Langmuir 1994,10, 3705-3713
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Electrostatic and van der Waals Contributions to Protein Adsorption: 2. Modeling of Ordered Arrays C. A. Johnson, P. Wu, and A. M. Lenhofl Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received November 15, 1993. In Final Form: August 11, 1994@ To understand and predict the adsorption behavior of proteins at high surface coverages, an accurate descriptionof lateral interactions among adsorbed molecules is imperative. We model protein adsorption as the formation of an ordered array of charged particles at a surface, considering both protein-protein and protein-surface interactionswithin the array. Energies are calculated as a combination of electrostatics, based on the solution of the linearized Poisson-Boltzmann equation, and van der Waals interactions, evaluated by painvise additive summation. A method is developed for the construction of adsorption isotherms on the basis of a Gibbs surface excess approach. The sensitivity of the system energetics to parameters describing the protein, the surface, and the solvent is assessed, and results indicate that double-layer screening in the electrolyte medium plays a predominant role in the adsorption process. Theoretical isotherms are generated for a spherical approximation of lysozyme which show trends that are consistent with experimental measurements. The model predicts the coexistence of multiple states of adsorption for certain conditions. Multiple equilibria in the theoretical isotherms provide possible explanations for protein adsorption phenomena such as the increase in surface coverage at high ionic strength and the appearance of irregularities in experimental isotherms of some proteins.
Introduction The adsorption behavior of proteins has been widely studied, motivated largely by its essential role in such applications as biomaterials selection and chromatographic separations. Optimal design procedures in each of these applications would be aided by capabilities to model adsorption behavior predictively, preferably using realistic mechanistic descriptions free of adjustable parameters. Unfortunately the complexity of both the protein molecules and, in many cases, the adsorbent surfaces has made this goal an elusive one. As a result, many models used to characterize adsorption behavior are descriptive rather than mechanistic, the most obvious example being the widespread use of the Langmuir formulation, which, as has often been observed,lZ2deviates in several key respects from known protein adsorption characteristics. While the parameters extracted using such models may be useful for qualitative and semiquantitative discussions of parametric effects on adsorption behavior, they lack the predictive capabilities that represent a more desirable objective in most cases. A variety of approaches have been used in the quest for realistic mechanistic models. These are reviewed in a companion paper,3 where the principles of colloid science have been applied as the basis for using protein and surface properties for predicting adsorption equilibria. That work, which included treatment of electrostatic and van der Waals interactions, considered only protein-surface interactions, so that the results are applicable only to adsorption at relatively low surface coverages. This limitation is shared by almost all other efforts to model protein adsorption, and it is an important one to relax, in view of the high-coverage adsorption encountered in applications such as overload and displacement chromatography. The modeling of adsorption behavior at high coverages requires a consideration of lateral interactions between adsorbed protein molecules in addition to the proteinAbstract published inAdvanceACSAbstracts, October 1,1994. (1)Norde, W. Adu. Colloid Interface Sci. 1986, 25, 267. (2)Andrade, J. D.; Hlady, V. Adu. Polym. Sci. 1986, 79, 1. (3) Roth, C. M.; Lenhoff, A. M. Langmuir l998,9, 962.
surface interactions treated previously. The most obvious lateral interactions are simple excluded volume effects that determine the upper limit of monolayer coverage, albeit these interactions are more complicated to characterize for proteins than for the more commonly used hard-sphere models. The importance of more specific lateral interactions has been demonstrated by prior experimental work that has sought to elucidate molecularlevel characteristics of adsorbed protein molecules. For example, Lee and Belfort4s5used a surface forces apparatus to follow the adsorption of ribonuclease A on mica. They interpreted their measurements as indicating a change in the orientation of adsorbed molecules as the surface coverage increased, with the protein-protein interactions at high surface coverages causing the initial orientation to become suboptimal. Other evidence of lateral interactions is the observation by scanningtunnelling microscopy (STM) that lysozyme forms two-dimensional arrays with a high degree of order when adsorbed on graphite.6 The goal of the work described here is to extend to high surface coverages the model developed previously to determine adsorption equilibrium constants at low surface coverages. The methodology is thus the same as that used previously, namely a colloidal description in which the protein molecules are treated individually, but other system features (e.g., solvent and electrolyte) are described within a continuum formulation in which electrostatic interactions and van der Waals attractions are considered. In practical terms, extending the model to high coverages requires including protein-protein interactions in the calculations, and this requires knowledge of the locations and orientations of the adsorbates a t the solid-liquid interface. The positioning of adsorbates at the surface has been suggested, a t least for one experimental system, by STM experiments, which indicate that lysozyme adsorbed on graphite self-assembles into two-dimensional arrays of quite regularly spaced protein molecules.6 In addition, observed salt effects suggest that electrostatics
@
(4) Lee, C.-S.;Belfort, G.Proc.Natl.Acad. Sci. U S A . 1989,86,8392. ( 5 )Belfort, G.; Lee, C.-S.Proc.Natl. Acad. Sci. U S A . 1991,88,9146. (6) Haggerty, L.; Lenhoff, A. M. Biophys. J. 1998, 64, 886.
