Hindered Convection of Ficoll and Proteins in Agarose Gels

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Hindered Convection of Ficoll and Proteins in Agarose Gels Scott T. Johnston† and William M. Deen*,†,‡ Department of Chemical Engineering and Division of Bioengineering and Environmental Health, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

The hindered convection of rigid macromolecules in hydrogels was studied by measuring the sieving coefficient (Θ, the ratio of filtrate concentration to retentate concentration) of various sizes of Ficoll for agarose membranes of varying concentrations. Samples of filtrate and retentate were fractionated by size-exclusion chromatography to obtain results for Ficolls with specific Stokes-Einstein radii (rs) ranging from 3.0 to 7.0 nm, and the volume fraction of agarose (φ) was varied from 0.04 to 0.08. Immunoglobulin G (IgG, rs ) 5.2 nm) was also employed as a test solute. The values of Θ decreased with increasing rs or increasing φ, as expected. That is, transport was increasingly hindered as steric solute-gel interactions became more prominent. The sieving coefficient for Ficoll was very similar to that for IgG of the same rs, as well as that for other globular proteins studied previously, indicating that molecular shape and charge were not important variables. The calculated values of the convective hindrance factor (Kc, the ratio of solute velocity to fluid velocity) tended to increase above unity and then to decrease toward zero as rs was increased, a trend that is qualitatively consistent with predictions from a previously reported theory for the hindered convection of rigid spheres through regular arrays of parallel fibers. Introduction The convection and diffusion of macromolecules in a variety of porous or fibrous media, including synthetic membranes, hydrogels, and various body tissues, is typically hindered relative to transport in a free solution. The reduced fluxes are a consequence of steric and hydrodynamic interactions between the mobile macromolecule and the fixed structures (pore walls or fibers), which become increasingly prominent as the characteristic dimension of the liquid-filled spaces (pore radius or interfiber spacing) approaches that of the macromolecular solute. The observed hindrances might also be influenced by colloidal (electrostatic or van der Waals) forces. For an isotropic material, the solute flux (N) can be expressed as

N ) -KdD∞∇C + KcvC

(1)

where C is the solute concentration, D∞ is the free (bulk) solution diffusivity, and v is the fluid velocity. For gels, which are the materials of interest in this study, C is based on the total volume (including solids), and v is the superficial velocity. The hindrances to diffusion and convection are expressed by the coefficients Kd and Kc, respectively. The diffusive hindrance factor is the diffusivity in the confined phase relative to that in free solution (i.e., Kd ) D/D∞), whereas the convective hindrance factor is the ratio of the solute velocity to the fluid velocity in the absence of diffusion. The objective of hindered transport theory is to predict the values of Kd and Kc from the solute size and shape and the structure of the porous or fibrous material. * Address for correspondence: William M. Deen, Department of Chemical Engineering, 66-572, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139. Telephone: (617) 253-4535. Fax: (617) 258-8224. E-mail: [email protected]. † Department of Chemical Engineering. ‡ Division of Bioengineering and Environmental Health.

Although this objective has been largely accomplished for the case of rigid, spherical solutes moving through pores of uniform cross section,1 much work remains to be done for flexible macromolecules and/or for media with more complex structures. The disordered structure of various synthetic or natural hydrogels suggests that they be modeled as arrays of randomly oriented fibers with fluid-filled interstices. This approach has provided the basis for theories that describe the equilibrium partitioning of rigid2-4 and flexible5 macromolecules between fibrous media and bulk solutions and for recent predictions of Kd for rigid spheres in fibrous media.6-8 However, the only comparable theory for Kc is restricted to parallel arrays of fibers.9 The transport of macromolecules in hydrogels has been investigated using a variety of experimental techniques. Here again, though, most of the available data are for diffusion.6,10-13 The only studies from which one can derive values of Kc appear to be those of Kapur et al.,14 who measured transport rates of two proteins in large pores filled with polyacrylamide gels, and Johnston and Deen,15 who measured sieving coefficients for three proteins in membranes made from agarose gels. The objective of the present study was to supplement that information by examining the sieving of polydisperse Ficoll across agarose gels of varying concentrations. Ficoll, a cross-linked copolymer of sucrose and epichlorohydrin, has been found to diffuse like a rigid, neutral sphere,16,17 making it a nearly ideal test solute. By using size-exclusion chromatography to fractionate samples of the filtrate and retentate, data were obtained for a wide range of molecular radii in each Ficoll experiment. To extend the previously studied range of protein sizes, the sieving of immunoglobulin G (IgG) was also examined. The results permit a critical evaluation of certain approaches that have been proposed to predict convective hindrance factors (or equivalently, reflection coefficients) in gels or other fibrous media.

