Ind. Eng. Chem. Res. 2005, 44, 8213-8217
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MATERIALS AND INTERFACES Mesh Size of Charged Polyacrylamide Hydrogels from Partitioning Measurements Shawn M. Russell and Giorgio Carta* Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22904-4741
The equilibrium partition coefficient of fluorescently labeled dextrans with molecular masses from 10 to 70 kDa and of two fluorescently labeled proteins was measured for anionic and cationic polyacrylamide gels with 5% bisacrylamide cross-linking. The gels were synthesized using 2-acrylamido-2-methylpropanesulfonic acid and (3-(methacryloylamino)propyl)trimethylammonium chloride monomers within fused silica capillaries with a square section. Partition coefficients were obtained by measuring the fluorescence intensity in the gel-filled capillaries equilibrated with dilute solutions of each probe. The partition coefficients show an exponential decay with gel concentration in agreement with the Ogston model for partitioning in fibrous gels. Comparison of the data for 10 kDa dextran with this model yields fiber radii of 0.79 and 0.99 nm for the anionic and the cationic gels, respectively, which are similar in magnitude to values reported in the literature for neutral polyacrylamide gels. Mesh sizes calculated on the basis of a singlepore-radius model vary between about 11 and about 6 nm for gels with gel concentrations between 0.1 and 0.26 g/cm3. Predictions of the dependence of the partition coefficient on the size of the solute based on the Ogston model are, however, only approximate. Better predictions are obtained with an empirical fit based on a model that considers the flexibility of the polymer chains. Introduction Quantifying the mesh size of charged hydrogels is an important step in determining how these matrixes function in a variety of applications ranging from drug delivery to separations. Considerable work has been done on neutral gels (e.g. see Tanaka1). However, much less is known about charged hydrogels, such as polyacrylamide-based materials, despite their widespread use.2 Previous work has shown that oppositely charged macromolecules such as proteins are very favorably adsorbed in these gels at low ionic strengths.2-5 On the other hand, uncharged macromolecules or macromolecules having the same charge as the gel are nearly completely excluded.6 This suggests that these gels have a mesh size that is close to the size of these macromolecules so that nearly complete size exclusion prevails in the absence of a favorable electrostatic interaction. The diffusion behavior of oppositely charged macromolecules in these gels is also unusual. Macroscopic measurements have shown adsorption rates that are faster than could be obtained by molecular diffusion in the bulk liquid phase with an equivalent concentration driving force.3-5 Moreover, microscopic observations of diffusion patterns of colored and fluorescently labeled proteins in these gels have revealed that diffuse profiles are established and that the adsorbed protein molecules retain diffusional mobility within the gel.7-10 It has been postulated that this behavior is due to the highly * To whom correspondence should be addressed. Tel.: (434) 924-6281. FAX: (434) 982-2658. E-mail:
[email protected].
favorable but reversible partitioning of these molecules in the gel. Accordingly, protein molecules are thought to diffuse while continuously interacting with oppositely charged functional groups through a tight-fitting polymer mesh. In this conceptual transport model the high protein concentration in the gel gives rise to a large diffusional driving force, which, in turn, results in a high rate of mass transfer.8 A measurement of the actual pore or mesh size of these gels is helpful in determining the validity of this conceptual model of transport as well as potential applications of these materials to broader ranges of molecules. Ordinary methods like mercury porosimetry or nitrogen adsorption are obviously not suitable for water-swollen gels. Thus, indirect methods such as quasielastic light scattering, tunneling electron microscopy of freeze-etched hydrogels, and electrophoretic mobility analysis have been used for these materials.11-13 Unfortunately, however, the results appear to vary depending on the method used. For example, Park et al.12 obtained a mesh size of 5.5 nm for a neutral 5% cross-linked polyacrylamide hydrogel with a 0.10 g/cm3 gel concentration using laser light scattering, while for the same gel Stellwagen13 obtained a mesh size of 21 nm using electrophoretic methods. Thus, methods where partitioning of macromolecules is measured directly are probably preferable if the goal is to determine the mesh size in relationship to size exclusion properties. Various approaches are available to describe partitioning of nonadsorbed solutes in gels. A widely used approach is based on the Ogston model14 according to which the partition coefficient, K ) q/C, defined by the
10.1021/ie050079m CCC: $30.25 © 2005 American Chemical Society Published on Web 09/29/2005
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ratio of solute concentrations in the gel, q, and in solution, C, is given by
[ ( )]
K ) exp -φ 1 +
rs rf
2
(1)
where φ is the polymer volume fraction in the gel, rs is the solute radius, and rf is the radius of the polymer fibers. Tong and Anderson15 found that protein partitioning in neutral polyacrylamide gels could be described by this equation using a fitted fiber radius of 0.65 nm. Different extensions of the basic Ogston model have also been proposed. Bosma and Wesselingh16 have modified this model by assuming that the gel fibers can overlap, obtaining the following result:
[
(1 -1 φ) (1 + r ) ] ) (1 - φ) rs
K ) exp - ln
2
[1 + (rs/rf)]2
(2)
Figure 1. Monomers used to prepare anionic and cationic gels: (a) AMPS; (b) MAPTAC.
