Langmuir 1997, 13, 6761-6768
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Influence of Structural Details in Modeling Electrostatically Driven Protein Adsorption Dilipkumar Asthagiri and Abraham M. Lenhoff* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received June 9, 1997. In Final Form: September 11, 1997X Mechanistic modeling of protein adsorption has evolved to include increasingly detailed descriptions of protein structure in an effort to capture experimentally observed behavior. This has been especially true of electrostatically driven adsorption, for which colloidal models have been used frequently. These efforts have focused on adsorption of proteins to oppositely charged surfaces and often capture the experimental trends even with gross simplification of protein structure. As a more stringent test of model sensitivity to structural details, we have modeled the patch-controlled adsorption of basic proteins on anion-exchange surfaces, where a small number of negative charges on the protein surface lead to a net attraction between the net positively charged protein and the positively charged surface. We account in detail for the protein shape and charge distribution and examine the role of the assumed surface description. A model assuming a uniformly charged surface is unable to predict electrostatically driven adsorption observed experimentally, whereas models accounting for the discreteness of charge on the adsorbent are able to explain some of the anomalous experimental trends. Although our results show that fine details of the models are crucial in correctly describing adsorption behavior under these unusual conditions, they also suggest that when the protein and the surface are oppositely charged, model calculations can be quite robust to model idealizations.
1. Introduction Studies of protein adsorption in which electrostatic effects are dominant are typically performed under conditions analogous to those in ion-exchange chromatography, with the net charge of the proteins concerned being opposite in sign to those of the adsorbent surfaces. There have been numerous theoretical studies of such systems in which the principles of colloidal theory have been applied, making use of varying amounts of protein structural information to account for the configurational dependencies of the interaction energy between the protein and the surface.1-7 Other efforts to model adsorption have used atomistic details of the protein and/or adsorbent (polymeric) surface,8-11 although these approaches are not completely satisfactory with regard to their treatment of electrostatic interactions in that pairwise Coulombic interactions were assumed and electrolyte effects were either neglected or accounted for by using effective dielectric constants. Systems in which the electrostatics are dominant and modeled within the framework of double-layer theory appear to be quite forgiving in not requiring an extremely detailed description of the geometry and charge distribution. Perhaps the best demonstration of this is the ability of the planar model of Ståhlberg et al.1,2 to capture quite well, even quantitatively, the extent of protein retention * To whom correspondence should be addressed: fax, +1-302831-4466; e-mail,
[email protected]. X Abstract published in Advance ACS Abstracts, November 15, 1997. (1) Ståhlberg, J.; Jo¨nsson, B.; Horva´th, C. Anal. Chem. 1991, 63, 1867. (2) Ståhlberg, J.; Jo¨nsson, B.; Horva´th, C. Anal. Chem. 1992, 64, 3118. (3) Roth, C. M.; Lenhoff, A. M. Langmuir 1993, 9, 962. (4) Grant, M. L.; Saville, D. A. J. Phys. Chem. 1994, 98, 10358. (5) Roush, D. J.; Gill, D. S.; Willson, R. C. Biophys. J. 1994, 66, 1290. (6) Roth, C. M.; Lenhoff, A. M. Langmuir 1995, 11, 3500. (7) Ben-Tal, N.; Honig, B.; Peitzsch, R. M.; Denisov, G.; McLaughlin, S. Biophys. J. 1996, 71, 561. (8) Lu, D. R.; Park, K. J. Biomater. Sci., Polym. Ed. 1990, 1, 243. (9) Lu, D. R.; Park, K. J. Biomater. Sci., Polym. Ed. 1992, 3, 127. (10) Noinville, V.; Vidal-Madjar, C.; Se´bille, B. J. Phys. Chem. 1995, 99, 1516. (11) Juffer, A. H.; Argos, P.; de Vlieg, J. J. Comput. Chem. 1996, 17, 1783.
S0743-7463(97)00608-2 CCC: $14.00
in ion-exchange chromatography and its dependence on ionic strength. Other idealized models can also demonstrate reasonably good agreement with experiment, e.g., treating the protein as a sphere with its net charge placed at the center.6 One reason for the insensitivity of the modeling to structural details is that many models contain one or more parameters that can be adjusted to fit experimental data. Another reason, specific to the electrostatic nature of the interactions, is that colloidal models invariably incorporate dependence on the Debye parameter κ, which is proportional to the square root of the ionic strength and thus captures the influence of electrolyte concentration on the interactions. In these models the screening effect of increasing ionic strength that is observed experimentally is clearly mediated by κ, but there is ambiguity regarding the precise dependence. For instance, the same experimental data that were correlated on a log-log plot of capacity factor versus ionic strength12 can also be correlated on a log-square root plot.1 Whatever the case, though, the behavior expected is that adsorption is strong at low ionic strength and decreases with increasing ionic strength; such behavior is also what is expected from colloidal models dependent on κ, with relatively little variation with model details. Several sets of calculations have been described in which the full three-dimensional structure of the protein has been considered. These have reported calculated energetics, as a function of orientation and separation distance, for interactions of a protein with a uniformly charged surface13 or a surface represented atomistically.5,7,10 Such calculations provide much detailed information, although all the different models retain some idealizations in such areas as the underlying physics, the boundary conditions, and the geometry and the charge distribution of the adsorbent surface, which are rarely known in detail. In addition, how well these models describe experimental data is often difficult to establish because of the computational expense of very extensive exploration of the relevant parameter space. Where comparisons have been (12) Melander, W. R.; El Rassi, Z.; Horva´th, C. J. Chromatogr. 1989, 469, 3. (13) Yoon, B. J.; Lenhoff, A. M. J. Phys. Chem. 1992, 96, 3130.
