Importance of Biopolymer Molecular Flexibility in Ultrafiltration

Aug 15, 2008 - Ultrafiltration is used extensively for the purification of therapeutic biomolecules, with the behavior of proteins well-described usin...
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Ind. Eng. Chem. Res. 2009, 48, 2395–2403

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Importance of Biopolymer Molecular Flexibility in Ultrafiltration Processes David R. Latulippe, Jessica R. Molek, and Andrew L. Zydney* Department of Chemical Engineering, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802

Ultrafiltration is used extensively for the purification of therapeutic biomolecules, with the behavior of proteins well-described using available hard-sphere models. The objective of this study was to examine the role of molecular flexibility in the ultrafiltration of plasmid DNA and PEGylated proteins, two important classes of next generation therapeutics. Plasmid DNA transmission was strongly dependent on the filtrate flux with very high transmission (>60%) obtained at the highest flux even though the radius of gyration of the plasmid was an order of magnitude larger than the membrane pore size. This behavior was consistent with models for infinitely flexible polymers accounting for elongation/deformation in the flow-field above the pore, although the elongation became significant at very small Deborah numbers using the available theory. In contrast, the transmission of PEGylated proteins was controlled by hard-sphere interactions at low filtrate flux, with the covalently attached polyethylene glycol (PEG) chains increasing the effective size of the biomolecule. However, there was clear evidence of elongation at high flux, particularly for PEGylated proteins linked to larger molecular weight polyethylene glycol chains. These results provide important insights into the role of biopolymer flexibility on the ultrafiltration characteristics of these second-generation biotherapeutics. Introduction Ultrafiltration is used extensively for the purification and concentration of a wide range of biomolecules including therapeutic proteins, industrial enzymes, natural protein products, and diagnostic antibodies. These proteins typically have a dense hydrophobic core, giving them a highly globular structure with relatively little molecular flexibility. Protein transport through semipermeable membranes has been very effectively described using theoretical models for hard spheres accounting for the steric, hydrodynamic, and long-range (electrostatic) interactions between the protein and the pore. The protein sieving coefficient (Sa) at high filtration velocities is expressed as the product of the partition coefficient (φ), which describes the thermodynamic partitioning between free solution and the pore space, and the convective hindrance coefficient (Kc), which describes the hydrodynamic interactions that govern protein motion through the pore:1 Sa ) φKc 1

(1) 2

Deen and more recently Dechadilok and Deen have provided a thorough review of available theoretical models for the hydrodynamic hindrance factors for rigid particles in both cylindrical and slit-shaped pores. The resulting equations have been successfully used to describe the effects of membrane pore size,3,4 solution pH,5 solution ionic strength,6-8 and membrane surface charge density9 on protein transport through a variety of semipermeable ultrafiltration membranes. In contrast, there is extensive experimental evidence suggesting that several important second generation biotherapeutics have a significant degree of molecular flexibility and cannot be effectively described using models for rigid spheres. For example, there is considerable interest in using plasmid DNA as an advanced biotherapeutic for both gene therapy applications and DNA-based vaccines. Plasmids are circular double-stranded extrachromosomal DNA containing the genetic information needed to express a desired protein/antigen in targeted cells. There have been over 200 human clinical trials of plasmid DNA* To whom correspondence should be addressed. Phone: 814-8637113. Fax: 814-865-7846. E-mail: [email protected].

based therapies over the past decade,10 and recently, the U.S. Department of Agriculture issued a conditional license for a plasmid DNA vaccine to treat canine melanoma.11,12 Most studies have shown that the supercoiled topology provides higher cell transfection efficiencies and gene expression levels compared to either the open-circular (i.e., nicked) or linear forms. Supercoiled plasmids for gene therapy and DNA vaccination behave as flexible polymer chains in solution since their total length, typically from 3 to 20 kilo-base pairs (kbp), is many times greater than the persistence length of DNA (typically 0.15 kbp), which is a measure of the stiffness of the double helix. The flexibility of DNA has been directly observed via fluorescence microscopy tracking of individual DNA molecules in various flow fields,13-15 and it has been accurately predicted using Monte Carlo simulations.16 Latulippe et al.17 have recently shown that plasmid DNA transmission through ultrafiltration membranes is strongly dependent on the filtration velocity due to the elongation/deformation of the large plasmid within the converging flow field above the membrane pores. PEGylated proteins, which are formed by covalent attachment of one or more long polyethylene glycol (PEG) chains to the native protein, have an even more complex morphology. This new class of biotherapeutic molecules has increased half-life and stability, leading to significant reductions in dosing level and frequency compared to conventional therapeutic proteins. PEGylated proteins are currently used to treat hepatitis C,18 neutropenia,19 and acromegaly.20 Although PEGylated proteins appear to behave as rigid solutes in applications of size exclusion chromatography,21 there is strong evidence suggesting that these biomolecules can elongate in response to the fluid flow during ultrafiltration.22 Vonarbourg et al.23 showed that PEGylated lipid nanocapsules caused very low levels of complement activation, which was attributed to the molecular flexibility of the attached PEG chains. The molecular flexibility of long (isolated) PEG chains has been well established in a variety of experimental systems24-26 including work on the ultrafiltration of PEG with molecular weights above 35 kDa.27,28 The objective of this study was to examine the role of molecular flexibility in the transport of these novel biomolecules through semipermeable ultrafiltration membranes. Ultracel

10.1021/ie8005337 CCC: $40.75  2009 American Chemical Society Published on Web 08/15/2008

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composite regenerated cellulose membranes were used in all experiments since these membranes are currently used in a wide range of bioprocessing applications.29 Experiments were performed with a supercoiled plasmid and with a series of PEGylated proteins. Data were analyzed using theoretical models for both flexible polymers and rigid spheres, with the results providing important insights into the effects of molecular flexibility on the behavior of these complex biomolecules during ultrafiltration. Theoretical Background In contrast to the behavior of rigid particles, the transport characteristics of highly flexible polymers have focused on the analysis of the flow-induced elongation/deformation that occurs in the vicinity of the pore entrance.1 The degree of polymer deformation is characterized by the Deborah number (De): τ ) τγ (2) γ-1 which is equal to the time scale for polymer relaxation (τ) divided by the time scale characteristic of the fluid flow. The relaxation time is the longest time required for a polymer to return to its natural equilibrium state after elongation, which is typically calculated using the chainlike bead-spring model originally developed by Rouse30 and modified by Zimm.31 Alternatively, values of τ can be evaluated experimentally using transient analysis of the poststretching recovery using electric birefringence,32 flow dichroism,33 creep recovery,34 or microscopy.14 The timescale for the flow is typically set equal to the inverse shear rate (γ-1), which is a function of the polymer location within the detailed flow field. For converging flow into a single isolated pore, the local shear rate depends on the distance from the pore entrance (x) to the third power.24 Daoudi and Brochard35 argued that the critical distance from the pore entrance where deformation effects become significant should be evaluated using the polymer radius of gyration (RG) giving the following: De )

