Influence of Charge Regulation in Electrostatic Interaction

May 1, 1996 - In this paper, the theory for charge regulation is applied to the original slab model, and simple algebraic equations are developed in o...
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Anal. Chem. 1996, 68, 1536-1544

Influence of Charge Regulation in Electrostatic Interaction Chromatography of Proteins Jan Sta˚hlberg*

Department of Analytical Chemistry, Stockholm University, S-106 91 Stockholm, Sweden Bengt Jo 1 nsson

Division of Physical Chemistry 1, Chemical Center, University of Lund, S-211 00 Lund, Sweden

Recently, the “slab model” was proposed to describe the interaction between a protein and the charged stationary phase surface in electrostatic interaction chromatography. The model is based on the solution of the linearized Poisson-Boltzmann equation for a system consisting of two charged planar surfaces in contact with an electrolyte solution. In the model it is assumed that the charge densities of both the protein and the stationary phase are constant during the adsorption process. However, as the protein comes close to the oppositely charged stationary phase surface, the protein net charge will change due to the electrical field from the stationary phase. In this paper, the theory for charge regulation is applied to the original slab model, and simple algebraic equations are developed in order to include the effect of charge regulation on the capacity factor. A large body of retention data are reanalyzed with the new model, and it is found that there is good agreement between the chromatographically and titrimetrically obtained protein net charge. An interesting consequence of charge regulation is that it gives a contribution to the retention of proteins with zero net charge and even to proteins with the same sign of charge as the stationary phase. In ion-exchange chromatography of proteins, the stationary phase consists of charged groups chemically bound to a hydrophilic surface, and the mobile phase is usually a buffer solution containing an eluting salt. When the protein and stationary phase are of opposite charge, and for low salt concentrations in the eluent, the protein is strongly adsorbed, resulting in long elution times. On increasing the salt concentration in the eluent, the adsorption strength decreases rapidly, leading to elution of the protein. In practical analytical work, salt gradients are therefore often used. An important practical and theoretical aspect of the adsorption properties of these surfaces is that the protein often maintains it biological activity after desorption. Ion-exchange chromatography is a major technique for both analytical and preparative separations, and because of the mild separation conditions, the biological activity of the species is often maintained. Even though ion-exchange chromatography is a standard technique for separation and purification of proteins, there is a lack of understanding of the basic adsorption mechanism of proteins to these surfaces. A deeper understanding of the physicochemical factors which determine the adsorption strength is of fundamental interest not only for analytical chemistry and preparative work 1536 Analytical Chemistry, Vol. 68, No. 9, May 1, 1996

but also for the general understanding of protein adsorption. The traditional stoichiometric description of ion-exchange chromatography of proteins was introduced by Boardman and Partridge1 and is bases on a process where the counterions bound to the stationary phase are stoichiometrically exchanged by the protein. One of several proposed formulations2 for this process is

PCNC + NSSh a P h + NSS + NCC

(1)

where P is the protein in the mobile phase, C is the counterion to the protein, which is assumed here to be monovalent, and S represents the co-ions bound to the stationary phase. NC and NS are the numbers of salt ions involved in the exchange process. Overbars indicate that the species are bound to the stationary phase. This model gives, for isocratic elution, a linear relationship between the logarithm of the protein capacity factor and the logarithm of the eluting salt concentration. The slope of such plots is formally equal to the number of bound counterions displaced by the protein as it adsorbs and is referred to as the Z value. A stoichiometric description implies short-range interactions between the interacting species and is therefore not appropriate for the long-range electrostatic interactions, which determine the retention of proteins. To overcome the limitations of the stoichiometric model, two different “nonstoiochiometric” models, both based on Manning’s condensation theory, were recently proposed.3,4 More recently, theories used in colloid and surface chemistry to describe electrostatic and other interactions have been applied to describe retention properties of proteins in ionexchange chromatography. In these theories, the electrostatic interaction is often calculated from solutions of the PoissonBoltzmann equation for a system of given geometry. Lenhoff and co-workers solved the equation numerically in order to study the electrostatic interaction between a spherical protein and a charged planar surface.5,6 For the same purpose, Sta˚hlberg and Jo¨nsson solved the linearized Poisson-Boltzmann equation for calculating the electrostatic interaction between a sphere and a planar (1) Boardman, N. K.; Partridge, S. M. Biochem. J. 1955, 59, 43. (2) Velayudhan, A.; Horva´th, Cs. J. Chromatogr. 1986, 367, 160. (3) Melander, W. R.; El Rassi, Z. J.; Horva´th, Cs. J. Chromatogr. 1989, 469, 3. (4) Mazsaroff, I.; Varady, L.; Mouchawar, G. A.; Regnier, F. E. J. Chromatogr. 1990, 499, 63. (5) Yoon, B. J.; Lenhoff, A. M. J. Phys. Chem. 1992, 96, 3130. (6) Roth, C. M.; Lenhoff, A. M. Langmuir 1993, 9, 962. 0003-2700/96/0368-1536$12.00/0

