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Langmuir 1993,9, 962-972
Electrostatic and van der Waals Contributions to Protein Adsorption: Computation of Equilibrium Consthnts Charles M.Roth and Abraham M.Lenhoff Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received September 11, 1992. In Final Form: January 25, 1993
Although protein adsorption has been much described and exploited, little effort has been directed toward the developmentof models for ita a priori prediction. Here we describe a methodologythat allowa for the computationof protein-surface equilibrium constants K (i.e., equilibriaat low surface coverages) based on protein molecular structure and surface properties. &e crux of the model is the computation of the electrostatic and van der Waals energies of interaction between a colloidal protein molecule and a planar, charged surfaceat a fixed distancefrom and orientation with respect to it. Resulta for the protein lysozymeare presented;however,these calculationsare computationallyvery time-intensive. Consequently, we utilize a simplified description of the protein as a low dielectric sphere with ita net charge placed at the center. It is used in particular to compute the relationship,relevant to ion-exchangechromatography, between ionic strength and Kq,due to the strong effect which salt exerts on electrostatic interactions. The physical properties that affect the value of the equilibrium constant are protein and surface net charges, Hamaker constant, and protein size; the first two influence electrostatic interactions, the third characterizes dispersion forces, and the last affects both types of interactions. In addition to allowing a priori prediction of adsorption equilibria, the construct presented in this paper can allow for improved understanding and interpretation of electrostatic and dispersive mechanisms for protein adsorption.
Introduction A large body of research today is directed toward elucidating the relationship between molecular structure of chemical constituents and expressed function on a ".croscopic, observablescale. The types of functionwhich are sought include kinetic properties, such as reactivity and catalyticactivity,and thermodynamicphase behavior, including solubility and, of primary interest to us, extent of adsorption. While this type of research has been ongoing for many types of molecules, application to the behavior of proteins has lagged due to their relative molecular complexity. An understanding of the molecular events involved in protein adsorption would be of great utility in many scientific contexts as well as industrial applications. For example, the ability to predict protein adsorption is relevant to the design of biomaterials, and of common and practical interest is the desire to predict and control ("tune") chromatographicseparations in order to increase their accuracy and efficiency. Few models exist for the prediction of observable behavior of proteins. Only recently have some studies emerged in which protein functional properties such as phase partitioning1I2and osmotic pressure3v4have been modeled using molecular properties of proteins, but even in these cases the extent of molecular information used was minimal. Previous models of protein adsorption have been as simple as a mass-action model for a strict ionexchange between protein and surface in which the protein is considered as a multivalent ion with a valency corresponding to the number of ions displaced on the ~urface.~ This idea was extended to include the counterions in solution and the idea of multiple binding sites on the p r ~ t e i n . ~These ? ~ models have generally been used a (1) Mahadevan, H.; Hall, C. K. AIChE J. 1990,36, 1517. (2) Vlachy, V.; Blanch, H. W.; Prausnitz, J. M. AIChE. J.1993,39,215. (3) Haynes, C. A.; Tamura, K.; KBrfer, H. R.; Blanch, H. W.; Prausnitz, J. M. J. Phys. Chem. 1992,96, 905. (4) Vilker, V. L.; Colton, C. K.; Smith, K. A. J. Colloid Interface Sci. 1981, 79, 548. (5) Boardman, N. K.; Partridge, S. M. Biochem. J. 1955, 59, 543.
0743-7463193/2409-0962$04.00/0
posteriori for analysis of chromatographic data, and, containingparameters without clear physical significance, are not suited for predictive purposes. Detailed incorporation of both protein and surface characteristicsinto models of proteinaurface interactions has been accomplished in limited applications with recent advances in computational capabilities. Atomic models of proteins and polymer surfaces have been used to calculate adsorption energies819and also to investigate dynamic effectdo for the adsorption of proteins onto polymer surfaces. While these modelshave provided some insight into the relative contributions of various types of interactions (e.g., electrostatic, dispersion, solvation) to protein-polymer interactions, the detail incorporated necessitates exclusionof other features suchas the presence of solvent and electrolyte and treatment of electrostatics in more than a pairwise Coulombic sense. The continuum principles of colloid science provide a basis for incorporating some of the relevant features of proteinaurface interactions without involving atomic detail and omitting solvent completely. Among the interactions which contribute to the adsorption of proteins at interfaces are electrostatics, dispersion forces, and solvationforces.ll Additionally,the conformationof many proteins has been found to be adaptable, leading to the possibilityof structural rearrangements as a protein nears the surface.11-13 Among these, the solvation and conformational effects are very difficult to describe, but electrostatics and dispersion forces are better understood in a colloidal context. Not only is the general nature of these (6) Kopaciewicz, W.; Rounds, M. A.; Fausnaugh, J.; Regnier, F. E. J. Chromatogr. 1983,266, 3. (7)Melander, W. R.; El Rassi, Z.; Horvlth, C. J. Chromatogr. 1989, 469, 3.
(8)Lu, D. R.; Park, K. J . Biomater. Sci., Polym. Ed. 1990, 1, 243.
0
(9) Lu, D. R.; Lee, S. J.; Park, K. J. Biomater. Sci., Polym. Ed. .""
1991,
0, 111.
(10) Lim, K.; Herron, J. N. In Biomedical Applications of Polyethylene Glycol Chemistry; Harris, J. M., Ed.; Plenum: New York, 1991. (11) Norde, W. Ado. Colloid Interface Sci. 1986, 25, 267. (12) Kondo, A.; Oku, S.; Higashitani, K. J. ColloidInterface Sci. 1991, 143, 214. (13) Norde, W.; Favier, J. P. Colloid Surf. 1992, 64, 87.