0743-746319412410-3705$04.50100 1994 American Chemical Society
3706 Langmuir, Vol. 10, No. 10, 1994
are involved in determining the arrangement of molecules in the arrays. We develop our model of protein adsorption from these observationsby using a two-dimensionalordered array to define the locations of protein molecules a t a solid, planar surface. The electrostatic and van der Waals interactions involved in the formation of the ordered array at the solidliquid interface are evaluated, allowing for interactions among particles on the surface, as well as the interactions of each particle with the surface. By varying the periodicity of the array, the model allows the calculation of adsorption energetics at low surface coverages, where interactions among adsorbed protein molecules are negligible, and at high surface coverages, where proteinprotein interactions play a critical role in determining the observedbehavior. The effects of variations in surface and solvent conditions on the adsorption energetics are investigated over the full range of surface coverages. The energy calculations are combined with a Gibbs surface excess method for computing equilibrium constants for particle adsorption in order to predict full adsorption isotherms. Because of the assumption of perfect ordering as well as additional simplificationsmade in the interests of tractability, the model is a conceptual rather than a rigorously realistic one. Nevertheless, the results display many of the key features seen experimentally and suggest possible explanations for observed behavior not fully explained previously.
Theory Model Description. The adsorption model is an idealized system comprised of a solid, planar surface, a liquid medium, and an assembly of identical, colloidal particles. Protein moleculescan be accuratelyrepresented by the model particles using atomic coordinates, available for many proteins from the Brookhaven Protein Data Bank.7 In this work, however, we have simplified the calculations by representing each protein molecule as a sphere with the net charge placed at the center. Although the symmetry of the spherical idealization precludes any evaluation of possible anisotropiceffects in the adsorption process, earlier work3 has shown that the quantitative accuracy of the more detailed model is mostly retained with the spherical approximation for protein molecules with relatively uniform charge distributions. Thus while the accuracy of the model is somewhat compromised with the spherical particles, the computational complexity is greatly reduced, thereby allowing other important parameters in the adsorption process to be examined more closely. The locations of the spherical particles in the model are defined by a two-dimensional, ordered array positioned parallel to the planar surface, with a nearest distance z between the array and the surface (Figure 1). In this work, the shape of the unit cell of the array is restricted to be any rhombus; i.e., the four sides of the unit cell are equal in length. Thus the array is defined by a unit cell shape and an interparticle spacingd defined as the nearest distance between adjacent particles. The interparticle spacing can range from zero, for a close-packed array, to very large values, for noninteracting particles a t low surface coverages. Although the requirement of an ordered array is certainly unrealistic at very low coverages, where particle-particle interactions should be weak and a random placement of protein molecules is more likely, the adsorption energeticswill still be adequately predicted ( 7 ) 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.
Johnson et al.
lnflnlte planar surface wlth constant surface charge denslty, o
Figure 1. Schematic diagram of the model for an ordered array of adsorbed particles. Protein molecules are approximated as monodisperse spheres, with separation d between surfaces of adjacent spheres and nearest distance z between the planar surface and the spheres.
since the energy per particle is similar to that for an isolated particle. As the coverage increases, the lateral interactions begin to influence the energy calculations, and two-dimensional order becomes more likely; this ordering can occur by virtue of interparticle repulsion, as is well known for three-dimensional suspensions of charged particles.8 Although it is not clear under what conditions the ordered array assumption is truly valid, we apply it throughout for consistencybecause it facilitates the calculation of energies at all coverages. , The colloidal particles are treated as distinct species, but in accordance with the continuum approach of the model, other species in the medium, i.e., water and salt molecules, are lumped together and treated as a continuous phase, the properties of which are defined by the ionic strength I , the pH, and the dielectric permittivity. The planar surface is also treated as a continuous phase with properties defined by a surface charge density 0. Additional material properties accounted for are the Hamaker constants characterizing particle-particle and particle-surface interactions in an aqueous medium. The continuum description becomes questionable for very short-range interactions, but we apply it throughout in the interest of tractability; even a t very small gap distances, the physical picture should be captured at least approximately. With this continuum colloidal formulation,the energetics are calculated for the electrostatic and van der Waals interactions of each particle with the adsorbent surface and with the other particles in the ordered array, which leads to the calculation of the Helmholtz free energy change for the incorporation of a single particle from the bulk solution to the ordered array. Since the colloidal particles in the model are consideredto be at fixed positions with no Brownian motion, entropy and thermal motion are not considered in the calculations, but their effects are evaluated in a qualitative sense by comparison of the thermal energy of a particle, kT,with its free energy. Thus the macroscopic adsorptionprocess is described by treating the protein solution as a complex fluid, with the microscopic Helmholtz free energy providing the potential that allows the calculation of the adsorption equilibrium. Electrostatic Interactions. Electrostatic effects in proteins and colloidal particles have been described quantitatively through different approaches, and reviews of these works are a ~ a i l a b l e . ~ -Our l ~ treatment of the (8)Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (9) Matthew, J. B. Ann. Rev. Biophys. Biophys. Chem. 1985,14,387. (10) Harvey, S.C. Proteins 1989, 5, 78. (11)Rogers, N.K.Progr. Biophys. Mol. Biol. 1986,48, 37.
Adsorption of Ordered Protein Arrays
Langmuir, VoE. 10, No. 10, 1994 3707
electrostatics is an extension of a previous modePJ3 that describes in a continuum formulation the interaction between a single, charged colloidal particle and a charged planar surface with an aqueous electrolyte as the intervening medium. In both models, the planar surface is assumed to be homogeneous and of constant surface charge density, u. The dielectric constant of the solid surface is taken to be much smaller than that of the solvent, so that the potential variations within the adsorbent solid can be ignored. A diffuse double layer of mobile ions is assumed to form around the charged particle and charged surface. The electrostatic interaction energy is calculated by solving numerically for the electrostatic potential, @, throughout the continuum system. Within each protein molecule, @ is described by the Poisson equation: V2Q.'= -
LL EIEO
(1)
where e' represents the charge distribution within the particle, is the dielectric constant of the protein interior, and EO is the dielectric permittivity of free space. In the intervening medium, @ is described by the linearized Poisson-Boltzmann equation: V2Qe = K2Qe
(2)
for a Z:Z electrolyte solution, where K~ = 2e2Z2gw1ce~&T, e is the electronic charge, 2 is the ion valency, is the dielectric constant of the electrolyte medium, k is the Boltzmann constant, T i s the temperature, and e" is the bulk concentration of the electrolyte before dissociation. This equation in its linearized form is strictly valid only for systemswith potentials much less than 1kTIe; however, studies have shown good agreement between the linear and nonlinear equations for potentials up to several kTl e.14216 With the complex system geometry, a numerical solution to the governing equations is necessary. We have employed a boundary element technique using the Green function formulation of the problem, which is described elsewhere.13,16 The modifications made to this calculation to accommodate the periodicity of the ordered array are shown in the Appendix. The free energy of the doublelayer system is calculated for the linearized description from12J3
."