© 1996 American Chemical Society

surface.7 From the microscopic structure of the protein and the surface, Noinville et al.8 calculated the interaction energy by summing pairwise the electrostatic and dispersive interactions using a modified version of the AMBER force field. In these calculations, they were able to reproduce the retention difference between calcium-loaded and -depleted R-lactalbumin. Recently, we proposed a “slab model”, where the interaction between the protein and the charged stationary phase surface is calculated by solving the linearized Poisson-Boltzmann equation for a system consisting of two charged planar surfaces in contact with an electrolyte solution.9 From this model, the following equation for the dependence of the logarithm of the capacity factor on the eluent salt concentration can be derived:

ln k )

Apσp2

1 + ln Φ F(2RT0r)1/2 xI

(2)

where Ap is the interacting area between the protein and the stationary phase, σp is the protein charge density, and I is the eluent ionic strength. The model predicts a linear relationship between the logarithm of the capacity factor and the reciprocal square root of the mobile phase ionic strength, and such plots show good linearity for a large body of retention data. From its slope, the protein net charge is estimated by using only fundamental physicochemical constants. The chromatographically measured charges compare well with those obtained from titrimetry. In accordance with the slab model, Malmquist and Lundell10 found by principal component analysis that retention data for nine proteins and 29 different eluting salts were best rationalized by using the total ionic strength of the mobile phase and that the type of salt is of minor importance. Besides the assumptions made in the linearized PoissonBoltzmann equation, several simplifications are made in the slab model; i.e., the geometry of the interaction is that of two planar surfaces, the interaction area is set to half the protein area, and both the effect of nonelectrostatic interactions and the effect of charge regulation on the electrostatic interaction are neglected. In two later papers, some of these assumptions are studied: in one, the interaction geometry is changed to that between a charged sphere and a charged surface,7 and in the second paper, the van der Waals interaction is included in the slab model to describe the role of nonelectrostatic interactions on retention at very high salt concentrations in the eluent.11 In the present paper, the slab model is complemented with a thermodynamic theory describing the effect of charge regulation. Charge regulation implies that the net charge of the adsorbed protein is different from its net charge in free solution. In the original slab model,9,12 it is assumed that the charge density of the two slabs, i.e. the protein and the stationary phase, is constant during the adsorption process and independent of whether the protein is far away from or close to the surface. However, as the protein comes closer to the oppositely charged stationary phase surface, it will change its net charge. The reason for this is that the electrical field from one surface penetrates the (7) Sta˚hlberg, J.; Jo ¨nsson, B. J. Colloid Interface Sci. 1995, 176, 397. (8) Noinville, V.; Vidal-Madjar, C.; Se´bille, B. J. Phys. Chem. 1995, 99, 1516. (9) Sta˚hlberg, J.; Jo ¨nsson, B.; Horva´th, Cs. Anal. Chem. 1991, 63, 1867. (10) Malmquist, G.; Lundell, N. J. Chromatogr. 1992, 627, 107. (11) Sta˚hlberg, J.; Jo ¨nsson, B.; Horva´th, Cs. Anal. Chem. 1992, 64, 3118. (12) Parsegian, V. A.; Gingell, D. Biophys. J. 1972, 12, 1192.

intervening salt solution and reaches the opposite surface; thereby the electrostatic potential on the respective surface changes, leading to different surface charge densities. It is shown in this paper that for strong ion-exchangers with surface charge density higher than the protein charge density, simple algebraic equations can be used to describe the retention of proteins as a function of eluent salt concentration. It is also shown that the adsorption energy is higher than that in the case of no charge regulation and that it depends on the slope of the pH titration curve of the protein. Since charge regulation derives from elementary electrostatics, it must, in principle, be included in any physical model describing protein adsorption; the purpose of this paper is to discuss the effect and to demonstrate its significance. The basic electrostatic equations describing the effect of charge regulation are essentially the same as those recently presented by Carnie and Chan,13,14 though some quantities are defined differently. THEORY The slab model9 calculates the change in free energy as a function of separation distance, L, between two charged planar surfaces when the charge density on both surfaces is constant and independent of L, i.e. σs ) σs(L f ∞) and σp ) σp(L f ∞); the equation is a starting point in the thermodynamic analysis of charge regulation. If the charged surface groups are in chemical equilibrium with ions in the electrolyte, e.g. H+ ions, the surface charge density and electrostatic surface potential are coupled to each other. This implies that the surface charge density on both surfaces is a function of L, σs(L) and σp(L), respectively, and that the interaction energy between a protein and a charged surface is different from that obtained from the slab model. The interaction free energy per unit area, ∆G(L)/Ap, is the difference in free energy between the state when the surfaces are separated by a distance L and infinitely separated:

∆G(L)/Ap ) [G(L) - G(∞)]/Ap

(3)

where G(L) depends on the unknown parameters σs(L) and σp(L). Since the free energy is a function of state, its value is independent on the executed path. The first step in the thermodynamic analysis consists of the charging of the two surfaces from its value at infinite separation, σs(L f ∞), to its value at separation distance L, σs(L), and σp(L f ∞) to σp(L), respectively. This charging take place when the surfaces are infinitely separated, and the free energy per unit area required for this charging process for surfaces s and p, respectively, is to a first approximation described by eqs 4:13,14

∆Gch,p [σp(L) - σp(∞)]2 ) A Kpκ0r

(4a)

∆Gch,s [σs(L) - σs(∞)]2 ) A Ksκ0r

(4b)

where σ(∞) is an abbreviation for σ(Lf∞), Kp and Ks are constants for each surface at a given ionic strength, and κ is the inverse Debye length. When the value for the constants is zero, the energy needed to change σ becomes infinite, and the surface (13) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 161, 260. (14) Carnie, S. L.; Chan, D. Y. C.; Gunning, J. S. Langmuir 1993, 10, 2993.