CO 1993 American Chemical Society
Computation of Protein Adsorption Equilibrium
forces well understood, as described by DLVO theory,14J5 but more sophisticatedtheorieshave been developedwhich can include details of protein structure. The DLVO theory has been applied to a kinetic model of protein adsorption in which the protein molecules were treated as spheres and adsorption occurred for proteins and surfaces of like charge by crossing over a potential energy barrier.16J7 An equilibrium model was recently proposed by SttMberget al.,18who developed an expression for the capacity factor (proportional to the adsorption equilibrium constant) in ion-exchange chromatography, based on the electrostatic free energy between two flat parallel plates. Despite the simplistic treatment of geometry, they found good correlation between protein charges backed out by applying their theory to published ion-exchange data and protein charges given by titration data. With the inclusion of the van der Waals interaction energyfor two flat plates, they were able to fit ion-exchange chromatographicdata over a wide range of ionic strengths,l9 albeit with the use of multiple fitting parameters. Another continuum approach, employingcomputation of the mean surface potential of proteins from their three-dimensional structure and charge distribution, has been applied to ionexchange data,20with a strong correlation found between mean surface potential and retention time. Because of the lack of sorbent characteristics in this latter model, however, its predictive applicability is negligible. The rapidly increasing amount of protein structural information becoming available makes this type of modeling possible. Detailed structural information exists for over 850 proteins,in the form of atomic coordinate files from crystallographicdata.21r22Althoughcurrently applied only to polypeptides and small proteins, NMR holds promise for determining protein structures in solution.23 Furthermore, basic information about size and charge can be obtained from scattering, electrophoresis,and titration experiments, or, for proteins produced by recombinant DNA techniques, directly from the primary structure derived from the gene. In this paper we outline a model for protein adsorption which utilizes the tenets of colloidal theory in such a way as to emphasize the electrostatic effects, but to include also van der Waals interactions. By including the structural information on the protein, the model accounts for configurational dependencies of the interaction between protein and surface; protein-protein interactions are not included, so applicability is limited to low surface coverages. The emphasis of this work is on the formulation of the model and on the dependence of the computed Kes on ionicstrength, for salt strength has a profound influence on electrostatic interactions, is easily altered experimentally, and can be used to manipulate the effects described herein for engineeringpurposes. A subsequent paper will assess the ability of the model to describe and predict experimental data obtained on planar charged surfaces corresponding to those described in this theory. (14) Derjaguin, B. V.; Landau, D. L. Acta Physicochim. URSS 1941, 14, 633. (15) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (16) Ruckenstein, E.; Prieve, D. C. AIChE J. 1976,22,276. (17) Prieve, D. C.; Ruckenstein, E. In Colloid and Interface Science; Kerker, M., Ed.; Academic Press: New York, 1976; Vol. 4, p 73. (18) StAlberg, J.; Jonsson, B.; Horvdth,C. Anal. Chem. 1991,63,1867. (19) StAlberg, J.; Jonsson, B.; Horvdth,C. Anal. Chem. 1992,64,3118. (20) Haggerty, L.; Lenhoff, A. M. J. Phys. Chem. 1991,95,1472. (21) Bernstein, F. C.; Koetzle, T. F.; Williams, G. J. B.; Meyer, E. F.;
Brice, M. D.; Rodgers, J. R.; Kennard, 0.; Shimanouchi, T.; Tasumi, M. J. J . Mol. Biol. 1977, 112, 535. (22) Protein Data Bank Quarterly Newsletter 59, January 1992. (23) Wiithrich, K. Science 1989,243, 45.
Langmuir, Vol. 9, No. 4, 1993 963
Negatively charged surface
Figure 1. Electrostatic model of interaction of a protein molecule with a charged surface.
Theory A. Electrostatics. Electrostatic interactions in proteins arise from ionizable sites on amino acids, usually residing near the surface of the folded molecule. Many current descriptions of protein electrostatics build on the model of K i r k ~ o o dwho , ~ ~ represented the protein as a sphere of low dielectric permittivity immersed in a solvent of higher dielectricpermittivity, and obtained an analytical expression for the work of charging the protein. Since that time, numerical approaches have been developed which can account for arbitrary protein shape and charge distribution, as well as nonlinear effects and the presence of a charged planar surface.2s28 Reviews are available el~ewhere.~~-~l Previous application of protein molecular electrostatics calculations has been to interpret the electrostatics of ion binding,32pKa ~ h i f t s , enzyme ~~,~~ activity,35and cofactor electr~chemistry~~ and also to correlateprotein mean surfacepotential with ion-exchange chromatographic data20 and to compute electrostatic interaction energies between proteins and charged surfaces.28 The electrostatic system comprises a protein at a fixed location and orientation with respect to a charged surface with an aqueous electrolyte as the intervening medium (Figure 1).The surface is assumed to be planar, embodying a homogeneous and constant surface charge density, Q. Within the protein, which is treated as a continuum colloidalparticle, charges arise from the titration of certain of its constituent amino acid residues. Mobile ions in the solution form a diffuse double layer around the molecule and the surface. The electrostatic potential 0'within the protein is described by the Poisson equation
where pi represents the charge distribution and ci the (24) Kirkwood, J. G. J. Chem. Phys. 1934,2,351. (25) Gilson, M. K.; Sharp, K. A.; Honig, B. H. J.Comput. Chem. 1987, 9, 327. (26) Jayaram, B.; Sharp, K. A.; Honig, B. H. Biopolymers 1988,28, 975. (27) Yoon, B. J.; Lenhoff, A. M.J. Comput. Chem. 1990.11, 1080. (28) Yoon, B. J.; Lenhoff, A. M. J. Phys. Chem. 1992,96,3130. (29) Matthew, J. B. Annu. Rev. Biophys. Biophys. Chem. 1985,14, 387. (30) Rogers, N. K. Prog. Biophys. Mol. Biol. 1986,48, 37. (31) Harvey, S. C. Proteins 1989,5,78. (32) Matthew, J. B.; Richards, F. M. Biochemistry 1982,21,4989. (33) Bashford, D.; Karplus, M. Biochemistry 1990,29, 10219. (34) Bashford, D.; Karplus, M. J. Phys. Chem. 1991,95,9556. (35) Klapper, I.; Hagstrom, R.; Fine, R.; Sharp, K.A.; Honig, B. Proteins 1986, 1, 47. (36) Gunner, M. R.; Honig, B. Proc. Natl. Acad. Sci. U.S.A. 1991,88, 9151.
