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Protein transport through porous membranes: effects of colloidal interactions
Colloids and Surfaces A: Physicochemical and Engineering Aspects 138 (1998) 133–143
Protein transport through porous membranes: effects of colloidal interactions Andrew L. Zydney *, Narahari S. Pujar Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA
Abstract Traditional analyses of membrane transport have generally considered only steric (size-based ) interactions between the solute and pores. Recent experimental work has clearly demonstrated the importance of longer-range colloidal interactions on the rate of solute transport. These interactions can dramatically affect membrane performance, changing the membrane from being almost completely permeable to completely retentive for a given solute. This manuscript discusses several applications of colloidal interaction theory to membrane systems, highlighting the tremendous insights that have been obtained into the performance of real membrane systems as well as the limitations of available theoretical descriptions. New calculations are presented for the transport of charged solutes through porous membranes which explicitly account for the effects of charge regulation arising from the dissociation of specific ionizable surface groups. The importance of non-equilibrium phenomena in membrane systems is also examined. Particular emphasis is placed on the use of membrane systems for protein separations, including the importance of protein structure and biochemistry on membrane selectivity. © 1998 Elsevier Science B.V. Keywords: Electrostatics; Membrane; Protein; Separations; Ultrafiltration
1. Introduction Traditional analyses of membrane transport have generally considered only steric (size-based) interactions between the solute and pores. Steric interactions can be effectively used to separate (1) macromolecular solutes from solvent, and (2) cells or large particles from macromolecules. Membrane systems in the biotechnology industry have thus been used extensively for protein harvest from cell suspensions, protein concentration (i.e. buffer removal ), and buffer exchange. It has not, however, been possible to obtain high selectivity for protein–protein separations using purely steric interactions [1], with the net result that membrane * Corresponding author. Tel: (+1) 302 831 2399; Fax: (+1) 302 831 1048; e-mail: [email protected] 0927-7757/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0 9 2 7 -7 7 5 7 ( 9 6 ) 0 3 92 8 - 3
systems have not been developed for actual protein purification. Recent experimental work has demonstrated that it is possible to improve dramatically the selectivity of membrane systems by exploiting longrange colloidal interactions between the solutes and membrane pores. For example, Saksena and Zydney [2] obtained separation factors of more than 30 for the separation of albumin (BSA) and immunoglobulins (IgGs) by operating the system at pH 4.7 and low salt concentration. The high selectivity obtained under these conditions was attributed to the electrostatic ‘‘exclusion’’ of the charged IgG molecules from the membrane pores. It was even possible to reverse the selectivity in this system, obtaining greater throughput of the larger IgG molecules, by operating at the isoelectric point of IgG so that there were strong electro-
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static interactions between the charged BSA molecules and the membrane pores. Higuchi et al. [3] have also shown good separation of BSA–IgG mixtures, with the high selectivity attributed to non-electrostatic interactions between the proteins and hydrophobic groups on the surface-modified polysulfone membranes. Nakatsuka and Michaels [4] examined the separation of myoglobin from BSA using both sorptive and non-sorptive membranes. In this case, the high selectivity obtained with the non-sorptive membranes was attributed to the different extents of protein adsorption caused by the combined electrostatic and van der Waals interactions. In order to develop more effective membrane systems for protein separations, it is critical to have a fundamental understanding of the nature of the colloidal interactions that govern protein transport through porous membranes. This manuscript discusses several applications of colloidal interaction theory in membrane systems, highlighting the tremendous insights that have been obtained into the performance of real membrane devices as well as the limitations of the available theoretical descriptions. New calculations are presented for the transport of charged solutes through narrow pores which explicitly account for the effects of charge regulation arising from the dissociation of ionizable groups on the protein surface. Particular emphasis is placed on those factors which affect the overall membrane selectivity, including a discussion of how to exploit these colloidal interactions in the development of improved membrane processes.
2. Theoretical background Solute transport in most membrane systems is dominated by convection. The convective solute flux across the membrane can be written as [5,6 ] N =wK V (1) 9C s c w where w is the equilibrium solute partition coefficient between the solution immediately upstream of the membrane and that in the membrane pores, K is the hindrance factor for convective solute c transport through the pore, V 9 is the average fluid
velocity through the membrane pore, and C is w the solute concentration in the solution immediately upstream of the membrane. The rate of solute transport through the membrane is thus governed by both thermodynamic and hydrodynamic interactions. The partition coefficient for a spherical solute in a cylindrical pore can be expressed in terms of the energy of interaction between the solute and the pore boundary as [5,7] w=
P
A
−Y rp total exp r2 kT p 0 2
B
r dr
(2)
where r is the pore radius, r is the radial coordip nate within the pore, k is Boltzmann’s constant, and T is the absolute temperature. The total interaction potential has contributions from steric (hard-sphere), electrostatic, and van der Waals forces: Y
total
=Y +Y +Y HS E VDW
(3)
The hindrance factor for convection can also be expressed as an integral over the pore radius [5,7]: rp−rs K= c
P
GV exp
0 rp−rs Vr dr 2
A P 0
A
rp−rs
−Y −Y E VDW kT
BC P A 0
exp
B
r dr
B D
−Y −Y E VDW r dr kT
(4) where V is the local fluid velocity in the pore and G is the lag coefficient which accounts for the additional hydrodynamic drag on the solute associated with the presence of the pore wall. The integrals in Eq. (4) are evaluated from r=0 to r=r −r to account for the steric (hard-sphere) p s exclusion of the solute from the region within one solute radius of the pore wall. Most previous studies have evaluated K using the centerline c approximation, in which both G(r) and Y +Y are assumed to be constant at their E VDW centerline (r=0) values [8]. Under these conditions, the exponential terms in the numerator and the denominator can be factored out of the integ-
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rals with the net result that K is independent of c Y and Y . The validity of this approximation E VDW for systems with complex interaction potentials has yet to be determined. In addition, although theoretical expressions for G(r) have been available for systems with purely hard-sphere interactions for more than 20 years [8,9], there are currently no corresponding calculations that account for the effects of long-range (e.g. electrostatic) interactions on the hydrodynamics. Pujar and Zydney [10] have obtained expressions for the enhanced drag coefficient K and the diffusional hindrance factor K for spherical solutes in a concentric spherical d cavity, but these results cannot be directly extended to systems with cylindrical pores or to the evaluation of the hindrance factor for convection K c because of the complex coupling between the electrostatics and hydrodynamics in these systems. 2.1. Hard-sphere interactions The equilibrium partition coefficient and convective hindrance factor for a system with purely hard-sphere interactions can both be expressed as unique functions of the ratio of the solute to pore radius, l=r /r . The partition coefficient is evals p uated directly from Eq. (2), yielding w=(1−l)2
(5)
which accounts for the excluded volume in the outer (annular) region of the cylindrical pore. The corresponding result for a membrane with slitshaped pores is w=1−l
(6)
where l is now the ratio of the solute radius to the slit half-width. Bungay and Brenner [9] developed an expression for K (l) for a hard sphere in c a cylindrical pore using matched asymptotic expansions for both small and close-fitting spheres. Corresponding results for slit-shaped pores are discussed by Deen [8]. The product wK , often referred to as the asympc totic membrane sieving coefficient S , is shown as 2 a function of the scaled solute radius in Fig. 1. Results are shown for a membrane with a single cylindrical pore radius (s/r:=0), for a membrane with a single slit-shaped pore, and for two mem-
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Fig. 1. Asymptotic sieving coefficients for membranes with different pore size distributions evaluated assuming purely hardsphere interactions. Results are plotted as a function of the scaled solute radius, where R* is defined as the radius of the solute for which S =0.5. Dashed line is for a membrane with 2 uniform slit-shaped pores. Open symbols represent experimental data for dextran transport through a polyethersulfone membrane [11].