f
It can be seen that at low values of φ the difference between this equation and the original Ogston model vanishes. However, larger differences exist at high values of φ when fiber overlapping presumably becomes significant. Johansson and Lo¨froth17 derived a more general form of the model by including a scaling factor, ν, to describe the fiber stiffness. The resulting equation, based on an empirical fit of Brownian dynamics simulations of hard spheres in polymer networks of wormlike chains, is
[ ( )]
K ) exp -φ 1 +
rs rf
ν
(3)
Obviously, this equation reduces to the Ogston model when ν ) 2. Partitioning data for nonadsorbing macromolecules in charged polyacrylamide gels with defined gel concentrations are currently not available. Thus, in this work, we determine the partition coefficient of fluorescently labeled dextrans in anionic and cationic polyacrylamide gels as well as the partition coefficient of fluorescently labeled, negatively charged proteins in anionic gels for a range of polymer concentrations. In both cases, the probe molecules are not adsorbed. Two functional characteristics of the polymer gels are determined from these measurements: the polymer fiber radius, rf, and the mesh size of the gels, ξ. The former is determined for different gel concentrations by comparing the experimental partition coefficients to the Ogston model and its different extensions using the fiber radius as an adjustable parameter. The mesh size is obtained on the basis of a single-radius pore model, where the partition coefficient is dependent on the pore radius, rp, according to
( )
K ) (1 - φ) 1 -
rs rp
2
(4)
We assume that the mesh size is equal to twice the pore radius determined from this equation. It should be emphasized that this is merely a functional definition of mesh size and that it does not necessarily reflect the actual geometrical arrangement of polymer fibers in the gel. Materials and Methods Preparation of Gels. Optically clear, cationic and anionic gels were synthesized within the lumen of fused
silica capillaries having a square section as described by Lewus and Carta7 and Russell et al.9 As in our previous work, 1 cm long sections of capillaries with 100 and 300 µm inside and outside dimensions (Polymicro Technologies, Phoenix, AZ) were stripped of their polymer coating and treated with Bind-Silane ((γ-(methacryloxy)propyl)trimethoxysilane) from Amersham Biosciences, Piscataway, NJ). The treated capillaries were then filled with a solution containing a water-soluble functionalized acrylamido monomer, a cross-linker (N,N′methylenebisacrylamide, MBA), an initiator (ammonium persulfate, AP), and a promoter (N,N,N′,N′tetramethylethylenediamine, TEMED). The monomer was (3-(methacryloylamino)propyl)trimethylammonium chloride (MAPTAC) for the cationic gels, and 2-acrylamido-2-methylpropanesulfonic acid (AMPS) for the anionic gels (Figure 1). Except for the anionic monomer, which was obtained from Lancaster (Pelham, NH), these chemicals were from Sigma Chemical Co. (St. Louis, MO). Gels with a polymer concentration of 0.16 g/cm3 and 5% cross-linking were prepared using 0.15 g of monomer, 0.0075 g of MBA, 0.0075 g of AP, and 1 µL of TEMED per cm3 of solution in distilled, deionized, degassed water. The corresponding charge density is 682 and 724 µmol/cm3 for the cationic and the anionic gel, respectively. Charged gels with gel concentrations of 0.10, 0.21, and 0.26 g/cm3 were prepared keeping the same proportions of the different components. Polymerization occurs rapidly after mixing and loading into the capillaries and is complete in less than an hour with a virtually complete incorporation of the monomer.9 Following polymerization, the gel-filled capillary sections were stored in a 10 mM, pH 6.5 Na2HPO4 buffer. An advantage of using gels synthesized within the constraints of these capillaries is that the gel volume remains constant. Thus, gel concentration and charge density are