Langmuir 2008, 24, 5991-5995
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Multiscale Modeling of Protein Uptake Patterns in Chromatographic Particles A. M. Lenhoff* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, UniVersity of Delaware, Newark, Delaware 19716 ReceiVed February 6, 2008. ReVised Manuscript ReceiVed April 28, 2008 A parallel diffusion model is presented to explain an apparent transition in uptake mechanism seen in experimental observations of protein uptake into porous adsorbents. While such models have been invoked previously, this mesoscopic description is augmented here by miscroscopic models for representing surface diffusion by ”hopping” and adsorption within a Gibbs surface excess formulation. These contributions lead to a relation for the apparent protein diffusivity as a function of adsorption conditions, which can be used predictively with knowledge of a few readily measured physical quantities. The approach can be useful in seeking optimal conditions for preparative protein chromatography separations.
Introduction The rate of diffusion in porous media depends quantitatively on numerous factors, but the qualitative characteristics of the transport processes are generally not significantly affected. This situation changes appreciably, however, if transport is coupled with adsorption in which the adsorption isotherm is nonlinear. Under these conditions, the relative contributions to the overall solute flux of the free solute in the pore lumen (pore diffusion) and of the adsorbate (variously referred to as solid or homogeneous diffusion)1 may give rise to differences in predicted uptake patterns. For pore diffusion with a near-rectangular isotherm, the concentration front moving into the adsorbent is extremely sharp, giving rise in the limit to the shrinkingcore model.2,3 If diffusion of the adsorbed material is dominant, however, the uptake behavior is effectively Fickian, and the uptake front is diffuse. The reliable prediction of transport behavior requires knowledge of qualitative factors such as which transport mechanism is dominant and quantitative methods of estimating the relevant transport coefficients. A model for such predictions is presented here for the specific case of protein uptake into chromatographic particles, for which the practical implications are important in view of the major role of chromatographic steps in protein purification in biotechnology. Despite the pronounced differences in predicted behavior for the two limiting uptake models, it is often difficult to distinguish them macroscopically in typical experimental systems such as packed columns or batch uptake measurements in stirred tanks. This occurs because the models give rise to similar profiles of fractional uptake observable externally.4,5 More recently, however, concentration profiles consistent with the different models of uptake have been observed directly by confocal microscopy6–10 and optical * Phone: (302) 831-8989. Fax (302) 831-4466. E-mail:
[email protected]. (1) Hall, K. R.; Eagleton, L. C.; Acrivos, A.; Vermeulen, T. Ind. Eng. Chem. Fundam. 1966, 5, 212–223. (2) Teo, W. K.; Ruthven, D. M. Ind. Eng. Chem. Process Des. DeV. 1986, 25, 17–21. (3) Pinto, N. G.; Graham, E. E. React. Polym. 1987, 5, 49–53. (4) Weaver, L. E.; Carta, G. Biotechnol. Prog. 1996, 12, 342–355. (5) Chang, C.; Lenhoff, A. M. J. Chromatogr., A 1998, 827, 281–293. (6) Wesselingh, J. A.; Bosma, J. C. AIChE J. 2001, 47, 1571–1580. (7) Dziennik, S. R.; Belcher, E. B.; Barker, G. A.; DeBergalis, M. J.; Fernandez, S. E.; Lenhoff, A. M. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 420–425. (8) Hubbuch, J.; Linden, T.; Knieps, E.; Thommes, J.; Kula, M. R. J. Chromatogr., A 2003, 1021, 105–115. (9) Dziennik, S. R.; Belcher, E. B.; Barker, G. A.; Lenhoff, A. M. Biotechnol. Bioeng. 2005, 91, 139–153.