Roth and Lenhoff
964 Langmuir, Vol. 9, No. 4, 1993 dielectric permittivity of the protein interior, with its dielectric constant generally taken to be in the range from 2 to 4.31,37-39The internal charge distribution consists of the charged amino acid sites, Le.
in which the locations of the charges are xk, 6(x)represents the Dirac delta function, and qk gives the magnitude of charge k. For a z:z electrolyte, the exterior electrostatic potential (Pe is described by the linearized PoissonBoltzmann equation p c p e = &le (3) where K~ = 2e2z2pm/&T, e is the electronic charge, z the ionvalency, k the Boltzmann constant, T the temperature, and p" the bulk concentration of electrolyte before dissociation. The linearization is strictly valid only for potentials much less than 1 kT/e; however, researchers have found surprising agreement between results from the linear and nonlinear equations for potentials up to several k T/ The boundary condition at the planar surfaceis assumed to be that its charge density is constant, so that the normal potential gradient at the surface is given by e.*a4
(4)
where n is the unit outward vector normal to the surface. A constant potential boundary condition could be used as an alternative.% At the dielectric boundary between pr6tein and electrolyte solution, the potential and the normal component of the electric displacementvector must be continuous, i.e. cp' = cpe
(54
and
When a protein molecule is involved, the solution to this set of equations must be obtained numerically, owing to the complex geometry involved. A procedure employing a boundary element technique which utilizes the Green function formulation of the problem has been described p r e v i ~ u s l y . ~The ~ - ~solution ~ is obtained in the form of potentials and their normal derivatives at a set of nodal points on the discretized dielectric boundary. From these, the potential at any interior or exterior location can be readily computed. The potentials allow calculation of the free energy of the double layer system. When the linearized PoissonBoltzmann equation is used for the electrolyte solution, the free energy of the system can be expressed a s 2 8 9 4 2
molecule from free solution to a given position and orientation in the vicinity of a surface can be computed. B. van der Waals Interactions. The original description of van der Waals interactions between macroscopic bodies was given by Ha1naker,4~who treated them by summing potential energies pairwise between constituent molecules. However, in condensed bodies, manybody effects have a significant influence on the overall interaction. Using quantum field theory, Lifshitz4' more rigorously described the van der Waals interaction of macroscopic bodies; many features of his result coincide with the simplified treatment of Hamaker. A roughly equivalent expression can be formulated b a d on the more readily recognizable tenets of electromagnetic theory.45 With these theories, subsequent workers have been able to compute Hamaker constants for biological materials from scant spectroscopic e v i d e n ~ e . ~ ? ~ ~ Incorporation of the detailed Lifshitz theory for computation of protein-surface dispersion forces is beyond the scope of this work. Instead, we proceed with the Hamaker approach extended to nonplanar geometries, essentially via integration. In the absence of a treatment accountingfor protein shape,the molecule is approximated as a sphere; this should be reasonable for the globular proteins used in this work, especially given the uncertainties in the material properties involved. The expression for the van der Waals interaction energy between a sphere and an infinite plane, utilizing this approach, is48
where R is the radius of the sphere, z is the nearest distance between the sphere and the plane, andAl32is the Hamaker constant, essentiallya material property, for the interaction of bodies 1 and 2 through medium 3. C. Thermodynamic Integration. Previous models of protein adsorption isotherms have generally been based on the Langmuir model or simple extensionsto it.49 These models assumethe presence of identical and independent adsorption sites. Instead, we employ here the Gibbs approach to adsorption, involving a surface excess and a distribution of concentrations from the surface out to the bulk solution. In this approach, which has been used to study adsorption of gasesw and in a similar theory for electrostatic interactions in chromatography,18the probability of finding an adsorbate molecule in a particular orientation with respect to and distance from the surface is given by a Boltzmann distribution with respect to the interaction energy. The surface excess n. is the amount of solute localized near the surface relative to that which would exist in the absence of the surface, which in terms of concentrations is
where c is the solute concentration anywherein the volume and Cb the bulk solute concentration (Le., far from the surface). Since we assume throughout that the surface is where aP represents the planar, charged surface. With this expression,the free energy change in bringing a protein (37) Takaehima, S.; Schwan, H. P. J. Phys. Chem. 1965,69, 4176. (38) Pennock, B. D.; Schwan, H. P. J. Phys. Chem. 1969, 73, 2600. (39) Harvey, S. C.;Hoekstra, P. J . Phys. Chem. 1972, 76, 2987. (40) Hogg, R.; Healy, T. W.; Fuerstenau, D. W. Trans. Faraday SOC. 1966,62, 1638. (41) Overbeek, J. Th. G. J. Chem. SOC., Faraday Trans. I 1988,84, 3079. (42) Sharp, K. A.; Honig, B. J. Phys. Chem. 1990,94,7684.
(43) Hamaker, H. C. Physica 1937,4,1058. (44) Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, L. P. Ado. Phys. 1961, IO, 165. (45) McLachlan, A. D. Discuss.Faraday SOC.1966,40,239. (46) Ninham, B. W.; Paraegian, V. A. Biophys. J. 1970,10,648. (47) Hough, D. B.; White, L. R. Ado. Colloid Interface Sci. 1980,14, 3. (48) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: London, 1986; Vol. 1. (49) Andrade, J. D.; Hlady, V. Adu. Polym. Sci. 1986, 79, 1. (50) Barker, J. A.; Everett, D. H. Trans. Faraday SOC.1962,58,1608.
Computation of Protein Adsorption Equilibrium
Langmuir, Vol. 9, No. 4, 1993 965
planar and uniform, the dimensionalityof integration space is reduced from three to one, but in general the interaction energy will also be a function of molecular orientation, which must also be included. The surface excess per area of surface is merely the surface concentration cs, so we have
where Q refers to orientational space and 20 is a cutoff distance, which is incorporated to account for the dominance of steric hindrance at short range. Incorporating the Boltzmann expression for c, we can divide through by Cb and obtain the equilibrium constant as
The energy of interaction, bF(O,z), is assumed to be the sum of contributions from electrostatic and van der Waals interactions. It is this equilibrium constant that characterizesthe equilibrium interaction of protein and surface at low concentrations, when contributions from steric and solvation forces, as well as from interactions among neighboring adsorbed molecules, are negligible.
