Theoretical Model To Predict the Diffusion Coefficients of Enzymes on

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Ind. Eng. Chem. Res. 2007, 46, 7410-7416

Theoretical Model To Predict the Diffusion Coefficients of Enzymes on Adsorption Processes Based on Hard-Core Two-Yukawa Potential Ruth Gutie´ rrez, Eva M. Martı´n del Valle,* and Miguel A. Gala´ n Department of Chemical Engineering, UniVersity of Salamanca, P/Los Caidos S/N 37008, Salamanca, Spain

A theoretical model based on the hard-core two-Yukawa potential, and taking into account the surface diffusion and molecular diffusion, has been developed to predict the diffusion coefficient of enzymes on adsorption processes. The theoretical diffusion coefficients predicted by the model have been compared to the experimental diffusion coefficients of dilute aqueous solutions of catalase and Cu,Zn-superoxide dismutase (Cu,Zn-SOD) at different pH values, as determined by dynamic methods for immobilized metal-ion affinity chromatography (IMAC). For SOD, the experimental and the theoretical data show an increase in the relative diffusion coefficient as the enzyme net charge is increased. This behavior is due to the presence of unscreened charge in the enzyme surface. However, for catalase, the experimental the theoretical data show a decrease in the relative diffusion coefficient as the enzyme net charge is increased. This behavior is a consequence of a hydrodynamic expansion of the molecule produced by changes in the pH of the medium. 1. Introduction Dispersion systems such as micelle, colloid, and microemulsion systems are of great importance in a variety of technological applications. Enzymes are charged macromolecular compounds that can be treated as colloid particles, and they have an important role in the physiological reactions of human and animals. An increasingly important operation in biotechnology industry is the separation and purification of proteins, where the degree of separation, purity, and yield of a particular protein is highly influenced by the media properties, and the diffusion coefficient has a significant role in these properties. Moreover, it is crucial to have some knowledge of diffusion dependence on pH and protein concentration, because the biological fluids in such process are concentrated solutions under a wide range of pH values. However, the diffusion coefficients are studied much less than is needed, because of the difficulty both in theory and experiment. Generally, enzymes in solution are composed of solvent, dispersed particles, and electrolytes. The dispersed particles immersed in the continuum are surrounded by an electric double layer, where one layer is formed by the charge in the surface of the particles and the other layer is formed by the excess of oppositely charged ions in the solution.1 Several methods have been used to determine experimentally the diffusion coefficient of proteins in solution. Dynamic light scattering (DLS) is one of the experimental methods often used in the determination of diffusion coefficients of globular proteins in aqueous solutions.2 However, due to the opacity of concentrated suspensions and of multiple light scattering effects, it fails at high volume fractions.3 Another method, which is greatly used in chromatography for dilute solutions, is the dynamic method of impulse-response and the analysis of the experimental results by the moment’s theory.4 Theoretically, the diffusion coefficients of charged globular proteins in aqueous electrolyte solutions can be predicted using the generalized Stokes-Einstein equation.1,3,5 In this method, the sedimentation coefficient and the osmotic pressure may be known in advance; however, they can also be determined by * To whom correspondence should be addressed. E-mail address: [email protected].