10.1021/ie010085s CCC: $22.00 © 2002 American Chemical Society Published on Web 06/29/2001

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Methods Macromolecules. Bovine IgG and a polydisperse preparation of Ficoll with a weight-average molecular weight of 70 000 (type 70) were obtained from Sigma (St. Louis, MO) and used without further purification. The Ficoll was labeled with 5-([4,6-dichlorotriazin-2-yl]amino) fluorescein (DTAF) (Sigma) using the procedure described by De Belder and Granath.18 Unreacted DTAF was removed by elution through 10 mL disposable desalting columns (Econo-Pac 10 DG, Bio-Rad, Hercules, CA). Labeled Ficoll was concentrated in a 200-mL ultrafiltration cell (model 8200, Amicon, Beverly, MA) with a regenerated cellulose membrane (PLCC 062 10, Millipore, Bedford, MA) that had a 5-kD molecular weight cutoff. Concentrated samples were freeze-dried until use. Fresh aqueous solutions were prepared by dissolving the macromolecules in a buffer consisting of 0.01 M sodium phosphate and 0.1 M KCl at pH 7.0. The solute concentration in all solutions was 2 mg/mL, and solutions were filtered using a 0.22 µm syringe filter (MillexGV Filter Unit, Millipore) prior to use in sieving experiments. Gel Membranes. Agarose membranes of 70-µm thickness were prepared on polyester mesh supports, as done previously.6,15 Agarose powder (type VI, high gelling temperature, Sigma) was added to a sodium phosphate/KCl buffer. The slurry was placed in an oven at 90 °C to dissolve the agarose and was shaken periodically by hand to ensure adequate mixing. Membranes were cast by placing a 25-mm-diameter piece of polyester mesh (Spectra/Mesh Polyester Filters, Spectrum Medical Industries, Inc., Houston, TX) on a glass plate (also heated to 90 °C) and pouring the hot agarose slurry onto it. A second hot glass plate was placed on top, and any air was squeezed out. The plates were clamped together, placed in buffer, and stored overnight at 4 °C. Membranes were prepared with agarose concentrations of 4, 6, and 8% (w/v). The volume fraction of agarose was obtained by dividing the mass fraction by 1.025.10 Darcy Permeability Measurements. The Darcy permeability of each agarose sample was determined by measuring the hydraulic permeability of the meshsupported membrane, as detailed previously.15 Briefly, membranes were placed in a stirred cell filled with sodium phosphate/KCl buffer. A transmembrane pressure difference (∆P) was applied using compressed nitrogen and was monitored with a pressure transducer. The values of ∆P were chosen so that the filtrate velocities would be ∼10-4 cm/s in all experiments. The values used were 1.9, 5.2, and 15 kPa for the 4, 6, and 8% gels, respectively. Corrections were made to account for hydrostatic pressure. The flow rate of solution through the membrane was determined by weighing the filtrate. The Darcy permeability was calculated as

κ)

