5134
J. Phys. Chem. 1996, 100, 5134-5139
Surface Site Heterogeneity and Lateral Interactions in Multipoint Protein Adsorption R. D. Johnson,† Z.-G. Wang, and F. H. Arnold* DiVision of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125 ReceiVed: August 15, 1995; In Final Form: NoVember 1, 1995X
Studies carried out using engineered proteins have demonstrated that protein adsorption to functional surfaces involves multiple interactions between specific groups on the protein and complementary binding sites distributed on the surface. A consequence of multipoint interactions is that protein binding affinity should depend strongly on the distribution of surface binding sites. In this investigation we present a thermodynamic framework for multipoint protein binding to a random arrangement of surface binding sites that also includes lateral interactions among adsorbed protein molecules. This framework results in reversible adsorption behavior analogous to that predicted by the Temkin model and chromatographic behavior analogous to that predicted by the “stoichiometric displacement” model (SDM). Using this framework we can now interpret the semiempirical parameters obtained using these models for protein binding in chromatographic systems in terms of thermodynamic parameters for protein-surface interactions. We show a correlation between Temkin model parameters for a series of cytochrome c variants in immobilized metal affinity chromatography (IMAC) that is consistent with protein adsorption to a nonuniform arrangement of surface binding sites. Lateral interactions among adsorbed protein molecules are shown to be insignificant for this system.
Introduction Protein adsorption to functional surfaces often involves multiple interactions between a single protein and a number of individual binding “sites” distributed on the surface.1-4 As protein binding affinity will strongly depend on the arrangement of surface binding sites, multipoint interactions can be expected to result in heterogeneous binding to a surface with a random arrangement of binding sites. A protein will show the highest affinity for surface arrangements which best match its own distribution of functional groups and will show lower affinity for less favorable arrangements. The net binding energy will therefore decrease with increasing protein coverage, as less favorable sites become more highly populated.5 Such negatively cooperative binding has been frequently observed in chromatographic systems1-4 and suggests that protein binding to these materials is indeed heterogeneous.5 However, there is an alternative source for negatively cooperative binding: unfavorable interactions among bound protein molecules. If the lateral interactions among proteins adsorbed at adjacent surface sites are unfavorable, then even for homogeneous surfaces binding affinity will decrease as neighboring sites become more populated.6 Although it is, in general, difficult to distinguish between the two mechanisms,7 identifying the dominant mechanism would have important consequences for quantitative descriptions of protein separations and for efforts to design new materials for protein recognition via specific multiple-site binding.8 To elucidate the molecular mechanisms of protein adsorption to chromatographic materials, we have studied the binding behavior of a series of engineered proteins in immobilized metal affinity chromatography (IMAC).4,5 Two unique features of IMAC make it an excellent system for studying protein adsorption to functional surfaces.9 First, we can manipulate the * To whom correspondence should be addressed. Telephone: 818-3954162. FAX: 818-568-8743. † Present address: Genentech Inc., 460 Point San Bruno Blvd., South San Francisco, CA 94080. X Abstract published in AdVance ACS Abstracts, February 15, 1996.