© 1997 American Chemical Society
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performed,6,7 rather simple models have provided comparable predictions of trends in adsorption equilibrium to the detailed models. Part of the reason for this may lie in the fact that calculation of the adsorption equilibrium involves Boltzmann-weighted averaging of the configuration-dependent energies, which tends to smooth out the orientational variations among what are predominantly attractive configurations. A better test of how well the models describe experimental behavior can be performed by seeking to model a system in which the qualitative behavior is less predictable. An example of this is an experimentally observed anomaly in protein adsorption, namely, the adsorption of some proteins with a net charge of the same sign as the surface,14 which has been attributed to discrete regions or patches of opposite charge on the protein.15 Because of the Boltzmann weighting, the small number of attractive orientations in this situation dominates the adsorption equilibrium whereas the repulsion observed in most configurations contributes to exclusion. A particularly extensive data set describes retention in anion-exchange chromatography of four basic proteinsslysozyme, R-chymotrypsinogen A, cytochrome c, and ribonuclease Asas a function of pH.16 The capacity factor k′ was close to zero for low pH values (pH 4.0 and pH 5.5) and low ionic strength (up to about 1 M). At higher pH values (pH 7.0 and pH 8.0) the capacity factor increased with decreasing ionic strength in the low ionic strength regime. This latter behavior is a signature effect of electrostatically dominated adsorption. Because of the pH effect, the authors concluded that negatively charged patches on the protein surfaceswhich become dominant at higher pH, where basic residues become deprotonatedsare responsible for the adsorption. We have sought to model this behavior using full structural details of the protein molecules. An element that is particularly important is the description of the adsorbent surface, the detailed three-dimensional structure of which is generally unknown at the molecular level. Three different levels of idealization are used to explore the sensitivity of the prediction of anomalous adsorption to the structure assumed for the adsorbent surface. Some aspects of the analysis that are applicable even in cases where the protein and surface are of opposite charge are also discussed. 2. Modeling and Theory Electrostatic interactions in proteins arise from the ionizable sites on amino acid residues, with the charge at each of these sites being dependent on the solution pH. Since aqueous electrolyte solutions are generally used as solvents for proteins, modeling of electrostatics must also account for the presence of mobile ions in the surrounding solvent. The resulting effects have been described quantitatively using different approaches, several reviews of which are available, e.g., refs 17 and 18. For the adsorption problem it is necessary to calculate the interaction energy between a single protein molecule and the absorbent surface. The treatment of electrostatics adopted here employs a continuum formulation similar to that used in most previous efforts to account in detail for protein structure in modeling adsorption.3,5,13 Central to the task is the calculation of the three-dimensional potential φ throughout the system. (14) Kopaciewicz, W.; Rounds, W. A.; Fausnaugh, J.; Regnier, F. E. J. J. Chromatogr. 1983, 266, 3. (15) Regnier, F. E. Science 1987, 238, 319. (16) Lesins, V.; Ruckenstein, E. Colloid Polym. Sci. 1988, 266, 1187. (17) Harvey, S. C. Proteins 1989, 5, 78. (18) Davis, M. E.; McCammon, J. A. Chem. Rev. 1990, 90, 509.
Asthagiri and Lenhoff
The potential φi within the protein is described by the Poisson equation
∇2φi ) -
Fi io
(1)
where i is the dielectric constant of the protein interior (taken to be 4 in our calculations), 0 is the permittivity of free space, and Fi is the charge density due to the charged residues. Thus Fi ) ∑kqkδ(x - xk), with qk the charge at site xk inside the protein, and δ the three-dimensional delta function. In the medium external to the protein, the charge density Fi is due to the mobile ions in solution, and the potential φe is described by the Poisson-Boltzmann equation, of which we use the linearized form (the Debye-Hu¨ckel approximation)
∇2φe ) κ2φe
(2)
κ, the reciprocal Debye length, is a measure of the screening of the electrostatic potential. It is proportional to the square root of ionic strength, the main solution property used to manipulate ion-exchange retention, and it thus plays an important role in our comparisons with experimental data. Although linearization may distort the computed potential field under some of the conditions examined here, the interaction free energies are likely to be affected much less strongly.19 The boundary conditions at the molecular surface are
φi ) φe and i
∂φi ∂φe ) e ∂n ∂n
(3)
representing the continuity of the potential and of the normal component of the electric displacement vector. Here n denotes the outward normal vector, and e is the dielectric constant of the electrolyte solution. As noted earlier, the description of the adsorbent is an important issue studied in this work. A real ion-exchange adsorbent usually comprises a bonded phase synthesized on a base matrix that may be an inorganic material, a synthetic polymer, or a natural polymer (usually a polysaccharide). For example, the experiments of Lesins and Ruckenstein16 employed a stationary phase in which the functional surface was a cross-linked poly(ethylenimine) (PEI) bonded phase, with the charges residing on protonated imine groups. Considerable complexities of both topography and charge distribution are thus possible, and efforts to account for such details have been made in two previous sets of calculations of protein-surface electrostatics.5,10 In those studies the ligand structures were represented atomistically and charge distributions were determined either by using CHARMM parameters5 or by semiempirical methods of quantum chemistry.10 Even so, the geometry was somewhat idealized in that the ligands were placed in planar square arrays. In view of the inevitable uncertainties that remain, and our desire to examine explicitly the effect of different surface representations, we have not included atomistic information in our surface description, but we have explored three different idealizations. The simplest representation, which is typical of that used in colloidal systems, is one that treats the surface as a plane of uniform continuous charge density.3,11,13 Mathematically this enters the description of the uniform charge model (UCM) via a boundary condition at the adsorbent surface; we make the assumption of constant charge density σ, leading to (19) Zhou, H.-X. J. Chem. Phys. 1994, 100, 3152.