γ≈

( )( ) Qp

RG3

)

Jvrp2

εRG3

(3)

where Qp is the volumetric flow rate through the pore, Jv is the filtrate flux (volumetric flow rate normalized by the total membrane area), rp is the membrane pore radius, and ε is the membrane porosity. In contrast, Long and Anderson36 assumed that the critical distance was equal to the membrane pore radius giving the following: γ≈

()( ) Qp

3

)

rp

Jv εrp

(4)

which has a totally different dependence on both the pore and polymer radius. Equations 3 and 4 were developed for a membrane having a uniform distribution of pores of a single pore radius. These equations have been applied to membranes have a pore size distribution, typically using the mean pore radius for rp, although this provides only an estimate for an “average” shear rate. Daoudi and Brochard35 used eqs 2 and 3 to describe the effects of chain deformation on the transport of a large linear polymer through a single cylindrical pore, with the polymer relaxation time evaluated using the Zimm model as follows: τ≈

µRG3 kBT

(5)

where µ is the fluid viscosity, kB is the Boltzmann constant, and T is the absolute temperature. Daoudi and Brochard35 hypothesized a sharp transition in polymer transmission at a critical value of the filtrate flux corresponding to a Deborah number of approximately 1. This type of sharp transition has been experimentally observed using a special double-layer membrane to minimize interactions between the flow-fields of adjacent pores.37 Subsequent to the work of Daoudi and Brochard,35 Keh38 included a more detailed analysis of the position dependence in the flow-field with the resulting analysis predicting a gradual transition in polymer transmission with increasing filtrate flux. This behavior is consistent with experimental data for polystyrene transmission through both tracketch36 and immersion-cast39 ultrafiltration membranes. A similar gradual increase in transmission was seen by Latulippe et al.17 for ultrafiltration of plasmid DNA, although they attributed this effect to the distributions in membrane pore size and plasmid morphology. In contrast to these studies of very large polymers, there has been relatively little work on the effects of flow-induced deformation on the ultrafiltration of flexible polymers that are smaller than the membrane pore radius. Long and Anderson36 obtained data for polystyrene transmission through track-etched membranes with pores that were slightly larger than the radius of gyration of the polymer and found clear deviations from the classical hard-sphere theory when De g 0.1. However, there was no attempt to develop a quantitative model for polymer transmission under these conditions. Materials and Methods A stock solution of a 3.0 kbp pBluescript II KS+ plasmid, commercially available from Stratagene (La Jolla, CA) was prepared by Aldevron (Fargo, ND) as previously described.17 The plasmid (>90% supercoiled as determined by agarose gel electrophoresis) was suspended in 10 mM Tris-base with 1 mM EDTA, divided into 110 µL aliquots, and stored at -20 °C. Plasmid solutions with a final concentration of 250 ng/mL were prepared by diluting the thawed stock solution with Tris-EDTA (TE) buffer made by diluting a Tris-EDTA concentrate (SigmaAldrich, St. Louis, MO) with deionized distilled water obtained from a NANOpure Diamond water purification system (Barnstead International, Dubuque, IA) with a resistivity greater than 18 MΩ cm. The ionic strength of the buffer solution was adjusted by adding appropriate amounts of sodium chloride (VWR, West Chester, PA) to obtain a final NaCl concentration of 150 mM. The solution pH (7.7 ( 0.1) and conductivity were measured using a 420APlus pH meter (Thermo Orion, Beverly, MA) and 105APlus conductivity meter (Thermo Orion), respectively. PEGylated forms of R-lactalbumin were produced in our laboratory as previously described.22 The protein was PEGylated with mPEG-succinimidyl propionate (mPEG-SPA) having a nominal weight average molecular weight of 5 kDa, 10 kDa, or 20 kDa, each with a polydispersity less than 1.05 (Nektar Therapeutics, Huntsville, AL). The diol concentration was less than 4% as given by the manufacturer. PEGylation reactions were performed by dissolving the protein of interest in a 150 mM phosphate buffer or a 150 mM TE buffer at a 1:5 molar ratio of protein to mPEG-SPA. The mPEG-SPA was added slowly to the solution until the total concentration of PEG was 10 g/L. The reaction mixture was stirred at room temperature (21-24 °C) for a minimum of 8 h to allow the reaction to go essentially to completion. The resulting solutions were mixed with 150 mM NaCl-TE or phosphate buffer to yield a total

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PEGylated protein concentration of approximately 4 g/L. The 20 kDa PEGylated protein feedstock contained approximately 30% PEGylated protein, 5% native protein, and 65% free PEG. The 10 kDa sample contained approximately 66% PEGylated protein, less than 2% native protein, and 32% free PEG. The 5 kDa sample contained approximately 88% PEGylated protein, less than 2% native protein, and 10% free PEG. The small variations in composition between different feed mixtures had no observable effect on the measured transmission. The final solutions were prefiltered through a 0.2 µm Acrodisc syringe filter (Pall Corporation, Ann Arbor, MI) to remove any particulate matter and larger aggregates prior to use. Matrix assisted laser desorption ionization time of flight (MALDI-TOF) mass spectrometry (Waters) was used to confirm the identity and evaluate the molecular weight of the PEGylated proteins.22 Plasmid solutions were used once and then discarded. The feed solution for the PEGylated proteins was reused for multiple experiments with the protein concentration, pH, and/or conductivity adjusted to the desired values by an ultrafiltration/ diafiltration using a 10 kDa Ultracel membrane.17,22 Ultrafiltration Experiments. Ultrafiltration experiments were conducted using Ultracel composite regenerated cellulose membranes (Millipore Corp., Billerica, MA) having nominal molecular weight cutoffs of 30 (Lot 101904BCT-5), 100 (Lot 112204BCH-1), and 300 kDa (Lot K021805ACM-4). Membrane discs (25 mm diameter) were cut from large flat sheets using a specially designed cutting device. All membranes were initially soaked in isopropyl alcohol and then flushed with at least 100 mL of water to remove residual storage agents and to ensure thorough wetting of the pore structure. Prior to being used in the filtration experiments, all membrane discs were soaked overnight in a solution containing the PEGylated protein or plasmid to eliminate any transient effects associated with biomolecule adsorption. The hydraulic permeability (Lp) of the membrane after protein/plasmid adsorption was indistinguishable from the value obtained for the fresh membrane indicating that there was minimal adsorption to the very hydrophilic Ultracel membranes. Data were obtained using 10 mL (effective membrane area ) 4.1 cm2) stirred ultrafiltration cells (Millipore) placed on a magnetic stir plate (VWR 205 Autostirrer). The stirrer speed was evaluated using a Strobotac type 1531-AB phototachometer (General Radio Co., Concord, MA). The stirred cell was connected to a reservoir that was pressurized with compressed air and fitted with a pressure gauge. Sieving data were obtained over a range of transmembrane pressure (TMP) from 0.5-83 kPa (corresponding to 0.07-12 psi). The stirred cell was filled with the appropriate solution, the stirrer was turned on, the feed pressure was set to the appropriate value (to achieve the desired filtrate flux), and the filtrate line was then unclamped. The filtrate flux (Jv) was determined by timed collection using a digital balance (Mettler Toledo, Columbus, OH). Filtrate samples were obtained after collection of at least 1 mL of filtrate to flush the dead volume beneath the membrane. The filtrate line was then clamped, the stirred cell was disassembled, and a “post-sieving” sample was taken directly from the solution in the stirred cell. All experiments were performed at room temperature, with samples stored at 4 °C until analysis. Additional details on the ultrafiltration experiments are provided elsewhere.17,22 The dynamic response of the plasmid DNA was examined as follows. The stirred cell was assembled as above, with the transmembrane pressure set using the hydrostatic head (no air pressurization) at a constant value of approximately 0.5 kPa. Filtrate samples were obtained at 5 min intervals over a period