0022-3654/96/20100-5134$12.00/0
number of interactions between a protein and the surface by controlling the density of surface metal sites and by controlling the number of metal-coordinating groups on the protein4 (e.g., histidines and/or deprotonated amines in the case of copper). Second, metal-to-ligand interactions are characterized by relatively large binding energies (∼5 kcal/mol for copper-imidazole8); therefore each microscopic interaction between a protein and a metal complex on the surface can be expected to have a significant effect on macroscopic protein binding phenomena of adsorption isotherms and chromatographic retention. In this investigation we present a thermodynamic description of multipoint protein binding to a random arrangement of surface binding sites and include lateral interactions among adsorbed protein molecules. We derive the conditions under which these adsorption mechanisms result in behavior analogous to the Temkin and “stoichiometric displacement” models previously used to describe protein adsorption and chromatography in IMAC5,10,11 and other chromatographic systems.12-14 We also show that the binding behavior of a series of protein variants is sufficient to distinguish between the roles played by these two mechanisms. These results demonstrate that heterogeneous adsorption of cytochrome c in IMAC is due primarily to a nonuniform arrangement of surface binding sites. There does not appear to be a significant contribution from lateral interactions among adsorbed protein molecules. Model Multipoint Binding with No Lateral Interactions. To develop a general thermodynamic model for protein adsorption by multipoint binding, we first divide the surface into individual “cells” as illustrated in Figure 1A. Division of the surface into cells is the simplest and most convenient way to incorporate the excluded volume interactions between adsorbed proteins. This treatment is valid for low to moderate surface coverages but becomes inadequate at high coverages. A more complete treatment of the excluded volume effects will necessarily involve significant mathematical complexity to describe the packing of (generally) irregularly-shaped objects on a two-dimensional © 1996 American Chemical Society
Surface Heterogeneity in Protein Adsorption
J. Phys. Chem., Vol. 100, No. 12, 1996 5135
Figure 1. Illustration of the thermodynamic model for multipoint protein binding . (A) The surface is divided into individual “cells,” each containing some number of individual binding sites (filled circles). Each cell can be either empty or occupied by protein, with the binding energy for protein adsorption dictated by the quality of the match between binding sites on the surface and functional groups on the protein. (B) The net protein binding energy depends on both proteinprotein and protein-surface interactions. Each protein-surface interaction is described by an energy Ep. The net contribution from proteinsurface interactions depends on the number of individual binding sites on the surface that match the location of functional groups on the protein; the cell on the left has three while the cell on the right has only two. The net contribution from protein-protein interactions depends on the number of neighboring cells occupied with protein and is described by an energy w.
surface.15 The cell construction thus provides a convenient, though crude, method for describing the specificity in the interaction between the groups on the protein and complementary binding sites on the surface, one of the main focuses of this study. Each cell contains some number of individual binding sites (e.g., immobilized CuIDA) and therefore supports a specific number of protein-surface interactions with a particular net binding energy. These cells can be either empty or occupied by protein. A protein occupies a cell by binding to the surface sites through m′ out of a maximum of m functional groups (e.g., histidines or amines), with the energy for each individual interaction given by Ep (Ep > 0 for a favorable interaction), illustrated in Figure 1B. At this stage we will assume that there is no interaction between different cells (w ) 0). Considering a particular cell j to be an open system with the protein having a chemical potential µp, the partition function for the cell is given by
Zj ) 1 + exp(βµp + βEpm′j)
(1)
where m′j is the number of surface sites within cell j that bind to the protein (m′j e m) and β is 1/kT. The measured amount of adsorbed protein Q (moles of protein/milliliter of IMAC support) is the spatial average over all cells of the probability that the cell is occupied times the total number of cells Np (moles/milliliter of support) on the surface. If the number of cells is sufficiently large, then this average is the same as an average over m′ for any given cell. Assuming that the unnormalized probability distribution for m′ is given by P(m′), then the amount of protein adsorbed to the surface Q (moles of protein/milliliter of support) is given by m
βEpm′
(cp/c0)e
Q ) Np ∑ P(m′) m′)0 1 + (cp/c0)eβEpm′