Methods The most detailed descriptions of protein structure are in the form of atomic coordinates, which are available for many proteins throughthe Brookhaven Protein Data Bank (PDB).22 These coordinates are obtained from crystallographic data, which should represent a reasonable approximation to the solution structure for the proteins considered in this work. The two proteins used in this work, lysozyme and chymotrypsinogen A, belong to the category of ‘hard” proteins,51which are characterized by low adiabatic compressibilitiesand a tendencyto maintain their native conformation under most ~onditions.l~v5~ Using a protein’s atomic coordinates, a triangulated protein boundary was generated.53 Point charges were placed in the locations suggested by the ionization constants of the constituent amino acids and at the N-and C-termini.These charges comprise the charge density pi in the Poisson equation and hence the sources for electrostatic potential. Because of the computationaleffort associated with the large number of triangles needed to represent a protein molecule, an alternative model was employed in which the protein molecule was treated as a sphere. For the computationspresented herein, a boundaryfor the protein hen egg-white lysozyme (molecular mass 14 kDa) was generated, and an equivalent sphere comprised of 240 triangles was also used (Figure 2). For another protein, chymotrypsinogen A (molecular mass 26 kDa), only an equivalent sphere was constructed. The radius of the equivalent sphere was computed based on the volume of the triangulated protein interior. This was preferred to one computed on the basis of area, because the surface area is highly dependent on the degree of tesselation of the surface. In this model, the net charge as computed from the protein structure was placed at the center of the sphere. Additionally, two charges were added in selected cases such that the dipole moment of the protein charge distributionwas mimicked,with the dipole charges residing nine-tenths of the distance from the sphere center to its boundary. (51) Arai, T.; Norde, W. Colloid. Surf. 1990, 51, 1. (52) Cekko, K.; Hasegawa, Y. Biochemistry 1986,,?5, 6563. (53) Connolly, M. L. J . Appl. Crystallogr. 1986, 18, 499.
Figure 2. Triangulated surfaces used for boundary element calculations of lysozyme molecular electrostatics: left, surface reflecting crystal structure; right, equivalent sphere. The boundary element program used to solve the governing eq~ations~’9~~ was implemented on a Sun 4/110 for the sphere computations. To solve for the nodal potentials and to compute the free energy at one orientation with respect to and distance from the surface requires 6 to 7 min of CPU time for a sphere of 240 elements. The discretized protein boundary, however, contains 2454 nodes, for each of which two equations must be formed and their coefficients stored in a matrix. Since even using single-precision arithmetic this requires approximately 100 Mbytes of storage, the protein computations were performed on the University of Delaware IBM 3090. To solve for nodal potentials and compute the free energy in this case required ca. 2000 s of CPU time per configuration. In order to extract an equilibrium constant from the free energies, integrations with respect to orientation and distance are required. Slightly different approaches are required depending on the dimensionality required, i.e., one-dimensional integration for spherelmonopole cases, two-dimensional for spherelmonopole+ dipole cases, and three for full protein cases. In all cases, however, the distance-dependence of the integration was handled in the sameway. Because the electrostaticfree energy decays roughly exponentially with distance, with a decay length equal to the Debye length, the exponent in the integrand in eq 10 itself behaves like an exponential function with respect to 2. In order to alleviate the difficulty of integrating such a steep integrand, a substitution of the form u = e-xrwas made in eq 10. The result is an integral over a finite domain which has a sufficientlysimple shape, with little stiffness, that its behavior resembles that of a low order polynomial. With this transformation, it is possible to use Gaussian quadrature to perform the integration,” obtaining an accurate estimate of the integral with very few integration points. Note that the actual free energy profile need not match the assumed exponential form exactly; so long as the general form is correct, the method will give a good approximation to the integral. Therefore, it can be used for both the electrostatic and combined electrostatic/van der Waals interactions. The primary drawback to Gaussian quadrature is the fact that increasing the number of integration points by one changes all of the points. As a result, it is difficult to check for convergence without using a large number of integration points, each of which is the result of a laborious computation in this work. We have thus used an extension to the Gaussian method which adds extra points most efficiently and maximizes the concomitant gain in p r e c i ~ i o n . ~ ~ (54) Stroud, A. H.; Secrest, D. Gaussian Quadrature Formulas; Prentice-Hall: Edgewood Cliffs, NJ, 1966. (55) Patterson, T. N. L. Math. Comput. 1968,22,&17.
966 Langmuir,
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VoZ.9, No. 4,1993
For this work, the three-point Gaussian formulas were used, with the four-point Patterson extension (to 7 points) and second extension (to 15 points) also employed. Convergence was reached after just three points for all electrostatic cases, but for combined electrostatic and van der Waals interactions, the assumed form of the integrand becomes less effective and the 15-point extension sometimes deviated from the 7-point extension by 10-20% or more. The most difficulty was found to occur in cases where large negative van der Waals energiescombinewith large negativeelectrostatic energies to produce a situation where the dependence of free energy on distance exhibits a strong, but not exponential, dependence on distance, as assumed. However, in these cases, the equilibrium constants are very large, and on the logarithmic scales to be shown, a 10-20% error is within the size of the symbols on the figures. For the two-dimensional sphere computations, the amount of computational effort required to obtain the free energy at one configurationis fairly small. As a result, the two-dimensional integrations were performed by repeated use of Gaussian quadrature. For this case, the integral in eq 10 becomes
with the orientation represented by the angle 8 between the dipole axis and the normal to the plane surface. : The distance integration was handled as above, while the interior transcendental portion of the integrand was transformed by a substitution of u = cos 8, and the resulting integral handled with the same Gaussian/Patterson integration technique as described for the distance-dependent integration. It was found that the orientational integrals converged with the minimum three points, but the seven-point extension was still used in all cases. Repeated application of this methodology to the threedimensional integration required for arbitrary protein shape and charge distribution would.require at least 7 X 7 X 15 = 735 computations of the free energy for the evaluation of one equilibrium constant. As the complete protein computationsare the most costly, computationally speaking, this is most infeasible. Since the three-point Gaussian formula was sufficient to compute the electrostatic equilibrium constants, it was used here in order to reduce the number of computations required. As a consequence, only an estimate for an electrostatic equilibrium constant (i.e., van der Waals interactions are not included) is presented. For the orientational integration, 12 points were chosen in the transformed orientational space (u = cos 8 as above and the azimuthal coordinate 4 ranging from 0 to 27r) with roughly equal intervals. The free energies computed at these points were then used to calculate the equilibriumconstant using a trapezoidal rule. As the accuracy of this cruder method has yet to be established, this value should be regarded as only an estimate of the equilibrium constant for the full protein.
