Aspects of weak interactions Dudley H. Williams* and Martin S. Westwell† Cambridge Centre for Molecular Recognition University Chemical Laboratory Lensfield Road Cambridge UK CB2 1EW Weak interactions such as those non-covalent interactions that occur in biological systems are less well characterised than their strong covalent counterparts. Here we discuss associations between two or more molecules and consider the effect of interactions with solvent molecules (particularly water) and changes in the internal structure of the associating molecules on binding. We go on to discuss some of the progress that has been made in the estimation of binding constants. 1 Introduction In this article we use the definition that ‘weak interactions’ are those involving bonds which are comparable to thermal energies.In such systems equilibrium constants between two possible states can frequently be varied (often drastically) in the temperature range 0–100 °C. The study of weak interactions is a topic of great current interest because it is these non-covalent interactions which determine the stability for example of DNA duplexes of the folded states of proteins of enzyme–substrate complexes and of ligand–receptor interactions. A deeper understanding of weak interactions is highly desirable not simply because of the intellectual drive to understand the above systems but also because of the wish of the pharmaceutical industry to ‘rationally’ design new drugs. We largely discuss an approach developed in our laboratory which builds on the earlier work of others (particularly Jencks).1,2 We first discuss associations where in the ideal case the associating entities do not interact strongly with solvent nor do they change their structures significantly upon association.Second we go on to consider associations in polar media particularly water and where the associating molecules may modify their internal structures upon binding. Lastly we discuss some of the progress that has been made in the estimation of binding constants. † Current Address Dyson Perrins Laboratory University of Oxford South Parks Road Oxford UK OX1 3QR. Martin Westwell Dudley Williams Dudley Williams has bachelor and PhD degrees in chemistry and organic chemistry respectively from the University of Leeds in 1958 and 1961.He subsequently studied as post-doctoral fellow at Stanford University from 1961–64. Since 1964 he has worked at the University of Cambridge where he is a Fellow of Churchill College and Professor of Biological Chemistry. He was elected a Fellow of the Royal Society in 1983. 2 Model 1 Where A and B do not interact strongly with solvent nor do they extensively modify their structures upon association Perhaps the most commonly used expression for the representation of a reversible association is eqn. (1). (1) A + B"A·B This kind of formalism has arisen because it is very useful in considering the association of for example two water molecules to give a hydrogen bonded dimer.To a good approximation each water molecule (one designated A and the other B) retains the same internal structure in the dimer as in the dissociated state. Thus as a useful approximation the bonding between A and B in A·B can be represented as a property of the interface between A and B. The approximation works well because the bonds within A and B are strong compared to the bonding between A and B. In such circumstances we have previously argued3,4 that the restriction of motion which occurs when the association of eqn. (1) takes place (measured in terms TDS where T is the temperature 298 K and DS is the overall loss of entropy of translation and rotation) will be related to the exothermicity of the association (DH) by a curve of the general form shown in Fig.1. The direction of curvature arises because motional restriction reaches a limit of TDS Å 50 to 60 kJ mol21 for the immobilisation of a molecule of mass of ca. 100–300 at room temperature and this limiting entropy loss is approached much before covalent bond strengths are reached. The general shape of this curve has subsequently received theoretical support,5 and we reproduce here experimental data for associations occurring in methylene chloride solution (Fig. 2).4 As a good approximation the species involved in these associations do not interact strongly with the solvent nor do they modify their structures on association. Since DG = 2RTlnK = DH 2 TDS the direction of curvature satisfies the requirement that in these simple systems the equilibrium constant for association will increase with increasing exothermicity of association.More importantly the curve qualitatively emphasises how the adverse entropy of Martin Westwell was born in 1971 in Lancashire. He obtained his undergraduate and doctoral degrees from Churchill College Cambridge University. After a further year working in the Chemistry Department with Professor Williams Dr Westwell has moved to Oxford University where he is the Glaxo Junior Research Fellow in Biological or Medical Sciences at Lincoln College. 