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Flux-dependent transmission of supercoiled plasmid DNA through ultrafiltration membranes
Journal of Membrane Science 294 (2007) 169–177
Flux-dependent transmission of supercoiled plasmid DNA through ultrafiltration membranes David R. Latulippe, Kimberly Ager, Andrew L. Zydney ∗ Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802, United States Received 1 November 2006; received in revised form 6 February 2007; accepted 18 February 2007 Available online 21 February 2007
Abstract Although previous studies have demonstrated the potential of using membrane-based processes for purification of plasmid DNA, there is considerable discrepancy regarding both the physical mechanisms governing DNA transmission and the effects of membrane pore size and operating conditions on the DNA sieving coefficient. The objective of this work was to obtain quantitative data on the transmission of a 3.0 kbp supercoiled plasmid DNA through Ultracel ultrafiltration membranes with different nominal molecular weight cut-offs over a range of filtrate flux. The extent of plasmid transmission was a very strong function of the filtrate flux, with the sieving coefficient increasing from essentially zero to nearly one as the flux increased. Data were analyzed in terms of available filtration models to examine the contributions of DNA elongation, shear deformation, and concentration polarization on the observed plasmid transmission. The results clearly indicate the importance of flow-induced elongation or deformation on DNA transmission, with the data consistent with a modified version of the elongational flow model using the plasmid superhelix radius as the characteristic dimension. The results provide important insights into the factors controlling DNA transmission through semipermeable ultrafiltration membranes. © 2007 Elsevier B.V. All rights reserved. Keywords: Ultrafiltration; Plasmid DNA; Sieving coefficient; DNA deformation; DNA elongation
1. Introduction Recent progress in the development of gene therapies [1,2] and DNA-based vaccines [3] has created a need for new separations technologies suitable for the large-scale production of highly purified plasmid DNA. Plasmids are circular doublestranded extrachromosomal DNA that are produced by many bacteria, often at high copy numbers. Large-scale production of plasmid DNA is performed in an appropriate bioreactor, with the plasmid released into the fluid medium by controlled cell lysis. Downstream purification is needed to remove cellular debris, host cell proteins, genomic DNA, RNA, and endotoxins. Although size exclusion and ion exchange chromatography are used extensively in current separation systems [4], these methods are expensive and time-consuming due to the low binding capacity and significant mass transfer limitations associated with the very large size of the plasmid DNA [5].
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Corresponding author. Tel.: +1 814 863 7113; fax: +1 814 865 7846. E-mail address: [email protected] (A.L. Zydney).
0376-7388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2007.02.033
Membrane processes have great potential for large-scale plasmid purification since they are easily scalable and are only weakly affected by diffusional limitations. Hancher and Ryon [6] published the first study on nucleic acid (RNA) filtration through polymeric ultrafiltration membranes. RNA transmission increased with increasing membrane molecular weight cut-off as expected, with the data providing the first demonstration of the feasibility of using ultrafiltration for the concentration and desalting of RNA solutions. Data were also obtained for the separation of a small tRNA from bovine serum albumin, although no significant separation was achieved over a range of filtration conditions. Hirasaki et al. [7] examined DNA filtration through membranes with mean pore diameters of 15 and 35 nm that were originally developed for virus filtration. DNA transmission decreased with increasing size of the DNA, with the fractional transmission for the DNA being significantly larger than that for globular proteins of comparable molecular weight. The high transmission of the DNA was attributed to the elongation or deformation of the DNA molecules due to the shear stress associated with the filtrate flow. Transmission of large linear
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DNA (>45 kbp = kilo base pairs) increased with increasing transmembrane pressure (TMP), which the authors attributed to the increased elongation or deformation at the higher shear. Subsequent work by Higuchi et al. [8] using the same membrane and filtration conditions gave very different results, with significant retention of a 23 kbp DNA by the 15 nm pore size membrane. No explanation was provided for this large difference in filtration behavior. Kahn et al. [9] investigated the sieving behavior of several plasmids (5.6–7.9 kbp) during tangential flow filtration (TFF) through a 500 kDa ultrafiltration membrane. Plasmid transmission into the filtrate increased with increasing TMP, although the authors did not provide a physical explanation for this behavior. Eon-Duval et al. [10] extended these studies to examine the effects of TMP, membrane pore size, and solution conductivity on the retention of a 7.7 kbp plasmid during TFF. Plasmid retention was very high, with significant removal of RNA through the membrane. Kepka et al. [11] used hollow fiber polysulfone membranes to concentrate and buffer exchange plasmids with sizes from 2.6 to 11.5 kbp. Recovery of the 6.1 kbp plasmid after the ultrafiltration/diafiltration process was only 78%, which was apparently due to plasmid adsorption to the polysulfone membrane. Interestingly, the yields obtained with the 2.6 and 11.5 kbp plasmids were greater than that for the 6.1 kbp plasmid, although this might have been due to the different isoforms, and thus, the different extent of adsorption, for the 6.1 kbp plasmid. In addition to these studies of DNA ultrafiltration, several investigators have