branes with different log-normal distributions of cylindrical pores (with s and r: the standard deviation and mean for the log-normal distribution) [12]. In each case, the solute radius has been made dimensionless by R*, which is defined as the radius of the solute for which S =0.5. This non-dimen2 sionalization highlights the effect of the pore shape (cylinder vs. slit) and pore size distribution on the sieving characteristics, and it also facilitates comparison with experimental data. The asymptotic sieving coefficient decreases with increasing r , s attaining a value of zero when r /R*=2.40 for a s membrane with a single cylindrical pore size and when r /R*=1.67 for a membrane with a slits shaped pore. The curve for the slit-shaped pore is significantly steeper than the results for a cylindrical pore at large r /R* due to the different depens dences of the partition coefficient and hindrance factor on l. The asymptotic sieving coefficients for the membranes with the log-normal pore size distributions approach zero asymptotically as r 2 due to the presence of a finite number of s very large pores in the log-normal distribution. S for the larger solutes (r /R*>1) increases with 2 s an increase in the breadth of the pore size distribu-
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tion (increasing s/r:), with this effect becoming quite pronounced at large r . This is due to the s very significant contribution to solute transport from the largest pores in the distribution arising from the very high fluid velocity in those pores [12]. The results in Fig. 1 suggest that the asymptotic sieving coefficients for small solutes are actually greater in membranes with narrow pore size distributions. This behavior is really just an artifact of plotting the results as a function of r /R*. For s example, if the S values had been plotted as a 2 function of r (or r /r:) the asymptotic sieving s s coefficients would increase monotonically with increasing s/r: for all values of the solute radius. The open symbols in Fig. 1 represent experimental data obtained by Mochizuki and Zydney [11] for the transport of polydisperse neutral dextrans ( long-chain polysaccharides) through an asymmetric polyethersulfone membrane with a nominal molecular weight cut-off of 100 000. The data are well represented by the theoretical curve with s/r:= 0.5, which is also consistent with independent estimates of the breadth of the pore size distribution in this type of polymeric membrane [12]. The good agreement between the model and data indicate that the cylindrical pore model provides a reasonable description of solute transport through these asymmetric membranes. However, these results should not be considered as proof that this membrane actually has a log-normal pore size distribution of cylindrical pores, since many different types of pore size distribution could potentially produce sieving curves very similar to those shown in Fig. 1. The results in Fig. 1 can be used to evaluate the membrane selectivity, which is simply defined as the normalized solute flux (N /C ) for a given s w solute divided by the corresponding flux for a second solute. For example, a membrane with a single pore size (s/r:=0) would have asymptotic sieving coefficients of S =0.2 and 0.8 (correspond2 ing to a selectivity of four) for two solutes with a size ratio of about 2.7. This would correspond to two proteins having a molecular weight ratio of approximately 10, which has generally been regarded as the minimum size difference required for effective protein separation using a purely sizebased membrane separation. Even greater size
ratios would be required for membranes with broader pore size distributions. For example, the membrane with s/r:=0.5 would yield S =0.2 and 2 0.8 for two solutes which differ in size by a factor of 3.6, while the membrane with s/r:=1 would provide an equivalent selectivity only if the two solutes differed by at least a factor of five in radius (corresponding to a factor of 100 in molecular weight). It is important to note that the selectivity could be increased significantly by using a membrane with a smaller average pore size, although this would be at the expense of a much smaller overall throughput. For example, a membrane with a single pore size would yield S =0.05 and 0.2 2 (which also corresponds to a selectivity of four) for two solutes that differ in radius by only a factor of 1.3. The low throughput obtained with this small pore membrane can be largely compensated for by employing a diafiltration cascade, in which the protein with the larger S is effectively 2 washed away from the more highly retained species. This type of separation process is discussed in more detail by van Reis et al. [13]. 2.2. Electrical interactions Smith and Deen [14,15] developed expressions for Y (r), the electrostatic potential energy of E interaction, by solving the linearized Poisson– Boltzmann equation for the ion concentrations and electrical potential around a single spherical solute in an infinitely long cylindrical pore. The total energy of interaction between the charged sphere and the pore, assuming constant surface charge density boundary conditions, can be expressed as the sum of three terms [14,15]: Y =A s2 +A s2 +A s s (7) E 1 pore 2 solute 3 pore solute where s and s are the surface charge densipore solute ties of the pore wall and solute respectively, and A , A , and A are all positive coefficients which 1 2 3 depend upon the solution ionic strength, the pore radius, and the solute radius. The first two contributions to the total energy of interaction arise from the deformation of the electrical double layer surrounding the pore wall and the solute respectively. These interactions always increase, the total free energy of the system, leading to a reduction
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in the partition coefficient and, in turn, the rate of protein transport through the membrane. The final term in Eq. (7) is associated with charge–charge interactions between the solute and pore wall. This term is positive when the solute and pore have like charge and negative when the solute and pore have opposite charge. It is important to note that the second term in Eq. (7) provides the dominant contribution to the electrostatic interaction potential if s is small, which can lead to a net pore repulsive-type electrostatic interaction even when the solute and pore have opposite charge. The effects of electrical interactions on membrane transport are examined in Fig. 2. The calculations were performed assuming that K is c unaffected by Y , although the validity of this E approximation has not been clearly established. Results are shown using constant surface charge density boundary conditions with s =−0.015 C m−2 (similar to the surface charge s on BSA) and s =0 (top panel ) and p s =−10−4 C m−2 (bottom panel ). Calculations p are shown at kr =0.1, 1, and 10, where k−1 is the p ˚ . The results double layer thickness and r =46 A p are again plotted as a function of the scaled solute radius. In this case, R* increases with increasing kr due to the reduction in the magnitude of the p electrostatic interactions at high salt concentrations. The asymptotic sieving coefficients at kr =10 are essentially identical to those seen in p Fig. 1 due to the strong electrostatic shielding provided by the supporting electrolyte under these conditions. The breadth of the sieving curve for s =0 (top panel ) decreases significantly with p decreasing kr due to the much larger relative p increase in the free energy associated with the deformation of the electrical double layer surrounding the larger solutes. The net result is a significant increase in the membrane selectivity at low solution ionic strength. For example, the membrane shown in the top panel of Fig. 2 would yield S =0.2 and 0.8 for solutes differing by a factor 2 of 2.7 in radius when kr =10, but would yield the p same sieving coefficients for two solutes differing by only a factor of 1.74 in radius when kr =0.1. p These calculations suggest that it should be possible to increase significantly the selectivity for the separation of protein oligomers (e.g. dimers from
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Fig. 2. Effect of electrostatic interactions on the asymptotic sieving coefficients. Top panel shows results for an uncharged pore while bottom panel shows results for s =−10−4 C m−2. p Calculations were performed using constant surface charge density boundary conditions at several values of kr with ˚ . Results are plotted as pa funcs =−0.015 C m−2 and r =46 A s p tion of the scaled solute radius, where R* is defined as the radius of the solute for which S =0.5. 2
monomers) using an electrically ‘‘neutral’’ membrane by operating at low salt concentrations and at a pH away from the protein isoelectric point. This behavior is consistent with experimental studies by van Reis et al. [13] for the separation of BSA monomers and dimers. The theoretical results when s =−10−4 C m−2 p
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(bottom panel ) are markedly different. In this case, the sieving profiles at kr =0.1 and kr =10 p p are almost identical, with only a slightly higher selectivity obtained at kr =1. The potential energy p of interaction when s =−10−4 C m−2 is deterp mined largely by the charge–charge interactions between the solute and the pore wall. These repulsive interactions cause a similar percentage reduction in w for both large and small solutes; thus, the sieving profile (when plotted vs. r /R*) shows s a very weak dependence on the solution ionic strength. The slight increase in membrane selectivity at the intermediate salt concentration (kr =1) p is due to the increase in the potential energy of interaction associated with the deformation of the double layer surrounding the solute molecule, with this effect being more significant for the larger solutes. The effects of solution ionic strength on the asymptotic sieving coefficient are examined in more detail in Fig. 3. The filled symbols represent experimental data obtained by Pujar and Zydney [7] for BSA transport through an asymmetric polyethersulfone membrane at pH 7. The asymptotic sieving coefficient decreases by more than two
Fig. 3. Asymptotic sieving coefficient as a function of the dimensionless pore radius for calculations using constant surface charge density, constant surface potential, and charge regula˚ and r =46 A ˚ . Filled tion boundary conditions with r =30 A s p symbols represent experimental data for BSA transport through a polyethersulfone membrane [7].
