Ind. Eng. Chem. Res. 1996, 35, 3015-3026
3015
Thermodynamics of Precipitation of Globular Proteins by Nonionic Polymers Rong Guo, Meining Guo, and Ganesan Narsimhan* Biochemical and Food Process Engineering, Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, Indiana 47907-1146
Flexible random-coil structure of polymer molecules and protein-polymer and polymer-solvent interactions were accounted for in the evaluation of the interaction potential between two globular protein molecules which consisted of attraction because of depletion of polymer segments in the vicinity of protein and electrostatic repulsion because of the net charge of protein. The Gibbs free energies of protein in the solid and liquid phases, evaluated using a second-order perturbation theory around hard sphere as the reference, were equated to predict the phase diagram. Protein solubility was found to be lower and the concentration of protein in the solid phase was higher for higher polymer concentrations, larger protein sizes, higher molecular weight of polymer, weaker protein-polymer interactions, poorer solvent, higher ionic strengths, and pH values closer to the isoelectric point. Experimental measurements of solubility of bovine serum albumin, human serum albumin, and ovalbumin for different molecular weights of poly(ethylene glycol) agreed well with the model predictions. 1. Introduction Protein precipitation represents one of the most important operations for the industrial-scale recovery and purification of proteins. These include vegetable and microbial food proteins, human and animal blood plasma proteins, and enzymes for analytical and industrial application. The technique of protein precipitation is usually carried out in the early stage of protein purification procedures, followed by chromatography. The advantages of using precipitation for concentration and purification are that the operation is easily adapted to a large scale, can be done continuously, and requires only simple equipment. Precipitation is effected by altering the solubility of proteins using various precipitating agents. Precipitants are known to be (i) organic solvents such as methanol, ethanol (Watt, 1972), and acetone, (ii) salts such as ammonium sulfate (Foster et al., 1971), (iii) uncharged water-soluble polymers such as poly(ethylene glycol) (Zeppezauer and Brishammar, 1965) and dextran (Laurent, 1963a,b; Laurent and Ogston, 1963; Kroner et al., 1982), and (iv) polyelectrolytes such as (trimethylamino)poly(ethylene glycol) (Johansson et al., 1970, 1973). Although protein precipitation can be done by adding these precipitants, there are limitations for some methods: (i) rigid temperature and buffer control are required when ethanol is used as a precipitating agent, (ii) denaturation may occur by using salts or organic solvents, and (iii) polyelectrolyte as a precipitating agent is expensive and is restricted to irreversible precipitation. The protein fractionation using uncharged-linear polymers has been under investigation for several decades. The most commonly used polymer is poly(ethylene glycol) (PEG) which has unique advantages in protein precipitation (Polson et al., 1964; McPherson, 1976). The increasing significance of protein precipitation with PEG was discussed in many reports (Polson et al., 1964; Zeppezauer and Brishammar, 1965; Gamble, 1971; Polson and Ruiz-Bravo, 1972; Foster et al., 1973; Ho¨nig and Kula, 1976; McPherson, 1976; Ingham, 1978; * To whom correspondence should be addressed.
S0888-5885(95)00745-7 CCC: $12.00
Miekka and Ingham, 1978; Atha and Ingham, 1981; Hasko´ and Vaszileva, 1982; Mahadevan and Hall, 1990, 1992a,b). Advantages of using PEG include (1) simple and rapid operations, (2) least denaturation of proteins, (3) reversible precipitation, (4) negligible temperature control, and (5) a relatively small amount of precipitants required. In addition to PEG, dextran with high molecular weight is also considered as a proper precipitant in terms of safety and cost. Its application is especially important in the food area. These attractive features account for the considerable interest in developing the use of PEG for large-scale purification of proteins from human plasma and other sources as well as intracellular enzymes. Polson et al. (1964) first tested the protein fractionating properties of five linear polymers: PEG, dextran, nonylphenol-ethoxylate, poly(vinyl alcohol), and poly(vinylpyrolidone) and found that PEG appeared to be the most suitable protein precipitant because its solutions were less viscous and caused virtually no denaturation at room temperature. Zeppezauer and Brishammar (1965) later confirmed and extended Polson’s work by comparing PEG, poly(propylene glycol), and poly(vinylpyrolidone) in precipitating human serum