examined the use of microfiltration for clarification of DNA solutions. For example, Kendall et al. [12] reported complete transmission of both 6.9 and 20 kbp plasmids through 0.45 m nitrocellulose membranes, with no indication of any shear damage up to a TMP of 400 Pa. More recently, Kong et al. [13] examined the effect of solution conditions on sterile filtration of both plasmid DNA (6–29 kbp) and bacterial artificial chromosomes (33–116 kbp). DNA transmission was greatest for the smallest DNA molecules, a result that was attributed to the increase in deformation of the smaller DNA associated with the elongational flow field near the membrane pores. No quantitative analysis of the data was provided. Although these studies have demonstrated the potential of using membrane-based processes for plasmid purification, there is still considerable discrepancy regarding the effects of membrane pore size, filtrate flux rate, and sieving mechanisms on DNA transmission. The objective of this work was to obtain quantitative data on the transmission of a well-defined supercoiled circular plasmid through a series of ultrafiltration membranes with different nominal molecular weight cut-offs over a range of filtrate flux. The data were analyzed in terms of available filtration models to identify the contributions of DNA elongation, shear deformation, and concentration polarization on the observed plasmid transmission. 2. Materials and methods Ultrafiltration experiments were performed using a 3.0 kbp pBluescript II KS+ plasmid commercially available from Stratagene (La Jolla, CA). A stock solution of the plasmid in 10 mM
Tris-base–1 mM ethylenediaminetetraacetic acid (EDTA) was prepared by Aldevron (Fargo, ND) as follows. The plasmid was transfected into a DH5␣ bacterial strain, with a single colony used to inoculate a small starter culture. The starter culture was allowed to grow for approximately 7 h at 37 ◦ C and was then used to inoculate 2 l of a nutrient-rich media, which was grown overnight in a shaker flask. The resulting bacterial mass was harvested and lysed, with the lysate processed by anion exchange. Agarose gel electrophoresis verified that >95% of the circular plasmid was in the desired supercoiled (SC) topology. The stock solution was divided into 110 l aliquots and stored at −20 ◦ C. Plasmid solutions were prepared by diluting thawed stock solution with Tris–EDTA (TE) buffer made by mixing a TE concentrate (Sigma–Aldrich, 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 M cm. The ionic strength of the buffer solution was adjusted with sodium chloride (VWR, West Chester, PA) to obtain a final NaCl concentration of 10 mM. The solution pH (7.7 ± 0.1) was measured using a 420APlus pH meter (Thermo Orion, Beverly, MA), and the solution conductivity (1920 ± 25 S/cm) was measured using a 105APlus conductivity meter (Thermo Orion). Plasmid concentrations were determined by fluorescence detection using the nucleic acid stain PicoGreen (Invitrogen, Carlsbad, CA). Samples were analyzed in duplicate using a 96well GENios FL microplate reader (TECAN, Research Triangle Park, NC). Each well was initially filled with a 100 l sample plus 100 l 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. Filtration experiments were conducted using Ultracel composite regenerated cellulose membranes (Millipore Corp., Billerica, MA) having nominal molecular weight cut-offs of 100 kDa (Lot 112204BCH-1), 300 kDa (Lot K021805ACM4), and 1000 kDa (Lot 080304ACX-3). Limited data were also obtained using Durapore polyvinylidene fluoride membranes (Millipore) with pore size of 0.22 m. 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 insure thorough wetting of the pore structure. 2.1. Plasmid adsorption The extent of non-specific adsorption of the plasmid to the Ultracel and Durapore membranes was examined by immersing a membrane in a sealed plastic jar filled with 30 ml of a 250 ng/ml plasmid solution. The jar was placed on a Gyrotory Model G2 shaker table (New Brunswick Scientific Co., Edison, NJ) and agitated for 24 h at room temperature. A second jar containing
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an equal volume of the plasmid solution without a membrane was used as a control. Samples were taken from both jars at 0, 12, 18, and 24 h to evaluate the extent of adsorption from the change in plasmid concentration in the bulk solution. The membrane hydraulic permeability (Lp ): Lp =
JV P
(1)
was also evaluated before and after exposure to the plasmid solution using the apparatus described below. The filtrate flux (Jv ) was determined by timed collection, and the transmembrane pressure difference (P) was determined by differential pressure measurement using a digital pressure gauge (Omega, Stamford, CT). 2.2. Plasmid sieving experiments Sieving experiments were conducted 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 1531AB phototachometer (General Radio Co., Concord, MA). The stirred cell was connected to a polycarbonate feed reservoir that was pressurized with compressed air. The reservoir was fitted with a ball valve for pressure relief and a DPG1203 digital differential pressure gauge (Omega, Stamford, CT). Sieving data were obtained over a range of transmembrane pressure from 0 to 62 kPa (corresponding to 0–9 psig). The stirred cell was filled with the appropriate plasmid solution and a sample was taken directly from the stirred cell (before sealing the assembly). The stirred cell was then connected to the feed reservoir and the entire system was pressurized. The stirrer was turned on, the filtrate line was unclamped, and the feed pressure set to the appropriate value to achieve the desired filtrate flux. Samples were collected after a minimum of 2 min and after collection of at least 1 ml of filtrate. 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 samples were stored at 4 ◦ C until analysis. The stirred cell was then reassembled for a second experiment or emptied and filled with buffer solution to measure the membrane hydraulic permeability. All experiments were performed at room temperature.