several empirical correlations.5 The sedimentation coefficient for completely ordered suspensions of spheres can be estimated, among others, by the correlations developed by Zick and Homsy6 and by Happel and Brenner,7 both showing great accuracy. The osmotic pressure for proteins or other electrostatically stabilized colloids may be calculated accurately via the use of an integral equation, the Poisson-Boltzmann (PB) cell model, based on the extended Derjaguin-Landau-VerweyOverbeek (DLVO) theory,8 which allows the prediction of the osmotic pressure for the hole concentration range for a given set of physicochemical conditions; or on the Yukawa potentials,9 using the hypernetted chain approximation or mean spherical approximation (MSA). The advantage of MSA is that it can give an analytical solution and an explicit equation of state (EOS).9 In the PB cell model,8 the osmotic pressure includes the contributions from electrostatic interactions, London-van der Waals forces and configurational entropy. The purpose of this work is to establish a predictive theoretical method for the prediction of diffusion coefficients of proteins in adsorption processes as immobilized metal-ion affinity chromatography (IMAC), as a function of the main physicochemical conditions and its dependence on the pH. To do that, we have applied the generalized Stokes-Einstein equation, where the required concentration dependence of the osmotic pressure is described by the MSA solution of the hard-core twoYukawa potential. Thus, the theoretical diffusion coefficients predicted by the model have been compared to the experimental diffusion coefficients of dilute aqueous solutions of catalase and SOD at different pH values, as determined using dynamic methods for IMAC chromatography. 2. Theoretical Background The Stokes-Einstein equation describes the Brownian motion of a single sphere in a liquid medium that is due solely to the thermal fluctuations of the molecular movements around the particle:

D0 )

10.1021/ie0700415 CCC: $37.00 © 2007 American Chemical Society Published on Web 04/21/2007

kbT 6πµahyd

(1)

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where D0 is the Brownian diffusion coefficient, ahyd the hydrodynamic radius of the particle, µ the viscosity of the fluid, kb the Boltzmann’s constant, and T the absolute temperature. However, particles are not alone in the medium, and their diffusion in concentrated suspensions is influenced by the presence of their neighbors. Moreover, there are two diffusion processes that must be distinguished: one known as selfdiffusion (Ds) and another one known as the gradient of bulk diffusion (DM). The self-diffusion describes the fluctuating trajectory of a tracer particle among others. The gradient or bulk diffusion is the macroscopic flux of particles produced by Brownian motion in the presence or a gradient in the total number density of particles. This latest coefficient one that appears in Fick’s Law of diffusion. Both the self-diffusion and gradient diffusion coefficients coincide with the Brownian one at infinite dilution but differ in more-concentrated suspensions. In the present paper, the concentration dependence of the gradient diffusion coefficient of electrostatically stabilized particles is investigated as a function of the physicochemical parameters. This problem is of great practical importance as the bulk diffusion coefficient is a key parameter in a large number of processes involving globular proteins, such as affinity chromatography. The theoretical calculation of the bulk diffusion coefficient is a difficult task because it is necessary to calculate multiparticle interactions, which is complicated and not entirely resolved. Several methods, based on the generalized Stokes-Einstein equation,10 have previously been used to predict the mutual diffusion coefficient,11-13

K(φ) DM(φ) ) D0‚ S(φ)

(2)

where φ is the volume fraction (φ ) πFσ3/6), K(φ) the hydrodynamic interaction coefficient, and S(φ) is the thermodynamic coefficient. This equation is valid over the entire range of volume fractions. 2.1. Generalized Stokes-Einstein Equation. The dilutelimit value of the Brownian diffusion coefficient is modified, at finite concentrations, by the fluid colloidal and hydrodynamic interactions. The combined action of these interactions may be described by the generalized Stokes-Einstein equation,10 which is valid for the complete range of volume fractions. The coefficient, S(φ), in eq 2 accounts for its thermodynamic properties. This coefficient may be directly determined from the osmotic pressure, Π(φ), as5

[

]

4πahyd3 ∂Π(φ) 1 ) 3kbT ∂φ S(φ)

(3)

The osmotic pressure of a colloidal suspension is a function of particle concentration, because of the combined effects of double-layer repulsion, van der Waals forces, and configurational entropy.8 Thus, prediction of the osmotic pressure allows the calculation of S(φ). The hydrodynamic coefficient, K(φ), accounts for the fact that, in concentrated suspensions, the drag force exerted on a single particle deviates from Stokes’ Law, because of the presence of the neighboring particles. This coefficient describes the sedimentation velocity on an assemblage of spheres:

K(φ) )

U(φ) U0

(4)

where U0 is the Stokes settling velocity of an isolated particle and U(φ) is the mean fall velocity of a particle in suspension. For systems of charged spheres where electrostatic repulsion is the dominant colloidal interaction, both S(φ) and K(φ) decrease with volume fraction. However, instead of their having cancelling effects on the diffusion coefficient, the resulting concentration dependence of DM(φ) can be significant in many cases. Prediction of the gradient diffusion coefficient relies on the accurate determination of both S(φ) and K(φ). 2.1.1. Yukawa Equation of State (EOS). The Yukawa potential may be used to describe the attractive and repulsive interaction of molecules. The mathematic representation of hardcore Yukawa potential can be expressed as follows:

u(r) )

{

∞ -

(for r e σ)

 exp-λ[r/(σ - 1)] (for r > σ) r/σ

(5)

where  is the energy parameter, σ the hard-sphere diameter of a molecule, r the center-to-center distance of two molecules, and λ an adjustable range parameter that represents the range size of interaction. Based on the mean spherical approximation (MSA), Duh and Mier-Y-Tera´n9 presented an explicit and analytical expression for the Helmholtz free energy in the one-component Yukawa fluid (DMT EOS), as

[

]

R0 dF(y) λ3 A - Ahs F(x) - F(y) - (x - y)‚ ) - β NkbT φ0 6φ dy

(6)

The expressions for x, y, R0, φ0, and the functions for F(x), dF(y)/dy are given in Appendix A. The detailed derivation of eq 6 is taken from Wu and Prausnitz.14 The corresponding compressibility factor can be derived as follows:

Z)F

[

ZYukawa ) ) -φ

[

]

]

∂(A/NkbT) ∂T

p - phs FkbT

β ∂R0 R0∂φ0 + φ0 ∂φ φ0∂φ

[ { [

(7)

N,T

3

]

dF(y) λ F(x) - F(y) - (x - y) 6φ dy

]

}

d2F(y) λ3 ∂x dF(x) dF(y) ∂y - (x - y) 6 ∂φ dx dy ∂φ dy2

(8)

There are no adjustable empirical parameters needed in eq 8, which is called the DMT EOS. Garnett et al.15 compared the compressibility factor calculated from the DMT EOS with Monte Carlo (MC) simulation data and MSA results obtained via the energy route, at λ ) 1.8, 2.0, 3.0, and 4.0. At λ ) 1.8 the DMT EOS can give good results that are consistent with MC simulations under all conditions covered. 2.1.2. Osmotic Pressure for Protein Solution. To develop an expression for the osmotic pressure of the protein solutions, we have considered the aqueous protein solution as a pseudoone-component system. Therefore, similarly charged protein molecules are considered as the only component, and the water and the micro-ions are treated as continuous medium. In the one-component model, aqueous solutions of globular proteins are represented with an assembly of spherical macro-ions that interact via effective solvent-dependent potentials. The effective

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potential, u(r), between protein molecules (with diameter σ and net charge zp) is given by the sum of three terms:

u(r) ) uhs(r) + ucc(r) + udis(r)

(9)

where r is the center-to-center distance. Here, uhs(r) is the hardsphere potential, and ucc(r) is the screened Coulomb potential that is due to electrostatic repulsion. Correspondingly, based on the McMillan-Mayer solution theory, the osmotic compressibility factor in protein electrolyte solutions can be expressed as

2.1.3. Donnan Equilibrium. The Donnan effect is the additional osmotic pressure brought by the micro-ions on both sides of an osmotic membrane. This extra pressure is due to the impossibility of proteins penetrating the membrane during the establishment of the osmotic equilibrium, which implies that the concentrations of micro-ions on both sides of the membrane are unequal, to maintain electroneutrality and equilibrium. The contribution of the Donnan effect to the osmotic compressibility factor (ZDοnnan) can be expressed as Donnan