µQL βA∆P

(2)

where µ is the viscosity of the buffer, Q is the volumetric flow rate through the membrane, L is the membrane thickness, β is a factor that accounts for the presence of the polyester mesh, and A is the exposed membrane area. For the polyester meshes used, it was determined that β ) 0.510.15 Sieving Measurements. Following the determination of the Darcy permeability for a particular mem-

brane, the ultrafiltration cell was emptied and refilled with sodium phosphate/KCl buffer containing 2 mg/mL of IgG or polydisperse Ficoll. The stirring rate, calibrated using a strobe, was set at 220 rpm, and pressure was applied again. After a delay of 1 h to minimize transient effects and purge the collection line, the filtrate was collected for 1 h. The retentate was then removed from the cell, which was rinsed and again filled with protein-free buffer. A second measurement of the Darcy permeability was performed following the sieving experiment, and data from an experiment were retained only if the hydraulic permeability changed by 1 only for the smallest molecule studied (lactalbumin). Otherwise, Kc decreases with increasing solute size, down to an average of 0.51 for the largest Ficoll. Also plotted in Figures 4-6 are theoretical predictions for Kc for transport perpendicular to a parallel array of fibers.9 Note that the data and theory exhibit the same qualitative behavior; in particular, at small φ both indicate that Kc > 1 for the smaller solutes. This is because the finite size of the solute prevents it from sampling regions near solid boundaries where the fluid velocity is lowest. Both the data and the theory show that, as molecular size and/or gel concentration increases, Kc decreases to values below 1. This is a result of the hindering effect of the fibers, which causes the solute velocity to fall below the average fluid velocity. However, quantitative agreement between the theory and data is poor. This is not surprising, because the theory is for parallel fibers arranged on a regular lattice, whereas agarose fibrils have a more random orientation.6 In a hypothetical material with regularly spaced, parallel fibers, Kc ) 0 when the solute size exceeds the dimension of the gaps between adjacent fibers. Accordingly, the theory predicts a sharp cutoff as rs is increased. For a randomly oriented array of fibers there will be a broad distribution of gap sizes and a more gradual decline in Kc with increasing solute size, consistent with the agarose data. Uncertainties in the estimates for Φ and Kd were found to have only modest effects on the calculated

Figure 7. Reflection coefficient (σ) as a function of partition coefficient (Φ) in 4% agarose. The pore theory curve is from eq 11, derived by Anderson and Malone.26

Figure 8. Reflection coefficient (σ) as a function of partition coefficient (Φ) in 6% agarose. The pore theory curve is from eq 11, derived by Anderson and Malone.26

values of Kc. Increasing Φ by 10% results in an average decrease in Kc of 9.9%, and decreasing Φ by 10% results in an average increase in Kc of 12%. The convective hindrance coefficient is much less sensitive to changes in Kd. Increasing Kd by 10% results in an average decrease in Kc of only 1%, and decreasing Kd by 10% results in an average increase in Kc of just 0.9%. The current results can be compared with previous work on hindered convection of solutes in pores by considering the convective reflection coefficient, σ, which is given by

σ ) lim (1 - Θ) ) 1 - ΦKc Pef∞

(10)

Anderson and Malone26 derived an expression for the osmotic reflection coefficient for neutral spheres in cylindrical pores, and Anderson27 found that result to be a good approximation also for the convective reflection coefficient in cylindrical pores and for either reflection coefficient in straight pores of noncircular cross section. The expression is

σ ) (1 - Φ)2

(11)

Curry and Michel28 assumed that this simple result is also applicable to fiber-matrix membranes. To test whether that is true for agarose, the present results are compared with the predictions of eq 11 in Figures 7-9. Equation 11 does a remarkably good job of predicting values of σ for solutes in 4% agarose gels (Figure 7). However, as φ increases, the agreement worsens (Figures 8 and 9). This trend was not evident in our earlier

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{

[( ) ] }

1 - σ ) 1 + 2400

Figure 9. Reflection coefficient (σ) as a function of partition coefficient (Φ) in 8% agarose. The pore theory curve is from eq 11, derived by Anderson and Malone.26

rs φ rf

4 -1

(13)

Kapur et al.14 also found differences between their diffusive hindrance coefficients in polyacrylamide and those of Johnson et al.6 in agarose. In conclusion, data were obtained for the convective hindrance factors of IgG and several sizes of Ficoll in agarose gels. Relative to previous data for hindered convection of three other proteins (lactalbumin, ovalbumin, and BSA) in agarose, these results expanded the range of solute sizes and provided a comparison between globular proteins and Ficoll, a neutral solute that has been found to behave hydrodynamically as a hard sphere. The present results show that the key solute property is the Stokes-Einstein radius, and they confirm our previous conclusion that convective hindrances in agarose gels are not predicted well by available theories either for parallel arrays of fibers or for straight pores. These findings should help motivate the development of a more complete hindered transport theory for randomly oriented arrays of fibers. Acknowledgment This work was supported by Grant DK20368 from the National Institutes of Health. Notation