orders of magnitude as the solution ionic strength is reduced from 0.15 to 0.0015 M (i.e. as kr is p reduced from 5.9 to 0.59). This change in solution environment thus converts the membrane from one which is reasonably permeable to BSA (S >0.1) to one which is almost completely reten2 tive for this protein (S <0.001). The solid curves 2 in Fig. 3 represent the model calculations using constant surface potential, constant surface charge density, and charge regulation boundary conditions. The charge regulation model is discussed in more detail below. The values of the surface potential f and surface charge density s in free solution s s were evaluated at each ionic strength by calculating the degree of ionization of the different dissociable groups in the BSA molecule (a-carboxyl, b,c-carboxyl, imidazole, a-amino, e-amino, phenolic, and guanidine) using the model developed by Tanford et al. [16 ], with the extent of Cl− adsorption determined using a modified form of the three-site binding model developed by Scatchard et al. [17]. ˚ and All calculations are shown using r =30 A s ˚ r =46 A [7]. The asymptotic sieving coefficients p evaluated using the constant surface charge density boundary condition decrease sharply with decreasing kr due to the strong electrostatic repulsion s between the like-charged solute and pore. In contrast, the constant surface potential model predicts an increase in S with decreasing kr due to the 2 p development of an induced (positive) charge on the pore wall for interactions at constant potential. All previous analyses of electrostatic interactions in membrane systems have used either the constant surface charge density or constant surface potential boundary conditions [7,8,14,18]. In actuality, both the surface charge and surface potential of the protein will vary as the protein enters the pore: this is due to the alteration in the electrochemical environment around the protein and the corresponding change in the extent of ionization of the different dissociable groups. A rigorous analysis of this problem is extremely complex due to the non-linear nature of the chemical equilibria. We have analyzed solute partitioning in this system using the linearized charge regulation model developed by Carnie and Chan [19] in which the surface charge s and the surface potential f are solute solute
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assumed to be related as =S−Kf (8) solute solute where S and K are the charge regulation coefficients for the solute. Note that Eq. (8) reduces to a constant surface charge boundary condition when K0 and to a constant surface potential boundary condition as K2 (with f =S/K=constant as K2). solute The potential energy of interaction for the charged protein was evaluated by solving the linearized Poisson–Boltzmann equation using the general approach developed by Smith and Deen [14,15], but with the boundary condition given by Eq. (8). The final result can be expressed in a form similar to Eq. (7): s
C
D C
D
S2 S2 pore solute +B 2 +D )2 (K +D )2 pore 1 solute 2 S S pore solute (9) +B 3 (K +D ) (K +D ) pore 1 solute 2 where S and K are the charge regulation constants i i for the solute and pore and the B and D are i i complex functions of the solute and pore size and the solution ionic strength [20,21]. The parameters S and K were evaluated at each ionic solute solute strength from the chemical equilibria in free solution for the different ionizable groups on the surface of BSA as described by Carnie and Chan [19] and Pujar [20]. It was not possible to use a complete charge regulation model for the pore boundary owing to the absence of experimental data on the dissociation equilibria and ion adsorption characteristics of the surface. Instead, the surface charge density for the pore was assumed to be constant (i.e. K =0). A more detailed pore discussion of the theoretical analysis is provided elsewhere [20,21]. The predicted values for S 2 using the charge regulation model lie only slightly above the constant surface charge density results under these conditions, indicating that the surface charge remains nearly constant as the protein enters the pore. Results at pH near the pK of the i different ionizable groups show much larger differences between the charge regulation calculations and those using either constant surface charge
Y =B E 1 (K
C
D
139
density or constant surface potential boundary conditions due to the greater sensitivity of the protein charge to small changes in the ionic environment under these conditions [20]. The theoretical results using the charge regulation model are in very good agreement with the experimental data for BSA transport, demonstrating that a proper description of the detailed chemical equilibria governing the dissociation reactions and ion binding (accounted for via the linearized charge regulation model ) is required to obtain an accurate model for the electrostatic interactions in this system. Future work in this area will clearly need to account for the charge regulation characteristics of the pore boundary as well. Electrostatic interactions can be very effectively exploited for protein separations by operating under conditions at which one of the proteins is essentially uncharged (i.e. at a pH near the isoelectric point of that protein) while the other protein has a relatively strong charge. Fig. 4 shows experimental data for the selectivity for the separation of BSA and hemoglobin (Hb) at pH 7 and different solution ionic strengths [22]. The selectivity was
Fig. 4. Selectivity for separation of Hb and BSA in different ionic strength solutions. Calculations were performed using constant surface charge density, constant surface potential, and ˚ and charge regulation boundary conditions with r =30 A s ˚ . Filled symbols represent experimental r =46 A data for p Hb–BSA fractionation using a polyethersulfone membrane [22].
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evaluated as the ratio of the Hb sieving coefficient to that of BSA at the optimal filtrate flux, i.e. at the flux which gave the highest selectivity. The selectivity at 0.1 M ionic strength (kr =4.8) was p relatively low, reflecting the similar size of BSA (molecular weight, 69 000) and Hb (molecular weight 67 000). The selectivity increased dramatically at lower salt concentrations due to the strong electrostatic exclusion of the charged BSA from the membrane pores. The solid curves in Fig. 4 represent the theoretical predictions for the intrinsic selectivity (defined as the ratio of the asymptotic sieving coefficients for the two proteins) for a system with two proteins with identical size ˚ ). In this case, Hb was assumed to be (r =30 A s completely uncharged, while the surface charge and potential for BSA were calculated using the charge regulation model discussed previously. The results using the constant surface charge density boundary condition tend to overestimate the selectivity, whereas the calculations using a constant surface potential actually predict a sharp reduction in selectivity at very low kr due to the developp ment of an induced charge on the pore wall. The calculations using the linearized charge regulation boundary condition (Eq. (8)) are in very good agreement with the data over the entire range of ionic strength. It is important to note that the selectivities in this system are much greater than those found for purely hard-sphere interactions (Fig. 1), even though the proteins were assumed to have identical size. These calculations clearly demonstrate that electrostatic interactions have the potential to provide much greater selectivity than would be possible using purely steric interactions. Although the charge regulation analysis presented above provides a much more realistic description of electrostatic interactions in membrane systems, the model still assumes a spherical solute with a uniform distribution of the charge over the entire surface. Thus, this approach neglects the detailed three-dimensional geometry of the protein as well as the actual locations of the different ionizable groups. Yoon and Lenhoff [23] have evaluated the detailed electrostatic potential around an isolated molecule of ribonuclease A based on the known three-dimensional crystallographic protein structure (which includes the loca-
tion of specific ionizable groups on the individual amino acids). These calculations demonstrated that ribonuclease A had significant regions of negative electrical potential, even when the protein as a whole has a significant net positive charge, due to the uneven distribution of the different ionizable groups. This non-uniform charge distribution dramatically altered the electrostatic interaction energy for ribonuclease A near a charged planar surface [22]. The effects of this type of nonuniform charge distribution on the equilibrium partition coefficient for a charged protein in a membrane pore remain to be determined. 2.3. Van der Waals interactions Bhattacharjee and Sharma [24] have recently analyzed the van der Waals interactions for a spherical particle in a cylindrical pore. The energy of interaction at the pore centerline was found to be Y
VDW
=−
pA
l3
3 (1−l2)3/2
(10)