Analytical Chemistry, Vol. 68, No. 9, May 1, 1996

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charge density remains constant at all separation distances. For this extreme case, the charge regulation model equals the model for fixed charges. In the next step in the thermodynamic analysis, the two surfaces with charge density σs(L) and σp(L), respectively, move from infinity to a distance L. The change in free energy per unit surface area for this step is given by eq 4c:9

(

)

-κL 2 2 ∆Gi(L) + 2σp(L)σs(L) 1 [σp (L) + σs (L)]e ) κL -κL Ap κ0r e -e

(4c)

The total interaction free energy is the sum of the free energy change for these two steps:

(

-κL 2 2 + 2σp(L)σs(L) ∆G(L) 1 [σp (L) + σs (L)]e ) + κL -κL Ap κ0r e -e

)

[σp(L) - σp(∞)]2 [σs(L) - σs(∞)]2 + (5) Kp Ks As the two surfaces approach each other, they will take the path that minimizes the free energy for each separation distance L, i.e., σs(L) and σp(L) take for each distance L a value which is a balance between the free energy of interaction and the free energy required to change the charge of respective surface. These values of σs(L) and σp(L) are found by setting the partial derivative of ∆G(L)/Ap in eq 5 with respect to σs(L) and σp(L) to zero for a given L,

|

|

∂(∆G(L)/Ap) ∂σs(L)

L

)0

|

|

∂(∆G(L)/Ap) ∂σp(L)

)0

(6)

L

and solving for σs(L) and σp(L) from the two equations. The result is σp(L) ) σp(∞) - (Kp - 1)σs(∞)e-κL - [σs(∞) - (Ks - 1)σp(∞)e-κL]e-κL 1 - (Ks - 1)(Kp - 1)e-2κL

(7) The corresponding surface charge density, σs(L), is obtained by interchanging p and s. Inserting the expressions for σp(L) and σs(L) into eq 5 gives the free energy per unit area for the system as a function of the separation distance and the surface charge density of the respective surface at infinite separation distance, σp(∞) and σs(∞), respectively: ∆G(L) ) Ap 2σp(∞)σs(∞)e-κL - [(Ks - 1)σp2(∞) + (Kp - 1)σs2(∞)]e-2κL κ0r(1 - (Kp - 1)(Ks - 1)e-2κL)

(8) In Figures 1-4, eqs 7 and 8 are illustrated, where the values for the interaction area and surface charge density are chosen to be representative for ion-exchange chromatography of proteins. The assumptions made in the calculations are the same as in previous papers, i.e., that the protein can be considered as a sphere 1538

Analytical Chemistry, Vol. 68, No. 9, May 1, 1996

with smeared out charges giving a uniform surface charge density and that half the total surface area of a spherical protein participates in the interaction. In ion-exchange chromatography, the stationary phase surface charge density, σs, is usually higher than that of the protein, σp. When the interaction free energy is calculated according to the slab model with fixed charge density, it is found that the surface with the lowest charge density entirely determines the strength of the interaction and also that, at a certain separation distance, the free energy goes through a minimum. The position of the minimum is determined by the difference in surface charge density between the two oppositely charged interacting surfaces and also by the ionic strength of the electrolyte.9 When the two surfaces regulate the charge density, the numerical difference decreases as the distance between them decreases, and the minimum in free energy consequently occurs at a smaller separation distance and may eventually disappear. This is illustrated in Figure 1a, where ∆G(L)/RT is plotted as a function separation distance, L, for various σp(∞) values and a given set of conditions (σs(∞) ) 0.2 C/m2, 1/κ ) 1 × 10-9 m, Ks ) 0.288, Kp ) 0.308). Relatively small energies are required to charge the surfaces for these Ks and Kp values, and the chosen conditions are therefore favorable for charge regulation. The interaction area, Ap, is set to 2500 Å2, which corresponds to a protein radius of 20 Å, assuming a spherical protein and that the interaction area is half the total protein area. The chosen values for the protein charge density, σp, correspond to protein net charges in free solution of -10, -11, and -12, respectively. As shown below, this Kp value corresponds to a change in protein net charge of 2 units per pH unit in a titration curve for a protein with 20 Å radius. In the figure, the interaction free energy calculated for the case of no charge regulation is also plotted for the same set of parameters, and the net charge -12. As expected, the surface charge regulation increases the interaction compared to the case of fixed charge density. It is also seen in the figure that, although charge regulation occurs, there is still a distinct minimum in the free energy versus separation distance plot; however, it has moved to a smaller separation distance compared to the case of fixed charges. The change in protein net charge due to charge regulation as it approaches the opposite surface is calculated from eq 7 and shown in Figure 1b for the same set of parameters as in Figure 1a. With the chosen Kp value, the protein increases its net charge by approximately 4 units when it moves from infinity to the distance of free energy minimum; for smaller separation distances, the net charge increases strongly. The influence of the Kp value on the free energy versus separation distance curve is shown in Figure 2a for a protein with radius 20 Å carrying net charge -10. The different Kp values correspond to changes of 1, 2, 3, and 4 unit charges of the protein per unit change in pH value in a titration curve. The rest of the parameters are the same as in Figure 1, except for the Ks value, which equals 0.001, implying a surface with essentially constant charge density. The figure shows that there is a strong influence on the free energy curve when the Kp value is varied from 0.154 (1 charge/pH unit) to 0.618 (4 charges/pH unit). A higher Kp value gives a stronger interaction and a free energy minimum at a smaller separation distance. The corresponding change in protein net charge as a function of separation distance is seen in Figure 2b, illustrating how the increase in protein net charge, due to the increase in Kp values, is accompanying the increased