Protein Adsorption Electrostatics
-0e
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∂φe )σ ∂n
(4)
The dielectric constant of the bulk adsorbent is thus implicitly assumed to be very small. Our second approach is more realistic in recognizing the discreteness of charge, while retaining the planarity of the surface. For mathematical convenience the charges are located on a square lattice at a low-dielectric surface, so the boundary condition for this discrete charge model (DCM) still takes the form of eq 4, but with σ now in the form
σ)
qδ(x - x˜ ) ∑ s,t
where x˜ ) si + tj
(5)
Figure 1. Schematic of a protein interacting with the polymeric adsorbent. The “end-group” involved in the ion-exchange interaction is explicitly depicted.
F)
1
1
n
∫ σφe(x) dA(x) + 2 ∑ qkφi(xk) 2 ∂P
(7)
k)1
i and j are unit vectors in the x and y directions, respectively, and x is the position vector in the plane of the adsorbent surface. The potential above the surface in the electrolyte solution in this case is20
φ)
q
∑ 2π a s,t e o
[ x( ) ( ) ( ) ]
exp -κa
x
a
2
-s
y
+
a
2
-t
z
+
a
x( ) ( ) ( ) x
a
y
2
-s
+
a
z
2
-t
2
2
+
a
(6)
q ()1e, where e is the proton charge) is the charge at each point on the surface, a is the lattice constant, and the summation is over all integers s and t, which specify the coordinates of each lattice point with respect to the reference point (x,y) ) (0,0). The third model seeks to account for the irregular topography of the adsorbent, specifically the fact that in most adsorbents the charged groups effectively appear on spacer arms rather than at flat surfaces. In this model, then, the protein-surface interaction is treated as a protein molecule interacting with the charged group alone (Figure 1), a formulation henceforth referred to as the protein end-group model (EGM). The end-group is surrounded by the high dielectric constant solvent, including electrolyte, but the dielectric constant of the end-group is set equal to that of the protein to make the model equations more tractable; boundary conditions 3 thus apply here too. Physically the dielectric constant of the end-group should be small, and for the present analysis it is reasonable to set it equal to that of the protein. This simplistic description of the polymeric end-group is dictated primarily by our inadequate knowledge of the surface, but the limitations of available computational resources are also a factor. Once the potential distribution φ is available everywhere, it can be used to calculate the free energy of interaction of the protein and the surface. The interaction free energy is computed as the difference in free energy between that for a given configuration (separation and orientation) and the value at infinite separation. Under the Debye-Hu¨ckel approximation, the Helmholtz free energy F of the double-layer system for any configuration of protein and surface is given by integration over all the fixed charges in the system; this leads, for the first two models, to (20) Nelson, A. P.; McQuarrie, D. A. J. Theoret. Biol. 1975, 55, 13.
where ∂P is the planar surface. The integral term is due to the surface,21 whereas the summation term is due to the charges within the protein molecule.22,23 For the DCM, the integral becomes a sum over the discrete charges, so that
F)
1
∑
2 s,t
qφe(x˜ ) +
1
n
∑ qkφi(xk)
2k)1
(8)
where x˜ is given in eq 5. For the EGM, the fixed charges on the “surface” are those on the end-group. The end-group can then be treated as a second particle mathematically similar to the protein, giving
F)
1
2
ni
∑ ∑ qk φ(xk )
2i)1ki)1
i
i
(9)
where there are ni charges of magnitude qki inside the ith particle, representing either the protein or the end-group.24 3. Methods The electrostatic equations were solved for the potential φ using a boundary element method (BEM).13,25-28 The procedure is, briefly, that the governing equations (1 and 2) are written as boundary integral equations for the potential and the normal derivative, making use of the boundary conditions. If the molecular surface is described by discretizing it into a large number of small triangles, the boundary integral equations can be written in terms of sums of integrals over the individual elements. By assuming a suitable functional form for the potentials and their normal derivatives within each elementslinear in this workseach integral over one element can be written instead in the form of a sum over the values of potentials and their normal derivatives at a discrete number of nodes. This leads to a system of linear equations that is solved for the nodal potentials and their normal derivatives, using which the potential anywhere in the system can be evaluated. Specifically, the potentials needed in calculating the free energies can be computed. An important advantage of the BEM is that the dielectric boundary, the (21) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the stability of lyophobic colloids; Elsevier: Leiden, 1948. (22) Kirkwood, J. G. J. Chem. Phys. 1934, 2, 351. (23) Sharp. K.; Honig, B. J. Phys. Chem. 1990, 94, 7684. (24) Zhou, H.-X. Biophys. J. 1993, 64, 1711. (25) Zauhar, R. J.; Morgan, R. S. J. Mol. Biol. 1985, 6, 815. (26) Yoon, B. J.; Lenhoff, A. M. J. Comput. Chem. 1990, 11, 1080. (27) Juffer, A. H.; Botta, E. F. F.; van Keulen, B. A. M.; van der Ploeg, A.; Berendsen, H. J. C. J. Comput. Phys. 1991, 97, 144. (28) Zhou, H.-X. Biophys. J. 1993, 65, 954.
