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Incorporating water-release and lateral protein interactions in modeling equilibrium adsorption for ion-exchange chromatography
Journal of Chromatography A, 1126 (2006) 304–310
Incorporating water-release and lateral protein interactions in modeling equilibrium adsorption for ion-exchange chromatography Marvin E. Thrash Jr. 1 , Neville G. Pinto ∗ Department of Chemical & Materials Engineering, University of Cincinnati, Cincinnati, OH 45221-0012, USA Available online 18 July 2006
Abstract The equilibrium adsorption of two albumin proteins on a commercial ion exchanger has been studied using a colloidal model. The model accounts for electrostatic and van der Waals forces between proteins and the ion exchanger surface, the energy of interaction between adsorbed proteins, and the contribution of entropy from water-release accompanying protein adsorption. Protein-surface interactions were calculated using methods previously reported in the literature. Lateral interactions between adsorbed proteins were experimentally measured with microcalorimetry. Water-release was estimated by applying the preferential interaction approach to chromatographic retention data. The adsorption of ovalbumin and bovine serum albumin on an anion exchanger at solution pH > pI of protein was measured. The experimental isotherms have been modeled from the linear region to saturation, and the influence of three modulating alkali chlorides on capacity has been evaluated. The heat of adsorption is endothermic for all cases studied, despite the fact that the net charge on the protein is opposite that of the adsorbing surface. Strong repulsive forces between adsorbed proteins underlie the endothermic heat of adsorption, and these forces intensify with protein loading. It was found that the driving force for adsorption is the entropy increase due to the release of water from the protein and adsorbent surfaces. It is shown that the colloidal model predicts protein adsorption capacity in both the linear and non-linear isotherm regions, and can account for the effects of modulating salt. © 2006 Elsevier B.V. All rights reserved. Keywords: Protein; Chromatography; Modeling; Non-linear; Ion-exchange
1. Introduction The mechanism of protein adsorption is not totally understood, and a generally applicable quantitative thermodynamic adsorption model is not available. This is primarily because the free energy contributions of underlying processes cannot be correctly estimated. Thus, in the scale-up and optimization of chromatographic systems for protein purification, empirical and semi-empirical isotherm models have to be utilized [1]. For ionexchange adsorbents, mass-action models [2–4] are often used to estimate protein isotherms. These models describe the process of protein adsorption as a stoichiometric exchange of surface counter-ions for one protein molecule. Mass-action models are often inadequate because they do not account for major nonideal effects associated with protein adsorption, or do so in a thermodynamically inconsistent manner. A second type of protein isotherm model found in the literature uses the colloidal ∗
Corresponding author. Tel.: +1 513 556 2770; fax: +1 513 556 0128. E-mail address: [email protected] (N.G. Pinto). 1 Present address: Department of Paper and Chemical Engineering, Miami University, 240 Gaskill Hall, Oxford, OH 45056-3629, USA. 0021-9673/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2006.06.058
method. Here, the protein is treated as a sphere possessing a fixed potential at the surface. The potential of the adsorbent surface is also assumed to be constant for a given set of process conditions. This approach was used by Bowen [5] to calculate protein distribution coefficients under linear conditions. Oberholzer and Lenhoff [6] expanded this approach and successfully calculated protein isotherms under linear and overloaded conditions. The main disadvantages of the colloid method are that the protein is assumed to be a spheroid, and the determination of the surface potentials is difficult at salt concentrations greater than 100 mM. Complex interactions arise when proteins are adsorbed on ion exchangers. Calorimetric studies have shown that protein adsorption onto these materials is often endothermic. The presence of endothermic heats indicates that other subprocesses are present besides the expected (highly favorable) ion-exchange interaction. Endothermic heats can arise from a number of sources such as repulsive interactions between surface molecules, repulsive interactions between hydrophobic groups on the protein surface and hydrophilic moieties on the adsorbent surface, repulsive interactions between charged groups on the protein’s surface and charged surface sites on the adsorbent
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possessing the same charge, release of water from the adsorbent surface and the contact surface of the protein, conformational changes in the protein’s three dimensional structure, reorientation of surface proteins or some combination of these effects [7,8]. The presence of these non-ideal events complicates the development of isotherm models. The primary focus of the work presented in this paper is to examine the effect of water-release on protein adsorption. Prior calorimetric studies suggest that the free energy associated with the release of water from the contact surface of the protein and the adsorbent may be sufficient to overcome the unfavorable enthalpic contributions associated with protein adsorption [8,9]. In order to investigate this effect, the energetics of water-release was incorporated into the colloidal modeling framework developed by Oberholzer and Lenhoff [6]. Model predictions were then compared to experimental isotherms to assess the impact of water-release on adsorption capacity. The probe proteins used in this study are bovine serum albumin (BSA) and ovalbumin (OVA), and the adsorbent was polyethyleneimine (PEI 1000-10), a commercial chromatographic anion exchanger. This system was chosen because the adsorption of both proteins onto PEI 1000-10 has previously been shown to be endothermic [15]. Furthermore, additional studies have shown that the release of water associated with the adsorption of each protein on PEI is significant [17].
K is the colloidal equilibrium constant, z0 is the distance of closest approach, z is the distance between the surface and the sphere, and k is the Boltzmann constant. For high affinity adsorption, Eq. (1) can be simplified to the following form. ∞ K= (e−(Gtotal )/(kT ) ) dz (2)
2. Protein adsorption model
U el (h) = Bps e−κah kT Aps 1 1 h vdw U (h) = − + + ln 6kT h h + 2 h+2
As shown in Fig. 1, the colloidal approach, in its simplest form, treats the protein as a charged sphere and the adsorbent as a charged surface. However, as mentioned earlier, there are other events, in addition to ion-exchange, associated with protein adsorption on an ion exchanger. Repulsive interactions between surface proteins, water-release and van der Waals forces all influence adsorption capacity. These non-ideal events can be readily incorporated into the colloidal modeling framework, if the free energy contribution of each is known. The colloidal approach is based on the fundamental thermodynamic relationship shown in Eq. (1) ∞ K= (e−(Gtotal )/(kT ) − 1) dz (1) z0
Fig. 1. Schematic representation of colloidal approach for modeling protein adsorption on an ion exchanger.