Figure 1. (a) Plots of the Gibbs free energy, ∆G(L), in RT units, as a function of the distance between the stationary phase surface and a protein, L, with protein net charge as the parameter. The results are calculated according to the charge-regulating slab, model for a protein of 20 Å radius carrying net charges of -10, -11, and -12, interacting with a surface with charge density 0.2 C/m2 in an electrolyte solution with ionic strength 0.1 mol/dm3. The charge-regulating constant of the protein, Kp is 0.308, and that of the surface is 0.288. As a reference is also shown ∆G(L) according to the slab model with fixed charge density for a protein of net charge -12; the rest of the parameters are the same. (b) Protein net charge as a function of L, q(L), for the three proteins shown in (a), calculated from eq 7.

interaction shown in Figure 2a. It should be emphasized, however, that for a protein of 20 Å radius, the Kp value usually lies in the region 0.2-0.5 for pH values in the interval 4-10. Retention of Isoelectric Proteins. A puzzling observation in ion-exchange chromatography of proteins is that retention often occurs for mobile phase pH values equal to its isoelectric point. It has even been observed that proteins carrying the same sign of charge as the stationary phase are retained.15 The conclusion (15) Kopaciewicz, W.; Rounds, M. A.; Fausnaugh, J.; Regnier, F. E. J. Chromatogr. 1983, 266, 3.

Figure 2. (a) Plots of the Gibbs free energy, ∆G(L), in RT units, as a function of the distance between the stationary phase surface and a protein, L, with the charge-regulating constant of the protein, Kp, as the parameter. The results are calculated according to the chargeregulating slab model for a protein of 20 Å radius with Kp values 0.154, 0.308, 0.462, and 0.616, interacting with a surface with charge density 0.2 C/m2 and Ks ) 0.001 in an electrolyte solution with ionic strength 0.1 mol/dm3. The net charge of the protein is -10. (b) Protein net charge as a function of L, q(L), for the three Kp values shown in (a), calculated from eq 7.

of these observations is that it is not the protein net charge that determines the retention, and Kopaciewicz et al. suggested that these deviations may result from asymmetrical distribution of charges in the protein. Although this explanation is probably correct in most cases, an interesting consequence of charge regulation is that it gives an additional mechanism for retention in these cases. Furthermore, also when retention occurs because of charge asymmetry, the charge regulation mechanism enhances the retention further because this physical mechanism always operates when the protein approaches a charged surface. In Figure 3a is shown the free energy of interaction as a function of Analytical Chemistry, Vol. 68, No. 9, May 1, 1996

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performed for a surface with very low charge density and a protein with a high Kp value. However, the conclusion still remains that charge regulation is a mechanism for retention of proteins with no net charge. Simplification of the Charge-Regulating Slab Model. An interesting behavior of the slab model with fixed and opposite charge on the two interacting surfaces is that the interaction free energy depends only on the surface with the lowest surface charge density. When there is a difference in charge density between the two surfaces, the interaction free energy goes through a minimum at a certain separation distance. A consequence of the presence of a minimum is that the integral representing the surface excess and the corresponding capacity factor can be approximated as a product of the phase ratio, Φ, and exp(∆Gm/ RT).9 Even though the existence of a minimum in the free energy as a function of separation distance is not as evident in the chargeregulating slab model as in the simple slab model, the K values and charge densities in most ion-exchange systems are such that a minimum is formed. It is therefore of practical interest to develop the corresponding simplification of the charge-regulating slab model. By setting the derivative of the right-hand side of eq 5 with respect to L to zero, we can express the separation distance for the free energy minimum by

a

b

Lm ) -

Lm ) -

( (

) )

σs(∞) 1 ln κ σp(∞)(Ks - 1)

when -σp(∞) > σs(∞)

σp(∞) 1 ln κ σs(∞)(Kp - 1)

when -σp(∞) < σs(∞)

(9a)

(9b)