57 Chemical Society Reviews 1998 volume 27 Fig. 1 The general form of the extent of the exothermicity of association (DH°) A + B?A·B as a function of the entropic cost (DS°) at a temperature of 298 K.There is a limit in the price in entropy to be paid (due to loss of translational and rotational freedom) and this limit is approached before covalent bond strengths are approached. Fig. 2 Enthalpy (DH°) vs. entropy (298DS°) for the association of macrocycles with neutral molecules in dichloromethane. Data collected by Izatt et al.39 Very weak exothermic associations have favourable entropies due to desolvation [an effect which is absent for an idealised non-polar solvent (or gas phase) (Fig. 1)]. association increases gradually as a function of increased bonding of the associating entities i.e. how increased bonding gradually restricts the dynamic motion in A·B. 3 Problems in trying to obtain the free energy of binding of specific groups in networks of weak interactions Much effort has gone into attempts to estimate free energies of binding for common types of weak interactions [e.g.the hydrophobic effect (Å22) amide–amide hydrogen bonds and salt bridges] as they occur as parts of a network of weak interactions. To understand the problems inherent in such an Chemical Society Reviews 1998 volume 27 58 approach let us first consider binding in a non-polar solvent where there are no internal rotations to be restricted on binding. A favourable free energy contribution to binding (DGp) can occur for any pair of electrostatic interactions (e.g. amide– amide hydrogen bond formation) found in the binding site. It is possible to consider the sum of all such pairs of interactions (SDGp).These binding terms are opposed by the entropic cost of reducing the overall motion of the ligand when it binds to its receptor. This is described as DGt+r the adverse free energy change due to loss of entropy of translation and overall rotation. Thus if the observed free energy of binding is DGobs an attempted partitioning can consider the approximation eqn. (2). (2) DGobs = DGt+r + SDGp The reason for believing that eqn. (2) might serve as a useful approximation derives from a limiting case considered in 1981 by Jencks.2 Consider that two species X and Y can associate separately as at left or connected together by a strain free connection as at right into two distinct binding sites of a receptor (Scheme 1). In the ideal and limiting case X Y and X–Y all lose all their translational and (overall rather than internal) rotational entropy on binding to the receptor; this corresponds to the value of TDS Å 50 to 60 kJ mol21 for immobilisation referred to above.Thus in applying eqn. (2) to all three possible binding events (of X Y and X–Y to the receptor) the adverse value of DGt+r would be the same 50 to 60 kJ mol21 in all three cases. Scheme 1 Let us hypothetically (and unreasonably—see later) select the binding constants of X and Y as KX = KY = 103 dm3 mol21 and take the loss of entropy as the limit DGt+r Å 57 kJ mol21 (because each 5.7 kJ mol21 opposes binding by a factor of 10 at room temperature). The price in entropy only has to be paid once in the binding of X Y or X–Y.Applying eqn. (2) the intrinsic binding affinity (the limiting DGp value defined by Jencks2 as the binding affinity expressed when association occurs without adverse entropy) of both X and Y can be derived. When X (or Y) binds alone to the receptor we know that DGobs Å 217.1 kJ mol21 (K = 103 dm3 mol21) and DGt+r Å + 57.0 kJ mol21 such that eqn. (3) holds. DGp(X) = DGobs2D Gt+r = 217.1257 = 274.1 kJ mol21 (3) Now if we consider the binding of X–Y the lost entropy is again in the limit DGt+r Å + 57.0 kJ mol21 and the hypothetical DGobs from eqn. (2) is given by eqn. (4). DGobs(X–Y) = DGt + r + SDGp = D Gt+r = + DGp(X) + DGp(Y) (4) = + 57.0274.1274.1 = 291.2 kJ mol21 Thus if KX = KY = 103 dm3 mol21 then from the above analysis KX–Y Å 1016 dm3 mol21.Although this example illustrates the principle of an extreme case it is also clear that it involves an assumption which is physically completely unrealistic. Any association with K = 103 dm3 mol21 would not occur with complete loss of translational and rotational entropy but rather with only partial loss of this entropy. The contribution of Fig. 1 towards an understanding of the problem is that a semi-quantitated form would allow a crude guideline to the entropic cost (due to loss of translational and rotational entropy) as a function of the strength of the electrostatic interaction formed. That is for a gas phase association the horizontal axis of Fig. 1 is the DGt+r term and the vertical axis is the DGp (or SDGp) term (although not the limiting DGp term as defined by Jencks).We see how in qualitative terms the term opposing association (DGt+r) by restricting motion is played off against the term promoting association [DGp (or SDGp)] by favourable bonding interactions. The gas phase description is carried over to the case of association in non-polar solvents as a useful approximation. Since even a non-polar solvent will always interact finitely with the associating species that are dissolved in it the nature of the approximation is exposed by the fact that the plot does not pass through the origin in Fig. 2. Where X and Y associate with very low exothermicity the adverse DGt+r term for the association is very small and is more than offset by the favourable entropy of release of solvent.The curve therefore tails into the TDS > 0 of Fig. 2 very weakly exothermic associations occurring with a net favourable entropy. Let us now consider some physically plausible cases in terms of the entropy–enthalpy compensation