Z)

Π ) ZDonnan + Zhs + Zcc + Zdis FpkbT

Z (10)

Here, Π is the osmotic pressure and Fp is the number density of protein molecules. The superscripts “Donnan”, “hs”, “cc”, and “dis” represent the contribution of the Donnan effect, hardsphere repulsion, double-layer repulsive charge-charge interactions, and attractive dispersion, respectively. The hard-sphere and double-layer repulsion potentials between proteins (from DLVO theory) are represented by16

{

∞ u (r) ) 0 hs

ucc(r) ) -

(for r e σ) (for r e σ) (for r > σ) (for r > σ)

′ exp[-λ′[r/(σ - 1)]} r/σ

(11)

(for r > σ) (12)

where

zp2e2 ′ ) σD[1 + (κσ/2)]2 λ′ ) κσ σ is the diameter of a protein molecule, zpe is the charge on a protein, D is the dielectric constant of water, κ is the Debye screening parameter related to the number density (Fi), valence of the micro-ion i (zi), and the temperature (T):

κ ) 4π 2

Fie2zi2

∑i Dk T

(13)

b

where kb is the Boltzmann constant. It can be observed that the double-layer repulsive interaction is a Yukawa potential. Therefore, the electrostatic contribution to the compressibility factor (Zcc) can be obtained using eq 8. The hard-sphere contribution to the compressibility factor (Zhs) can be calculated with the Carnahan-Starling equation,1 which may be written as

Zhs )

1 + φ + φ 2 - φ3 (1 - φ)3

(14)

Now we use another Yukawa potential to express the dispersion attractive interactions between protein molecules as

udis(r) ) -

 exp{-λ[r/(σ - 1)]} r/σ

(for r > σ) (15)

where  > 0 is the energy parameter. We choose the value λ ) 1.8, which is claimed to yield results comparable with those obtained from the Lennard-Jones potential.16 The contribution of attractive dispersion to the compressibility factor (Zdis) can also be calculated with eq 8.

in out out Fin + + F- + F+ + F) Fp

(16)

where Fp, F+, and F- are the number densities of the protein, microcation, and microanion, respectively. Considering a single electrolyte, the equations of electroneutrality and the equal ionic concentration products on both sides of the membrane can be expressed as in zpFp + z+Fin + + z- F- ) 0

(17)

out z+Fout + + z- F- ) 0

(18)

z- in z+ out z- out z+ (Fin +) (F-) ) (F+ ) (F- )

(19)

where zp, z+, and z- are the charge numbers of the protein, microcation, and microanion, respectively. The superscripts “in” and “out” represent the protein side and the micro-ion side, respectively. In this study, the Donnan effect has been considered negligible, according to the results presented by Fu et al.1 2.1.4. Hydrodynamic Coefficient for Protein Solution. Because of the importance in practical applications of sedimentation of a suspension of particles under the action of gravity, it has been extensively studied, mainly the monodisperse spheres sedimentation velocity dependence on concentration at very small particle Reynolds numbers. This concentration dependence reflects both the hydrodynamic interactions and the interparticle forces, which determine the equilibrium microstructure. The influence of colloidal forces on hydrodynamic interactions affects the sedimentation of charged spheres, which can be enhanced by attractive colloidal interaction and slowed by repulsive interactions at a given volume fraction. For electrostatically stabilized enzymes, if the electrolyte concentration in solution is not high (as the case in this study), the interaction between two enzyme molecules is repulsive and the sedimentation coefficient always decreases as the volume fraction is increased. For Brownian hard spheres, many experimental data can be correctly described up to φ ) 0.05 by the Felderhof approach,17 which coincides with the Bachelor’s result in the dilute limit:

K(φ) ) 1 - 6.54φ

(20)

This correlation gives a satisfactory description of the sedimentation of a disordered suspension of hard spheres. 2.1.5. Surface Diffusion Coefficient. Surface diffusion has an important role in the mass transfer inside the intraparticle space of porous adsorbents,18 and its contribution to the total effective diffusion is quite important when the adsorbent intraparticle surface is high, as is the case for our chromatographic support, agarose. Therefore, it is necessary to determine the experimental surface diffusion coefficient to compare the experimental and theoretical relative diffusion coefficients properly.