where A is the effective Hamaker interaction constant between the solute and pore in the particular solvent. The interaction energy is generally negative, corresponding to an attractive potential and an increase in the equilibrium partition coefficient. Typical values of the Hamaker interaction constant are A=1.96kT for BSA and poly(methyl methacrylate) and A=0.18kT for BSA and the much more hydrophobic polytetrafluoroethylene [25]. Theoretical calculations for the asymptotic sieving coefficient accounting for van der Waals interactions are shown in Fig. 5. The sieving coefficients (S =wK ) were evaluated by numerical integration 2 c of Eqs. (2) and (4) with Y (r) given by VDW Bhattacharjee and Sharma [24] and Y =0. A E Born repulsion was also included in the calculations; thus, the integrals were evaluated from r= ˚ [24]. The 0 to r=r −r −d with d taken as 1.57 A p s integration in Eq. (4) was performed using the centerline approximation for G, consistent with the approach used previously. The van der Waals interactions cause a significant increase in S due 2 to the increase in the equilibrium partition coeffi-
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Fig. 5. Effect of van der Waals interactions on the asymptotic sieving coefficient. Results are plotted vs. the scaled solute radius, where R* is defined as the radius of the solute for which S =0.5 for the case with purely hard-sphere interactions. 2 ˚. Calculations were all performed using r =46 A p
cient w caused by the attractive potential. This effect is most pronounced for the larger solutes, with S eventually becoming much greater than 2 one for any A/kT>0. The strong attractive interactions in these simulations would likely lead to significant membrane fouling (unless they were compensated for by a strong repulsive electrostatic interaction); thus, it is unclear whether the behavior shown in Fig. 5 would ever be seen experimentally. 2.4. Non-equilibrium considerations All of the calculations presented in the last few sections assumed implicitly that thermodynamic equilibrium exists between the solution in the membrane pore and that immediately upstream of the membrane surface. For a membrane with ˚ and w=0.1, this thermodynamic analysis r =25 A p predicts that there would be fewer than ten solute molecules in a pore at any given time (assuming a pore length of 0.5 mm, a bulk protein concentration of 1 g l−1, and a protein molecular weight of 70 000). In addition, the residence time for each solute molecule within the pore would be less than
141
0.01 s (assuming an average filtrate flux of 50 l m−2 h−1 and a membrane porosity of 0.2, values which are typical of existing membrane systems for protein separations). These results clearly bring into doubt the validity of the equilibrium approximation, and they suggest that a dynamic analysis of solute transport might be more appropriate for describing the performance of these membrane systems. There have been relatively few dynamic analyses of solute transport through membrane pores. Yan et al. [26 ] analyzed the problem of particle entrainment into a small circular side pore fed from a large main tube. Particles are entrained into the pore only if they are located on a streamline within the capture tube, i.e. on a streamline which actually enters the pore. The particle concentration in the pore can be considerably less than that in the bulk solution since the particles are sterically excluded from the region within one particle radius of the tube wall. The ratio of the particle concentration in the pore to that in the bulk solution was given approximately as C
pore =(1−al2)3/2 C bulk
(11)
where a is related to the ratio of the fluid velocity through the pore to that down the main tube. Eq. (11) predicts a sieving curve qualitatively similar to that seen in Fig. 1, although the actual location of the curve depends on the magnitude of the filtrate flux and the crossflow velocity. This type of fluid skimming model has not been applied to the analysis of protein transport through porous membranes. Bowen and Sharif [27] examined the hydrodynamic and electrostatic interactions governing the motion of a charged sphere approaching a single pore in a charged planar surface. A Galerkin finite element scheme was used to solve the non-linear Poisson–Boltzmann equation for the electrostatic potential, with the fluid flow evaluated from the governing Navier–Stokes equation. For highly charged particles (at relatively low ionic strength), the repulsive electrostatic force was greater than the hydrodynamic drag on the particle associated with the fluid flow. Under these conditions the
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particle was completely excluded from the membrane pore, even when the particle radius was considerably smaller than the pore radius. Although this analysis provides considerable insight into the ‘‘dynamic’’ exclusion of charged particles from membrane pores, the calculations neglect the effects of Brownian diffusion which would allow the particles to cross the fluid streamlines and enter the pore even in the presence of these strong repulsive interactions. Future work along these lines will be required to bridge the gap between the purely hydrodynamic and purely thermodynamic analyses of particle transport through porous membranes.
3. Discussion Long-range colloidal interactions can have a dramatic effect on the rate of protein transport through porous membranes. In order to most effectively exploit these effects in the actual separation of protein mixtures, the membrane systems should be operated under conditions in which the magnitude of the colloidal forces are very different for the different proteins. This is easily achieved for electrical interactions by operating at (or near) the isoelectric point of one of the proteins, and this is the approach that was used by Saksena and Zydney [2] in their study of BSA–IgG separations. Very high selectivities were obtained by adjusting the solution to pH 4.7 (near the isoelectric point of BSA) and using very low salt concentrations (small kr ). IgG was almost completely s rejected by the membrane under these conditions (S <0.001), while BSA transport remained rela2 tively high since BSA has minimal net charge at this pH. Similar results were obtained by Eijndhoven et al. [22] for the separation of BSA and Hb, with the optimal selectivity obtained at pH 7 (near the isoelectric point of Hb) due to the strong electrostatic exclusion of BSA. It is also possible, at least in principle, to exploit differences in van der Waals interactions in membrane systems due to the different hydrophilic/hydrophobic character of different proteins. Higuchi et al. [3] attributed the high selectivity that they obtained for BSA–IgG mix-
tures to specific interactions between hydrophobic groups on the proteins with corresponding groups on the surface-modified polysulfone membranes. Additional studies are required to confirm this hypothesis. The importance of steric, electrostatic, and van der Waals interactions in these membrane systems is in many ways analogous to the use of these same intermolecular interactions to affect separations in size-exclusion, ion exchange, and reverse-phase chromatography. The development of these successful chromatographic processes has been greatly facilitated by the application of colloidal theory to describe the underlying intermolecular interactions, with the properties of the chromatographic media very carefully chosen so as to most effectively exploit these interactions. Future progress in the development of high selectivity membrane processes will require similar application of colloidal interaction theories. Future membrane systems will likely use much more sophisticated membrane materials (e.g. polymers with carefully chosen surface charge and hydrophilicity) and device operating conditions (e.g. buffers with appropriate electrolytes and stabilizing agents) that have been specifically chosen to exploit the full range of colloidal interactions that govern protein transport in these systems.
Acknowledgment The authors would like to acknowledge the financial support provided by Millipore Corp. and Genentech, Inc.
References [1] E. Flaschel, C. Wandrey, M.R. Kula, Adv. Biochem. Eng. Biotechnol. 26 (1983) 73. [2] S. Saksena, A.L. Zydney, Biotechnol. Bioeng. 43 (1994) 960. [3] A. Higuchi, S. Mishima, T. Nakagawa, J. Membrane Sci. 57 (1991) 175. [4] S. Nakatsuka, A.S. Michaels, J. Membrane Sci. 69 (1992) 189. [5] J.L. Anderson, J.A. Quinn, Biophys. J. 14 (1974) 130. [6 ] W.S. Opong, A.L. Zydney, AIChE J. 37 (1991) 1497.
A.L. Zydney, N.S. Pujar / Colloids Surfaces A: Physicochem. Eng. Aspects 138 (1998) 133–143 [7] N.S. Pujar, A.L. Zydney, Ind. Eng. Chem. Res. 33 (1994) 2473. [8] W.M. Deen, AIChE J. 27 (1987) 952. [9] P.M. Bungay, H. Brenner, Int. J. Multiphase Flow 1 (1973) 25. [10] N.S. Pujar, A.L. Zydney, AIChE J. 42 (1996) 2101. [11] S. Mochizuki, A.L. Zydney, J. Membrane Sci. 68 (1992) 21. [12] L. Zeman, A.L. Zydney, Microfiltration and Ultrafiltration, Dekker, New York, 1996. [13] R. van Reis, S. Gadam, L. Frautschy, S. Orlando, E.M. Goodrich, S. Saksena, R. Kuriyel, C.M. Simpson, S. Pearl, A.L. Zydney, Biotechnol. Bioeng. (in press) [14] F.G. Smith, III, W.M. Deen, J. Colloid Interface Sci. 78 (1980) 444. [15] F.G. Smith, III, W.M. Deen, J. Colloid Interface Sci. 91 (1983) 571. [16 ] C. Tanford, S.A. Swanson, W.S. Shore, J. Am. Chem. Soc. 77 (1955) 628. [17] G. Scatchard, J.S. Coleman, A.L. Shen, J. Am. Chem. Soc. 79 (1957) 12.