z0
The details of this model are found in [6,10,11]; a brief description is presented here for convenience. In the absence of entropic effects, the total dimensionless free energy between the protein and the surface is expressed as: Gtotal = U el (h) + U vdw (h) + Hpp
(3)
Here h is the ratio of z/a, where a is the protein radius. Therefore, h is the dimensionless gap between the protein and the adsorption surface. Uel (h) is the electrostatic particle surface interaction energy, and Uvdw (h) is the van der Waals interaction energy. Hpp is the enthalpic contribution from repulsive interactions between surface proteins. In Eq. (3), the following assumptions were made (1) Hpp = Gpp ; (2) Uel (h) = Gel (h); (3) Uvdw (h) = Gvdw (h). The Yukawa form [6] was used to calculate the electrostatic interaction energy and the Hamaker equation was used to calculate van der Waals interaction energy for the adsorption of BSA. These terms are calculated from: (4) (5)
Bps is a dimensionless parameter calculated by the method presented in reference [11]. ys + 4γΩκa 4πkTεε0 a Ψs Bps = 4 tanh (6) 2 e 1 + Ωκa 4 where γ = tanh
Ψs 4
and
Ω=
ys − 4γ 2γ 3
(7)
Ψ s and ys are the dimensionless electrical potentials of the adsorption surface and the protein surface. Ψ s can be experimentally measured or calculated from Stahlbergs method [12]. ys can be calculated from the method of Sader [10] using the net protein charge. However, the second bracketed term in Eq. (6) represents effective protein potential. This value was experimentally measured using zeta potential instrumentation. The experimental value of the zeta potential is scaled by kT/e to be consistent with the central bracketed term in Eq. (6). ε is the dielectric constant of the solution, ε0 is the dielectric permittivity of free space, e is the electronic charge and κ is the Debeye screening length. Aps is the Hamaker constant, which is taken to be 1 × 10−20 J [10] for proteins. Although Eq. (3) properly accounts for repulsive interactions between surface proteins, it does not account for entropic effects such as water-release. As a result, the total adsorptive free energy (Eq. (3)) was redefined
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to incorporate the free energy contribution arising from waterrelease. Gtotal (h) = Gps (h) + Gpp + Gwr
(8)
Gpp and Gwr are the dimensionless free energy contributions of repulsive interactions and water-release from the contact surface of the protein and the adsorbent surface. Gps is the sum of the electrostatic energy and van der Waals energy. Incorporating these effects and rewriting Eq. (2) in terms of h, the following relationship shown in Eq. (9) is obtained: h∞ cs = cb (e−(Gpp +Gwr )/kT )a e−Gps /kT dh (9) h0
Gpp and Gwr are surface effects, and considered independent of h. Cs is the surface concentration and Cb is the bulk concentration. Eq. (9) is of the same general form used by Oberholzer and Lenhoff [6], with some differences. As originally defined, Gpp is the free energy change arising from repulsive interactions between surface proteins, and as part of the original model it was assumed that Gpp was approximately equal to Hpp . In the original model, molecular simulations determined Hpp as a function of protein surface concentration. Here Hpp is estimated from measurements of the heat of adsorption as a function of adsorbed protein concentration. As previously stated, h is the dimensionless gap distance between the protein and the surface. Specifically, the distance of closest approach was estimated from the literature [7,8] to be approximately 0.1 nm. h∞ is the gap distance at which the protein-surface interaction energy is approximately zero. Gwr was calculated from the relationship developed by Nemethy and Scheraga [13]. From this equation, it was determined that the free energy reduction from waterrelease is approximately 12 mJ/m2 . Eqs. (3)–(9) were used to simulate protein adsorption isotherms on ion exchangers. This model assumes monolayer surface coverage. It also assumes the protein is a sphere of fixed diameter calculated from the radius of gyration. Although it is understood that conformational changes, orientational changes, steric effect or charge redistribution might be present, the free energy contribution resulting from these effects was assumed to be negligible. 3. Experimental
The salts were purchased from the Fisher Scientific Company (Hanover Park, IL, USA). Piperazine was purchased from the Eastman Company (Kingsport, TN, USA). HPLC grade water was obtained from an ultrafiltration unit purchased from the Millipore Corporation (Bedford, MA, USA). 3.2. Flow microcalorimetry The FMC (Gilson Instruments, Westerville, OH) is operated similar to a liquid chromatograph. The calorimeter sample cell is enclosed in an insulating metal block. The cell temperature is controlled with a block heater. Temperature changes resulting from protein adsorption are monitored by two highly sensitive thermisters mounted in the wall of the sample cell containing the adsorbent. The output of the thermisters is routed to the FMC control unit. The control unit converts the analog input of the thermisters into a digital signal that is sent to the computer while simultaneously transmitting an analog signal to a strip chart recorder. The column or cell volume is 0.171 ml. The flow rate through the cell is controlled by precision syringe micropumps. The flow rate used in these experiments was 3.30 ml/h. The operating temperature was 298 K and pH 6.2. As in a chromatograph, the FMC is equipped with a configurable injection loop to accommodate different injection volumes. The effluent was collected and analyzed with a Milton Roy Spectrophotometer. The FMC is initially filled with a specified volume of adsorbent. The next step is the evacuation of the cell to a vacuum pressure of 30 in.Hg. Once the cell has been successfully evacuated, the contents are “wetted” with the carrier fluid. The effluent from the cell is filtered with a polyethersulfone membrane. Following wetting, the syringe pumps are turned on and the adsorbent is equilibrated with the carrier solution. Once the system has reached thermal equilibrium, the sample (dissolved in the carrier fluid) is loaded into the injection loop, and introduced into the cell by switching a multiport valve. The adsorption of the sample onto adsorbent surface causes a change in cell temperature, which is converted to a heat signal by the FMC through an experimentally determined calibration factor. The calibration factor was obtained using the 50 m PEI particles (PEI 1000-50). Once the mass in the effluent is quantified with the spectrophotometer, a simple mass balance is performed to determine the quantity of sample adsorbed. From these data the specific heat of adsorption is calculated.
3.1. Materials and apparatus 3.3. Isotherms BSA and OVA was purchased from Sigma (St. Louis, MO, USA) and used without further purification. The IEC chromatographic support used (PEI-1000-10) is a functionalized silica particle with a mean diameter of 10 m and an average pore ˚ The BET surface area is 30 m2 /g. The surface is size of 1000 A. activated with cross-linked polyethyleneimine ligands. The IEC support was purchased from the Millipore Corporation (Bedford, MA, USA). The carrier fluid for the IEC experiments was 10 mM piperazine (pH 6.2). Sodium chloride, potassium chloride and lithium chloride were used as modulators for all IEC experiments.