curve. In the situation we consider first (solid lines in Fig. 3) the simplifying assumption is made that the exothermicity of the association of X–Y with the receptor is simply the sum of the exothermicities when X and Y bind separately (DHX–Y = DHX + DHY). Suppose X alone binds with an exothermicity of 20 kJ mol21 and Y with an exothermicity of 50 kJ mol21. On the basis of Fig. 3 the respective adverse TDSt+r terms would be approximately 20 and 38 kJ mol21 giving rise to DGX and DGY of 0 and 12 kJ mol21 (KX = 1 and KY = 1.3 3 102 dm3 mol21 at 298 K).Taking the exothermicity of binding of X–Y as 70 kJ mol21 from Fig. 3 the cost in TDSt+r is 46 kJ mol21 giving DGX–Y as 24 kJ mol21 (KX–Y = 1.6 3 104 dm3 mol21). Thus in this case of X binding with a relatively small exothermicity and Y binding with a moderate exothermicity (case 1 Table 1) the enhancement of binding constant of X–Y relative to its separate components (expressed as KX–Y/KXKY) is ca. 102. In contrast if the exothermicities of binding of X and Y are respectively 50 and 70 kJ mol21 then the respective TDSt+r values for X and Y are estimated as 38 and 46 kJ mol21 giving rise to DG DG X and Y of 12 and 24 kJ mol21 (KX = 1.3 3 102 and KY = 1.6 3 104 dm3 mol21 at 298 K); X–Y binds with an exothermicity of 120 kJ mol21 and an adverse TDSt+r value of 50 kJ mol21 (TDSt+r is approaching its limit) giving DGX–Y as 70 kJ mol21 (KX–Y = 1.9 3 1012 dm3 mol21).In this case of X binding with Fig. 3 Entropy–enthalpy compensation curve illustrating the enthalpy and related entropy values for the association of X and Y with a substrate where DHX = 20 kJ mol21 DHY = 50 kJ mol21 DHX–Y = 70 kJ mol21 (solid lines). When X and Y are tethered together the binding of X will enhance the binding of Y and vice versa. This is represented by the dashed lines. a moderate exothermicity and Y binding with a larger exothermicity (case 3) the enhancement of binding constant of X–Y relative to its separate components (expressed as KX–Y/ KXKY) is ca.9 3 105 dm3 mol21. The most important of these data along with the analysis of one where the X and Y exothermicities are respectively small and large (case 2) are summarised in Table 1. Whatever a more precise form of Fig. 3 may be given only the generality that more exothermic interactions approach a limiting cost in entropy it is seen that this expression of cooperativity (the classical chelate effect) is greatest where each of the associations of X and Y are quite strongly exothermic. (It is for this reason that the hypothesis KX = KY = 103 dm3 mol21 made at the beginning of this section for the purposes of illustration of an intrinsic binding affinity is an unreasonable one if X and Y lose essentially all their entropy in binding to a receptor the binding into the receptor sites would have to be very strong and KX and KY would have to much greater than 103 dm3 mol21).Table 1 First approximation for estimating some hypothetical relative magnitudes of chelate effects as a function of exothermicities of association K K X–Y b Y b X a DHY a KX b Case DH KX–Y/ KXKY c 1 1 2 3 220 220 250 1.3 3 102 1.6 3 104 1.2 3 102 1 1.6 3 104 2.3 3 107 1.4 3 103 1.3 3 102 1.6 3 104 1.9 3 1012 9.0 3 105 250 270 270 a kJ mol21. b dm3 mol21. c mol dm23. We can in fact make a simple refinement of the situation considered thus far. The consequences of this refinement will be considered for case 1 in Table 1. For each of the X and Y interactions only a part of the total theoretical maximum translational and rotational entropy which could be lost is lost (cf.Fig. 3). This is because in a weak interaction the theoretical maximum bonding (which could be expressed at 0 K) is not expressed at room temperature due to the opposing entropic advantage of residual motion resulting in an average position in the enthalpic well corresponding to an exothermicity of 20 kJ mol21 when X binds alone and 50 kJ mol21 when Y binds alone. However if the restriction of motion is aided by a neighbouring exothermic interaction then the average position in the enthalpic well will correspond to some larger exothermicity. This is because the binding of the X part of X–Y is aided by the binding of the Y part of X–Y and vice versa.Thus in the general and real case it is to be expected that the exothermicity of binding of X–Y will be greater than the sum of the exothermicities with which X and Y separately bind. The effect of this second cooperative factor which is quite distinct from that described to give the numbers in Table 1 is illustrated by a hypothetical example in Fig. 3. The solid horizontal lines indicate the enthalpies of association of X Y and X–Y (20 50 and 70 kJ mol21 respectively) for case 1 (Table 1) analysed with the previous simple assumption that the exothermicity of association of X–Y is equal to the sum of X and Y. The dotted lines indicate the corresponding analysis allowing for the effect now being considered whereby X–Y binds more exothermically than the sum of its parts since in X–Y X helps to anchor Y and Y helps to anchor X.The increase in binding energy of X–Y due to this effect is DDH 2 298DDS (Fig. 3). Fig. 3 has been drawn to illustrate the physically likely consequence that a weakly exothermic interaction (of X) is likely to be strengthened more by the assistance of a stronger adjacent interaction (of Y) than vice versa. The