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7413 Table 1. Parameters Used To Evaluate the Mutual Diffusion Coefficient for Aqueous SOD and Catalase Solutions Value parameter

SOD

catalase

Mp σ λ T /kb zp pH 7.00 pH 7.40 pH 7.80

32500 Da 2.8 nm 1.8 298 K 91.6 K

250000 Da 8.5 nm 1.8 298 K 91.6 K

-3.12 -4.33 -5.18

2.79 -0.66 -2.80

Mass transfer inside porous adsorbents is frequently taken into account by assuming that the total effective diffusion is explained by parallel contributions of bulk and surface diffusion,19 with

(DM)t ) DM + FsefKADS

(21)

where the bulk diffusion coefficient, DM, can be calculated from the gyration radius, Rg:20

DM )

1.69 × 10-9 Rg

Rg ) 0.359Naa3/5 + 7.257

(22) (23)

with Naa being the number of amino acids. 3. Results and Discussion According to the theories and models described before, we have determined the effect that the enzyme concentration and the media pH have on the diffusion coefficient. The first step necessary to apply the Stokes-Einstein equation in the determination of the diffusion coefficient is to calculate the different coefficients involved in the osmotic compressibility factor (Zhs, Zdis, and Zcc), using the Yukawa potential (eq 8).

The values of the parameters used in the calculation of the collective diffusion coefficient are collected and listed in Table 1. Figure 1 shows that increasing the volume fraction (directly related to concentration) implies an increase in the osmotic compressibility factor, Z(φ), for both enzymes. It is also possible to see that the osmotic compressibility factor is dependent on the net charge of the enzyme, given that the contribution of the charge-to-charge compressibility factor to the osmotic compressibility factor increases as the enzyme’s net charge is increased. Comparing the net-charge dependence of the two enzymes, it can be concluded that the higher the enzyme’s molecular weight, the smaller the effect of the net charge on it, given that it is also possible to consider as negligible the charge interactions between catalase molecules to be negligible. As a direct consequence of this, the osmotic compressibility factor for catalase is smaller than that for SOD. When the osmotic compressibility factor has been determined, using the third term of eq 10, the effect of the concentration on the osmotic pressure, Π(φ), can be determined using the second term of eq 10, and, therefore, the Stokes-Einstein thermodynamic coefficient (S(φ)) can be determined by eq 3. Figure 2 represents the results obtained. In this figure, it can be observed that increasing the enzyme concentration decreases the thermodynamic coefficient, which is consistent with Bowen’s results.5 It can also be observed that decreasing the enzyme’s net charge also increases the thermodynamic coefficient. The other parameter necessary in the diffusion coefficient determination is the hydrodynamic coefficient, K(φ). As it has been explained previously, this coefficient has been determined by the Felderhof approach (eq 20). After all the parameters that are needed have been determined, the next step is to calculate the theoretical bulk diffusion coefficient as a function of volume fraction, according to the generalized Stokes-Einstein equation (eq 2). To be able to compare the experimental effective diffusion coefficients to the theoretical values, it is necessary to determine the effective theoretical diffusion coefficients, for which one must first know

Figure 1. Dispersion, hard-core, charge-to-charge, and global osmotic compressibility factor versus volume fraction for SOD and catalase at different pH values. Conditions: 298 K and 50 mM potassium phosphate buffer.

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Figure 2. Thermodynamic coefficient versus volume fraction for SOD and catalase at different pH values. Conditions: 298 K and 50 mM potassium phosphate buffer.