Protein isotherms for BSA and OVA were measured at selected modulator concentrations in 10 mM piperazine buffer at 298 K (pH 6.2) by the batch method. PEI-1000-10 was weighed into test tubes, and protein solution of a known concentration was pipetted into the tube. The test tubes were sealed with parafilm, placed in a shaker, and agitated at 200 rpm for 24 h at 298 K. Preliminary experiments established that equilibrium is effectively reached in 8–10 h. After equilibration, the slurry solution was allowed to stand for 1 h before filtering with a 0.45-m polyethersulfone filter. The absorbance of the filtrate was mea-
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sured at 280 nm to obtain the equilibrium solution concentration. The equilibrium distribution was calculated from a mass balance. 3.4. Zeta potential measurements Zeta potential measurements were made on a ZetaPals (Brookhaven Instruments). A 10 mg/ml protein solution (pH 6.2) was added to a cuvette, which in turn was inserted into a temperature-controlled chamber located in an internal compartment. An electrode was placed inside the cuvette containing the protein solution to create an electric field for the purpose of inducing movement of the protein molecules. The direction of protein movement is dependent on the protein’s net potential. When the electrode is energized, a laser beam is fired through the solution. A portion of the beam is shifted as it travels though the protein solution. The magnitude and direction of this shift is proportional to the mobility of the protein molecules. Once the mobility is known, the zeta potential is calculated using the Smoluchowski equation [14].
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Table 1 Model parameters for protein adsorption on PEI 1000-10 at 298 K and pH 6.2 Parameter Protein radius (m) Protein zeta potential (mV) Bps Gwr /kT in NaCl Gwr /kT in LiCl Gwr /kT in KCl Adsorbent potential (mV) Adsorbent surface area (m2 /g)
BSA 29.8 × 10−10
−4.0 −0.26 −16.5 −12.4 −22.4 9.65 30
OVA 22.6 × 10−9 −7.2 −0.35 −10.5 −6.6 −15.1 9.65 30
3.5. Isocratic elutions Capacity factor data for BSA and OVA were obtained on an HP1100 chromatograph unit with a 25 cm × 0.46 cm I.D. column at 25, 33 and 40 ◦ C at pH 6.2 (piperazine buffer). The column was equilibrated with buffer at different concentration of salt and a flow-rate of 1.0 ml/min. Elution times were measured for the injection of 3 l of 2.0 mg/ml protein solution. The water-release was determined with the preferential interaction model [17]. The salt range was 300–1000 mM salt. All responses were monitored at 280 nm. 3.6. Measurement of adsorbent potential The potential of the PEI 1000-10 adsorbent was measured with an electro acoustic spectrophotometer (model DT1200) manufactured by Dispersion Technology. The adsorbent was immersed in a piperazine buffer solution (pH 6.2) and the resulting solution was placed in the DT1200. The double layer of the adsorbent particles is disturbed by an ultrasonic wave. This induced disturbance in the double layer ultimately produces an electric field that is proportional to the surface potential. The DT1200 senses the electric field and calculates the surface potential.
Fig. 2. Heat of adsorption of BSA on PEI 1000-10. Arrows show method for calculating Hpp at selected protein surface concentration (298 K; pH 6.2; 100 mM each salt).
zeta potential and adsorbent potential reported in Table 1. The values of Bps from Eq. (4) calculated using this approach are also shown in Table 1. The measurement of the change in heat of adsorption with surface concentration is the key to estimating Hpp in this approach. Heat-of-adsorption measurements made for BSA and OVA on PEI 1000-10 as a function of modulator salt and protein loading are reported in Figs. 2 and 3, respectively. The heat of adsorption measured in all cases is endothermic. This is not generally expected in ion-exchange systems, as has been discussed previously [15]. Also, the heat of adsorption increases with an increase in surface with surface concentration, indicating that the repulsive energy increase with surface coverage.
4. Results and discussion Adsorption isotherms of two proteins, BSA and OVA, were measured on the anion exchanger PEI-1000 at 298 K in three salt solutions 100 mM NaCl, 100 mM KCl, and 100 mM LiCl. Eq. (9) was used to predict the isotherms, and the predictions were compared to the experimental measurements. To calculate the adsorbed, equilibrium concentration, cs , from Eq. (9), at a selected solution concentration cb , requires values of Gps (h), Gpp , and Gwr . Gps (h) was calculated using Eqs. (4)–(6), as described earlier, and the measured values of protein
Fig. 3. Heat of adsorption of OVA on PEI 1000-10 (298 K; pH 6.2; 100 mM each salt).
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The presence of salt is seen to reduce repulsive energy between adsorbed proteins, as is expected. The data in Fig. 2 were previously reported [15], and are repeated to illustrate the calculation of Hpp . It should be noted that Hpp is a function of surface coverage. This term was incorporated into Eq. (9) assuming, as stated earlier, that Hpp = Gpp , and using an iterative calculation to match surface concentration Cs in Eq. (9) with the corresponding value of Gpp . Details of the iterative method are reported elsewhere [16]. For the calculation of Gwr in Eq. (9), the water-release upon protein adsorption must be known. This was obtained using the Preferential Interaction Model [17] on chromatographic retention data of BSA and OVA on PAE 1000-10, following the method reported in the literature [15]. The water-release as a function of modulating salt is shown in Table 2. These data show that water-release is greatest in the presence of KCl and smallest in the presence of LiCl for both proteins. Also observed is that the amount of water-release is smaller for OVA than for BSA. Ovalbumin is a smaller protein (MW = 43,500 Da), and consequently the surface footprint is smaller than for BSA (MW = 69,000 Da). The variation of water-release with molecular weight in these data is consistent with the trends presented by Perkins et al. [17]. From the water-release numbers in Table 2, Gwr values in Table 1 were calculated, following the method of Nemethy and Scheraga [13]. From their equation, the free energy reduction for water-release at 298 K was calculated to be −12 J/m2 . Using this with the recommended specific adsorbent surface coverage area for water of 7.1 × 10−20 m2 /molecule, the values in Table 2 are obtained. A comparison of experimentally measured isotherms and those predicted with the colloidal model for BSA and OVA are shown in Figs. 4 and 5, respectively. In general, both the linear and plateau regions are well characterized for both proteins. What is most significant is that differences in observed adsorption capacity due to changes in modulator are predicted by the model when water-release effects are incorporated. There appears to be a direct relationship between the differences in the amount of water-released and the differences in capacity. It is worth mentioning that predicting the effects of modulator on adsorbent protein capacity has been one of the major shortcomings of models in the literature. It is illustrative to break down the contributions of each of the major free energy terms in Eq. (9) to the overall isotherm prediction. Shown in Fig. 6 is the prediction of the colloidal model for BSA adsorption on PAE 1000-10 anion exchanger in 0.1 M NaCl if Gpp = 0 and Gwr = 0; i.e., only electrostatic
Fig. 4. Comparison of simulated and experimental isotherms for BSA (298 K; pH 6.2; 100 mM each salt).