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3. Results and discussion The extent of plasmid adsorption to the Ultracel and Durapore membranes was evaluated by solution depletion over a period of 24 h. Data obtained with a 250 ng/ml solution of the 3.0 kbp plasmid showed no measurable change in the solution concentration (compared to that for the control) over a 24-h static adsorption experiment at room temperature, even at the very low plasmid concentrations used in these studies. In addition, the hydraulic permeability of the membrane evaluated after adsorption was statistically indistinguishable from the value obtained for the fresh membrane (at a significance level of 0.01). These data indicate that there was minimal plasmid adsorption to either the Ultracel or Durapore membranes over the range of conditions examined in this study. Typical data for the observed sieving coefficient (So ) of the 3.0 kbp plasmid through the 300 kDa Ultracel membrane are shown in Fig. 1 as a function of the filtrate flux. Experiments were performed using a feed concentration of 250 ng/ml and a stirrer speed of 730 rpm. The observed sieving coefficient was evaluated as the ratio of the plasmid concentration in the filtrate solution to that in the feed, with the feed concentration determined as the arithmetic average of the concentrations evaluated from samples obtained from the stirred cell immediately before and after the sieving measurements. The error bars in the figure represent plus/minus one standard deviation as determined from the accuracy of the concentration measurements. The sieving coefficient was a strong function of filtrate flux, with So increasing by over 70-fold as the flux increased from approximately 20 to 125 m/s (corresponding to transmembrane pressures from 3.4 to 28 kPa). The sieving coefficient was negligible (So < 0.02) for filtrate flux below 20 m/s, with this value of the flux defined as the critical flux (Jcrit ) for plasmid transmission through the 300 kDa membrane under these experimental conditions. There was no evidence of any fouling in these experiments; the hydraulic permeability of the membrane after filtration was statistically indistinguishable from the per-
2.3. Agarose gel electrophoresis The topology of the plasmid was examined by agarose gel electrophoresis. Horizontal gels were made with 0.8% agarose in Tris–acetate–EDTA buffer containing ethidium bromide. Samples were prepared by mixing 20 l of the plasmid solution with 4 l of TrackIt buffer (Invitrogen). TrackIt 1 kbp DNA ladder (Invitrogen) was loaded into the first well of the gel as a reference. Gels were run at 60 V for 40–60 min in a Mini-Sub Cell GT (BioRad, Hercules, CA) with a PowerPac Basic power supply (BioRad). Gel images were captured using a UV transilluminator and AlphaImager software (Alpha Innotech Corp., San Leandro, CA).
Fig. 1. Effect of filtrate flux on the sieving coefficient of the 3.0 kbp plasmid through the Ultracel 300 kDa membrane at a plasmid concentration of 250 ng/ml and a stirring speed of 730 rpm.