From these equations, the conditions for the existence of a minimum can be inferred; since σs(∞) and σp(∞) are of opposite sign, it is necessary that Ki < 1 (i ) s or p), and since Lm must be positive, another condition is that σi/σj < Ki - 1. Inserting eq 9 into eq 8, we can express the minimum value of the free energy as Figure 3. (a) Plots of the Gibbs free energy, ∆G(L), in RT units, as a function of the distance between the stationary phase surface and a protein, L, with the net charge of the protein as the parameter. The results are calculated according to the charge-regulating slab model for a protein of 20 Å radius with net charges of 0, 0.5, and 1.0, interacting with a surface with charge density 0.02 C/m2 in an electrolyte solution with ionic strength 0.029 mol/dm3. The chargeregulating constant of the protein, Kp, is 0.77, and that of the surface is 0.001. (b) Protein net charge as a function of L, q(L), for the three proteins shown in (a), calculated from eq 7.

separation distance between a protein of radius 20 Å carrying a new charge of 0, 0.5, and 1 and a surface with surface charge density 0.02 C/m2. For low electrolyte ionic strengths, in the figure 0.029 mol/L, and high Kp values (0.77), there is an attractive interaction between the uncharged protein and the charged surface at all separation distances. When the protein carries the same sign of charge as the surface, repulsion occurs for long separation distances, while for shorter distances there is attraction for the lower charged protein. The corresponding protein net charge as function of separation distance is calculated from eq 7 and is shown in Figure 3b. Because of the limitations of the linearized Poisson-Boltzmann equation, these calculations are 1540

Analytical Chemistry, Vol. 68, No. 9, May 1, 1996

σs2(∞) ∆Gm )Ap κ0r(1 - Ks)

(10a)

σp2(∞) ∆Gm )Ap κ0r(1 - Kp)

(10b)

For ion-exchange chromatography of proteins, we assume that the condition σp(Lm) < σs(Lm) is fulfilled, and eq 10b is therefore used in the ensuing calculations. It should be noticed that, when the energy needed to change the charge of the protein surface becomes infinitely large, the value for Kp goes to zero, and eq 10b becomes the same as in the slab model. A thermodynamic definition of the capacity factor identifies it as an integral of excess protein concentration at the surface.9,11 When a minimum in the free energy curve exists, the capacity factor can be approximated as9

k ) (Asb/V0)e-∆Gm/RT

(11)

where b is the characteristic width of the absorption layer. In

under practical isocratic conditions, the ionic strength of the mobile phase is such that when retention occurs, the Kp value is nearly constant, giving an almost linear ln k vs 1/I1/2 (or, alternatively, 1/κ) plot. Since eq 13 can be used only when there is a minimum in the free energy curve, it cannot be used to describe the retention properties of proteins close to its pI value. For such calculations, numerical integration of the thermodynamically exact definition is necessary. Interpretation of the Kp Value. Consider a surface consisting of nAtot chargeable groups in protolytic equilibrium with a solution:

A- + H+ a AH

(14)

Thermodynamics gives that the differential change in free energy of the system, δG, when the mole number of charged groups on the surface is changed by δnA- is

( ) ∂G ∂nA-

1/I1/2

Figure 4. Comparison of ln k vs plots calculated by numerical integration of eq A6 in ref 9 combined with eq 7 and according to its simplified form eq 13 (dotted lines), using the same set of parameters as in Figure 1. Stationary phase surface area is set to 1 × 107 m2/ (m3 of column dead volume).

the Appendix in ref 9, the procedure for calculating the value of b is discussed, applied to this case, one obtains

(

(

b ) 2 (ln 2)RT0r

Ks - 1 2

σs

) )

Kp - 1 σp

2

σp Ap

( )

F ∂G A ∂σ

) µA- + µH+ - µAH

(15)

nAtot

where µA-, µH+, and µAH are the chemical potentials of the charged surface, the hydrogen ions in the solution, and the protonated surface, respectively, the A is the surface area (in m2). For a given pH value in the solution, the chemical potential of the hydrogen ion is constant, and under this condition we obtain from eq 15 that

( ) ∂2G ∂σ2

)-

σ(∞)

(

)

A ∂(µA- - µAH F ∂σ

(16) σ(∞)

1/2

/κAp

(12) From the definition of the Kp value, eq 4, it follows that

where Ap is the area of the protein interacting with the stationary phase. Comparing eq 12 with the corresponding equation for the slab model with fixed charge density, it is seen that as the Ki values goes to zero, the two expressions are equal. The simplified expression for the logarithmic capacity factor is obtained by combining eqs 11 and 10, and the final result is 2

)nAtot

( )

Asb ln k ) + ln 1/2 V0 F(2RT0r) (1 - Kp)xI

(13)

Equation 13 is a useful approximation since, for moderate changes in ionic strength, under practical conditions, e.g., from 0.05 to 0.5 M, the variation in ln(Asb/V0) with ionic strength is small compared to the variation in ∆Gm given by the first term in eq 13. A comparison between ln k calculated from numerical integration of the thermodynamically exact expression, eq A6 in ref 9, in combination with eq 8, and the simplified eq 13 combined with eq 12, is made in Figure 4 using the same set of data as in Figure 1 and under the assumption that Ap equals half the area of the corresponding sphere. The two different calculation procedures give almost the same result for both the absolute numerical value and the ionic strength dependence. The simplified set of equations will, however, somewhat underestimate the absolute value of the capacity factor. It can also be assumed that,