Figure 10. Sieving coefficient for pure convection (1 - σ) for Ficoll and globular proteins in agarose gels. The abscissa is the ratio of the solute radius to the fiber radius (rs/rf) times the fiber volume fraction (φ). The curves correspond to eq 12 (Kapur et al.14) and eq 13 (fit to present data).

study,15 because of the more limited range of solute sizes studied. It is not known whether eq 11 correctly describes the limiting behavior of σ at low agarose concentrations or whether its success at 4% was coincidental. Lower gel concentrations were not studied because of the difficulty in handling such gels and also the lack of data for Kd. Another problem with interpreting such experiments is that the measured reflection coefficients would be close to zero. Kapur et al.14 measured the reflection coefficients of ribonuclease A (RNAse) and BSA using poly(vinylidine fluoride) membranes whose pores were filled with polyacrylamide gel. The polyacrylamide fiber volume fraction was varied from 0.04 to 0.09. Estimating partition coefficients from hindered diffusion studies carried out using the same membranes, they found good agreement between their data for BSA and the prediction of eq 11. However, their data for RNAse deviated significantly from this model. Those authors presented the following empirical equation, which was a fit to their experimental data for both solutes:

{

1 - σ ) 1 + 127

[( ) ] } rs φ rf

4 -1

(12)

Equation 12 is compared with the present data in Figure 10. Although it appears that σ in agarose gels is a function only of (rs/rf)φ, as proposed, the agreement between the data and eq 12 is poor. As seen in the plot, a good fit to the present data is given by

A ) exposed membrane area C ) volume-average concentration in gel Cf ) concentration in filtrate Cm ) concentration at membrane surface Cb ) concentration in bulk retentate D ) apparent diffusivity in gel D∞ ) diffusivity in free (bulk) solution f ) adjusted fiber volume fraction (eq 6) Kc ) convective hindrance coefficient Kd ) diffusive hindrance coefficient L ) membrane thickness N ) local solute flux (vector) P ) pressure Pe ) membrane Pe´clet number Q ) volumetric filtration rate rs ) Stokes-Einstein radius of solute rf ) gel fiber radius v ) superficial fluid velocity (vector); magnitude ) v Greek Letters β ) polyester mesh correction factor ∆ ) transmembrane difference, upstream minus downstream (as in ∆P) Θ ) true sieving coefficient (corrected for concentration polarization) Θ′ ) measured or apparent sieving coefficient κ ) Darcy permeability µ ) viscosity σ ) convective reflection coefficient φ ) gel fiber volume fraction Φ ) partition coefficient

Literature Cited (1) Deen, W. M. Hindered Transport of Large Molecules in Liquid-Filled Pores. AIChE J. 1987, 33, 1409. (2) Ogston, A. G. The Spaces in a Uniform Random Suspension of Fibers. Trans. Faraday Soc. 1958, 54, 1754.