microscopy,11,12 allowing more careful characterization of transport behavior. The focus here is on protein uptake into ion-exchange adsorbents, in which adsorption is by electrostatic interactions between the protein and a charged adsorbent surface and the solution ionic strength is used to manipulate adsorption behavior. Several characteristic trends in intraparticle protein transport emerge from both macroscopic and microscopic observations. If macroscopic uptake data are fitted to the shrinking-core model regardless of the actual form of the concentration profiles, the apparent pore diffusivity tends to increase with decreasing solution protein concentration4,9 and with increasing salt concentration7,9 or, equivalently, with weaker retention.6 The latter effect is also apparent in the sharpness of column breakthrough fronts with increasing salt concentration.9,13 When microscopic observations of intraparticle concentration profiles are available, they indicate a change from sharp fronts typical of the shrinking-core mechanism at low salt concentration to diffuse fronts consistent with homogeneous diffusion at high salt concentration.7–9 The transition appears to be a function of how strongly the protein is bound to the surface,6,8,14 an observation that has been quantified in terms of the chromatographic retention factor k′: the transition occurs at log k′ values of around 3.5 to 414. Although these patterns of behavior have been observed quite widely, a general mechanistic basis has not yet been demonstrated. The pore diffusion analysis yields diffusivity values that are plausibly related to the free-solution diffusivities, and the lower values with decreasing ionic strength have been suggested to be due to electrostatic repulsion between free and adsorbed protein molecules.9,13 However, for small proteins on relatively widepore adsorbents this is likely to be significant only at very low ionic strengths. The diffusivity values obtained from the homogeneous diffusion model are less easily interpreted; one suggestion is that there is an electrokinetic contribution to transport7, but this is unlikely to explain the magnitude of the (10) Zhou, X. P.; Li, W.; Shi, Q. H.; Sun, Y. J. Chromatogr., A 2006, 1103, 110–117. (11) Russell, S. M.; Belcher, E. B.; Carta, G. AIChE J. 2003, 49, 1168–1177. (12) Stone, M. C.; Carta, G. J. Chromatogr., A 2007, 1160, 206–214. (13) Harinarayan, C.; Mueller, J.; Ljunglof, A.; Fahrner, R.; Van Alstine, J.; van Reis, R. Biotechnol. Bioeng. 2006, 95, 775–787. (14) Langford, J. F., Jr.; Xu, X.; Yao, Y.; Maloney, S. F.; Lenhoff, A. M. J. Chromatogr., A 2007, 1163, 190–202.
10.1021/la8004163 CCC: $40.75 2008 American Chemical Society Published on Web 05/16/2008
5992 Langmuir, Vol. 24, No. 12, 2008
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Figure 1. Effect of surface diffusion contribution on intraparticle adsorbed concentration profiles as a function of β. Labels indicate the value of dimensionless time τ. All results shown are for R ) 0.01 and R ) 100, but the results are quite insensitive to these two parameters as long as adsorption is highly favorable.
trends or to provide a straightforward means to estimate diffusivities a priori. The purpose of this letter is therefore to present a new interpretation, specific to the ion-exchange uptake of proteins, that is consistent with the experimental observations outlined above and provides a basis for predicting both the apparent uptake mechanism and the effective diffusivity values. The approach is based on combining models on different scales, viz., a mesoscopic (adsorbent particle level) model and a microscopic (adsorbent surface level) model. The former is a widely used analysis of uptake in adsorption to which the present analysis adds little, but combining this with a molecular interpretation of adsorption and diffusion yields a predictive basis for modeling protein uptake.
Although parallel diffusion has been widely used to describe uptake in preparative chromatography,6,16,18,19 the concentration profiles predicted by the model have only rarely been reported.6,20 Figure 1 shows a more systematic parametric examination of how the profiles vary with β for the case of R ) 0.01 and a nearly rectangular Langmuir isotherm in the form1,15
c ⁄ c0 q ) q0 R + (1 - R)(c ⁄ c0)
(3)
as well as any parameters appearing in the isotherm equation used. In eq 2, c0 is a characteristic solution-phase concentration, and q0 is the corresponding adsorbed concentration.