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meability of the fresh (un-used) membrane. In addition, the flux obtained with the plasmid solution was always within 10% of the value determined using the buffer. Agarose gel electrophoresis confirmed that there were no topological changes in the plasmid, with the DNA in the feed and filtrate solutions both migrating as supercoiled DNA. The increase in plasmid transmission with increasing filtrate flux seen in Fig. 1 is qualitatively consistent with the build-up of a high concentration of retained solute near the membrane surface due to the effects of concentration polarization [14]: So =
Sa (1 − Sa ) exp(−Jv /km ) + Sa
(2)
where Sa is the actual sieving coefficient, equal to the ratio of the solute concentration in the filtrate solution to that in the solution immediately upstream of the membrane surface, and km is the bulk mass transfer coefficient. Eq. (2) can be re-arranged as: 1 Jv 1 − 1 = ln −1 − (3) ln So Sa km indicating that a plot of ln(1/So − 1) should be linear in the flux assuming that the membrane Peclet number (Pem ) is large enough (typically greater than about 4) so that Sa is independent of the filtrate flux. It is difficult to obtain quantitative estimates for the membrane Peclet number for the plasmid DNA due to the absence of theoretical results for the hindrance factors describing the convective and diffusive transport of large (flexible) polymers through narrow pores. The most conservative estimate of Pem is obtained assuming that the hindrance factors for diffusion and convection are both negligible, in which case Pem is greater than 4 for filtrate flux above 32 m/s. A more realistic value for Pem can be obtained using expressions for the hindrance factors for a hard-sphere in a cylindrical pore [14] assuming that the characteristic size of the plasmid DNA in the membrane pore is one-half the pore radius. This gives Pem > 4 over the entire range of filtrate flux examined in this study. The experimental data in Fig. 1 were re-plotted in the form suggested by Eq. (3) with the results shown in Fig. 2. The data show a strong non-linear response over this range of filtrate flux, with the slope (equal to 1/km ) varying by more than a factor of 10 for data obtained at low and high filtrate flux. These results strongly suggest that the increase in So seen in Fig. 1 is not due to traditional concentration polarization effects, which is also consistent with the similarity in the filtrate flux obtained with solutions of the plasmid DNA and the buffer. Note that the observed dependence on the flux cannot be explained by any flux-dependence of the actual sieving coefficient; this phenomenon would cause the data to be concave down in contrast to the strong upward concavity seen in Fig. 2. Instead, the increase in sieving coefficient with increasing filtrate flux is more likely due to some type of flow-induced elongation or deformation of the DNA. This is discussed in more detail subsequently. In order to obtain additional insights into the possible effects of concentration polarization on the plasmid sieving coefficient, a series of experiments were performed over a range of stirring speeds corresponding to a range of bulk mass transfer
Fig. 2. Concentration polarization analysis of the sieving coefficient data in Fig. 1. Slope is equal to the reciprocal of the mass transfer coefficient if the flux dependence is due to concentration polarization.
coefficients in the stirred cell. Results are shown in Fig. 3 for the 300 kDa Ultracel membrane at three different values of the filtrate flux. The plasmid transmission was completely independent of the stirring speed, with values varying by less than 10% between 300 and 970 rpm even at the highest filtrate flux. The observed sieving coefficients at the lowest filtrate flux were all less than 0.004, with no measurable variation over the range of stirring speed examined in these experiments. These data provide additional confirmation that the flux dependence seen in Fig. 1 is not governed by concentration polarization. The experimental data in Fig. 1 indicate that the plasmid is able to pass relatively easily through the pores of the 300 kDa membrane at high values of the filtrate flux, with greater than 60% transmission at the highest flux. This high degree of transmission is surprising given the large size of the plasmid DNA. The radius of gyration (RG ) of the supercoiled plasmid DNA used in these experiments can be approximated using the ‘wormlike chain’ model originally developed by Kratky and Porod. The
Fig. 3. Effect of stirring speed on the plasmid sieving coefficients through the Ultracel 300 kDa membrane at a filtrate flux of 13 m/s (♦), 44 m/s (), and 120 m/s () with a feed concentration of 250 ng/ml.
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size of a linear 3.0 kbp DNA can be estimated as [15]: 1/2 a 2 2a L −L/a −1+ −2 (1 − e ) RG-Linear = a 3a L L
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ments for the shear-induced elongation of highly flexible chains approaching a single cylindrical pore as: (4)
The contour length of the DNA (L) is calculated by multiplying the number of base pairs by the axial rise per base pair (0.34 nm/bp). The persistence length (a) is proportional to the DNA stiffness and has a relatively constant value of 50 nm over a range of solution ionic strength [16]. The size of the supercoiled form is reduced compared to that of the linear DNA since the double-helix is covalently closed resulting in a self-coiling ‘plectonemic’ structure. Previous studies indicate that the size of the supercoiled DNA is slightly less than that for the opencircular (OC) form [17,18], with the latter estimated from the radius of gyration of the linear form as [19]: 1/2 1 2 RG-OC = (5) RG-Linear 2 Eq. (5) predicts a radius of gyration of 86 nm for a 3.0 kbp plasmid in the open circular form, with the radius of the supercoiled form estimated as 73 nm based on the data from Voordouw et al. [17] and Fishman and Patterson [18]. The radius of gyration for the supercoiled plasmid examined in this study is more than 8 times larger than the effective pore radius of the 300 kDa membrane, which was estimated as 8.6 nm using the Hagen–Poiseuille equation [20]: 8μδm Lp 1/2 rp = (6) ε assuming that the membrane is composed of a parallel array of uniform cylindrical pores. The hydraulic permeability of the 300 kDa membrane, determined from the buffer flow measurements, was 4.6 × 10−6 m/(s kPa). The membrane thickness, δm , was estimated as 0.5 m for the skin layer in the asymmetric composite regenerated cellulose membrane, and the membrane porosity, ε, was estimated as 50% [21]. Calculations using a range of membrane thickness (from 0.2 to 1 m) and porosity (from 20 to 80%) gave a membrane pore radius from about 4 to 20 nm, which is still significantly smaller than the radius of gyration of the plasmid DNA. Thus, the high degree of plasmid transmission obtained in these experiments can only occur if there is significant elongation or deformation of the plasmid during ultrafiltration. There is extensive previous work on the flow-induced deformation of flexible polymer chains. The extent of polymer deformation is typically analyzed in terms of the Deborah number (De), which is the ratio of the time-scale for polymer relaxation (τ) to the characteristic time for the fluid flow (γ −1 ) where γ is the shear-rate of the fluid flow [22]. Significant polymer deformation typically begins to occur when De = τγ ≥ 1. One of the earliest attempts to model the effects of chain deformation on the transport of large linear polymers into cylindrical pores was due to Daoudi and Brochard [22]. For very large polymers, the analysis predicts a sharp transition in polymer transmission as the filtrate flux increases, with the critical flux characterizing this transmission evaluated using scaling argu-
Jcrit =
εkB T rp2 μ
(7)
where kB is the Boltzmann constant, T the absolute temperature, μ the fluid viscosity, and rp is the pore radius. Long and Anderson [23] used Eq. (7) to analyze data for the transmission of linear polystyrene through track-etched mica membranes, with the results in good qualitative agreement with the model under conditions where the radius of gyration of the polystyrene was larger than the membrane pore radius. Eq. (7) gives a critical flux for plasmid transmission through the 300 kDa membrane of approximately 2.8 × 104 m/s, assuming a membrane porosity of 0.5 and a pore radius of 8.6 nm, which is more than three orders of magnitude larger than the critical flux seen in Fig. 1. This discrepancy is much larger than any uncertainty in the calculated value of the membrane pore radius. Eq. (7) also predicts that the critical flux varies with the reciprocal of the pore radius squared; thus, membranes with larger pore size should have a smaller critical flux. This effect is examined in more detail in Fig. 4, which shows data for plasmid transmission through three different nominal molecular weight cut-off Ultracel membranes. The hydraulic permeabilities of the 100 and 1000 kDa membrane were 2.5 × 10−6 and 8.7 × 10−6 m/(s kPa), which correspond to pore radii of 6.4 and 11.8 nm, respectively. The plasmid sieving coefficient at any given flux increases significantly with increasing membrane pore size. For example, the sieving coefficient through the 1000 kDa membrane at a filtrate flux of 50 m/s was about 100 times larger than that through the 100 kDa membrane at a comparable flux. Also shown for comparison is the plasmid transmission through a 0.2 m pore size Durapore membrane. The plasmid sieving coefficients for the Durapore membrane are above 0.9 over the entire range of filtrate flux examined in Fig. 4, indicating that there is nearly complete plasmid transmission through this membrane, consistent with the large pore size (200 nm) relative to the size of the plasmid DNA (73 nm).
Fig. 4. Plasmid sieving coefficients through the Ultracel 100 kDa (♦), Ultracel 300 kDa (), Ultracel 1000 kDa (), and Durapore 0.22 m () membranes at a feed concentration of 250 ng/ml and a stirring speed of 730 rpm.
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Fig. 5. Critical filtrate flux as a function of the membrane pore size. Solid line is prediction of the elongational flow model (Eq. (7)); dash-dot line is prediction of shear deformation model (Eq. (9)); dashed line is prediction of modified elongational flow model (Eq. (12)).
In order to examine the pore size dependence in more detail, the calculated values of the critical flux are shown as an explicit function of the membrane pore radius in Fig. 5. For each membrane, the critical flux was evaluated by linear regression of the sieving coefficient versus flux data in Fig. 4, using data obtained for sieving coefficients between 0.02 and 0.5, with Jcrit determined as the x-intercept (corresponding to So = 0) on the filtrate flux axis. The log–log plot in Fig. 5 yields a slope of −3.4, a significantly stronger dependence than that given by Eq. (7), and the predicted values of the critical flux using the elongation model (solid curve) are nearly three orders of magnitude larger than the experimental results. One possible reason for the large discrepancy between the values of the critical flux predicted by Eq. (7) and the experimental data is that the model neglects the effects of intermolecular interactions on the elongation of the plasmid DNA. Daoudi and Brochard [22] and Nguyen and Neel [24] extended this analysis to account for the reduction in polymer length caused by polymer–polymer interactions, with the critical flux now a function of the polymer concentration: Jcrit
εkB T = 2 rp μ
C∗ C
15/4 (8)
where C* is the polymer overlap concentration. Eq. (8) predicts that the critical flux is a strong inverse function of the polymer concentration. Experimental data for the plasmid sieving coefficient at feed concentrations of 250, 750, and 2500 ng/ml are shown in Fig. 6. In each case, the feed was prepared by diluting an appropriate amount of the 250 g/ml stock solution with 10 mM NaCl–TE buffer to obtain a final volume of 100 ml. The plasmid sieving coefficients were completely independent of the feed concentration over the entire range of filtrate flux, in sharp contrast to the nearly 4-order of magnitude variation in critical flux predicted by Eq. (8). The data clearly indicate that intermolecular interactions have no significant effect on plasmid transmission through the 300 kDa Ultracel membrane.