importance of this last analysis is in a case where X and Y bind in a strain free manner as X–Y through a direct connection of X and Y. The enhancement of binding should occur not simply through a classical entropy driven chelate effect but also as a consequence of an improved enthalpy of binding. It is this latter consequence that precludes the derivation of free energies of binding which are character- 59 Chemical Society Reviews 1998 volume 27 istic of common functional groups even in relatively simple systems.The binding energy obtainable from any specified weak interaction will always be context dependent. In sections 5 and 6 we present experimental data to support this view. 4 Model 2 Where associations occur in water Having considered weak associations occurring where the interaction with solvent is relatively weak we now turn to the more complex situation of associations in water. Much effort has gone into attempts to estimate free energies of binding for common types of weak interactions [e.g. the hydrophobic effect (Å22),6–9 amide–amide hydrogen bonds10–16 and salt bridges10,17–19] as they occur as parts of a network of weak interactions in water as solvent.For convenience we may consider two kinds of models in this area. First those systems where it might be a useful approximation to consider that the associating entities A and B retain their structures in the associated state except insofar as there might be restrictions of internal rotations of A and/or B. Second where A and B do not retain their internal structures in the same form; we reserve considerations of these cases until section 6. In the former case eqn. (5) would form a simple extension of eqn. (2) for strain-free systems eqn. (5). DGobs = DGt+r +DGr +DGh + SDGp (5) Here the contributions which promote binding are the familiar term SDGp and the new term DGh the contribution from the hydrophobic effect which corresponds to the favourable free energy of binding arising from the removal of hydrocarbon surface area from water upon association.It is a term which is conveniently separated from other binding terms for two reasons. First at room temperature it is essentially purely an entropy term—favourable because when hydrocarbon surface area is removed from exposure to water the water structure becomes more disordered.20 Second its magnitude is proportional to the surface area of hydrocarbon removed from exposure to water.6–9 This surface area is frequently conveniently measured from modelling studies. The terms which oppose binding are the familiar DGt+r term and the new term DGr due to loss of entropy associated with the restriction of any internal rotations which may occur on binding.If this loss of t+r entropy is DSr then at 298 K DGr = 2298DSr. A problem in applying eqn. (5) will presumably derive from the same source as in applying eqn. (2). For example if we add a polar group into a binding site and then measure the change in DGobs we will in the general case not obtain a true DGp value because the addition of the polar group increases the adjacent interactions in a manner analogous to the arguments already presented for the case in non-polar solvents. It is true that the addition of the polar group will also increase the adverse DG term but the crux of Fig. 1 and 3 is that the benefit to binding of the extra bonding will outweigh the extra cost in entropy.The key conclusion is that deletions (including mutation studies on proteins) of groups which contribute to binding in strain-free systems will normally be expected to give DGp values which are too large. In the following section we present data for binding in water to support this conclusion. 5 Attempts to obtain individual group contributions to binding and cooperativity at an interface In the early 1990s we selected glycopeptide antibiotics of the vancomycin group in their binding of bacterial cell-wall peptide analogues as a vehicle to test the application of the approximation represented by eqn. (5). Since we now know in what way this approximation is likely to err what have we been able to learn from the studies? Since the study involved restriction of internal rotations within the bacterial cell-wall peptide analogues upon binding to the antibiotics we required guides to DGr.This parameter was taken to lie in the range 2–5 kJ mol21; the lower end of this Chemical Society Reviews 1998 volume 27 60 range came from the restriction of internal rotations for the formation of crystals from the liquid state,6,21 and the upper end from the restriction of internal rotations in the formation of small rings from linear hydrocarbons.22 The smaller values (2–3 kJ mol21) presumably reflect the larger residual motions in crystals compared to tight ring structures and therefore are probably more appropriate to the associations commonly found in biology. We were able to delete methyl groups and amide–amide hydrogen bonds in the antibiotic binding sites and so evaluate the apparent binding energies associated with these entities.Our first attempts carried out the partitioning incorrectly and gave amide–amide hydrogen bond strengths which were far too high (ca. 20 kJ mol21 in water).23,24 A more appropriate partitioning gave these bond strengths in the range 0–7 kJ mol21 and a hydrophobic effect of 0.20 kJ mol21 Å22 (of hydrocarbon buried from water