe that their model is no more than a mixture model. The free energy is written as a sum of two terms, the van der Waals term and a new H-bond term.The new term is made up of two parts, weighted by the fraction fof the strong directional H-bonds. Compared with our ideas, which are rather similar'f would correspond to the fraction of open tetrahedral (strong H-bond) structure. The fractionfdepends in a Gaussian way on the total volume of water, being a maximum at a characteristic volume. Ultimately obtained is a three-parameter fit to water properties, which is not too good. The fit does produce the anomalies crudely. The most important aspect of this work is to show how the two different types of behaviour of the spinodal line and the temperature of the density maximum line can occur. None of the MD calculations mentioned by Dr. Luzar reproduce the density maximum consistently. It never occurs at the correct temperature, and, as recently seen by Sciortino and Sastryl for TIP4P water, it seems to be dependent on the system size.In the Poole et al. MD, carried out as in the old Stillinger-Rahman work2 using ST2 water, a density maximum is found at 39 "C and 82 MPa, instead of 27 "Cand near zero pressure as obtained in the old work,2 and at 4 "C and 0.1 MPa as in real water. The most important aspect of this paper was to show that Angel1 and co-workers was wrong in his guess about how the line of density maximum behaves at negative pressures. Baez and Clancy find a density maximum in the SPCE model at -38 "Cand 1 bar pressure. In Dr. Luzar's ref. 12, MD is carried out on ST2, TIP4P and SPCE, with the 'best developed'density maximum occurring for ST2.Thus, in all the MD computations to date, the density maximum is a very fragile thing: moving around, or being of the wrong shape or at the wrong temperature, or not appearing at all. Combined with the run-time problem mentioned by Dr. Luzar, in addition to the 'quantum effects' mentioned in the Stillinger and Rahman paper, and the system size effect found by Scioritino and Sastry, what all this means to me is that some very subtle, accidental and unknown feature can sometimes give rise to a density maximum in a water model. Certainly, no simple physical origin of the density maximum can be gained from any of this MD work. On the other hand, in our approach, a built-in propensity to create the correct outer structure in the liquid will totally overwhelm any fragile density-maximum-creating effects in the previous MD models and lead to a correct and stable density maximum in water for easily understood reasons.General Discussion 1 F. Sciortino and S. Sastry, J. Chem. Phys., 1994, 100, 3881. 2 F. H. Stillinger and A. Rahman, J. Chem. Phys., 1974,60,1545. Dr. Soper stated: I have several serious difficulties with this paper. First there is a vast literature on water and its interactions in chemical and biological systems. It is quite incorrect to claim, as the title of this paper suggests, that ‘hardly anyone wants to see (this problem) solved’. The authors are well aware of that literature, yet they are content to ignore it and assume that they should start from scratch and try to derive a new potential.The progress demonstrated in this paper is extremely limited, and the results produced in the final sections are, where they exist at all, of a preliminary, specu- lative nature. Secondly the attitude in this paper is that getting the density maximum correct in water is the main fundamental issue to be addressed. What is the justification for this assumption? Many of the empirical potentials for water produce a density maximum, albeit at the wrong temperature. Surely the highly angle dependent nature of the water potential is the one characteristic which distinguishes water so vividly from most other liquids? The preliminary results from this paper ignore the angle dependence of the potential [see eqn. (13)].Therefore of what value are they? Probably any number of parametrised potentials could be used to produce the density maximum at the correct temperature, but they would have nothing to do with water. Also why is it that D,O has a density maximum at a much higher temperature than H,O, although both have (classically) the same interaction potential? Finally, there is much discussion in this paper of a double-well potential for water, apparently on the grounds that the measured isochoric temperature derivative shows a peak at a different position to the usual near-neighbour distance for water (ca. 3.5 8, as opposed to ca. 2.8 A). However, owing to the complications arising from many-body correlations, the theory of liquids gives no justification for the assignment of a well in the interaction potential to a peak in g(r), particularly for a liquid as complicated as water.In fact, as shown by the work of Svischev and Kusalik,’ simple potentials such as SPC/E do produce a peak in the spatial distribution function at ca. 3.5 A in some directions (see also Fig. 3(a) of my paper). Hence a double-well potential is not necessar- ily needed to achieve this effect. Obviously I am not opposed to the quest for a better water potential, nor do I believe that present potentials are adequate, particularly as we move away from non- ambient conditions. Unfortunately, however, this paper appears to be stepping back- wards a few decades: the authors have apparently ignored much of the theory of liquids in attempting to set up their potential, and I would be truly amazed if they make any significant progress along the route they have chosen.1 I. M. Svischev and P. G. Kusalik, J. Chern. Phys., 1993,99,3049. Prof. Robinson replied: I am sorry that the title of my paper upset you and some others. Actually, it was meant to do just this, since I strongly believe that the current barriers to the ultimate resolution of the water problem are caused by too much empha- sis on first principle physics and on theories of (ordinary) fluids. To me, at least, follow- ing the lead of Mother Nature, rather than Father Physics, and looking at the intermolecular bonding properties in the more stable polymorphs (ice I,, ice 11, ice 111) seems more to the point.This is the approach we are taking. As far as the double-well potential and the lack of angular dependence in our paper, this was merely a very preliminary exercise to see if a realistic pressure-dependent density maximum could be produced in this way. Certainly, a correct angular depen- dence is required, as already emphasized at various points in our paper. The computa- tional model that we are currently developing, not mentioned in a detailed way in our General Discussion paper, is one whose angular dependence is designed to reduce the encroachment on the normal 3.4 A 0-0Lennard-Jones minimum by the 2.8 A hydrogen-bond interaction. The latter is more of a valence-type interaction, as pointed out many years ago by C.A. Coulson,l and thus should be very directional, having a much narrower angular depen- dence than given by present-day water-water interaction models. Such modifications, we believe, will allow ice 11-and ice 111-type structures to form in a natural way. No other interaction model has done this, yet I feel it is central to the ultimate understanding of the liquid water problem. Why is it important to achieve a really accurate representation of the density maximum in a computational model? I am a bit surprised that anyone working in water research should ask such a question. The density maximum signals structural effects peculiar to water that must be properly reproduced in any potential model that is meant to give realistic bulk and interfacial properties.If this is not exactly right, rather than merely being crudely or accidentally present, as described in my response to Dr. Luzar’s comments, nothing else can be really correct. Distortions caused by such an inadequacy might be particularly important near interfaces of biological interest, where varietal structural properties of the adjacent water molecules could give rise to a range of qualit-atively different effects. You asked about the density maxima in H20 and D20.A ‘thermal lag effect’ was mentioned on the second page of our paper, and a discussion concerning the isotope effect on the density has recently been published in our ref. 20. This effect has also been applied to the viscosity, and a paper on this is in preparation.The properties of D20 lag 6-7°C behind those of H20 because the liquid structure, the proportion of open to dense bonding in our picture, has such a lag. The lag is not classical, but very likely arises from zero-point effects on vibrational levels below an activation barrier, as is well known in chemical reaction theory. A discussion of most of your other comments can be found in our forthcoming book,2 including the Svischev and Kusalik result, using, by the way, an overly simple water model that would not be expected to produce accurately the actual subtle structural effect they report. I don’t see how you can disagree with the majority of the facts presented in our second section. Our philosophy for improving water models simply rests on these facts.We are considering not just a second-neighbour minimum near 3.4 A,but rather we are augmenting current models so that both this distance and the 4.5 A distance are produc- ed under appropriate thermodynamic conditions. The word ‘include’ in the last para- graph of this section is important here. We don’t say ‘at the exclusion of’. Our section on the Takahashi model should have included, as pointed out by Dr. Luzar, a description of the extensive work of Bell and co-workers on this topic. This description now forms part of my response to her. Briefly, our two-well model contains the essence of both temperature and pressure effects in water, while Bell’s one-well model (with steps) does not capture the correct pressure effects. In any case, such models are meant only as an illustration that the correct T and P dependence of the density can be obtained when the outer structure in the liquid is allowed to assume two possible configurations.This was fairly obvious on intuitive grounds, but we wanted to see it more definitively with a simple calculation. The Takahashi model that we present in our paper is certainly not, as you seem to have concluded, the central issue of our model, but only illustrates that we can confidently proceed to look for ways of incorporating this sort of feature into three-dimensional MD computations of water. Finally, coming back to the double-well potential, about which you seem to have strong objections, this is actually the true picture for water-water interactions.There is a hydrogen-bond interaction along the O-.-Hintermolecular direction with a relatively deep minimum near 2.8 A, and there is a van der Waals interaction, which is not so highly directional, with a shallower minimum near 3.4 A. We are simply trying to incorporate this truth into interaction models. We are therefore not starting ‘from General Discussion scratch’ to derive a new potential, but rather, as clearly stated near the end of our paper, we want first to optimize such modified interaction models, then ‘to start all over again applying such models to various interfacial states of water ’. 1 R. McWeeny, Coulsons Valence, Oxford University Press, Oxford, 3rd edn. 1979. 2 G. W.Robinson, S-B.Zhu, S.Singh and M. W. Evans, Water in Biology, Chemistry and Physics: Experi- mental Overviews and Computational Methodologies, World Scientific, Singapore, 1996. Dr. Soumpasis said: The problem is not constructing a water model which explains a few anomalies, but rather constructing a model which explains all the anomalies and yields thermodynamic, transport and structural data of sufficient accuracy. Several avail- able models of both the empirical and the ab initia type yield good results for many of these properties when used in high quality, careful simulations, although none is capable of describing all subtleties of liquid water and ice polymorphism. We have to see how well your model will survive the Procrustean bed of extensive testing.Prof. Robinson said in reply: Yes, it is totally agreed that all the properties of water must be reproduced by any model claiming to provide an accurate representation of this substance. The subtle details of the variations in the structure and dynamics of water near any surface or biological macromolecule cannot help but bear on the chemistry and the biological function of that system. Yet, as shown recently by Philpott and co- workers and described briefly in our book,’ the ‘model dependence’ (ST2, SPCE, TIP4P) of water structure near a surface can be huge. This means that from present models we cannot be confident about what is really going on. Our very precise empirical fits of experimental data as cited in our ref. 15, 17, 20 and 21, and the structural data themselves cited in ref.23 and 29 strongly suggest that it is the outer (rather than the nearest-neighbour) structure that is the important determining factor for all the proper- ties of water. Our goal is to try to incorporate these outer structural effects into water interaction models as described in my paper and my response to other comments. 1 G. W. Robinson, S-B.Zhu, S. Singh and M. W. Evans, Water in Biology, Chemistry and Physics: Experi- mental Overviews and Computational Methodologies, World Scientific, Singapore, 1996. Dr. Adya communicated: I completely disagree with the remarks made by Dr. Soumpasis that any model, for that matter even simple hard spheres (or dumbbells), can reproduce most of the results from neutron diffraction studies.It may be fortuitous in a specific case to get a good agreement even though the model may be completely unphysical but, in general, it is not true. It would be ideal if simulations could reproduce both microscopic (from structure determination such as neutron diffraction) and macro- scopic (from thermodynamic measurements, etc.) properties of the system. It is for this reason that some ambiguities exist in data interpretation if simulation data are com- pared with only one set of g(r) obtained from neutron diffraction. This stresses the importance of working by H/D substitution techniques to extract several g(r)s. Also, neutron scattering measurements should be made at several thermodynamic state points (T, P, etc.).Then, if a model can reproduce all the g(r)s, it will help in discarding spu- rious or unphysical models. In fact, the neutron diffraction with H/D substitution is the most powerful technique for looking at the atomic scale structure by separating the solute-solute, solute-solvent and solvent-solvent partial pair correlation functions. Prof. Robinson communicated in response: The remarks of Dr. Adya are relevant to some of the comments about my own paper. Because of the vast improvements during the past 15 years in the accuracy and the extent of experimental data (supercooled temperatures) on bulk and perturbed water, the standards for computational modelling General Discussion of water are correspondingly being raised. The new experimental results are creating a real challenge for MD, but, at the same time, are making it easier, as Dr.Adya says, to discard incorrect models. Dr. Sournpasis also communicated in reply: Dr. Adya did not correctly understand what I was trying to say. There is no doubt that g(r)s obtained from good scattering experiments or computed from good simulations are a valuable source of information. We use them and compute them all the time. However, it is also clear that being only spherical averages of the pair distributions of scattering (or interacting) sites they do not provide enough information to decide unambiguously which structural model is better. For instance, for water, the experimental g(r)s can be fitted not only by means of empiri- cal three- and four-site monomer water models but also via a model where the scattering units are five-water clusters (four 0s are the vertices of a tetrahedron, with one at the centre) and monomer~.’-~ In addition, important questions like H-bonding in water cannot be answered using g(r)s alone but can be addressed using the triplet correlations we have computed for several water models.The fact that many features of g(r)s can be fitted by considering packing effects alone (e.g., hard-sphere mixtures, dumbbells, etc.) is known to many com- putational physicists. In summary, g(r)s are necessary but not sufficient. 1 N. Ohtomo, K. Tokiwano and K. Arakawa, Bull. Chem. SOC.Jpn., 1981,54,1802. 2 N. Ohtomo, K. Tokiwano and K. Arakawa, Bull. Chem. SOC.Jpn., 1982,!