Figure 1. Plasmid sieving coefficients as a function of the filtrate flux for the Ultracel 100 kDa (2) and 300 kDa (b) membranes.

of 25 min to evaluate both the flux and plasmid concentration. The TMP was then rapidly increased to 8.1 kPa by air pressurization of the stirred cell, with filtrate samples obtained at 30 s intervals over approximately 15 min. The TMP was then returned to its original value for another 30 min. No samples were taken directly from the stirred cell during this experiment. Assays. Plasmid DNA concentrations were determined by fluorescence detection using the nucleic acid stain PicoGreen (Invitrogen, Carlsbad, CA). Samples were analyzed in duplicate using a 96-well GENios FL microplate reader (TECAN, Research Triangle Park, NC). Each well was initially filled with a 100 µL sample plus 100 µL of the PicoGreen reagent, with the latter prepared by a 200:1 dilution of the stock reagent with TE buffer. The solution was mixed by repeated aspiration, the plate was then shaken for 3 min at 36 °C, and the fluorescence intensity was measured at 535 nm (excitation at 485 nm). Actual concentrations were determined by comparison with a standard calibration curve. DNA concentrations could be accurately measured as low as 0.25 ng/mL. Concentrations of the PEGylated proteins were determined using size exclusion chromatography (SEC) as described previously.22 Briefly, an Agilent 1100 Series HPLC system was used with a Superdex 200, 10/300 gel permeation column (13 µm particle size, 1 × 105 MW exclusion limit) obtained from GE Healthcare (Uppsala, Sweden). The mobile phase was a 50 mM phosphate buffer at pH 7 containing 0.15 M NaCl at a flow rate of 0.3 mL/min; 100 µL samples were injected at a rate of 200 µL/min and sample detection was performed using an Agilent 1100 series refractive index detector and an Agilent 1200 series UV-vis detector at 280 nm, with the two detectors operated in series. The unreacted PEG was effectively invisible to the UV detector, providing much more accurate measurements of the protein concentrations. Concentrations could be accurately measured to within 0.02 g/L. Calibration curves for all species were constructed using samples of known concentration, with the peak areas determined by numerical integration. Overlapping peaks were simply split at the location of the minimum. Results and Discussion Plasmid DNA. Experimental data for the observed sieving coefficient of the 3.0 kbp plasmid through the 100 and 300 kDa Ultracel membranes are shown in Figure 1 as a function of the filtrate flux. Ultrafiltration data were obtained at a constant stirring speed of 730 rpm. The observed sieving coefficient (So) was evaluated as the ratio of the plasmid concentration in the

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filtrate solution to that in the feed, with the feed concentration evaluated as the arithmetic average of the concentrations in samples obtained from the stirred cell immediately before and after the filtrate measurement. The error bars represent plus/ minus one standard deviation as determined using standard propagation of error analysis based on the estimated accuracy of the plasmid concentration measurements. The radius of gyration of the 3.0 kbp supercoiled plasmid was estimated as RG ) 61 ( 3 nm as follows. First, RG was calculated for a linear and open-circular 3.0 kbp plasmid using the “wormlike chain” model and the flexible circular chain model, respectively. The radius of gyration for the supercoiled plasmid was then evaluated based on experimental data for the ratio of RG for the supercoiled to linear or open-circular forms at the same buffer conditions used in these experiments. Additional details on these calculations are provided by Latulippe and Zydney.40 The pore sizes of the 100 and 300 kDa Ultracel membranes were calculated as 6.4 and 8.6 nm from measurements of the hydraulic permeability (Lp), calculated from data for the filtrate flux obtained with the pure TE buffer for the membranes after overnight adsorption in the plasmid solution (Lp ) 2.5 × 10-6 and 4.6 × 10-6 m/(s kPa), respectively), using the HagenPoiseuille equation:

(

)

8µδmLp 1⁄2 (6) ε Calculations were performed assuming a membrane skin layer thickness of δm ) 1.0 µm and a membrane porosity of ε ) 0.5.17 Zydney and Xenopoulos41 estimated the mean pore size of the 100 kDa Ultracel membrane as 6.0 nm based on dextran retention tests, in very good agreement with the value determined from the hydraulic permeability data. The plasmid transmission increased with increasing filtrate flux, going from a value less than 0.01 at a flux of 7 µm/s (25 L/(m2 h)) to more than 0.90 at a flux of 66 µm/s for the 300 kDa membrane. The plasmid transmission at high filtrate flux approached 100% (i.e., So ) 1) for the 300 kDa membrane even though the radius of gyration of the plasmid is 7 times the nominal pore size for this membrane. High transmission was also obtained with the 100 kDa membrane, although this required considerably higher filtrate flux. The increase in plasmid transmission with increasing filtrate flux seen in Figure 1 was not due to concentration polarization effects. Experiments performed over a range of stirring speeds showed no dependence of the observed sieving coefficient on the stirring speed.17 Agarose gel electrophoresis confirmed that there were no changes in the supercoiled plasmid structure due to either the stirring conditions or passage of the DNA through the membrane.17 The experimental data in Figure 1 were replotted as a function of the Deborah number, calculated from eqs 2, 3, and 5, with the results shown as the filled symbols in Figure 2. Although the use of the Deborah number does bring the data closer together, the results for the 100 kDa membrane still lie well to the right of the data for the 300 kDa membrane. Note that using eq 4 to evaluate the characteristic time for the fluid flow would have shifted the data for the 300 kDa membrane much further to the left of the data for the 100 kDa membrane, expanding (instead of reducing) the differences seen in Figure 1. The data for the 100 and 300 kDa membranes could be artificially collapsed to a single curve (open circles and filled triangles) using a nominal pore size of 12 nm for the 300 kDa membrane (instead of the value of 8.6 nm as calculated from the permeability data). This larger pore size is consistent with a membrane having a porosity of 26%, which is within the quoted rp )