Fig. 6. Plasmid sieving coefficients through the Ultracel 300 kDa membrane at feed concentrations of 250 ng/ml (♦), 750 ng/ml (), and 2500 ng/ml () and a stirring speed of 730 rpm.
In addition to solute deformation due to the elongational flow near the pore entrance, the local shear stress at the pore wall can cause an orientation and elongation of the polymer chain [25]. The shear rate in the pore can be evaluated assuming Hagen–Poiseuille flow with the critical flux again estimated as occurring at De = 1 yielding: kB T εrp Jcrit = (9) 4 μR3G The dash-dot curve in Fig. 5 shows the model predictions for the critical flux based on Eq. (9). This shear flow model predicts a critical flux of 11 m/s for the 300 kDa membrane (with rp = 8.6 nm), which is in good agreement with the experimental data. However, the shear-flow model predicts that the critical flux increases with increasing membrane pore size due to the reduction in the wall shear rate in the larger pores, in sharp contrast to the behavior observed experimentally. Thus, Eq. (9) would give a critical flux of 16 m/s for the 1000 kDa membrane and 260 m/s for the 0.2 m pore size Durapore membrane, while the experimental data in Fig. 4 are consistent with Jcrit values of 8 m/s and below 6 m/s, respectively. Note that the application of this model to the Durapore membrane should be done with caution since the radius of gyration of the plasmid is actually smaller than the membrane pore size, eliminating the requirement for plasmid elongation/deformation for entry into the pore. The development leading to Eqs. (7) and (9) both used the Zimm ‘non-free draining’ model to evaluate the time-scale for polymer relaxation, τ [26]. Rouse [27] developed an alternative model to describe the dynamics of polymer molecules in dilute solution that ignored the effects of intramolecular hydrodynamic interactions on polymer dynamics. Berne and Pecora [28] derived the following expression for the ‘free-draining’ relaxation time in terms of experimentally measurable quantities: τ=
R2G π2 D
(10)
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where D is the translational diffusion coefficient. Lewis and Pecora [29] evaluated the relaxation time of a 2.3 kbp DNA fragment by birefringence experiments as τ = 680 s. This value is in much better agreement with predictions of the ‘free-draining’ model by Eq. (10) compared to the ‘non-free draining’ model. Light scattering studies of a slightly larger 3.7 kbp supercoiled plasmid yielded a value of D = 4 × 10−12 m2 /s and this value was used for this work [18]. Incorporation of Eq. (10) into the elongational flow model gives: εR3G π2 D (11) Jcrit = rp2 R2G which predicts a critical flux of 1.9 × 104 m/s for the 300 kDa membrane. This value is still three orders of magnitude larger than the experimental data in Fig. 1; thus, the difference between the model and data is not simply related to the choice of a model for the deformation of a flexible polymer chain (either freedraining or non-free-draining) to evaluate the relaxation time for the plasmid DNA. The elongational flow model developed by Daoudi and Brochard [22] was based on flow through a single cylindrical pore located within an infinite flat (non-porous) plane. The critical flux was then evaluated by assuming that polymer deformation becomes significant when the polymer is located at a distance of one radius of gyration above the pore entrance. Although this physical picture might be appropriate for a nonporous plane with a single pore, this model will break down for a real membrane since the fluid streamlines from adjacent pores will begin to interact at a distance above the surface of the membrane approximately equal to the pore–pore spacing, which is much less than the 73 nm radius of gyration for the plasmid DNA examined in this study. Given the geometry of the supercoiled plasmid, it is likely that the plasmid would be able to approach the membrane surface to within a distance characterized by the radius of the plasmid superhelix, RS , which is the average distance between the superhelix axis and the axis of the DNA double helix [30]. Thus, it seems appropriate to evaluate the critical flux for DNA transmission using a modified form of the elongation flow model presented by Daoudi and Brochard, in which the critical distance from the membrane surface is evaluated as the radius of the superhelix and the relaxation time for the DNA is evaluated using the expression for a ‘free-draining’ polymer chain (Eq. (10)) giving: εR3S π2 D Jcrit = (12) rp2 R2G Boles et al. [30] estimated the superhelix radius of 3.5 and 7.0 kbp supercoiled plasmid DNA as approximately 10 nm based on image analysis of electron micrographs. The dashed curve in Fig. 5 shows the model predictions for the critical flux based on Eq. (12). For the 300 kDa membrane, the predicted critical flux of 50 m/s is a factor of 2.5 larger than the experimental data (Jcrit = 20 m/s), although this small discrepancy is not significant since the original theoretical development pre-