where aP represents the charged, planar surface and Qk and xk represent the magnitude and location, respectively, of charge k in the particle. This calculation is typically performed for one particle in the lattice. van der Waals Interactions. The Hamaker method of pairwise summation of forces between macroscopic bodies17-19 is used to determine the van der Waals contribution to the free energy of formation of the ordered array. For protein molecules that are approximated as (12) Sharp, K. A.; Honig, B. J.Phys. Chem. 1990, 94, 7684. (13)Yoon, B. J.; Lenhoff, A. M. J. Phys. Chem. 1992,96,3130. (14)Hogg, R.; Healy, T. W.; Fuerstenau, D. W. Trans.Faraday SOC. 1966, 62, 1638. (15) Overbeek, J. Th. G . J. Chem. SOC.,Faraday Trans. 1 1988,84, 3079. (16)Yoon, B. J.; Lenhoff, A. M. J. Comput. Chem. 1990, 11, 1080. (17) Hamaker, H. C. Physica 1937,4, 1058. (18)Hunter, R. J. Foundations of Colloid Science, Vol. 1 ; Oxford University Press: London, 1986. (19) Israelachvili,J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992.
spheres, the van der Waals interaction energy of each molecule with the infinite planar surface is's
where R is the radius ofthe sphere, z is the nearest distance between the sphere and the plane, andA132is the Hamaker constant for the interaction of bodies 1 and 2 through medium 3. The van der Waals interaction among particles in the array is found by calculating the interaction of a single sphere with each sphere in the adjacent rows around it, summing the individual energies, and dividing by 2 to correct for double-counting. Each sphere-sphere interaction energy id8 hFsphere/sphere
2R2 - -6 d2+4Rd
2R2
+
d2+4Rd+4R2
In( d2 d2 4Rd
+
+
+
+ 4R2)} ( 5 )
where d is the nearest distance between the two spheres andA131is the Hamaker constant for two bodies ofmaterial 1in medium 3. Equations 4 and 5 are most accurate in the absence of many-body forces, which could slightly reduce the van der Waals attraction between two macrobodies in the vicinity of other macrobodies.20 The effect of many-body forces is neglected to maintain simplicity in the van der Waals calculations. In addition, electrolyte screening ofvan der Waals forces is neglected in the model. Opposing views exist as to whether such an effect is appreciable,18~21~22 but as the latter authors point out, the effect occurs only through the screening of the zerofrequency contribution to the Hamaker constant and is appreciable only for the interaction of similar materials having strong overlap of their W spectra. A finite limit must be placed on the allowable separations between macroscopic bodies in the model to mimic the short-range repulsive forces, such as Born repulsion and steric hindrance, that arise during contact of the surfaces of real protein molecules. Without such limits in the model system, the free energy drops to --oo as either d or z approaches zero, as a result of the singularities that arise in eqs 4 and 5. The choice of cutoff distances could have a significant effect on the predicted adsorption behavior since the sphere-sphere cutoff determines the highest allowable surface coverage in the model and the sphere-surface cutoff defines the lower limit of the thermodynamicintegration in the calculation of isotherms (discussed below). Both cutoff distances, zo and do, should represent the distances at which short-range repulsion begins to dominate over the relatively longer range interactions that are considered in this model. A value of 1A is somewhat arbitrarily chosen for both cutoffs. The individual van der Waals components are summed and added to the electrostatic free energy to give the total interaction of a charged particle in an ordered array of adsorbed particles. The assumption that the electrostatic and van der Waals interactions are additive is the basis of the classic DLVO t h e ~ r y ,and ~ ~although , ~ ~ there is some (20) McLachlan, A. D. Discuss. Faraday SOC. 1966,40, 239. (21) Parsegian, V. A. Physical Chemistry: Enriching Topics from Colloid and Surface Science; Theorex: La Jolla, CA, 1975. (22) Mahanty, J.; Ninham, B.W. Dispersion Forces;Academic Press: New York, 1976. (23) Dejaguin,B.V.;Landau,D.L.ActaPhysicochim. U.R.S.S. 1941, 14, 633. (24)Verwey, E. J. W.; Overbeek, J. Th. G . Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948.