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(3) Fanti, L. A.; Glandt, E. D. Partitioning of Spherical Particles into Fibrous Matrices 1. Density Functional Theory. J Colloid Interface Sci. 1990, 135, 396. (4) Lazzara, M. J.; Blankschtein, D.; Deen, W. M. Effects of Multisolute Steric Interactions on Membrane Partition Coefficients. J. Colloid Interface Sci. 2000, 226, 112. (5) White, J. A.; Deen, W. M. Equilibrium Partitioning of Flexible Macromolecules in Fibrous Membranes and Gels. Macromolecules 2000, 33, 8504. (6) Johnson, E. M.; Berk, D. A.; Jain, R. K.; Deen, W. M. Hindered Diffusion in Gels: Test of the Effective Medium Model. Biophys. J. 1996, 70, 1017. (7) Clague, D. S.; Phillips, R. J. Hindered Diffusion of Macromolecules through Dilute Fibrous Media. Phys. Fluids 1996, 8, 1720. (8) Phillips, R. J. A Hydrodynamic Model for Hindered Diffusion of Proteins and Micelles in Hydrogels. Biophys. J. 2000, 79, 3350. (9) Phillips, R. J.; Deen, W. M.; Brady, J. F. Hindered Transport in Fibrous Membranes and Gels: Effect of Solute Size and Fiber Configuration. J. Colloid Interface Sci. 1990, 139, 363. (10) Johnson, E. M.; Berk, D. A.; Jain, R. K.; Deen, W. M. Diffusion and Partitioning of Proteins in Charged Agarose Gels. Biophys. J. 1995, 68, 1561. (11) Gibbs, S. J.; Lightfoot, E. N.; Root, T. W. Protein Diffusion in Porous Gel-Filtration Chromatography Media Studied by Pulsed Field Gradient NMR Spectroscopy. J. Phys. Chem. 1992, 96, 7458. (12) Kosar, T. F.; Phillips, R. J. Measurement of Protein Diffusion in Dextran Solutions by Holographic Interferometry. AIChE J. 1995, 41, 701. (13) De Smedt, S. C.; Meyvis, T. K. L.; Demeester, J.; Van Oostveldt, P.; Blonk, J. C. G.; Hennink, W. E. Diffusion of Macromolecules in Dextran Methacrylate Solutions and Gels as Studied by Confocal Scanning Laser Microscopy. Macromolecules 1997, 30, 4863. (14) Kapur, V.; Charkoudian, J.; Anderson, J. L. Transport of Proteins through Gel-Filled Porous Membranes. J. Membr. Sci. 1997, 131, 143. (15) Johnston, S. T.; Deen, W. M. Hindered Convection of Proteins in Agarose Gels. J. Membr. Sci. 1999, 153, 271. (16) Bohrer, M. P.; Patterson, G. D.; Carroll, P. J. Hindered Diffusion of Dextran and Ficoll in Microporous Membranes. Macromolecules 1984, 17, 1170.

(17) Davidson, M. G.; Deen, W. M. Hindered Diffusion of WaterSoluble Macromolecules in Membranes. Macromolecules 1988, 21, 3474. (18) De Belder, A. N.; Granath, K. Preparation and Properties of Fluorescein-Labeled Dextrans. Carbohydrate Res. 1973, 30, 375. (19) Johnston S. T.; Smith, K. A.; Deen, W. M. Concentration Polarization in Stirred Ultrafiltration Cells. AIChE. J. 2001, 47, 1115. (20) Vilker, V. L.; Colton, C. K.; Smith, K. A. The Osmotic Pressure of Concentrated Protein Solutions: Effect of Concentration and pH in Saline Solutions of Bovine Serum Albumin. J. Colloid Interface Sci. 1981, 79, 548. (21) Potschka, M. Universal Calibration of Gel Permeation Chromatography and Determination of Molecular Shape in Solution. Anal. Biochem. 1987, 162, 47. (22) Djabourov, M.; Clark, A. H.; Rowlands, D. W.; RossMurphy, S. B. Small-Angle X-ray Scattering Characterization of Agarose Sols and Gels. Macromolecules 1989, 22, 180. (23) Johnson, E. M. Partitioning and Diffusion of Macromolecules in Charged Gels. Ph. D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1995. (24) Solomentsev, Y. E.; Anderson, J. L. Rotation of a Sphere in Brinkman Fluids. Phys. Fluids 1996, 8, 1119. (25) 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. (26) Anderson, J. L.; Malone, D. M. Mechanism of Osmotic Flow in Porous Membranes. Biophys. J. 1974, 14, 957. (27) Anderson, J. L. Configurational Effect on the Reflection Coefficient for Rigid Solutes in Capillary Pores. J. Theor. Biol. 1981, 90, 405. (28) Curry, F. E.; Michel, C. C. A Fiber Matrix Model of Capillary Permeability. Microvasc. Res. 1980, 20, 96.

Received for review January 26, 2001 Revised manuscript received May 14, 2001 Accepted May 17, 2001 IE010085S