6764 Langmuir, Vol. 13, No. 25, 1997
distribution of point charges within the protein, and the effectively unbounded nature of the solution can be represented accurately. The specifics of the BEM implementation are very similar for the three models. For the UCM and the DCM, the potential distribution can be found by a simple modification of the method for an isolated molecule.13 However, for the EGM, the boundary integral equations for the protein and the end-group are coupled, as should be expected since the presence of the end-group will influence the potential distribution on the protein surface and vice versa. The coupled set of equations is solved to yield the potentials and their normal derivatives for both the protein and the end-group.24,28 The interaction free energy is then obtained as described above for the respective models. Although Lesins and Ruckenstein16 presented experimental results for four proteins, we have concentrated our analysis on the experimental data for ribonuclease A (bovine pancrease; PDB file 1rtb) and cytochrome c (horse heart; PDB file 2hrc), which are similar in size but differ somewhat in their electrostatic properties. At neutral pH, cytochrome c has a significantly higher net charge than ribonuclease, but ribonuclease has a much higher dipole moment. The other pair of proteins examined by Lesins and Ruckenstein, chymotrypsinogen A and lysozyme, show a similar contrast in their electrostatic properties, although chymotrypsinogen has almost double the molecular weight of the other three. To compare with experimental data, the ionic strengths that were used in the calculations were 0.05 and 0.1 M for ribonuclease and 0.1 and 0.2 M for cytochrome c. The crystallographic structures were obtained from the Brookhaven Protein Data Bank.29 The surface discretization (tessellation) was performed using the Molecular Surface Package.30 For ribonuclease a surface discretization consisting of 2996 nodal points and 5988 triangles was used, and for cytochrome c the tessellation was 3892 nodal points and 7780 triangles. For any given pH, the intrinsic pK values of the amino acids were used in the calculation of the protein charge distribution.31,32 Formal charges were assigned to all arginine (Arg), lysine (Lys), histidine (His), aspartate (Asp), and glutamate (Glu) residues and to unmodified Nand C-termini. The positions of the charged groups were defined as the coordinates of the N and NZ atoms for N-terminal and Lys residues, respectively, the mid-point of the carboxyl oxygen atoms for C-terminal, Asp and Glu residues, the mid-point of the NH1 and NH2 atoms for Arg residues, and the mid-point of the ND1 and NE2 atoms for the His residues. In the case of cytochrome c the protoporphyrin IX (which chelates Fe(III)) also contributes to the charge distribution. In this case a charge of -1e was placed at the mid-point of each pair of propionate carboxyl oxygen atoms (O1A and O2A; O1D and O2D), and a charge of 3e was placed at the position of Fe(III). For the DCM, a lattice constant of 30 Å was used in the absence of detailed knowledge of the surface charge distribution. The primary motivation for this was that it would minimize the combinatorial problem of dealing with the situation where a protein molecule might interact with multiple surface charges simultaneously; for patchcontrolled adsorption this would be quite unlikely. Thus even if the mean charge density is higher than the value (29) Bernstein, F. C.; Koetzle, T. F.; Williams, G. J. B.; Meyer, E. F.; Brice, M. D.; Rodgers, J. R.; Kennard, O.; Shimanouchi, T.; Tasumi, M. J. Mol. Biol. 1977, 112, 535. (30) Connolly, M. J. Appl. Crystallogr. 1983, 16, 548. (31) Zubay, G. Biochemistry; Addison-Wesley: New York, 1984. (32) Stryer, L. Biochemistry; W. H. Freeman: New York, 1988.
Asthagiri and Lenhoff
Figure 2. Schematic of protein interacting with a charged surface. The residue labeled A is “optimally” oriented, whereas residue B is “non-optimally” oriented.
we have assumed, we expect patch-controlled adsorption to occur primarily at locations where interaction with only one ligand is possible. We have, however, examined the sensitivity of the results to the assumed value of the lattice constant. Corresponding to the lattice constant of 30 Å, the surface charge density for the UCM was taken as 1.8 µC/cm2, which is quite low relative to the ion-pairing capacity of PEI materials.33 Although the ion-pairing capacity would be expected to be higher than the effective surface charge density for protein adsorption, the value of σ may still be somewhat low and can thus be expected to underestimate the extent of repulsion due to the like net charge of the protein and the surface. For the EGM, the end-group was assigned a charge of 1e, and its radius was set equal to that of an amine (1.7 Å) in terms of the radius set used in the Molecular Surface Package (version 2.9).30 A slightly different radius for the end-group is not expected to affect the results appreciably. The discretized description of the molecular surface and the charge distribution were then used in the integral formulation to compute the interaction free energy (IFE). For calculation of the IFE, the separation and orientation relative to the surface (or the end-group) must be specified. The separation in the case of the protein-surface calculation is given by the gap distance, the nearest separation between the protein van der Waals surface and the adsorbent surface, for the specified orientation (Figure 2). Because of the appreciable computational task involved, extensive configurational exploration was not possible, so a more limited set of cases was chosen based on the physical situation of interest. Each orientation studied was such that a specific acidic residues contributing the negative chargeswas as close to the positively charged surface as possible for the given gap distance; steric factors would often preclude a closer approach. For orientations that we term “optimal”, the residue under study was in fact the closest residue to the surface (Figure 2), and the charge on the residue (residue charge) was a distance of about (1.7 Å + gap) away from the surface. The 1.7 Å results from the observation that this distance is approximately the mean position of the negative charges inside the van der Waals surface, the precise value being dependent on the radius set used in the MSP program. Gap distances of 2 and 4 Å were used, which for optimal orientations give residue-charge-tosurface separations of approximately 3.7 and 5.7 Å, respectively. In the case of the EGM, the separation between the protein and the end-group was defined as follows. The position of the residue charge was taken as the origin and an outwardly directed ray (piercing the molecular surface) was sought that was roughly perpendicular to the mo(33) Alpert, A. J.; Regnier, F. E. J. Chromatogr. 1979, 185, 375.