( ) ∂2G ∂σ2

)

σ(∞)

2A Kpκ0r

(17)

The general thermodynamic interpretation of the Kp value is obtained by combining eqs 16 and 17:

(

)

κ0r ∂(µA- - µAH) 1 )Kp 2F ∂σ

σ(∞)

(18)

In this work we are mainly interested in obtaining the Kp value for a protein from its pH titration curve. At a given point on the titration curve, there is an equilibrium such that

µA- + µH+ - µAH ) 0

(19)

where µH+ is obtained from the definition of pH:

µH+ ) µ0H+ + RT ln aH+ ) µ0H+ - RT ln 10 pH (20) Combining eqs 18-20 gives the Kp value at a point on the titration curve at which the charge density is σ(∞). From the relation between σ and the number of elementary charges, q, on a protein with the surface area, A0p, the equation for Kp becomes Analytical Chemistry, Vol. 68, No. 9, May 1, 1996

1541

Kp ) -

∂q 2F2 0 ∂pH κ0rApRT ln 10

(21)

From this equation, we see that the Kp value decreases when the protein area increases and when the change in protein charge per pH unit decreases, i.e., when the slope in a protein titration curve decreases. RESULTS AND DISCUSSION From the examples in the Theory section, it is clear that charge regulation has a significant effect on the magnitude of the electrostatic interaction between a protein and a charged surface. It is also shown that, for ion-exchange chromatography, it is reasonable to assume that the interaction is determined by the surface with the lowest charge density, i.e., the protein charge density, which in most cases implies that there is a minimum in the ∆G(L) curve. For these cases, the charge-regulating slab model predicts that the salt dependence of the capacity factor for a protein follows eq 20. The difference between this expression and the corresponding equation for the slab model with fixed charge density is the factor (1 - Kp) in the denominator. Although the Kp value for a given protein is different at different ionic strengths, the linearity of the ln k vs 1/I1/2 plot is still a useful approximation (see, e.g., Figure 4), especially if the Kp value is calculated at the middle of the ionic strength range used in the experiments. In an earlier paper,9 a large set of retention data were analyzed according to the slab model with fixed charged density, and it was found that the ln k vs 1/I1/2 plots exhibited good linearity: for 25 of 28 reported plots, the correlation coefficient exceeded 0.99. In this paper, the same set of experimental chromatographic data are used and reanalyzed according to the charge-regulating slab model, i.e., eq 13. The titrimetric conditions used for obtaining the titration curves are summarized in Table 1. From the slope of these curves, dqtitr/ dpH, at the pH of the chromatographic experiment, the Kp value is calculated according to eq 21. The titrimetric net charge of the protein, qchr, is obtained from the titration curve in combination with the pI value of the protein. The physical data used in the calculation of the spherical protein molecule radius are the molecular weight and the density; the data used are compiled in ref 9. A critical test of the proposed theory is to calculate the protein net charge from the slope of the ln k vs 1/I1/2 plot after correction for charge regulation using an appropriate Kp value and then to compare the result with the protein net charge from pH titration experiments. When calculating the characteristic charge of the protein from eq 13, we assume that the protein can be considered as a sphere with uniformly distributed charges and that only half the total area of the spherical protein participates in the interaction, i.e., Ap ) 1/2A0p. According to eq 13, the slope of a ln k vs 1/I1/2 plot is proportional to (qchr/A0p)2, and the chromatographically (16) Kenchington, A. W. In A Laboratory Manual of Analytical Methods of Protein Chemistry; Alexander, P., Block, R. J., Eds.; Pergamon Press: Oxford, U.K., 1960. (17) Tanford, C.; Wagner, M. L. J. Am. Chem. Soc. 1954, 76, 3331. (18) Lehman, L. D.; Hanania, G. I. H.; Gurd, F. R. N. Biochem. Biophys. Res. Commun. 1978, 81, 416. (19) Tanford, C.; Hauenstein, J. D. J. Am. Chem. Soc. 1956, 78, 5287. (20) Cannan, R. K.; Palmer, A. H.; Kibrick, A. C. J. Biol. Chem. 1942, 142, 803. (21) Tanford, C.; Swanson, S. A.; Shore, W. S. J. Am. Chem. Soc. 1955, 77, 6414. (22) Tanford, C. J. Am. Chem. Soc. 1950, 72, 441.