Results Potential fields for the electrostatic interaction of ribonuclease A with a negatively charged surface using the complete protein structure and charge distribution have been presented previously28 and show that the potential distribution is highly perturbed by inclusion of molecular detail and highly dependent on the orientation of the protein molecule with respect to the surface. A further result of this study was a striking dependence of the electrostatic energy of interaction on orientation, i.e.,
-0.85
-1.51
surface charge = -2.2 pC/cm *
Figure 3. Polar plot showing orientational dependence of electrostaticinteractionenergyof lysozyme with a charged surface ( u = -2.2 pC/cm2)at pH 7,O. 1M ionic strength, for a gap distance of 7.66 A.
3
0
5
10
15
20
Gap distance (A)
25
0
Figure 4. Dependenceon gap distanceof electrostaticinteraction energy between lysozyme and a charged surface for full protein model (shaded area) and sphere model (solid line). Parameter values as in Figure 3.
that some orientations result in favorable energies of interaction while other orientations exhibit unfavorable energies of interaction with the negatively charged surface. For lysozyme, the dependence of electrostatic energy on orientation is shown in Figure 3 for an ionic strength of 0.1 M and a surface charge density Q = -2.2 pC/cm2.As with ribonucleaseA, there exists a significant dependence on orientation, with values ranging from -0.80 to -1.91 kT,but in all cases the interaction is attractive. Although these interaction energiesare not large relative to k T, they are strongly dependent on the protein-surface gap size and are much more significant for smaller gaps. Figure 4 shows the electrostatic interaction energy as a function of distance from the surface for the sphere model results under the same conditions as Figure 3, with the range of the results from the orientation-dependent model also shown. It can be seen that the sphere model falls within the range of the orientation-dependent model and toward its more favorable end. The equation constantsaccountingfor electrostatic interactions only-for the two models compare favorably, with values of 73.9 A for the sphere model and 74.2 A for the orientational model. As mentioned previously, the detailed protein computations are memory- and time-intensive computationally, and integrating with respect to orientation necessitates many more free energy computations to calculatean equilibrium constant than for a sphere model containing a charge distribution with intrinsic symmetry. Another reason to simplify the model is the fact that for many proteins the complete crystallographicstructure is unknown, but their sizes and net charges can be estimated from solution experimental techniques. And since the results from the sphere with monopole are close to those for the complete lysozyme shape and charge distribution, all subsequent
Computation of Protein Adsorption Equilibrium __....... .... ....
0 ,
-
1
I
2
5
0
10
Langmuir, Vol. 9, No. 4, 1993 967
9 15
20
Gap distance (A)
1 25
33
Figure 5. Dependence on gap distance of interaction energy between lysozyme and a charged surface showing electrostatic and van der Waals contributions. Electrostatics based on sphere model with parameter values as in Figure 3, except surface charge density u = -4.6 &/cm2; A = 1.0 X 10-2'JJ.
3
iX1o7 1x106
.*2
1x104
r M
lxld
1
....
'%.
1x102
II
1x101 l X 1 ~ l
1
1x100 0.05
'
'
'
'
41
I
0.1
Ionic strength (M)
Fkure 6. Predicted equilibrium constant for lysozyme adsorption on a charged surface including electrostatic and van der Waals contributions. Parameter values as in Figure 5, with zo = 1 A.
results involve the sphere approximation, with only the monopole moment of the charge distribution represented unless otherwise noted. From the electrostatic potential distribution, the electrostatic component of the interaction energy can be computed from eq 6; the van der Waals contribution is given directly by eq 7. Typical profiles as a function of distance for the electrostatic, van der Waals, and total interaction energies are shown in Figure 5. As demonstrated earlier, the electrostatic energy falls off roughly exponentially with distance, although the dependence is weaker at short distances; the physical basis for this weakened dependence is discussed later. The van der Waals energy is significant only at small separations of sphere and surface, with the dispersion interaction increasing sharply in magnitude when they are brought into proximity with one another. The total energy curve is the one which is used to compute the equilibrium constant in subsequent results, with the assumption that the electrostatic and van der Waals interaction energies are additive. W i l e there is some debate as to the validity of this assumption, it forms the basis for the classic DLVO the0ryl4J5and is used here in the absenceof strong evidence against it. We are primarily interested in the relation between the equilibriumadsorbed amount and physical characteristics of the protein, surface, and solvent environment. Utilizing the same conditions as above except for a variable ionic strength, the relation between the computed equilibrium constant and salt strength is given in Figure 6. The equilibrium conatant decreases with ionic strength to a degree that its effect must be plotted on log-log coordinates
iX1o7
-0-
*+a
1x106
--*.
Q=+6
0.05
I
0.1
1
Ionic strength (M)
Figure 7. Effect of protein charge on predicted equilibrium constant for adsorption on a charged surface including electrostatic and van der Waals contributions. Other parameter values as in Figure 6.
in order that the relationship between the two variables can be seen. When more salt ions are present in solution, the electrostaticattraction between the protein ahdsurface is screened and thus diminished. While there is no simple explanation for the nearly linear dependenceof log K, on log1(I = ionic strength, moVL) in Figure 6, the electrostatic potential is known to fall off exponentially with distance in various geometries, with the decay length being the Debye length, which is proportional to the inverse square root of the ionic strength, so it is not surprising that the effect of increased ionic strength is so strong. In order to understand more fully how the physical interactions are reflected in the protein-surface equilibrium constant and its variation with ionic strength, the relevant properties and parameters were varied from the values used to construct Figure 6. First and foremost is the protein net charge; it is in this quantity that intuition leads one to believe that different proteins are moat likely to distinguish themselvesin terms of adsorptiveproperties. Typically proteins will have a net charge of a few elemental charges, rarely greater than plus or minus ten a t pH 7.b6 Figure 7 shows the influence of protein charge on the computed equilibrium constant. For weakly charged proteins, the computed equilibriumconstant doea not vary strongly with ionic strength. Since the model does not include screeningeffects on dispersion forces,there is little electrostatic interaction and van der Waals attraction is the predominant contributor to adsorption. At higher net charges, an enhancementin the extent of adsorption results from stronger electrostatic interactions, which are mast pronounced at low ionic strength. When the protein charge is zero or of the same sign as the surface, the opposite trend is observed. In this case, electrostatic repulsion counteracts the short-range van der Waals attraction. At low ionic strengths this becomes manifest as a surface depletion, or negative equilibrium constant by the definitions in eqs 9 and 10; these points cannot be plotted because of the logarithmic Kw axis. When more salt is added to the solution, this repulsion is screened and so the equilibrium constant increases toward an asymptote governed by dispersion forces only. As the complementto protein net charge,higher absolute values of the surface charge density, when of sign opposite to the protein net charge, also act to enhance attractive electrostaticinteractionsand consequentlythe equilibrium constant between protein and surface (Figure 8). When protein and surface charge are of the same sign, electrostatic repulsion results, along with surface depletion when the electrostatic repulsion outweighs the van der Waals (56) Barlow, D.J.; Thornton, J.