Figure 3. Theoretical and experimental relative diffusion coefficient versus volume fraction for SOD and catalase at different pH values. Conditions: 298 K and 50 mM potassium phosphate buffer.

the surface diffusion coefficients. The latter values have been calculated from the experimental effective diffusion coefficients and equilibrium constants, according to eq 21. Figure 3 shows the experimental and the theoretical relative diffusion coefficients, as a function of the volume fraction for SOD and catalase. In this figure, it can be observed that the theoretical data are in good agreement with the experimental data. It can also be observed that, in the case of SOD in 50 mM potassium phosphate solution and 298 K, the theoretical model predicts an increase in the relative diffusion coefficient when increasing the enzyme concentration, and when increasing the pH of the medium. This increase in the relative diffusion coefficient, depending on pH, can be explained taking into account that SOD acquires a large negative net charge as the

pH is increased over the isoelectric point, a part of which is neutralized by the strong binding of K+ ions. At high pH, an SOD molecule can carry a charge up to 100 electronic units and the electrostatic (repulsive) energy between two unscreened charged molecules has an important role in the diffusive motion.21 Because the medium ionic strength has been kept constant for the three experimental pHs, the unscreened charge of the SOD molecules increases as the medium pH is increased, i.e., as the medium pH moves away from the isoelectric point of the enzyme. Thus, it can be concluded that the medium ionic strength is not enough to totally screening the SOD’s charge, remaining an increasing number of unscreened charges in the molecules surface (as the medium pH is moved away from the SOD pI), which enhances the SOD’s diffusion due to the

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7415

dF(x) x(1 - 3x + 3x2) ) dx (1 - 2x)(1 - x)2

increase in the repulsion among the enzyme molecules. This behavior totally agrees with the studies done by Raj and Flygare.21 According to Raj and Flygare,21 the diffusion coefficient is not dependent on concentration when the enzyme net charge is completely shielded, corresponding the changes in the diffusion coefficient at different pHs to conformational changes in the enzyme structure. In the case of catalase, the relative diffusion coefficient slightly decreases when the enzyme concentration increases, because it decreases in a sensitive way when increasing the pH of the medium. Taking into account the theoretical and experimental results, and considering that the catalase net charge is low, it can be considered that its charge is completely shielded, and, also, that an increase in the medium pH, i.e., an increase in the enzyme net charge, implies an hydrodynamic expansion of the molecule, which involves a decrease in the diffusion coefficient.

d2F(x) dx2

Acknowledgment This research was supported by funds from the Ministerio de Educacion y Ciencia. Ms. R. Gutie´rrez gratefully acknowledges a fellowship from the same organization (MEC).

1 - 5x + 11x2 - 9x3 (1 - 2x)2(1 - x)3

R0 ) φ0 )

L(λ) λ (1 - φ)2 2

exp(-λ)L(λ) + S(λ) λ3(1 - φ)2 ω)6

( ) φ φ02

λ2(1 - φ)2(1 - exp(-λ)) 12φ(1 - φ){1 - (λ/2) [1 + (λ/2)] exp(-λ) } φ) exp(-λ)L(λ) + S(λ)

4. Conclusions A theoretical model to predict the effective diffusion coefficient of enzymes in adsorption processes as immobilized metalion affinity chromatography (IMAC) has been proposed. The model is based on the generalized Stokes-Einstein equation, taking into account the contribution the surface diffusion has on the adsorption process. The hard-core two-Yukawa model with mean spherical approximation is introduced to evaluate the influence of thermodynamic interactions on the effective diffusion coefficient. From the comparison with the experimental effective diffusion coefficients, as determined in IMAC by dynamic methods, it can be concluded that the hard-core twoYukawa model coupled with the sedimentation coefficient and with the surface diffusion contribution is a good theory to predict the behavior of the effective diffusion coefficient of dilute enzyme solutions in electrolyte suspensions related to the pH of the medium.