Protein Adsorption Electrostatics
Langmuir, Vol. 13, No. 25, 1997 6765 Table 2. Interaction Free Energy (in kbT) for Ribonuclease A, Shown as a Function of pH, Ionic Strength, Gap Distance, and Model Forma UCM
DCM
I ) 0.05 M
I ) 0.1 M
I ) 0.05 M
I ) 0.1 M
2Å
2Å
2Å
2Å
4Å
4Å
4Å
4Å
pH 8
Glu 49 -0.1 -0.3 -0.3 -0.4 -1.7 -1.3 -1.7 -1.3 Asp 53 -0.1 -0.7 -0.1 -0.6 -4.0 -2.6 -3.8 -2.4 pH 5.5 Glu 49 1.6 1.1 0.6 0.3 0.1 0.2 -0.8 -0.5 Asp 53 1.2 0.5 0.5 0.0 -2.6 -1.4 -3.1 -1.8 a
Figure 3. R -Carbon trace of ribonuclease A with His 48, Glu 49, and Asp 53 shown in spacefill format. Table 1. Residue-Charge-surface charge separation and the corresponding interaction free energy for ribonuclease Aa residue
∆F (kbT)
charge-surface distance (Å)
Glu 9 Asp 38 Glu 49 Asp 53 Glu 86 Glu 111
1.5 5.7 -1.7 -4.0 5.8 1.1
6.1 3.9 7.1 3.5 8.7 7.3
a The DCM is used for the surface and the ionic strength is 0.05 M: gap distance, 2 Å; lattice constant, 30 Å. Note that although Glu 49 is “nonoptimally” placed the interaction free energy (∆F) is negative, because of the proximity to Asp 53.
lecular surface. The center of the end-group (also the location of the end-group charge) was then placed on this ray at the desired separation from the residue charge. Separations of 3.7 and 5.7 Å were used, allowing comparison with the protein-surface calculations. 4. Results We first present the results for ribonuclease and then compare them to those for cytochrome c. On the basis of exploratory calculations and visual inspection of the molecules using Rasmol,34 we identified six possible residues for each protein that are likely to be involved in the patch-controlled adsorption, given the experimental trends observed. The criteria for choosing a residue were the closeness to other acidic residues and the remoteness from the nearest basic residues. An additional specific criterion was based on the observation that appreciable changes in retention between pH 8 and pH 5.5 can be ascribed to a rather small number of effects. The most important of these are protonation and thus charging of the N-terminus and His residues, with protonation and thus uncharging of acidic residues perhaps beginning to have a small effect at pH 5.5. In the case of ribonuclease, we identified Glu 49 and Asp 53 as a possible negative patch. The reasons for this choice were (a) Asp 53 is optimally oriented in the sense defined earlier, that is, the protein can be oriented such that this residue is closest to the surface, (b) Glu 49 and Asp 53 are located in close proximity (Figure 3), and (c) exploratory calculations for the DCM at 2 Å gap spacing (Table 1) show that Glu 49 and Asp 53 are the only two of the six tentatively identified patch residues for which ∆F < 0 is predicted at pH 8; ∆F is given in units of kbT, (34) Sayle, R.; Milner-White, J. E. Trends Biochem. Sci. 1995, 20, 333.
For the DCM the lattice constant is 30 Å.
where kb is the Boltzmann constant and T is the absolute temperature. The other residues evaluated for the protein surface interaction were non optimal and/or energetically unfavorable (see Table 1). Further reinforcement of the importance of Glu 49 and Asp 53 came from the calculations for the transition from pH 8 to 5.5. Charging using the intrinsic pK values gives the net charge on ribonuclease to be 3.9e at pH 8 and 7.4e at pH 5.5; these values are in good agreement with experimental titration data.35 The major change here is titration of the histidines, which occur in ribonuclease at residues 12, 48, 105, and 119. His 48 is close to Glu 49 and Asp 53 (Figure 3), the putative negative patch, and the three residues together display an appreciable change in charge over the above range in pH: the net charge of the three residues changes from -2e to -1.2e. As will be seen, this has a significant effect on the interaction free energy. On the basis of these results, we pursued further calculations for ribonuclease at planar surfaces using only the Glu 49 and Asp 53 orientations. Table 2 shows the effects of various parametersspH, ionic strength, gap distance, and model form (UCM or DCM; the EGM is discussed below)son the interaction free energy (IFE) for ribonuclease in the two preferred orientations. The results at pH 8 are the easiest to interpret. Firstly, the IFEs for the UCM are all greater than -1kbT, meaning that the interaction between the surface and the protein molecule is not strong enoughs compared to thermal fluctuationssto drive adsorption. Secondly, the distance dependence of the IFE indicates that for the UCM the IFE becomes larger (less attractive) with decreasing gap distance. The DCM, in contrast, yields IFEs that are significantly lower than -1kbT, implying that these are very favorable and strong enough to drive adsorption. The dependence on ionic strength, with the IFEs becoming less negative as the ionic strength is increased, is in agreement with the experimental observation that increasing ionic strength results in decreased adsorption in the low ionic strength regime, where electrostatics are expected to be dominant in driving adsorption. Finally, the distance dependence indicates that the more closely the protein molecule approaches the surface, the stronger the attraction, consistent with the intuitive expectation for electrostatically dominated adsorption. The pH 5.5 behavior has some interesting features. The UCM gives IFEs that are positive, implying an unfavorable configuration of the protein with respect to the surface, and the distance dependence shows the repulsion to increase as the protein approaches the surface. Surprisingly, the DCM still gives IFEs smaller than -1kbT, at least with Asp 53 oriented toward the surface, but the ionic strength dependence indicates that the attraction is stronger at higher ionic strengths. This is clearly due to the screening of repulsive contributions to the interaction. (35) Tanford, C.; Hauenstein, J. D. J. Am. Chem. Soc. 1956, 78, 5287.