%, 2788.3 N. Karmakar and K. Joarder, Phys. Reo. E, 1993,47,4215. Prof. Finney communicated : I have several comments relating to Prof. Robinson’s paper : (1) Temperature derivative pair correlation function data must be interpreted with care. The plots of Fig. 1 of our paper show increases in the population of distances between ca. 2.9 A and 3.8 K at the expense of both shorter and longer distances. A simple interpretation might be in terms of some temperature-induced lengthening of the first-neighbour hydrogen-bonded molecules (the loss of shorter distances) together with an increase in bond bending (the reduction of some shorter distances). We should not talk about a 3.4 A peak, as is done in the caption to Fig. 1, because there isn’t one. There was an old controversy in the late 1960s, based upon the early X-ray work of Narten and the Oak Ridge team,’ which originally showed a peak at this distance.Narten subsequently stated categorically that this feature was an artifact of the trunca- tion of the Fourier transform. Subsequent neutron work has failed to see a similar feature. (2)Increased bond bending as temperature increases has been accepted for decades as leading to structure densification as second neighbours can, through the bond bending, approach slightly closer on the average. The ‘correct’ degree of bond bending should come out as a natural consequence of assigning the ‘correct’ degree of softness to the parameters in the assumed water-water potential that relate to bond bending.This is the approach taken by nearly all potential function designers, who prefer, as far as possible, to build reasonable physics into their models, rather than add particular pre- scriptions to ‘patch up’ specific defects in a particular model. This is the philosophy we tried to follow in the early 1970s in trying to build up a polarisable model based upon the (as far as possible) experimentally determined multi- pole moments and polarisabilities.2 (See also similar work at the same time of Berendsen and van der Velde?) It is also the philosophy which more recently led us to argue in favour of using non-spherical repulsions on the basis of experimental results from crys- tallographic observation^,^ which seemed to be in agreement with high quality quantum mechanical calculations on the water molecule electron distribution. From these physi- cally reasonable properties, we should be able to build a model that contains the essen- General Discussion tial physics, and reproduce the experimental structure.In fact, in terms of the orientational structure (that which is quantified by the next-nearest neighbour configu- rations central to Robinson’s paper) the non-spherical repulsion does pretty well, as do those water models that use non-spherical repulsions (q.,MCY). The approach taken in this paper is rather different, in that it picks on the temperature-induced shift in next-nearest neighbour distance distributions, and uses this to put a particular driving force into a potential function to produce a particular result.Rather than build a model from the basic physics (bond bending as temperature is raised), an artificial device (a second well at a distance of 3.4 A) is introduced to try to reproduce one aspect of the structural results. While such a model might be para- metrised to give reasonable agreement with experiment, we will have learned no new physics from introducing the device, and we will not have confidence that this way of introducing bond bending (for that is what the device is) is sufficiently physically realistic to deal with other situations away from the conditions used to parametrise the 3.4 A second well. In the simplest terms, we could consider the second-neighbour configu- rations as resulting from an interplay between the softness of the bond bending and the shape and depth of the ‘non-hydrogen-bonded’ interaction, which itself is, as some of the work mentioned above4 suggests, non-isotropic.(3) The use of two potential wells, one relating to the 4.5 A peak in goo(r), the second to a 3.4 A distance, is reminiscent of the classical two-component ‘chemical’ mixture models used until the early 1970s. The 4.5 8, well relates to a ‘bulky’ phase, the 3.4 8, well to a ‘dense’ phase. Mix the two together, and you have all you need to reproduce the maximum density behaviour of water, as was done many years ago by Henry Frank (a dense and a bulky phase), and George Nemethy and Harold Scheraga (0, 1, 2, 3 and 4-bonded molecules).Such mixture models, after appropriate parametrisation, can, in fact must, reproduce the essential volume-temperature behaviour of water. The trouble is, they don’t give us much help in understanding the all-important structure-property relationships that we need to understand if we are to make sensible progress towards understanding the more complex situations that occur under pressure, in supercritical fluids or at complex macromolecular interfaces. 1 A. H. Narten, M. D. Danford and H. A. Levy, 1967. 2 P. Barnes, J. L. Finney, J. D. Nicholas and J. E. Quinn, Nature (London), 1979,282,459. 3 H. J. C. Berendsen and G. A. van der Velde, Proceedings of CECAM Workshop on Molecular Dynamics Calculations on Water, 1974, p. 63. 4 H. F. J.Savage and J. L. Finney, Nature (London), 1986,322, 717. Prof. Robinson communicated in reply : I have already considered some of your com- ments in my responses to Dr. Soper and Dr. Luzar. We all agree, I think, that water is a non-standard (anomalous) liquid. It has more structural forms in its crystalline and amorphous phases than any other molecule of anywhere near a comparable size. These two truths, I strongly believe, are closely related, and non-standard interpretational methods must therefore be used for the liquid water problem. (i) In the case of diffraction studies, the standard methods used for the interpretation of ordinary liquids have often given ambiguous or controversial results for water. The well known controversy over the 3.4 8, peak in the radial distribution function is but one example.You mention the flawed interpretation in the Narten et al. work, but you do not mention the earlier X-ray study of Morgan and Warren,’ which also indicated a peak in this region. Morgan and Warren studied the liquid at 1.5, 13, 30, 62 and 83 “C. In their Fig. 6, a non-standard method for interpreting their data was used. An RDF for an ice-like structure having four neighbours at 2.85 A, 13 (instead of 12) at 4.5 A and nine at 5.30 A was constructed. Subtracting this ice-like RDF from the liquid RDF showed a clear peak at ca. 3.6 A. Morgan and Warren, being guided by the now invali- General Discussion dated ideas of Bernal and Fowler,2 were clearly not happy with this peak, but they couldn’t get it to go away.See their Fig. 7. A point to be gained from this paper is that the nearest-neighbour count out to the first RDF minimum in the liquid is always greater than the expected <4 because of intrusion by 3.4-3.6 A structure. A related point is that the first minimum in the liquid RDF becomes much less prominent com- pared with that in ice I,, because of this intrusion. In fact, one need only compare the depth of the first minimum in the liquid RDF with that in ice to see that the liquid RDF could very well be ‘hiding something’. Barclay Kamb in ref. 30 of our paper showed, non-standardly, that the liquid RDF of Morgan and Warren could be interpreted as a composite of two types of bonding in the liquid, open ice Ih tetrahedral bonding with a second-neighbour distance of 4.5 A and the type of bonding that occurs in the moder- ately dense polymorphs such as ice I1 and 111, with a non-nearest-neighbour distance of 3.4 A.However, because of the understandably crude early liquid data and the fixation (based on what?) developed over recent years against mixture models, Kamb’s interpre-tation never held up. What happens if you do as Morgan and Warren, and Kamb, and try to resolve non-standardly the much improved modern liquid RDFs into an ice I component plus an additional component? This could be definitive one way or the other. More than a decade ago, Bosio, Chen and Teixiera, as cited in our paper, introduced another non-standard approach for liquid water :isochoric temperature differential (ITD) studies.While these authors were careful to avoid the word ‘peak’, very likely because of the past controversies about this topic, and speak rather of a radual shift with decreasing temperature towards the tetrahedral structure, their 3.4 1differential feature (our Fig. 1) remains sharp without appreciable shift and increases in intensity with increasing temperature differential. It seems to me then that it is correct to classify this as a ‘peak’ just as much as the 2.8 A and 4.5 A features, which show up no more prominently in the ITD data, but are both interpreted as ‘peaks’ in standard diffraction interpretations. In these standard interpretations, the 3.4 A feature is not considered a ‘peak’ because it is being overwhelmed by the decreasing intensity of the normal tetra- hedral RDF at this distance.Let me quote four observations from the paper of Bosio et al. that I believe to be very important. (1) ‘A major structural rearrangement at the outer-neighbour shell is identified from the interpretation of the ITD’; (2) ‘the first coordination shell is less strongly affected by the temperature variation than farther neighbour shells’; (3)‘This is a manifestation that 0-0-0 angular correlations are at the origin of the density fluctuations and, consequently, also of the thermal anomalies of water’; and (4)‘we see that although their distances are slightly contracted, the number of first nearest neigh- bours is not significantly affected (by temperature)’. As far as pressure is concerned, in our paper, we cited a comment by Bellisent-Funel and Bosio about its effect on the nearest compared with outer neighbours in the liquid.In future computational studies, not only these pressure effects, but also ITD diffraction data, must be -modelled, a start in this direction having already been made by Corongiu and Clementi.3 Our mission then in creating a modified water-water potential is merely to use these very important and revealing experimental conclusions to construct a model for liquid water that behaves properly with temperature and pressure changes, not only in the first-neighbour shell, as has been done many times before, but also in outer-neighbour shells, which has never been done.As described in my reply to Prof. Lynden-Bell, I believe it to be a vain hope to rely on first principles of physics, such as quantum theory, the theory of fluids or cluster experiments, to achieve a correct non-nearest-neighbour picture, and I believe it to be an even more distant hope to expect the correct features to fall out ‘naturally’ from any current water-water potential model. In any case, this has not yet happened, unless one is merely satisfied, as no one will be for very long (see my remarks to Dr. Adya), with an argon-like liquid that has a few water-like properties. General Discussion (ii) With respect to ‘reasonable physics’, ‘essential physics’ or ‘basic physics’, I have dealt with this point already in my answers to other comments.I simply regard the known characteristics of hydrogen bonding and hydrogen-bond bending in the ice poly- morph structures as more reasonable physics than obtaining this information from nec- essarily oversimplified quantum mechanics based on few-body concepts. (iii) It is certainly true that our model is a ‘mixture model’. For various reasons, we have called it a ‘modern mixture model’, since the basic idea, as a ‘physical’ but not as a ‘chemical’ mixture, goes back over a century. An attempt was made by Kauzmann4 to discredit such models by suggesting that they cannot reproduce the isothermal com- pressibility. We have recently shown this not to be true. Just because it is old and simple, why look away from a concept that is consistent with the structures of the more stable ice polymorphs and with all the experimental facts on the liquid to within the 4-6 decimal point accuracy of their measurement? This transparently simple idea has also suggested a correct interpretation of H-D-T isotope effects in liquid water, where none has previously existed.Compare ref. 20 and 22 in our paper. Incorporation of this concept into a liquid water computational model, as we are now attempting to do, is no more of an ‘artificial device’ than any other of the previous attempts to improve water interaction models. In fact, past work has strained to make certain that the ice I, opentetrahedral structure is well represented in the liquid, but you now object to this same procedure being used to include ice I1 and ice I11 bonding. It should be realized by everyone that it is the outer structure that distinguishes normal ice I,, from ice I1 and ice 111, all of these polymorphs having essentially the same nearest-neighbour tetrahedral configurations.This sets the stage exactly for the structure in the liquid. As far as ‘the physics that we learn’ is concerned, the ‘modern mixture model’ has not only exhibited a predictive value in the isotope-effect problem, but has also provided a clear and easily understandable picture for accurately describing the anomalies of water. Do we really need to replace this with chaos? 1 J. Morgan and B. E. Warren, J. Chem. Phys., 1938,6,666. 2 J. D.Bernal and R. H. Fowler, J. Chem. Phys., 1933,1,515. 3 G.Corongiu and E.Clementi, J. Chem. Phys., 1992,97,2030. 4 W. Kauzmann, Colloq. Znt. C.N.R.S., L’Eau Syst. Biol., 1975,246,63. Prof. Zundel communicated: (a) It has already been shown 30 years ago by the near IR studies of Prof. W. Luck that in liquid water free OH groups are present (at 10°C cu. 7%). All these results have been published in Ber. Bunsen-Ges. Phys. Chen~.’-~It could confirm this result in the middle IR. In solutions of H20 in D20 a weak shoulder is observed at the high wavenumber slope of the v(0H) band. With the ATR technique no free OH groups are observed since the water is structured at the silicon surface. (b) It was also shown by Prof. Palma-Vittorelli in the near IR that ice-like regions are present in liquid water.4 (c) Recently it was shown by Prof.Lauberaux in Munich by hole burning experi- ments in the broad v(0H) stretching vibration of liquid water that various types of hydrogen bonds contribute to this band. All these results should be taken into account by a model! 1 W. Luck, Ber. Bunsen-Ges. Phys. Chem., 1962,66,766. 2 W. Luck, Ber. Bunsen-Ges. Phys. Chem., 1963,67, 186. 3 W. Luck, Ber. Bunsen-Ges. Phys. Chem., 1965,69,627. 4 G.Andaloro, P. Chirico, G. Guzzio, M. Leone and M. B. Palma-Vittorelli, J. Chem. Phys., 1977,66,335. Prof. Robinson concluded : The word ‘understanding’ is most important. Simply bun- dling various potential models into MD simulations and sometimes seeing a density maximum, as a few computations have done, does not give any molecular-level under- standing whatsoever of this property, at least that I am aware of.Our motive is to create General Discussion such an understanding, as we think we have done through emphasis on non-nearest- neighbour bent hydrogen-bonding properties in the liquid. Prof. Yarwood opened the discussion on Dr. Luzar’s paper: (1) Are the elementary ‘diffusion times’ given [l ps for H20 and 6-10 ps (for H20 near DMSO)] the same times as those measured using QENS, for example (usually interpreted as ‘time between jumps ’) ? (2) You use a particular criterion for ‘breaking of hydrogen bonds’ in the first coor- dination shell via eqn. (5). Does this mean that, for extremely short observation times (<0.1 ps), we ought to see ‘non-hydrogen-bonded’ water molecules in our spectra? We do not see such non-hydrogen-bonded molecules at the levels shown in Fig.2. Could it be that the language used is ambiguous? (3) Your ‘residence time’ seems to be defined differently from other groups who use 7,,, k: THB. This arises from the explicit inclusion of the diffusion process (which seems sensible), but on the other hand I don’t understand how ‘diffusion’ can be separated conceptually from HB ‘breaking’ and ‘making’ in liquid water. Dr. Luzar replied : The elementary diffusion time, T, is the time to diffuse through an elementary distance in the model. This elementary distance is the width of the domain within which two tagged molecules can form an H bond and is of the order of a molecu- lar radius.Its value, less than ca. 0.15 nm in pure water, is not expected to change for water in the mixture. The self-diffusion constant for water in 1DMSO : 2H20,however, is ca. 3 times below the corresponding value in pure water. Therefore, the elementary diffusion time, which scales as 1/D, is ca. 1.5 ps in the mixture, compared with ca. 0.5 ps in pure water. I define the characteristic residence time in my model as the average time a molecule remains within a narrow domain in which a bond with another molecule can exist. It is therefore equal to the time to break a bond, l/k, plus the time needed for a molecule to then escape from the first coordination shell, 7. According to the values in Table 2 of my paper, it is 1.8[1 f0.41 ps in pure water and 6[1 f0.61 ps for a water-water pair in the mixture, and 11 [1 f0.61 ps for a water-DMSO pair in the mixture.This time might be compared but not identified with the ‘time between jumps’ obtained from QENS (further discussion is given below). Such measurements have been carried out in pure water, but not yet in the mixture. As I already pointed out in my paper, it is not always legitimate to equate T,,, with 7HB. In a liquid, where H bonds lead to the formation of dimers, like formic acid, the residence time and the H-bond lifetime are not very different. This is also apparently the situation in 2 : 1 water-DMSO mixture, where DMSO on average forms only two H bonds with water molecules.’ However, in situations, where each molecule is involved in several hydrogen bonds, e.g., liquid water, the residence time is larger than the H-bond lifetime.What I consider in the model is a coupled reaction, e.g., a hydrogen-bond breaking/ makingdiffusion problem. Therefore, diffusion is not conceptually separated from hydrogen-bond dynamics in liquid water. I do, however, conceptually differentiate between rotational diffusion, involving H-bond breaking/making and translational diffu- sion which carries molecules apart after the H bond has been broken. Before the bond breaks, the motion of the pair of molecules under consideration has predominantly rota- tional character. When the bond is broken, the translational component becomes rela- tively more significant. Experimentalists do not measure these times directly.Instead, they have to relate the measured values (intensities, frequencies, linewidths, etc.) to the parameters of a simpli- fied model they use to interpret the data. For example, for the interpretation of quasi-elastic neutron scattering (QENS) data, experimentalists are forced to assume a General Discussion decoupling of rotational and translational motions of the molecules in order to obtain a tractable analytical model for diffusion of the water molecule. Translational and rota- tional parts of individual water molecules are modelled, respectively, by a jump diffusion and rotational diffusion model. The residence time, associated with translational diffu- sion, and rotational correlation time, related to hydrogen-bond lifetime, are used as fitting parameters of this simplified modeL2 Neither the QENS nor the MD data can be convincingly described by simple diffu- sion model^.