M.Biopolymers
1986,25, 1717.
Roth and Lenhoff
968 Langmuir, Vol. 9,No. 4, 1993 1x109
43-
lxloB
-9.2pUcmA2
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0.05
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Figure 8. Effect of surface charge density on predicted equilibriumconstantfor adsorptionon a charged surfaceincluding electrostatic and van der Waals contributions. Other parameter values as in Figure 6.
1
Ionic strength (M)
1
Figure 9. Effect of Hamaker constant on predicted equilibrium constant for adsorption on a charged surface including electrostatic and van der Waals contributions. Other parameter values a8 in Figure 6.
attraction. Whereas the charge of the protein can be determined from its amino acid sequenceor experimentally from titration data, determining the charge on surfaces in the planar geometry indicated by the model is more difficult. Zeta potential measurements are possible, but the relation between the experimentally determined quantity and surface charge is still a subject of active research.57 For two types of surfaces commonly used in adsorption studies, fused quartz slides and mica, the zeta potentials correspond to charges of -1 to -2.5 p C / ~ m ~ , ~ 8 * 5 ~ with the actual surface potentials likely to be greater in 1x1001 ' ' 0.05 0.1 magnitude since the zeta potential is that at some distance Ionic strength (M) from the surface. When the surface exists as a dispersion of (nonpolar)particles, much more surface area is exposed Figure 10. Effect of cutoff distance on predicted equilibrium constant for adsorption on a charged surface including electroand electrophoretic and titration data might give a static and van der Waals contributions. Other parameter values reasonable estimate of surface charge. For polystyrene as in Figure 6. dispersions, surface charges ranging from -2.3 to -15.5 pC/cm2 have been reported.60 Because of the wide Opposing views exist as to whether the effect is signifidiscrepancies among various techniques, more research is ~ant,48$631~ with the latter authors pointing out that the needed in this area for accurate modeling of actual proteinzero-frequency contribution to the Hamaker constant is sorbent systems. the only one which is screened, and it is significant only As the charges on protein and surface affect the for the interaction of similar organic materials with strong electrostatic interactions exclusively, so the Hamaker overlap in their UV spectra. constant solely reflectsthe dispersion forcesacting between The cutoff distance zo represents the distance at which protein and surface. Experimental and theoretical values the short-range steric repulsion between the protein and of Hamaker constants are emerginel but scant, and most the surface becomes dominant over the relatively longer available data are for the interaction of a substance with range effects considered here. While it is not clear what itself through a medium. For example, in water, fused the value of this distance should be, it should be on the quartz interacts with itself with a Hamaker constant of order of the size of a solvent molecule; a value of 1 A was (0.63-0.83)X 10-20J and mica with itself at a value of rather arbitrarily chosen for most of this work. As can be (2.0-2.2) X JF2whereas proteins are described by a seen in Figure 10, the effect is quite strong, even in the J.6* The Hamaker constant in the range of (1.0-2.2) X range from 1 to 3 A. Because of the steep increase at short effect of the Hamaker constant within ita likely range is distances in van der Waals attraction, it is most affected shown in Figure 9. As opposed to varying electrostatic by a change in cutoff distance. Consequently, the effect properties, which affect the shape of the log Kw vs log I on the equilibrium constant appears quite similar to that plot (in particular its degree of steepness), changing the of variation in the Hamaker constant. Since values of Hamaker constant does not change the shape of the both the Hamaker constant and the cutoff distance are relation significantly but increases the equilibrium connot easily accessible experimentally or theoretically, a stant by roughly equal (logarithmic) amounts across the completely predictive molecular model would require a range of ionic strengths. We have not taken into account better understanding of these parameters. the effect of electrolyte screening on the Hamaker constant. Whereas the influence of the above properties was (57) Scales, P. J. Streaming Potential/Ionization Behavior of Water manifest either in van der Waals interactions or in Insoluble Surface Probe Molecules. Ph.D. Dissertation, University of electrostatics,but not both, the size of the protein molecule Melbourne, 1988. is an important part of each of these interactions. For a 158) Scales, P. J.; Grieser, F.; Healy, T. W.; White, L. R.; Chan, D. Y. c . Langmuir 1992,8,965. given protein net charge, increasing size translates into a '
'
I
(59) DePalma, V. A. Reu. Sci. Instrum. 1980,52, 1390.
(60)Norde, W.; Lyklema, J. J . Colloid Interface Sei. 1978, 66, 257. (61)Afshar-E+d, T.; Bailey, A. I.; Luckham, P. F.; MacNaughtan, W.; Chapman, D. Btochtm. Btophys. Acta 1987, 925, 101.
(62) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992.
(63)Parsegian, V. A. In Physical Chemistry: Enriching Topics from Colloid and Surface Science; Theorex: La Jolla, CA, 1975. (64)Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: New York, 1976.
Langmuir, Vol. 9, No. 4, 1993 969
Computation of Protein Adsorption Equilibrium
-
5
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r=WA
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r=20A
-
r=156A
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Ionic strength (MI
Figure 11. Effect of sphere radius on predicted equilibrium constant for adsorption on a charged surface including electrostatic and van der Waals contributions. Other parameter values as in Figure 6.