/
∑ P(m′)
P(m′) ) pm′(1 - p)m-m′
(2)
(3)
High Binding Site Density: Homogeneous Surface. We can consider two limiting cases for the probability distribution P(m′). At very high binding site density, p approaches 1 (ln(p/1 - p) > 1/m) and the dominant term in P(m′) becomes m′ ) m. Under these conditions we obtain the following expression for the adsorption isotherm,
(cp/c0)eβEpm Q ) Np 1 + (cp/c0)eβEpm
(4)
This expression can be recognized as the Langmuir isotherm for adsorption to homogeneous surfaces,6 where saturation coverage corresponds to occupation of all the surface cells (Np), and the equilibrium constant (exp(βEpm)/c0) corresponds to m individual protein-surface interactions. Therefore at high binding site density, each of the m functional groups on the protein has no difficulty finding an appropriate match among the dense array of binding sites on the surface; the surface is effectively homogeneous. Moderate Binding Site Density: Heterogeneous Surface. At more moderate binding site densities, p ∼ 0.5 (|ln(p/1 - p)| < 1/m) and the probability distribution P(m′) is approximately independent of m′. If we approximate the summation in eq 2 by an integral, we then obtain a closed-form expression for the adsorption isotherm,
Q)
m
m′)0
where cp is the protein solution concentration (moles/liter) corresponding to a chemical potential µp ) kT ln(cp/c0), and c0 is the standard state protein solution concentration of unit activity (moles/liter).16 The problem is now reduced to determining the form of the probability distribution for P(m′). We can distinguish between the two limiting cases of spatially redundant surface sites and spatially distinct surface sites. In the case of spatially redundant surface sites, the functional groups on the protein are able to reach and bind to all of the surface binding sites within the cell. The net binding energy is therefore dictated by the total number of binding sites within each cell. In this case P(m′) can be shown to follow binomial or Poisson distributions, which have been used previously to model protein adsorption in affinity chromatography.2 If surface sites are spatially distinct, then the net protein binding energy is dictated instead by the number of binding sites on the surface which match the location of complementary functional groups on the protein, as illustrated in Figure 1B. We consider this case to be more physically relevant for IMAC because of the directed nature of metal-ligand coordination and the limited mobility of both surface binding sites and protein functional groups. We can then consider the cell to be a lattice17 in which the likelihood that a particular lattice position is occupied by a binding site is given by the probability p. If m′ groups on the protein are bound to the surface, then their positions correspond exactly to m′ binding sites on the surface arranged in a specific spatial pattern, while the remaining m m′ protein groups lie on top of lattice positions in which there are no binding sites. Assuming that there is no correlation in the spatial distribution of the binding sites on the surface, we may write the probability distribution P(m′) of finding m′ sites in a given spatial arrangement as18
[
]
1 + (cp/c0)eβEpm Np ln βEpm 1 + (cp/c0)
(5)
This expression can be recognized as the Temkin model6,19 for
5136 J. Phys. Chem., Vol. 100, No. 12, 1996
Johnson et al.
adsorption to heterogeneous surfaces with binding energies ranging from 0 to mEp. Saturation coverage again corresponds to the total number of cells (Np). Therefore at intermediate binding site densities, individual “cells” match the arrangement of functional groups on the protein to differing degrees, resulting in binding that is inherently heterogeneous. Combining Lateral Interactions and Surface Heterogeneity. Next we will consider the possibility that proteins can interact between adjacent cells on a heterogeneous surface. We will use the Bragg-Williams approximation20 to include an interaction energy w (w > 0 for a favorable interaction) between proteins occupying neighboring cells, as illustrated in Figure 1B. In this case the general isotherm for protein adsorption becomes
(cp/c0)eβ(Epm′+ZwQ/Np)
m
/
Q ) Np ∑ P(m′) m′)0 1 + (cp/c0)eβ(Epm′+ZwQ/Np)
m
∑ P(m′) m′)0
(6)
For intermediate surface site densities which result in heterogeneous adsorption, we again assume the probability distribution P(m′) to be independent of m′ and evaluate the summation in eq 6 by an integral. Under these conditions we obtain a closedform expression for the adsorption isotherm,
Q)
[
Np 1 + (cp/c0)e ln βEpm 1 + (cp/c0)eβ(ZwQ/Np)
and Ki is the equilibrium binding constant for competitor binding to a single surface site (liters/mole). Protein partitioning behavior in chromatography is often described by the “stoichiometric displacement” (SDM) model, originally applied to ion exchange systems.12-14 In this interpretation, protein binding to the chromatographic support (given by the “capacity factor” k′ at very low protein loading) displaces a number of equilibrium competitors equal to the number of individual protein-surface interactions. Presuming that protein adsorption is in local equilibrium, the capacity factor at low protein coverage is equal to the initial slope of the equilibrium binding isotherm.24 Expressing Q in eq 9 as a Taylor series expansion about cp ) 0, we then obtain the following expression for chromatographic experiments conducted at low protein loading,