3708 Langmuir, Vol. 10,No.10, 1994
debate as to its validity, we use the assumption in the absence of a more plausible theory. Thermodynamic Integration and Modeling of Adsorption Isotherms. Most existing models of protein isotherms are based on the Langmuir model, which assumes the presence of identical and noninteracting sites is not of adsorption and thus, as discussed realistic for the situation considered here. Other mode l ~ are ~ , based ~ ~ on , the ~ ~Gibbs surface excess approach to adsorption, which assumes that there is a smooth concentration profile of adsorbate molecules from the surface of adsorption to the bulk solution. With the Gibbs model, the probability of finding an adsorbate molecule at a certain location and orientation relative to the surface is given by a Boltzmann distribution with respect to the energy of interaction of the molecule with the surface. As discussed previ~usly,~ this model leads to a method of thermodynamic integration that relates surface concentration cs to bulk concentration c b in the form
where z is the distance of the particle from the surface, and the orientation dependence included previously has been neglected because the present work is applied to spherical particles only. In eq 6, AF is the free energy of a solute particle, going from the bulk solution to a distance z from the planar surface, andzo is the minimum distance of approach of the particle to the surface, as discussed above. In the limit of large d (very low surface coverages, where lateral interactions among adsorbed particles are negligible), there is no d dependence, and the situation corresponds t o that discussed previ~usly.~ Equation 6 can be used to yield the equilibrium constant, Keq= C&b, the ratio of the surface to the bulk concentration. (Kenis termed the equilibrium constant, although its value vanes if high surface coverages are considered.) To allow for the effect of d at higher surface coverages, Keq must account for the lateral interaction among particles in the adsorbed array, such that
where AFIat(df)is the change in free energy per particle due to lateral interactions occurring when an adsorbed array ofvery wide interparticle spacing, i.e., with negligible interaction between particles, is compressed laterally to a final interparticle spacing of df at a fixed distance zo from the surface. Equation 7 is used to find Keqfor a given set of system parameters and array dimensions, and c, is determined strictly by the array dimensions. Thus for a given array, Keqand cs are found, from which cb is calculated as c$Keq,allowing the full isotherm to be generated. The configurational integral (eq 7) for the equilibrium constant can also be formulated in alternative ways, but the one other case that has been examined yields essentially the same results as are described here.27
Methods We use a boundary element method to calculate the electrostatic potentials in the model. The methodology requires a mathematical description of the protein mol(25) Barker, J. A.; Everett, D. H. Trans. Faraday Soc. 1962,58,1608. (26) Sthhlberg, J.; Jtinsson, B.; Horvath, Cs. Anal. Chem. 1991,63, 1867. (27) Johnson, C. A. Protein Adsorption: Modeling of Ordered Arrays and Observations by Atomic Force Microscopy. M.Ch.E. Thesis. University of Delaware, Newark, DE, 1994.
Johnson et al. ecules in which each molecule is discretized into a finite number of small boundary elements. The calculations were performed on an IBM RISC 6000/560. For a spherical particle discretized into 320 boundary elements, about 1 min of CPU time was required to calculate the nodal potentials and to compute the free energy per particle in the ordered array. To generate adsorption isotherms, equilibrium constants were extracted from the free energy data through the numerical integration of eq 7. In similar ~ t u d i e s , ~ this integration was performed using Gaussian quadrature after a polynomial transformation of the integrand was made in order to minimize the number of integration points. In our work, however, the computational time was short enough to allow the calculation of the free energy for a large number of configurations. Hence, the step size was made small enough (0.25 A in the region of steepest variation) to provide an accurate estimate through numerical integration using Simpson's rule.
Results and Discussion Predictions of the model are explored for parameters on the basis of the following reference conditions. Molecules of the protein lysozyme are selected as the model adsorbate particles, and each is approximated as a sphere of diameter 30 A with a point charge of $8 electron charge units (the net charge on lysozyme at pH 7) at its center. The interior of each sphere is assigned a dielectric constant of ci = 4. A hexagonal lattice is assumed for the ordered array under the reference conditions. The aqueous electrolyte medium is assigned an ionic strength of 0.1 M, which for a 1:lelectrol e such as NaCl corresponds to a The assigned ionic strength Debye length of 9.6 influences the solution to the electrostatic model through the value of the parameter K , the inverse of the Debye length, appearing in the Poisson-Boltzmann equation. The dielectric constant of the medium is set at ce = 80, a value appropriate for an aqueous medium. The planar adsorbent surface is chosen to be negatively charged, with a surface charge density u equal to -4 pClcm2, which is on the order of the charge densities reported for charged polystyrene28 and silica surfaces.29 Surface potentials resulting from these conditions are calculated with the Grahame equation to be about 2kTle, a magnitude low enough to maintain reasonable accuracy with the linearized model. The Hamaker constants, A131 and A132, are both assigned avalue of J, which is within the range expected for dispersion forces involving proteins.30 The free energy of the ordered array is first examined as a continuous function of two characteristic parameters of the array, the interparticle spacing d and the vertical distance z between the planar surface and the array. The results for the reference conditions are plotted as a threedimensional surface diagram (Figure 2) in which the two horizontal axes represent the array parameters d and z , and the vertical axis represents the free energy AF of a particle in an array defined by the corresponding values of the variables d and z . Some general trends in the behavior of A F in the model are suggested by the curved surface of Figure 2. Where both d and z are large (as for point A in Figure 21, A F is calculated to be zero, as one would expect when each particle is too far from all other particles and from the planar surface to feel the effects of electrostatic or van der Waals forces. If the vertical distance z between the isolated particle and the planar
2
(28) Norde, W.; Lyklema, J. J. Colloid Interface Sci. 1978,66,257. (29) Scales, P.J.; Grieser, F.; Healy, T. W.; White, L. R.; Chan, D. Y. C. Langmuir 1992,8, 965. (30)Af'shar-Rad, T.; Bailey, A. I.; Luckham, P. F.; MacNaughtan, W.;Chapman, D. Biochim. Biophys. Acta 1987,915,101.