proteins and lipoproteins. Dextran is another nonionic polymer commonly used with PEG in two-phase aqueous separation (Kroner et al., 1982; Albertsson, 1986). Because it is uncharged, it was also selected to investigate the mechanism of protein precipitation using nonionic polymer to avoid any charge interaction (Laurent, 1963a). Evidence supporting the steric exclusion hypothesis was presented by Laurent (1963a). He used dextran of molecular weights 153 000, 450 000, 6.9 × 106, 13 × 106, and 50 × 106 to precipitate glucose and several proteins such as fibrinogen, γ-globulin, and albumin at various pH and ammonium sulfate concentrations. At high ionic strength, approximately from 1 to 3 M, the solubility of proteins was found to decrease logarithmically with ionic strength. Results indicated that no glucose was precipitated in the presence of dextran. However, the solubility of proteins was sharply decreased with increasing dextran concentrations. It was also found that there was a strong dependence of protein sizes. As the © 1996 American Chemical Society
3016 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
size increases, protein solubility decreases due to more volume exclusion. A steric exclusion of dextran chain segments from macromolecules (proteins) was believed to be the mechanism of precipitation, and this effect was found to be more pronounced for larger molecules. In the results presented by Atha and Ingham (1981), the size of protein molecules significantly affects protein solubility. In general, as protein size increases, protein solubility decreases. Among all the tested proteins, human fibrinogen and γ-globulin (larger proteins) exhibited lower solubility but lysozyme and R-lactalbumin (smaller proteins) gave higher solubility. The effect of the molecular weight of polymers on protein precipitation was discussed by Ingham (1977, 1978). In general, molecular weights of PEG between 4000 and 8000 (concentrations between 1 and 25%) were most commonly used due to relatively high effectiveness and relatively low viscosity (Polson et al., 1964; Ingham, 1977), although using a lower molecular weight of PEG might improve the specificity of separating proteins in a mixture (Ho¨nig and Kula, 1976). The effect of pH on protein solubility was due to the changes in the net charge of protein molecules with different pHs (Tanford, 1955). pH therefore directly affected electrostatic protein-electrolyte ion and proteinprotein interactions. When the pH was away from the isoelectric point, the experimental results showed that protein solubility became greater (Foster et al., 1973; Ingham, 1978). At the isoelectric point, protein exhibited a minimum solubility (Ingham, 1978). Hasko´ and Vaszileva (1982) conducted experiments for measuring plasma protein solubility in the presence of PEG-4000 at various PEG concentrations and pH. Their results showed that protein solubility did not change with increasing PEG concentrations but was affected by pH. Protein solubility was found to be higher at pH away from the isoelectric point. One of the advantages of using PEG was the thermal stability of PEG on protein precipitation (Juckes, 1971; Foster et al., 1973; Knoll and Hermans, 1981; Ingham, 1978; Atha and Ingham, 1981). Juckes (1971) found that the precipitation of bovine albumin by PEG-6000 shifted only about 1% between 4 and 22 °C. In the results of Foster et al. (1973), when temperature changed from 6 to 30 °C, the shift of PEG concentrations in precipitating invertase was less than 2%. Atha and Ingham (1981) conducted experiments on the melting temperature of ribonuclease (around 45 °C) in the presence of PEG-4000 and PEG-400 up to 30% and found that neither PEG-4000 nor PEG-400 had a significant effect on the melting temperature. On the other hand, ethanol at 25% concentration lowered the temperature by -8 °C. Recently, Lee and Lee (1987) studied thermal unfolding of ribonuclease, lysozyme, chymotrypsinogen, and β-lactoglobulin in the absence and presence of PEG. Results showed that the melting temperatures of all the tested proteins except ribonuclease decreased with increasing PEG concentrations from 0 to 30%. All the melting temperatures under investigation were above 55 °C. Protein precipitation using nonionic polymer was believed to be due to the steric exclusion of polymer chains from the vicinity of protein molecules (Laurent, 1963a,b; Ingham, 1977; Atha and Ingham, 1981; Hermans, 1982; Knoll and Hermans, 1983; Arakawa and Timasheff, 1985). Traditionally, only phenomenological models were employed for the precipitation of protein solubility in the presence of polymer at various condi-