]
β(Epm+ZwQ/Np)
(7)
k′ )
∂Q
|
∂cp
) cp)0
eβEpm′ P(m′) ∑ c0 m′)0 (1 + Kici)m′
Np
m
/
m
∑ P(m′)
(10)
m′)0
Because of the choice for the probability distribution P(m′), the summation of eq 10 can be expressed analytically for all p. For relatively strong binding,25 this expression becomes
k′ )
KmaxNp
(
)
am(a - 1)
(1 + Kici)m am+1 - 1
(11)
where a ) p/(1 - p) and Kmax ) exp(βEpm)/c0. Results and Discussion
This expression is easily inverted to express the protein concentration cp in terms of the amount of adsorbed protein Q (the resultant expression has been omitted for brevity). However, for the majority of the adsorption isotherm,21 eq 7 simplifies to
Q)
Np β(Epm - Zw)
ln[1 + (cp/c0)eβ(Epm)]
(8)
Therefore, comparing eq 8 to eq 5, over most of the adsorption isotherm the net effect of including lateral interactions is to divide the apparent saturation coverage by a constant factor of 1 - Zw/Epm. A similar principle has been demonstrated previously on statistical thermodynamic grounds by Yang.22 Chromatographic Behavior. To predict protein binding behavior in chromatographic experiments, it is necessary to include the effect of an equilibrium competitor that can bind to a single surface site (e.g., imidazole in IMAC). In that case a cell can be either empty, occupied by protein, or occupied by competitor at any of the m′ sites needed for protein binding. This modifies the partition function of eq 1, replacing the leading term of unity with a factor (1 + Kici)m′, where ci is the competitor solution concentration23 (moles/liter) and Ki is the equilibrium binding constant for competitor binding to a single surface site (liters/mole). As a result, the general protein adsorption isotherm in the presence of an equilibrium competitor is given by m (cp/c0)eβ(Epm′+ZwQ/Np) Q ) Np ∑ P(m′) m′)0 (1 + Kici)m′ + (cp/c0)eβ(Epm′+ZwQ/Np) m
/
∑ P(m′) m′)0
(9)
where ci is the competitor solution concentration23 (moles/liter)
Multipoint Binding in Chromatography. In the interpretation of the “stoichiometric displacement” (SDM) model,12-14 protein binding to the chromatographic support displaces a number of equilibrium competitors equal to the number of individual protein-surface interactions. Recent application of this model to IMAC10,11 has shown that protein partitioning to the IMAC support (given by the “capacity factor” k′ at very low protein loading) is inversely related to the concentration of the equilibrium competitor, imidazole (ci),
ln k′ ) ln k′0 - z ln(1 + Kici)
(12)
The coefficient z represents the number of molecules of imidazole displaced per molecule of protein adsorbed, Ki (liters/ mole) is the equilibrium constant for imidazole binding, and k′0 (milliliters of eluent/milliliter of support) represents the protein partitioning in the absence of imidazole. When the SDM expression (eq 12) is compared to the expression derived for multipoint binding to a random arrangement of surface sites (eq 11), it can be seen that the protein capacity factor k′ is described by the stoichiometric displacement model with z ) m. Therefore the number of displaced competitors z is given by the maximum number of proteinsurface interactions m and is independent of surface site density. This result is consistent with the results of ion exchange chromatography experiments performed using cytochrome c and lysozyme, in which the coefficient z was found to be insensitive to the charge density on the support.26 On the other hand, it is also clear that protein binding affinity (k′0 of eq 12) will be strongly dependent on the surface site density, as given by
k′0 ) KmaxNp(am(a - 1)/(am+1 - 1))
(13)
The dependence of k′0 on a (related to the surface site density, a ) p/(1 - p)) is shown in Figure 2. At high surface site densities
Surface Heterogeneity in Protein Adsorption
Figure 2. Dependence of the initial slope of the adsorption isotherm on the surface site density as predicted by eq 13. At high surface site densities the initial slope approaches a constant value of KmaxNp. At lower site densities, the initial slope increases as the site density raised to the mth power.
Figure 3. Effect of the copper site density on protein binding in IMAC. The initial slope of the equilibrium binding isotherm (b) for horse cytochrome c for TSK-IDACu prepared at decreased copper loading was measured previously.4 The density of accessible copper sites Cuacc was determined by the capacity of the support to bind N-acetylhistidine.8 Error bars represent the standard error of the measurement. The solid line shows the best fit to eq 13 (a ) p/(1 - p)) with Kmax Np ) 175 mL/mL TSK support, m ) 3.3, and p ) 0.033Cuacc.
