Experimental and Theoretical Aspects of Hydration Isothermsfor BiomoleculesB Y PETER R. C. GASCOYNE AND RONALD PETHIG"School of Electronic Engineering Science, University College of North Wales, Bangor,Gwynedd, WalesReceived 5th May, 1976A resonating quartz crystal microbalance technique has been used to obtain room temperaturewater sorption isotherms for cytochrome-c, DNA, lecithin, lysozyme and serum albumin. Theresults compare favowably with earlier work using more conventional techniques. A completelygeneral formula describing the sorption isotherms is derived. With the assumption of identical andnon-interacting primary sorption sites, this general formula gives exact values for the monolayersite capacity and the thermodynamic activities of all the hydration states.However, it is shown thatat least one of these assumptions is not valid for the materials studied here, accordingly only limitedinformation can be derived using this theory. This restricted usefulness also applies to other sorptiontheories described in the literature, most of which are based on purely modelistic or kinetic con-siderations. The significance of a sorption activity parameter having values greater or less thanunity is discussed for the materials examined.The experimental and theoretical aspects of protein hydration have receivedconsiderable attention, and are the subject of recent reviews.l* A recent advance inthe measurement technique for obtaining sorption isotherms of biological materialsis the use of a resonating quartz crystal mi~robalance.~~ This technique has beenused for the studies described here and confirms the earlier conclusions 39 regardingits advantages over more conventional methods.Numerous theoretical treatmentshave been proposed to describe the hydration processes of proteins. Many of thesetheories can, however, be considered to be based on models of doubtful physicaljustification. Guggenheim has shown that the original isotherm formulae ofLangmuir,6 Freundlich,' and Brunauer, Emmett and Teller (B.E.T.),8 which arebased entirely on kinetic considerations, may be derived from general statisticalmechanical treatments using Grand Partition Functions. The use of such fuiictionsis extended here to derive a completely general equation describing sorption isotherms,and the applicability and usefulness of this treatment, and of others described in theliterature, is discussed.EXPERIMENTALThe following materials were used: Bovine Serum Albumin (Fraction V), SigmaChemical Co. ; Cytochrome-c (Horse heart), Type VI, Sigma Chemical Co.; L-a-Lecithin(egg yolk), Sigma Chemical Co. ; Lysozyme (egg white), Sigma Chemical Co. ; NaDNA (calfthymus), BDH.A thin film (approximate area 20 mm2 x 25 pm thick) of the material under investigationwas applied to the centre of one side of a fundamental AT-cut 2 MHz quartz crystal oscillator(Cathodeon Crystals). For such a crystal of mass m, the addition of a small mass Am willcause the resonant frequency f to change by an amount Af according to the relationshipA f / f = - K h / m (1 )17172 HYDRATION ISOTHERMS FOR BIOMOLECULESwhere K is a mechanical constant for the crystal and normally has a value very close to unity.A full discussion of the validity of eqn (1) has been given recently by P ~ l k e r .~The test crystal was enclosed inside a vacuum cell within a constant temperature environ-ment, and connected to a frequency comparator circuit as outlined in fig. 1. This electricalcircuit provided the necessary conditions for the crystal to resonate in its fundamental mode.The resulting oscillating signal was mixed with that from a standard 2 MHz oscillator andthe resultant beat frequency detected. This beat frequency was measured to within 0.1 Hzusing a digital frequency counter (Venner Type TSA 6636/2M).If the crystal with no filmattached has a resonant frequency of exactly 2 MHz, then this beat frequency of the loadedcrystal corresponds to Af of eqn (1) with Am being the test film mass. In practice, exactmatching of the standard 2 MHz oscillator with clean 2 MHz crystals was not possible, anda small correction was required to account for this.rnanom t prFIG. 1 .