lower surface charge density and lessened electrostatic interactions. Van der Waals interactions, on the other hand, increase with increasingsize of the molecules. These competing trends are demonstrated in Figure 11,in which the higher surface charge density resulting from small sphere size produces dominant electrostatic interactions at low salt strength and in which the increase in van der Waals interactions with size is dominant at high salt strength. All the curves intersect at about 0.1 M salt concentration. The energy profiles at this ionic strength are shown in Figure 12. The relative contributions of van der Waals and electrostatic forces change with increasing radius, but the shape and magnitude of the total energy profile are roughly the same throughout. On the basis of computations at higher Hamaker constants (data not shown),this common intersection point does not seem to be a general feature of the model. In some instances of oppositely charged bodies, electrostatic repulsion can actually occur. Since this effect is the result of mismatched, overlapping double layers, it is most significant when the surface charge densities of protein and surface are much different in magnitude. As one example, Figure 13a shows a case in which the degree of electrostaticattraction is diminished close to the surface on account of this effect; as a result the electrostatic energy curve exhibits a minimum at around 5 A. Within this range the van der Waals attraction is very strong and the total energy curve is not qualitatively affected. When the surface charge from Figure 13a is doubled, the more extreme case depicted in Figure 13b results, in which the electrostatic energy displays its minimum as far out as 12 A and becomes repulsive at distances less than about 3 A. In this case the total energy curve possesses a local maximum. The secondary minimum is due to the minimum in electrostatic energy, since dispersion forces are weak at long distances. At very short distances, a second turning point occurs due to the close-range van der Waals attraction, which outweighs the slightly repulsive electrostatic energy. When the charges are as mismatched as in this latter case, the effect on the log KW vs log I curve is profound, as in Figure 14. Here, a shallow minimum exists at about 0.2 M for u = -9.2 pC/cm2, with greater equilibrium constants at both higher and lower ionic strengths. Yet, ignoring van der Waals contributions results in the more usual, nearly linear relationship between log Kes and log I with negative slope. For further amplification of this phenomenon, compare this to the same conditions, except with the surface charge density reduced by a factor of 2. The equilibrium constant based purely on electrostatics is nearly the same for the two different charge densities.
For combined electrostatic and van der Waals interactions, however, the equilibrium constants are actually less at u = -9.2 pC/cm2than those at u = -4.6 pC/cm2. As a next approximation to estimating the magnitude of electrostatic effects without doing three-dimensional protein computations, the dipole moment of the protein charge distribution was added to the monopole moment in computations for lysozyme and chymotrypsinogen A. For lysozyme (Q = +8, dipole moment = 72.2 D as computed via methods similar to those of Barlow and Thornton%),the incorporation of the dipole has little effect on the computed equilibrium constant (Figure 15a). For chymotrypsinogen A (Q = +4, dipole moment = 516 D), however, the effect is much more significant (Figure 15b). In this case, the favorable energies for the dipole oriented toward the surface are amplified in the calculation of the equilibrium constant by the exponential Boltzmann factor.
Discussion The chromatographiccommunity often presents its ionexchange results in the form of log k' vs log I, where k' is a capacity factor that for a linear adsorption isotherm is given by the product of a geometric phase ratio and the equilibrium constant. Thus Figure 6 and similar plots convey essentially the same information. What is often reported is a linear fit between the log k' and log I values; however, the span of salt strengths employed is usually quite ~ m a l l . ~In J Figure 6, there is a degree of curvature to the log Kes vs log I relation which becomes more pronounced at lower ionic strengths. Since the span of salt strengths here runs the gamut from 0.05 to 1.0 M, this is not inconsistentwith the chromatographic experimental results. While Figure 6 represents the typical relationship between equilibrium constant and ionic strength due to electrostaticeffects, more unusual and interesting features can arise for certain reasonable combinations of protein and surface properties. For instance, although oppositely charged bodies usually experience electrostatic attraction, under some circumstances such bodies can be repelled. Specifically, infinite flat plates with opposite charge densities that are not exactly equal undergo electrostatic repulsion when the distance between them is sufficiently ~ m a l l . l * The * ~ ~results of Figure 13 suggest that this type of repulsion may occur for the plane-sphere geometry as well. The physical reason for this behavior is that the more strongly (negatively) charged surface carries a large body of counterion (positive) charge in its double layer which is of the same sign as the weakly charged protein. As a result, the potential at the surface of the protein increases to greater than zero to reflect the neighboring sources. The entropy of the system, proportional to the integral of charge density multiplied by potential,decreases because potential is now uniformly negative across the region between surface and sphere with positive charges present throughout that volume at greater than bulk density. Because the doublelayer is extendedover a larger volume at lower ionic strengths, it is in this regime that repulsion first becomes apparent. With this in mind, we can proceed to explain the behavior exhibited in Figure 14. At very low salt strength the repulsion due to mismatched surface charges occurs over the entire range (i.e., distances from the surface) for which van der Waals interactions are significant; consequently, the combined interaction asymptotically approachesthat for electrostatics only. Note that because the doublelayer (65) Parsegian, V. A.; Gingell, D. Biophys. J. 1972, 12, 1192.
970 Langmuir, Vol. 9, No. 4, 1993
I
Roth and Lenhoff
/-......... Electrostatics
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Figure 12. Dependence on gap distance of interaction energy at 0.1 M ionic strength for sphere radii corresponding to curves shown in Figure 11.
0.11 . 0.05
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1 7 ................
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decreasing salt concentration. At some higher ionic strength (here ca. 0.1 M),the double layer repulsion is confiied to a distance small enough that the repulsive electrostatic effects are screened over a significantportion of the range in which attractive dispersion forces act. Ae a result, the equilibrium conatant for combined interactions becomes significantly greater than that for pure electrostatics. At yet higher salt concentration, the electrostatic interactionsare confined to a region very cloae to the surface, and the equilibrium is dominated by dispersion forces. At this point, the equilibrium constant
Computation of Protein Adsorption Equilibrium
Langmuir, Vol. 9, No. 4, 1993 971
* dipole Ly2,monopolet
1.1.1 1x10'
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Figure 15. Effect of inclusion of dipole in sphere model of protein interaction with a charged surface for (a) lysozyme and (b) chymotrypsinogen A.
approaches a constant value determined by the van der Waals interactions only. A consequence of this result is the situation that increasing the magnitude of the charge of an oppositely charged surface to the protein can indeed diminish the adsorption of the protein onto it, as evidenced by Figure 14b. This perhaps surprising result was implied by Parsegian and Gingell,65who found that for two flat plates carrying opposite charges, the electrostatic energy of interaction is independentof the value of the greater charge density at large ratios of charge densities. The combined equilibrium constant is actually reduced at low ionic strengths upon an increase in surface charge density due to the introduction of electrostatic repulsion in the domain over which van der Waals attraction occurs. An increase in adsorption with ionic strength has been observed by many chromatographers' as an often sharp turnaround in the curve at high ionic strength. This behavior has generally been attributed to solvation effects, but this electrostatic repulsion could account for part of the effect. The sharp turnaround which is observed correlates more strongly with solubility effectss than with the shallow upturn observed here. It is not possible to determine the mechanistic basis for this behavior, and the issue remains unresolved at this time. It is important to point out some of the limitations of the model which we have presented and some ways in which it might be improved. First of all, the linearized form of the Poisson-Boltzmann equation has been used. Although the linearized version is generally considered to be reasonable for potentials up to severalW e ,electrostatic potentials at low ionic strengths can be quite large and the nonlinear Poisson-Boltzmann equation should be employed for modeling at lower ionic strength ranges than (66) Melander, 200.