(a > 1), k′0 approaches a constant value of Kmax Np. This is the regime under which adsorption follows the Langmuir model (eq 4). At moderate surface site densities (ln a < 1/m), k′0 increases rapidly with the mth power of the surface site density. This is the regime under which adsorption follows the Temkin model (eq 5). These two extremes of binding behavior are in excellent agreement with results for binding of phosphorylase b to hydrophobic alkylagarose prepared at different densities of surface hydrophobic groups.1 At very high densities of hydrophobic groups, the amount of protein adsorbed was independent of the site density; at lower densities, the logarithm of the initial slope of the adsorption isotherm increased linearly with the logarithm of the surface site density. As shown in Figure 3, the predicted binding behavior is also in good agreement with our results for adsorption of horse cytochrome c to an IMAC support prepared with different densities of immobilized copper.4 The best fit of these adsorption data to eq 13 corresponds to an equilibrium constant of 4.6 × 107 M-1 for cytochrome c adsorption when all lattice sites are occupied by copper ions.27 This is a reasonable estimate for protein adsorption Via approximately three immobilized copper sites, considering that the equilibrium constant for N-acetylhistidine
J. Phys. Chem., Vol. 100, No. 12, 1996 5137 adsorption Via a single copper site is approximately 103 M-1.11 We have also assumed a linear relationshp between surface copper ion density (Cuacc) and the probability p that a lattice site is occupied by a binding site. While this relationship is reasonable for low to moderate copper loading, it may not hold true at very high loading. Nevertheless, this relationship predicts that approximately 50% of the lattice sites are “occupied” at a copper loading of 15 µmol Cu/mL of the support (TSK-IDA). Chromatographic behavior at low loading is not sensitive to either of the mechanisms proposed for negatively cooperative binding. Heterogeneity in protein-surface interactions will have little effect on adsorption behavior at very low loading, because retention is strongly biased by those surface sites with the highest net binding energy and therefore the maximum number of protein-surface interactions. Protein-protein interactions will also have little impact on adsorption behavior at very low loading, because lateral interactions depend on the likelihood that adjacent sites are occupied. Although the stoichiometric displacement model has been generally successful in predicting the effects of multipoint binding on chromatographic behavior at low protein loading, it cannot be expected to lead to a description of the binding heterogeneity observed in protein adsorption at higher coverages. Multipoint Binding in Equilibrium Adsorption. A consequence of multipoint interactions is that a protein will show the highest affinity for surface arrangements which best match its own distribution of functional sites. As a result, any description of protein adsorption must consider a heterogeneous population of surface binding sites, resulting in a range of binding energies. Using a series of 10 variants of yeast iso-1cytochrome c differing in number and placement of surface histidines, we have found that reversible protein adsorption to metal affinity supports is best described by the Temkin isotherm.5 The Temkin isotherm characterizes protein binding as a uniform distribution in binding energies over the population of surface sites, in which the amount of protein adsorbed Q is a logarithmic function of the solution concentration cp,
Q(cp) ) qT ln[1 + KTcp]
(14)
where KT (liters/mole) is the maximum binding affinity and qT (moles of protein/milliliter of IMAC support) is the differential increase in the limiting capacity for protein adsorption with increasing protein binding affinity. When the Temkin model (eq 14) is compared to the expression derived for multipoint binding to a random arrangement of surface sites (eq 8), it can be seen that protein adsorption is described by the Temkin model with
KT ) Kmax
(15)
qT ) NpkT/(Epm - Zw)
(16)
Because unfavorable protein-protein interactions (w < 0) and the heterogeneity inherent in multipoint binding to a random arrangement of surface sites (Ep > 0) both lead to the Temkin isotherm, the adsorption isotherm alone is not sufficient to distinguish which of these two mechanisms is responsible for the negatively cooperative binding.28 Therefore, in the absence of other evidence, multipoint protein adsorption includes the possibility of a heterogeneous population of surface binding sites and the possibility of lateral interactions. However, an important distinction between the two mechanisms can be seen in the interdependence of two parameters of the Temkin model, KT and qT. Increasing the number or magnitude of protein-surface interactions (mEp) has the effect
5138 J. Phys. Chem., Vol. 100, No. 12, 1996
Johnson et al.