known precisely. Because of the covalent attachment to the capillary wall via the Bind-Silane, the capillary-supported gels are stable. Measurement of Partition Coefficients. Partition coefficients in the charged gels were determined using fluorescently labeled dextrans and proteins. Tetramethylrhodamine dextrans with molecular masses of 10, 40, and 70 kDa were obtained from Molecular Probes (Eugene, OR). Unfortunately, these samples are not monodispersed. For example, according to the manufacturer, the 70 kDa dextran contains polymers with molecular masses in the range of 60-90 kDa. Our
Ind. Eng. Chem. Res., Vol. 44, No. 22, 2005 8215 Table 1. Size of the Dextran Probes and Proteins Used in This Work probe
rs (nm)
10 kDa dextran 40 kDa dextran 70 kDa dextran ovalbumin BSA
2.7 18 5.0 18 6.3 18 2.8 25 3.6 25
measurements do not explicitly take into account the molecular weight distribution of these samples and thus have to be regarded as averages. The experiments were conducted with 0.05-2 mg/cm3 dextran solutions using greater concentrations for the larger dextrans in order to be able to obtain a consistent signal-to-noise ratio in the measurements of partitioning. The Stokes radii of the dextran probes were calculated from the empirical relationship of Oliver et al.18 and are given in Table 1. It should be noted that theories for partitioning of linear, coiled polymers in fiber matrixes suggest that the radius of gyration may be a better measure of dextran partitioning.19 Nonetheless, in this work we use the Stokes radius to define the size of the probe molecules since it has been suggested that globular proteins and dextrans are partitioned similarly for similar Stokes radii.20 Since our intent is to characterize charged polyacrylamide gels used in protein chromatography applications, this approach seems appropriate for predicting partitioning of spherical solutes.20 Partitioning experiments were also performed for the anionic gels with ovalbumin (Mr ∼ 45 000) labeled with rhodamine green and bovine serum albumin (BSA, Mr ∼ 65 000) labeled with tetramethylrhodamine isothiocyanate. The former was prepared as described by Russell and Carta,10 while the latter was obtained from Sigma. Solutions containing native and labeled protein (1 mg/cm3 total protein; degree of labeling, 68% for BSA and 15% for ovalbumin) were prepared in 10 mM sodium phosphate with 500 mM NaCl at a pH value of 6.5. Since the pI values of BSA and ovalbumin are 4.9 and 4.7, respectively, both proteins have a negative net charge at this pH.21 Partitioning experiments were conducted by placing the gel-filled capillaries in solutions for 1 or 2 days. After reaching equilibrium, the gels were placed on a glass slide under a fluorescence microscope (Nikon, Mod. Eclipse E200) at 100× magnification, and the fluorescence intensity in the gel was measured as described by Russell and Carta.10 The fluorescent intensity of solution-filled capillaries was also measured and found to be linearly correlated to the solute concentration. Results The partition coefficient of 10 kDa dextran in different gels is shown in Figure 2. More extensive data were obtained for this size dextran since, due to its smaller size, it could be used to explore broader ranges of gel concentrations. Moreover, it is closer in size to the proteins used in this work. Partitioning is obviously a strong function of gel concentration but is similar for anionic and cationic gels at the same gel concentration. The specific volume of the polymer, νpol, is needed in order to evaluate the polymer volume fraction, φ ) νpolCpol. Munk et al.22 have reported a value νpol ) 0.70 cm3/g for neutral polyacrylamide, but this value is not available for our charged gels. We thus measured the density of aqueous solutions of each monomer with a
Figure 2. Partition coefficient of 10 kDa dextran in anionic and cationic polyacrylamide gels. Lines are fitted with the Ogston model.