)

[(

L(λ) ) 12φ 1 +

φ λ + 1 + 2φ 2

)

}

S(λ) ) (1 - φ)2λ3 + 6φ(1 - φ)λ2 + 18φ2λ - 12φ(1 + 2φ) Appendix B. Calculation of Net Charge of a Protein at a Given pH In this paper, the net charge (zp) of catalase and SOD was calculated from the Henderson-Hasselbalch equation:

pH ) pKa + log

( ) [A-] [HA]

(B1)

where [A-]/([A-] + [HA]) is the percentage of charged elements. To calculate the net charge of catalase and SOD at a given pH, we must follow the next scheme: (1) List the catalase and SOD ionizable residues and their respective pKa (see Table B1). (2) Determine the percentage of charged elements (chargei). (3) Determine the total charge contributed by each amino acid,

chargeAA ) chargei × ni

(B2)

Appendix A. Expressions and Functions The following equations describe and define the expressions and functions mentioned in the main text of this paper.

1 1 3 F(x) ) - ln(1 - 2x) - 2 ln(1 - x) - x +1 4 2 1-x

()

x)

[

]

(1 + λφ)ω

y)

λ2

β

where ni is the number of residues. (4) Determine the net charge of catalase and SOD:

zp )

∑ ni × chargei + chargeN-term + chargeC-term

The equation obtained, following this procedure for catalase and SOD net charge, as a function of pH, is

zp ) cys

(

asp

( (

his

()

1 T kb

) ( ) ( ) ( )

)

-10pH-8.5 -10pH-4.5 + glu + 1 + 10pH-8.5 1 + 10pH-4.5

(ωφλ)β

β)

(B3)

arg

(

) )

-10pH-4.5 -10pH-10 + tyr + 1 + 10pH-4.5 1 + 10pH-10 106.5-pH 1010-pH + lys + 1 + 106.5-pH 1 + 1010-pH

1012-pH + chargeN-term + chargeC-term (B4) 1 + 1012-pH

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Table B1. Ionizable Residues of Superoxide Dismutase (SOD) and Catalase and the Corresponding pKa Values Ionizable Residue cysteine glutamic acid aspartic acid tyrosine C-term histidine lysine arginine N-term

SOD

catalase

pKa

3 8 11 1 1 8 10 4 1

4 26 37 21 1 24 28 32 1

8.5 4.5 4.5 10 3.5 6.5 10 12 7.5

Nomenclature A ) Helmholtz free energy ahyd ) hydrodynamic radius of the particle D ) dielectric constant of water D0 ) Brownian diffusion coefficient Dc ) mutual diffusion coefficient Ds ) self-diffusion coefficient e ) electron charge kb ) Boltzmann constant K(φ) ) hydrodynamic interaction coefficient N ) Avogadro’s number Naa ) number of amino acids Rg ) radius of gyration (Å) r ) center-to-center distance of two molecules S(φ) ) thermodynamic coefficient T ) absolute temperature (K) U0 ) Stokes settling velocity of an isolated particle U(φ) ) mean fall velocity of a particle in suspension u(r) ) effective potential between protein molecules Z(φ) ) osmotic compressibility factor zi ) valence of the micro-ion i zpe ) charge on a protein Greek Letters φ ) volume fraction  ) energy parameter F ) number density κ ) Debye screening parameter λ ) range size of interaction parameter Π(φ) ) osmotic pressure σ ) hard-sphere diameter of a molecule µ ) viscosity of the fluid Literature Cited (1) Fu, D.; Lu, J. F.; Wu, W.; Li, Y. G. Study on osmotic pressure and liquid-liquid equilibria for micelle, colloid and microemulsion systems by Yukawa potential. Chin. J. Chem. 2004, 22, 627-637.