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[18] C. Causserand, M. Meireles, P. Aimar, Trans. Inst. Chem. Eng. 74 (1996) 113. [19] S.L. Carnie, D.Y.C. Chan, J. Colloid Interface Sci. 161 (1993) 260. [20] N.S. Pujar, Ph.D. Thesis, University of Delaware, 1996. [21] N.S. Pujar, A.L. Zydney, J. Colloid Interface Sci. (accepted ). [22] R.H.C.M. van Eijndhoven, S. Saksena, A.L. Zydney, Biotechnol. Bioeng. 48 (1995) 406. [23] B.J. Yoon, A.M. Lenhoff, J. Phys. Chem. 96 (1992) 3130. [24] S. Bhattacharjee, A. Sharma, J. Colloid Interface Sci. 171 (1995) 288. [25] C.M. Roth, B.L. Neal, A.M. Lenhoff, Biophys. J. 70 (1996) 977. [26 ] Z.-Y. Yan, A. Acrivos, S. Weinbaum, J. Fluid Mech. 229 (1991) 1. [27] W.R. Bowen, A.O. Sharif, in: W.R. Bowen, R.W. Field, J.A. Howell (Eds.), Proc. Euromembrane ’95, University of Bath, Antony Rowe Ltd., Chippenham September 18–20, 1995, vol. 1, p. 35.
Protein transport through porous membranes: effects of colloidal interactions Andrew L. Zydney *, Narahari S. Pujar Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA
Abstract Traditional analyses of membrane transport have generally considered only steric (size-based ) interactions between the solute and pores. Recent experimental work has clearly demonstrated the importance of longer-range colloidal interactions on the rate of solute transport. These interactions can dramatically affect membrane performance, changing the membrane from being almost completely permeable to completely retentive for a given solute. This manuscript discusses several applications of colloidal interaction theory to membrane systems, highlighting the tremendous insights that have been obtained into the performance of real membrane systems as well as the limitations of available theoretical descriptions. New calculations are presented for the transport of charged solutes through porous membranes which explicitly account for the effects of charge regulation arising from the dissociation of specific ionizable surface groups. The importance of non-equilibrium phenomena in membrane systems is also examined. Particular emphasis is placed on the use of membrane systems for protein separations, including the importance of protein structure and biochemistry on membrane selectivity. © 1998 Elsevier Science B.V. Keywords: Electrostatics; Membrane; Protein; Separations; Ultrafiltration
1. Introduction Traditional analyses of membrane transport have generally considered only steric (size-based) interactions between the solute and pores. Steric interactions can be effectively used to separate (1) macromolecular solutes from solvent, and (2) cells or large particles from macromolecules. Membrane systems in the biotechnology industry have thus been used extensively for protein harvest from cell suspensions, protein concentration (i.e. buffer removal ), and buffer exchange. It has not, however, been possible to obtain high selectivity for protein–protein separations using purely steric interactions [1], with the net result that membrane * Corresponding author. Tel: (+1) 302 831 2399; Fax: (+1) 302 831 1048; e-mail: [email protected] 0927-7757/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0 9 2 7 -7 7 5 7 ( 9 6 ) 0 3 92 8 - 3
systems have not been developed for actual protein purification. Recent experimental work has demonstrated that it is possible to improve dramatically the selectivity of membrane systems by exploiting longrange colloidal interactions between the solutes and membrane pores. For example, Saksena and Zydney [2] obtained separation factors of more than 30 for the separation of albumin (BSA) and immunoglobulins (IgGs) by operating the system at pH 4.7 and low salt concentration. The high selectivity obtained under these conditions was attributed to the electrostatic ‘‘exclusion’’ of the charged IgG molecules from the membrane pores. It was even possible to reverse the selectivity in this system, obtaining greater throughput of the larger IgG molecules, by operating at the isoelectric point of IgG so that there were strong electro-
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static interactions between the charged BSA molecules and the membrane pores. Higuchi et al. [3] have also shown good separation of BSA–IgG mixtures, with the high selectivity attributed to non-electrostatic interactions between the proteins and hydrophobic groups on the surface-modified polysulfone membranes. Nakatsuka and Michaels [4] examined the separation of myoglobin from BSA using both sorptive and non-sorptive membranes. In this case, the high selectivity obtained with the non-sorptive membranes was attributed to the different extents of protein adsorption caused by the combined electrostatic and van der Waals interactions. In order to develop more effective membrane systems for protein separations, it is critical to have a fundamental understanding of the nature of the colloidal interactions that govern protein transport through porous membranes. This manuscript discusses several applications of colloidal interaction theory in membrane systems, highlighting the tremendous insights that have been obtained into the performance of real membrane devices as well as the limitations of the available theoretical descriptions. New calculations are presented for the transport of charged solutes through narrow pores which explicitly account for the effects of charge regulation arising from the dissociation of ionizable groups on the protein surface. Particular emphasis is placed on those factors which affect the overall membrane selectivity, including a discussion of how to exploit these colloidal interactions in the development of improved membrane processes.
2. Theoretical background Solute transport in most membrane systems is dominated by convection. The convective solute flux across the membrane can be written as [5,6 ] N =wK V (1) 9C s c w where w is the equilibrium solute partition coefficient between the solution immediately upstream of the membrane and that in the membrane pores, K is the hindrance factor for convective solute c transport through the pore, V 9 is the average fluid
velocity through the membrane pore, and C is w the solute concentration in the solution immediately upstream of the membrane. The rate of solute transport through the membrane is thus governed by both thermodynamic and hydrodynamic interactions. The partition coefficient for a spherical solute in a cylindrical pore can be expressed in terms of the energy of interaction between the solute and the pore boundary as [5,7] w=
P
A
−Y rp total exp r2 kT p 0 2
B
r dr
(2)
where r is the pore radius, r is the radial coordip nate within the pore, k is Boltzmann’s constant, and T is the absolute temperature. The total interaction potential has contributions from steric (hard-sphere), electrostatic, and van der Waals forces: Y
total
=Y +Y +Y HS E VDW
(3)
The hindrance factor for convection can also be expressed as an integral over the pore radius [5,7]: rp−rs K= c
P
GV exp
0 rp−rs Vr dr 2
A P 0
A
rp−rs
−Y −Y E VDW kT
BC P A 0
exp
B
r dr
B D
−Y −Y E VDW r dr kT
(4) where V is the local fluid velocity in the pore and G is the lag coefficient which accounts for the additional hydrodynamic drag on the solute associated with the presence of the pore wall. The integrals in Eq. (4) are evaluated from r=0 to r=r −r to account for the steric (hard-sphere) p s exclusion of the solute from the region within one solute radius of the pore wall. Most previous studies have evaluated K using the centerline c approximation, in which both G(r) and Y +Y are assumed to be constant at their E VDW centerline (r=0) values [8]. Under these conditions, the exponential terms in the numerator and the denominator can be factored out of the integ-
A.L. Zydney, N.S. Pujar / Colloids Surfaces A: Physicochem. Eng. Aspects 138 (1998) 133–143
rals with the net result that K is independent of c Y and Y . The validity of this approximation E VDW for systems with complex interaction potentials has yet to be determined. In addition, although theoretical expressions for G(r) have been available for systems with purely hard-sphere interactions for more than 20 years [8,9], there are currently no corresponding calculations that account for the effects of long-range (e.g. electrostatic) interactions on the hydrodynamics. Pujar and Zydney [10] have obtained expressions for the enhanced drag coefficient K and the diffusional hindrance factor K for spherical solutes in a concentric spherical d cavity, but these results cannot be directly extended to systems with cylindrical pores or to the evaluation of the hindrance factor for convection K c because of the complex coupling between the electrostatics and hydrodynamics in these systems. 2.1. Hard-sphere interactions The equilibrium partition coefficient and convective hindrance factor for a system with purely hard-sphere interactions can both be expressed as unique functions of the ratio of the solute to pore radius, l=r /r . The partition coefficient is evals p uated directly from Eq. (2), yielding w=(1−l)2
(5)
which accounts for the excluded volume in the outer (annular) region of the cylindrical pore. The corresponding result for a membrane with slitshaped pores is w=1−l
(6)
where l is now the ratio of the solute radius to the slit half-width. Bungay and Brenner [9] developed an expression for K (l) for a hard sphere in c a cylindrical pore using matched asymptotic expansions for both small and close-fitting spheres. Corresponding results for slit-shaped pores are discussed by Deen [8]. The product wK , often referred to as the asympc totic membrane sieving coefficient S , is shown as 2 a function of the scaled solute radius in Fig. 1. Results are shown for a membrane with a single cylindrical pore radius (s/r:=0), for a membrane with a single slit-shaped pore, and for two mem-
135
Fig. 1. Asymptotic sieving coefficients for membranes with different pore size distributions evaluated assuming purely hardsphere interactions. Results are plotted as a function of the scaled solute radius, where R* is defined as the radius of the solute for which S =0.5. Dashed line is for a membrane with 2 uniform slit-shaped pores. Open symbols represent experimental data for dextran transport through a polyethersulfone membrane [11].