Fig. 5. Comparison of simulated and experimental isotherms for OVA (298 K; pH 6.2; 100 mM each salt).
and van der Waal interactions between the ion-exchange surface and the protein are considered. In this case, the model predicts a linear isotherm. In the absence of a constraint such as repulsive interactions between surface proteins, the isotherm would show a linear increase in surface concentration until the maximum adsorbent capacity has been reached. Based on the available surface area and assuming monolayer coverage, a saturation capacity of 180 mg/g was additionally imposed. This gives an
Table 2 Average water-release for protein adsorption on PEI 1000-10 (pH 6.2) Modulating salt
LiCl NaCl KCl
Average water-release (mol/mol adsorbed protein) BSA
OVA
60 80 108
32 51 73
Temperature range: 298–313 K; salt range: 0.3–0.9 M.
Fig. 6. Effects of free energy change of protein surface (Gps ), protein–protein (Gpp ) and water-release (Gwr ) contributions on predictions of BSA isotherms on PAE 1000-10 anion exchanger (298 K, pH 6.2, 100 mM NaCl).
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isotherm consisting of two linear regions (Fig. 6) with a sharp discontinuity at the point of intersection. If protein–protein interactions are added to protein-surface contributions the predicted isotherm the prediction shifts as shown in Fig. 6. It should be noted that there is no saturation capacity additionally imposed, and the effect of water-release are not included; i.e., Gwr = 0. The model underestimates the experimental isotherm in this case. This result is consistent with those reported with the NISS model earlier [8]. This earlier work was one of the first modeling efforts to identify the importance of considering changes in entropy associated with protein adsorption on ion exchangers. Upon incorporation of the water-release term in Eq. (9), the isotherm prediction is as shown in Fig. 4. For clarity this is also shown in Fig. 6. In this particular case, the measured average release of 80 water molecules per molecule of adsorbed protein (Table 2) was used to estimate Gwr . It is observed that while the prediction generally matches well with the experimental isotherm there are regions, particularly in the transition from the linear to the plateau, where the data are under-predicted. This is true for all the cases studied (Figs. 4 and 5), and could originate from a number of sources is suspected to be due to the use of an average value of water-release. The Preferential Interaction Model gives only the average number of water molecules released over a range of modulator concentrations at protein loadings in the linear region. If a method could be developed to estimate the water-release as a function of protein loading the prediction is expected to improve. Other effects that may underlie these differences are changes in orientation and/or conformation of the protein with loading (though the latter is less likely for the specific cases studied). While this is not explicitly accounted for in the model, it would in part be indirectly incorporated if a method was available for measuring the amount of water released with protein loading. 5. Conclusions This paper illustrates the importance of accounting for entropic contributions to the free energy change associated with protein adsorption on ion exchangers. It is often assumed that electrostatic and van der Waal forces dominate the determination of isotherm capacity in protein ion-exchange. Using a colloidal model in combination with heat of adsorption measurements, it has been shown that protein capacity on ion exchangers is strongly limited by repulsive interactions between the adsorbed proteins. Attractive electrostatic and van der Waal forces between the ion exchanger and the protein, and the entropy increase accompanying water-release from the protein and adsorbent surface drive against the repulsive protein–protein surface forces. The reduction in free energy from the waterrelease can be significant in relation to the other energetic changes occurring during adsorption, as has been shown for the adsorption of two proteins, ovalbumin and bovine serum albumin, on a commercial anion exchanger. In fact, for all cases studied, heat of adsorption of the net negatively charged proteins on the positively charged ion exchanger is endothermic, and it is the increase in entropy that makes the adsorption energetically
309
favorable. It has further been shown for these cases that the modulating salt (NaCl, LiCl or KCl) has a strong influence on the repulsive interactions between adsorbed proteins as well as on the amount of water released due to protein adsorption. Methods have been presented to effectively estimate these effects to obtain good predictions of isotherm behavior with the colloidal model. 6. Nomenclature
a protein radius Aps Hamaker constant Bps dimensionless parameter defined by Eq. (6) Cb bulk or solution protein concentration Cs surface protein concentration G free energy Gpp /kT dimensionless free energy change of repulsion between adsorbed proteins Gps /kT sum of dimensionless free energy change of electrostatic and van der Waal interactions between protein and adsorbent surface Gtotal /kT dimensionless free energy change due to waterrelease upon adsorption Gwr /kT dimensionless free energy change due to waterrelease upon adsorption h dimensionless gap distance (z/a) H enthalpy change of adsorption Hpp enthalpy change of adsorption from repulsion between adsorbed proteins k Boltzmann constant K colloidal equilibrium constant SAX strong anion exchanger T absolute temperature U internal energy Uel internal energy change from electrostatic interactions between protein and adsorbent Uvdw internal energy change from van der Waals interactions between protein and adsorbent ys dimensionless protein surface potential z distance between surface and sphere z0 distance of closest approach Greek letters ε dielectric constant of solution dielectric constant in free space εo γ γ = tanh((ys )/(4)) dimensionless parameter used to calculate potential Ψs dimensionless adsorbent surface potential κ Debye screening length Ω Ω = (ys − 4γ)/(2γ 3 ) dimensionless parameter used to potential References [1] G. Guiochon, S.G. Shirazi, A.M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, Boston, 1994. [2] C.A. Brooks, S.M. Cramer, AIChE J. 38 (1992) 1969. [3] Y. Li, N.G. Pinto, J. Chromatogr. A 702 (1995) 113.
310 [4] [5] [6] [7] [8] [9] [10] [11]
M.E. Thrash Jr., N.G. Pinto / J. Chromatogr. A 1126 (2006) 304–310 P. Raje, N.G. Pinto, J. Chromatogr. A 760 (1997) 89. W.R. Bowen, L.-C. Pan, A.O. Sharif, Colloids Surf. A 143 (1998) 117. M.R. Oberholzer, A.M. Lenhoff, Langmuir 15 (1999) 3905. W.R. Bowen, D.T. Hughes, J. Colloid Interface Sci. 158 (1993) 395. P. Raje, N.G. Pinto, J. Chromatogr. A 796 (1998) 141. D.S. Gill, D.J. Roush, R.C. Wilson, J. Colloid Interface Sci. 167 (1994) 1. J.E. Sader, J. Colloid Interface Sci. 188 (1997) 508. M.R. Oberholzer, N.J. Wagner, A.M. Lenhoff, J. Chem. Phys. 107 (1997) 9157.