-Outline of the experimental arrangement used for determining the water sorption isotherms.The water vapour used in the hydration studies was provided from a water reservoircontaining triply distilled de-ionised water which had been degassed by redistilling andfreezing under vacuum. The water vapour pressure was determined usicg a calibratedsilicon-oil manometer allowing an accuracy of better than 13 Pa.Before a hydration runwas commenced, the system was evacuated to about Pa and the partial pressure ofwater was further reduced using liquid nitrogen and P205 traps. Temperatures to within40.15 K were determined using a thermocouple, and temperature stability to within 0.5 Kwas achieved by immersing the vacuum cell in an oil bath. The vacuum cell accommodatedfour crystals which could be independently switched into circuit, so allowing for simultaneousobservations of different test materials.It follows from eqn (1) that for a small mass increase am associated with a uniformadsorption of water molecules on the test film, the corresponding change af in the crystalfrequency is given bywhich is independent of any crystal constants.Apart from lecithin, which was suppliedin hexane solution, the biological materials were dissolved in doubly distilled de-ionisedwater and applied to the crystal surface using a clean nylon brush. The mass of the depositedfilm, when dry, was such that on average the crystal resonant frequency decreased by about3 kHz, and was typically reduced further by about 1 kHz as a result of water adsorbed atpartial pressures of the order of 0.8. Thus the maximuni change in crystal resonant frequencydue to the attached hydrated sample was about 4 kHz, well within the accepted range ofvalidity of eqn (1).Effects associated with mechanical coupling to the ambient atmosphere, and with theadsorption of water on the quartz crystal surfaces themselves, were found to be negligible,corresponding at most to a 6 Hz shift at partial pressures of the order 0.8.Eqn (2) is validd f l A f = dm/Am (2P . R . C . GASCOYNE AND R . PETHIG 173so long as the mechanical coupling between the test film and the crystal remains constant.At the higher relative humidities (- 0.9) the DNA samples became gelatinous and it wasconsidered doubtful that the mechanical coupling had remained unchanged compared withthe dry and moderately hydrated condition. For this reason the sorption isothermsdescribed here were restricted to an upper limit of relative humidity of 0.9. With therelatively unsophisticated experimental arrangement described here, these sorption isothermswere determined to resolutions of 5 parts in lo5 change in mass and 4 x partial pressure,which compares very favourably with other more conventional techniques.THEORETICALFollowing the treatments of Hill l o and Guggenheim,' the Grand Partitionfunction for Ns identical primary sorption sites may be writtenwhere x is the activity of the sorbed gas which may, without loss of generality, beidentified with the relative pressure of the gas.The coefficients ni refer to the activityof sorbed molecules in the ith layer. The total sorption O(x) will be the sum of theoccupations of all layers of sites, when, writing the partition function for a singlesite as.f(x) = F(x)'JNSwe have, as shown by Dole,ll thatF(x) = (1 +alx+ala2x2 +a1a2a3x3 + . . . )Ns (3)Dividing both sides of this equation by x and integrating, and noting that for anypartition function describing sorptionf(0) = 1, we may write 1; F) dx = In f ( x ' ) .Now O(x) js the ratio of the total sorption, v(x), to the monolayer sorption capacityof the primary sites, u,, hence finallyThis equation, which to our knowledge has not previously been derived, may bzwritten in the alternative formd( 5 ) v(x) = v,x - In f ( x ) .axSubstitution for f (x) in eqn (5) allows the corresponding sorption isotherm toFor example, setting the activities of sorption sites in the second andSubstitutionbe established.subsequent layers to zero leads to the partition functionf(x) = 1 +ax.into eqn ( 5 ) yields the isotherm equationv,ax1+axu(x) = ___.This is the well known Langmuir equation for monolayer adsorption.An interesting case, previously analysed by Anderson,12 Hill l3 and Halsey,I4occurs when the activities of the second and subsequent layers are equal.Theresulting partition function is a geometrical progression(7) f(x) = 1 + abx+a(bxI2 + a ( b ~ ) ~ + . . 174 HYDRATION ISOTHERMS FOR BIOMOLECULESwhere (ab) is the activity of gas sorbed in the first layer, and b is the activity in allsubsequent layers. Substitution of f(x) in eqn (5) yields the isotherm equationv,abx(1 - bx)[l +(a- b)x]'v(x) =In the special case of the activity in the second and subsequent sorption layers beingequal to that of the bulk condensed gas, b becomes unity. In this case eqn (8) reducesto the well known and much used B.E.T. isotherm equation *av,x(1 - x)[ 1 + ( a - 1)xl'v(x) = (9)Eqn (4) allows the partition function for the sorption sites to be determined directlyfrom sorption isotherm data.The treatment is completely general and, unlike othertheoretical treatments, involves no assumptions regarding the form of f(x). Byusing, for example, an orthogonal Chebyshev polynomial curve fitting technique theactivities of all sorption layers may be obtained directly from the partition