branes with different log-normal distributions of cylindrical pores (with s and r: the standard deviation and mean for the log-normal distribution) [12]. In each case, the solute radius has been made dimensionless by R*, which is defined as the radius of the solute for which S =0.5. This non-dimen2 sionalization highlights the effect of the pore shape (cylinder vs. slit) and pore size distribution on the sieving characteristics, and it also facilitates comparison with experimental data. The asymptotic sieving coefficient decreases with increasing r , s attaining a value of zero when r /R*=2.40 for a s membrane with a single cylindrical pore size and when r /R*=1.67 for a membrane with a slits shaped pore. The curve for the slit-shaped pore is significantly steeper than the results for a cylindrical pore at large r /R* due to the different depens dences of the partition coefficient and hindrance factor on l. The asymptotic sieving coefficients for the membranes with the log-normal pore size distributions approach zero asymptotically as r 2 due to the presence of a finite number of s very large pores in the log-normal distribution. S for the larger solutes (r /R*>1) increases with 2 s an increase in the breadth of the pore size distribu-
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tion (increasing s/r:), with this effect becoming quite pronounced at large r . This is due to the s very significant contribution to solute transport from the largest pores in the distribution arising from the very high fluid velocity in those pores [12]. The results in Fig. 1 suggest that the asymptotic sieving coefficients for small solutes are actually greater in membranes with narrow pore size distributions. This behavior is really just an artifact of plotting the results as a function of r /R*. For s example, if the S values had been plotted as a 2 function of r (or r /r:) the asymptotic sieving s s coefficients would increase monotonically with increasing s/r: for all values of the solute radius. The open symbols in Fig. 1 represent experimental data obtained by Mochizuki and Zydney [11] for the transport of polydisperse neutral dextrans ( long-chain polysaccharides) through an asymmetric polyethersulfone membrane with a nominal molecular weight cut-off of 100 000. The data are well represented by the theoretical curve with s/r:= 0.5, which is also consistent with independent estimates of the breadth of the pore size distribution in this type of polymeric membrane [12]. The good agreement between the model and data indicate that the cylindrical pore model provides a reasonable description of solute transport through these asymmetric membranes. However, these results should not be considered as proof that this membrane actually has a log-normal pore size distribution of cylindrical pores, since many different types of pore size distribution could potentially produce sieving curves very similar to those shown in Fig. 1. The results in Fig. 1 can be used to evaluate the membrane selectivity, which is simply defined as the normalized solute flux (N /C ) for a given s w solute divided by the corresponding flux for a second solute. For example, a membrane with a single pore size (s/r:=0) would have asymptotic sieving coefficients of S =0.2 and 0.8 (correspond2 ing to a selectivity of four) for two solutes with a size ratio of about 2.7. This would correspond to two proteins having a molecular weight ratio of approximately 10, which has generally been regarded as the minimum size difference required for effective protein separation using a purely sizebased membrane separation. Even greater size
ratios would be required for membranes with broader pore size distributions. For example, the membrane with s/r:=0.5 would yield S =0.2 and 2 0.8 for two solutes which differ in size by a factor of 3.6, while the membrane with s/r:=1 would provide an equivalent selectivity only if the two solutes differed by at least a factor of five in radius (corresponding to a factor of 100 in molecular weight). It is important to note that the selectivity could be increased significantly by using a membrane with a smaller average pore size, although this would be at the expense of a much smaller overall throughput. For example, a membrane with a single pore size would yield S =0.05 and 0.2 2 (which also corresponds to a selectivity of four) for two solutes that differ in radius by only a factor of 1.3. The low throughput obtained with this small pore membrane can be largely compensated for by employing a diafiltration cascade, in which the protein with the larger S is effectively 2 washed away from the more highly retained species. This type of separation process is discussed in more detail by van Reis et al. [13]. 2.2. Electrical interactions Smith and Deen [14,15] developed expressions for Y (r), the electrostatic potential energy of E interaction, by solving the linearized Poisson– Boltzmann equation for the ion concentrations and electrical potential around a single spherical solute in an infinitely long cylindrical pore. The total energy of interaction between the charged sphere and the pore, assuming constant surface charge density boundary conditions, can be expressed as the sum of three terms [14,15]: Y =A s2 +A s2 +A s s (7) E 1 pore 2 solute 3 pore solute where s and s are the surface charge densipore solute ties of the pore wall and solute respectively, and A , A , and A are all positive coefficients which 1 2 3 depend upon the solution ionic strength, the pore radius, and the solute radius. The first two contributions to the total energy of interaction arise from the deformation of the electrical double layer surrounding the pore wall and the solute respectively. These interactions always increase, the total free energy of the system, leading to a reduction
A.L. Zydney, N.S. Pujar / Colloids Surfaces A: Physicochem. Eng. Aspects 138 (1998) 133–143
in the partition coefficient and, in turn, the rate of protein transport through the membrane. The final term in Eq. (7) is associated with charge–charge interactions between the solute and pore wall. This term is positive when the solute and pore have like charge and negative when the solute and pore have opposite charge. It is important to note that the second term in Eq. (7) provides the dominant contribution to the electrostatic interaction potential if s is small, which can lead to a net pore repulsive-type electrostatic interaction even when the solute and pore have opposite charge. The effects of electrical interactions on membrane transport are examined in Fig. 2. The calculations were performed assuming that K is c unaffected by Y , although the validity of this E approximation has not been clearly established. Results are shown using constant surface charge density boundary conditions with s =−0.015 C m−2 (similar to the surface charge s on BSA) and s =0 (top panel ) and p s =−10−4 C m−2 (bottom panel ). Calculations p are shown at kr =0.1, 1, and 10, where k−1 is the p ˚ . The results double layer thickness and r =46 A p are again plotted as a function of the scaled solute radius. In this case, R* increases with increasing kr due to the reduction in the magnitude of the p electrostatic interactions at high salt concentrations. The asymptotic sieving coefficients at kr =10 are essentially identical to those seen in p Fig. 1 due to the strong electrostatic shielding provided by the supporting electrolyte under these conditions. The breadth of the sieving curve for s =0 (top panel ) decreases significantly with p decreasing kr due to the much larger relative p increase in the free energy associated with the deformation of the electrical double layer surrounding the larger solutes. The net result is a significant increase in the membrane selectivity at low solution ionic strength. For example, the membrane shown in the top panel of Fig. 2 would yield S =0.2 and 0.8 for solutes differing by a factor 2 of 2.7 in radius when kr =10, but would yield the p same sieving coefficients for two solutes differing by only a factor of 1.74 in radius when kr =0.1. p These calculations suggest that it should be possible to increase significantly the selectivity for the separation of protein oligomers (e.g. dimers from
137
Fig. 2. Effect of electrostatic interactions on the asymptotic sieving coefficients. Top panel shows results for an uncharged pore while bottom panel shows results for s =−10−4 C m−2. p Calculations were performed using constant surface charge density boundary conditions at several values of kr with ˚ . Results are plotted as pa funcs =−0.015 C m−2 and r =46 A s p tion of the scaled solute radius, where R* is defined as the radius of the solute for which S =0.5. 2
monomers) using an electrically ‘‘neutral’’ membrane by operating at low salt concentrations and at a pH away from the protein isoelectric point. This behavior is consistent with experimental studies by van Reis et al. [13] for the separation of BSA monomers and dimers. The theoretical results when s =−10−4 C m−2 p
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A.L. Zydney, N.S. Pujar / Colloids Surfaces A: Physicochem. Eng. Aspects 138 (1998) 133–143
(bottom panel ) are markedly different. In this case, the sieving profiles at kr =0.1 and kr =10 p p are almost identical, with only a slightly higher selectivity obtained at kr =1. The potential energy p of interaction when s =−10−4 C m−2 is deterp mined largely by the charge–charge interactions between the solute and the pore wall. These repulsive interactions cause a similar percentage reduction in w for both large and small solutes; thus, the sieving profile (when plotted vs. r /R*) shows s a very weak dependence on the solution ionic strength. The slight increase in membrane selectivity at the intermediate salt concentration (kr =1) p is due to the increase in the potential energy of interaction associated with the deformation of the double layer surrounding the solute molecule, with this effect being more significant for the larger solutes. The effects of solution ionic strength on the asymptotic sieving coefficient are examined in more detail in Fig. 3. The filled symbols represent experimental data obtained by Pujar and Zydney [7] for BSA transport through an asymmetric polyethersulfone membrane at pH 7. The asymptotic sieving coefficient decreases by more than two
Fig. 3. Asymptotic sieving coefficient as a function of the dimensionless pore radius for calculations using constant surface charge density, constant surface potential, and charge regula˚ and r =46 A ˚ . Filled tion boundary conditions with r =30 A s p symbols represent experimental data for BSA transport through a polyethersulfone membrane [7].