^ Specifically, the diffusive motion of water molecules is not well described by the random jump diffusion, used to extract residence times from experiments.The MD simulations tend to enforce a description of a fairly continuous diffusive dynamics in water. The remarks I have made in reference to Dr. Bellisent-Funel’s paper provide pertinent references. The phenomonology developed in this work identifies elementary processes govern- ing H-bond dynamics in water. Comparison with the computed reactive flux correlation functions indentifies the rate constants for these processes.The model also provides a degree of predictive power, relating H-bond dynamics to translational and rotational diffusion. The next step is to relate this phenomenological model with correlation func- tions that experimentalists measure, for example the intermediate scattering function, obtained from neutron scattering. This is the subject of future investigation. Until then, any direct comparison between times estimated in this work, and those, determined indirectly from QENS data, are not very conclusive. Regarding the ‘non-hydrogen-bonded’ water molecules in the first coordination shell, which cannot be seen in the spectra: the non-zero value of n(t) in our calculation does not point to the existence of water molecules without hydrogen bonds.It implies the absence of a hydrogen bond between a tagged pair of adjacent water molecules, regardless of the status of other bonds the two molecules engage in. 1 I. I. Vaisman and M. L. Berkowitz, J. Am. Chem. SOC.,1992, 114, 7889; A. Luzar and D. Chandler, J. Chem. Phys., 1993,98,8160. 2 S. H. Chen, in Hydrogen-Bonded Liquids, ed. J. C. Dore and J. Teixeira, NATO AS1 Series, C329, Kluwer, Dordrecht, 1991, p. 289. 3 D. D. Cola, A. Deriu, M. Sampoli and A. Torcini, J. Chem. Phys., 1996,104,4223. Prof. Finney said: You comment in your paper that ‘a significant amount of bonds . . . break, but remain nearest neighbours’. It strikes me that this could merely be a consequence of taking what must essentially be a semi-arbitrary criterion for ‘bond breaking’ for a potential function that is essentially smooth and continuous.Although your criterion says that a particular bond is broken, the energy of the interaction may be only slightly less attractive than that between these two molecules in a slightly different configuration which your criterion says satisfied the bonding criterion. Unless there is a natural energy-justified criterion for bond breaking, as in the ST2 potential, I do not find models based on broken bonds particularly helpful physically. The device may allow you to set up models and perform calculations, but if these models require such an artificial device, do they give us any helpful physical insight? These kinds of model bring back memories of the old broken hydrogen-bond models that computer simulation techniques have led to be shelved in the past 20 years.Dr. Soumpasis then stated: I do not think that the interaction commonly called a hydrogen bond is very clearly defined. Some people use geometrical criteria (distances and angles) to define it while others (e.g., Geiger and Mausbach) prefer interaction energy levels. Similar problems arise in defining clusters, networks, etc., and hamper the evaluation of many computer experiments as well as structural studies and spectroscopic data. 1 A. Geiger and P. Mausbach, in Hydrogen Bonded Liquids, ed. J. Dore and J. Teixeira, NATO AS1 Series, C329, Kluwer, Dordrecht, 1991, pp. 171-183. General Discussion Dr. Luzar responded : It seems appropriate to answer Professors Finney and Soum- pasis’ questions together, since they have a number of points in common.The existence of an H bond between two molecules is not fundamentally a ‘yes or no’ proposition. The H-bond phenomenon is connected with continuous spatial varia- tion in interaction energies, and does not discontinuously ‘click on’ at a unique distance. However, this observation should in no way be interpreted as minimizing the impor- tance of the H-bond concept for chemistry, since this concept conveys specific quantitat- ive information about potential surfaces.’ In connection with computer simulations of water, it is useful to introduce a conven- tion for H bonds, which can be applied to an arbitrary given configuration of N mol-ecules, in order to establish what pattern of H bonds exists.Although such a convention must necessarily involve some element of arbitrariness, it can serve to legitimize an important class of questions about the topological patterns of H bonds existing in liquid water and aqueous solutions, as well as about their kinetic properties. Having selected a particular criterion, it is possible to classify molecules according to the number of H bonds in which they simultaneously participate. Computer simulations, including the present one, reveal that this distribution has a single maximum, at CQ. 3.5 bonds per molecule,2 showing that 10-15% of protons on average do not participate in a bond. This value is consistent with Luck’s measurement^.^ My paper is not concerned with time-average properties, e.g., the equilibrium frac- tion of broken bonds, but deals instead with the dynamics of an H bond between a particular tagged pair of molecules.If one can accept the concept of an H-bond lifetime, than one has also to accept the concept of a broken bond, as we all know that H-bond lifetime is not infinite, but of the order of 1 ps, on average. Eventually, every bond breaks. The concept of a broken bond can be viewed in terms of a standard picture of water as a distorted random tetrahedral network of H bonds. Each water molecule optimally participates in four H bonds. At standard conditions, however, about half of the water molecules use only three of the four bonding sites. Thus, in the vicinity of an H bond, one or more neighbouring water molecules can receive an additional bond.The average water environment provides ample opportunity for a tagged H bond to disap- pear as water molecules switch allegiance^.^ This process thus involves the breaking and making of H bonds in liquid water. It has been argued’ that conventional energetic and geometric definitions of H bonds used in a simulation are not suitable for studying the dynamical properties of H bonding. Indeed, the fast and ample librational motions (hindered rotations) of the water molecules introduce a fast component in the time evolution of the interaction energy, or relative orientations, with a consequent artificially high breaking and reform- ing rate. Therefore a broad range of H-bond lifetimes have been reported in computer simulations, depending on how these fast librational motions have been accounted for.Sciortino and Fornili were the first to introduce a more appropriate combined energetic, geometric and temporal definition of an H bond to study H-bond dynamics in a com- puter simulation.6 We adopt an alternative approach. The first attempt to determine H-bond lifetimes in hydrogen-bonded liquids by using a reactive flux correlation function approach has been given by us recently.’ The reactive flux method provides information about the short-time dynamics, i.e., transient relaxation, as well as the long-time relaxation. It determines the transition-state theory estimate of the H-bond breaking rate constant, as well as the lifetime of the bond.An assortment of motions leading to H-bond breaking is evident from the reactive flux correlation function, k(t) (Fig. 1 of my paper). At short times, it quickly changes from its initial value, indicating many recrossings in and out of the bonding region on a timescale of less than 0.1 ps. The dynamics on this timescale is primarily due to librations. In analysing the trajectories, two water molecules are considered to be H bonded only if their interoxygen distance is less than 3.6 A, and simultaneously the angle General Discussion between the 0-0 axis and one of the 0-H bonds is less than 30". There exists an assortment of reasonable alternative choices corresponding to different surfaces in con- figuration space dividing bonded from non-bonded regions.The long-time relaxation, referred to in Fig. 1 of my paper, is, however, invariant with respect to the specific definition of an H bond. To understand this fact, consider two different but reasonable choices of H-bond definition. The surfaces in configuration space dividing bonded and non-bonded states are different for the two different definitions. But if both definitions are physically reasonable, the dividing surfaces will lie close to each other. Trajectories pass quickly between nearby surfaces. Alternation of the dividing surface will thus affect the short-time transient decay and the amount of recrossings, but not the longer time reactive flux. Thus, the post-transient relaxation illustrated in Fig. 1 is not an artifact of the specific H-bond definition employed in the calculation. It is therefore a property of liquid water.The assignment of ca. 0.2-0.3 ps as the end of the transient period would coincide with the arbitrary temporal definition of an H bond, used by Sciortino and Fornili6 The population n(t) represents the conditional probability that a tagged (labelled) bond between a pair of water molecules is broken at time t and the oxygen of the second molecule stays within the first coordination shell of the first molecule, given that the bond was intact (formed) at time zero. For the reasons already discussed, this function depends on the H-bond definition only within the transient time <0.3ps. Therefore, the time dependence of this population shows (Fig.2 of my paper) that there is quite a notable number of bonds that break, but the molecules that previously had a common bond remain nearest neighbours over some significant period of time. This dynamical behaviour is reflected in the equilibrium properties. The possibility of finding adjacent molecules which do not share a mutual H bond has been confirmed by experiment. A recent analysis of bond-angle distributions from neutron diffraction data' points to a significant number of non-H-bonded molecules in the first coordination shell of a speci- fied water molecule (of course, these molecules have H bonds with other neighbours). The result is consistent with the oxygen coordination number in pure water being greater than four.g This observation is in agreement with simulation results; all empiri- cal pair potentials developed for water give a number greater than four, when integrated up to the first minimum in goo(r).The analysis of the angular distribution of water molecules around a central molecule deduced by computer simulation'' points to the same conclusion. 1 F. H. Stillinger, Ado. Chem. Phys., 1975,31, 1. 2 A. Luzar and D. Chandler, J. Chem. Phys., 1993,98,8160. 3 W.A. P. Luck, Ber. Bunsen-Ges. J. Phys. Chem., 1963,67, 186. 4 F. H. Stillinger, Science, 1980, 209,451. 5 F. Sciortino, in Correlations and Connectioity, ed. H. E. Stanley and N. Ostrowsky, Kluwer Academic, Dordrecht, 1990. p. 214. 6 F. Sciortino and S. L. Fornili, J.Chem.Phys., 1989,90, 2786. 7 A. Luzar and D. Chandler, J. Chem. Phys., 1993, 98, 8160; A. Luzar and D. Chandler, in HydrogenBond Networks, ed. M. C. Bellisent-Funel and J. C. Dore, Kluwer Academic, Dordrecht, 1994, p. 239. 8 A. K. Soper, Chem. Phys., 1996,202,295; A. K. Soper and A. Luzar, J. Phys. Chem., 1996,100, 1357. 9 A. K. Soper and M. G. Phillips, Chem. Phys., 1986,107,47. 10 I. M. Svischev and P. G. Kusalik, J.Chem. Phys., 1993,99, 3049. Prof, Markhal commented: This comment is related to the concept of 'broken hydrogen-bonds' in liquid water and aqueous systems which has just been evoked and has been said to be poorly defined. Experimentally Luck could unequivocally recognize 'free OH' groups in the NIR (near infrared) spectra of liquid water.' He estimated the concentration of such groups to be of the order of 10-15% at room temperature.In the conventional IR region no such groups could be detected,2 which indicates that assump- tions made to estimate their numbers from NIR spectra should be somewhat modified and that their concentration at 1 atm and below 100°C is less than 5%. General Discussion This low concentration of ‘broken hydrogen-bonds’ in liquid water implies that the environment of each H20 molecule in water is primarily tetrahedral even if distorted. The coordination number remains four. It then raises the question of the origin of the fluidity of water: how can it be a liquid with almost every H20 molecule remaining tetra-coordinated? The answer from IR spectroscopy (in the conventional region) is that rotations of an appreciable number of H20molecules around their symmetry axes acquire a diffusive character with a relaxation time shorter than 1 x s.It implies that hydrogen bonds in water below 100°Care for many of them bent or distorted but cannot be considered as broken. However, this short relaxation time remains to be correlated with experimental values of the viscosity of water. In my opinion this molecu- lar origin of the fluidity of water may be the main exceptional property of water that is still currently not understood: one can understand, at least qualitatively if not yet quan- titatively, how the exceptionally great number of hydrogen bonds in water (they are as numerous as covalent bonds) give this species exceptional physical and chemical proper- ties.One does not see clearly how it can be a liquid. 1 P.Luck, Ber. Bunsen-Ges. Phys. Chem., 1963,67, 186. 2 Y. Marhhal, in Hydrogen Bond Networks, ed. M. C. Bellissent-Funel and J. C, Dore, NATO AS1 Series C435, Kluwer, Dordrccht, 1994; J. Mol. Struct., 1994,322, 105. Dr. Luzar said in reply: For water to be fluid it is essential that bonds break and form all the time. The frequency of these events, rather than the equilibrium fraction of broken bonds, is responsible for the fluidity of water. In my paper I quantify this fre- quency, i.e. the frequency of switching allegiance. The origin and mechanism of this process are thoroughly described in the paper. Prof.Haymet asked: What evidence is there that the dynamics of any of the simple models of water (e.g., Lennard-Jones plus fractional point charges) are related to the dynamics of real water. Dr. Luzar responded :Despite their simplicity, the effective pair potentials (ST2, SPC, TIP4P, for example) reproduce the properties of liquid water surprisingly well. This holds for the thermodynamics and equilibrium properties as well as for the dynamics. A detailed comparison of various models with experimental data has been given by Wata- nabe and Klein.’ The dynamical experimental data include self-diffusion coefficient, rela- tive permittivity, dielectric relaxation times and NMR rotational correlation times. This comparison rules out any concern that the dynamics of real water cannot be qualit- atively reproduced in a computer, using any of the simple models of water.One has to bear in mind, however, that any purely classical description of the high- frequency vibrational motion of water will be limited in accuracy by the exclusion of quantum effects. These motions, which we are leaving out, happen on a different time- scale (of the order of 10 fs) from the one we are exploring. Prof. Haymet may worry about protons being quantum mechanical: however the interplay between H-bond dynamics and diffusion is still present, whether simulating quantum or classical water. In other words, simulating quantum rather than classical water would not affect the pertinent physics, namely that the source of the complex relaxation in liquid water lies in the virtually self-evident coupling that exists between translational diffusion and H-bond dynamics.Along with resolving the curious complex kinetics, the analysis is also of quantitative use as it succeeds in demonstrating the exis- tence of rate constants that characterize the making and breaking of H bonds. These rate constants are material properties. For example, the values of k and k depend on the temperature and concentration of the solutes in the system (like DMSO),and will also be sensitive to H/D isotope variation. Similarly, for simulations, the values of k and k’ can depend on the potential model used. We are aware of the limitations of intermolecular pair potentials like SPC to produce correct absolute values of the known General Discussion experimental dynamical properties. However, we aim to understand the connection between different dynamical processes and to compare the relative dynamics.Irrespec- tive of the system and the model used, however, eqn. (6) in my paper provides the connection between rigorous microscopic dynamics and H-bond rate constants. 1 K. Watanabe and M. L. Klein, Chem. Phys., 1989,131, 157. Prof. Robinson communicated: In your abstract you mention breaking and forming hydrogen bonds in different environments. I would like to know why you think that this is an important issue, since experiments have definitively shown that hydrogen bonds are not broken to any great extent in liquid water, perhaps only in supercritical water or in water undergoing vaporisation.You continue with this rather outdated theme in the first sentence of your Introduction. Though I do agree with the possible importance of the presence of coupling between translational diffusion and hydrogen-bond dynamics, my own feeling is that hydrogen-bond breaking is not at the heart of the liquid water problem. In your paper you say you use the rigid SPC model in your simulations. I personally do not believe this highly oversimplified computational model has any chance at all of catching the essence and subtleties of real liquid water or of DMSO solutions. So, in conclusion, for many reasons, I believe your results to be no more than some sort of theoretical game, of which we have already had too many in the problem of water.Would you comment on this? Dr. Luzar communicated in reply : Prof. Robinson’s belief that hydrogen-bond dynamics has little to do with the dynamics of hydrogen-bonded liquids is not some- thing I understand or can comment upon. The experimental data on the equilibrium fraction of broken bonds is obviously irrelevant to the dynamical issue, which I address in my paper, and not inconsistent with our model or findings. Models like SPC reproduce a huge amount of water data (including the temperature of maximum density) and it is not worth spelling all that out in this response since it is generally very well known. The remark I have made in reference to Prof. Haymet, as well as to his own paper, Prof.Robinson might find illuminating. Dr. Tromp opened the discussion of Dr. Soper’s paper: What does the concept of a ‘hydrogen bond’ mean when it does not result in a peak in gHH(r), such as is the case at 400°C and d = 0.6 in water? Dr. Soper replied : The hydrogen bond can be defined in various ways, namely if you are an experimentalist measuring g(r), then you will very likely define it in terms of a specific peak in g(r), whereas if you are a spectroscopist you might define it in terms of a particular shift in frequency or change in peak shape in the vibrational spectrum. Com- puter simulators have the luxury of being able to make a precise definition of the hydro- gen bond in terms of peaks in g(r), binding energies, and most importantly in terms of specific local molecular orientations.Not surprisingly, therefore, all these different defini- tions do not always agree as to the extent and nature of hydrogen bonding in a fluid, as has been argued by Chialvo and Cummins.’ According to the EPMC view of the high- temperature water data, the geometrical definition of the hydrogen bond leads to a residual degree of hydrogen bonding, albeit with considerable disruption of the H-bond network, even though the distinct peak in gOH(r)has disappeared. 1 A. A, Chialvo and P. T. Cummings, J. Ckem. Phys., 1994,101,4466. Dr. Hummer said: In your approach, interaction potentials are developed based on a fit to diffraction data. However, diffraction data are not necessarily free from significant General Discussion systematic errors, like oscillations of g(r) at short distances.To what extent can such systematic errors distort the fitted potentials and the overall structure obtained based on the fitted potentials? Dr. Soper responded: The spurious oscillations at small r arise primarily from trun- cation errors. I use the minimum noise method of extracting g(r) which helps to keep these errors to an absolute minimum. The systematic errors are, however, more serious. The one thing in our favour, however, is that because we use isotope substitution to extract the partial g(r)s, then it is likely that a systematic error that causes an overly positive excursion in one g(r) will cause a negative excursion in another, as has been demonstrated.' Indeed it was as a result of that analysis that we took another look at the process of extracting g(r) from diffraction data, and came up with a revised set of g(r)s, which are the ones used in this paper.The revised analysis will be published else- where.2 The EPMC technique provides a useful test for the existence of such errors since the extracted g(r)s cannot then be fitted by physical distributions of molecules. By paying careful attention to the details of the data analysis it is now possible to keep such systematic errors reasonably small, but there is admittedly always an uncertainty in this regard. If, after performing the EPMC simulation on a variety of data from different thermodynamic state points, one finds the EPMC potential has a similar form for all state points, or else shows a systematic trend with state conditions, then I would have greater confidence both that the data are sensible, and that the EPMC potential is useful.At the present time that full test of the procedure is not complete. 1 G. Lofller, H. Schreiber and 0.Steinhauser, Ber. Bunsen-Ges. Phys., 1994,98, 1575. 2 A. K. Soper, M. A. Ricci and F. Bruni, J. Chem.Phys., 1996, in press. Mr. Helms said: Could you, by deriving EPMC models that fit neutron diffraction data at different temperatures, give us an idea by how much empirical force field models should be varied for simulations at different T? Dr. Soper said in reply: I would hope so, but refer also to my reply above. The systematic errors in this experiment are not trivial, and might mask the subtle changes to the effective potential with thermodynamic state condition. Dr.Adya said: Although the empirical potential Monte Carlo (EPMC) simulation technique is analogous to the reverse Monte Carlo (RMC) simulation, I feel that the EPMC should be much more robust because it has some scientific base. It is based on the potential of mean force (PMF) between sites a and j3 in the fluid as compared with the RMC which assumes no potentials whatsoever. The EPMC, therefore, should not have the pitfalls of RMC and so there is an additional advantage in using the PMF to indicate how to modify an initial potential to perform computer simulations in order to reproduce the experimentally measured site-site pair correlation functions accurately.What was your motivation behind developing this technique and did you have a chance to compare the performance of EPMC uis u uis RMC? I know that in certain cases RMC fails miserably and it would be interesting to see how the EPMC performs in such cases. In this regard, it would be very convenient to have a user friendly version of the EPMC. Do you intend to produce such a version? Dr. Soper answered: In fact I did indeed start by doing reverse Monte Carlo simula- tion, but found, particularly for systems involving molecules, it had a great propensity to get stuck in a local minimum and not change appreciably however long I ran it. It was necessary to do artificial things like randomly disordering the system or raising the temperature by a large amount, and then cooling it again, in order to get the simulation to proceed on a vaguely sensible random walk.I understand I am not alone in this General Discussion regard. If a simulation is proceeding normally it should proceed on a random walk so that correlation functions become smoother and smoother the longer you run the simu- lation, in the same way that the statistics in the neutron experiment can be improved by longer running. I believe there are formal reasons why RMC fails in this regard, but I will not elaborate on these here. Even so it should be emphasized that the basic idea behind RMC (and EPMC for that matter), i.e. that one is trying to apply realistic con- straints on possible distributions of atoms or molecules, is fundamentally the correct way forward in interpreting diffraction data from disordered materials.As for providing a user friendly version, that will require effort, which only time and probably money can resolve, but certainly the aim is to do this at some stage. Prof. Bellissent-Funel asked: In your paper you mentioned some previous experi- ments about the structural studies of supercritical water (ref. 10 and 11). From the analysis of the data you claimed that the H bond disappears; your arguments were based on the observation of the absence of a peak in the pair correlation function of H and 0 at ca. 1.9 A. The presence of a peak was considered by researchers doing either experiments or computer simulations and by yourself as the signature of H bonding in water.You propose some sophisticated EPMC method which you use to reanalyse your old data and this leads to some different conclusions in the sense that the H bond is now present. Can you comment about the limit of the method and the influence of the inter- atomic potential energy on the final result. How do you explain the difference in your respective conclusions? How do you propose to estimate the level of hydrogen bonding in water? Dr. Soper responded: I have already largely answered this question in reply to com- ments by Dr. Tromp and Dr. Hummer. The reliability of the method depends primarily on the number of distinct site-site pair correlation functions which can be extracted from the measured data by isotope (or other labelling).The more of these that are available then the more reliable will be the final reconstruction of the liquid. Note that in the most recent version of the programme, both the pressure and internal energy are constrained, making the reconstruction more realistic. In reply to the question about the level of hydrogen bonding in the liquid, I would say that the EPMC (or RMC) technique allows us to go a stage beyond simply measur- ing the height of a peak in go&) Using this approach one can now use an angular definition for the hydrogen bond, or even impose an energy condition as well. Since no one in my experience can actually define what they mean by a hydrogen bond, other than by some more or less arbitrary definition to be used for analysing the results from a computer simulation, then I do not propose to do so either! Prof.Robinson communicated: How does the number of nearest 0..O neighbours* in the liquid vary with temperature? Starting at very low temperatures and going up to, say, 50°C, why isn't this number four or perhaps a bit less? Structural workers keep getting 4.5 or greater by integrating out to the first minimum. My belief is that this first 0.6 a0minimum is contaminated with a weak 3.4 A .O. -peak, replacing the 4.5 A of the open tetrahedral bonding, thus apparently giving a greater number of nearest neigh- bours. Is there any way to correct for this? Perhaps you could put in exactly four neighbours in your computer analysis, subtract this from the experimental g(r),and see if the 3.4 A part of the differential X-ray data of Bosio et a1.l is reproduced.1 L. Bosio,S-H.Chen and J. Teixeira, Phys. Rev. A, 1983,27, 1468. Dr. Soper communicated in response: Most computer simulations of the spatial dis- tribution function of water, including the EPMC simulation shown in this paper (see General Discussion Fig. 3a), do indeed show a weak peak at ca. 3.4 A in some directions, and as you correctly state, since this coincides with the first minimum of the 00 g(r), it is no doubt the reason why conventional estimates of the coordination number around a central water molecule are greater than four, instead of less than four. The most realistic way to correct for this is to inspect the spatial distribution functions for water, as obtained from a simulation of the data, and integrate only those peaks which correspond truly to H bonding. In fact, by integrating over the first peak in the go&) function one obtains a local coordination in the region of 3.2-3.8 water molecules in H-bonding positions at 25°C.Although I do not have any data over the temperature range you mention, I would expect this number to fall slightly with increasing temperature, but the real changes to the H-bond coordination number will not happen until we get to 300°Cand above. Prof, Zubdel opened the discussion of Mr. Lee's paper: I would like to know which conformation corresponds to the two minima by which the lowest dispersion region is explained. Concerning the dispersion region at 10-8 Hz:Mg" ions are strongly bound to polynucleotides in aqueous solutions.They turn the phosphate groups toward the base residues and induce in this way right-handed helices. In contrast to this result, Na' ions are only loosely bound and the phosphate groups are turned away from the base residues. Thus one has two different backbone conformations as a function of the cations. These results have been obtained by IR studies.'" These results should be helpful in the explanation of this dispersion region. 1 K. Ktilkenbeck and G. Zundcl, Biophys. Struct. Mechanism, 1975,1,203. 2 R. Herback and G. Zundel, Biochim. Biophys. Acta, 1976,418,52. Mr. Lee answered: Archer et al.' assume that the cations are located around the phosphate group.At low hydrations, theoretical studies' show that the most plausible sites for the location of the counter-ions are (a) the bidentate position in which the cation is situated on the bisectrix of the 0-P-0 angle and (b) the monodentate site in which the cation is situated on the external site of the 0-P-0 angle making a 120" angle with it. At higher hydration levels, through-water interactions between cations and the phosphate group are favoured3 see Fig. 1. In structure (ii) the metal ion can be located with equal probability at a position adjacent to either of the anionic oxygens, thus producing the double-minimum potential well stated. If the DNA has two different backbone confirmations, as a function of the cation employed, then the results of IR studies may well prove to be useful in understanding the differences in the observed experimental results between Na+ and Mg" data.From the hydration isotherm work presented in the paper (Fig. 3 and Table 1) however, it can be seen that any change in conformation between the two samples has a relatively small effect on the amount of water bound to the primary hydration shell of the DNA (4.3-4.8water molecules per base pair for Na' and Mg' + DNA, respectively). (1) (9 Fw 1 Bidentate position for (i) a dehydrated and (ii) a hydrated phosphate group General Discussion 1 G. P. Archer, S. Bone and R.Pethig, J. Mol. Electron., 1990, 6, 199. 2 B. Pullmann, N.Gresh and H. Berthod, Theor. Chim. Acta, 1975, 44, 71; A. Pullmann, B. Pullmann, N. Gresh and H. Berthod, Theor. Chim. Acta, 1977,44, 151. 3 A. Pullmann, Stud. Biophys., 1980,84, 17; A. Pullmann and H. Berthod, Chem. Phys. Lett., 1977,46,249. Dr. Bruni asked: There is a minimum number of molecules required to transfer protons along adjacent H,O molecules. Therefore, the observed lack of dependence of the amplitude of the dispersion on sample hydration is expected, and cannot be used to rule out this phenomenon from the list of potential contributors to the dielectric disper- sion observed. Additionally, there are no hydration effects on the position of the peak in Fig. 2 of your paper, i.e., the process is not getting slower with decreasing hydration. What, then, is the role of hydration on the results shown? Mr.Lee answered: The point made regarding the minimum number of water mol- ecules required before proton translocation can occur is a valid one. However, conduc- tion studies of proton pathways in proteins such as lysozyme indicate that there is a rapid increase in protonic conductivity in the hydration range that corresponds to binding of water to lysozyme that is irrotationally bound by two or more hydrogen bonds thus forming on this timescale a relatively long-lived structure and possibly pro- viding links in the proton pathways between hydrogen-bonded side-chain segments. Hydration above this range, by singly hydrogen-bonded water molecules, does not appear to influence the level of protonic conduction.The experimental data presented here show the opposite trend to this and hence we feel that proton translocation may not be responsible for the observed dispersion. The position of the dispersion centred around lo8 Hz and hence the relaxation fre- quency does not significantly alter with hydration (the enthalpy of activation, AH, was found to be less than 0.3) unlike the magnitude of the dispersion which increases with hydration. Hydration therefore facilitates a process which has a small enthalpy of activa- tion but which increases in magnitude with hydration. 1 S. Bone, Biochim. Biophys. Acta, 1991, 1078, 336. Prof. Goodfellow asked: Should not the counter-ions be associated with the negative potential within the major groove of B-DNA rather than directly with phosphate groups? Mr.Lee replied: It is convenient to associate the positively charged counter-ions with the negatively charged phosphate groups. I fully support the idea that the counter- ions are located in the major groove and I would like to investigate this further in the future. I know that, amongst others, Dr. Soumpasis has computed ionic distributions around DNA. Dr. Soumpasis said : One does not need Manning’s counter-ion condensation theory in order to understand the fact that alkali-metal halides (except Li’) do not bind to DNA but form a delocalised cloud around the polyion. This behaviour is primarily due to the electronic structure (noble-gas configuration) of these ions, which does not favour inner-sphere complexation to DNA sites, unlike transition-metal cations where appre- ciable binding is observed.(For a discussion of these topics see ref. 1.) Manning’s theory, an infinite dilution, infinite uniformly charged wire in a sea of point ions description, is completely out of its regime of validity for the condensed phase DNA systems you study. In addition, delocalized ‘territorial binding’ of counter-ions is not a derived but an essentially postulated property. The problem of determining counter-ion distributions around charged biomolecules taking into consideration finite ion size, ionic correlations, molecular structure and supramolecular organization (e.g. General Discussion crystals, fibres in the regime 0.1-5.0 mol dmv3 NaCl can be effectively solved using our PMF strategy (see ref.2). 1 D. M. Soumpasis, in Biomolecular Stereodynamics ZZI,ed. R. Sharma and M. Sharma, Academic Press, New York,1986, pp, 47-62. 2 R. Klement, D. M. Soumpasis and T. M. Jorin, Proc. Natl. Acud. Sci. USA, 1991,88,4631. Mr. Lee answered: Mannings theory has been extensively applied to DNA systems (for example see ref. 1). The major weakness of applying his model to this particular study is that the DNA is in the hydrated solid state. An assumption in using Manning’s model here is that when the DNA is hydrated the counter-ions are mobile. The advan- tage of using Manning’s model is that it can be applied (in this case using the theory postulated by Mande12) to high frequency dielectric measurements. All models have limitations, for example, in the model to which you refer PMF (potentials of mean force framework) seem to be valid only for monovalent salt concentrations, and we are con- tinually looking at ways to improve the models we have available at present.1 A. Benincentre, M. Ma-, F. Muzzei, A. Minoprio and F. Pedone, Biochim. Biophys. Acta, 1993,1171, 288; B. Sait, R. K. Mohr, C. J. Montrose and T. A. Litovitz, Biopolyrners, 1991,31,1171. 