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Ind. Eng. Chem. Res. 2003, 42, 890-896
Evaluation of Three Kinetic Equations in Models of Protein Purification Using Ion-Exchange Membranes Heewon Yang† and Mark R. Etzel*,†,‡ Department of Chemical Engineering, 1415 Engineering Drive, and Department of Food Science, 1605 Linden Drive, University of Wisconsin, Madison, Wisconsin 53706
The Langmuir, steric hindrance, and spreading equations were evaluated separately in a mathematical model of protein purification using ion-exchange membranes. The spreading equation provided the best fit to experimental breakthrough curves (BTCs) for R-lactalbumin (ALA) and thyroglobulin (THY), followed by the steric hindrance equation, and finally the Langmuir equation. The intrinsic rate of protein adsorption to the membrane was found to be rate-limiting, whereas effects of liquid-phase mass transfer and flow nonidealities were negligible. An adsorption rate constant that decreased with increasing surface coverage was required to fit BTCs that were sharp initially and then broadened dramatically as the membrane approached saturation. Predictions using the spreading equation agreed with literature reports on the spreading of blood proteins such as fibrinogen on polymer surfaces. Introduction Membrane chromatography is a promising new separation technology used for biomolecules such as proteins, plasmid DNA, and viruses.1-10 Membrane chromatography can eliminate the disadvantages of traditional bead-based chromatography such as slow diffusion for large beads and high pressure drops for small beads. For large biomolecules having megadalton molecular weights such as viruses and plasmid DNA, membrane chromatography has a greater capacity than traditional chromatography.3,11 Membranes can be designed to eliminate the significance of slow mass transfer and nonideal flow on broadening of breakthrough curves (BTCs) during capture of a target biomolecule. Sharp BTCs would be expected to result. However, asymmetric BTCs are not uncommon, even for well-designed membrane systems.12-14 Initial breakthrough can be quite sharp, but BTCs broaden dramatically as the membrane approaches saturation. An adsorption rate constant that decreases as surface coverage increases has been suggested to explain this result.14 The Langmuir model does not allow the rate constant to decrease as surface coverage increases and fails to describe experimentally observed asymmetric BTCs. The steric hindrance model is a modification of the Langmuir model wherein the adsorption rate constant is multiplied by a term that decreases with increasing surface coverage.15 The premise of the steric hindrance model is that large molecules adsorb in a random pattern on the adsorption surface such that the maximum surface coverage at equilibrium is less than monolayer coverage, because the space between randomly adsorbed solutes is often too small for adsorption of another large molecule. The effect is greatest for large solutes and for surfaces with an excess of binding sites. * Corresponding author. E-mail:
[email protected]. Telephone: (608) 263-2083. Fax: (608) 262-6872. † Department of Chemical Engineering. ‡ Department of Food Science.
The latter condition is precisely the case for chromatographic membranes, which typically have a very high site density, so that one small protein molecule covers many dozens of sites. For large biological macromolecules such as viruses, plasmid DNA, and very large proteins, it would not be surprising to observe steric hindrance effects. The steric hindrance model has been shown to describe well asymmetric BTCs in traditional chromatography,15 but there are no reports of its application to membrane chromatography. The spreading model was developed to explain observations of anomalous protein adsorption onto polymer surfaces.16 Particularly in the blood-compatibility literature, proteins such as fibrinogen are reported to exhibit a maximum in the amount bound over time.17 This is known as the Vroman effect. Furthermore, the strength of attachment to the polymer surface increases with increasing time spent by the protein on the surface. The spreading model postulates that biomolecules can change orientation or conformation after adsorption, thus occupying a larger area. Because spreading is slow compared to initial adsorption, the spreading model can also describe an accelerated decrease in the rate of adsorption as a surface approaches saturation. However, there are no reports of the application of the spreading model to membrane chromatography. The goal of this work was to evaluate the Langmuir, steric hindrance, and spreading equations in a mathematical model of protein capture using ion-exchange membranes. Specifically, we examined the ability of each kinetic equation to describe the experimental asymmetric BTCs for ALA and THY. A broader aim was to develop a deeper understanding of the impact of anomalous adsorption in models on the performance of membrane chromatography systems. Mathematical Models Adsorption of protein in the membrane was modeled using the continuity equation for the flowing liquid
10.1021/ie020561u CCC: $25.00 © 2003 American Chemical Society Published on Web 01/21/2003
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phase, which, in dimensionless form, is18,19
∂C ∂C 1 ∂2C + ) -R ∂τ ∂ζ Pe ∂ζ2
(1)
with the initial and boundary conditions
C ) Cs ) 0 1 ∂C C)1 Pe ∂ζ ∂C )0 ∂ζ
for ζ g 0, τ ) 0 for ζ ) 0, τ > 0 for ζ ) 1, τ > 0
(2) Figure 1. Schematic illustration of the adsorption of a protein molecule: (a) steric hindrance model and (b) spreading model.