[12] J. Stahlberg, Anal. Chem. 66 (1994) 440. [13] G. Nemethy, H.A. Scheraga, J. Chem. Phys. 36 (1962) 1773. [14] R.J. Hunter, Zeta Potential in Colloid Science: Principles and Applications, third ed., Academic Press, London, 1981. [15] M.E. Thrash, J.M. Phillips, N.G. Pinto, Adsorption J. 10 (2004) 299. [16] M.E. Thrash, Doctoral dissertation, University of Cincinnati, Cincinnati, OH, 2003. [17] T.W. Perkins, D.S. Mak, T.W. Root, E.N. Lightfoot, J. Chromatogr. A 766 (1997) 1.
Incorporating water-release and lateral protein interactions in modeling equilibrium adsorption for ion-exchange chromatography Marvin E. Thrash Jr. 1 , Neville G. Pinto ∗ Department of Chemical & Materials Engineering, University of Cincinnati, Cincinnati, OH 45221-0012, USA Available online 18 July 2006
Abstract The equilibrium adsorption of two albumin proteins on a commercial ion exchanger has been studied using a colloidal model. The model accounts for electrostatic and van der Waals forces between proteins and the ion exchanger surface, the energy of interaction between adsorbed proteins, and the contribution of entropy from water-release accompanying protein adsorption. Protein-surface interactions were calculated using methods previously reported in the literature. Lateral interactions between adsorbed proteins were experimentally measured with microcalorimetry. Water-release was estimated by applying the preferential interaction approach to chromatographic retention data. The adsorption of ovalbumin and bovine serum albumin on an anion exchanger at solution pH > pI of protein was measured. The experimental isotherms have been modeled from the linear region to saturation, and the influence of three modulating alkali chlorides on capacity has been evaluated. The heat of adsorption is endothermic for all cases studied, despite the fact that the net charge on the protein is opposite that of the adsorbing surface. Strong repulsive forces between adsorbed proteins underlie the endothermic heat of adsorption, and these forces intensify with protein loading. It was found that the driving force for adsorption is the entropy increase due to the release of water from the protein and adsorbent surfaces. It is shown that the colloidal model predicts protein adsorption capacity in both the linear and non-linear isotherm regions, and can account for the effects of modulating salt. © 2006 Elsevier B.V. All rights reserved. Keywords: Protein; Chromatography; Modeling; Non-linear; Ion-exchange
1. Introduction The mechanism of protein adsorption is not totally understood, and a generally applicable quantitative thermodynamic adsorption model is not available. This is primarily because the free energy contributions of underlying processes cannot be correctly estimated. Thus, in the scale-up and optimization of chromatographic systems for protein purification, empirical and semi-empirical isotherm models have to be utilized [1]. For ionexchange adsorbents, mass-action models [2–4] are often used to estimate protein isotherms. These models describe the process of protein adsorption as a stoichiometric exchange of surface counter-ions for one protein molecule. Mass-action models are often inadequate because they do not account for major nonideal effects associated with protein adsorption, or do so in a thermodynamically inconsistent manner. A second type of protein isotherm model found in the literature uses the colloidal ∗
Corresponding author. Tel.: +1 513 556 2770; fax: +1 513 556 0128. E-mail address: [email protected] (N.G. Pinto). 1 Present address: Department of Paper and Chemical Engineering, Miami University, 240 Gaskill Hall, Oxford, OH 45056-3629, USA. 0021-9673/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2006.06.058
method. Here, the protein is treated as a sphere possessing a fixed potential at the surface. The potential of the adsorbent surface is also assumed to be constant for a given set of process conditions. This approach was used by Bowen [5] to calculate protein distribution coefficients under linear conditions. Oberholzer and Lenhoff [6] expanded this approach and successfully calculated protein isotherms under linear and overloaded conditions. The main disadvantages of the colloid method are that the protein is assumed to be a spheroid, and the determination of the surface potentials is difficult at salt concentrations greater than 100 mM. Complex interactions arise when proteins are adsorbed on ion exchangers. Calorimetric studies have shown that protein adsorption onto these materials is often endothermic. The presence of endothermic heats indicates that other subprocesses are present besides the expected (highly favorable) ion-exchange interaction. Endothermic heats can arise from a number of sources such as repulsive interactions between surface molecules, repulsive interactions between hydrophobic groups on the protein surface and hydrophilic moieties on the adsorbent surface, repulsive interactions between charged groups on the protein’s surface and charged surface sites on the adsorbent
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possessing the same charge, release of water from the adsorbent surface and the contact surface of the protein, conformational changes in the protein’s three dimensional structure, reorientation of surface proteins or some combination of these effects [7,8]. The presence of these non-ideal events complicates the development of isotherm models. The primary focus of the work presented in this paper is to examine the effect of water-release on protein adsorption. Prior calorimetric studies suggest that the free energy associated with the release of water from the contact surface of the protein and the adsorbent may be sufficient to overcome the unfavorable enthalpic contributions associated with protein adsorption [8,9]. In order to investigate this effect, the energetics of water-release was incorporated into the colloidal modeling framework developed by Oberholzer and Lenhoff [6]. Model predictions were then compared to experimental isotherms to assess the impact of water-release on adsorption capacity. The probe proteins used in this study are bovine serum albumin (BSA) and ovalbumin (OVA), and the adsorbent was polyethyleneimine (PEI 1000-10), a commercial chromatographic anion exchanger. This system was chosen because the adsorption of both proteins onto PEI 1000-10 has previously been shown to be endothermic [15]. Furthermore, additional studies have shown that the release of water associated with the adsorption of each protein on PEI is significant [17].
K is the colloidal equilibrium constant, z0 is the distance of closest approach, z is the distance between the surface and the sphere, and k is the Boltzmann constant. For high affinity adsorption, Eq. (1) can be simplified to the following form. ∞ K= (e−(Gtotal )/(kT ) ) dz (2)
2. Protein adsorption model
U el (h) = Bps e−κah kT Aps 1 1 h vdw U (h) = − + + ln 6kT h h + 2 h+2
As shown in Fig. 1, the colloidal approach, in its simplest form, treats the protein as a charged sphere and the adsorbent as a charged surface. However, as mentioned earlier, there are other events, in addition to ion-exchange, associated with protein adsorption on an ion exchanger. Repulsive interactions between surface proteins, water-release and van der Waals forces all influence adsorption capacity. These non-ideal events can be readily incorporated into the colloidal modeling framework, if the free energy contribution of each is known. The colloidal approach is based on the fundamental thermodynamic relationship shown in Eq. (1) ∞ K= (e−(Gtotal )/(kT ) − 1) dz (1) z0
Fig. 1. Schematic representation of colloidal approach for modeling protein adsorption on an ion exchanger.