2 F. Van der Touw and M. Mandel, Biophys. Chem., 1974, 2, 218; M. Mandel. Ann. NY Acud. Sci., 1977, 303,74. Prof. Yarwood asked: Have there been measurements of ‘ionic’ relaxation rates near a protein surface (e.g.,NMR data) which would allow you to assess whether or not your relaxation times are relevant to the processes proposed? Mr.Lee replied: An NMR relaxation investigation of water mobility in aqueous bovine serum albumin solutions by Grosch and Noack’ indicated the presence of three distinct water environments: free water (2, x 10-l1 s), translationally hindered water (rHT =s lo-’ s) and rotationally bound water (T x~~ s). The lo8 Hz dispersion observed in this work would relate to the translationally hindered water (rHTx s)in the NMR study. In our discussion we discounted the probability of bound water due to the small activation energy observed and a difference in the dielectricf(&) graph from that expected for bound water. A number of assumptions were, however, made: (i) in relation to the dielectric f(~)graph, the line representing bound water was calculated using the dipole moment of normal water and (ii) the activation energy of bound water is equal to the breaking of a single hydrogen bond.It is possible that either or both of these assumptions may be incorrect and therefore the possibility that the relaxation process is due to the rotational relaxation of bound water cannot be completely ruled out and indeed would be supported by NMR data. 1 L.Grosch and F. Noack, Biochim. Biophys. Acta, 1976,453,218. Dr. Garcia commented: Someone in the audience asked if there have been other measurements of the low frequency optical properties of DNA. Lindsay and collabo- rators, from Arizona State University, have studied the optical properties of DNA fibres.Two papers which show results that may be relevant to your presentation are ref. 1 and 2. I understand that the techniques used were different, but there is some overlap in the measured frequencies. 1 T. Weidlich, S. M. Lindsay and A. Ruppretch. Biopolymers, 1987,26,439. 2 S. A. Lee, Biopolymers, 1987,26,1637. Mr. Lee communicated in response: The work to which the question refers is similar to our dielectric work in as much as it is concerned with investigation of hydration General Discussion effects on the DNA molecule and analysis of the results is capable of producing comple- mentary information. However, the work cited employed optical techniques in the optical frequency range, whereas the work in this study used dielectric techniques over the 1 MHz to 10 GHz frequency range.Dr. Hummer opened the discussion on Dr. Simonson’s paper: Can one quantify the effects of the 12 A cut-off in your calculations of relative permittivities? The work of Neumann and Steinhauser showed that both in finite’ and periodic systems,2 a cut-off grossly distorts angular correlations, among other effects. Could that explain the overall shape of the distance-dependent Kirkwood factor, G(r),in Fig. 2 of your paper, in partic- ular the minimum at 25 I$? Also, is there an explanation of the strong drift with time of G(r) at larger distances? Using the methods of ref. 1 and 2, it should be possible to calculate G(r)in the limit of continuum electrostatics with a spherical cut-off.I would also like to bring a paper by Prof. Berendsen to your attention, who studied dipole moment fluctuations in spherical geometries equivalent to yours.3 In addition, I want to point out a paper by Dr. Garcia: who introduced the analysis of collective motions in protein dynamics by projection on normal coordinates. 1 M. Neumann and 0.Steinhauser, Mol. Phys., 1980,39,437. 2 M. Neumann, Mol. Phys., 1983,50,841. 3 H. J. C. Berendsen, in CECAM Report: Molecular Dynamics and Monte Carlo Calculations on Water, 1972, p. 29. 4 A. E. Garcia, Phys. Rev. Lett., 1992,68, 2696. Dr. Simonson replied: Cut-off effects are an important concern in calculations of dielectric properties. Unfortunately, an analytical correction for cut-off effects on G(r) does not exist for this geometry.However, two elements suggest that our 12 A cut-off does not overly distort the dielectric properties of the system. First, we have explicitly compared pure water spheres of the same size, simulated with and without cut-off. No-cut-off simulations predict a relative permittivity of 82, in good agreement with the relative permittivity of bulk water. 12 A cut-off simulations yield a relative permittivity of 110. Therefore, while the cut-off effect is significant, it does not affect the order of magnitude of the relative permittivity in the pure water case. Secondly, we have analysed 1-2 ns simulations of five different proteins, carried out with a range of cut-off distances, in droplet geometries or periodic geometries, with several unit cell sizes and shapes.’ All five systems exhibit the same qualitative behaviour, reported here for cytochrome c: a low internal relative permittivity, and a much larger overall relative permittivity, arising from the charged side chains at the protein surface.We agree of course that the detailed behaviour of the system is affected by the use of a cut-off, and it may indeed affect the distance dependence of G(r).The drift of G(r)with time at large distances is probably a sign of incomplete convergence. Values of G(r) at large distances are likely to be more sensitive to motions along the first few principal components of the system, which are imperfectly sampled in 1 ns. 1 T. Simonson and C. L. Brooks, J. Am. Chem.Soc., 1996,35,8452. Dr. Garcia commented: There are a few errors of interpretation in your analysis. First, to confuse the principal component analysis with the quasiharmonic approx- imation is incorrect. Although the mathematics involved in both methods is quite similar (Lev,diagonaiization of the displacement covariance matrix), the quasiharmonic approximation makes other assumptions (and errs). By assuming that the fluctuations described by the second moment (covariance) are those of a harmonic system, two major properties are assigned to the system: (1) energy equipartition and (2) uncoupling of modes. Those two properties are known not to be satisfied as soon as you include small anharmonic perturbations (this was observed in the first ‘MD’ simulation ever by General Discussion Fermi, Pasta and Ulam’).The equipartition of energy is used to map the fluctuations to vibrational frequencies. How representative is your simulation of a ‘quasiharmonicsystem’? Not at all. Fig. 5 of your paper shows the distribution of displacements along these directions. Notice that, even though you fitted them to Gaussians, they are not Gaussian or unimodal. The PCA is purely geometric and the principal directions can be obtained by mini- mizing a functional describing the mean-square distance between conformations.’ The optimization of this functional gives a set of directions that systematically ranks the goodness of the representation of the fluctuations by one vector according to the corre- sponding eigenvalue, nothing else is assumed.However, if you look at the characteristics of the motions you may see that these are diffusive rather than vibrational. In your text you say ‘Such an analysis does not assume that the protein dynamics are actually harmonic’, but then you contradict yourself. In your description of the results you resort to terms that are proper to harmonic systems, for example, ‘eigenvalues are ordered by decreasing magnitude, that is by increasing frequency’, this implies energy equipartition. Later on you say ‘where the first few principal components deviated strongly from a Gaussian distribution. It may be that the probability distributions of the first few normal coordinates would converge to a Gaussian shape given a longer simulation time.’ I read this as saying, the system is quasi-harmonic but the oscillation timescales are much longer than the simulation time; if we sample more we get everything in agreement with a harmonic system.I believe that a diffusive model in a space that contains many basins of attraction, where the system gets trapped for long periods of time, is more appropriate to describe the dynamics of a protein. As to what will happen when the simulation time increases, maybe what you see over 1 ns will look as one of the basins that you now see, except that the displacements and timescales will be larger. This model will be consistent with the landscape picture proposed by Frauenfelder and collaborators, while the quasi- harmonic system exists only in the mind of theorists that have ignored the last 40 yearsof progress in the understanding of non-linear dynamics.1 E. Fermi, J. R. Pasta and S. M. Ulam, Studies of Non-linear Problems, Vol. 1, Los Alamos Rep., LA-1940, 1955; E. Fermi, Collected Works of E. Fermi, Vol. 2, University of Chicago Press, Chicago, IL, 1965, pp. 978-988. 2 A. E. Garcia, Phys. Rev. Lett., 1992,68,2696. Dr. Simonson responded: We are perfectly aware that cytochrome c like other pro- teins is extremely anharmonic. The principal component analysis we carry out does not make any assumptions, harmonic or otherwise, about the dynamics of the system. Quite the contrary, our analysis of the principal components makes it very obvious that the system is strongly anharmonic.This is seen, for example, in the dynamics of the first 300 principal components, which disagree completely with a harmonic oscillating picture, but are much more suggestive of diffusive wandering from one energy well to another. (For a more detailed analfiis of hopping between energy wells or substates, see, for example, ref. 1.) The probability distributions of the first few principal components are indeed not Gaussian on the nanosecond timescale, although they may converge to a Gaussian shape over longer timescales. The Gaussian probability distributions seen for the other principal components have little to do with the system being harmonic (which it is not). Liquid water obviously has very anharmonic dynamics, yet the fluctuations of the elec- trostatic potential on a small solute particle are nearly Gaussian (at least in the ther- mally accessible range) (e.g., ref.2). These are consequences of the central limit theorem, not harmonic dynamics. General Discussion Principal component analysis has many flavours. One other possibility is to mass- weight the atomic covariance matrix, and this flavour is in fact the quasiharmonic analysis. Again, the ‘quasiharmonic’ coordinate system can be used whether or not the system is harmonic. Results (not shown in our paper) are almost identical to the prin- cipal components calculated without mass-weighting (for the first few hundred eigenvalues). The anharmonicity of the system is just as striking, when one examines the motions along the quasiharmonic eigenvectors, as with other choices of generalized coordinates.As you point out, other choices of principal components are possible, and some of these may be even more illuminating. It will be interesting in the future to see which type of analysis can best visualize the many levels of complexity in protein dynamics. 1 A. Hodel, T., Simonson, R. Fox and A. T. Briinger, J. Phys. Chem., 1993,97,3409. 2 G. Archontis, T. Simonson and M. Karplus, unpublished results. Dr. Ludemann said: It was shown’ that (M2) calculated from a trajectory where Coulomb interactions were treated by simple cut-off simulations is an order of magni- tude too small compared with (M’) derived from simulations done with the reaction field method.Although in Fig. 2 of your paper you show that G(r) has converged, how can you be sure that the values for the variance (AM2) of the total protein dipole moment and hence the permittivity of the protein E,, ,are realistic. 1 H. Schreiber and 0.Steinhauser, J. Mol. Biol., 1992,228, 909. Dr. Simonson replied: The answer to your question is contained within my earlier response to Dr. Hummer. Prof. Berendsen asked: I refer to your figure (Fig. 2) which shows a strong anti- correlation between the dipole fluctuation due to water and the fluctuation due to the protein. Since you simulate a droplet in vacuum in which the total dipole moment fluctuation is quenched, it is likely that this causes such an anticorrelation. Could then the anticorrelation be an artefact of the chosen geometry of a sphere in vacuum rather than a real effect of a protein in solution? Dr.Simonson responded: It is certainly true that the dipole moment fluctuation of the entire droplet is quenched by the surrounding vacuum. This could indeed artificially enhance the correlations between fluctuations ofthe electrostatic potential due to the water and that due to the protein. Unfortunately, we do not have data for this (or any other) protein in a true bulk medium at the present time; it would be very interesting to compare with such data. Prof. Goodfellow asked: Regarding Fig. 10 of your paper and the correlation of side-chain torsion angles with X-ray crystallographic data: (1) have you considered com- paring your side-chain motion with the temperature factors for yeast cytochrome c and (2) are you aware of correlations of the torsion angle with secondary structure? Dr.Simonson said in reply: Our calculated b-factors agree qualitatively with experi- mental b-factors, similar to other molecular dynamics simulations. The interesting point in our side-chain statistics is that the torsional statistics are very similar from one lysine to another, yet the different lysines make very different contributions to the overall G-factor : these variations must therefore result from different motions of the backbone. Dr. Moore said: In your system you haven’t included counter-ions. Will this cause problems with the electric field generated, i.e., the protein is sitting in a fluctuating field? (I might expect some type of fluctuating double layer at the protein surface), General Discussion electrostatic potential on 22 Fig.2 Electrostatic potential on C, of Lys 22 from molecular dynamics as a function of time. Protein contribution, 5 (black), solvent contribution, V, (grey) and contribution from charged protein side chains, V,(offset by -50 kcal mol-' e-' for clarity). Do you see any unusually low dielectric/low polarizability regions near the active sites, i.e., would you expect a fixed local arrangement of dipoles near active sites? Dr. Simonson answered: We do indeed neglect counter-ions, since we do not feel we can treat them accurately. It is unclear what counter-ion concentration to assume, and it is unlikely that we can accurately sample their available conformations within nanose- cond simulations.This should not affect our calculation of the protein relative permit- tivity very much; in particular the theory we use does not require (in this particular geometry) that the system be neutral. Also Poisson-Boltzmann calculations indicate that the solvation free energy of charges located within the protein are completely insensitive to weak ionic strengths. We do indeed see unusually low polarizability of the cytochrorne c interior in the vicinity of the active site, i.e., the centre of the haem group. The local relative permit- tivity is at a minimum there, and so is the microscopic susceptibility in response to perturbing point charges. Dr.Warwicker commented: In relation to the use of relative permittivities in contin- uum electrostatic calculations, I would like to ask about the significance of the spatial variation of charged side-chain fluctuations. The flexible surface lysines give the largest contributions of polar fluctuations and increments to the relative permittivity. This may not make much change for continuum calculation using relative permittivities of 2-4 for protein and cu. 80 for solvent water, since electrostatic interactions with largely solvent accessible groups are dominated by solvent rather than protein. For continuum calcu- lations, the behaviour of buried or partially buried, charged side-chains may be of more interest since such groups make the largest energetic contributions.Would the author like to comment? General Discussion Dr. Simonson replied : True, continuum models of proteins are somewhat insensitive to the exact relative permittivity assumed for the external, solvent-exposed, parts of the protein. Charged residues at the protein surface are in any case strongly screened by solvent, even when one assumes a low protein relative permittivity throughout. Never- theless, the assumption of a low relative permittivity throughout is not completely benign. We have considered simple analytical models, such as a pair of charges near a planar or a spherical surface, separating a low and a high dielectric medium. The charge solvation energies and interaction energies vary quite a bit depending on whether the charges are placed in the low or the high dielectric medium, at least for moderate charge separations.For real applications, it remains to be seen whether a continuum model that includes a position-dependent relative permittivity within the protein will actually have more predictive value than the usual implementations. Dr. Doster asked: Why were the oscillations in Fig. 6 of your paper (underdamped motions on a timescale of hundreds of picoseconds) not observed? Dr. Simonson said in response: The motions along the first principal components of the protein, in our simulations, correspond to infrequent hopping between energy wells. These energy wells are only sampled a few times on a nanosecond timescale. The under- damped oscillations we see on a timescale of hundreds of picoseconds could therefore be influenced and possibly provoked by insufficient sampling. They could also be influ- enced by the presence of a vacuum around our 24 A radius simulation sphere. To my knowledge this is the first time such a long protein trajectory has been subjected to principal component analysis. Therefore it is difficult to assess exactly what features of the molecular dynamics model are likely to produce artefacts in these degrees of freedom. Dr. Soumpasis commented : The problem with the continuum electrostatics of pro- teins is that we simply do not know whether potentially charged groups are actually charged especially when they are buried in the interior of the protein. Dr. Simonson concluded by saying: In cytochrome c, as in other proteins, there is some ambiguity about which histidines are or are not charged, and there is some uncer- tainty for the two propionic groups attached to the haem: they are partially buried in the structure, and the experimental data on their titration appears unclear. We believe they are ionized, based on the polarity of their environment; this assumption has also been made in other theoretical studies of cytochrome c. In any case, modelling them as neutral would almost certainly have a negligible effect on the dielectric properties we have calculated for cytochrome c. On the other hand it would affect the detailed redox properties, which we have not analysed here.