with R ) 0.01. The solutions were obtained by numerical integration using function pdepe in Matlab. Because all of the plots are shown as a function of dimensionless time τ ) tDp/r02, where r0 is the particle radius, increasing β corresponds to increasing values of Ds. For small β, the concentration profiles are extremely sharp, consistent with the shrinking-core model, but with increasing β, the profiles become more diffuse and uptake becomes faster, as expected from the increasing values of Ds. The transition from sharp to diffuse fronts as observed by microscopy methods appears to occur around β ≈ 1. These results change little with changes in the isotherm as long as a sharp, favorable isotherm is used, as usually applies in protein uptake in ion exchange, and changing R primarily changes the time scale as long as R . 1. The predictions of uptake based on the parallel diffusion model can be compared with corresponding ones using alternative models,15 for which the shrinking-core model is convenient in view of its simplicity and the availability of solutions for singleparticle and batch uptake as well as column breakthrough. The predictions using the two models, in terms of fractional uptake vs dimensionless time, are shown in Figure 2 for β ) 1 and the other parameters as in Figure 1. The shrinking-core model with the apparent pore diffusivity Da/Dp ) 1.75 is in excellent agreement with the parallel diffusion model despite the clear inapplicability of the shrinking-core model assumption of perfectly sharp fronts at such a high β value (Figure 1). Indeed, similarly good agreement is obtained for β values up to at least 10. The good agreement in terms of the “lumped” measure of fractional uptake despite the obvious mismatch in intraparticle concentration profiles presumably results from the unimodal
(15) Yoshida, H.; Yoshikawa, M.; Kataoka, T. AIChE J. 1994, 40, 2034–2044. (16) Hunter, A. K.; Carta, G. J. Chromatogr., A 2000, 897, 81–97. (17) Suzuki, M. Adsorption Engineering; Kodanshar: Tokyo, 1990.
(18) Natarajan, V.; Cramer, S. Sep. Sci. Technol. 2000, 35, 1719–1742. (19) Farnan, D.; Frey, D. D.; Horvath, C. J. Chromatogr., A 2002, 959, 65–73. (20) Pritzker, M. D. Chem. Eng. Sci. 2003, 58, 473–478.
Mesoscopic Model: Parallel Diffusion The mesoscopic model used to describe protein uptake is a straightforward combination of pore and surface diffusion contributions into a parallel diffusion model.6,15,16 The concentrations c of the protein in the pore and q of adsorbed protein as a function of time t and radial position r are governed by
εp
∂c ∂q 1 ∂ ∂c 1 ∂ ∂q + Ds 2 r2 + ) εpDp 2 r2 ∂t ∂t r ∂r ∂r r ∂r ∂r
( )
( )
(1)
and associated boundary and initial conditions, as well as an adsorption isotherm relating q to c. Here, εp is the effective particle porosity, Dp is the pore diffusivity, and Ds is the surface diffusivity. The method of moments can be used to extract parallel diffusion parameters from chromatographic data obtained under linear isotherm conditions,17 but this approach is inapplicable under the higher adsorbate loads and stronger binding used in preparative chromatography. For the latter situation, scaling of the governing conservation equations15 shows that uptake behavior is determined by two dimensionless parameters
R)
q0 Ds and β ) R εpc0 Dp
(2)
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Langmuir, Vol. 24, No. 12, 2008 5993
However, because Ds is treated largely as an adjustable parameter in this analysis, it is difficult to reach definitive mechanistic conclusions or, more importantly, to make quantitative predictions of uptake. It is for this reason that microscopic models of adsorption and transport at the adsorbent surface are considered.
Microscopic Modeling: Adsorption and Surface Diffusion
Figure 2. Calculated fractional uptake as a function of dimensionless time for the parallel diffusion model with β ) 1 and other parameter values as in Figure 1 (solid line), compared with predictions of the shrinking-core model using an apparent pore diffusivity of Da ) 1.75Dp (dashed line).
Figure 3. Values of Da/Dp obtained by least-squares fits of uptake profiles such as those in Figure 2 for different values of β (solid line). The dashed line shows eq 4.
character of the uptake curves predicted by both models, allowing the parametrization of uptake predictions in terms of either model. Therefore the shrinking-core model with an appropriately adjusted pore diffusivity Da can provide an excellent description of uptake even for the more complex parallel diffusion mechanism. Figure 3 shows that the apparent diffusivity can be related straightforwardly to β; these results are well described by the simple relation
Da ) 1 + 0.6β Dp
(4)
Again, these results are with the other parameter values as shown in Figure 1, but they vary little if those values are changed as long as the isotherm remains highly favorable and R. 1. The factor of 0.6 is the same as that obtained in relating a homogeneous diffusivity to the shrinking-core pore diffusivity for a set of experimental data.4 That this factor is