Adsorption of Ordered Protein Arrays
Langmuir, Vol. 10, No. 10, 1994 3709 10
!-I 5 LA
Q
-10
spaclng, d
(A)
25
separation, z
(A)
Figure 2. Three-dimensionaldiagram showingthe free energy of an ordered array of charged spheres as a function of two
variables, the interparticle spacing d and the sphere-surface separation z. Particle parameters: diameter = 30 A, point ~ 9.61 charge = +8, ci = 4. Solvent parameters: I = 0.1 M ( 1 / = A>,co = 80. Surface charge: o = -4 pC/cm2.A131=A132 = J. The lattice is hexagonal. surface is reduced while d is kept constant, which corresponds to moving along the plot from point A to point B in Figure 2, the free energy drops monotonically, with its steepest dope a t very small values ofz. This behavior is attributed to both electrostatic attraction between opposite charges and van der Waals attraction at small sphere-surface separations, and it corresponds to the low coverage adsorption situation examined previ~usly.~ The dependence of hF on the interparticle spacing d at large values of z is revealed by the path between point A and point C in Figure 2. This corresponds to the conceptual process of forming a two-dimensional array in free solution, far from the surface. As the interparticle spacing is first reduced, AF rises gradually due to electrostatic repulsion among the similarly-charged spheres in the array. Then at very small interparticle spacing, the van der Waals attraction among the spheres becomes dominant, causing AF to reach a peak and to drop sharply as d approaches zero. In the region of the plot where both d and z have small values, a convolution of these two basic trends occurs. The behavior of hF in this region, where each particle in the array interacts strongly with the surface and with other particles, is most important to the model at high coverages, and although this part of the plot is obscured in Figure 2, it is examined more thoroughly later as a function of the system parameters. In view of our primary interest in the state of adsorbed particles, the remaining energy diagrams in this work are shown for a constant sphere-surface separation of 1 while the interparticle spacing is varied continuously, rather than both distances being varied. The influence of various system parameters on the energetics of the model was investigated by repeating the computations while the system parameters were independently varied from their base values. The energetics of the ordered array at surfaces of several different charge densities are shown in Figure 3. In this series of energy curves, the surface charge density ranges from o =0, for an uncharged surface, to o = -8 pC/cm2, the approximate charge density of a mica surface in a 0.1 M aqueous KN03 s01ution;~lthe validity of the linearized Poisson-Boltzmann equation (31)Israelachvili,J. N.; Adams, G.E.J.Chem. SOC.,Faraday Trans. 1 1978, 74, 975.
0
5
10
15
20
25
Interparticlespacing, d (A)
Figure 3. Influenceof the surface charge density 0 on the free
energy of the ordered array at the planar surface. z is held constant at 18.Other parameter values as in Figure 2.
is, however, questionable at the higher charge densities. Each curve represents the free energy of adsorption for a single particle in the ordered array as a function of the interparticle spacing. The shape of the energy curve changes very little as a function of 0,but as o becomes more negative, the entire energy profile shifts to a lower energy level because of the greater electrostatic attraction between the positively charged particles of the array and the negatively charged surface. At very large interparticle spacing, the curves level off at values equal to the energy of interaction of a single charged particle with the planar surface. The asymptotic value for the top curve, i.e., for o = 0, is still well below zero, even though the surface is uncharged, which reveals mainly the extent to which van der Waals attraction occurs between the adsorbed particle and the planar surface. Our base conditions,as discussed previously,correspond to an aqueous medium with an ionic strength of 0.1 M. The calculations were repeated over a range of ionic strengths to analyze the screening effect of salt at concentrations commonly encountered in protein adsorption situations. The energetics are affected dramatically by changes in ionic strength, as shown by the series of curves in Figure 4. The plateau levels of these curves rise significantly with increasing ionic strength. This plateau energy is the free energy of interaction of a single spherical molecule with the charged surface at a separation of zo. This weakening of the adsorption affinity is attributed to greater double-layer screening of the electrostatic attraction between each sphere and the surface at higher ionic strengths. A phenomenon that is perhaps more relevant to the formation of ordered arrays is the screening of electrostatic repulsion among positively charged molecules at small interparticle spacing, which is seen in Figure 4 as the dramatic reduction of the energy barrier with increasing ionic strength. As this potential barrier becomes smaller, the state of densely packed adsorbed molecules, corresponding to the primary minimum on the energy curve, becomes more readily accessible. The screening of repulsion among molecules also lowers the primary minimum on the energy curve, making the state of high surface concentration thermodynamically more favorable. To examine the dependence of the adsorption energetics on the size of the adsorbate, calculations were performed for a series of particle sizes. The spherical shape of the
Langmuir, Vol. 10, No. 10, 1994
Johnson et al.
Q
0.2
E
er8
0.1
8 0.05 0
10
20
30
40
lnterpartlcle speclng, d (A) Figure 4. Influence of the solvent ionic strength I on the free energy of the ordered array. z is held constant at 1A. Other parameter values as in Figure 2.
0
10
20
30
0
50
40
lnterpartlcle spaclng, d (A)
Figure 5. Influence of particle size on the free energy of the ordered array.z is heldconstantat 1A. Other parameter values as in Figure 2. particle and the monopolar charge of +8 ecu were retained, while the diameter of the sphere was changed from 30 A, for the spherical approximation of lysozyme, to 45 and 60 A. These larger diameters correspond to proteins with molecular weights of about 48 and 115 kDa, assuming a molecular density equivalent to that of lysozyme. Energy profiles for the three particle sizes are shown in Figure 5, which indicates that the energy barrier is rapidly reduced as the particle size increases. Additionally, the plateau energy level at wide interparticle spacing is lower for larger particles. Both ofthese trends can be attributed to the dramatic increase in van der Waals interactions for a sphere of larger radius. The energy barrier is also attenuated by a decrease in electrostaticrepulsion between spheres due to the reduced charged density in the larger spheres. Variations of the Hamaker constants and the unit cell shape have also been explored,27but since the effects are minor compared to those of the parametric variations examined above, these results are not presented here. The adsorption behavior of a protein can be described quantitatively in the form of an adsorption isotherm, which
0
5 10 15 Bulk concentration (mg/ml)
20
Figure 6. Adsorption isotherms calculated for the ordered array model at different ionic strengths.Other parameter values as in Figure 2.