ISSN:1359-6640    

DOI:10.1039/FD9960300091

出版商:RSC    

年代:1996

数据来源: RSC

摘要:

Faraday Discuss., 1996,103, 1 17- 124 Hydrophobicity reinterpreted as ‘minimisation of the entropy penalty of solvation ’ A. D. J. Haymet,” Kevin A. T. Silversteid and Ken A. Dill‘ a School of Chemistry, University of Sydney, NSW 2006, Australia’Graduate Group in Biophysics, University of CaliJornia, San Francisco, CA 94118, USA Department of Pharmaceutical Chemistry, University of California, San Francisco, CA 94118, USA Alkanes are simple examples of hydrophobic molecules. Hydrophobic, loosely translated, means ‘water hating’. Many fascinating physical pheno- mena are today explained by appealing to this concept. The facts about alkanes in water are simple. At room temperature and pressure, simple alkanes are CQ. 100 times less soluble in water than in benzene.Many people say that this is because the alkanes are ‘hydrophobic’, but do the alkanes really ‘hate’ the water? In the view of this author, the experimental data are unequivocal on this point: the simple alkanes ‘like’ the water. It is the water which ‘hates’ the alkanes. This view is supported further by the high solu- bility of water in liquid methane and simple two-dimensional models. 1 Hydrophobicity To explain the importance of the concept of hydrophobicity in rationalizing many complex systems, one can do no better than cite the definitive work of 1980 by Professor Arieh Ben-Naim,’ who pointed out that the concept of hydrophobic interactions has been used frequently in the chemical and biochemical literature, to explain a whole range of phenomena, including ‘conformational changes of a biopolymer, the binding of a substrate to an enzyme, the association of subunits to form a multisubunit enzyme, and processes involving high levels of aggregation such as the formation of biological membranes and the organisation of biological molecules to form a functional unit in a living system.’ The words ‘hydrophobic’ and ‘hydrophobicity ’ remain popular in the literature.Dif- ferent authors may disagree over the progress in assembling proof that the above pheno- mena are in fact due to hydrophobic interactions, rather than some other property of water or the solute. In fact, as is common in science, the definition of a concept (such as hydrophobicity) has become confused with its (alleged) consequences.There are a number of recent significant developments in this field. In 1993 Blokzijland Engberts2 published a definitive review article on hydrophobic effects. In 1995 Guillot3 performed a series of simulations using faithful models of rare gases and water, and determined solubilities along the liquid-gas coexistence line. In 1992, Smith and co-workers4~ showed that, at room temperature, hydrophobic association of unchargedsolutes in water is almost entirely an entropy effect. This is consistent with the ther- modynamic views of Stillinger’’ and Hildebrand,21*22 emphasised by Dillz3 and Haymet,6 and the experimental enthalpy data cited by Privalov and Gill.24 The experi- mental data show that methane ‘likes’ water about as well as it ‘likes’ any other solvent 117 118 Hydrophobicity and entropy of soluation from the point of view of enthalpy.This view is underlined by the experimentally high solubility of water in cryogenic solvents such as liquid methane. In this case, hydropho- bicity, in fact, arises from the entropy penalty incurred by the water. The methane is not hydrophobic: the water is ‘metho-phobic’. In this paper, we review the experimental data on the solubility of simple alkanes, discuss the origin of the entropy penalty and its potential spatial location, and introduce preliminary data from a two-dimensional model of simple solutes in water. 2 Solubility of simple alkanes in water: Experimental data There is much to be gained from studying the experimental solubility of simple alkanes in water.The following data have been obtained from the IUPAC ~eries.~-~ The rele- vant definitions of solubility may be summarised as follows. At temperature T and total pressure P, let p1 denoted the number density of solvent, and p2 the number density of solute. Then the mole fraction solubility of the solute is defined to be x2 = P2,liquid P 1. liquid + P2, liquid The Ostwald solubility is defined to be P2,liquid Pz. gas The mole fraction solubility of course appears in the famous Henry’s ‘Law’ (1803) for the partial pressure of the solute in the gas phase above the solution, P2 = k,x2 (3) where k, is Henry’s constant. Lewis corrected this law for non-ideal systems to read f2 = ktG2 (4) where f2 is the so-called ‘fugacity’ of the solute and is simply the quantity which makes the above equation true.Thermodynamics does not provide us with an estimate for Henry’s constant. Both the experiments and computer simulations discussed in this paper have as part of their aim the determination of Henry’s constant, which is not easy to predict from first principles. For simple alkanes in water, it has been possible to fit the experimental mole fraction solubility x2 at an alkane partial pressure of 0.103 25 MPa using a function of the form In x2 = A + B/z + C In z + Dz (5) where z = T/lOO and A, B, C and D are fitted parameters. The parameters for methane, ethane, propane and (over a limited temperature range) butane are listed in the IUPAC table^.^-^ The thermodynamic quantities for the transfer of the solute from the vapour phase at 0.101 325 MPa to the (hypothetical) solution phase of unit mole fraction may be calcu- lated from the above equation.In Fig. 1 is displayed the Gibbs energy of transfer, AG, as a function of temperature, for the four solutes methane, ethane, propane and butane, and an additional curve which represents an extrapolation of the butane, propane and ethane data to a hypothetical alkane with one carbon atom, denoted below as C,- alkane. The difference between this extrapolated value and the true Gibbs energy of transfer for methane is shown at the bottom of the figure. In Fig. 2 are shown the corresponding enthalpy of transfer AH, and the entropy of transfer TAS multiplied by the temperature, for the four liquids plus the hypothetical C,-alkane. A.D. J. Haymet et al. 0.0 260 280 300 320 340 360 380 temperaturelK Fig. 1 Gibbs energy of transfer (kJ mol- ') for alkanes from the vapour phase into solution as a function of absolute temperature. From top to bottom, the data are for butane, propane and ethane (solid lines), hypothetical C,-alkane (long dashed line). The data for methane (short dashed line) crosses the three solid lines. From these data a number of features emerge: Alkanes are highly insoluble in water, ca. 100 times less soluble than in organic solvents. At room temperature, from the point of view of enthalpy, these alkanes 'like' to be dissolved in water.In fact, the enthalpy of transfer to water is as much 'downhill' as for transfer to most organic solvents. Hence, the anomalous solubility of alkanes in water is due to the entropy penalty incurred upon dissolution. This penalty may be interpreted as 'structuring' the solvent (water) in some way, although of course thermodynamics gives no information on the nature or spatial location of this ordering (see below). 0-260 280 300 320 340 360 380 temperawren< Fig. 2 Enthalpy of transfer (kJ mol- ') (upper part of figure) and the entropy of transfer multiplied by the temperature (kJ mol-') (lower part of figure), for alkanes from the vapour phase into solution as a function of absolute temperature. From top to bottom, the data are for methane, ethane, propane and butane (solid line); hypothetical C,-alkane (long dashed line).120 Hydrophobicity and entropy of solvation At a temperature of ca. 365 K, the enthalpies of transfer for simple alkanes attain a common value, and this value is in fact zero. This observation has been noted before, but to the authors’ knowledge has not been explained. Note that the fitted data for the solubility of butane in water do not appear to be accurate at higher temperatures, but the low-temperature trend is consistent with this observation. Simple alkanes, except methane, become modestly less soluble as the carbon number increases, and as the tem- perature increases. Methane itself is anomalous in a second-order way, in that it is even less soluble than an extrapolation of the other simple alkane data would suggest, as shown in Fig.1 and 2. The extrapolated data labelled C,-alkane follow trends in harmony with the other simple alkanes. However, for example at 298 K, the methane mole fraction solubility is x2 = 2.55 x less than 60% of the hypothetical C,-alkane solubility, 4.33 x The additional, weakly temperature dependent, contribution to the Gibbs energy of transfer, which decreases from 1.5 to 0.6 kJ mol-’ over the temperature range shown, has not been explained at the molecular level, to our knowledge. This paper tries to address the question of the entropy penalty for dissolving alkanes in water. The last observation above is important in the sense that, for valid reasons of simplicity, most of the computer simulations of hydrophobicity have focussed on the simplest alkane, methane, and it should be remembered that it has an additional 5% contribution to the Gibbs energy of solvation at room temperature, as shown in Fig.1. It is worth restating the facts in the other limit, namely dissolving water in liquid methane. From FTIR spectroscopy in a vacuum-insulated absorption cell, Rebiai et al.” concluded that the mole fraction solubility of water in methane at 112 K is 6 x and hence that ‘this figure is many orders of magnitude greater than that expected which is to 10-l8mole fraction’. This supports the view that methane ‘likes’ water about as well as it ‘likes’ any other molecule from the point of view of enthalpy.The difference is entropy. Smith and Haymet have investigated whether entropy also plays the major role in the interaction of hydrophobic solute^,^^^^ by calculating the entropy and energy (enthalpy) components of the Gibbs energy of association of two methane molecules in water using three methods. The ‘fluctuation’ formula for the entropic contribution is convenient, where TS(r,) is an unimportant additive constant, since it requires only a single simula- tion and may be incorporated directly into existing Gibbs energy codes. As the tem- perature increases, the magnitude of the attraction at short separations increases, and the contact well in the potential of mean force becomes deeper. The entropy contribu- tion, TS(r),is found to drive the two solute molecules together, in agreement with earlier speculations. 3 Where is the penalty incurred? If one accepts that hydrophobicity is an entropy effect at room temperature, it is reason- able to ask where in the liquid the penalty is incurred.This has been done frequently in the literature. One explanation is the so-called ‘clathrate model’, for which convincing thermodynamic evidence has been assembled by Soren~en.~’ In the opinion of the authors, the answer is not yet clear. The data are perhaps not yet totally definitive, but there is no universally accepted microscopic evidence that the ‘simple’ explanation, namely that the water molecules are more ordered (either translationally or rotationally or both) around the solute, is correct.If this view is correct, then one is forced to look to higher-order correlation functions for the explanation. A. D.J. Haymet et al. 121 Fortunately, a formalism is available for this investigation, although the numerical data are not complete. This topic has a long, rich history. The entropy, S, cannot be written simply in terms of the pair correlation function, unlike most thermodynamic quantities. In the course of investigating approximations for the entropy, we have exploited the entropy functional in the rand ensemble as a starting point for generating systematic expansions for the entropy. lk14 For a homogeneous liquid with Y components, volume V and temperature T, the total grand potential f2 = -PV can be constructed from the total energy, entropy and chemical potential, V CnCgil), d$),8:;;*I= W(gt!.}I = EC{~!!.}I (7)TSC(g!!.}I pa Nagi$$, --5 a-1 For a total potential energy, u, which is the sum of an external potential (below set to zero) and two-, three- and four-body potential energies, where a, fl, y and S are component types.The average total energy E is given straightfor- wardly by the functional where pa is the partial number density of component a, and pt = pa is the total number density. The chemical potential terms also cause no problem, as shown below. The entropy is not so simple. The work of Green et al. and Ravechk coupled with Hernando’s result for the ring entropy,” leads to the expansion for the entropy in the grand canonical ensemble.V SC{&!.)I/Vk = 5 pt -CPa Wan3 a=l 122 Hydrophobicity and entropy of solvation where I is the v + v identity matrix, l?(k)ap = pi/2baa(k)pi/2is the total correlation func- tion matrix, Tr represents a ‘trace’ and Skix(i)is the multicomponent analogue of the remainder term in Hernando’s expansion. Minimization of the resulting grand potential functional pi2 at constant temperature and chemical potential, with respect to g$Ld(ra,ra, ry,rd),g$\(r,, ra, ry),g$)(r~,Jand the component densities yields (a) closed-form, consistent equations for the chemical poten- tial terms in eqn. (7) above, (b) the consistent equation for the equilibrium pressure and (c) integral equations which at the simplest level are familiar, and at higher order are the natural generalisations of the standard text-book theories. This expansion for the entropy is being investigated term-by-term to exhibit the source of the entropy penalty of solvation.4 Preliminary data from a two-dimensional water model Many of the above ideas may be investigated thoroughly and exactly via a simulation of a simple model, in this case ‘two-dimensional’ (d = 2) water. There are a number of canonical ways to make a model of water in d = 2, including the model of Nose and co-workersl6l1’ in which the molecule has two ‘arms’ at an angle of 90”. However, here we adopt the model of Ben-Naim,’ amplified by Andaloro and Sperandeo-Mineo,” which in fact has three identical ‘arms’ separated by 120”.The reason for this construct is to mimic the open, tetrahedral lattice forms of ice found in real d = 3 water by a honeycomb d = 2 lattice, which is the low-temperature ground state of this model. The water molecules in the model interact via a pairwise additive potential energy which is the sum of two terms: an orientationally independent Lennard-Jones term, which defines the size of the molecules; and an orientation-dependent term which favours the alignment of the arms of neighbouring molecules, with a hydrogen-bond energy parameter denoted E, . Full details of the model potential energy are given by Andaloro and Sperandeo-Mineo.Ig Owing to the d = 2 nature of this model, highly detailed analyses of fully equili- brated solvent and solvent-solute systems are possible.Rough preliminary data for the thermodynamics of transfer of a structureless solute molecule into the water phase are shown in Fig. 3. These data show qualitatively the same trends as the experimental data for transfer of argon from the pure vapour phase to the aqueous phase. 0.2 0.0 -0.2 9r g -Om4 0) -0.6 -0.8 -1 .o 0.16 0.18 0.20 0.22 kTIEh Fig. 3 Helmholtz energy of transfer (upper line), energy of transfer (middle line) and the entropy of transfer multiplied by the temperature (lower line), in units of the hydrogen-bond energy E,, as a function of dimensionless temperature (kT/E,) for a simple solute in the two-dimensional water model A. D. J. Haymet et al. Fig.4 Snapshot of the equilibrium ice phase at low temperature, for the pure two-dimensional water model Fig. 5 Snapshot of the equilibrium water phase at the dimensionless temperature (kT/E,) = 0.18, for the pure two-dimensional water model. Note the expansion of the disordered water phase compared with the ordered solid phase in the previous figure. 124 Hydrophobicity and entropy of solvation Typical ice-like and water-like configurations from the simulations are shown in Fig. 4 and 5. The latter figure shows characteristic water structure, namely no lattice order, some ring structures, clusters of various sizes and sizable density fluctuations. Further analysis of these simulations is underway. 5 Conclusions The hypothesis advanced in this article, that hydrophobicity (at room temperature) is an entropy effect, may be tested by analysis of experimental data and by computer simula- tions.However, it should be noted that in developing force fields for the simulation of large molecules in solution, equal priority should be given to modelling the entropy changes in the solvent and development of energy parameters for the solute and solvent. In Australia this research was supported, in part, by the Australian Research Council (ARC) grant No. A29530010. One of us (K.A.T.S.) is supported by a National Science Foundation Graduate Fellowship. References 1 A. Ben-Naim, Hydrophobic Interactions. Plenum, New York, 1980. 2 W. Blokzijl and J. B. F. N. Engberts, Angew. Chem., Znt. Ed.Engl., 1993,32, 154. 3 B. Guillot, in Physical Chemistry of Aqueous Systems, ed. H. J. White, Jr., J. V. Sengers, D. B. Neumann and J. C. Bellows, Begell House, New York, 1995. 4 D.E.Smith, L. Zhang and A. D. J. Haymet, J. Am. Chem. SOC.,1992,114,5875. 5 D.E.Smith and A. D. J. Haymet, J. Chem. Phys., 1993,98,6445. 6 A. D.J. Haymet, Ann. New York Acad. Sci., 1994,715, 146. 7 H. L. Clever and C. L. Young, ZUPAC Solubility Data Series, Methane, Pergamon Press, New York, 1987,vol. 27/28. 8 W. Hayduk, IUPAC Solubility Data Series, Ethane, Pergamon Press, New York, 1982,vol. 9. 9 W. Hayduk, I UPAC Solubility Data Series, Propane, Butane and 2-Methylpropane, Pergamon Press, New York, 1986,vol. 24. 10 R. Rebiai, R. G. Scurlock and A. J. Rest, in Advances in Cryogenic Engineering, ed.R. Rebiai, New York, 1984,vol. 29,pp. 1005-1012. 11 D. E. Smith and A. D. J. Haymet, J. Chem. Phys., 1992,96,8450. 12 B.B. Laird, J. Wang and A. D. J. Haymet, Phys. Rev. E, 1993,47,2491. 13 B. B. Laird, J. Wang and A. D. I.Haymet, Phys. Rev. E, 1993,48,4145. 14 A. G.Schlijper and R. Kikuchi, J. Stat. Phys., 1990,61, 143. 15 J. A. Hernando, Mol. Phys., 1990,69, 327. 16 K. Okazaki, S.Nose, Y. Katoka and T. Yamamoto, J. Chem. Phys., 1981,7!5,5864. 17 Y. Katoka, H. Hamada, S. Nose and T. Yamamoto, J. Chem. Phys., 1982,77,5699. 18 A. Ben-Naim, J. Chem. Phys., 1973,59,6535. 19 G. Andaloro and R. M. Sperandeo-Mineo, Eur. J. Phys., 1990,11,275. 20 F.H. Stillinger, Science, 1980,209, 451. 21 J. H. Hildebrand, J. Phys. Chem., 1968,72, 1841. 22 J. H.Hildebrand, Proc. Natl. Acad. Sci., 1979,76, 194. 23 K. A. Dill, Biochemistry, 1990, 29, 7133. 24 P. L.Privalov and S. J. Gill, Adv. Protein Chem., 1988,39, 191. 25 C.M.Sorensen, in Physical Chemistry of Aqueous Systems, ed. H. J. White, Jr., J. V. Sengers, D. B. Neumann and J. C. Bellows, Begell House, New York, 1995,pp. 308-316. Paper 6/00164E;Received 8th January, 1996