W.;Horvath, C. Arch. Biochem. Biophys. 1977, 183,
those employed in this work or when significantly larger charge densities are involved. While no electrostatic computations have been performed for the nonlinear Poisson-Boltzmann equation with a surface present, the methodology exists27and has been incorporated for the same boundary element technique in the absence of a surface!7 the extension of which to an incorporated charged surface is straightforward. In addition to the low-salt restrictions present for our model due to the linearized electrostatics employed, an upper limit also exists due to the increased significance of hydration effects as ionic strength is increased. ABsalt is added to a solution,not only are any fixed charges within it screened from one another, but hydration becomes an issue due to the number of water molecules which orient themselves around each salt ion. A consequence of this effect is the commonlyobserved salting-outbehavior which affects protein solubility as well as ad~orption.7@'3*@~~9 Hydration effects due to hydrophobicity at the adsorbing surface could also be significant9 over the entire ionic strength range, but for hydrophilic surfaces this should be less of a consideration. While the overall charge of the protein seems to be an important indicator of the adsorption equilibrium of a protein on an oppositely charged surface,18evidenceexists that suggests the importance of the charge distribution in determining adsorption b e h a v i ~ r . ~Since J ~ proteins tend to fold with their charged groups to the exterior, both to minimize charge-charge repulsions as well as to maximize the shielding of hydrophobic residues in the interior of the protein, one concern regarding the sphere model may be the placement of the charge at the center of the sphere rather than near the dielectric boundary as occurs in folded proteins. In fact, however, when calculations were performed in which the net charge was smeared out into 240 portions (one for each element), the value of the equilibrium constant was practically unaffected over the entire range of ionic strength. Since net charge is not considered to be a sufficient indicator of electrostatic adsorption in proteins, the dipole moment was added to the sphere model in order to investigate the effect of charge polarity. Lysozyme has an unusually uniform distribution of charge and a small dipole moment,SBso the incorporation of dipole moment hardly affects its computed equilibrium constants. The uniform attraction experienced by lysozyme for a negatively charged surface and the good agreement between the sphere model and the three-dimensionalrepresentation are also in line with this observation. Chymotrypsinogen A, on the other hand, is a much more polar molecule and is expected from Figure 15bto have much more orientationspecific electrostaticinteractions with a negatively charged surface than would lysozyme. Furthermore, the dipole calculations suggest that chymotrypsinogen A would be likelyto exhibit strongeradsorption to a negatively charged surface than would be expected from the sphere model with monopole only. For future work, it remains to be seen how the orientation-dependent protein free energies for chymotrypsinogen A compare to those predicted from the monopole and dipole models and whether the dipole moment is a sufficient correction to the sphere model to account for heterogeneities in the charge distribution. It (67) Vorobjev, Y. N.; Grant, J. A.; Scheraga, H. A. J. Am. Chem. SOC. 1992, 114,3189. (68) Melander, W.R.; Corradini, D.; Horvath, C. J . Chromatogr. 1984, 31 7. 67. (69) Shibata, C. T.; Lenhoff, A. M. J. Colloid Interface Sci. 1992, fa, 469. (70) Lee, C. S.;Belfort, G.R o c . Natl. Acad. Sci. U.S.A. 1989,86,8392.
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972 Langmuir, Vol. 9, No. 4, 1993 also remains to be determined to what extent the shape of the protein molecule determines the electrostatic potential and consequently the interaction energy and adsorption extent.
Summary A method has been described which allows for the computation of electrostatic and van der Waals contributions to the equilibrium constant between protein and surfaceat low surface coverages based on protein molecular structure and charge density of the surface. In addition to ita utility for a priori estimation of the extent of adsorption, the method can also serve as a construct for interpretation and understanding of how electrostatics and dispersion forces contribute to the equilibrium constant. Since the chromatographic capacity factor k’ is directly proportional to the adsorption equilibrium constant, the application of this methodology could possibly aid in the rational design of chromatographicseparations in which these interactions are controlling. Results for the protein lysozyme suggest that the threedimensional protein shape and charge distribution can be represented by a simplified model, in which the protein shape is represented by a triangulated sphere and ita charge distribution is represented by ita monopolemoment placed at the sphere center. As a result, the sphere model is used to show the effect of the relevant physical properties on the equilibrium constant. Results from thia model show that the linear relationship between equilibrium constant and ionic strength (on a log-log scale) breaks down under most conditions when a wide range in ionic strength is considered, but that the nonlinearity is usually weak and could be taken as linear over a small range. Parametric studies indicate that the protein and surface charge
Roth and Lenhoff
densities act to affect the steepness of the curve; the equilibrium constant is most affected at low ionic strengths. The Hamaker constant characterizes the van der Waals interactions and generally only shifts curves up or down, without affecting their shape. The size of the colloidal particle is seen to be important in both electrostatic and van der Waals interactions, with an increase in size lessening the contribution of the former and increasing that of the latter. Some counterintuitive behavior can be explained in terms of entropic effects, which can result in seemingly favorable electrostatic conditions becoming repulsive. As a consequence, the equilibrium constant can actually increase with increasing ionic strength. The modeling of the electrostatic effects requires the greatest amount of computational effort. Hence, the issue of appropriate yet computationally reasonable models is of great importance. Utilizing the experimentally measurable properties of size and net charge results in a good approximation for lysozyme, but this is not expected to hold as well for other proteins. Incorporation of the protein net dipole into the model is the next simplest approximation which more realistically represents the physical situation within a protein. Since results for the proteins studied show a significant effect in the one case (chymotrypsinogen A) and not in the other (lysozyme),it seems reasonable that this inclusion is a realistic improvement in the model, but testa against the complete orientational model and against experimental isotherms remain to be done before ita value can truly be assessed.
Acknowledgment. We are grateful for support received from the National Science Foundation under Grant Number CTS-9111604.