Figure 2. Plasmid sieving coefficients as a function of the Deborah number for the Ultracel 100 kDa (2) and 300 kDa (b,O) membranes. The open symbols (O) represent data for the 300 kDa membrane using rp ) 12 nm as described in the text.

range of porosities for ultrafiltration membranes42 although there is no independent evidence for a significant difference in porosity for the 100 and 300 kDa Ultracel membranes. The need for a larger pore size is also consistent with results obtained by Latulippe et al.,17 with the critical filtrate flux for plasmid DNA transmission correlated using a power-law dependence on membrane pore size with a 3.4 power (instead of the second power as predicted by eq 3). The plasmid sieving coefficient is essentially equal to zero at small Deborah numbers and becomes greater than about 0.1 at De ≈ 0.001. This indicates that the critical value of the Deborah number (Decrit) corresponding to significant plasmid deformation/elongation is about 0.001, which is nearly 3 orders of magnitude smaller than the value obtained from direct measurements of the elongation of linear DNA fragments using video fluorescence microscopy.14,43 In addition, a previous study of polystyrene ultrafiltration through low porosity (double-layer) membranes clearly suggest that Decrit is of order 1,37 consistent with the original development by Daoudi and Brochard.35 This large discrepancy in the values of the critical Deborah number may be related to the very high porosity (ε ≈ 0.5) of the Ultracel membranes used in this study. Under these conditions, the flow field above each pore will be strongly influenced by the presence of the adjacent pores at a distance of approximately one pore radius above the membrane (rp ) 6.4 nm for the 100 kDa membrane). In contrast, Daoudi and Brochard35 assumed that the critical distance for polymer elongation was approximately equal to the radius of gyration of the polymer (RG ) 61 nm for the 3.0 kbp plasmid), which is nearly 10 times the pore radius in these experiments. Note that if the characteristic time for the fluid flow had been evaluated at a distance x ) 0.1RG above the pore, the data in Figure 2 would have been shifted by 3 orders of magnitude due to the x3 dependence of the flow-field. This would result in a critical Deborah number of approximately 1, in good agreement with independent measurements of DNA elongation. It is also possible that this small critical distance is related to the unique geometry of the supercoiled plasmid in which the “ends” of the polymer are connected to form a circular piece of DNA that twists to reduce the resulting strain in the DNA double helix. This phenomenon is discussed in more detail by Latulippe et al.17 The dynamic response of the plasmid to changes in the TMP (and thus the filtrate flux) is examined in Figure 3. The plasmid

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Figure 3. Dynamic response of plasmid concentration in the filtrate solution (O) to changes in TMP during filtration of a 3.0 kbp plasmid through the Ultracel 300 kDa membrane. The right-hand axis shows the corresponding values of the filtrate flux ((). The dashed line at V ) 3 mL marks a TMP increase from 0.5 to 8.1 kPa; the dashed line at V ) 20 mL marks a TMP decrease from 8.1 to 0.5 kPa.

concentration in the filtrate solution is plotted as a function of the cumulative filtrate volume (V) during an ultrafiltration experiment performed using a 300 kDa membrane with a TMP of approximately 0.5 kPa (corresponding to the hydrostatic head) for V < 3 mL and for V > 20 mL and a TMP of 8.1 kPa for 3 < V < 20 mL. Also shown on the right-hand axis are the measured values of the filtrate flux. The plasmid concentration in the filtrate solution at the low pressure was essentially undetectable, corresponding to a sieving coefficient of less than 0.01. When the pressure was increased to 8.1 kPa, giving a filtrate flux of about 38 µm/s, the plasmid concentration in the filtrate increased rapidly to approximately 260 ng/mL. This corresponds to a sieving coefficient of approximately 0.8, accounting for the estimated increase in the plasmid concentration within the stirred cell from 250 to 310 ng/mL during the ultrafiltration at low pressure. The plasmid concentration decreased slightly during the ultrafiltration at 8.1 kPa, which is probably due at least in part to the slight reduction in filtrate flux (from 38 to 35 µm/s) associated with the variation in the air pressure overlay used to establish the TMP. Note that this effect would be at least partially compensated for by the slight increase in plasmid concentration within the stirred cell due to plasmid retention by the membrane during the ultrafiltration. The origin of the small but fairly sharp reduction in filtrate concentration at V ≈ 16 mL is unclear. When the TMP was decreased to 0.5 kPa at V > 20 mL, the plasmid concentration in the filtrate solution decreased from 210 to 4 ng/mL during filtration of the next 1.5 mL of filtrate volume. This transient response is consistent with the time required to wash out the 1 mL dead volume beneath the membrane and in the short filtrate tubing; there is no evidence for any transient response associated with a change in plasmid elongation in response to the change in flux. This behavior is consistent with the very small time constant for plasmid elongation, τ ∼ 5 × 10-5 s, using eq 5. The plasmid concentration in the filtrate at V ) 22.5 mL was less than 2 ng/mL, similar to what was obtained during the initial low pressure ultrafiltration. PEGylated Proteins. Figures 4 and 5 show the observed sieving coefficient of PEGylated R-lactalbumin with 20 kDa of total attached PEG using either four 5 kDa PEG chains, two 10 kDa PEG chains, or a single 20 kDa PEG. Several previous studies have demonstrated that attachment of PEG chains to a variety of proteins has no effect on the three-dimensional

Figure 4. Observed sieving coefficient of PEGylated R-lactalbumin with four 5 kDa PEG chains (b) and one 20 kDa PEG chain (0) through an Ultracel 100 kDa membrane at a stirring speed of 1200 rpm. The solid curve is the model calculation as described in the text.

Figure 5. Observed sieving coefficient of PEGylated R-lactalbumin with four 5 kDa PEG (b) and two 10 kDa PEG (0) chains through an Ultracel 30 kDa membrane at a stirring speed of 600 rpm. The solid curve is the model calculation as described in the text.

conformation of the protein.44 In addition, Molek and Zydney22 and Fee45 showed that PEGylated proteins having different size PEG chains, but with the same total molecular weight of attached PEG, have identical sizes as determined by size exclusion chromatography. For example, the PEGylated R-lactalbumin with four 5 kDa PEG chains has Reff ) 5.27 nm compared to Reff ) 5.23 nm for the PEGylated protein with one 20 kDa PEG chain as determined by SEC.22 This 1% difference in radius is due to the very small difference in total molecular weight of the attached PEG groups (21.0 versus 20.7 kDa) associated with the slight variations in molecular weight of the mPEG-SPA used to form the PEGylated proteins. Figure 4 shows results obtained using two separate 100 kDa Ultracel membranes, both cut from a single flat sheet of membrane. At low filtrate flux, the sieving coefficients of the two PEGylated proteins are similar, with So ≈ 0.1 at a filtrate flux of 7 µm/s (25 L/(m2 h)), which is consistent with the very similar values of the effective radii. In both cases, the sieving coefficient increases with increasing filtrate flux due to concentration polarization, but the magnitude of this increase is much more pronounced for the protein with one 20 kDa PEG chain. Thus, at a filtrate flux of 30 µm/s (108 L/(m2 h)) the sieving coefficient of the PEGylated R-lactalbumin with one