function.To complete the analysis, however, it will be noted that a value for vm is required ineqn (4). It is clear that any partition function for multilayer sorption will approachthe form f(x) = 1 +ax at sufficiently small values of x. All sorption isotherms forthe case of identical primary sites will, therefore, be described by eqn (6) when x isvery small.A plot of x/u(x) against x will have a gradient 1 /v, at very low hydrations.The factor vm can, therefore, without loss of generality be defined fromprovided a 9 1.Eqn (4) and (5) apply only to sorption isotherms of materials whose primarysorption sites are identical. It may readily be demonstrated that an extended formof eqn (4) which describes sorption by materials with N different types of primarysites isN n [fj(x')]um~ = exp {Jr dx) j = 1where vmj are the primary site capacities of the N different types of sites whoserespective partition functions are fj(x).Hydration isotherms for materials such as the naturally occurring polypeptidesare described by eqn (1 l), since several different types of water sorption sites will bepresent.Unfortunately, this equation is insoluble unless information regarding thesorption sites is available. On the other hand it is useful to note that the left handside of eqn (11) may be approximated to the form of eqn (7). It must be stressed,however, that because the form of the partition function is assumed, an analysis interms of eqn (7) represents only an approximation to the true isotherm equation.For this reason the value obtained for v, will be an effective value and unlikely tobe identical with vmj, the true primary site monolayer capacity. The valueobtained for the activity of the sites will also be no more than an effective value.However, it can be seen from eqn (1 1) that if members of the set of functions fj(x)are sufficiently similar, the treatment may yield reasonable approximations to thetrue values.Nj = P.R. C . GASCOYNE AND R. PETHIG 175RESULTS AND DISCUSSIONA typical water sorption isotherm for the materials studied is that of BovineSerum Albumin (BSA) illustrated in fig. 2.Mpartial pressure, x0, This work (296.5 K).(298 K).FIG. 2.-Water sorption isotherm for BSA. @, Isotherm data of Bull l 6To analyse the experimental data it was found convenient to write eqn (8) in theformwhereA = (v,ab)-l; B = a-2b and C = b(a-b). (1 3)A least squares computer routine was used to fit eqn (12) to the experimenta1sorption isotherms obtained for the various biological materials. The values of v,,u and b are given in table 1, together with results obtained from conventional B.E.T.analyses of the isotherms.It can be seen that the results obtained here for BSA,DNA and lysozyme using the crystal microbalance technique give B.E.T.-derivedvalues for v, and (ab) in good agreement with earlier work using more conventionalsorption techniques. We cannot explain the disparity between our B.E.T. values forcytochrome-c and lecithin and those derived from the isotherms given in ref. (17),but wish to make the comment that for cytochrome-c at least, close inspection of theisotherm of ref. (17) reveals deviations from the characteristic shape commonly foundfor proteins. Our use of lecithin prepared from hexane solution may account forthe observed differences for this material.For all the freshly prepared samples, ourresults consistently gave the same values for v, (k 0.1) and ab (& 0.2) for each material.There are two cases of interest regarding the activity parameter b. A value of bgreater than or equal to unity implies that the activity of sorption sites in the secondand subsequent sorbed layers is respectively greater than or equal to that of the sorbat176 HYDRATION ISOTHERMS FOR BIOMOLECULESin its pure bulk state. Eqn (8) implies that materials with such a property will attaininfinite hydration at a vapour pressure below p o (x less than unity). The onlymaterial investigated which yielded a value for b greater than unity was DNA, andit is interesting to note that the sample films of this material exhibited deliquescentproperties at a partial pressure of about 0.9.Conversely, a value of b less than unityTABLE VALUES OF THE PERCENTAGE HYDRATION urn AND ACTIVITIES (ab) FOR THE FIRSTFOR THE PARAMETER b REQUIRED TO GIVE THE BEST LINEAR PLOTS AS FOR FIG. 3 AND 4.VALUES DERIVED FROM THE CONVENTIONAL B.E.T. GRAPHICAL ANALYSIS ARE INCLUDED FORBOUND MONOLAYERS DERIVED FROM A COMPUTER FIT OF EQN (12), TOGETHER WITH THE VALUESCOMPARISONB.E.T. ( b = 1)- [eqn ( 1 311material Cln ( ~ b ) b Crn ( ~ b )BSA 7.87 9.63 0.81 6.5 12.26.7 12.56.7 11.3cytochrome-c 8.27 13.4 0.88 7.5 9.86.3 14.1"DNA 11.5 17.3 1.05 12.1 13.312.2* 13.6"12.5" 12.9*4.6 6.2*7.2* 1 8 9lecithin 7.18 3.79 0.88 5.2 8.51 y