(3) (4)
Here, R is the dimensionless rate of protein binding to the membrane surface. When both the liquid-film masstransfer rate (RL) and the adsorption rate (mRA) are included, the rate and additional equations are
R ) RL ) nL(C - Ci)
(5)
dCi ) RL - mRA(Ci,Cs) dτ
(6)
dCs ) RA(Ci,Cs) dτ
(7)
When the liquid-film mass transfer can be ignored, the rate is
R ) mRA(C,Cs)
(8)
where C is substituted for Ci. Solution of the above equations requires information regarding the rate of adsorption of protein to the membrane surface, which will be discussed later. Experimental flow systems exhibit mixing and deadvolume effects external to the membrane itself. A simple method to describe these effects is the serial combination of the model of the membrane with those for a continuously stirred tank reactor (CSTR) and an ideal plug-flow reactor (PFR).18,19 Thus, the system volume is
Vsys ) VCSTR + VPFR
(10)
The PFR model shifts the system response by a delay time. The first temporal moment method can be used to determine VCSTR and VPFR.20 Because the PFR volume includes the void volume in the membrane (Vvoid), the pure delay time is due to the remaining PFR volume (VPFR - Vvoid). Langmuir Model. The simplest and most widely used adsorption kinetic model was proposed by Langmuir
RA )
n n C (1 - Cs) C m i m(r - 1) s
RA )
n C (1 - Cs)[1 - B1(R)Cs - B2(R)Cs2]2 m i n C (12) m(r - 1) s
where the coefficients B1 and B2 are functions of R
B1(R) )
0.7126R + 1.404R1.5 1 + 3.4363R + 2.4653R1.5
(13)
B2(R) )
0.07362R + 0.1204R1.5 1 + 0.5443R + 0.2725R1.5
(14)
Solutes are represented by hard spherical particles of diameter σ. The relative binding site density is the average number of sites contained within an area equal to one solute cross section
(9)
The CSTR model with the initial condition cout(t)0) ) 0 is
dcout Q ) (c - cout) dt VCSTR in
a binding site by an adsorbed molecule is assumed not to affect adsorption onto other sites of newly adsorbed molecules. This occurs when the binding site density is so low that the average distance between adjacent sites is larger than the diameter of the adsorbed molecule. Steric Hindrance Model. The Langmuir model is not valid for adsorption systems involving large molecules such as proteins, because occupation of a binding site excludes some adjacent empty sites from binding because of steric hindrance effects (Figure 1a). As binding approaches saturation, steric hindrance effects become substantial. A kinetic equation including the steric hindrance effect was proposed by Jin et al.15
(11)
This model is based on the assumption of a homogeneous adsorption surface. Furthermore, occupation of
R)
πσ2 F 4 site
(15)
In essence, the steric hindrance model is a modification of the Langmuir model such that the forward rate constant is multiplied by a term (1 - B1Cs - B2Cs2)2 that decreases with increasing surface coverage. The maximum surface coverage at equilibrium is less than monolayer coverage because a “jamming limit” is reached wherein all of the binding sites that remain unoccupied are too small for adsorption of an additional molecule. Steric hindrance is greatest for the adsorption of large molecules onto surfaces with high binding site densities (R ≈ ∞). Spreading Model. In both the Langmuir and steric hindrance models, proteins are not allowed to change conformation after adsorption. In reality, proteins can change orientation or conformation after adsorption. Initially, a protein might be captured by loosely binding to one or more sites on a surface and then binding to more adjacent sites while undergoing a conformational change such as flattening or spreading (Figure 1b).