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20 kDa PEG chain is almost twice as large as that for the PEGylated protein with four 5 kDa PEG chains. The increase in sieving coefficient with increasing filtrate flux seen with the PEGylated R-lactalbumin is due in part to the effects of concentration polarization in the stirred cell. The filtrate flux drags protein toward the membrane, where it begins to accumulate due to retention by the membrane. The extent of accumulation is determined by the rate of mass transfer away from the membrane and back into the bulk solution, which is a function of the protein diffusion coefficient and the stirring speed. This phenomenon is typically analyzed using the stagnant film model, with the observed sieving coefficient expressed in terms of the actual sieving coefficient (Sa), the bulk mass transfer coefficient (km), and the filtrate flux (Jv) as follows:

() ()

Sa exp So )

Jv km

Jv 1 - Sa + Sa exp km

(7)

The bulk mass transfer coefficient in the stirred cell was calculated using the correlation developed by Smith et al.:46 Sh ) β(Re)d(Sc)0.33 (8) 2 where Sh ) kmr/D∞ is the Sherwood number, Re ) ωr /ν is the Reynolds number, Sc ) ν/D∞ is the Schmidt number; r is the radius of the stirred cell, ω is the stirring speed, ν is the kinematic viscosity, and D∞ is the protein diffusion coefficient in free solution. The value of the pre-exponential term β is a function of the stirred cell geometry. Colton47 used data for the dissolution of benzoic acid under turbulent flow to evaluate β ) 0.285, while data obtained by Opong and Zydney3 for the flux of bovine serum albumin solutions through fully retentive membranes gave β ) 0.23. The exponent d varies from 0.567 for laminar flow (Re < 10 000) to 0.746 for fully turbulent flow (Re > 60 000). The experimental conditions in Figure 4 correspond to a Reynolds number of 22 000 giving d ) 0.59.47 The diffusion coefficients of the PEGylated proteins were calculated using the Stokes-Einstein equation with the equivalent hydrodynamic radius determined from the correlation presented by Fee45 based on the retention volume in size exclusion chromatography: Reff )

2 A 1 + R 2 + RPEG 6 3A PEG 3

(9)

A ) [108Rpro3 + 8RPEG3 + 12(81Rpro6 + 12Rpro3RPEG3)1/2]1/3 (10) where Rpro and RPEG are the radii of the unmodified protein and the free PEG chain, respectively. This gives Reff ) 5.15 nm and km ) 9.9 µm/s (using D∞ ) 4.6 × 10-11 m2/s). The solid curve in Figure 4 is a model calculation developed using eq 7 with the best fit value of Sa ) 0.022 determined by minimizing the sum of the squared residuals. The model is in fairly good agreement with the data for the PEGylated R-lactalbumin with four 5 kDa PEG chains, describing the approximately 5-fold increase in the observed sieving coefficient with increasing filtrate flux. However, the experimental data for the observed sieving coefficient vary almost linearly with the filtrate flux compared to the exponential dependence given by eq 7. The origin of this discrepancy is unclear, although it could be related to the use of the simple stagnant film model to evaluate mass transfer in the stirred cell or it might reflect a change in mass transfer coefficient associated with a flowinduced conformational change or alignment of the PEGylated

protein with increasing flux. In contrast to the data with the 5 kDa PEGylated protein, the model significantly underpredicts the observed sieving coefficient data for the PEGylated R-lactalbumin with one 20 kDa PEG chain. This effect is most pronounced at the highest values of the filtrate flux. For example, the model predicts So ) 0.37 at a flux of 32 µm/s for the 100 kDa membrane, which is a factor of 2 smaller than the experimental value (So ) 0.75). These results clearly suggest that the 20 kDa PEG chains elongate at the higher filtrate flux, reducing the effective size of the PEGylated protein and increasing its transmission through the ultrafiltration membrane. This phenomenon is discussed in more detail subsequently. Figure 5 shows corresponding data for R-lactalbumin with four 5 kDa PEG chains and two 10 kDa PEG chains using a 150 mM phosphate buffer with a 30 kDa membrane at a stirring speed of 600 rpm. The experiments with the R-lactalbumin having two 10 kDa PEG chains were performed in duplicate while those with the 5 kDa PEG chains were performed in triplicate. In both cases, the error bars represent the standard deviation in these repeat measurements. Similar to the results in Figure 4, the sieving coefficients increase with increasing filtrate flux due to concentration polarization, with this effect being more pronounced for the R-lactalbumin with two 10 kDa PEG chains. At a filtrate flux of 16 µm/s the sieving coefficient of the PEGylated R-lactalbumin with two 10 kDa PEG chains is an order of magnitude higher than the PEGylated protein with four 5 kDa PEG chains. The solid curve is the model calculation using eq 7 with the best fit values of the actual sieving coefficient (Sa ) 8.9 × 10-4) determined by least-squares optimization using a mass transfer coefficient of km ) 5.3 µm/s as calculated from eq 8 with d ) 0.567 assuming laminar flow at the lower stirring speed. The best fit value of actual sieving coefficient through the 30 kDa membranes is more than a 100-fold smaller than that through the 100 kDa membrane, consistent with the smaller pore size of the 30 kDa membrane. The model is in good agreement with data for the PEGylated R-lactalbumin with four 5 kDa PEG chains, but dramatically under predicts the results for the PEGylated protein with two 10 kDa PEG chains even though these species have identical size (as determined by SEC) and should thus have very similar diffusivities and mass transfer coefficients. The effects of the number and size of the PEG chains on the sieving characteristics of the PEGylated R-lactalbumins are examined in more detail in Figure 6. Results are shown for the observed sieving coefficients for PEGylated R-lactalbumin with one to four 5 kDa PEG chains and two to four 10 kDa PEG chains through the 30 kDa Ultracel membrane at a filtrate flux of 14 µm/s. Also shown for comparison are literature data for the observed sieving coefficients for a series of unmodified (i.e., nonPEGylated) proteins (R-lactalbumin, ovalbumin, and bovine serum albumin) through the same pore size membrane at the same filtrate flux.22 The data are plotted as a function of the effective protein radius with the radius of the PEGylated proteins calculated using eqs 9 and 10. The sieving coefficients decrease with increasing effective radius as expected, with the results for the unmodified proteins in good agreement with model calculations for hard spheres accounting for a log-normal membrane pore size distribution.22 The data for PEGylated R-lactalbumins lie uniformly above those for the unmodified proteins, with this discrepancy being most pronounced for the R-lactalbumin PEGylated with the 10 kDa mPEG-SPA. For example, the PEGylated R-lactalbumin with two 10 kDa PEG chains had a sieving coefficient of 0.16, which is an order of magnitude larger than the sieving coefficient for the PEGylated R-lactalbumin with four 5 kDa PEG groups. The sieving

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2401

Figure 6. Observed sieving coefficient through the 30 kDa Ultracel membrane at a filtrate flux of 14 µm/s for PEGylated R-lactalbumin formed with 5 (0) and 10 kDa (b) PEG chains. Data for a series of globular proteins (2), R-lactalbumin, ovalbumin, and bovine serum albumin, are shown for comparison. Results are plotted as a function of the effective protein radius determined by size exclusion chromatography.