sozyme 8.15 10.5 0.82 7.4 14.9ref.this work15t16this work17this work1718this work17this work19t Bovine plasma albumin (BPA) ; * calculated from B.E.T.plots derived from isotherm datagiven in reference. Measurements for " this work " were made at 296.5 (kO.1) K.implies finite hydration at the saturated vapour pressure po. Such a value of b wa5obtained for all the materials investigated with the exception of DNA. This findingis in agreement with the observation that when the sample films of the proteins wereleft in a saturated water vapour atmosphere deliquescence did not occur.A consistent trend seen in the results derived from the computer fit of eqn (12)is that the values obtained for the monolayer hydration capacities exceed the B.E.T.-derived values when b is less than unity, whereas for DNA, with b greater than mity,the value of urn is reduced.Eqn (8) may be rearranged to allow convenient graphical representation of theexperimental results. Theresults obtained for BSA using the quartz crystal microbalance technique are showntogether with those of Bull obtained using a conventional weighing-bottle technique.A straight line fit to the data over the entire partial pressure range is obtained withan appropriate value for b found from the computer fit of eqn (12).The effect ofvarying the parameter b is clearly demonstrated in fig. 3. The plot for b = Icorresponds to the conventional B.E.T. plot, and the characteristic deviation from astraight line plot, as remarked upon by a number of workers,l* 2o occurs because ofthe assumption that water sorbed in the second and subsequent hydration layersbehaves as pure bulk water. Such an assumption leads to the prediction that morewater is sorbed than is measured experimentally for BSA.It can be seen that onlya single value for b results in a good linear plot of the experimental data, over thewhole partial pressure range, as can also be seen in fig. 4 for the results of DNA.Fig. 3 shows a plot of x/[u(x)( 1 /b -x>] against x for BSAP . R. C. GASCOYNE AND R. PETHIGi ’177partial pressure, xFIG. 3.-Plots of the function x / [ u ( x ) . (l/b-x)] against x for BSA, where x is the partial pressurep / p o , for various values of the parameter 0. 1, h = 1.0; 2, b = 0.81 ; 3, 0.67. 0 This workIsotherm data of Bull l 6 (298 K).(296.5 K).partial pressure, xFIG. 4.-Plots as for fig. 3 for DNA (296.5 I<). 1, b = 1.17 ; 2, b = 1.05 ; 3, b = 0.95.Eqn (4) represents the most general description of the partition function f(x) forbound water in all the multilayers, with Ihe restriction that all the primary monolayersorption sites are identical and non-interacting. Such a restriction is in fact thebasis for all the sorption theories reviewed by Kuntz and Kauzmann.l A test fo178 HYDRATION ISOTHERMS FOR BIOMOLECULESthe validity of this assumption was made by using a Chebyshev orthogonal curvefitting computer technique for the experimental isotherm data to check for theplausibility of the derived activity parameters. The results of such a curve fittingprocedure gave negative values for some of the activity coefficients, which can haveno physical meaning.Typically, consistent values were obtained for the activitiesof the primary sorption layers, negative values for the second layers and unreasonablylarge values for the third layers.CONCLUSIONSWe consider that the isotherm results obtained here confirm the earlier conclusionsof Kennerley regarding the useful applicability of the resonating quartz crystalmicrobalance technique for hydration studies. Even with a relatively unsophisticatedexperimental arrangement the sensitivity and speed that the method affords comparesvery favourably with the more conventional ‘‘ weighing bottle and salt solution ”and gravimetric methods, for example.-3c Ipartial pressure, xFIG.5.-The mean deviation A from the best linear plots of the form of fig. 3 and 4, for all thematerials investigated here (296.5 K).The partition functionf(x) described by eqn (7), which is similar to that proposedby G~ggenheim,~ can be regarded as a geometrical series approximation to the moregeneral description off(x) given by eqn (4) derived here. We find that a geometricseries does not exactly fit the experimentally derived sorption data. Close inspectionof fig. 3 for example indicates that the experimental data tend to deviate from the beststraight line plots in a consistent periodic manner. The form of such periodicdeviations is illustrated more clearly in fig. 5, which shows the mean deviation fromthe best straight line plots ofeqn (8) for all the biological materials studied here.Thisshows that the treatment of eqn (7), and hence of Guggenheim, is only an approxima-tion, and that the values given in table 1 should only be treated as such. The B.E.T.