ISSN:1359-6640    

DOI:10.1039/FD9960300117

出版商:RSC    

年代:1996

数据来源: RSC

摘要:

Faraday Discuss., 1996,103, 125-139 Hydrophobic interactions :Conformational equilibria and the association of non-polar molecules in water Shekhar Garde; Gerhard Hummerb and Michael E. Paulaitis"*t Centerfor Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA Theoretical Biology and Biophysics T-10, MS K710, and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Recently developed proximity approximations have been used to calculate inhomogeneous water density profiles around non-polar molecular solutes. Relative Helmholtz energies of hydrophobic hydration are calculated from these density profiles using two inherently different approaches : Helmholtz energy perturbation and a multiparticle correlation function expansion. Entropic contributions to the hydration Helmholtz energy are also calcu- lated using the multiparticle correlation function expansion for the entropy truncated at the level of pair correlations.We show that the proximity approximations describe water structure around a tetramethylammonium ion in good agreement with neutron diffraction experiments, and provide an accurate description of water structure around simple alkanes and benzene as reflected in their entropies of hydration. Further, we reproduce two important features of hydrophobic interactions : a highly favoured contact minimum and a solvent separated minimum in the PMFs for methane- methane and neopentane-neopentane association in water.Our calculations also show that the more compact conformations of n-butane and n-pentane are favoured in water, as expected based on traditional ideas regarding hydrophobic interactions. Since the iceberg hypothesis by Frank and Evans,' the prediction of the inhomogeneous structure of water around non-polar molecules has been central to a molecular-level description of hydrophobic phenomena. Molecular details of water organization have been incorporated in previous studies either directly or indirectly, and have led to an understanding of the underlying physics of hydrophobic hydration and interactions for model solute^.^-'^ Although detailed molecular simulations of the hydration of model solutes can provide exact structural information on solutewater correlations, it becomes computationally more expensive to simulate more complex solutes.More efi- cient approaches are, therefore, needed to resolve water structure around complex mol- ecules, as well as methods for deriving reliable entropies and Helmholtz energies of hydrophobic hydration and association on the basis of this information. In this paper, we calculate the inhomogeneous water density around non-polar molecular solutes assuming that water organization is only locally sensitive to the struc- tural details of these solutes. Helmholtz energies of hydrophobic hydration and associ- ation are calculated on the basis of this water structure using the Helmholtz energy perturbation (HEP) technique,14 and by the direct calculation of entropy using a corre- lation function expansion truncated at the level of solute-water pair correlation^.'*^ -f Present address: Department of Chemical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA.125 Hydrophobic interactions Solvent contributions to the potential of mean force (PMF) are calculated for methane- methane association and for neopentane-neopentane association along two different tra- jectories. Solvent contributions to the Helmholtz energy and entropy of hydration for different conformations of n-butane and n-pentane are also calculated. For n-pentane, we report the complete hydration Helmholtz energy surface and the corresponding surface of entropic contributions to this Helmholtz energy.Theory Proximity approximations The inhomogeneous one-particle density of water molecules surrounding a molecular solute fixed at an arbitrary position in space and in a specific conformation is given by the conditional probability of finding a water molecule at rw, given that the n sites defining the solute are fixed at (rl, ...,r,),15*16 where g(") is the n-particle correlation function related to the probability of finding the solute in the specified conformation, g(,+ is the water-solute (n + l)-particle corre- lation function and po is the bulk water density. The n-particle correlation function is related, by definition, to the n-particle potential of mean force, Wn)(rl,..., r,) = -k, T ln[g(')(r,, ...,r,)] and similarly for the water-solute (n + 1)-particle correlation function.This (n + 1)-par-ticle PMF can be decomposed in terms of lower order PMFs to get the conditional water-solute PMF: I rl, .. . , r,) = w("+ ...,r,)Wn~l)(rw l)(rW,rl, ..., r,) -w(,)(rl, n = 1W(2)(rw,Ti)+ C[W3)(rW,ri ,rj) -W2)(ri,rj) i= 1 i, ji> j Combining eqn. (1) and (2) while retaining terms up to three-particle PMFs in eqn. (3), we obtain the following expression for the one-particle water density in terms of two-particle and three-particle correlation functions, Truncating eqn. (3) at the level of two- and three-particle PMFs corresponds to the Kirkwood' and the Fisher-Kopeliovich' superposition approximations, respectively, for the three- and four-particle correlation functions in eqn. (4).Eqn. (4) has been successfully applied to strongly associating inhomogeneous fluids using two- and three-particle correlation functions derived from molecular simulations. These applications include ion density distributions near nucleic acids,' the icelwater interfaceI5 and the hydration of biomolecules.' More recently, Garde et af." used6v20-21 eqn. (4) to calculate water density profiles around various non-polar molecules, as well as non-bonded clusters of methane molecules. For molecular solutes consisting of over-lapping (covalently bonded) sites, water densities in the first hydration shell were found to be substantially overestimated if all pair and triplet correlation functions are included mfi ,;...,1,min,= S.Garde et al. 127 in eqn. (4). Alternative one-site and two-site proximity approximations were proposed based on the assumption that only the local structural details of the solute contribute significantly to the inhomogeneous one-particle density of water. In the i-site proximity approximation, this density is given by correlations with only the i proximal sites of the solute (i < n). The one- and two-site proximity approximations are thus the lowest order approximations. For the one-site proximity approximation : p(,J'(r, I r1, ... ,r,) x po g'2'(rw,rj) (5) with j such that I rw -rjI = mink= ...,,I r, -rkI, and for the two-site proximity approx- imation : with j and k such that I rw -rjl = rnin,=1, ..., "1 rw -rlI and I rw -rk I = I rw -I, I.When compared with water density profiles obtained directly from simulations, these approximations were found to reproduce both the water-oxygen and water-hydrogen density profiles around non-polar molecular solutes with different degrees of structural complexity and site-site overlap." We apply these approximations here to develop water density profiles around non-polar molecular solutes, which are then used to calculate entropies and Helmholtz energies of hydrophobic hydration and association, as described below. Correlation function expansion for the entropy The entropy of a classical fluid can be expressed as an expansion in terms related to one-particle, two-particle and higher-order correlations22-28 that allows direct entropy calculations from the intermolecular structure of the fluid.In a recent series of pub-lications, we have shown that this correlation function expansion truncated at the level of solute-water pair correlations can be used to estimate entropies of hydration of simple non-polar solutes in ~ater.~.~ Contributions to the hydration entropy arising from higher-order solute-water correlations (e.g. solute-water-water triplet correlations) and due to perturbations in water-water correlations in the presence of the solute (solvent reorganization) were neglected, since the high dimensionality of these corre- lations makes their calculation impractical. Moreover, at least to first order, the solvent reorganization contribution to the entropy is equal in magnitude to the solvent reorga- nization en erg^,^*^^ and thus, the two contributions cancel exactly in the expression for Helmholtz energy.The standard entropy of hydration3' given within the context of the correlation function expansion truncated at the level of solute-water pair correlations is6 AS:= -kBTao + kBpOfi2 --gga In &a dr do -s (ggk -1) dr do] (7) where gca is the water-solute pair correlation function, ao is the coefficient of thermal expansion of pure water, and fir is the infinite-dilution partial molar volume of the solute in water. Assuming entropy-energy compensation of the solvent reorganization contributions, the infinite-dilution chemical potential of the solute is given by6 pr = kB T h(pAh:) + -ggk In ygk dr do -(gga -1) dr do] (8)kBTposz u s where, yck = g#, exp(+,Jk, T) is the cavity distribution function, #WA is the solute- water pair interaction energy, and pA is the solute number density.Since entropy and internal energy calculations are independent of one another, the relative importance of Hydrophobic interactions entropic contributions to the Helmholtz energy of hydration can be evaluated within the context of the truncated correlation function expansion for the entropy. For molecular solutes of arbitrary shape, the solute-water pair correlation function will depend on solute-water separation, relative orientation and solute conformation. The additional degrees of freedom required to describe solute-water orientations and solute conformations make a rigorous evaluation of this pair correlation function practi- cally intractable, even for relatively simple molecular solutes.These complexities are avoided, however, by invoking the proximity approximation for the pair correlation function, which leads to the following expression for the standard entropy of hydration,6 rr r 1 where pogKy is the one-particle density of water oxygens at Y and pGz"(0, x) is the orientational distribution function with respect to the nearest solute site a. Eqn. (9) reflects the same factorization for gpy(r,8, x) applied previously to simple, spherically symmetric non-polar solutes and to hydrophobic chain molecules. That is, orientational correlations are independent of solute site-water separation within the first hydration shell of the solute site, and are negligible beyond the first hydration shelL5 The contribu- tion due to orientationally-averaged translational correlations in eqn.(9) (third term) can be calculated using either the one-site or two-site proximity approximation described in the previous section. Within the one-site proximity approximation, the con- tribution due to orientational correlations in eqn. (9) (last term) is directly proportional to the coordination number of site a,poV:h. Helmholtz energy calculations The closely related techniques of Helmholtz energy perturbation and thermodynamic integration31 have been used extensively in studies of hydration to calculate Helmholtz energies of hydration as a function of solute c~nformation,~~-~~ the PMF for methane- methane association' and relative Helmholtz energies of hydration of different sol~tes~~-~~In HEP, the change in Helmholtz energy between two states of the system, defined by different values of the coupling parameter, i,,is given by AAij = Aj -Ai = -kR T In(exp( -+ij/kB T))Ai (10) where 4ij= U(jLj)-U(Ai)and U(i)is the total potential energy of interaction for a given value of the coupling parameter.The canonical ensemble average denoted by (a . .)li is carried out in state i corresponding to Ri, and is typically evaluated by molecular simu- lations. Although the above expression for the Helmholtz energy change is exact, its implementation in practical calculations requires the two states of the system to be similar.When the states corresponding to ijand ELi are not very different, the exponen- tial in eqn. (10)can be expanded to estimate the Helmholtz energy change as follows, AAij (4ij>ii (1 1) From the Gibbs-Bogoliubov inequality, the right-hand side of eqn. (11) is an upper bound to AAij, approaching the true value when states i andj are only infinitesimally far apart. Thus, to insure an accurate value for AA, the trajectory corresponding to an overall change in ;I is divided into many closely spaced increments in A. For a given state of the system, we apply the two-site proximity approximation described above to estimate the average structure of water around non-polar molecular solutes in a given conformation.This average structure is then used to obtain the ensem- ble average in eqn. (11) for each step along the desired trajectory. S. Garde et al. 129 Simulation method and computational details Water density profiles around non-polar solutes defined as a collection of covalently bonded methane (Me) sites were estimated on the basis of the one- and two-site proxim- ity approximations in eqn. (5) and (6). This calculation requires Me-water oxygen (hereafter denoted 0)pair and Me-Me-0 triplet correlation functions, which have been obtained from simulations of a single Me site and Me-Me dimers at fixed separations in SPC water." Once this database has been generated, the estimation of inhomogeneous water densities around molecular solutes of arbitrary shape and size, and in different molecular conformations, is straightforward.For orientational correlations, the one-site proximity approximation is applied as described elsewhere.6 Within this approximation, a reasonable estimate for the orientational entropy per water molecule is 0.35 cal mol-' K-1 6 Monte Carlo simulations of the Me monomer and Me-Me dimers in water were performed at a temperature of 298 K and a constant density of 33.33 molecules nm-3. The united-atom representation was used for Me, which was treated as a spherically symmetric Lennard-Jones particle with oMe-Me = 0.373 nm and &Mule-Me = 0.294 kcal mol-'.39 The SPC model was used for water.40 Lennard-Jones parameters for Me- water interactions were calculated using Lorentz-Berthelot mixing rule^.^' The Ewald summation technique was used for Coulomb interaction^.^^ MeMe-0 triplet corre- lation functions were obtained at Me-Me separations from 1.2 to 6.6 8 at an interval of 0.2 A.Equilibration consisted of 50-60 K passes (one pass corresponds to one attempt- ed move for each water molecule), and production runs following equilibration amount- ed to 60-100 K passes with averaging done once every 5 passes. Further details are given elsewhere. ' To apply HEP, water-oxygen-atom densities were determined at uniformly spaced grid points around a given solute conformation using either the one-site or the two-site proximity approximation. Perturbations were made in both the forward (A + AA/2) and reverse (A -AA/2) directions along the trajectory of interest.For each of these pertur- bations, the energy of interaction with the solvent was calculated using the water density profile corresponding to the unperturbed state. The Helmholtz energy difference between these two states was subsequently calculated using eqn. (11). We found these Helmholtz energy profiles to be insensitive to the spacing of grid points for values below 0.4 A, to perturbations in either the Me-Me or neopentane-neopentane centre-to-centre distances GO.2 8, and to perturbations in the torsional angles for n-butane and n-pentane <5". Results and Discussion Water structure around the TMA ion In previous work," we have shown that the proximity approximations can reproduce the density profiles of water oxygen atoms and hydrogen atoms around model non- polar solutes derived from explicit simulations of these systems.We make the direct comparison here with actual experiments to provide an additional evaluation of these approximations, recognizing that the potential models used in our analysis are only approximate representations of the force fields for the systems of interest. The experi- mental determination of water structure around hydrophobic solutes is extremely diffi- cult owing to the low solubility of these solutes in water. However, tetraalkylammonium salts are highly soluble in water, and are thought to exhibit apolar behaviour character- istic of hydrophobic hydrati~n.