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obtained net charge of the protein can be calculated from the following expression:

qchr )

x

S(1 - Kp)A0p 135

(22)

where A0p is the protein surface area (in Å2/molecule), S is the slope of the ln k vs 1/I1/2 plot (when I is in mol/dm3), and the constant 135 includes numerical values of the physical constants and conversion factors from SI units. The conditions used in the chromatographic experiments and the protein charge calculated from eq 2 are presented in Table 2. The Kp(chr) value in Table 2 is obtained by adjusting the corresponding Kp value from titration shown in Table 1 to the new ionic strength used in the chromatographic experiments. The Kp(chr) value is calculated in the middle of the reciprocal square root of the ionic strength (1/ I1/2), interval used in the experiments. In Table 2, a comparison between the two sets of protein charges is made by taking their ratio, qchr/qtitr, and it is seen that there is, in general, a good agreement between the protein charge determined by the two methods. In the table is also shown the protein charge calculated from the slab model with fixed charge density, q*chr, and also the corresponding q*chr/qtitr ratio. The trend is that the chromatographically obtained charge after compensation for charge regulation is lower than that determined titrimetrically; this contrasts to the result from the slab model with fixed charge density, where the numerical q*chr values usually are higher. The explanation for this difference between the two sets of qchr values is the inclusion of charge regulation in the model, which has a tendency to overcompensate the result. The Kp value obtained from a titration curve is a point value which, in principle, is valid only for a large separation distance between the protein and the stationary phase surface. In the model, this Kp value is used for all the different separation distances between the protein and the stationary phase, which in extreme cases may give erroneous results; e.g., when the kp value is calculated at a pH value where the titration curve is extremely steep, the obtained Kp value then becomes high but is not representative for the whole interaction process, which, in principle, involves a whole set of Kp values. Ovalbumin at pH ) 5.5 has a very steep titration curve, and this effect probably is the reason for the large deviation between the two calculated charges in this case. It is also seen in Table 2 that large deviations are found for human serum albumin at both pH ) 6.5 and 9.6. An explanation for the deviation at pH ) 9.6 may be that the charge density of human serum albumin is extremely high (-0.06 C/m2) and can be close to the charge density of the stationary phase; if this is the case, the model is not valid. For 23 of the 28 chromatographically obtained protein charges, qchr, shown in Table 2, the agreement with the protein charge found titrimetrically is within 30%. Exceptions are the above-mentioned three cases, together with those of ovalbumin at pH ) 9.6 and cytochrome c at pH ) 6.0 (WCX). Since a heterogeneous charge distribution on the protein surface is not specificially included in the model, deviations due to this may also be a possible explanation for strong deviations from the model. (23) Hearn, M. T. W.; Hodder, A. N.; Stanton, P. G.; Aguilar, M. I. Chromatographia 1987, 24, 769. (24) Parente, E. S.; Welaufer, D. B. J. Chromatogr. 1986, 355, 29. (25) Stout, R. W.; Sivakoff, S. I.; Ricker, R. D. J. Chromatogr. 1986, 353, 439.

Table 1. Titration Data for Proteins protein

pI

pH

dqtitr/dpH

ovalbumin

4.58

lysozyme

11.10

cytochrome c

10.10

ribonuclease

9.60

myoglobin β-lactoglobulin

8.30 5.09

bovine serum albumin human serum albumin

5.55 5.3

5.5 6.0 6.5 7.0 7.5 7.8 8.0 9.6 4.9 6.0 6.4 4.9 6.0 6.0 6.4 6.0 7.0 8.0 7.8 6.5 7.5 9.6

-8.7 -4.7 -3.6 -3.5 -2.1 -1.6 -1.0 -9.0 -2.4 -1.3 -1.2 -2.5 -1.3 -4.0 -4.0 -2.9 -6.5 -5.5 -6.5 -5.8 -6.0 -18.0

salt

I, M

qtitr

Kp

radius, Å

ref

KCl KCl KCl KCl KCl KCl KCl KCl KCl KCl KCl

0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.15 0.15 0.15 0.10 0.10 0.15 0.15 0.01 0.135 0.135 0.15 0.15 0.15 0.15

-9.6 -13.1 -15.3 -17.1 -18.0 -18.8 -19.3 -22.2 +10.6 +9.0 +8.6 +8.9 +7.2 +6.1 +5.4 +4.8 -13.5 -18.5 -15.0 -9.8 -16.2 -34.8

0.57 0.31 0.24 0.23 0.14 0.11 0.07 0.59 0.47 0.26 0.24 0.60 0.31 0.83 0.83 1.87 0.69 0.59 0.43 0.38 0.39 1.18

23.3 23.3 23.3 23.3 23.3 23.3 23.3 23.3 15.7 15.7 15.7 15.7 15.7 15.2 15.2 17.0 21.8 21.8 26.9 27.1 27.1 27.1

16 16 16 16 16 16 16 16 17 17 17 18 18 19 19 18 20 20 21 22 22 22

KCl KCl KCl KCl KCl NaCl NaCl NaCl

Table 2. Chromatographic Conditions and Calculated Net Charge of Proteins, qchr, from Chromatographic Experiments and a Comparison between qchr and qtitr protein ovalbumin

lysozyme

cytochrome c ribonuclease myoglobin β-lactoglobulin bovine serum albumin human serum albumin

eluent pH

stationary phase

eluting salt

(1/xI)av (mol/dm3)-1

Kpchr

qchr

qchr/qtitr

q*chr

q*chr/atitr

ref

5.5 6.0 6.5 7.0 7.5 7.8 7.8 8.0 9.6 4.9 6.0 6.0 6.4 6.4 4.9 6.0 6.0 6.0 6.4 6.4 6.0 7.0 8.0 7.8 7.8 6.5 7.5 9.6