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sented by Daoudi and Brochard was based on scaling arguments and would thus, be expected to give only order of magnitude agreement with the data. The predicted values of the critical flux for the 100 kDa (90 m/s) and 1000 kDa (26 m/s) membrane are also in good agreement with the experimental results of 64 and 8 m/s, respectively, particularly considering the approximations involved in both the model development and the evaluation of the membrane pore radius. In addition, the predicted value of the critical flux for the 0.2 m PVDF membrane (Jcrit = 0.1 m/s) is consistent with the very high degree of DNA transmission seen in Fig. 4 (although the plasmid DNA probably does not need to elongate to enter the pores of this microfiltration membrane). 4. Conclusions The experimental results obtained in this study provide some of the most quantitative data currently available for the transmission of plasmid DNA through semipermeable ultrafiltration membranes. The extent of plasmid transmission is a very strong function of the filtrate flux, with the sieving coefficient increasing from essentially zero to nearly one as the flux increases. This large increase in sieving coefficient with increasing flux was not due to concentration polarization effects; the functional dependence on the flux was inconsistent with predictions of the polarization model and the sieving coefficient showed no dependence on the stirring speed. Instead, the flux dependence was likely due to some type of deformation or elongation of the plasmid during filtration, consistent with the ability of the large plasmid (radius of gyration > 70 nm) to pass through the 8.6 nm pores of the 300 kDa Ultracel membrane examined in these experiments. The plasmid sieving coefficient was also a strong function of the membrane pore size, with the sieving coefficient through the 1000 kDa membrane being nearly two orders of magnitude larger than that through the 100 kDa membrane at a filtrate flux of 50 m/s. The critical flux decreased significantly with increasing membrane pore size, which is consistent with predictions of the elongation flow model developed by Daoudi and Brochard [22], but the magnitude of the critical flux was approximately three orders of magnitude smaller than the predicted values. In contrast, the shear flow deformation model gave a reasonably good prediction of the critical flux for the 300 kDa membrane, but this model predicts an increase in critical flux with increasing pore size, exactly opposite of the trend found experimentally. A modified form of the elongational flow model using the plasmid superhelix radius as the characteristic dimension and the freedraining model for the plasmid relaxation time was shown to be in significantly better agreement with the experimental data than any of the published models for polymer deformation in membrane systems. According to the elongation flow model, the plasmid transmission should increase rapidly from zero to essentially one when the flux exceeds the critical flux. However, the experimental results obtained with the Ultracel membranes show a much broader variation in plasmid transmission, with the sieving coefficient increasing from 0 to 1 over about an order of magnitude
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range in the filtrate flux. Similar behavior was seen by Long and Anderson [23] for the transmission of large linear polystyrene molecules through microporous track-etched mica membranes, which the authors attributed to possible discrepancies in the velocity flow field above the rhomboidal pores of the mica membrane. The gradual increase in the plasmid sieving coefficient observed in this study could be due to a number of phenomena. First, Perkins et al. [31] showed that there is a very large heterogeneity in the stretching of individual DNA molecules as determined from direct observations of fluorescently labeled DNA in an elongational flow field. This heterogeneity could lead to a gradual variation in DNA transmission with increasing filtrate flux. Second, the Ultracel membranes have a significant pore size distribution. Plasmid transmission would initially become significant when the filtrate flux exceeds the critical flux in the largest pores, with the gradual increase in transmission reflecting the variation in the critical flux due to the variation in the pore radius throughout the distribution. The presence of a broad pore size distribution might also explain the experimental observation that the critical flux decreases with increasing pore size with a power law dependence of approximately rp−3.4 , which is a much larger exponent than predicted by the elongational flow model. Since the critical flux was defined at the onset of significant plasmid transmission, Jcrit would be governed by the largest pores in the distribution, which may not be properly described using the mean pore size evaluated from the hydraulic permeability data. Future work will be required to fully characterize the nature of the flow/deformation that governs the sieving characteristics of plasmid DNA during filtration through semipermeable ultrafiltration membranes. Acknowledgements The authors would like to acknowledge Millipore Corporation for donation of the Ultracel membranes. The authors would also like to thank Aldevron for providing technical details on the process used to prepare the plasmid DNA.