on binding).6 Interestingly the values are in good accord with the apparent binding energies obtained by protein engineering experiments (1–8 kJ mol21 and a hydrophobic effect of 0.23 kJ mol21 Å22).7 Despite this agreement the arguments presented in sections 2 and 3 suggest that these values should be larger than the true local binding energies.We now present two of our most recent studies which support this conclusion. O H O– CH3 O– H3C N H3C O NHAc O H O N H3C CH3 H H3C H N H (i) We have examined binding of the ligands 1 to 4 into the binding site of vancomycin group antibiotics. As the network of interactions which increases along the series 1 to 4 is extended (to the left as displayed in Scheme 2) the strength of the hydrogen bond between the carboxylate oxygen atom of the ligands and the antibiotic NH (designated w2) gradually increases.25 Thus the carboxylate anion is bound more strongly into the pocket which receives it as it is aided in this binding by adjacent interactions which help to restrict the ligand motion.This result indicates that if amide–amide hydrogen bond free energies are inferred from free energies of binding in the series 1 ? 2 ? 3 then the derived hydrogen bond strengths will be too large for the addition of these hydrogen bonds to the network increases the strengths of adjacent interactions. (ii) We have also examined the ligand series 3 5 6 and 7. In every case it is observed that a change in the ligand of a Gly O H O H N H3C N H N-AcDA 2 N H O H H H H O H H O acetate 1 H3C O Di- N-AcKDADA 4 O– N H N-AcGDA 5 H3C N CH3 H O O H O N-AcDAG 6 H O H N CH3 O– N CH3 H O O O H N-AcDADA 3 H CH3 O– H CH3 O– O H N O– N H H O H N-AcGG 7 NHAc CH O H O 3 N N N O – w2 H O H CH O H 3 N H N Di- N-Ac-KDADA H O H N C N antibiotic binding pocket Scheme 2 to an Ala increases the strength of the hydrogen bond from carboxylate oxygen to w2 the antibiotic NH.26 Thus the apparent increased binding energies from the Gly ? Ala ‘mutations’ must reflect in part a contribution from the increase in strength of this adjacent hydrogen bond.This cooperative strengthening of adjacent interactions can account for the fact that the values cited earlier in this section for the hydrophobic effect (0.20 and 0.23 kJ mol21 Å22)7,21 are larger than the solvent transfer value9 (0.125 kJ mol21 Å22 which cannot benefit from the type of cooperativity described).While it is of course also possible that the conformational bias of the Alacontaining peptides (relative to Gly-containing) might improve binding site affinity and so contribute to the strengthening of the adjacent hydrogen bond the larger apparent value of the hydrophobic effect from the ‘binding site’ vs. ‘solvent transfer’ experiments is understandable from a common viewpoint the addition of binding affinity or the restriction of motion at one point in the binding site can improve the binding affinity in an adjacent site. In summary real systems will always have residual motion. The addition of an extra interaction will typically (in a strainfree system) reduce adjacent motions and improve the free energy of binding of adjacent groups.Therefore attempts to estimate the binding affinities of specified groups will in strain free systems tend to give values that are too large. This consequence has been exemplified for the hydrophobic effect above and also implies that the apparent amide–amide bond strengths in water of 0–8 kJ mol21 are probably benefiting from other cooperative interactions and are therefore likely to be over-estimates. This conclusion is consistent with recent calculations which suggest that the enthalpy of formation of peptide hydrogen bonds is close to zero.27 6 Model 3 Binding where A and B can adjust their structures on association In the light of the formalism of eqn. (1) it has perhaps been inevitable that many studies of weak interactions have sought the origins of experimental binding energies by an examination of the interactions of A and B with solvent and of the interface between these two entities.This approach will be invalid if A and B change their internal structures upon association. Therefore if one or more of the associating components is a folded polypeptide (essentially all biological receptors) or a polymer of DNA or RNA which is involved in duplex or folded structures then the binding energy of the two components cannot reliably be sought at the binding interface even after consideration of the energetics of desolvation of this interface. If we temporarily ignore the interactions with solvent then in such cases a more appropriate form of the equilibrium constant would be given by eqn.