pycnometer and derived specific volumes of 0.69 and 0.87 cm3/g for AMPS and MAPTAC, respectively. We assume that these values can be taken as representative of the polymer density in the corresponding gels. Since the probe radius is known (see Table 1), rf can then be estimated from a least-squares fit of the data with the Ogston model. We obtain rf ) 0.79 ( 0.02 nm for the anionic gels and rf ) 0.99 ( 0.04 nm for the cationic gels. The larger rf value obtained for the cationic gel may be attributed to the bulkier trimethylammonium chloride pendant group (see Figure 1). The partition coefficient for the 10 kDa dextran was also determined for cationic gels with a gel concentration of 0.16 g/cm3 at a 400 mM ionic strength and was found to be 0.19 ( 0.03. The corresponding rf value is 1.08 ( 0.08 nm, which suggests that there is little dependence of partitioning on ionic strength in these constrained gels. The rf values determined for our charged gels are similar in magnitude to those obtained for neutral polyacrylamide gels by other authors by fitting with the Ogston model. For example, Ogston et al.23 and Tong and Anderson15 obtained values of rf ) 0.52 ( 0.06 nm and rf ) 0.65 nm, respectively, on the basis of protein partitioning in neutral cross-linked polyacrylamide gels, while Williams et al.24 obtained rf values in the range of 0.75-0.86 nm, based on partitioning of dextrans. Farnan et al.6 also used the Ogston model to describe protein partitioning in Q-HyperD-F chromatography particles. These particles consist of large-pore silica beads filled with a cationic polyacrylamide gel reportedly similar to gels used in our work. These authors determined a fiber radius of 0.5 nm. However, the actual gel concentration in the Q-HyperD-F particles was not known and these authors’ determination was based on an assumed polymer volume fraction of 0.05. The data in Figure 2 could also be fitted with the model of Bosma and Wesselingh.16 The rf values obtained in this case were 0.84 ( 0.02 and 1.07 ( 0.04 nm, for the anionic and cationic gels, respectively. These values are clearly not significantly different from those obtained with the Ogston model. Figure 3 shows the mesh size calculated from eq 4 with the single-pore-radius model in comparison with the estimates of Park et al.12 and Stellwagen13 for neutral polyacrylamide gels. Our values for charged gels are intermediate between the two. The differences can be ascribed in part to the different methods used.
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Figure 3. Mesh size calculated from Ogston model for anionic and cationic gels using 10 kDa dextran as a probe. Results for neutral polyacrylamide gels are from Park et al.12 and Stellwagen.13 Table 2. Comparison of Experimental K-Values with Model Predictions Based on the 10 KDa Dextran Data for Anionic Gelsa probe
Cgel (mg/cm3)
Kexptl
40 kDa dextran 70 kDa dextran ovalbumin BSA
0.16 0.10 0.16 0.10
0.024 ( 0.006 0.021 ( 0.003 0.060 ( 0.010 0.080 ( 0.010
KOgstonb KB&Wc KJ&Ld 0.004 0.004 0.118 0.119
0.005 0.006 0.129 0.136
0.012 0.023 0.100 0.151
a The Johansson and Lo ¨ froth model was fitted to the dextran data in Figure 4. b Ogston model, eq 1. c Bosma and Wesselingh model, eq 2. d Johansson and Lo¨froth model, eq 3.
Stellwagen determined the mesh size by observing the gel concentration where the electrophoretic mobility of a probe is reduced to half of the free solution value. Park et al., on the other hand, determined the mesh size using photon correlation spectroscopy. In any case, these results indicate that the mesh size of these gels is similar in magnitude to the size of typical globular proteins with diameters in the 6-8 nm range. The results for partitioning of dextrans with different molecular masses and for partitioning of ovalbumin and BSA are shown in Table 2 for different anionic gels. The table also shows the predicted K-values based on the Ogston and Bosma and Wesselingh models using the parameters derived from the 10 kDa dextran results. It can be seen that the predictions are reasonable for the proteins but inaccurate for the larger dextrans with 40 and 70 kDa molecular masses. The dextran data were thus refitted with the Johansson and Lo¨froth model (eq 3) using both rf and ν as regressed parameters. The corresponding fitted values are rf ) 0.30 ( 0.05 nm and ν ) 1.29 ( 0.08. A parity plot comparing experimental and calculated values is shown in Figure 4, and values of K are shown in Table 2. The maximum relative error is around 50% for the 40 kDa dextran and is much smaller for the other probes. Discussion and Conclusions The partitioning of nonadsorbing solutes in charged polyacrylamide gels is a strong function of gel concentration and solute size. The polymer fiber radii estimated on the basis of the Ogston model from the 10 kDa dextran partitioning are around 0.7 nm for the anionic gels and around 0.9 nm for the cationic gels. These