tions. The interactions of protein-protein, proteinpolymer segment, segment-solvent, and proteinelectrolyte ion (at pH * pI) were usually represented by several interaction constants (Edmond and Ogston, 1968; Polson and Ruiz-Bravo, 1972; Atha and Ingham, 1981; Hasko´ and Vaszileva, 1982; King et al., 1988; Haynes et al., 1989a,b). The thermodynamic theory developed by Edmond and Ogston (1968) and employed by Atha and Ingham (1981) described protein solubility S as
ln S ) k - aC
(1)
where C is the polymer concentration, a is the proteinpolymer interaction parameter, and k is related to protein solubility in the absence of polymer. A similar formalism for protein solubility was developed based on volumetric displacement theory (Ogston and Phelps, 1960; Juckes, 1971) which gives
ln S ) K - βC
(2)
In the above equation, the interaction constant β can be regarded as a measure of the volume mutually excluded by neighboring molecules. If both protein and polymer molecules have spherical shape,
β)
( )
vj rs + rr 2.303 rr
3
(3)
where rs and rr are the cross-sectional radius of the polymer and the radius of the protein molecule, respectively. vj is the partial specific volume of the polymer. Further discussion of mutual excluded volume of polymers and proteins with different geometric shapes such as cylindrical rod, random coil, and finite and infinite chains was presented by Hermans (1982). Atha and Ingham (1981) compared the interaction constants resulting from the experimental observation with the volumetric displacement model and found that the former was much less than the latter. Mahadevan and Hall (1990, 1992a,b) extended the work of Gast et al. (1983) to predict protein precipitation in the presence of nonionic polymers. In their approach, the pair interaction potential between protein molecules was divided into two parts, i.e., the interactions only due to volume exclusion and the electrostatic interactions between charged protein molecules. The potential due to volume exclusion as proposed by Asakura and Oosawa (1958) was employed to account for the effect of polymer molecules on protein-protein interactions (due to volume exclusion). The electrostatic potential due to an electrical double layer was discussed by Verwey and Overbeek (1948). This potential was a function of particle size and surface potential or surface charge density. Essentially, the electrical potential was related to the pH that altered the protein net charge. For the binary interaction, a perturbation theory to hard-sphere potential was applied to evaluate the Gibbs free energy and subsequently the phase diagram for protein precipitation. Their model considered the flexible random-coil polymer molecules to be spherical particles. Since the model did not account for specific protein-polymer and polymer-solvent interactions, it may not be able to describe different precipitation actions of polymers of the same size (molecular weight) but different specific interactions with the protein and solvent molecules. The model was
Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 3017
only able to describe the trends of the effect of different variables on protein solubility. Recently, Yu et al. (1994) proposed a model for estimating protein solubility in aqueous polymer solutions. Their model considered the random-coil nature of polymer molecules and accounted for proteinpolymer interaction through a simple scaling analysis. However, they assumed that the chemical potential of protein in the solid phase was constant. The comparison of their model prediction with the data of Atha and Ingham (1981) and Mahadevan and Hall (1992a) was reasonable. In this paper, a statistical mechanical model for the precipitation of globular proteins using nonionic polymers was proposed. We follow the same framework proposed by Gast et al. (1983) in that the protein solution was considered as a pseudo-one-component system. The effects of polymer and solvent were accounted for through protein-protein interaction potential which consisted of attractive interactions due to polymer depletion in the vicinity of protein molecules and electrostatic repulsion due to the net charge of protein molecules. The Gibbs free energies in the liquid and the solid phases were evaluated using a secondorder perturbation theory with hard sphere as the reference. In the evaluation of the interaction potential between two protein molecules, the flexible random-coil structure of polymer molecules and protein-polymer and polymer-solvent interactions were accounted for. The interaction energy between two protein molecules was evaluated from the interaction energy of two parallel plates using Derjaguin’s integration. The segment density distribution of polymer molecules between two interacting plates and protein-polymer interaction, entropy, and enthalpy of mixing between the polymer segments and solvent were accounted for in the evaluation of interaction energy between two plates. Protein concentrations in the coexisting equilibrium solid and liquid phases were then evaluated by equating the Gibbs free energy of a protein molecule in both the phases. The model description is presented in the next section. Materials and Methods followed by the effect of different variables on the predicted phase diagram and finally comparison of experimental data with model predictions are presented in the subsequent sections. 