Figure 7. Scaled sieving coefficients of PEGylated R-lactalbumin (0) and plasmid DNA (2) as a function of Deborah number for the Ultracel 100 kDa membrane.

coefficient of immunoglobulin G, a globular protein with approximately the same effective size as these biomolecules, was undetectable under these conditions (So < 0.001). Similar results were obtained using ovalbumin PEGylated with 5 kDa or 10 kDa branches.22 These results provide further evidence for the elongation of the PEGylated protein at high filtrate flux, an effect that appears to be most pronounced for PEGylated species formed with higher molecular weight (i.e., longer) PEG chains As discussed previously, the large increase in sieving coefficient at high filtrate flux for the PEGylated R-lactalbumin with the longer PEG chains is likely due to the elongation of the PEG in the converging flow field above the membrane pore. The experimental data for the PEGylated R-lactalbumin with one 20 kDa PEG chain (Figure 4) have been replotted in Figure 7 as the scaled sieving coefficient (χ): χ)

So - S1 S2 - S1

(11)

S1 is defined as the observed sieving coefficient for an undeformed protein at the same flux, which was evaluated directly from the model calculations (solid curve) in Figure 4.

S2 is defined as the observed sieving coefficient for the PEGylated protein with fully elongated PEG chains, which was evaluated from corresponding data for the unmodified R-lactalbumin assuming that the fully elongated PEG chains have no influence on protein transmission. Also shown for comparison are the calculated values of the scaled sieving coefficient for the plasmid DNA through the same 100 kDa membrane. In this case, the sieving coefficient of the undeformed plasmid, S1, was assumed to be zero since the plasmid is much larger than the membrane pore. The sieving coefficient of the fully elongated plasmid, S2, was taken as one based on the results in Figures 1 and 2. In both cases, the results are plotted as a function of the Deborah number, which was calculated using eqs 2, 3, and 5 with the pore radius determined from hydraulic permeability measurements. The radius of gyration for the PEGylated proteins were evaluated as follows. The hydrodynamic radius for the free PEG chain was calculated as a function of the PEG molecular weight using the correlation presented by Singh.48 The value of RG was taken as 1.5 times the hydrodynamic radius of the PEG based on previous studies for similar polymers in good solvents.49 This analysis implicitly assumes that the elongation of the PEG chain is unaffected by the attachment to the protein and that the deformation of the PEGylated protein is solely due to the PEG. The scaled sieving coefficients for the plasmid DNA and the PEGylated R-lactalbumin increase with increasing Deborah number, consistent with the elongation of the plasmid and PEG associated with the fluid flow (corresponding to the increase in γ). The variation of χ with De looks qualitatively similar for both biomolecules, suggesting that the gradual increase in sieving coefficient with increasing filtrate flux is due to the properties of the membrane and/or flow-field as opposed to some variability in the biomolecules (as had been previously suggested by Latulippe et al.17 for plasmid DNA). The value of the critical Deborah number for the PEGylated protein, defined as the value of De corresponding to a significant increase in χ, is around 10-4 which is an order of magnitude smaller than the critical Deborah number for the plasmid. This would seem to suggest that the PEG chains in the PEGylated protein are more flexible, i.e. elongate more easily in response to the flow field, than the supercoiled plasmid. However, it is important to recognize that there are several significant uncertainties in the calculation of the Deborah number for the PEGylated protein. First, the critical distance above the membrane pore in eq 3 was set equal to the radius of gyration of the free PEG chain, completely neglecting the effect of the R-lactalbumin and the detailed geometry of the PEGylated protein. Second, the concentration of PEGylated protein at the membrane surface can be quite large due to concentration polarization effects, which could significantly alter both the solution viscosity (in eq 5) and the PEG relaxation time (if the local concentration exceeds the polymer overlap concentration).28 This phenomenon would not have occurred for the plasmid due to the very low feed concentration (250 ng/mL for the plasmid DNA compared to 4 mg/mL for the PEGylated R-lactalbumin). Alternatively, the small value of Decrit for the PEGylated protein could be directly associated with the covalent attachment of the PEG chain to the R-lactalbumin. Perkins et al.14,50 showed that attachment of a 144 kbp linear DNA to a 0.3 µm rigid bead significantly increased the relaxation time of the DNA due to the restricted motion of the attached end. A similar phenomenon with the PEG chains would cause a shift in the Deborah numbers for the PEGylated proteins to uniformly larger values of De, potentially bringing the results for the PEGylated

2402 Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009

protein and the plasmid into much better agreement. It is also possible that the hydrodynamic drag on the R-lactalbumin causes a preferential orientation of the PEGylated protein, potentially increasing the extent of elongation of the PEG chain within the flow field above the membrane pore.

of molecular flexibility and elongation on the ultrafiltration characteristics of these important second-generation biotherapeutics.

Conclusions

The authors would like to acknowledge Millipore for donation of the Ultracel membranes, the National Science Foundation for financial support, and Kimberly Ager for assistance with some of the experiments.