Figure 4. Comparison of experimental and calculated partition coefficients for dextrans and proteins in anionic gels on the basis of the model of Johansson and Lo¨froth.17
values are similar to those estimated for neutral polyacrylamide gels with the same degree of cross-linking. The mesh size in these charged gels varies between about 11 and about 6 nm for gel concentrations in the range of 0.1-0.26 mg/cm3. Similar values are obtained using the model of Bosma and Wesselingh. However, the K-values predicted by both the Ogston and the Bosma and Wesselingh models are consistently lower than the experimental ones for the larger probes, suggesting that the charged polymer gels are actually capable of accommodating larger solutes than expected. Both of these models assume that the polymer chains are rigid. On the other hand, the Johansson and Lo¨froth model could describe approximately both the dependence on φ and the dependence on solute size. This model takes into account the flexibility of the polymer chains through the scaling parameter ν. It appears that accounting for the flexibility of the polymer chains may have to be taken into account in order to predict the effect of probe size on partitioning, especially when the size of the probe approaches the gel mesh size. The partition coefficients for the proteins are somewhat overestimated by the model. This could result from the inherent difficulty of predicting the partitioning of charged globular molecules from the behavior of the neutral dextrans. Nonetheless, despite the quantitative uncertainties, an important conclusion can be reached regarding the implications of the small mesh sizes obtained for our charged polyacrylamide gels on protein adsorption rates. Previous work had hypothesized that mass transfer of oppositely charged proteins in these gels occurs through a tight-fitting polymer network with the protein molecules continuously interacting with multiple charged functional groups as they diffuse through the gel matrix. According to this hypothesis, high rates of mass transfer are obtained because of the large driving force for diffusion afforded by the high protein concentration in the gel. For example, Russell and Carta9 observed adsorption rates of negatively charged myglobin (rs ∼ 2 nm) in a 0.21 g/cm3 cationic gel that were 20-30 times faster than could be obtained by diffusion in a free solution with the same concentration gradient driving force. On the basis of the results of the present work, this gel has a mesh size of about 7 nm, which is less than double the diameter of the
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diffusing protein. Accordingly, diffusion is expected to be highly hindered in this gel, and indeed we found9 the diffusion coefficient of myoglobin in this gel to be about 8 × 10-8 cm2/s or only about 1/13th of its free solution diffusivity.25 On the other hand, the rate of adsorption depends on the product of the diffusion coefficient and the driving force. For this system, the diffusion coefficient is small because of the small mesh size, but the driving force is very large because of the very favorable partitioning of myoglobin in this cationic gel at low ionic strengths (K ∼ 300 at 1 mg/cm3 protein concentration in 50 mM Tris buffer9). As a final result the rate of adsorption is large. Similar results have been found for other proteins in anionic polyacrylamide gels8 and for commercial chromatography particles incorporating similar gels.3-6 Acknowledgment This research was supported by NSF Grants CTS0079334 and 0414143. Nomenclature C ) solute concentration in the liquid phase, mg/cm3 Cpol ) gel concentration in the gel, g/cm3 q ) solute concentration in the gel, mg/cm3 K ) partition coefficient rf ) fiber radius, nm rp ) pore radius, nm rs ) solute radius, nm νpol ) specific volume of polymer fibers, cm3/g ν ) scaling factor in eq 3 φ ) volume fraction of polymer fibers in gel ξ ) mesh size of gel, nm
Literature Cited (1) Tanaka, T. Gels. Sci. Am. 1981, 244, 124. (2) Boschetti, E. Advanced Sorbents for Preparative Protein Separation Purposes. J. Chromatogr., A 1994, 658, 207. (3) Fernandez, M. A.; Carta, G. Characterization of Protein Adsorption by Composite Silica-Polyacrylamide Gel Anion Exchangers. I. Equilibrium and Mass Transfer in Agitated Contactors. J. Chromatogr., A 1996, 746, 169. (4) Weaver, L. E.; Carta, G. Protein Adsorption on Cation Exchangers: Comparison of Macroporous and Gel-Composite Media. Biotechnol. Prog. 1996, 12, 342. (5) Farnan, D.; Frey, D. D.; Horvath, Cs. Surface and Pore Diffusion in Macroporous and Gel-Filled Gigaporous Stationary Phases for Protein Chromatography. J. Chromatogr., A 2002, 959, 65. (6) Farnan, D.; Frey, D. D.; Horvath, Cs. Intraparticle Mass Transfer in High-Speed Chromatography of Proteins. Biotechnol. Prog. 2002, 13, 429, (7) Lewus, R. K.; Carta, G. Protein Diffusion in Charged Polyacrylamide Gels: Visualization and Analysis. J. Chromatogr., A 1999, 865, 155.