orders of magnitude as the solution ionic strength is reduced from 0.15 to 0.0015 M (i.e. as kr is p reduced from 5.9 to 0.59). This change in solution environment thus converts the membrane from one which is reasonably permeable to BSA (S >0.1) to one which is almost completely reten2 tive for this protein (S <0.001). The solid curves 2 in Fig. 3 represent the model calculations using constant surface potential, constant surface charge density, and charge regulation boundary conditions. The charge regulation model is discussed in more detail below. The values of the surface potential f and surface charge density s in free solution s s were evaluated at each ionic strength by calculating the degree of ionization of the different dissociable groups in the BSA molecule (a-carboxyl, b,c-carboxyl, imidazole, a-amino, e-amino, phenolic, and guanidine) using the model developed by Tanford et al. [16 ], with the extent of Cl− adsorption determined using a modified form of the three-site binding model developed by Scatchard et al. [17]. ˚ and All calculations are shown using r =30 A s ˚ r =46 A [7]. The asymptotic sieving coefficients p evaluated using the constant surface charge density boundary condition decrease sharply with decreasing kr due to the strong electrostatic repulsion s between the like-charged solute and pore. In contrast, the constant surface potential model predicts an increase in S with decreasing kr due to the 2 p development of an induced (positive) charge on the pore wall for interactions at constant potential. All previous analyses of electrostatic interactions in membrane systems have used either the constant surface charge density or constant surface potential boundary conditions [7,8,14,18]. In actuality, both the surface charge and surface potential of the protein will vary as the protein enters the pore: this is due to the alteration in the electrochemical environment around the protein and the corresponding change in the extent of ionization of the different dissociable groups. A rigorous analysis of this problem is extremely complex due to the non-linear nature of the chemical equilibria. We have analyzed solute partitioning in this system using the linearized charge regulation model developed by Carnie and Chan [19] in which the surface charge s and the surface potential f are solute solute
A.L. Zydney, N.S. Pujar / Colloids Surfaces A: Physicochem. Eng. Aspects 138 (1998) 133–143
assumed to be related as =S−Kf (8) solute solute where S and K are the charge regulation coefficients for the solute. Note that Eq. (8) reduces to a constant surface charge boundary condition when K0 and to a constant surface potential boundary condition as K2 (with f =S/K=constant as K2). solute The potential energy of interaction for the charged protein was evaluated by solving the linearized Poisson–Boltzmann equation using the general approach developed by Smith and Deen [14,15], but with the boundary condition given by Eq. (8). The final result can be expressed in a form similar to Eq. (7): s
C
D C
D
S2 S2 pore solute +B 2 +D )2 (K +D )2 pore 1 solute 2 S S pore solute (9) +B 3 (K +D ) (K +D ) pore 1 solute 2 where S and K are the charge regulation constants i i for the solute and pore and the B and D are i i complex functions of the solute and pore size and the solution ionic strength [20,21]. The parameters S and K were evaluated at each ionic solute solute strength from the chemical equilibria in free solution for the different ionizable groups on the surface of BSA as described by Carnie and Chan [19] and Pujar [20]. It was not possible to use a complete charge regulation model for the pore boundary owing to the absence of experimental data on the dissociation equilibria and ion adsorption characteristics of the surface. Instead, the surface charge density for the pore was assumed to be constant (i.e. K =0). A more detailed pore discussion of the theoretical analysis is provided elsewhere [20,21]. The predicted values for S 2 using the charge regulation model lie only slightly above the constant surface charge density results under these conditions, indicating that the surface charge remains nearly constant as the protein enters the pore. Results at pH near the pK of the i different ionizable groups show much larger differences between the charge regulation calculations and those using either constant surface charge
Y =B E 1 (K
C
D
139
density or constant surface potential boundary conditions due to the greater sensitivity of the protein charge to small changes in the ionic environment under these conditions [20]. The theoretical results using the charge regulation model are in very good agreement with the experimental data for BSA transport, demonstrating that a proper description of the detailed chemical equilibria governing the dissociation reactions and ion binding (accounted for via the linearized charge regulation model ) is required to obtain an accurate model for the electrostatic interactions in this system. Future work in this area will clearly need to account for the charge regulation characteristics of the pore boundary as well. Electrostatic interactions can be very effectively exploited for protein separations by operating under conditions at which one of the proteins is essentially uncharged (i.e. at a pH near the isoelectric point of that protein) while the other protein has a relatively strong charge. Fig. 4 shows experimental data for the selectivity for the separation of BSA and hemoglobin (Hb) at pH 7 and different solution ionic strengths [22]. The selectivity was
Fig. 4. Selectivity for separation of Hb and BSA in different ionic strength solutions. Calculations were performed using constant surface charge density, constant surface potential, and ˚ and charge regulation boundary conditions with r =30 A s ˚ . Filled symbols represent experimental r =46 A data for p Hb–BSA fractionation using a polyethersulfone membrane [22].