6766 Langmuir, Vol. 13, No. 25, 1997
Asthagiri and Lenhoff
Table 3. Interaction Free Energy (in kbT) for Ribonuclease A with the Protein End-Group Modela I ) 0.05 M pH 8
pH 5.5
Glu 9 Asp 38 Glu 49 Asp 53 Glu 86 Glu 111 Glu 9 Asp 38 Glu 49 Asp 53 Glu 86 Glu 111
Table 4. Interaction Free Energy (in kbT) for Cytochrome ca
I ) 0.1 M
3.7 Å
5.7 Å
3.7 Å
5.7 Å
-2.0 1.6 -3.4 -3.0 0.2 -0.3 -1.5 2.1 -2.5 -2.5 0.7 0.6
-1.1 2.1 -2.3 -1.9 0.8 0.6 -0.7 2.5 -1.6 -1.5 1.2 1.4
-2.0 0.9 -3.1 -2.7 -0.3 -0.6 -1.5 1.3 -2.4 -2.3 0.1 0.0
-1.1 1.4 -2.0 -1.6 0.4 0.3 -0.8 1.7 -1.4 -1.4 0.6 0.8
a The distance shown is that between the residue charge and the end-group charge.
UCM
DCM
I ) 0.1 M I ) 0.2 M pH 7.0 Glu 4 Asp 50 Glu 62 Glu 66 Glu 69 Glu 104 pH 5.5 Glu 4 Glu 104 a
2Å
4Å
2Å
4Å
1.2 2.3 2.7 5.3 2.9 1.4 1.8 2.3
0.7 1.7 1.8 3.4 2.0 0.9 1.2 1.6
0.4 0.9 1.4 2.4 1.5 0.6 0.6 0.9
0.1 0.5 0.8 1.7 0.8 0.2 0.3 0.5
I ) 0.1 M
I ) 0.2 M
2Å
2Å
Table 5. Interaction Free Energy (in kbT) for Cytochrome c for the Protein End-Group Model
pH 7.0
pH 5.5
Similarly, the orientation with Glu 49 toward the surface results in repulsion at an ionic strength of 0.05 M but very weak attraction at 0.1 M. For the EGM, results are presented for all six of the residues originally identified as candidates for being involved in patch-controlled adsorption (Table 3). At pH 8, we see that in addition to Glu 49 and Asp 53, the orientation corresponding to Glu 9 also leads to IFEs less than -1kbT. The distance and ionic strength dependence for all three of these residues are consistent with electrostatically dominated adsorption. Even at pH 5.5, the orientations corresponding to Glu 49 and Asp 53 both lead to IFEs less than -1kbT, and the ionic strength dependence indicates that increasing the ionic strength increases the IFE. However, now the orientation corresponding to Glu 9 indicates screening of repulsion with increasing ionic strength, a behavior similar to that of Asp 53 with the DCM at pH 5.5. In the case of cytochrome c, identification of a particularly plausible negative patch of the kind found for ribonuclease was not possible, and we performed calculations for orientations corresponding to all six of the acidic residues considered likely to interact favorably with the surface. Of these six residues, Glu 62, Glu 66, and Glu 69 are close together, but they are surrounded by basic residues (Figure 4). Glu 4 and Glu 104, though not as close together, have fewer basic residues in proximity. Furthermore only Glu 4, Glu 104, and Asp 50 are optimally oriented. The relevant experimental data for cytochrome c are at pH 7 and pH 5.5, and ionic strengths of 0.1 and 0.2 M;16 the higher ionic strengths used presumably reflect the larger net charge of cytochrome c relative to that of
4Å
The lattice constant for the DCM is 30 Å.
I ) 0.1 M
Figure 4. R-Carbon trace of cytochrome c with relevant residues shown in spacefill format.
4Å
-1.3 -0.5 -1.9 -0.9 0.4 0.8 -0.9 -0.3 1.3 1.0 0.2 0.2 3.9 2.8 1.8 1.1 2.2 1.6 0.9 0.5 -0.8 -0.2 -1.5 -0.7 -0.3 0.1 -1.6 -0.7 0.5 0.7 -0.8 -0.3
Glu 4 Asp 50 Glu 62 Glu 66 Glu 69 Glu 104 Glu 4 Asp 50 Glu 62 Glu 66 Glu 69 Glu 104
I ) 0.2 M
3.7 Å
5.7 Å
3.7 Å
5.7 Å
-1.4 -0.6 -1.3 -0.4 -0.5 -1.3 -1.2 -0.5 -1.1 -0.2 -0.2 -0.8
-0.6 0.0 -0.6 0.2 0.0 -0.6 -0.5 0.1 -0.4 0.3 0.1 -0.2
-1.3 -0.8 -1.3 -0.7 -0.7 -1.3 -1.2 -0.7 -1.1 -0.5 -0.5 -0.9
-0.6 -0.2 -0.6 -0.1 -0.2 -0.6 -0.6 -0.2 -0.5 0.0 -0.1 -0.4
ribonuclease at neutral pH, necessitating stronger screening. The interaction free energies for cytochrome c are given in Table 4. For the UCM the IFEs are not favorable to drive adsorption, and the distance dependence shows increased repulsion as the protein approaches the surface. For the DCM, only orientations corresponding to Glu 4 and Glu 104 lead to negative IFEs at both the ionic strengths studied, and the ionic strength dependence is not consistent with electrostatically dominated adsorption. Because of these results, calculations at pH 5.5 were performed only for the orientations corresponding to Glu 4 and Glu 104. The UCM again yields IFEs that are positive and that increase as the molecule approaches the surface, whereas the DCM behavior indicates screening of repulsion with increasing ionic strength. For the EGM (Table 5), however, attractive interactions are much more prevalent, although the IFEs are not much different from -1kbT. The orientations corresponding to Glu 4 and Glu 62, and possibly Glu 104, are the most plausible candidates for the negative patch, whereas all other residues show screening of repulsion. The situation at pH 5.5 is similar to that at pH 7.0. For this case the EGM is not able to discriminate the change in net charge with change in pH. 5. Discussion The trends shown in the results are quite consistent in that the UCM is least likely to predict electrostatically driven adsorption, followed by the DCM and the EGM, respectively. This sequence is what would be expected physically, as electrostatically controlled adsorption would be expected to result when the positive charge on the adsorbent is most effectively focused on the negatively charged (acidic) residues on the protein. The DCM is clearly capable of doing so for residues that we have termed optimal, but the EGM can do so even for residues that are sterically hindered from approaching an opposite charge on a planar surface. This result illustrates the potential