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Table 1. Values of the Parameters Used in Fitting the Breakthrough Curves11 Q (mL/min)
L (cm)
dp (cm)
d (cm)
1.0
0.098
6.5 × 10-5
2.5
0.7
D (cm2/s)
c0 (M)
Vsys (mL)
VPFR (mL)
10-6
10-6
3.5
2.2
1.1 × (ALA) 2.5 × 10-7 (THY)
3.5 × (ALA) 7.6 × 10-8 (THY)
A model accounting for adsorption and changes in conformation was proposed by Lundstro¨m.16 If Cs1 and Cs2 are the adsorbed protein concentrations in states 1 and 2, respectively, in Figure 1b, then the kinetic equations are
RA )
dCs1 dCs2 + dτ dτ
(16)
n1 n2 dCs1 n1 ) Ci(1 - Cs1 - βCs2) C C dτ m m(r1 - 1) s1 m2 s1 (1 - Cs1 - βCs2) (17) n2 dCs2 n2 C (1 - Cs1 - βCs2) C ) dτ m2 s1 m(r2 - 1) s2
(18)
where β is the ratio of the area on the surface of state 1 to that of state 2. Parameter Estimation. To solve the equations for the model of ion-exchange membrane performance, the PDASAC (partial-differential-algebraic sensitivity analysis code) software package21 was used. The model equations were fitted to experimentally determined BTCs11 using a general regression software package.22 The fitting results are in terms of the most likely parameter value with a (95% confidence interval. Values of the membrane and flow system parameters used in simulating the BTCs are reported in Table 1. Results BTCs for capture of a small and a large protein, R-lactalbumin (ALA, 0.0144 MDa, 3.5-nm diameter) and thyroglobulin (THY, 0.66 MDa, 20-nm diameter), respectively, by an anion-exchange membrane were determined experimentally (Figure 2).11 The flat-sheet quaternary-amine (Q) membranes were prepared by immobilization of 2-aminoethyltrimethylammonium chloride (AETMA) to acylimidazole-activated microporous poly(vinylidene difluoride) membranes. The membranes were 140 µm thick and 25 mm in diameter. A stack of seven Q-membrane disks sandwiched between four blank disks was placed into a membrane holder, and feed solution consisting of 0.05 mg/mL protein in 50 mM Tris, pH 8.3, was loaded into the stack at a flow rate of 1 mL/min. Both BTCs were asymmetric, first rising sharply toward C ) 0.6-0.8 and then rising slowly toward, but never reaching, C ) 1.0. Even after 500 membrane volumes () τ) of feed solution had been loaded, ALA rose to C ) 0.986 and THY rose to C ) 0.924. The slow approach to saturation was somewhat less for ALA than THY. Breakthrough (C ) 0.1) occurred sooner for ALA (τ ) 93.6) than for THY (τ ) 232), corresponding to a 60% smaller dynamic binding capacity for the smaller protein. Initially, a small amount of early breakthrough was observed in each BTC. This was attributed to trace
Figure 2. Experimental11 and fitted BTCs for (a) ALA and (b) THY using an ion-exchange membrane.