2. Model for the Prediction of Phase Diagram Consider a saturated solution of globular protein in the presence of nonionic polymer in equilibrium with the precipitate phase. The equality of chemical potential (Gibbs free energy) of protein in the two phases in equilibrium can be invoked in order to relate the protein composition in the two phases for the prediction of the phase diagram. A statistical mechanical formulation (Gast et al., 1983; Mahadevan and Hall, 1990) is employed for the evaluation of Gibbs free energy of a protein molecule in both the phases. An equivalent pseudo-one-component system of globular protein molecules is considered for the evaluation of Gibbs free energy. The presence of nonionic polymer and solvent molecules is implicitly accounted for through their effect on the interaction between globular protein molecules. In addition to the equality of Gibbs free energy, mechanical equilibrium of the pseudo-one-component liquid and solid phases would also imply equality of pressure. Pressure in the pseudo-one-component system will be equivalent to the osmotic pressure of a protein solution. Gibbs free energy and pressure are related to Helmholtz
free energy A through (Gast et al., 1983)
∂ FA G ) kT ∂F kT
(4)
FG FA P ) kT kT kT
(5)
( )
where F, the number density of the pseudo-one-component system, is related to the protein volume fraction, φ2, and diameter d2 through
F)
6φ2 N ) V πd 3
(6)
2
At equilibrium
Gliquid ) Gsolid
Pliquid ) Psolid
(7)
Knowledge of A at different F should enable us to evaluate G and P from eqs 4 and 5. 2.1. Helmholtz Free Energy. Helmholtz free energy is related to the interaction potential between protein molecules through the configurational integral. The evaluation of the interaction potential is discussed in section 2.3. If the interaction potential is expressed as a perturbation around hard-sphere potential, a second-order perturbation expansion can be employed to give (Reed and Gubbins, 1973; Gast et al., 1983)
Ahs λF ∞up(r) A ) + g (r) 4πr2 dr NkT NkT 2 d2 kT hs 2 ∞ up(r) λ2F ∂F kT d ghs(r) 4πr2 dr (8) 2 4 ∂p hs kT
∫
()
∫
[ ]
where λup(r) is the perturbing interaction potential between protein molecules, ghs(r) is the radial distribution function for hard spheres, Ahs is the Helmholtz free energy for hard spheres, N is the number of particles in the system, F is the number density of protein molecules, (∂F/∂p)hs is the compressibility for hard spheres, k is the Boltzmann constant, and T is the temperature. The perturbation expansion assumes pairwise additivity of interactions. It is applicable only when the interaction between two protein molecules (as a result of polymer depletion and electrostatic interactions) is significant at separations only slightly greater than the diameter of the protein molecule. It will be shown in section 2.3.1 that the above condition is indeed satisfied for perturbation expansion to be valid. For liquids,
(∂p∂F)
hs
kT )
(1 + φ)4 3 (1 + 2φ)2 - φ3(4 - φ) 2
(9)
where φ is the volume fraction. For solids,
(∂p∂F)
hs
kT )
1 Zhs + φ(∂Zhs/∂φ)
(10)
where Zhs, the compressibility factor for a hard-sphere solid, is given by (Hall, 1972)
3018 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
Zhs )
3 + 2.557696 + 0.1253077Γ + 0.1762393Γ2 R 1.053308Γ3 + 2.818621Γ4 - 2.921934Γ5 + 1.118413Γ6 (11)
In the above equation, Γ ) 4(1 - φ/φ0) and R ) φ0/φ 1, where φ0, the close-packing volume fraction, is (x2π)/6 ()0.74) for face-centered-cubic (fcc) packing. Hard-sphere Helmholtz free energies for liquids and solids are given by (Kang et al., 1985; Weis, 1974), liq Ahs 4φ - 3φ2 ) ln F - 1 + NkT (1 - φ)2
(12)
and sol Ahs 2R ) ln F0 - s0 - 3 ln + NkT 3
( )
∫0Γ[Zhs - ZLJD] Γ dΓ -4 (13)
where s0 ) -0.24 ( 0.04 (Alder et al., 1968), and ZLJD, the compressibility factor for Lennard-Jones-Devonshire cell theory, is given by
ZLJD ) 3 + 3/R
(14)
2.2. Radial Distribution Functions of Hard Sphere for Liquids and Solids. In a hard-sphere system, the radial distribuion functions of hard sphere for liquids, ghs(x)liq, and solids, ghs(x)sol, were solved by several groups (Wertheim, 1963; Throop and Bearman, 1965; Barker and Henderson, 1967; Smith and Henderson, 1970). For liquids, Smith and Henderson (1970) obtained an explicit analytic solution to the PercusYevick equation for the hard sphere given by ∞