Protein Adsorption Electrostatics
Langmuir, Vol. 13, No. 25, 1997 6767
Table 6. Effect of Lattice Constant in the DCM on the Interaction Free Energy of Ribonuclease Aa I ) 0.05 M a (Å) pH 8 pH 5.5
a
30 20 10 30 20 10
2Å
4Å
I ) 0.1 M 2Å
4Å
Table 7. Interaction Free Energy for Ribonuclease A for the DCM, for 10o Changes in Orientation about the Nominal Orientation with Asp 53 Facing the Surfacea orientation 10o,
-4.0 -4.2 -4.0 -2.6 -0.6 2.8
-2.6 -3.1 -3.4 -1.4 0.0 2.3
-3.8 -4.1 -4.6 -3.1 -2.2 -1.0
-2.4 -2.8 -3.6 -1.8 -1.3 -0.7
The orientation corresponds to Asp 53 facing the surface.
importance of adsorbent topography on adsorption energetics, including the likely role of spacer arms, and highlights the paucity of detailed information regarding the geometry and charge distribution of typical adsorbents. The results change quantitatively but not qualitatively with various computational parameters. For the UCM, the consistent failure of the results to show strong attraction was obtained despite the use of a very low surface charge density, and in view of the smeared interactions over the whole molecular footprint, increasing the charge density will not lead to prediction of stronger attraction. For the DCM, the situation is more complex, and we have explicitly considered the effect of the lattice constant a. From the form of the potential (eq 6), it is clear that decreasing a increases the potential near the surface. At the same time, decreasing a will cause more of the protein molecule to be affected by the lattice of charges. Thus, as the lattice constant is decreased there will be a trade-off between increased dominance of the local interaction and the larger number of positive charges in the protein that are affected by the positive surface charges. Table 6 shows results for ribonuclease for various lattice constants. The above mentioned trade-off is most evident for the case of ionic strength 0.05 M and pH 8.0, with ∆F first decreasing, then increasing, as a changes from 30 to 10 Å. From the preceding arguments it is clear why the UCM is inadequate in explaining patch-controlled adsorption. The DCM and EGM are somewhat satisfactory for ribonuclease, but even the most attractive interactions are relatively weak (≈-4 kbT). These values depend on numerous factors, with the dependence on gap distance especially important for the present discussion. Some enhancement of the attractive energy would be expected at smaller gap distances, although the validity of the physical description becomes more questionable there. A related possibility is that patch-controlled interactions may be enhanced by conformational adaptation of the protein molecule, in contrast to the rigid structure assumed in the calculations. The electrostatic calculations may also be affected by the fact that the experiments were all performed using ammonium sulfate, and we have neglected the 1:2 nature of the electrolyte as well as effects such as sulfate ion binding, which could lead to a lower net charge, and thus influence the adsorption calculations. Sulfate ion bridging may also facilitate linkage of cationic groups on the protein and the adsorbent. An alternative viewpoint is that elimination of solvent from the gap, leaving a low dielectric medium, can enhance the interaction quite appreciably.7 In the EGM, the addition of an extended low-dielectric medium to represent the support matrix would also affect the results. Furthermore, we have not considered other contributions to the attractive interactions, e.g., van der Waals interactions; these might be expected to add O(kbT) at very close range. However, it is important to stress again the clear picture that emerges from the experimental data, namely, that electrostatic interactions are controlling.
10o
φ+ θ+ θ + 10o, φ -10o o θ - 10 , φ + 10o θ - 10o, φ - 10o
pH 8.0
pH 5.5
-2.0 -1.5 -2.3 -1.3
-0.9 -0.4 -0.9 -0.1
a θ, φ represents the orientation with Asp 53 toward the surface. The ionic strength is 0.05 M. Gap distance was 4 Å; lattice constant was 30 Å.