(6). (6) A + B"AA·BA Eqn. (6) recognises that once A and B have associated then typically they no longer exist. Rather they have been replaced by modified entities AA and BA. Crucially the formalism of eqn. (6) emphasises that the binding energy between the two entities that come together is not simply a property of the interface between them but also is dependent upon the modifications of the internal structures of A and B (A ? AA and B ? BA). Although it might be argued that so much is self-evident reference to the literature indicates that this is so for many authors but equally not so for many others. It is a common practice to rationalise the observed binding energy between A and B by examination of the interface between them and to ignore the consequences of the changes A?AA and/or B?BA.These changes may at one extreme take the form of obvious structural modifications but at the other extreme may in principle involve essentially no structural reorganisation but simply a ‘tightening’ (or a ‘loosening’) of the internal structure of A when it is modified to AA (or of B when it is modified to BA). Where these consequences can be considered it may be possible to make semi-quantitative adjustments for the reorganisation B?BA where B is a small substrate but so far as we are aware never for A ? AA where A is a large receptor. Calculations might attempt to account for the free energy change and even to include the effect of solvent but the forcefields currently in use are not sufficiently accurate to give reliable free energy changes for systems involving large receptors.‡ So the binding affinities of greatest interest cannot be readily understood in molecular terms.We present below a relatively simple example of this complication. 7 Cooperativity beyond the interface While some glycopeptide antibiotics show no measurable propensity to dimerise (e.g. teicoplanin) some do strongly (e.g. eremomycin).29 In general the antibiotics dimerise more strongly in the presence of bacterial cell-wall mucopeptide precursor analogues than in their absence.29 For example the dimer of the glycopeptide antibiotic eremomycin has Kdim = 3 3 106 dm3 mol21 in the absence of di-N-acetyl-Lys-d-Alad-Ala but Kdim = 3 3108 dm3 mol21 in its presence.It follows from a thermodynamic cycle that di-N-acetyl-Lys-d-Ala-d-Ala is bound by a factor of 10 more strongly by the dimer than by the monomer (each site of the dimer binds cell-wall analogue with the same affinity.30) The basis for at least part of this cooperativity can be seen from the structure of the ligand-bound dimer [Fig. 4(a)]. The antibiotics showing the largest dimerisation constants have in addition to the four hydrogen bonds at the dimer interface [heavy dashed lines in Fig. 4(a)] two additional hydrogen bonds. These two bonds are from the alkylammonium ions of the amino sugars (which are unique to the strongly dimerising antibiotics) to amide carbonyl groups in the other half of the dimer [Fig.4(b)]. The alkyl ammonium ions one at each end of the head-to-tail dimer are brought into the proximity of the cell-wall analogue carboxylate anion [one in each half of the dimer; Fig. 4(b)]. The resulting Coulombic attraction can tighten binding at both the ligand–antibiotic and dimer interfaces (Fig. 4). One way of looking at this cooperativity is that strong dimer formation also makes for the formation of a salt bridge which is mediated through an intervening amide bond [arrowed in Fig. 4(b)]. The evolution of this sophisticated interaction suggests that dimers may work more efficiently than monomers in antibacterial action and indeed this has been shown to be the case.31 However in the present context the relevant questions are (i) in the promotion of the dimerisation constant of eremomycin from Kdim = 3 3 106 to Kdim = 3 3 108 dm3 mol21 by ligand can we infer anything useful about the origin of the extra binding energy? ‡ The difference in binding affinities (DDG) of two closely related ligands to a common receptor of moderate size had been calculated with impressive accuracy,28 but this accuracy is only possible because relatively large systematic errors in the calculation are removed by difference.61 Chemical Society Reviews 1998 volume 27 Fig. 4 (a) The structure of the antibiotic dimer (backbone only) bound to ligand (N-acetyl-d-Ala-d-Ala). The hydrogen bonds at the dimer interface are represented as broad dashed lines and those in the ligand binding pocket as narrower dashed lines.(b) The strongly dimerising antibiotics have an amino sugar on residue 6 which has a Coulombic interaction with the carboxylate of the ligand. This is essentially a salt bridge mediated by the peptide bond between residues 2 and 3. (ii) similarly in the promotion of the binding of di-N-acetyl- Lys-d-Ala-d-Ala to the dimer over the monomer by a factor of 10 can we infer anything useful about the origin of the extra binding energy? These questions have not yet been addressed by experiment. Yet what seems to be a physically reasonable model representing dimer by 8 ligand-bound monomer by 9 and ligand-bound dimer by 10 gives plausible insights (in these diagrams the + signs represent the ammonium ions of the amino sugar and the 2 signs represent the negative charge of the carboxylate ion of the cell-wall analogue ligand).Since there is little doubt that at least part of the cooperativity is due to the Coulombic attraction between these two opposite charges the models 8 to 10 suggest the following. In the case of question (i) part of the extra stability of the dimer structure 10 over the dimer structure 8 lies in the stronger binding of the ligand in 10 relative to its binding in 9. That is when two of the entities shown in 9 come together to give one of 10 then compared to the formation of 8 from two antibiotic monomers part of the increased binding energy will come from Chemical Society Reviews 1998 volume 27 62 the fact that ligand is bound more tightly in 10 than 9.In other words in the increase of the dimerisation constant by a factor of 100 some of the favourable free energy