(8) Lewus, R. K.; Carta, G. Protein Transport in Constrained Anionic Hydrogels: Diffusion and Boundary Layer Mass Transfer. Ind. Eng. Chem. Res. 2001, 40, 1548. (9) Russell, S. M.; Belcher, E. B.; Carta, G. Protein Partitioning and Transport in Supported Cationic Acrylamide-Based Hydrogels. AIChE J. 2003, 49, 1168. (10) Russell, S. M.; Carta, G. Multicomponent Protein Partitioning and Transport in Supported Cationic Acrylamide-Based Hydrogels. AIChE J. 2005, 51, 8. (11) Ruchel, R.; Steere, R. L.; Erbe, E. F. TransmissionElectron Microscopic Observations of Freeze-Etched Polyacrylamide Gels. J Chromatogr., A 1978, 166, 563. (12) Park, I. H.; Johnson, C. S.; Gabriel, D. A. Probe Diffusion in Polyacrylamide Gels As Observed by Means of Holographic Relaxation Methods: Search for a Universal Equation. Macromolecules 1990, 23, 1548. (13) Stellwagen, N. C. Apparent Pore Size of Polyacrylamide Gels: Comparison of Gels Cast and Run in Tris-acetate-EDTA and Tris-borate-EDTA buffers. Electrophoresis 1998, 19, 1542. (14) Ogston, A. G. The Spaces in a Uniform Random Suspension of Fibres. Trans. Faraday Soc. 1958, 54, 1754. (15) Tong, J.; Anderson, J. L. Partitioning and Diffusion of Proteins and Linear Polymers in Polyacrylamide Gels. Biophys. J. 1996, 70, 1505. (16) Bosma, J. C.; Wesselingh, J. A. Partitioning and Diffusion of Large Molecules in Fibrous Structures. J. Chromatogr., B: Biomed. Sci. Appl. 2000, 743, 169. (17) Johansson, L.; Lo¨froth, J. E. Diffusion and Interaction in Gels and Solutions. 4. Hard Sphere Brownian Dynamics Simulations. J. Chem. Phys. 1993, 98, 7471. (18) Oliver, J. D.; Anderson, S.; Troy, J. L.; Brenner, B. M.; Deen, W. M. Determination of Glomerular Size-Selectivity in the Normal Rat with Ficoll. J. Am. Soc. Nephrol. 1992, 3, 214. (19) White, J. A.; Deen, W. M. Equilibrium Partitioning of Flexible Macromolecules in Fibrous Membranes and Gels. Macromolecules 2000, 33, 8504. (20) Hagel, L.; O ¨ stberg, M.; Andersson, T. Apparent Pore Size Distributions of Chromatography Media. J. Chromatogr., A 1996, 743, 33. (21) Righetti, P. G.; Caravaggio, T. Isoelectric Points and Molecular Weights of ProteinssA Table. J. Chromatogr., A 1976, 127, 1. (22) Munk, P.; Aminabhavi, T. M.; Williams, P.; Hoffman, D. E. Some Solution Properties of Polyacrylamide. Macromolecules 1980, 13, 871. (23) Ogston, A. G.; Preston, B. N.; Wells, J. D. On the Transport of Compact Particles Through Solutions of Chain. Proc. R. Soc. London, Ser. A 1973, 333, 297. (24) Williams, J. C.; Mark, L. A.; Eichholtz, S. Partition and Permeation of Dextran in Polyacrylamide Gel. Biophys. J. 1998, 75, 493. (25) Tyn, M. T.; Gusek, T. W. Prediction of Diffusion Coefficients of Proteins. Biotechnol. Bioeng. 1990, 35, 327.
Received for review January 19, 2005 Revised manuscript received June 14, 2005 Accepted August 29, 2005 IE050079M