There is growing interest in the development of next generation biotherapeutics, including both plasmid DNA (for gene therapy and DNA-based vaccines) and polymer-conjugated proteins (for enhanced bioactivity). In contrast to the behavior of globular proteins, the sieving characteristics of these second generation products show significant elongation effects at high filtrate flux, analogous to the ultrafiltration behavior of flexible polymers. These elongation effects are minimal at low flux, conditions that would allow one to use these ultrafiltration membranes for concentration and buffer exchange in the formulation of the plasmid DNA or polymer-conjugated protein. This is particularly true for plasmid DNA, which is fully retained by the ultrafiltration membranes at low filtrate flux where plasmid elongation is negligible but is easily transmitted through the membrane at very high filtrate flux due to elongation of the plasmid within the flow field. The experimental and theoretical results presented in this study can provide a useful guide for the selection of membrane pore size and filtrate flux for ultrafiltration of these complex biomolecules using commercial ultrafiltrationmembranesappropriateforbioprocessingapplications. In contrast to prior results with polystyrene and other linear polymers, the critical Deborah number for plasmid transmission, evaluated using a scaling model for the fluid flow (eq 3) and the Zimm model for polymer relaxation time (eq 5), is around 0.001 (versus Decrit ≈ 1 for polystyrene). This discrepancy appears to be related to the location used to evaluate the characteristic time for the fluid flow. Previous analyses assumed that this critical distance was equal to the radius of gyration, but this implicitly assumes that the flow field above each pore remains unaffected by the flow through adjacent pores for x < RG, an assumption that will not be valid for the high porosity ultrafiltration membranes examined in this study. Calculations performed using a modified Deborah number, with γ evaluated at a distance of 0.1RG from the membrane pore, yielded a critical Deborah number around 1, which is consistent with independent measurements of the elongation of individual plasmid DNA molecules. The behavior of the PEGylated proteins was even more complex due to the unique structure of the polymer-protein conjugate. Elongation of the PEG chain was negligible at low filtrate flux, with the sieving coefficient of the PEGylated protein being very similar to that of a globular protein (i.e., hard sphere) under these conditions. The extent of PEG elongation increased with increasing filtrate flux, with a greater elongation seen for PEGylated proteins conjugated with longer PEG chains. Thus, the sieving coefficient for PEGylated R-lactalbumin with one 20 kDa PEG chain was significantly larger than that for the PEGylated R-lactalbumin with four 5 kDa chains (Figure 4) even though these molecules have essentially identical size as determined by size exclusion chromatography. The data for PEGylated R-lactalbumin show significant elongation effects at very low values of De, which may be directly related to the covalent attachment of one end of the PEG chain to the protein. In particular, the results are consistent with an early orientation (and elongation) of the PEG chain due to the strong hydrodynamic drag on the protein part of the polymer-protein conjugate. Additional studies will be needed to fully understand the effects

Acknowledgment

Nomenclature D∞ ) free solution diffusion coefficient (m2/s) De ) Deborah number (-) Decrit ) critical Deborah number (-) Jv ) filtrate flux (m/s) kB ) Boltzmann constant (J/K) km ) bulk mass transfer coefficient (m/s) Kc ) hindrance factor for convection (-) Lp ) membrane hydraulic permeability (m/(s kPa)) Qp ) volumetric flow rate through pore (m3/s) r ) stirred cell radius (m) rp ) membrane pore radius (m) RG ) radius of gyration (m) Reff ) effective protein radius (m) RPEG ) radius of free PEG (m) Rpro ) radius of unmodified protein (m) Re ) Reynolds number (-) Sa ) actual sieving coefficient (-) So ) observed sieving coefficient (-) Sc ) Schmidt number (-) Sh ) Sherwood number (-) T ) absolute temperature (K) x ) distance from pore entrance (m) γ ) inverse of characteristic time for fluid flow (s-1) δm ) membrane thickness (m) ε ) membrane porosity (-) ν ) kinematic viscosity (m2/s) φ ) partition coefficient (-) µ ) fluid viscosity (Pa · s) τ ) polymer relaxation time (s) χ ) scaled sieving coefficient (-) ω ) stirring speed (s-1)

Note Added after ASAP Publication: The version of this paper that was published on the Web August 15, 2008 had incorrect versions of Figures 1 and 5, as well as an error involving the symbol used to denote the distance from the pore entrance in the Nomenclature section. The corrected version of this paper was reposted on the Web August 21, 2008. Literature Cited (1) Deen, W. M. Hindered Transport of Large Molecules in LiquidFilled Pores. AIChE J. 1987, 33 (9), 1409–1425. (2) Dechadilok, P.; Deen, W. M. Hindrance Factors for Diffusion and Convection in Pores. Ind. Eng. Chem. Res. 2006, 45 (21), 6953–6959. (3) Opong, W. S.; Zydney, A. L. Diffusive and Convective Protein Transport through Asymmetric Membranes. AIChE J. 1991, 37 (10), 1497– 1510. (4) Meireles, M.; Bessieres, A.; Rogissart, I.; Aimar, P.; Sanchez, V. An Appropriate Molecular-Size Parameter for Porous Membranes Calibration. J. Membr. Sci. 1995, 103 (1-2), 105–115. (5) Burns, D. B.; Zydney, A. L. Effect of Solution pH on Protein Transport through Ultrafiltration Membranes. Biotechnol. Bioeng. 1999, 64 (1), 27–37.

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2403 (6) Saksena, S.; Zydney, A. L. Effect of Solution pH and Ionic-Strength on the Separation of Albumin from Immunoglobulins (IgG) by Selective Filtration. Biotechnol. Bioeng. 1994, 43 (10), 960–968. (7) Causserand, C.; Meireles, M.; Aimar, P. Proteins Transport through Charged Porous Membranes. Chem. Eng. Res. Des. 1996, 74 (A1), 113– 122. (8) Zydney, A. L.; Pujar, N. S. Protein Transport through Porous Membranes: Effects of Colloidal Interactions. Colloids Surf., A 1998, 138 (2-3), 133–143. (9) Mehta, A.; Zydney, A. L. Effect of Membrane Charge on Flow and Protein Transport During Ultrafiltration. Biotechnol. Prog. 2006, 22 (2), 484–492. (10) Gene Therapy Clinical Trials Worldwide. http://www.wiley.co.uk/ genmed/clinical/ (accessed Mar 27, 2008). (11) Bergman, P. J.; Camps-Palau, M. A.; McKnight, J. A.; Leibman, N. F.; Craft, D. M.; Leung, C.; Liao, J.; Riviere, I.; Sadelain, M.; Hohenhaus, A. E.; Gregor, P.; Houghton, A. N.; Perales, M. A.; Wolchok, J. D. Development of a Xenogeneic DNA Vaccine Program for Canine Malignant Melanoma at the Animal Medical Center. Vaccine 2006, 24 (21), 4582– 4585. (12) Merial. USDA Grants Conditional Approval for First Therapeutic Vaccine to Treat Cancer. 2007. (13) Perkins, T. T.; Smith, D. E.; Larson, R. G.; Chu, S. Stretching of a Single Tethered Polymer in a Uniform Flow. Science 1995, 268, 83–87. (14) Perkins, T. T.; Smith, D. E.; Chu, S. Single Polymer Dynamics in an Elongational Flow. Science 1997, 276, 2016–2021. (15) Smith, D. E.; Babcock, H. P.; Chu, S. Single-Polymer Dynamics in Steady Shear Flow. Science 1999, 283, 1724–1727. (16) Larson, R. G.; Perkins, T. T.; Smith, D. E.; Chu, S. Hydrodynamics of a DNA Molecule in a Flow Field. Phys. ReV. E 1997, 55 (2), 1794– 1797. (17) Latulippe, D. R.; Ager, K.; Zydney, A. L. Flux-Dependent Transmission of Supercoiled Plasmid DNA through Ultrafiltration Membranes. J. Membr. Sci. 2007, 294 (1-2), 169–177. (18) Harris, J. M.; Martin, N. E.; Modi, M. Pegylation - a Novel Process for Modifying Pharmacokinetics. Clin. Pharmacokinet. 2001, 40 (7), 539– 551. (19) Piedmonte, D. M.; Treuheit, M. Formulation of Neulasta (Pegfilgrastim). AdV. Drug DeliVery ReV. 2008, 60, 50–58. (20) Fee, C. J.; Van Alstine, J. M. PEG-Proteins: Reaction Engineering and Separation Issues. Chem. Eng. Sci. 2006, 61, 924–939. (21) Fee, C. J.; Van Alstine, J. M. Prediction of the Viscosity Radius and the Size Exclusion Chromatography Behavior of PEGylated Proteins. Bioconjugate Chem. 2004, 5 (6), 1304–1313. (22) Molek, J. R.; Zydney, A. L. Ultrafiltration Characteristics of PEGylated Proteins. Biotechnol. Bioeng. 2006, 95 (3), 474–482. (23) Vonarbourg, A.; Passirani, C.; Saulnier, P.; Simard, P.; Leroux, J. C.; Benoit, J. P. Evaluation of PEGylated Lipid Nanocapsules Versus Complement System Activation and Macrophage Uptake. J. Biomed. Mater. Res., Part A 2006, 78A (3), 620–628. (24) Nguyen, Q. T.; Neel, J. Influence of the Deformation of Macromolecular Solutes on the Transport through Ultrafiltration Membranes. J. Membr. Sci. 1983, 14, 111–128. (25) Lentsch, S.; Aimar, P.; Orozco, J. L. Separation Albumin-PEG: Transmission of PEG through Ultrafiltration Membranes. Biotechnol. Bioeng. 1993, 41, 1039–1047. (26) Pradanos, P.; Arribas, J.; Hernandez, A. Hydraulic Permeability, Mass Transfer and Retention of PEGs in Cross-Flow Ultrafiltration through a Symmetric Microporous Membrane. Sep. Sci. Technol. 1992, 27 (15), 2121–2142. (27) Zeman, L.; Wales, M. Steric Rejection of Polymeric Solutes by Membranes with Uniform Pore Size Distribution. Sep. Sci. Technol. 1981, 16 (3), 275–290. (28) Odell, J. A.; Keller, A.; Miles, M. J. Assessment of Molecular Connectedness in Semi-Dilute Polymer Solutions by Elongational Flow. Polymer 1985, 26, 1219–1226. (29) van Reis, R.; Zydney, A. Bioprocess Membrane Technology. J. Membr. Sci. 2007, 297 (1-2), 16–50.