should come from the ‘tightening’ of the ligand–antibiotic interface upon dimerisation. The Coulombic attractions which are unique to 10 (and indicated by double-headed arrows in this structure) can be expected to lead to strengthening of the weak interactions at all three interfaces present in 10. It is for this reason that we have schematically inferred bond-shortening at all three interfaces in 10 relative to 8 and 9. In the case of question (ii) the analysis is of course equally relevant in illustrating how the increased affinity of the ligand for the dimeric receptor (10) over the monomeric receptor (8) cannot be simply ascribed to the strengthening of the weak interactions at the ligand–receptor interface in 10 relative to 9.It will in part be also due to the strengthening of the weak interactions at the dimer interface. These models illustrate in a simple way the potential origins of binding affinity which is remote from the binding interface. In large systems this makes the prediction of binding constants more difficult because interactions remote from the binding site may significantly contribute to the overall binding constant. 8 Can binding affinities be reliably predicted? The extensive and subtle changes which can affect binding affinities as outlined in the preceding sections suggest that relatively accurate de novo predictions of equilibrium constants for associations of extended networks are not likely to be achieved in the general case in the near future.Despite the problems in the search for a solution that is universally valid the use of eqn. (5)21,32 has recently been extended by Bohm and with a surprisingly good outcome for a limited data set.33 In this work the four terms (DGt+r + DGr + DGh + SDGp) of eqn. (5) are used but the last of these (for binding electrostatic attractions) is sub-divided into two terms—one for formally uncharged hydrogen bonds (DGhb) and a second for a polar interaction involving a charged entity (DGionic). A set of 45 interactions of experimentally known binding constants was then considered where ligands of relatively small molecular mass (66 to 1047) interact with proteins.Since the modified form of eqn. (5) has only five unknowns and the 45 interactions involve different combinations of these unknowns average values for the five parameters can be obtained. From this training set the average contribution from a neutral hydrogen bond of ideal geometry was 24.7 kJ mol21 from the corresponding ionic interaction was 28.3 kJ mol21 from the hydrophobic effect was 20.17 kJ mol21 Å22 and from the restriction of a rotatable bond was +1.4 kJ mol21. The average opposition to binding from DGt+r was +5.4 kJ mol21. This energy function reproduced the binding constants of the training set (which experimentally range from 40 to 2.5 3 1013 dm3 mol21) with a standard deviation of 7.9 kJ mol21 corresponding to 1.4 orders of magnitude of binding affinity.These results are approximations which may be very useful and merit some comment. First the values obtained for the polar interactions are in good agreement with those obtained by other approaches,11–13,16 although for the reasons presented earlier these should perhaps best be regarded as apparent binding energies12,34,35 rather than as localised bond energies. Second, the value for the hydrophobic effect is in reasonable accord with that from other work being intermediate between that from solvent transfer experiments (20.12 kJ mol21 Å22)9 and from the deletion of methyl groups in binding sites (20.20 to 20.23 kJ mol21 Å22).7,21 Third the cost of restricting a rotatable bond is somewhat less than that found for the melting of crystals (2 to 3 kJ mol21),6 and may reflect the fact that rotations are somewhat less restricted in these binding sites than they are in crystals.Fourth the average value of DGt+r of +5.4 kJ mol21 is remarkably small. It represents only about one tenth of the maximum theoretical entropy loss (corresponding to complete immobilisation of the ligand). If it is assumed that this estimate is realistic (Bohm notes that this parameter is the most uncertain of all those derived) then it may reflect small average exothermicities of association of these ligands to their protein receptors in water—a suggestion which seems physically plausible. Last it is noteworthy that one of the largest errors in the estimated binding constants is found for the streptavidin– biotin interaction (Kcalc = 1010.75 vs.Kexp = 1013.4 dm3 mol21). In a sense this is remarkably good agreement for such high affinity binding when computed by a simple method. But it should be noted that this measure of agreement was only obtained by regarding the ureido group of biotin as a charged entity (following the suggestion of Weber et al.),36 despite the fact that it is formally uncharged. This is perhaps an attempt to rationalise the origins of binding energy at an interface where in fact the origins are probably much more complex in this case.37 Despite such problems the application of eqn. (5) in the approach of Bohm gives impressively accurate predictions overall and as a pragmatic and simple approach it has much to commend it.It will be of great interest to see how it performs with a wider set of associations or possibly also after further refinement. Another promising approach to predict the binding affinity of