UV-absorbing impurities in ALA and THY, which were used as received from Sigma.11 Three different kinetic equations were used in the BTC model and sequentially fit to the experimental observations for ALA and THY: (1) the Langmuir equation, (2) the steric hindrance equation (Figure 1a), and (3) the spreading equation (Figure 1b). Comparing the three kinetic equations, the spreading model provided the closest match to the experimental observations for both ALA and THY (Figure 2). This model was able to capture precisely both the fast initial rise and the slow approach to saturation. The steric hindrance model was the next best match to the experimental observations. For ALA, the steric hindrance model was equal to the spreading model for describing the rapid initial rise of the BTC, but it was less able to describe the slow approach to saturation. For THY, the steric hindrance model was unable to describe either the rapid initial rise in the BTC or the slow approach to saturation. Nevertheless, the steric hindrance model was superior to the Langmuir model, which failed for both ALA and THY to describe the asymmetric nature of the BTCs. Values of the fitted parameters are listed in Table 2. The Langmuir equation was fitted to the data using two adjustable parameters: ka and cl. The data were also fitted by including the boundary-layer mass-transfer coefficient in the liquid phase (kl) and dissociation rate constant (kd). The fitted value for kl was so large (>107 cm/s) and the value for kd was so small (107 cm/s) were so large and the value for kd was so small (10 000×) compared to the rate constant for spreading of bound protein (ka2 ) 0.35 M-1 s-1 for ALA and 1.7 M-1 s-1 for THY). The time scales calculated on the basis of values of ka1 for the intrinsic rate of protein binding in the initial portion of the BTC, when surface coverage was low, were 60 s for ALA and 180 s for THY. The time scale for spreading (∼1/ka2cs1) of THY was 4 h. Compared to the time scale for convection (∼20 s), it would be expected that the BTCs would be reasonably sharp initially for ALA and THY and would then broaden tremendously as the membrane approached saturation. Predictions made using the spreading equation matched precisely the experimental observations. Spreading of proteins bound to polymer surfaces has been commonly observed in the past, especially in bloodcompatibility research.17 In the Vroman effect, the amount of fibrinogen adsorbed to a polymer surface exhibits a maximum with time. The strength of attachment of fibrinogen to the surface increases with increasing residence time on the surface. The longer the fibrinogen remains adsorbed to the surface, the lower the recovery during elution. These observations were successfully modeled using the Lundstro¨m spreading equation. Similar phenomena appear to be occurring with adsorption of proteins to anion-exchange membranes. One difference among the three models for adsorption kinetics is that only the spreading model can predict a maximum in the amount of protein adsorbed to the membrane surface over time. The spreading model is the only one of the three models capable of describing the classic Vroman effect in blood-compatibility research. When adsorption of ALA and THY to an ionexchange surface was examined experimentally using surface plasmon resonance, the amount of THY ad-
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sorbed passed through a maximum, whereas the amount of ALA did not.23 Judging from this observation, the steric hindrance model might be able to describe the observed asymmetric BTCs for ALA but not for THY. An alternative hypothesis to the spreading model was examined as an explanation of the slow approach to saturation observed in the BTCs, specifically for the THY BTC where the spreading model was required to fit the data. By this hypothesis, the THY sample contained an impurity that bound more strongly to the membrane than THY, causing the slow displacement of bound THY and the observed slow rise to saturation late in the BTC. This behavior has been observed for BSA samples contaminated with ∼20% dimer.24,25 THY is present in nature as a dimer. The equivalent situation to BSA dimer contamination of BSA monomer would be for THY to contain high (∼20%) levels of tetramers. However, there is no evidence of THY tetramers in the literature, and SEC HPLC analysis of our THY sample revealed a single peak (data not shown). This does not prove the nonexistence of THY tetramers however, because the molecular weight of THY tetramers would be 1.32 MDa, which is beyond the capability of SEC HPLC. It does provide evidence of the nonexistence of lower-molecular-weight impurities in the sample. Furthermore, as explained above, we observed a maximum in the amount of THY adsorbed to a similar anionexchange surface using surface plasmon resonance. This result cannot be explained by the postulated existence of THY tetramer in the sample. Displacement of THY dimer by the larger THY tetramer would increase, not decrease, the total amount of THY adsorbed to the surface. Spreading of adsorbed protein is needed to explain the observation of a maximum. For these reasons, this alternative hypothesis was rejected. Whether adsorption follows the steric hindrance model or the spreading model will make a difference in the design and operation of industrial process chromatography systems. In industrial process chromatography, loading of feed solution is usually stopped at the point of breakthrough (Cs ) 0.1) to avoid loss of the target protein. At this point, the surface contains mostly unspread THY (Figure 3). For ALA, the surface would contain even more unspread protein, because ALA spreads much more slowly than THY. Spread protein is more likely to be denatured and to bind more tightly to the membrane surface, preventing complete recovery. Direct evidence for this is that recovery of bound protein in the elution peak was found to be 97% for ALA and 77% for THY.11 It might be advantageous in industrial practice to minimize spreading by using short loading times, short of the point of breakthrough, and to quickly elute bound protein after loading to reduce the time available for spreading. If spreading does not occur and the steric hindrance model is applicable, then short loading times would unnecessarily reduce the dynamic capacity. Conclusions Membrane chromatography has several advantages compared to traditional bead-based column chromatography. Membranes can eliminate mass-transfer and pressure-drop limitations inherent in traditional chromatography. This work confirms earlier findings of the absence of mass-transfer limitations in membrane chromatography. Furthermore, the effect of flow nonidealities on the shape of the BTC was shown to be negligible.