140
A.L. Zydney, N.S. Pujar / Colloids Surfaces A: Physicochem. Eng. Aspects 138 (1998) 133–143
evaluated as the ratio of the Hb sieving coefficient to that of BSA at the optimal filtrate flux, i.e. at the flux which gave the highest selectivity. The selectivity at 0.1 M ionic strength (kr =4.8) was p relatively low, reflecting the similar size of BSA (molecular weight, 69 000) and Hb (molecular weight 67 000). The selectivity increased dramatically at lower salt concentrations due to the strong electrostatic exclusion of the charged BSA from the membrane pores. The solid curves in Fig. 4 represent the theoretical predictions for the intrinsic selectivity (defined as the ratio of the asymptotic sieving coefficients for the two proteins) for a system with two proteins with identical size ˚ ). In this case, Hb was assumed to be (r =30 A s completely uncharged, while the surface charge and potential for BSA were calculated using the charge regulation model discussed previously. The results using the constant surface charge density boundary condition tend to overestimate the selectivity, whereas the calculations using a constant surface potential actually predict a sharp reduction in selectivity at very low kr due to the developp ment of an induced charge on the pore wall. The calculations using the linearized charge regulation boundary condition (Eq. (8)) are in very good agreement with the data over the entire range of ionic strength. It is important to note that the selectivities in this system are much greater than those found for purely hard-sphere interactions (Fig. 1), even though the proteins were assumed to have identical size. These calculations clearly demonstrate that electrostatic interactions have the potential to provide much greater selectivity than would be possible using purely steric interactions. Although the charge regulation analysis presented above provides a much more realistic description of electrostatic interactions in membrane systems, the model still assumes a spherical solute with a uniform distribution of the charge over the entire surface. Thus, this approach neglects the detailed three-dimensional geometry of the protein as well as the actual locations of the different ionizable groups. Yoon and Lenhoff [23] have evaluated the detailed electrostatic potential around an isolated molecule of ribonuclease A based on the known three-dimensional crystallographic protein structure (which includes the loca-
tion of specific ionizable groups on the individual amino acids). These calculations demonstrated that ribonuclease A had significant regions of negative electrical potential, even when the protein as a whole has a significant net positive charge, due to the uneven distribution of the different ionizable groups. This non-uniform charge distribution dramatically altered the electrostatic interaction energy for ribonuclease A near a charged planar surface [22]. The effects of this type of nonuniform charge distribution on the equilibrium partition coefficient for a charged protein in a membrane pore remain to be determined. 2.3. Van der Waals interactions Bhattacharjee and Sharma [24] have recently analyzed the van der Waals interactions for a spherical particle in a cylindrical pore. The energy of interaction at the pore centerline was found to be Y
VDW
=−
pA
l3
3 (1−l2)3/2
(10)
where A is the effective Hamaker interaction constant between the solute and pore in the particular solvent. The interaction energy is generally negative, corresponding to an attractive potential and an increase in the equilibrium partition coefficient. Typical values of the Hamaker interaction constant are A=1.96kT for BSA and poly(methyl methacrylate) and A=0.18kT for BSA and the much more hydrophobic polytetrafluoroethylene [25]. Theoretical calculations for the asymptotic sieving coefficient accounting for van der Waals interactions are shown in Fig. 5. The sieving coefficients (S =wK ) were evaluated by numerical integration 2 c of Eqs. (2) and (4) with Y (r) given by VDW Bhattacharjee and Sharma [24] and Y =0. A E Born repulsion was also included in the calculations; thus, the integrals were evaluated from r= ˚ [24]. The 0 to r=r −r −d with d taken as 1.57 A p s integration in Eq. (4) was performed using the centerline approximation for G, consistent with the approach used previously. The van der Waals interactions cause a significant increase in S due 2 to the increase in the equilibrium partition coeffi-
A.L. Zydney, N.S. Pujar / Colloids Surfaces A: Physicochem. Eng. Aspects 138 (1998) 133–143
Fig. 5. Effect of van der Waals interactions on the asymptotic sieving coefficient. Results are plotted vs. the scaled solute radius, where R* is defined as the radius of the solute for which S =0.5 for the case with purely hard-sphere interactions. 2 ˚. Calculations were all performed using r =46 A p
cient w caused by the attractive potential. This effect is most pronounced for the larger solutes, with S eventually becoming much greater than 2 one for any A/kT>0. The strong attractive interactions in these simulations would likely lead to significant membrane fouling (unless they were compensated for by a strong repulsive electrostatic interaction); thus, it is unclear whether the behavior shown in Fig. 5 would ever be seen experimentally. 2.4. Non-equilibrium considerations All of the calculations presented in the last few sections assumed implicitly that thermodynamic equilibrium exists between the solution in the membrane pore and that immediately upstream of the membrane surface. For a membrane with ˚ and w=0.1, this thermodynamic analysis r =25 A p predicts that there would be fewer than ten solute molecules in a pore at any given time (assuming a pore length of 0.5 mm, a bulk protein concentration of 1 g l−1, and a protein molecular weight of 70 000). In addition, the residence time for each solute molecule within the pore would be less than
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0.01 s (assuming an average filtrate flux of 50 l m−2 h−1 and a membrane porosity of 0.2, values which are typical of existing membrane systems for protein separations). These results clearly bring into doubt the validity of the equilibrium approximation, and they suggest that a dynamic analysis of solute transport might be more appropriate for describing the performance of these membrane systems. There have been relatively few dynamic analyses of solute transport through membrane pores. Yan et al. [26 ] analyzed the problem of particle entrainment into a small circular side pore fed from a large main tube. Particles are entrained into the pore only if they are located on a streamline within the capture tube, i.e. on a streamline which actually enters the pore. The particle concentration in the pore can be considerably less than that in the bulk solution since the particles are sterically excluded from the region within one particle radius of the tube wall. The ratio of the particle concentration in the pore to that in the bulk solution was given approximately as C
pore =(1−al2)3/2 C bulk
(11)
where a is related to the ratio of the fluid velocity through the pore to that down the main tube. Eq. (11) predicts a sieving curve qualitatively similar to that seen in Fig. 1, although the actual location of the curve depends on the magnitude of the filtrate flux and the crossflow velocity. This type of fluid skimming model has not been applied to the analysis of protein transport through porous membranes. Bowen and Sharif [27] examined the hydrodynamic and electrostatic interactions governing the motion of a charged sphere approaching a single pore in a charged planar surface. A Galerkin finite element scheme was used to solve the non-linear Poisson–Boltzmann equation for the electrostatic potential, with the fluid flow evaluated from the governing Navier–Stokes equation. For highly charged particles (at relatively low ionic strength), the repulsive electrostatic force was greater than the hydrodynamic drag on the particle associated with the fluid flow. Under these conditions the
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particle was completely excluded from the membrane pore, even when the particle radius was considerably smaller than the pore radius. Although this analysis provides considerable insight into the ‘‘dynamic’’ exclusion of charged particles from membrane pores, the calculations neglect the effects of Brownian diffusion which would allow the particles to cross the fluid streamlines and enter the pore even in the presence of these strong repulsive interactions. Future work along these lines will be required to bridge the gap between the purely hydrodynamic and purely thermodynamic analyses of particle transport through porous membranes.
3. Discussion Long-range colloidal interactions can have a dramatic effect on the rate of protein transport through porous membranes. In order to most effectively exploit these effects in the actual separation of protein mixtures, the membrane systems should be operated under conditions in which the magnitude of the colloidal forces are very different for the different proteins. This is easily achieved for electrical interactions by operating at (or near) the isoelectric point of one of the proteins, and this is the approach that was used by Saksena and Zydney [2] in their study of BSA–IgG separations. Very high selectivities were obtained by adjusting the solution to pH 4.7 (near the isoelectric point of BSA) and using very low salt concentrations (small kr ). IgG was almost completely s rejected by the membrane under these conditions (S <0.001), while BSA transport remained rela2 tively high since BSA has minimal net charge at this pH. Similar results were obtained by Eijndhoven et al. [22] for the separation of BSA and Hb, with the optimal selectivity obtained at pH 7 (near the isoelectric point of Hb) due to the strong electrostatic exclusion of BSA. It is also possible, at least in principle, to exploit differences in van der Waals interactions in membrane systems due to the different hydrophilic/hydrophobic character of different proteins. Higuchi et al. [3] attributed the high selectivity that they obtained for BSA–IgG mix-
tures to specific interactions between hydrophobic groups on the proteins with corresponding groups on the surface-modified polysulfone membranes. Additional studies are required to confirm this hypothesis. The importance of steric, electrostatic, and van der Waals interactions in these membrane systems is in many ways analogous to the use of these same intermolecular interactions to affect separations in size-exclusion, ion exchange, and reverse-phase chromatography. The development of these successful chromatographic processes has been greatly facilitated by the application of colloidal theory to describe the underlying intermolecular interactions, with the properties of the chromatographic media very carefully chosen so as to most effectively exploit these interactions. Future progress in the development of high selectivity membrane processes will require similar application of colloidal interaction theories. Future membrane systems will likely use much more sophisticated membrane materials (e.g. polymers with carefully chosen surface charge and hydrophilicity) and device operating conditions (e.g. buffers with appropriate electrolytes and stabilizing agents) that have been specifically chosen to exploit the full range of colloidal interactions that govern protein transport in these systems.
Acknowledgment The authors would like to acknowledge the financial support provided by Millipore Corp. and Genentech, Inc.
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