novel ligands for receptors of known three-dimensional structure is currently being explored.38 This method known as VALIDATE gives an absolute average error of only 1.45 log units between experimental and computed binding constants for 11 thermolysin inhibitors which were not part of the training set (though other thermolysin–ligand interactions were in the training set). Additionally for a set of 14 inhibitors where neither the ligands nor the specific receptors were included in the training set the technique gave estimated binding constants that had an absolute average error of only 0.68 log units relative to the experimental values.Both of the above semi-empirical methods may lack an accurate physical description of binding processes but their importance lies in the reasonable success of their predictions. Conclusions In studying weak interactions we have considered the play-off or compensation between electrostatic bonding and the restriction in motion. These two terms form the basis of our attempt for gas phase interactions and those occurring in non-polar solvents to factorise the observed free energy of binding into its component parts. For associations in water we have developed an approach where the hydrophobic effect and internal rotations in the associating molecules are taken into account such that some prediction of binding constants and their component parts can be made.Further we note that in biological systems cooperativity both at the binding interface and remote from it can contribute significantly to the observed free energy; see also ref. 37. That is the sum of the isolated component parts is less than the whole observed binding energy. Because the entropic cost of an association is variable this complication cannot be removed by simply trying to factor out the entropic term. Motion and bonding are intrinsically linked. Despite these complications current semi-empirical approaches for the estimation of binding affinities are giving promising results. Weak interactions may still be ill understood but they are giving up their secrets.10 Acknowledgements We thank the EPSRC BBSRC and Glaxo Research and Development for financial support. Lilly Laboratories Indianapolis are thanked for the exchange of information and useful discussions regarding the vancomycin group antibiotics. 11 References 1 W. P. Jencks Adv. Enzymol. 1975 43 219. 2 W. P. Jencks Proc. Natl. Acad. Sci. USA 1981 78 4046. 3 M. S. Searle M. S. Westwell and D. H. Williams J. Chem. Soc. Perkin Trans. 2 1995 141. 4 M. S. Westwell J. Klein and D. H. Williams J. Phys. Chem. 1996 100 16000. 5 J. D. Dunitz Chem. Biol. 1995 2 709. 6 M. S. Searle and D. H. Williams J. Am. Chem. Soc. 1992 114 7 L. Serrano J.-L. Neira J. Sancho and A. R. Fersht Nature 1992 356 8 K. A.Sharp A. Nicholls R. Friedman and B. Honig Biochemistry 10690. 453. 1991 30 9686. 9 R. L. Baldwin Proc. Natl. Acad. Sci. USA 1986 82 8069. 10 A. Fersht Enzyme Structure and Mechanism W H Freeman and Company New York 2nd edn. 1985. 11 A. R. Fersht Trends Biochem. Sci. 1987 12 301. 12 A. R. Fersht J.-P. Shi J. Knill-Jones D. M. Lowe A. J. Wilkinson D. M. Blow P. Brick P. Carter M. M. Y. Waye and G. Winter Nature 1985 314 235. 13 B. A. Shirley P. Stanssens U. Hahn and C. N. Pace Biochemistry 1992 31 725. 14 R. L. Baldwin Trends Biochem. Sci. 1989 14 291. 15 K. A. Dill Biochemistry 1990 29 7133. 16 D. H. Williams M. S. Searle M. S. Westwell U. Gerhard and S. E. Holroyd Phil. Trans. R. Soc. London A 1993 345 11. 17 A. R. Fersht J. Mol. Biol.1972 64 497. 18 M. F. Perutz Nature 1970 228 726. 19 A. Bierzynski P. S. Kim and R. L. Baldwin Proc. Natl. Acad. Sci. USA 1982 79 2470. 20 N. Muller Acc. Chem. Res. 1990 23 23. 21 M. S. Searle D. H. Williams and U. Gerhard J. Am. Chem. Soc. 1992 114 10697. 22 M. I. Page and W. P. Jencks Proc. Natl. Acad. Sci. USA 1971 68 1678. 23 D. H. Williams Aldrichim. Acta 1991 24 71. 24 A. J. Doig and D. H. Williams J. Am. Chem. Soc. 1992 114 338. 25 P. Groves M. S. Searle M. S. Westwell and D. H. Williams J. Chem. Soc. Chem. Commun. 1994 1519. 26 G. J. Sharman M. S. Searle B. Benhamu P. Groves and D. H. Williams Angew. Chem. 1995 34 1483. 27 M. J. Sippl J. Mol. Biol. 1996 260 664. 28 W. C. Still and R. Liu Phil. Trans. R. Soc. London A 1993 345 97. 29 J. P. Mackay U. Gerhard D. A. Beauregard M. S. Westwell M. S. Searle and D. H. Williams J. Am. Chem. Soc. 1994 116 4581. 30 G. Batta M. F. Cristofaro G. J. Sharman and D. H. Williams Chem. Commun. 1996 101. 31 D. A. Beauregard D. H. Williams M. N. Gwynn and D. J. Knowles Antimicrob. Agents Chemother. 1995 39 781. 32 P. R. Andrews D. J. Craik and J. L. Martin J. Med. Chem. 1984 27 1648. 33 H.-J. Bohm J. Comp.-Aided Mol. Design 1994 8 243. 34 A. E. Mark and W. F. van Gunsteren J. Mol. Biol. 1994 240 167. 35 D. H. Williams M. S. Searle M. S. Westwell J. P. Mackay P. Groves and D. A. Beauregard Chemtracts 1994 7 133. 36 P. C. Weber J. J. Wendoloski M. W. Pantoliano and F. R. Salemme J. Am. Chem. Soc. 1992 114 3197. 37 D. H. Williams and M. S. Westwell Chem. Biol. 1996 3 695. 38 R. D. Head M. L. Smythe T. I. Oprea C. L. Waller S. M. Green and G. R. Marshall J. Am. Chem. Soc. 1996 118 3959. 39 R. M. Izatt J. S. Bradshaw K. Pawlak R. L. Bruening and B. J. Tarbet Chem. Rev. 1992 92 1261. Received 10th July 1997 Accepted 15th September 1997 63 Chemical Society Reviews 1998 volume 27