z0
The details of this model are found in [6,10,11]; a brief description is presented here for convenience. In the absence of entropic effects, the total dimensionless free energy between the protein and the surface is expressed as: Gtotal = U el (h) + U vdw (h) + Hpp
(3)
Here h is the ratio of z/a, where a is the protein radius. Therefore, h is the dimensionless gap between the protein and the adsorption surface. Uel (h) is the electrostatic particle surface interaction energy, and Uvdw (h) is the van der Waals interaction energy. Hpp is the enthalpic contribution from repulsive interactions between surface proteins. In Eq. (3), the following assumptions were made (1) Hpp = Gpp ; (2) Uel (h) = Gel (h); (3) Uvdw (h) = Gvdw (h). The Yukawa form [6] was used to calculate the electrostatic interaction energy and the Hamaker equation was used to calculate van der Waals interaction energy for the adsorption of BSA. These terms are calculated from: (4) (5)
Bps is a dimensionless parameter calculated by the method presented in reference [11]. ys + 4γΩκa 4πkTεε0 a Ψs Bps = 4 tanh (6) 2 e 1 + Ωκa 4 where γ = tanh
Ψs 4
and
Ω=
ys − 4γ 2γ 3
(7)
Ψ s and ys are the dimensionless electrical potentials of the adsorption surface and the protein surface. Ψ s can be experimentally measured or calculated from Stahlbergs method [12]. ys can be calculated from the method of Sader [10] using the net protein charge. However, the second bracketed term in Eq. (6) represents effective protein potential. This value was experimentally measured using zeta potential instrumentation. The experimental value of the zeta potential is scaled by kT/e to be consistent with the central bracketed term in Eq. (6). ε is the dielectric constant of the solution, ε0 is the dielectric permittivity of free space, e is the electronic charge and κ is the Debeye screening length. Aps is the Hamaker constant, which is taken to be 1 × 10−20 J [10] for proteins. Although Eq. (3) properly accounts for repulsive interactions between surface proteins, it does not account for entropic effects such as water-release. As a result, the total adsorptive free energy (Eq. (3)) was redefined
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to incorporate the free energy contribution arising from waterrelease. Gtotal (h) = Gps (h) + Gpp + Gwr
(8)
Gpp and Gwr are the dimensionless free energy contributions of repulsive interactions and water-release from the contact surface of the protein and the adsorbent surface. Gps is the sum of the electrostatic energy and van der Waals energy. Incorporating these effects and rewriting Eq. (2) in terms of h, the following relationship shown in Eq. (9) is obtained: h∞ cs = cb (e−(Gpp +Gwr )/kT )a e−Gps /kT dh (9) h0
Gpp and Gwr are surface effects, and considered independent of h. Cs is the surface concentration and Cb is the bulk concentration. Eq. (9) is of the same general form used by Oberholzer and Lenhoff [6], with some differences. As originally defined, Gpp is the free energy change arising from repulsive interactions between surface proteins, and as part of the original model it was assumed that Gpp was approximately equal to Hpp . In the original model, molecular simulations determined Hpp as a function of protein surface concentration. Here Hpp is estimated from measurements of the heat of adsorption as a function of adsorbed protein concentration. As previously stated, h is the dimensionless gap distance between the protein and the surface. Specifically, the distance of closest approach was estimated from the literature [7,8] to be approximately 0.1 nm. h∞ is the gap distance at which the protein-surface interaction energy is approximately zero. Gwr was calculated from the relationship developed by Nemethy and Scheraga [13]. From this equation, it was determined that the free energy reduction from waterrelease is approximately 12 mJ/m2 . Eqs. (3)–(9) were used to simulate protein adsorption isotherms on ion exchangers. This model assumes monolayer surface coverage. It also assumes the protein is a sphere of fixed diameter calculated from the radius of gyration. Although it is understood that conformational changes, orientational changes, steric effect or charge redistribution might be present, the free energy contribution resulting from these effects was assumed to be negligible. 3. Experimental
The salts were purchased from the Fisher Scientific Company (Hanover Park, IL, USA). Piperazine was purchased from the Eastman Company (Kingsport, TN, USA). HPLC grade water was obtained from an ultrafiltration unit purchased from the Millipore Corporation (Bedford, MA, USA). 3.2. Flow microcalorimetry The FMC (Gilson Instruments, Westerville, OH) is operated similar to a liquid chromatograph. The calorimeter sample cell is enclosed in an insulating metal block. The cell temperature is controlled with a block heater. Temperature changes resulting from protein adsorption are monitored by two highly sensitive thermisters mounted in the wall of the sample cell containing the adsorbent. The output of the thermisters is routed to the FMC control unit. The control unit converts the analog input of the thermisters into a digital signal that is sent to the computer while simultaneously transmitting an analog signal to a strip chart recorder. The column or cell volume is 0.171 ml. The flow rate through the cell is controlled by precision syringe micropumps. The flow rate used in these experiments was 3.30 ml/h. The operating temperature was 298 K and pH 6.2. As in a chromatograph, the FMC is equipped with a configurable injection loop to accommodate different injection volumes. The effluent was collected and analyzed with a Milton Roy Spectrophotometer. The FMC is initially filled with a specified volume of adsorbent. The next step is the evacuation of the cell to a vacuum pressure of 30 in.Hg. Once the cell has been successfully evacuated, the contents are “wetted” with the carrier fluid. The effluent from the cell is filtered with a polyethersulfone membrane. Following wetting, the syringe pumps are turned on and the adsorbent is equilibrated with the carrier solution. Once the system has reached thermal equilibrium, the sample (dissolved in the carrier fluid) is loaded into the injection loop, and introduced into the cell by switching a multiport valve. The adsorption of the sample onto adsorbent surface causes a change in cell temperature, which is converted to a heat signal by the FMC through an experimentally determined calibration factor. The calibration factor was obtained using the 50 m PEI particles (PEI 1000-50). Once the mass in the effluent is quantified with the spectrophotometer, a simple mass balance is performed to determine the quantity of sample adsorbed. From these data the specific heat of adsorption is calculated.
3.1. Materials and apparatus 3.3. Isotherms BSA and OVA was purchased from Sigma (St. Louis, MO, USA) and used without further purification. The IEC chromatographic support used (PEI-1000-10) is a functionalized silica particle with a mean diameter of 10 m and an average pore ˚ The BET surface area is 30 m2 /g. The surface is size of 1000 A. activated with cross-linked polyethyleneimine ligands. The IEC support was purchased from the Millipore Corporation (Bedford, MA, USA). The carrier fluid for the IEC experiments was 10 mM piperazine (pH 6.2). Sodium chloride, potassium chloride and lithium chloride were used as modulators for all IEC experiments.