In contrast to the situation for ribonuclease, we see significant discrepancies for cytochrome c. A possible computational problem here could be inaccuracy in describing the protein surface. A calculation of the depth of all the charges (below the van der Waals surface) using two different tessellations, one approximate and the other the most accurate, shows that there are slight discrepancies in the depth values. The most detailed surface description was not used for any of our calculations because of the significant increase in computing power that would be required for the numerous calculations that were performed in this study. Although the sequence UCM, DCM, EGM is qualitatively as expected, a direct quantitative comparison of theory and experiment is possible only in part. The reported experimental quantity is the capacity factor k′,16 which can be expressed as the product of the (Henry’s law) equilibrium constant for adsorption and the phase ratio (surface area per unit volume of mobile phase) of the sorbent as packed in the column. The value of the phase ratio is not available, but the equilibrium constant can be estimated from the orientation-dependent free energy via a Gibbs surface excess approach that involves averaging a Boltzmann-weighted quantity.3 The equilibrium constant for the UCM is given by
K)
∫z∞ dz ∫Ω[exp(-∆F(z,Ω)/kbT) - 1] dΩ 0
(10)
where Ω refers to orientational space, and z0 is a cutoff distance. When the DCM is used, there is also a need to perform a Boltzmann average over the unit cell on the surface. The orientational averaging must be performed with some care because of the relatively small attractive region associated with patch-controlled adsorption; on the other hand, a full orientational averaging is computationally infeasible. In order to probe the sensitivity of the interaction free energy to orientation, we have performed exploratory calculations for the DCM for small changes in the orientation of ribonuclease with Asp 53 facing the surface (Table 7). For each orientation shown, the molecule was rotated about the center of mass relative to the nominal orientation (θ,φ). The molecule was then translated perpendicular to the surface so as to keep the minimum gap distance at 4 Å. The Asp 53 residue was thus no longer immediately above the surface charge site, nor was it optimally oriented. Nevertheless, the IFEs change relatively little with changes in orientation, and they are all still negative. This dependence obviously affects the Boltzmann averaging, but it also indicates that the “patch” actually covers a range of configurations. However, the calculation of K will still be dominated by a very small fraction of the orientational configurations that yield negative interaction free energies. The emphasis here on anomalous adsorption may obscure the implications of the work for adsorption under more conventional ion-exchange conditions, where the
6768 Langmuir, Vol. 13, No. 25, 1997
Asthagiri and Lenhoff
The one basic residue that yields a positive IFE is an interesting case. This is Arg 33, which is close to the acidic residues Glu 9 and Asp 14. Thus when Arg 33 is oriented toward the surface, the IFE calculated using the DCM is more positive than that using the UCM, thereby demonstrating that when both models are similarly affected by local interactions, the DCM is more sensitive than the UCM. From these results we can infer that when the protein and the surface are oppositely charged, model calculations are much less sensitive to model idealizations than in the case of like charge. A quantitative comparison of the model predictions shows that the DCM is more sensitive to both the local and global charge characteristics than is the UCM, but the same trends, especially dependence on ionic strength, are predicted by both. 6. Conclusions
Figure 5. Comparison of the IFEs calculated for ribonuclease adsorbing onto a negatively charged surface. Filled squares correspond to orientations in which a basic residue is oriented toward the surface. Open squares correspond to one in which one of the six acidic residues (in table 1) is oriented toward the surface. Gap distance was 2 Å; ionic strength was 0.05 M; pH was 8.0.
protein and the surface have net charges of opposite sign. We have studied such a situation for ribonuclease at pH 8 adsorbing on a negatively charged surface, for both the UCM and the DCM. The lattice constant is maintained at 30 Å, but now the charge at each lattice point is -1e. Similarly, for the UCM, the surface charge density is -1.8 µC/cm2. The IFEs for corresponding orientations using the UCM and the DCM at a 2 Å gap are plotted against each other in Figure 5. The trend in the calculated IFE is similar for the two models, and in particular, both predict attraction for the same orientations. Thus both models will predict similar trends in adsorption behavior, although the calculated adsorption energetics and equilibria will differ quantitatively, with the DCM magnitudes generally higher. The differences in K may be substantial because of the Boltzmann weighting, but they may be masked if any adjustable parameters are included in the formulation. Furthermore, since most orientations give rise to attractive IFEs, calculation of K is less dependent on the IFE in specific orientations, and even highly simplified models can capture the correct trends. A further analysis of Figure 5 offers some additional insights into the UCM and the DCM for this situation. For the orientations corresponding to the three acidic residues that yield a negative IFE, the UCM value is more negative than the DCM value. This is a consequence of the fact that even though the global charge characteristics (namely protein net charge) yield a negative IFE, the local repulsion between the acidic residues and the surface is higher for the DCM than for the UCM. For the remaining acidic residues, the interaction free energies are positive, with the DCM value more positive, again indicating that the local repulsion is greater for the DCM than for the UCM. A similar trend is seen for the orientations corresponding to the basic residues, as a result of which the residues that yield a negative IFE lead to more negative values for the DCM than for the UCM.
We have calculated electrostatic interaction energies for ribonuclease and cytochrome c on anion-exchange surfaces. As has been done in several previous studies,3,5,8-10,13 we have accounted for structural details of the protein molecule (geometry and charge distribution). The distinguishing feature of the present work is that the calculations were performed for solution conditions under which both the protein molecules and the adsorbent have a net positive charge. As a result, the sensitivity of the calculations to structural details is not masked by the net opposite charges of the protein and the surface, as in previous studies, and any adsorption observed would be due largely to the heterogeneity of the protein surface and the adsorbent surface. We have thus been able to identify more effectively the contributions to adsorption of charge patches (including local geometry) on the protein and the sensitivity of the calculated adsorption behavior to different idealized representations of the adsorbent surface. Our calculations indicate that the uniform charge model is incapable of accounting for the local interactions between the positive surface charge and the negative charges on the protein and yields results that are largely dependent on the net charge of the protein molecule. The discrete charge and end-group models are significantly better and are able to account to varying degrees for the local and net charge effects on protein adsorption. These results indicate the potential importance in adsorption calculations of adsorbent structural details at a level that is largely inaccessible experimentally. Although our emphasis has been on conditions where the protein and the surface are of like charge, our results also have implications for the more commonly studied systems in which the protein and surface are of opposite net charge. We have shown that here the model predictions are much less sensitive to model details. The relative insensitivity of predicted behavior to structural details, coupled with the experimental inaccessibility of detailed adsorbent structural information, suggests that under these conditions there is much less justification for using detailed protein models to describe adsorption behavior and that simplified models that account mainly for the protein size and net charge may often be adequate. Acknowledgment. We are grateful for support from the National Science Foundation under Grant Number CTS-9321318. LA970608U