(2) Meechai, N.; Jamieson, A. M.; Blackwell, J. Translational Diffusion Coefficients of Bovine Serum Albumin in Aqueous Solution at High Ionic Strength. J. Colloid Interface Sci. 1999, 218, 167-175. (3) Yu, Y. X.; Tian, A. W.; Gao, G. H. Prediction of collective diffusion coefficient of bovine serum albumin in aqueous electrolyte solution with hard-core two-Yukawa potential. Phys. Chem. Chem. Phys. 2005, 7, 24232428. (4) Gutie´rrez, R.; Del Valle, E. M. M.; Gala´n, M. A. Adsorption and mass transfer studies of Catalase in IMAC chromatography by dynamics methods. Process Biochem. 2006, 41, 142-151. (5) Bowen, W. R.; Liang, Y.; Williams, P. M. Gradient diffusion coefficients-theory and experiments. Chem. Eng. Sci. 2000, 55, 23592377. (6) Zick, A. A.; Homsy, G. M. Stokes flow through periodic arrays of spheres. J. Fluid Mech. 1982, 115, 13-26. (7) Happel, J.; Brenner, H. Flow relative to assemblages of particles. In Low Reynolds Number Hydrodynamics; Noordhoff International Publishing: Leiden, The Netherlands, 1973; pp 358-430. (8) Bowen, W. R.; Williams, P. M. The osmotic pressure of electrostatically stabilized colloidal dispersions. J. Colloid Interface Sci. 1996, 184, 241-250. (9) Duh, D. M.; Mier-Y-Tera´n, L. An analytical equation of state for the hard-core Yukawa fluid. Mol. Phys. 1996, 3, 373-379. (10) Batchelor, G. K. Brownian diffusion of particles with hydrodynamic interaction. J. Fluid Mech. 1976, 74, 1-29. (11) Petsev, D. N.; Denkov, N. D. Diffusion of charged colloidal particles at low volume fraction: Theoretical model and light scattering experiments. J. Colloid Interface Sci. 1992, 149, 329-344. (12) Genz, U.; Klein, R. Collective diffusion of charged spheres in the presence of hydrodynamic interaction. Physica A 1991, 171, 26-42. (13) Bowen, W. R.; Mongruel, A. Calculation of the collective diffusion coefficient of electrostatically stabilized colloidal particles. Colloids Surf. A 1998, 138, 161-172. (14) Wu, J. Z.; Prausnitz, J. M. Osmotic pressures of aqueous bovine serum albumin solutions at high ionic strength. Fluid Phase Equilib. 1999, 155, 139-154. (15) Vilker, V. L.; Colton, C. K.; Smith, K. A. The osmotic pressure of concentrated protein solutions: Effect of concentration and pH in saline solutions of Bovine Serum Albumin. J. Colloid Interface Sci. 1981, 79, 548-566. (16) Lin, Y. Z.; Li, Y. G.; Lu, J. F. Study on osmotic pressures for aqueous Lysozyme and alpha-Chymotrypsin-Electrolyte solutions with two Yukawa potentials. J. Colloid Interface Sci. 2002, 251, 256-262. (17) Felderhof, B. U. Diffusion of interacting Brownian particles. J. Phys. A 1978, 11, 929-937. (18) Miyabe, K.; Guiochon, G. New model of surface diffusion in reversed-phase liquid chromatography. J. Chromatogr., A 2002, 961, 2333. (19) Ruthven, D. M. Principle of Adsorption and Adsorption Processes; Wiley: New York, 1984. (20) Tyn, M. T.; Gusek, T. W. Prediction of diffusion coefficients of proteins. Biotechnol. Bioeng. 2004, 35, 327-338. (21) Raj, T.; Flygare, W. H. Diffusion studies of Bovine Serum Albumin by Quasielastic Light Scattering. Biochemistry 1974, 13, 3336-3340.

ReceiVed for reView January 9, 2007 ReVised manuscript receiVed February 28, 2007 Accepted March 16, 2007 IE0700415