Mono Q (SAX) SynChropak Q 300 Mono Q (SAX) SynChropak Q 300 Mono Q (SAX) Zorbax Bio Series WAX 300 Zorbax Bio Series SAX 300 SynChropak Q 300 Mono Q (SAX) in-housemade WCX Zorbax Bio Series WCX 300 Zorbax Bio Series SCX 300 Zorbax Bio Series WCX 300 Zorbax Bio Series SCX 300 in-housemade WCX Zorbax Bio Series WCX 300 Zorbax Bio Series SCX 300 Zorbax Bio Series WCX 300 Zorbax Bio Series WCX 300 Zorbax Bio Series SCX 300 Zorbax Bio Series WCX 300 Synchropak SAX 300 Synchropak SAX 300 Zorbax Bio Series WAX 300 Zorbax Bio Series SAX 300 Mono Q (SAX) Mono Q (SAX) Mono Q (SAX)

NaCl NaCl NaCl NaCl NaCl (NH4)2SO4 (NH4)2SO4 NaCl NaCl Ca(OAc)2 (NH4)2SO4 (NH4)2SO4 NaOAc NaOAc Ca(OAc)2 (NH4)2SO4 (NH4)2SO4 (NH4)2SO4 (NH4)2SO4 (NH4)2SO4 (NH4)2SO4 NaCl NaCl (NH4)2SO4 (NH4)2SO4 NaCl NaCl NaCl

2.60 2.63 2.20 2.60 2.07 1.45 2.00 2.35 1.85 2.0 1.30 1.40 1.50 1.46 2.0 1.35 1.55 1.95 2.20 2.46 2.70 1.73 1.73 1.46 1.35 2.28 2.05 1.85

0.77 0.42 0.28 0.31 0.15 0.08 0.12 0.09 0.57 0.33 0.12 0.13 0.13 0.12 0.38 0.13 0.15 0.63 0.71 0.79 0.50 0.44 0.37 0.24 0.23 0.34 0.31 0.85

-5.6 -11.5 -14.8 -13.8 -16.0 -14.7 -12.5 -18.5 -13.0 +7.5 +9.9 +8.4 +10.2 +9.7 +6.8 +10.0 +7.9 +5.3 +5.1 +3.8 +4.9 -15.9 -16.8 -16.1 -13.0 -15.3 -16.7 -9.10

0.58 0.88 0.97 0.81 0.89 0.78 0.66 0.96 0.59 0.71 1.10 0.93 1.18 1.12 0.76 1.39 1.10 0.88 0.95 0.70 1.02 1.18 0.91 1.08 0.87 1.56 1.03 0.26

-11.7 -15.1 -17.5 -16.6 -17.4 -15.3 -13.3 -19.4 -19.9 +9.2 +10.6 +9.0 +10.9 +10.3 +8.6 +10.7 +8.6 +8.8 +9.5 +8.3 +6.9 -21.3 -21.2 -18.5 -14.8 -18.8 -20.1 -23.5

1.22 1.15 1.14 0.97 0.97 0.81 0.71 1.01 0.90 0.87 1.18 1.00 1.27 1.20 0.97 1.49 1.19 1.44 1.76 1.54 1.44 1.58 1.15 1.23 0.99 1.92 1.24 0.68

23 4 23 4 23 3 3 4 23 24 3 3 25 24 24 3 3 4 24 24 4 15 15 23 23 25 25 25

When comparing the two sets of chromatographically obtained values for the protein charge, qchr and q*chr, it is seen that, in most cases, there is a nonnegligible effect which in some individual cases is very strong. A comparison between the titrimetric protein charge and the value from the slab model with fixed charge density shows that eight values are found outside the 30% limit. Although this general agreement is of the same order as that when charge regulation is included, the awareness of the chargeregulating process and the factors that influence its magnitude may, in many cases, be crucial for the understanding of protein retention in ion-exchange chromatography.

CONCLUSIONS Charge regulation is an important physical mechanism when discribing the electrostatic interaction between a protein and a surface, an effect that hitherto has been neglected in models of protein adsorption and protein chromatography. In this paper, it is shown that the original slab model can be complemented with the effect of charge regulation and still retaining its simple algebraic form and interpretation. Besides the assumptions made in the slab model with fixed charge density, it is assumed that the value of the constant characterizing the charge regulation, Kp, is less than 1 and that the ratio σp/σs < Kp - 1. It is believed Analytical Chemistry, Vol. 68, No. 9, May 1, 1996

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that most ion-exchange chromatography experiments meet these requirements. This is demonstrated by the good agreement between chromatographically and titrimetrically obtained protein net charges obtained by reanalyzing the large data set originally used in ref 9. Retention of a protein at its pI in ion-exchange chromatography is usually explained by nonuniform charge distribution on the protein surface.15 An important implication of charge regulation is that it gives an additional physical mechanism for retention for

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Analytical Chemistry, Vol. 68, No. 9, May 1, 1996

these cases. More detailed studies based on a closer examination of the charge distribution on the protein surface will be needed in order to evaluate the relative contribution of these effects. Received for review October 3, 1995. Accepted February 6, 1996.X AC9509972 X

Abstract published in Advance ACS Abstracts, March 15, 1996.