Protein isotherms for BSA and OVA were measured at selected modulator concentrations in 10 mM piperazine buffer at 298 K (pH 6.2) by the batch method. PEI-1000-10 was weighed into test tubes, and protein solution of a known concentration was pipetted into the tube. The test tubes were sealed with parafilm, placed in a shaker, and agitated at 200 rpm for 24 h at 298 K. Preliminary experiments established that equilibrium is effectively reached in 8–10 h. After equilibration, the slurry solution was allowed to stand for 1 h before filtering with a 0.45-m polyethersulfone filter. The absorbance of the filtrate was mea-
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sured at 280 nm to obtain the equilibrium solution concentration. The equilibrium distribution was calculated from a mass balance. 3.4. Zeta potential measurements Zeta potential measurements were made on a ZetaPals (Brookhaven Instruments). A 10 mg/ml protein solution (pH 6.2) was added to a cuvette, which in turn was inserted into a temperature-controlled chamber located in an internal compartment. An electrode was placed inside the cuvette containing the protein solution to create an electric field for the purpose of inducing movement of the protein molecules. The direction of protein movement is dependent on the protein’s net potential. When the electrode is energized, a laser beam is fired through the solution. A portion of the beam is shifted as it travels though the protein solution. The magnitude and direction of this shift is proportional to the mobility of the protein molecules. Once the mobility is known, the zeta potential is calculated using the Smoluchowski equation [14].
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Table 1 Model parameters for protein adsorption on PEI 1000-10 at 298 K and pH 6.2 Parameter Protein radius (m) Protein zeta potential (mV) Bps Gwr /kT in NaCl Gwr /kT in LiCl Gwr /kT in KCl Adsorbent potential (mV) Adsorbent surface area (m2 /g)
BSA 29.8 × 10−10
−4.0 −0.26 −16.5 −12.4 −22.4 9.65 30
OVA 22.6 × 10−9 −7.2 −0.35 −10.5 −6.6 −15.1 9.65 30
3.5. Isocratic elutions Capacity factor data for BSA and OVA were obtained on an HP1100 chromatograph unit with a 25 cm × 0.46 cm I.D. column at 25, 33 and 40 ◦ C at pH 6.2 (piperazine buffer). The column was equilibrated with buffer at different concentration of salt and a flow-rate of 1.0 ml/min. Elution times were measured for the injection of 3 l of 2.0 mg/ml protein solution. The water-release was determined with the preferential interaction model [17]. The salt range was 300–1000 mM salt. All responses were monitored at 280 nm. 3.6. Measurement of adsorbent potential The potential of the PEI 1000-10 adsorbent was measured with an electro acoustic spectrophotometer (model DT1200) manufactured by Dispersion Technology. The adsorbent was immersed in a piperazine buffer solution (pH 6.2) and the resulting solution was placed in the DT1200. The double layer of the adsorbent particles is disturbed by an ultrasonic wave. This induced disturbance in the double layer ultimately produces an electric field that is proportional to the surface potential. The DT1200 senses the electric field and calculates the surface potential.
Fig. 2. Heat of adsorption of BSA on PEI 1000-10. Arrows show method for calculating Hpp at selected protein surface concentration (298 K; pH 6.2; 100 mM each salt).
zeta potential and adsorbent potential reported in Table 1. The values of Bps from Eq. (4) calculated using this approach are also shown in Table 1. The measurement of the change in heat of adsorption with surface concentration is the key to estimating Hpp in this approach. Heat-of-adsorption measurements made for BSA and OVA on PEI 1000-10 as a function of modulator salt and protein loading are reported in Figs. 2 and 3, respectively. The heat of adsorption measured in all cases is endothermic. This is not generally expected in ion-exchange systems, as has been discussed previously [15]. Also, the heat of adsorption increases with an increase in surface with surface concentration, indicating that the repulsive energy increase with surface coverage.
4. Results and discussion Adsorption isotherms of two proteins, BSA and OVA, were measured on the anion exchanger PEI-1000 at 298 K in three salt solutions 100 mM NaCl, 100 mM KCl, and 100 mM LiCl. Eq. (9) was used to predict the isotherms, and the predictions were compared to the experimental measurements. To calculate the adsorbed, equilibrium concentration, cs , from Eq. (9), at a selected solution concentration cb , requires values of Gps (h), Gpp , and Gwr . Gps (h) was calculated using Eqs. (4)–(6), as described earlier, and the measured values of protein
Fig. 3. Heat of adsorption of OVA on PEI 1000-10 (298 K; pH 6.2; 100 mM each salt).
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The presence of salt is seen to reduce repulsive energy between adsorbed proteins, as is expected. The data in Fig. 2 were previously reported [15], and are repeated to illustrate the calculation of Hpp . It should be noted that Hpp is a function of surface coverage. This term was incorporated into Eq. (9) assuming, as stated earlier, that Hpp = Gpp , and using an iterative calculation to match surface concentration Cs in Eq. (9) with the corresponding value of Gpp . Details of the iterative method are reported elsewhere [16]. For the calculation of Gwr in Eq. (9), the water-release upon protein adsorption must be known. This was obtained using the Preferential Interaction Model [17] on chromatographic retention data of BSA and OVA on PAE 1000-10, following the method reported in the literature [15]. The water-release as a function of modulating salt is shown in Table 2. These data show that water-release is greatest in the presence of KCl and smallest in the presence of LiCl for both proteins. Also observed is that the amount of water-release is smaller for OVA than for BSA. Ovalbumin is a smaller protein (MW = 43,500 Da), and consequently the surface footprint is smaller than for BSA (MW = 69,000 Da). The variation of water-release with molecular weight in these data is consistent with the trends presented by Perkins et al. [17]. From the water-release numbers in Table 2, Gwr values in Table 1 were calculated, following the method of Nemethy and Scheraga [13]. From their equation, the free energy reduction for water-release at 298 K was calculated to be −12 J/m2 . Using this with the recommended specific adsorbent surface coverage area for water of 7.1 × 10−20 m2 /molecule, the values in Table 2 are obtained. A comparison of experimentally measured isotherms and those predicted with the colloidal model for BSA and OVA are shown in Figs. 4 and 5, respectively. In general, both the linear and plateau regions are well characterized for both proteins. What is most significant is that differences in observed adsorption capacity due to changes in modulator are predicted by the model when water-release effects are incorporated. There appears to be a direct relationship between the differences in the amount of water-released and the differences in capacity. It is worth mentioning that predicting the effects of modulator on adsorbent protein capacity has been one of the major shortcomings of models in the literature. It is illustrative to break down the contributions of each of the major free energy terms in Eq. (9) to the overall isotherm prediction. Shown in Fig. 6 is the prediction of the colloidal model for BSA adsorption on PAE 1000-10 anion exchanger in 0.1 M NaCl if Gpp = 0 and Gwr = 0; i.e., only electrostatic
Fig. 4. Comparison of simulated and experimental isotherms for BSA (298 K; pH 6.2; 100 mM each salt).