relates the equilibrium surface concentrationcsto the bulk concentration Cb of protein in the solvent. Such quantitative information can be predicted with the present model using eq 7 and the thermodynamic integration method described earlier. Isotherms computed for the reference conditions and for two other ionic strengths are shown in Figure 6. The predicted isotherms are in good qualitative and reasonable quantitative agreement with experimental adsorption isotherms of lysozyme observed in various protein adsorption s t ~ d i e s . As ~ ~expected, - ~ ~ the isotherms are roughly linear at low surface coverages, where particle-particle interactions are negligible. As interparticle repulsion increases with increasing coverage, the familiar convex upward shape emerges in the isotherm; this happens in the absence of any assumptions regarding Langmuirian behavior. The sensitivity of the computed isotherms to ionic strengths is observed in the isotherms of Figure 6. The two predominant effects of increasing ionic strength are the reduction of the slope of the linear region of the isotherm and the reduction of the plateau level of the isotherm. Both of these trends, which are consistent with experimental 0bservations,6~~~,3~ can be attributed to the screening of the electrostatic attraction between the particles and the surface, which weakens the adsorption affinity by reducing the free energy change upon adsorption of a single particle. This predicted lowering of the adsorption isotherm with increasing ionic strength has previously been correlated with the increase in protein solubility with the addition of salt, i.e., saltingin.36-38 Under conditions where the lateral interactions in the ordered array are not dominated by electrostatic repulsion, quite different adsorption behavior may be predicted with (32) Shibata, C. T.; Lenhoff, A. M. J.Colloid Interface Sci. 1992,148, 469. (33) Lok, B. K. Protein Adsorption onto Cross-linked Polydimethylsiloxane Using Total Internal Reflection Fluorescence. Ph.D. Dissertation. Stanford University, Stanford, CA,1982. (34)Van Enckevort, H. J.; Dass, D. V.; Langdon, A. G. J. Colloid Interface Sci. 1984,98,138. (35) Gadam, S. D; Jayaram, G.; Cramer, S. M. J.Chromatogr. 1993, 630,37. (36) Melander, W.; Horvlth, Cs.Arch. Biochem. Biophys. 1977,183, 200. (37) Melander, W. R.; Corradini, D.; Horvlth, Cs. J. Chromatogr.
1984,317,67. (38)Melander, W.R.; El Rassi, Z.; Horvath, Cs. J.Chromatogr. 1989,
469,3.
Adsorption of Ordered Protein Arrays
0
2
4 6 8 Bulk concentration (mg/ml)
Langmuir, Vol. 10,No. 10,1994 3711
10
Figure 7. Adsorption isotherm calculated for spherical particles 45 A in diameter. Other parameter values as in Figure 2. The sigmoidal nature of the isotherm reveals that the model predicts multiple adsorbed states for these conditions.
the model. For example, when the model particles are 45
A in diameter, van der Waals attraction dominates in the
ordered array at small interparticle spacing, as shown by the very small energy barrier and the potential well in the corresponding energy profile of Figure 5. The isotherm computed for this system (shownin Figure 7) reveals more complicated adsorption behavior. In the region of low surface coverage, the isotherm is similar in shape to those discussed previously, but at high surface coverages, the isotherm is sigmoidal in shape. The uppermost section of the isotherm, shown by the horizontal dotted line, corresponds to the upper limit of csimposed by the cutoff distance for the interparticle spacing. The sigmoidal shape of the curve represents the prediction of multiple values of cs for a single Cb: e.g., a bulk concentration of 4 mg/mL is in equilibrium with three different surface coverages, marked as points A-C in Figure 7. The prediction of multiple adsorbed states in the model can be explained in terms of a balance of forces in a DLVO system. In the equilibrium adsorbed state at the lower surface concentration, labeled as state A in Figure 7,the electrostatic repulsion among the adsorbed particles in the array is counterbalanced by the large gain in free energy upon adsorption of the particles to the surface. In the equilibrium state at the higher surface concentration, labeled as state B, the interparticle spacing has been compressed over the energy barrier and van der Waals attraction among the particles compensates to some extent for the electrostatic repulsion. However, when state B is reached, thermal motion in the array should then drive the system into the potential well ofthe close-packed state, labeled as state C. In other words, once the potential barrier is overcome, the system is pulled down into the deep potential well of van der Waals attraction until it meets the vertical wall ofthe short-range repulsion defined by the cutoff separation do. Thus equilibrium state B, when attainable, is not stable. These stability considerations can be analyzed more rigorously within the framework of standard approaches for dealing with phase stability.39 Indeed, Figure 7 and similar isotherms (39) Sandler, S.I. Chemical and Engineering Thermodynamics, 2nd ed.; John Wiley & Sons: New York, 1989.