-derived values will be of very much greater inaccuracy due to the erroneous assumptionof having b = 1.Also, an attempt to make an exact fit of a polynomial of the formj-(X) = 1 + a x + ~ X 2 + y X 3 + . . .to the experimental data using a Chebyshev orthogonal curve fitting techniquP. R. C . GASCOYNE AND R. PETHIG 179resulted in physically unrealizable activity coefficients a, p, y, . . . This implies forthe samples studied here that either there are several different types of primarysorption sites, or that the primary sites interact with one another.Either of theseeffects negates the basic assumptions used to derive the various sorption theoriesdescribed in the literature and reviewed by Kuntz and Kauzmann. Hence withoutmore specific information regarding the partition functions fj(x) or the monolayerhydration capacities umj [see eqn (ll)] for the various types of sites, more detailedinformation regarding the sorption site activities cannot be derived from experimentalisotherm data. For biological materials it is reasonable to expect different types ofprimary sorption sites to be present ; the conclusions above serve to indicate that thisfact severely restricts the information that can be derived from the analysis of sorptionisotherms.For synthetic polypeptides however, where the different kinds of sorptionsites can be more clearly defined, analysis using eqn (4) can be expected to be morefruitful.A further complication which will effect the usefulness of any sorption theory canbe appreciated from eqn (3). The Grand Partition Function described by thisequation implicitly assumes that the activity of any one sorbed layer is not altered bythe sorption of subsequent laysrs upon them. This restriction can be overcome byrewriting eqn (3) in the formF(x) = (1+a,x+a;a,x2+a';a;a,x3+ . . .>Nswhich can be shown to lead to results of the same form as eqn (4) and (5). Eqn (3)and (1 4) are therefore thermodynamically equivalent so that without other physicaldata, sorption measurements cannot provide information regarding the individualactivities of sorption layers beyond the first.In applying eqn (4) it is important to be able to assess how experimental errorsin u(x) and x effect the accuracy Gf the derived activity coefficients.A detailedanalysis of this problem will not be given here, it being sufficient to indicate that forthe first few sorption layers errors in ai are comparable to, if not less than, the rootmean square experimental errors in determining u(x) and x.The observation that the activity parameter b of eqn (8) is less than unity for theproteins studied, but of value greater than unity for DNA, we consider to be ofsignificance regarding their behaviour in solution. The value of b less than unity forthe proteins implies that their outermost hydration layer is formed by water moleculesof lower activity than bulk water, so that extra layers of water are required to beassociated with the hydrated macromolecule before it can become fully accommodatedinto normal bulk water.For materials such as DNA, with a value b greater thanunity, the outermost hydration shell is already indistinguishable from bulk waterand no such additional modification of the bulk water is required. These considera-tions may be of direct relevance to such phenomena as salting-in and salting-out,and to other macroinolecular interactions in solution.Finally we wish to add that favourable attention is often given in the literatureregarding certain isotherm equations which give good descriptions of experimentalisotherm data.Such good agreement should not, we feel, be considered to lead toaccurate physical sorption parameters. For example eqn (12) is identical in form tothat obtained by Hailwood and Horrobin,21 and eqn (8) can be compared to thatobtained by Enderby.22 The restrictions described here regarding the derivation ofaccurate sorption parameters should also be considered pertinent to these theories.We acknowiedge the invaluable mathematical zdvice of Messrs. T. P. T. Williamsand D. Everett, and the S.R.C. for the award of a studentship to P. K. C. G1 so HYDRATION ISOTHERMS FOR BIOMOLECULESD. Kuntz and W. Kauzmann, Adu. Proteiri Chem., 1974, 28, 239.R. Cooke and I. D. Kuntz, Ann. Rev. Biophys. Bioeng., 1974, 3, 95.M. G. Kennerley, Polymer, 1969, 10, 833.M. M. Breuer and M. G. Kennerley, J. Colloid Interface Sci., 1971, 37, 124.chap. 11, pp. 186-206.' E. A. Guggenheim, Applications of Statistical Mechartics (Oxford, Clarendon Press, 1966)-' I. Langmuir, J. Amer. Chein. SOC., 1918, 40, 1361. ' H. Freundlich, Kapillarchernie (Leipzig, 1909).' H. K. Pulker, Thin Solid Films, 1976, 32, 27.S. Brunauer, P. 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