(30) Rouse, P. E. A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers. J. Chem. Phys. 1953, 21 (7), 1272– 1280. (31) Zimm, B. H. Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence, and Dielectric Loss. J. Chem. Phys. 1956, 24 (2), 269–278. (32) Thompson, D. S.; Gill, S. J. Polymer Relaxation Times from Birefringence Relaxation Measurements. J. Chem. Phys. 1967, 47 (12), 5008. (33) Callis, P. R.; Davidson, N. Hydrodynamic Relaxation Times of DNA from Decay of Flow Dichroism Measurements. Biopolymers 1969, 8 (3), 379. (34) Klotz, L. C.; Zimm, B. H. Retardation Times of Deoxyribonucleic Acid Solutions. II. Improvements in Apparatus and Theory. Macromolecules 1972, 54, 471. (35) Daoudi, S.; Brochard, F. Flow of Flexible Polymer Solutions in Pores. Macromolecules 1978, 11 (4), 751–758. (36) Long, T. D.; Anderson, J. R. Flow-Dependent Rejection of Polystyrene from Microporous Membranes. J. Polym. Sci. Polym. Phys. Ed. 1984, 22, 1261–1281. (37) Jin, F.; Wu, C. Observation of the First-Order Transition in Ultrafiltration of Flexible Linear Polymer Chains. Phys. ReV. Lett. 2006, 96 (23), 237801. (38) Keh, H. J. Hydrodynamic and Electrokinetic Characteristics of Transport of Macromolecules through Porous Media. Ph.D. Thesis, Carnegie-Mellon University, Pittsburgh, PA, 1984. (39) Beerlage, M. A. M.; Heijnen, M. L.; Mulder, M. H. V.; Smolders, C. A.; Strathmann, H. Non-Aqueous Retention Measurements: Ultrafiltration Behaviour of Polystyrene Solutions and Colloidal Silver Particles. J. Membr. Sci. 1996, 113 (2), 259–273. (40) Latulippe, D. R.; Zydney, A. L. Salt-Induced Changes in Plasmid DNA Transmission through Ultrafiltration Membranes. Biotechnol. Bioeng. 2008, 99 (2), 390–398. (41) Zydney, A. L.; Xenopoulos, A. Improving Dextran Tests for Ultrafiltration Membranes: Effect of Device Format. J. Membr. Sci. 2007, 291 (1-2), 180–190. (42) Zeman, L. J.; Zydney, A. L. Microfiltration and Ultrafiltration: Principles and Applications; Marcel Dekker: New York, 1996. (43) Wong, P. K.; Lee, Y.-K.; Ho, C.-M. Deformation of DNA Molecules by Hydrodynamic Focusing. J. Fluid Mech. 2003, 497, 55–65. (44) Digilio, G.; Bracco, C.; Barbero, L.; Chicco, D.; DelCurto, M. D.; Esposito, P.; Traversa, S.; Aime, S. NMR Conformational Analysis of Antide, a Potent Antagonist of the Gonadotropin Releasing Hormone. J. Am. Chem. Soc. 2002, 124 (13), 3431–3442. (45) Fee, C. J. Size Comparison between Proteins PEGylated with Branched and Linear Poly(Ethylene Glycol) Molecules. Biotechnol. Bioeng. 2007, 98 (4), 725–731. (46) Smith, K. A.; Colton, C. K.; Merrill, E. W.; Evans, L. B. Convective Transport in a Batch Dialyzer: Determination of True Membrane Permeability from a Single Measurement. Chem. Eng. Prog. Symp. Ser. 1968, 64 (84), 45–58. (47) Colton, C. K. Permeability and Transport Studies in Batch and Flow Dialyzers with Applications to Hemodialysis. Ph.D. Thesis, Massachusetts Institute of Technology: Boston, MA, 1969. (48) Singh, S.; Khulbe, K. C.; Matsuura, T.; Ramamurthy, P. Membrane Characterization by Solute Transport and Atomic Force Microscopy. J. Membr. Sci. 1998, 142, 111–127. (49) Kok, C. M.; Rudin, A. Relationship between the Hydrodynamic Radius and the Radius of Gyration of a Polymer in Solution. Macromol. Rapid Commun. 1981, 2 (11), 655–659. (50) Perkins, T. T.; Quake, S. R.; Smith, D. E.; Chu, S. Relaxation of a Single DNA Molecule Observed by Optical Microscopy. Science 1994, 264 (5160), 822–826.

ReceiVed for reView April 3, 2008 ReVised manuscript receiVed June 5, 2008 Accepted June 16, 2008 IE8005337