resemble van der Waals loops, albeit with the axes transposed and the cs axis inverted to represent a specific surface area. Two-phase equilibria of adsorbed molecules have been predicted in previous models accounting for lateral interactions among adsorbates, such as in the approach due to Fowler and Guggenheim,@.whichleads to a modified Langmuir equation, and in the two-dimensionalvan der Waals equation of state.41 In addition, molecular simulations and experimental studies of gas adsorption on solid surfaces both indicate the existence of multiple surface phases and the possibility of phase transitions; examples and overviews of such work are discussed in detail elsewhere.42The models used in those studies are in many respects similar to the one used here, with most of the prior work based on Lennard-Jones and similar potentials, which have the counterpart in our model of van der Waals attraction and the hard core repulsion implied by the cutoff distances do and 20. Since the presence of multiple adsorption states in the model is attributed to a favorable interaction among particles in the close-packed state, the qualitative similarities between the potentials described above are what lead to the similar predictions of multiple adsorbed states. In the present model, the cutoff distance do for particle-particle separations is a key parameter determining the energetics of the state of higher surface coverage. An important difference between our model and earlier descriptions is the inclusion of electrostatic interactions, which add the important feature of long-range repulsive interactions. They also add dependence on ionic strength, a key experimental variable, and it is indeed this dependence that provides much of the motivation for the work presented here. The balance of DLVO forces among charged particles is strongly affected by the ionic strength and can control the accessibility and energetics of the closepacked state. When the ionic strength is raised above a certain level, electrostatic screening should reduce the free energy of the close-packed array t o the point at which it becomes thermodynamically favored over the wider interparticle spacing. For the lysozyme model (30 fi spheres), this transition occurs at an ionic strength of about 0.3 M in our calculations when all other parameters are set to our base conditions. Thus a system that is first characterized by a monotonic isotherm could be characterized by a multistate isotherm at higher ionic strengths. This salt effect, which has similarities to the phenomenon of salting-out, would be observed as a sharp increase in surface coverage or protein adsorption affinity at high ionic strength. The effect has been linked theoretically in a previous work3s to the interplay of electrostatic and hydrophobic interactions, but o u r model predicts the effect in terms of only the DLVO contributions to the interaction of adsorbed particles. Since it is predicted that under certain conditions a system can attain two different equilibrium surface concentrations, one can envision cases where both states occur simultaneously at different areas on the adsorbing surface (in the presence of kinetic limitations). Macroscopically, this would be measured as a surface concentration somewhere between the two equilibrium values. Theoretically, such a bimodal state of adsorption could arise when the close-packed equilibrium state is physically (40) Fowler, R. H.; Guggenheim,E. A. Statistical Thermodynamics; Cambridge University Press: Cambridge, 1952. (41) Ross, S.;Olivier, J. P. On Physical Adsorption; John Wiley & Sons: New York, 1964. (42) Taub, H., Torzo, G., Lauter, H. J., Fain, S. C., Jr., Eds. Phase Transitions in Surface Films 2; NATO AS1 267; Plenum Press: New York, 1991.
3712 Langmuir, Vol. 10, No. 10, 1994 accessible (i.e., both points on the isotherm lie below the limit for close-packed particles), but the difference in free energy between the two predicted equilibria is small (on the order of KT). The drop in free energy would drive the formation of the close-packed array, but thermal and entropic factors would favor the wider spacing. The state that predominates would be determined by the bulk particle concentration, with higher concentrations favoring the close-packed array. This argument may explain the occurrence of “steps” or “kinks’) sometimes observed in the isotherms of different proteins at various surfaces.l The stepped isotherms have been attributed to phenomena such as molecular reorientation of the adsorbed layer: protein conformational changes, or the formation of a second layer, but our work suggests that the steps could be caused by a structural transition in the adsorbed layer as the system shifts from a population of mostly wide interparticle spacing to one of close packing. Our explanation is quite similar to the mechanism proposed by Fair and Jamieson t o explain the stepped isotherms observed for the adsorption ofbovine serum albumin and y-globulin onlatex.43 In their model at low concentrations, adsorption proceeds by a random uncorrelated irreversible mechanism while, at higher concentrations, a transition occurs to a cooperative adsorption mode in which a close-packed ordered monolayer of protein is formed. Their explanation for the phase transition is derived from kinetic principles: at low concentrations, the frequency of collision of a protein molecule with a nucleation site on the close-packed monolayer is too low to maintain growth of the surface crystal while, at high concentrations, the collision frequency is high enough that nucleation and growth of the ordered phase occur. One respect in which the explanation of Fair and Jamieson differs from our model is that they do not require any order in the adsorbed layer when the surface coverage is low. At low coverages,however, the lateral interactions, which in our model hold the particles in an ordered array, are weak in comparison to the system thermal energy, so while the interparticle spacing leads directly to the macroscopic surface coverage, a rigid surface structure should not be assumed. Another difference is that in the model of Fair and Jamieson adsorption is considered irreversible, while in ours a true thermodynamic equilibrium is assumed. Our experiments on charged surf a c e and ~ ~ ion-exchange ~ behavior in general show that protein adsorption on charged surfaces is generally a reversible process. Both our work44and that of Melander et al.38indicate that desorption occurs less readily at high salt concentrations, precisely the conditions under which a close-packed array would be expected. The cooperativity among particles in such an array would preclude their easy desorption.
Conclusions We have developed a mechanistic model to describe the adsorption of colloidal particles based on the electrostatic and van der Waals contributions to the free energy of adsorption. Through the assumption of two-dimensional periodicity in the positions of adsorbates at the surface, it is possible to calculate the lateral interaction in the adsorbed layer in addition to the particle-surface interaction, thus overcoming the limitation to very low surface coverages where particle-particle interactions are negligible. With a spherical, monopolar approximation of a lysozyme molecule as the model adsorbate, the energetics (43) Fair, B. D.; Jamieson, A. M. J. Colloid Interface Sei. 1960,77,
525. (44) Roth,
C.M.; Lenhoff, A. M. In preparation.