~~-~~ In particular, neutron diffraction experiments per- formed on aqueous solutions of tetramethlyammonium (TMA) salts show that waters of hydration adopt preferred orientations around TMA, which are likewise found in simu- lation studies of purely non-polar solutes in water.2 These preferred orientations are Hydrophobic interactions characterized by water oxygen atoms and hydrogen atoms that are, on average, at the same distance from the TMA ion.43 We have calculated the density profile of water oxygen atoms and hydrogen atoms around a TMA-like molecule represented by four Me sites in a tetrahedral geometry covalently bonded to a central nitrogen site with a C-N bondlength of 1.47 A.47Since the nitrogen atom in TMA is not exposed to the aqueous environment, our Me-0 pair and Me-Me-0 triplet correlation functions can be used with the proximity approx- imations to generate the water oxygen atom and hydrogen atom density profiles sur- rounding this solute.The water oxygen atom density profile calculated using the two-site proximity approximation is compared with the TMA-water oxygen atom pair correlation function obtained from neutron diffraction measurements in Fig. 1. The posi- tions of the two peaks in this density profile at ca. 4.7 and 7.5 8, are reproduced with reasonable accuracy, although the height of the first peak is underestimated. The higher first peak observed in the scattering experiments may be attributed, at least in part, to the polar nature of the TMA ion. Undoubtedly, another contributing factor is the high statistical uncertainties in the Me-Me-0 triplet correlation functions.The same two peaks in the water hydrogen atom density profile, which is shown in Fig. 1, are at essentially the same positions, indicating the preferred water orientations consistent with those inferred from the neutron diffraction results. Thus, the two-site proximity approx- imation appears to have indeed captured characteristic features of the water organiz- ation around this solute. Hydration entropies of simple non-polar molecules Translational entropy contributions to the standard entropy of hydration of simple alkanes and benzene have been calculated using the two-site proximity approximation for water-oxygen-atom densities around these solutes. The corresponding orientational entropy contribution has been calculated from the average number of water molecules in the first hydration shell, obtained by integrating these profiles up to a site-oxygen distance of 5.2 A, and using a value of 0.35 cal mol-' K-' for the orientational entropy per water molecule, as discussed above.These contributions are given in Table 1. Stan- dard entropies of hydration, calculated from eqn. (9), are also presented. The infinite- 0 12 3 4 5 6 7 8 9 10 11 r/A Fig. 1 TMA-water oxygen-atom pair correlation function (solid line), TMA-water hydrogen-atom pair correlation function (dashed line) calculated using two-site proximity approximation. The neutron diffraction result for TMA-water-oxygen-atom pair correlation function (points) is also shown.43 (Pair correlation functions are defined with respect to the centre of the TMA molecule.) S.Garde et al. Table 1 Translational and orientational contributions to the pair corre-lation entropies of hydration calculated using eqn. (9) AS: translational orientational solute entropy entropyb eqn. (9) experimental' ethane -15.7 -7.7 -18.0 -20.1 propane -19.9 -8.9 -21.7 -22.8 n-butaned -23.7 -9.1 -23.3 -26.0 benzene -26.6 -10.9 -28.8 -21.2 'The third term in eqn. (9). The last term eqn. (9). Ref. 50. n-butane is assumed to be a 56/44% translgauche mixture.34 Experimental standard entropies of hydration are also reported in the units of cal mol-' K-I. dilution partial molar volume of the solute required in these calculations was obtained from the water-oxygen-atom density profiles using the well-known Kirkwood-Buff inte- gral.5*48 This calculation is sensitive to the long-range behaviour of the density profile, and therefore, to the truncation of the Kirkwood-Buff integral.When this integral was truncated at ca. 10 A,the calculated and measured values of fiz were found to be in very good agreement for those cases in which experimental values of 172 were available.49 Therefore, all infinite-dilution partial molar volumes were calculated by integrating water-oxygen-atom density profiles up to ca. 10 A. The calculated standard entropies in Table 1 are large and negative, with the trans- lational entropy contribution approximately twice that of the orientational entropy con- tribution in all cases.These entropies are in very good agreement with experimental valuess0 for the alkanes, but a value significantly more negative than the experimental value is obtained for benzene. Note, however, that the description of benzene as six overlapping Me sites is a severe approximation for this molecule which neglects its complex electronic structure. Standard entropies of hydration for the normal alkanes have also been calculated from eqn. (9) using the water structure obtained directly from explicit simulations of these solutes in TIP4P water.6 The translational and orientation- a1 entropies from this study are compared in Table 2 with those values based on the two-site proximity approximation.The agreement in each case is very good, and indi- cates that the proximity approximation, in which water organization is assumed to be only locally sensitive to structural details of the solute, provides an accurate description of water structure around simple, non-polar molecules as reflected in their entropies of hydration. Methanemethane PMF in water We revisit here the simplest model system demonstrating hydrophobic interactions between two non-polar molecules in The solvent contribution to the Me-Me Table 2 Comparison of translational and orientational contributions to the pair correlation entropies of hydration calculated using eqn. (9) and from explicit simulations6 (units of cal mol- K-') eqn-(9) Ashbaugh and Paulaitis6 solute translational orientational translational orientational ethane -15.7 -7.7 -15.8 -7.8 propane -19.9 -8.9 -19.6 -9.4 n-bu tane -23.7 -9.1 -24.2 -11.2 Hydrophobic interactions 1 1 1.5 1 -1.51 ' ' ' ' ' ' ' ' ' ' ' I 0123456789101112 r/A Fig.2 (a) Solvent contribution to the Me-Me potential of mean force calculated using the two- site proximity approximation (solid line); the KSA (dashed line); and the direct entropic contribu- tion, -TAS, calculated using truncated entropy expansion (points). (b) Me-Me potential of mean force calculated using the two-site proximity approximation (solid line) and the KSA (dashed line). PMF in water, calculated using HEP, is shown in Fig. 2(a). In this case, the inhomoge- neous water density profile given by the two-site proximity approximation is exact.The PMF obtained using the Kirkwood superposition approximation is also depicted. The total Me-Me PMF is given in Fig. 2(b). A deep contact minimum in the total PMF is observed at ca. 3.8 A and a shallow, solvent separated minimum is observed at ca. 7.1 A. The barrier to the contact minimum is slightly higher than that reported previously by Smith and Ha~met,~ whereas a slightly lower barrier is obtained using the Kirkwood superposition approximation. The entropic contribution to the Me-Me PMF is also given in Fig. 2(a). Statistical uncertainties in the Me-Me-0 triplet correlations are evident in the noise observed in these results. Comparable uncertainties are also obtained when these contributions are calculated by subtracting the internal energy from the Helmholtz energy.' Nevertheless, it is apparent that this contribution dominates the Helmholtz energy in the vicinity of the contact minimum, but is not significant near the solvent separated minimum.Similar observations have been reported previously.' S. Garde et al. 133 Conformational equilibria of n-butane and n-pentane Numerous studies of the conformational equilibrium of n-butane in water based on computer simulation^,^^*^'*'^*^^ integral equation theory" and more recently informa- tion theory'' have demonstrated the importance of hydration contributions in stabili- zing the more compact gauche conformation in water, although quantitative details of these studies are widely disparate.We have calculated here the Helmholtz energy of hydration using both eqn. (8) and (1l), and the corresponding entropy of hydration using eqn. (9), for different conformations of n-butane in water. In these calculations, butane is represented by four covalently bonded Me sites, and the two-site proximity approximation is applied to obtain water structure as a function of butane conformation [defined by the C( l)-C(2)-C(3)-C(4) torsional angle]. The two Helmholtz energy profiles, shown in Fig. 3, are essentially the same, with the more compact gauche conformation favoured relative to the trans conformation. The stabilization of gauche butane in water is also dominated by entropic contributions to the Helmholtz energy of hydration.Only one study of n-pentane conformations in water has been reported to date, and this study was restricted to calculating Helmholtz energy profiles along specific trajec- tories of the two torsional angles needed to define the conformation^.'^ We have calcu- lated instead the complete hydration Helmholtz energy surface for n-pentane conformations in water using both eqn. (8) and (11). Since the two Helmholtz energy surfaces were found to be nearly identical, we report only those results from the HEP calculations, which are given in Fig. *a). Again, the two-site proximity approximation is used to obtain water structure as a function of pentane conformation, and n-pentane is defined as five covalently bonded Me sites.The entropic contribution to the conforma- tional Helmholtz energy of hydration is given in Fig. qb). The single (global) Helmholtz energy minimum, observed in Fig. qu), corresponds to the most compact (cis-cis) conformation of n-pentane. Moreover, for a fixed value of the torsional angle, 02, the more compact conformation of n-pentane corresponding to 8,x 0" (cis) is locally favoured. Similar observations for selected 8,8, trajectories have been reported by B~ckner.'~ From Fig. qb), the entropic contributions to the Helm- holtz energy of hydration also favour the compact conformations of n-pentane in water. 0.20*310.1 Fig. 3 Solvent contribution to the Helmholtz energy of the conformational equilibrium of n-butane (0 = 180 corresponds to trans conformer) calculated using the HEP (solid line), and eqn.(8) (dashes). The butanewater interaction energy (diamonds) and entropic contribution -TAS(crosses) are also shown. Hydrophobic interact ions 0-180 Fig. 4 (a)Solvent contribution to the Helmholtz energy of hydration of n-pentane as a function of conformation, calculated using the Helmholtz energy perturbation method. (8, and 8, are the two torsional angles for which 8 = 180 corresponds to trans conformer.) (b) Direct entropic contribu- tion (-TAS) to the Helmholtz energy calculated using the truncated entropy expansion. Neopentane-neopentane PMF in water Finally, we study the association of two neopentane molecules in water along two differ- ent trajectories (Fig.5). In the first, the two molecules approach each other along a face-to-face trajectory in which three Me sites bonded to the tertiary carbon site on one molecule are brought into contact with its mirror image, but in a staggered orientation such that the face of one molecule is rotated 60" with respect to the other. In the second, the two neopentane molecules approach each other along a staggered face-to-toe trajec- tory. We expect the two-site proximity approximation to give an accurate description of water structure in the vicinity of these solutes when they are close enough to each another that water is excluded from the region between them. However, when interstitial waters are present, the two solute sites proximal to the interstitial water molecule will almost always belong to the same solute.Thus, important features of the PMF, such as the solvent separated minimum, would not be captured by the two-site approximation unless additional contributions are included in the determination of water structure to account for the mutual influence of both solutes. This problem does not arise in the S. Garde et al. Face-to-face trajectory Face-to-toe trajectory Fig. 5 Trajectories for neopentane-neopentane association ---2.5 -+ 0 -E -3-+ 8 -2 -+ <, L-2.5 - Hydrophobic interactions J2 .-_II0 0 e 0 0 0 0 0 0 -4 1 L 4 5 6 7 8 9 r/A Fig. 7 Entropic and energetic contributions to the Helmholtz energy of association of two neo-pentane molecules along the face-to-face trajectory; -TAS (points), solute-water interaction energy (dashed line), and Helmholtz energy using eqn.(8)(solid line) determination of the Me-Me PMF since, by definition, only two solute sites are involved, and the two-site proximity approximation gives an exact account of their mutual influence on water structure. Such effects for molecular solutes could be incorp-orated into our formalism for estimating water structure by identifying additional important contributions to the water-solute PMF due to presence of another molecular solute. To this end, we propose the following modification to the two-site proximity approx-imation when the two neopentane molecules are separated by more than one solvent diameter (centre-to-centre distance > 7 A), Wwater-solutes - W2*l)(rWTi,rj) + W2*')(r,,,I I Ti,rk) (12) where i and j are the two proximal sites.The first term in eqn. (12) corresponds to the two-site proximity approximation. If i andj are sites on the same molecule, then k is the nearest site on the second moIecule; whereas if i andj are sites on different molecules, k is the next nearest site. This simple modification thus takes into account the influence of the second solute on the water-solute PMF, and reduces to the two-site proximity approximation when two solutes are sufficiently far apart. Solvent contributions to the neopentane-neopentane PMF calculated using this modified two-site proximity approximation are given in Fig.6 for the two trajectories described above. The PMF along each trajectory was calculated using both eqn. (8) and (11). In these calculations, neopentane is represented by four Me sites tetrahedrally bonded to a central (tertiary) carbon site. The interaction of this tertiary carbon site with water is neglected in the HEP calculations. Also included in Fig. 6 are the PMFs for the same two trajectories obtained from explicit simulations of two neopentane molecules in TIP4P water.5 The neopentane molecules were simulated using OPLS parameter^.^' We note that the PMFs calculated from simulations have large statistical uncertainties. For the sake of comparison, the PMFs are set to zero at a separation of 10 A, although the simulation results show a large positive value for the PMF at this distance. Differ-ences in the potential parameters between HEP calculations and computer simulations make these comparisons qualitative in nature.We also note that the PMF for the face-to-face trajectory, calculated by HEP, exhibits a small barrier to the solvent-separated minimum at larger separations. S. Garde et al. For the face-to-face trajectory, the two Helmholtz energy profiles calculated using the proximity approximation reproduce the simulation results. Further, all three profiles have qualitative features characteristic of hydrophobic interactions in the Me-Me PMF ; namely, the deep contact minimum and the solvent separated minimum [Fig. qa)].In contrast, a solvent separated minimum is not apparent in any of the three profiles for the face-to-toe trajectory.In this case, the PMF calculated by HEP is in good agreement with the simulation results, whereas a much lower barrier to the contact minimum is obtained using eqn. (8). This discrepancy is most likely an artifact of the approximations made in calculating the orientational entropy contribution for the face-to-toe trajectory. Separate energy and entropy contributions to the PMF are shown in Fig. 7 for the face-to-face trajectory. It is evident from these results that the short-range part of the PMF is dominated by entropy, whereas favourable energetic contributions give rise to the solvent separated minimum. Similar calculations for the face-to-toe trajectory suggest that the lack of a solvent separated minimum in the PMF for this trajectory is due to less favourable energetic contributions.Conclusions Our calculations support the traditional ideas of hydrophobic hydration and hydropho- bic interactions. Namely, that the large negative entropies of hydrophobic hydration observed at room temperature can be reconciled on the basis of water organization in the vicinity of hydrophobic solutes, as characterized by the inhomogeneous one-particle water density profile around these solutes. Further, that hydrophobic interactions involve a partial reversal of the hydration process, and likewise are entropy driven at room temperature. These interactions favour the association of hydrophobic solutes and stabilize the more compact conformations of flexible hydrophobic molecules in water. We also observe a solvent separated minimum in the neopentane-neopentane PMF for the face-to-face trajectory when the simple two-site proximity approximation is modified to account for the mutual influence of both solutes on water structure.Moreover, a solvent separated minimum in the PMF is not seen for the face-to-toe trajectory. Both observations are in agreement with simulation results. Prediction of the solvent separated minimum as a distinguishing feature of hydro- phobic interactions has been proposed as a stringent test of models for hydrophobic phen~mena.~~Thus, the ability of the proximity approximation to capture these features in the case of methane-methane and face-to-face neopentane-neopentane association and to predict no solvent separated minimum for the face-to-toe neopentane-neopentane trajectory indicates that water structure in the vicinity of similar non-polar molecules can be reasonably estimated on the basis of limited information contained in a relatively small number of well-characterized solute-water correlation functions. We have also found that the calculation of water structure using the proximity approx- imation is ca.lo4 times faster than conventional approaches for calculating water struc- ture; e.g. by explicit molecular simulations of water. The simplicity and computational efficiency of the proximity approximations also encourages their further applications to the hydration of more complex, conformationally labile solutes.We have used two inherently different approaches to calculate Helmholtz energies: HEP [eqn. (1l)] and a correlation function expansion formalism [eqn. (S)]. The inhomo- geneous one-particle density of water surrounding the solute is only indirectly involved in the HEP, whereas it is central to the correlation function expansion for the entropy. Further, HEP is in principle exact in that it incorporates higher order correlations in the calculation of Helmholtz energies, although it is approximate in our implementation using eqn. (11). In contrast, practical application of the correlation function expansion is intrinsically approximate in the sense that it necessarily requires truncation at the level of pair correlations. 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ISSN:1359-6640    

DOI:10.1039/FD9960300125

出版商:RSC    

年代:1996

数据来源: RSC