ghs(x)liq )
H(x-j) gj(x) ∑ j)0
(15)
where x ) r/dhs; H(n) is the Heaviside step function, which is 0 for n < 0 and 1 for n > 0, and j is an integer. ghs(x)liq is continuous except at x ) 1, gj(j) ) 0 for j * 1; thus, ghs(1)liq ) g1(1). The expression for gj(x) was given by Throop and Bearman (1965). Following the work done by Kincaid and Weis (Weis, 1974; Kincaid and Weis, 1977), the radial distribution function of hard sphere of solids, ghs(x)sol, was obtained by fitting the Monte Carlo data of Weis (1974) with the assumption that the structure defined by the hardsphere centers at close packing was a fcc lattice. The distribution function is given by
ghs(x)sol )
Q1 x
exp[-W12(x - x1)2 - W24(x - x1)4] + W
[∑
24ηxπ
∞
nni
i)2
xxi
]
exp[-W2(x - xi)2] (16)
where x is the reduced distance (x ) r/dhs) of two spherical centers. nni and xi are the number of ith neighbors and the lattice distance (in units of dhs) to the ith neighbor (Hirschfelder et al., 1954; Kincaid and Weis, 1977). For the fcc lattice, values of nni and xi are given in Hirschfelder et al. (1954). The parameters Q1, x1, W1, W2, and W were obtained by fitting the “exact” computer results (Weis, 1974; Kincaid and Weis, 1977).
2.3. Interaction Potential between Protein Molecules. As pointed out earlier, nonionic polymers modify the interaction between protein molecules. When the distance of separation between two protein molecules becomes comparable to the radius of gyration of the flexible polymer molecule, steric interactions would result in depletion of polymer segments in the intervening region between the two protein molecules. This depletion will be more pronounced if the polymer segments either are nonadsorbing or have unfavorable interactions with protein. Such depletion would result in attractive interactions between the protein molecules. In addition, when the protein molecule is charged, there will be repulsive interactions because of the overlap of the electrical double layers. Therefore, the net interaction potential uT(r) can be written as
uT(r) ) uA(r) + uR(r)
(17)
where uA and uR refer to attractive and repulsive interactions, respectively. 2.3.1. Potential Energy of Attraction and Segment Density Distribution. Unlike earlier treatment by Mahadevan and Hall (1990), the present analysis accounts for the random-coil nature of the polymer molecule. In addition, it also considers specific proteinpolymer, protein-solvent and polymer-solvent interactions. In other words, the present analysis can account for (i) solvent quality and (ii) specific protein-polymer interaction on the potential energy of interaction between protein molecules. Therefore, the present model can describe different precipitation actions of polymers with the same size (molecular weight) but exhibiting different specific interactions with proteins. The globular protein molecule is assumed to be spherical with uniform surface properties. The attractive interaction potential uA(y) between two spherical protein molecules separated by a minimum surface to surface distance y can be evaluated by employing Derjaguin’s integration to give
uA(y) ) πR kT
∫y∞
ufp(x) dx kT
(18)
where R is the radius of the protein molecule and ufp(x) is the interaction potential per unit area between two flat plates of the same surface property as that of protein, separated by a distance x. It will be shown later that the range of interaction between two protein molecules is smaller than the radius of the protein molecule and hence the use of Derjaguin’s approximation is justified. The treatment of Scheutjens and Fleer (1979, 1985) is employed to evaluate the interaction potential between two flat plates. The region between two flat plates is divided into quasicrystalline lattice layers parallel to the flat plate surfaces. Each lattice cell has the same volume as that of the solvent molecule. The polymer is assumed to be linear, homogeneous, nonionic, and uniform with respect to molecular weight and constitution. Any conformation of the flexible polymer molecule is described as a step-weighed random walk in the lattice. The conformation probabilities and the free energy of mixing are calculated under the BraggWilliams approximation. The model accounts for all the possible chain conformations. The polymer segments are interconnected by assigning a probability λ0 to each bond which lies within one layer and a probability λ1 to each bond crossing to one of the neighboring layers. The
Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 3019
Figure 1. Polymer segment density distribution at different distances of separation between the two plates. The distance between the two plates is shown as the number of lattice layers. The thickness of each lattice layer is 4 Å, χ ) 0.44, s ) 182, φ3,* ) 10-6, and χs ) 0.122.