Fig. 5. Comparison of simulated and experimental isotherms for OVA (298 K; pH 6.2; 100 mM each salt).
and van der Waal interactions between the ion-exchange surface and the protein are considered. In this case, the model predicts a linear isotherm. In the absence of a constraint such as repulsive interactions between surface proteins, the isotherm would show a linear increase in surface concentration until the maximum adsorbent capacity has been reached. Based on the available surface area and assuming monolayer coverage, a saturation capacity of 180 mg/g was additionally imposed. This gives an
Table 2 Average water-release for protein adsorption on PEI 1000-10 (pH 6.2) Modulating salt
LiCl NaCl KCl
Average water-release (mol/mol adsorbed protein) BSA
OVA
60 80 108
32 51 73
Temperature range: 298–313 K; salt range: 0.3–0.9 M.
Fig. 6. Effects of free energy change of protein surface (Gps ), protein–protein (Gpp ) and water-release (Gwr ) contributions on predictions of BSA isotherms on PAE 1000-10 anion exchanger (298 K, pH 6.2, 100 mM NaCl).
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isotherm consisting of two linear regions (Fig. 6) with a sharp discontinuity at the point of intersection. If protein–protein interactions are added to protein-surface contributions the predicted isotherm the prediction shifts as shown in Fig. 6. It should be noted that there is no saturation capacity additionally imposed, and the effect of water-release are not included; i.e., Gwr = 0. The model underestimates the experimental isotherm in this case. This result is consistent with those reported with the NISS model earlier [8]. This earlier work was one of the first modeling efforts to identify the importance of considering changes in entropy associated with protein adsorption on ion exchangers. Upon incorporation of the water-release term in Eq. (9), the isotherm prediction is as shown in Fig. 4. For clarity this is also shown in Fig. 6. In this particular case, the measured average release of 80 water molecules per molecule of adsorbed protein (Table 2) was used to estimate Gwr . It is observed that while the prediction generally matches well with the experimental isotherm there are regions, particularly in the transition from the linear to the plateau, where the data are under-predicted. This is true for all the cases studied (Figs. 4 and 5), and could originate from a number of sources is suspected to be due to the use of an average value of water-release. The Preferential Interaction Model gives only the average number of water molecules released over a range of modulator concentrations at protein loadings in the linear region. If a method could be developed to estimate the water-release as a function of protein loading the prediction is expected to improve. Other effects that may underlie these differences are changes in orientation and/or conformation of the protein with loading (though the latter is less likely for the specific cases studied). While this is not explicitly accounted for in the model, it would in part be indirectly incorporated if a method was available for measuring the amount of water released with protein loading. 5. Conclusions This paper illustrates the importance of accounting for entropic contributions to the free energy change associated with protein adsorption on ion exchangers. It is often assumed that electrostatic and van der Waal forces dominate the determination of isotherm capacity in protein ion-exchange. Using a colloidal model in combination with heat of adsorption measurements, it has been shown that protein capacity on ion exchangers is strongly limited by repulsive interactions between the adsorbed proteins. Attractive electrostatic and van der Waal forces between the ion exchanger and the protein, and the entropy increase accompanying water-release from the protein and adsorbent surface drive against the repulsive protein–protein surface forces. The reduction in free energy from the waterrelease can be significant in relation to the other energetic changes occurring during adsorption, as has been shown for the adsorption of two proteins, ovalbumin and bovine serum albumin, on a commercial anion exchanger. In fact, for all cases studied, heat of adsorption of the net negatively charged proteins on the positively charged ion exchanger is endothermic, and it is the increase in entropy that makes the adsorption energetically
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favorable. It has further been shown for these cases that the modulating salt (NaCl, LiCl or KCl) has a strong influence on the repulsive interactions between adsorbed proteins as well as on the amount of water released due to protein adsorption. Methods have been presented to effectively estimate these effects to obtain good predictions of isotherm behavior with the colloidal model. 6. Nomenclature
a protein radius Aps Hamaker constant Bps dimensionless parameter defined by Eq. (6) Cb bulk or solution protein concentration Cs surface protein concentration G free energy Gpp /kT dimensionless free energy change of repulsion between adsorbed proteins Gps /kT sum of dimensionless free energy change of electrostatic and van der Waal interactions between protein and adsorbent surface Gtotal /kT dimensionless free energy change due to waterrelease upon adsorption Gwr /kT dimensionless free energy change due to waterrelease upon adsorption h dimensionless gap distance (z/a) H enthalpy change of adsorption Hpp enthalpy change of adsorption from repulsion between adsorbed proteins k Boltzmann constant K colloidal equilibrium constant SAX strong anion exchanger T absolute temperature U internal energy Uel internal energy change from electrostatic interactions between protein and adsorbent Uvdw internal energy change from van der Waals interactions between protein and adsorbent ys dimensionless protein surface potential z distance between surface and sphere z0 distance of closest approach Greek letters ε dielectric constant of solution dielectric constant in free space εo γ γ = tanh((ys )/(4)) dimensionless parameter used to calculate potential Ψs dimensionless adsorbent surface potential κ Debye screening length Ω Ω = (ys − 4γ)/(2γ 3 ) dimensionless parameter used to potential References [1] G. Guiochon, S.G. Shirazi, A.M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, Boston, 1994. [2] C.A. Brooks, S.M. Cramer, AIChE J. 38 (1992) 1969. [3] Y. Li, N.G. Pinto, J. Chromatogr. A 702 (1995) 113.
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[12] J. Stahlberg, Anal. Chem. 66 (1994) 440. [13] G. Nemethy, H.A. Scheraga, J. Chem. Phys. 36 (1962) 1773. [14] R.J. Hunter, Zeta Potential in Colloid Science: Principles and Applications, third ed., Academic Press, London, 1981. [15] M.E. Thrash, J.M. Phillips, N.G. Pinto, Adsorption J. 10 (2004) 299. [16] M.E. Thrash, Doctoral dissertation, University of Cincinnati, Cincinnati, OH, 2003. [17] T.W. Perkins, D.S. Mak, T.W. Root, E.N. Lightfoot, J. Chromatogr. A 766 (1997) 1.