Static and dynamic capacities are therefore essentially equal for membrane chromatography systems. The slow approach to saturation, well after the point of breakthrough, is evidence of a decrease in the rate constant for binding as the membrane approaches saturation. This observation cannot be explained using the Langmuir equation, because this equation contains no mechanism for the rate constant to decline as the surface approaches monolayer coverage. The steric hindrance equation was much more successful in describing the observed behavior, because it includes a mechanism for the binding rate constant to decrease as the surface approaches the jamming limit where many of the remaining unoccupied binding sites are too small for adsorption of another molecule of protein. Yet, the steric hindrance equation also failed to precisely describe the experimental observations. Only the spreading equation was able to match the experimental observations. The rate of spreading was 5 times faster for THY than for ALA. Furthermore, the binding capacity was 3-4 times greater for THY than ALA, perhaps because larger proteins form thicker layers on the membrane surface.3,11 This work provides a theoretical framework for analyzing the impact of these effects on the performance of ion-exchange membranes and finding operating conditions and membrane designs that avoid undesirable phenomena while maintaining the inherent advantages of membrane chromatography. Acknowledgment Funding for this work was provided by the National Science Foundation (BES-9631962) and the University of Wisconsin College of Agricultural and Life Sciences. Symbols Latin Letters am ) 6/dp(1 - ) ) ratio of surface area to solid volume, cm-1 B1, B2 ) coefficients in the steric hindrance equation c ) solute concentration in the liquid phase, M ci ) interfacial solute concentration in the liquid phase, M cin ) input solute concentration for CSTR, M cout ) output solute concentration for CSTR, M cl ) membrane capacity based on the solid volume, M csj ) solute concentration in the solid phase based on the solid volume, M c0 ) solute concentration in the feed solution, M C ) c/c0 ) dimensionless solute concentration in the liquid phase Ci ) ci/c0 ) dimensionless interfacial solute concentration in the liquid phase Csj ) csj/cl ) dimensionless solute concentration in the solid phase D ) diffusion coefficient of the solute, cm2 s-1 d ) diameter of the membrane, cm dp ) average pore size of the membrane, cm kaj ) association rate constant, M-1 s-1 kdj ) dissociation rate constant, s-1 kl ) mass-transfer coefficient in the liquid boundary layer, cm s-1 Kdj ) kdj/kaj ) desorption equilibrium constant, M L ) length of the membrane, cm m ) (1 - )cl/(c0) ) dimensionless saturation capacity m2 ) (1 - )/ ) ratio of the membrane solid volume to void volume
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nj ) (1 - )clkajL/v ) dimensionless number of transfer units nL) (1 - )kLamL/v ) dimensionless number of transfer units in the liquid boundary layer Pe ) vL/ξ ) axial dispersion Peclet number Q ) flow rate, mL s-1 or mL min-1 rj ) 1 + c0/Kdj ) dimensionless separation factor R ) dimensionless reaction rate of protein to solid phase, RL or RA RA ) dimensionless reaction rate by adsorption RL ) dimensionless reaction rate by liquid-film mass transfer t ) time, s v ) interstitial liquid velocity, cm s-1 Vsys ) system volume, mL VCSTR ) CSTR volume, mL VPFR ) PFR volume, mL Vvoid ) void volume, mL z ) spatial coordinate, cm Subscripts j ) state index in the spreading model only Greek Letters R ) dimensionless binding site density β ) area ratio in the spreading equation ) membrane void fraction ζ ) z/L ) dimensionless spatial coordinate ξ ) axial dispersion coefficient, cm2 s-1 Fsite ) ion-exchange site density, cm-2 σ ) solute diameter, cm τ ) vt/L ) dimensionless time