Johnson et al. can be computed for a range of array dimensions to reveal the optimal configuration and any potential energy barrier for achieving that configuration. Double-layer screening is demonstrated to have a prevailing effect in determining the thermodynamically favored state of the adsorbed array. The simplifying assumptions of the model limit the accuracy of the calculated results, but the general trends in the adsorption behavior of the model agree with those of known experimental systems. Adsorption isotherms for the spherical approximation of lysozyme, found with the Gibbs surface excess approach, have the familiar convex upward shape with slopes and plateau values that vary with solvent conditions in accordance with experimental observations. Under some conditions, the model predicts multiple surface concentrations for a single bulk concentration of particles. The prediction of such multiple surface states provides an explanation for several aspects of protein adsorption, such as the observation of steps or kinks in isotherms. We propose that these steps between different plateau levels on an isotherm may be the observable result of a transition in the adsorbed layer from a state of widely-spaced particles kept apart by electrostatic protein-protein interactions to a state of close-packed particles held together by van der Waals forces. The model also predicts an increase in surface coverage at high ionic strengths when the close-packed surface structure becomes energetically favored. This phenomenon, seen previously in protein adsorption exp e r i m e n t ~ ,may ~ ~ be related to the rise in adsorption affinity at high ionic strengths in analytical chromatography. It has been attributed in the past to less well understood phenomena such as hydrophobic interact i o n ~ , but ~ * this work suggests that the mechanism may involve only electrostatic and van der Waals interactions. Further improvements to the model will include describing the protein molecule with an atomically detailed shape and charge distribution, rather than as a monopolar sphere. The asymmetric shape and charge distribution ofthe more detailed model are expected to introduce some degree of anisotropic behavior in the energy of adsorption, which could suggest a preferential protein orientation relative to the surface. Energy changes calculated for changes in protein-protein orientation at the surface could indicate whether the close-packed state is a layer of randomly oriented protein molecules or a highly ordered two-dimensional crystal. Such considerations are, of course, also affected by thermal contributions, which have been omitted here but which may be estimated by extending the present model.
Acknowledgment. We are grateful to Norman J. Wagner for providing insight into numerous key issues during the evolution of this work. Both he and Charles M. Roth provided useful comments on the manuscript. We are also pleased to acknowledge the financial support of the National Science Foundation (Grant CTS-9111604) and the National Aeronautics and Space Administration (Grant NAGW-2798). Appendix: Calculation of Electrostatic Free Energy of Array-Surface Interaction The electrostatic contributions to the interaction energies reported here were calculated using a boundary integral formulation that is a n extension of that described earlier.13 That method pertains to determining the potential distribution for different configurations of a single particle and a planar surface, and subsequent
Langmuir, Vol. 10,No. 10, 1994 3713
Adsorption of Ordered Protein Arrays
quadrature to find the Helmholtz free energy using eq 3. This Appendix describes extension of the potential calculations to allow the potential distribution to be found for an ordered array of particles and presents a computationally efficient approach to evaluating the integral in eq 3. The potential is found by matching, on the particle boundary aB, interior and exterior integral representations of the potential q5 and its exterior normal derivative a@/ an.16The interior and exterior representations are integral representations of the Poisson equation and the linearized Poisson-Boltzmann equation, respectively. Since the surface of each particle in an ordered array is treated separately in the boundary integral formulation, the interior integral representation is unaffected from that used previously,13J6and it is only the exterior representation that must be modified. For a single particle near a surface, the exterior representation provides the potential @ at a point x P aB in the form
Since ?$ represents the surface potential in the absence of particles, it is constant, and the first term becomes simply the product of ?$ and the pertinent surface area. In fact, in view of our usual interest in the free energy change AF,it is generally not necessary to evaluate this term at all. The other two terms involve expressions of the form
for the evaluation of which it is convenient to reverse the order of integration to
The outer integral is evaluated using the boundary element method, while the inner one can be evaluated very efficiently as follows. Let x = (xy,z)and x’= (x’y’,~’), with the plane aP representing thexy plane (z = 0). Then
+
[r2
where Ge,the Green function of the linearized PoissonBoltzmann equation in the presence ofthe surface, is given by
Here y’ is the mirror-image point of x’, and the positive (negative) sign correspondsto the assumption of constant surface charge (potential)on the planar surface. ?$ denotes the potential field of the surface in the absence of particles, which is given simply by the one-dimensionalsolution to the linearized Poisson-Boltzmann equation for an isolated planar surface. It is eq A.l that must accommodate the presence of additional particles in an ordered array. This is done by taking a lattice sum to change the Green function G”,defined in eq A.2, to the periodic Green function
G~(Ix- ~ ’ 1 ) =
C G“(~x- (x’+ %)I)
(A.3)
m
+
where x, = ml a m2 b, 1711,1722 = 0 , f l , f2, ..., with a and b the vectors determining the structure of the unit cell. It can be shown that Gi is convergent. The integral in eq 3 is combined with eq A. 1 to give, for constant surface potential conditions,
(A.7)
where the last equality is obtained by a transformation to cylindrical coordinates. The integration, too, can be expressed in cylindrical coordinates, and following substitution of eq A.2, the inner integrals can be evaluated analytically. Thus for constant surface charge conditions, the free energy of a system comprising a single particle and a charged surface becomes
n
where n,, is the z’ component of the outwardly directed unit normal at x’ on the particle surface. The same expression can be shown to apply to free energy calculations for a particle in an ordered array parallel to the surface, as follows. The inner integral in eq A.6 now has in the integrand the periodic Green function,eqA.3, and the domain is part of aP corresponding to a single unit cell, say aPo,for which ml = m2 = 0. Thus the inner integral is of the form