Figure 4. Relationship between Temkin model parameters KT and qT. Parameters (with standard error bars) are determined by nonlinear regression of the Temkin model to equilibrium binding data for yeast cytochrome c variants to CuIDA-TSK in the absence of imidazole.8 Data are shown for cytochrome c variants ranging from 1 to 3 surface histidines (b) and for horse cytochrome c at copper loadings ranging from 9 to 18 µmol/mL TSK (O). The solid line represents the best fit to eq 17 with Np ) 3.81 ( 0.67 µmol protein/mL TSK and Zw ) -1.99 ( 2.75kT (presuming a standard state protein concentration16 of 1 M).
of increasing the maximum binding constant KT and decreasing the differential binding capacity qT. On the other hand, increasing the magnitude of unfavorable protein-protein interactions (Zw, w < 0) has only the latter effect. Therefore the two mechanisms can in principle be distinguished by a process which changes protein-surface interactions but does not change protein-protein interactions. This principle is not limited to metal-affinity chromatography4,5 but could apply to any reversible adsorption process dominated by multipoint interactions.1-3 Under such conditions, the inverse of the parameter qT should be linearly related to the logarithm of the Temkin parameter KT,
1 1 1 ) ln KT + (ln c0 - Zw) qT kTNp kTNp
(17)
with the slope giving an estimate of the saturation coverage (Np) and the intercept giving an estimate of protein-protein interactions (-Zw). We have previously compared the equilibrium binding isotherms of variants of a small globular protein, yeast iso-1cytochrome c , on a polymer matrix derivatized with copper(II) iminodiacetate (TSK-IDACu). We manipulated the overall protein binding affinity through two processes:5 (1) changing the number of functional groups on the protein (histidines) and (2) changing the density of surface binding sites (immobilized copper ions). Presuming that single amino acid substitutions have no significant effect on the protein-protein interactions,29 neither process should affect the interaction energy Zw, while both processes should affect the net protein-surface interaction energy mEp. These changed the Temkin parameter KT over 5 orders of magnitude (KT ∼ 103-108 M-1), while the parameter qT decreased only slightly with increasing KT (qT ∼ 0.3 ( 0.1 µmol/mL support). As shown in Figure 4, this trend agrees very well with the behavior predicted by the thermodynamic description of multipoint binding. In fact, the best fit of these data to eq 17 provides an estimate of saturation coverage (Np ) 3.8 ( 0.7 µmol protein/mL support) in excellent agreement with our estimate of cytochrome c saturation coverage of this protein on
the IMAC material.5 While the positive intercept is consistent with slightly unfavorable protein-protein interactions (Zw/kT ) -2.0 ( 2.8, presuming a standard state protein concentration of 1 M), its value is not well determined and is effectively zero within experimental error. We therefore conclude that the negatively cooperative binding observed in this system arises from the surface heterogeneity inherent in multipoint binding between a protein (a fixed pattern of functional groups) and a surface (a random arrangement of binding sites). As a result, protein adsorption in IMAC will be dominated by those surface sites which optimize the number of interactions between the protein and the chromatographic support, particularly at the relatively low coverage typical of chromatographic applications. It should therefore be possible to target individual proteins by matching the distribution of binding sites on the surface to the spatial distribution of functional groups on the protein.8 Although protein-protein interactions did not play a significant role in these experiments, it is conceivable that they could be significant in other chromatographic systems, particularly those with lower binding energies for individual protein-surface interactions or with very high surface site densities. Both of these conditions would decrease the range of binding energies resulting from multipoint binding to a random arrangement of surface sites relative to that resulting from lateral interactions. Acknowledgment. This research is supported by the National Science Foundation. F.H.A. acknowledges an NSF PYI award and a fellowship from the David and Lucile Packard Foundation. R.D.J. acknowledges a predoctoral training fellowship from the U. S. National Institute of General Medical Sciences, Pharmacology Sciences Program. Note Added in Proof: While the manuscript was in press, a different theoretical approach to tackling multipoint binding to inhomogeneous surfaces was devised (Z.G.W.). This new approach, which lifts some of the assumptions and approximations of the current paper and yields results applicable to a wider range of experimental conditions, will be published shortly. References and Notes (1) Jennissen, H. P. J. Chromatogr. 