free energy per lattice site of two flat plates separated by a distance of M lattice layers is given by
2u1|s
2γa ) kT
kT
M
- 2χsφ3,1 +
[ ( ) φ1,i
φ1,i ln ∑ φ i)1
φ3,i ln pi - (φ1,i - φ1,*) -
(
1,*
)]
φ3,i - φ3,* s
M
χ
+
+
[φ1,i(〈φ3,i〉 - φ3,*) - φ1,*(φ3,i - φ3,*)] ∑ i)1
(19)
where a is the area of the lattice site, γ is the energy per unit area of plate, s is the length of the polymer molecule in lattice units, χ is the Flory-Huggins parameter, χs is defined as χs ) (u1|s - u3|s)/kT, with u1|s and u3|s being the adsorption energies onto the flat plate of solvent molecule and polymer segment, respectively, φ1,i and φ3,i refer to the volume fractions of solvent and polymer segment in layer i, respectively, φ1,* and φ3,* refer to the bulk volume fractions of solvent and polymer, respectively, and 〈 〉 indicates an average over three adjacent layers with probability factors λ1, λ0, and λ1. pi is the free segment probability in the ith layer. The procedure for the evaluation of segment density distribution and free segment probability was given by Scheutjens and Fleer (1985). The interaction potential ufp(y) between two flat plates is then given by
ufp(y) 2(γ(y) - γ∞) ) kT kT
(20)
where γ∞ is the free energy per unit area of an isolated plate immersed in the polymer solution. The effect of distance of separation on the segment density distribution of polymer segments between two flat plates is shown in Figure 1. For a relatively small χs ()0.122), protein-polymer interactions are not that favorable. Hence, there is a depletion of polymer segments between the two plates at a sufficiently small distance of separation. At a smaller distance of separation, this depletion is found to be more pronounced. As the two plates move away from each other, the density of polymer segments between the two plates increases, approaching that of bulk for sufficiently large distances of separation. Because of the depletion of polymer
Figure 2. Effect of molecular weight of polymer on the interaction potential between two protein molecules of diameter d2 ) 7.04 nm, χs ) 0.04, χ ) 0.44, φ3,* ) 0.17, and zp ) 0.
segments, there is attractive interaction between the two flat plates. The range of this interaction is found to be less than 20% of the diameter of the protein molecule (result not shown here). Since the range is smaller than the protein size, Derjaguin’s integration was employed to evaluate the attractive interaction between two protein molecules. A typical interaction potential energy between two protein molecules for different polymer molecular weights is shown in Figure 2. Low molecular weight (shorter chain) polymer molecules, being more flexible, can have more conformations between two protein molecules. Consequently, less depletion of polymer segments occurred for lower molecular weight polymers, thus leading to less attractive interactions between two protein molecules. It is to be noted that the range of interaction is only about 30% of the protein size. Consequently, perturbation expansion as given by eq 8 can be employed to evaluate the Helmholtz free energy. More depletion of polymer segments was found for smaller values of χs (or, equivalently, less favorable protein-polymer interactions). Consequently, the interaction between two protein molecules becomes more attractive for smaller χs values (results not shown). For less favorable solvents (i.e., for larger values of χ), more depletion of polymer segments is found to occur in the vicinity of protein molecules since it is easier to demix the polymer segments from the solvent in those cases. Consequently, protein-protein attractive interaction was higher (more negative potential energy) for worse solvents (results not shown). Because of more pronounced polymer depletion at higher polymer concentrations, the interaction was found to be more attractive at larger polymer concentrations. The depth of the protein-protein interaction potential was found to be roughly proportional to polymer concentration. 2.3.2. Potential Energy of Double-Layer Repulsion. For high κd2 (>20) and low constant surface potential (Ψ0 , kT/e), Derjaguin’s procedure can be applied to evaluate the potential energy of repulsive interaction for large particles with a relatively thin double layer to give (Verwey and Overbeek 1948)
uR(r) 2π0RΨ02 ln[1 + exp(-κx)] ) kT kT
(21)
where κ is the Debye-Hu¨ckel parameter, 0 is the permittivity of vacuum, is the dielectric constant of the medium, and zi is valence number of the symmetric
3020 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
electrolyte. Ψ0 is the surface potential which can be obtained from Gouy-Chapman theory for a wider range of potential values.
Ψ0 )
[
2kT σ* sinh-1 zie (80kTn0)1/2
]
(22)
where e is the elementary charge, n0 is the number concentration of electrolyte, and σ*, the surface charge density, can be related to the net charge of a protein molecule at a certain pH. For small κd2 (