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(9) Warner, T.; Kostel, P. Mustang Q Disposable Capsules and Cartridges. Genet. Eng. News 2000, 20, 1. (10) Zeng, X.; Ruckenstein, E. Membrane Chromatography: Preparation and Applications to Protein Separation. Biotechnol. Prog. 1999, 15, 1003. (11) Yang, H.; Viera, C.; Fischer, J.; Etzel, M. R. Purification of a Large Protein Using Ion-Exchange Membranes. Ind. Eng. Chem. Res. 2002, 41, 1597. (12) Dancette, O. P.; Taboureau, J.-L.; Tournier, E.; Charcosset, C.; Blond, P. Purification of Immunoglobulins G by Protein A/G Affinity Membrane Chromatography. J. Chromatogr. A 1999, 723, 61. (13) Kochan, J. E.; Wu, Y.-J.; Etzel, M. R. Purification of Bovine Immunoglobulin G via Protein G Affinity Membranes. Ind. Eng. Chem. Res. 1996, 35, 1150. (14) Serafica, G. C.; Pimbley, J.; Belfort, G. Protein Fractionation Using Fast Flow Immobilized Metal Chelate Affinity Membranes. Biotechnol. Bioeng. 1994, 43, 21. (15) Jin, X.; Talbot, J.; Wang, N.-H. L. Analysis of steric hindrance effects on adsorption kinetics and equilibria. AIChE J. 1994, 40, 1685. (16) Lundstro¨m, I. Models of protein adsorption on solid surfaces. Prog. Colloid Polym. Sci. 1985, 70, 76. (17) Slack, S. M.; Horbett, T. A. Changes in the Strength of Fibrinogen Attachment to Solid Surfaces: An Explanation of the Influence of Surface Chemistry on the Vroman Effect. J. Colloid Interface Sci. 1989, 133, 148. (18) Yang, H.; Bitzer, M.; Etzel, M. R. Analysis of Protein Purification Using Ion-Exchange Membranes. Ind. Eng. Chem. Res. 1999, 38, 4044. (19) Suen, S.-Y.; Etzel, M. R. A Mathematical Analysis of Affinity Membrane Bioseparations. Chem. Eng. Sci. 1992, 47, 1355. (20) Lightfoot, E. N.; Lenhoff, A. M.; Rodriguez, R. L. Use of Moments to Characterize Mass Transport in Steady Flows of Arbitrary Complexity. Chem. Eng. Sci. 1982, 37, 954. (21) Caracotsios, M.; Stewart, W. E. Sensitivity Analysis of Initial-Boundary-Value Problems with Mixed PDEs and Algebraic Equations. Comput. Chem. Eng. 1995, 19, 1019. (22) Stewart, W. E.; Caracotsios, M.; Sı´rensen, J. P. General Regression Software Package for Non-Linear Parameter Estimation, GREG; University of Wisconsin: Madison, WI, 1994. (23) Fischer, J. Separation of Nanometer-Sized Biological Particles Using Membrane Chromatography: An Analysis of Adsorption Kinetics. M.S. Thesis, University of WisconsinMadison, Madison, Wisconsin, 2000. (24) Weinbrenner, W. F.; Etzel, M. R. Competitive Adsorption of Alpha-Lactalbumin and Bovine Serum Albumin to a Sulfopropyl Ion-Exchange Membrane. J. Chromatogr. A 1994, 662, 414. (25) Hunter, A. K.; Carta, G. Effects of Bovine Serum Albumin Heterogeneity on Frontal Analysis with Anion-Exchange Media. J. Chromatogr. A 2001, 937, 13.
Received for review July 25, 2002 Revised manuscript received December 4, 2002 Accepted December 11, 2002 IE020561U