1978, 159, 71-83. (2) Dowd, V.; Yon, R. J. J. Chromatogr. 1992, 627, 145-151. (3) Gill, D. S.; Roush, D. J.; Willson, R. C. J. Colloid Interface Sci. 1994, 167, 1-7. (4) Todd, R. J.; Johnson, R. D.; Arnold, F. H. J. Chromatogr. A 1994, 662, 13-26. (5) Johnson, R. D.; Arnold, F. H. Biochim. Biophys. Acta 1995, 1247, 293-297. (6) Tompkins, F. C. Chemisorption of Gases on Metals, Academic Press: San Francisco, CA, 1978. (7) Levitzki, A. Receptors: A QuantitatiVe Approach, Benjamin/ Cummings: Menlo Park, CA, 1984. (8) Mallik, S.; Plunkett, S.; Dhal, P. K.; Johnson, R. D.; Pack, D.; Shnek, D.; Arnold, F. H. New J. Chem. 1994, 18, 299-304. (9) Johnson, R. D.; Arnold, F. H. Biotechnol. Bioeng. 1995, 48, 437443. (10) Vunnum, S.; Gallant, S.; Kim, Y.; Cramer, S. M. Chem. Eng. Sci., in press. (11) Johnson, R. D.; Todd, R. J.; Arnold, F. H. J. Chromatogr. A 1996, 725, 225-235. (12) Drager, R. R.; Regnier, F. E. J. Chromatogr. 1986, 359, 147-155. (13) Velayudhan, A.; Horvath, C. J. Chromatogr. A 1994, 663, 1-10. (14) Gerstner, J. A.; Bell, J. A.; Cramer, S. M. Biophys. Chem. 1994, 52, 97-106. (15) In the current analysis, the highest protein loading is less than half the saturation coverage (Np) predicted by the fit to the data (see Figure 4) and by our estimate of the limiting coverage based on the protein dimensions.5 An additional difficulty comes from the fact that the adsorbate can bind to two or more surface sites. An elegant discussion of the multisite effects for irreversible adsorption of homogeneous objects has been given by Jin et al. (Jin, X.; Wang, N.-H. L.; Tarjus, G.; Talbot, J. J. Phys. Chem. 1993, 97, 4256-4258). It is not clear, however, how this treatment can be
Surface Heterogeneity in Protein Adsorption generalized to the case of reversible adsorption of objects that have imbedded inhomogeneity (e.g., the distribution of histidines on a protein). (16) The standard state concentration can also be considered in terms of an equilibrium constant K0 (M-1) for nonspecific protein adsorption to the surface (K0 ) 1/c0). (17) The lattice size should not be taken literally as the true lattice size of the surface, but rather as some effective size corresponding to the “sphere of influence” of a binding site. (18) At very low p, eq 6 is not expected to be valid because of possible restrictions in translation, rotation, etc., due to artifacts arising from dividing the surface into discrete cells. (19) Bolis, V.; Morterra, C.; Fubini, B.; Ugliengo, P.; Garrone, E. Langmuir 1993, 9, 1521-1528. (20) Bragg, W. L.; Williams, E. J. Proc. R. Soc. A 1934, 145, 699730. (21) Equation 8 is valid for Q < |NpkT/Zw| or Q > NpkT/(Epm - Zw). (22) Yang, C.-H. J. Phys. Chem. 1993, 97, 7097-7101. (23) For simplicity we have presumed an imidazole solution standard state of 1 M.
J. Phys. Chem., Vol. 100, No. 12, 1996 5139 (24) Arnold, F. H.; Schofield, S. A.; Blanch, H. W. J. Chromatogr. 1986, 335, 1-12. (25) Equation 11 is derived with the assumption that a ) p/(1 - p) . (1 + Kici) exp(-βEp). (26) Wu, D.; Walters, R. J. Chromatogr. 1992, 598, 7-13. (27) This estimate presumes a standard state protein concentration of 1 M and Np ) 3.8 µmol protein/mL TSK (see Figure 4). (28) We disagree with Yang’s conclusion22 that the degree of surface heterogeneity is independent of temperature while lateral interactions are proportional to 1/kT. This conclusion is based on the assumption that the degree of surface site heterogeneity (a in ref 22) is temperature-independent. The degree of surface heterogeneity results from a range of binding energies;19 as a result it is also temperature-dependent. (29) These experiments were performed at high ionic strength (0.5 M NaCl) to minimize electrostatic interactions.4
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