Nomenclature a C C* D De Jcrit Jv kB km L Lp P Pem rp RG
DNA persistence length (m) polymer concentration (kg/m3 ) polymer overlap concentration (kg/m3 ) diffusion coefficient (m2 /s) Deborah number critical filtrate flux (m/s) filtrate flux (m/s) Boltzmann constant (J/K) bulk mass transfer coefficient (m/s) DNA contour length (m) membrane hydraulic permeability (m/(s kPa)) transmembrane pressure difference (kPa) membrane Peclet number membrane pore radius (m) radius of gyration (m)
RS Sa So T
superhelix radius (m) actual sieving coefficient observed sieving coefficient absolute temperature (K)
Greek symbols γ shear rate (s−1 ) membrane thickness (m) δm ε membrane porosity μ fluid viscosity (Pa s) τ relaxation time (s)
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Flux-dependent transmission of supercoiled plasmid DNA through ultrafiltration membranes David R. Latulippe, Kimberly Ager, Andrew L. Zydney ∗ Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802, United States Received 1 November 2006; received in revised form 6 February 2007; accepted 18 February 2007 Available online 21 February 2007
Abstract Although previous studies have demonstrated the potential of using membrane-based processes for purification of plasmid DNA, there is considerable discrepancy regarding both the physical mechanisms governing DNA transmission and the effects of membrane pore size and operating conditions on the DNA sieving coefficient. The objective of this work was to obtain quantitative data on the transmission of a 3.0 kbp supercoiled plasmid DNA through Ultracel ultrafiltration membranes with different nominal molecular weight cut-offs over a range of filtrate flux. The extent of plasmid transmission was a very strong function of the filtrate flux, with the sieving coefficient increasing from essentially zero to nearly one as the flux increased. Data were analyzed in terms of available filtration models to examine the contributions of DNA elongation, shear deformation, and concentration polarization on the observed plasmid transmission. The results clearly indicate the importance of flow-induced elongation or deformation on DNA transmission, with the data consistent with a modified version of the elongational flow model using the plasmid superhelix radius as the characteristic dimension. The results provide important insights into the factors controlling DNA transmission through semipermeable ultrafiltration membranes. © 2007 Elsevier B.V. All rights reserved. Keywords: Ultrafiltration; Plasmid DNA; Sieving coefficient; DNA deformation; DNA elongation
1. Introduction Recent progress in the development of gene therapies [1,2] and DNA-based vaccines [3] has created a need for new separations technologies suitable for the large-scale production of highly purified plasmid DNA. Plasmids are circular doublestranded extrachromosomal DNA that are produced by many bacteria, often at high copy numbers. Large-scale production of plasmid DNA is performed in an appropriate bioreactor, with the plasmid released into the fluid medium by controlled cell lysis. Downstream purification is needed to remove cellular debris, host cell proteins, genomic DNA, RNA, and endotoxins. Although size exclusion and ion exchange chromatography are used extensively in current separation systems [4], these methods are expensive and time-consuming due to the low binding capacity and significant mass transfer limitations associated with the very large size of the plasmid DNA [5].
∗
Corresponding author. Tel.: +1 814 863 7113; fax: +1 814 865 7846. E-mail address: [email protected] (A.L. Zydney).
0376-7388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2007.02.033
Membrane processes have great potential for large-scale plasmid purification since they are easily scalable and are only weakly affected by diffusional limitations. Hancher and Ryon [6] published the first study on nucleic acid (RNA) filtration through polymeric ultrafiltration membranes. RNA transmission increased with increasing membrane molecular weight cut-off as expected, with the data providing the first demonstration of the feasibility of using ultrafiltration for the concentration and desalting of RNA solutions. Data were also obtained for the separation of a small tRNA from bovine serum albumin, although no significant separation was achieved over a range of filtration conditions. Hirasaki et al. [7] examined DNA filtration through membranes with mean pore diameters of 15 and 35 nm that were originally developed for virus filtration. DNA transmission decreased with increasing size of the DNA, with the fractional transmission for the DNA being significantly larger than that for globular proteins of comparable molecular weight. The high transmission of the DNA was attributed to the elongation or deformation of the DNA molecules due to the shear stress associated with the filtrate flow. Transmission of large linear
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DNA (>45 kbp = kilo base pairs) increased with increasing transmembrane pressure (TMP), which the authors attributed to the increased elongation or deformation at the higher shear. Subsequent work by Higuchi et al. [8] using the same membrane and filtration conditions gave very different results, with significant retention of a 23 kbp DNA by the 15 nm pore size membrane. No explanation was provided for this large difference in filtration behavior. Kahn et al. [9] investigated the sieving behavior of several plasmids (5.6–7.9 kbp) during tangential flow filtration (TFF) through a 500 kDa ultrafiltration membrane. Plasmid transmission into the filtrate increased with increasing TMP, although the authors did not provide a physical explanation for this behavior. Eon-Duval et al. [10] extended these studies to examine the effects of TMP, membrane pore size, and solution conductivity on the retention of a 7.7 kbp plasmid during TFF. Plasmid retention was very high, with significant removal of RNA through the membrane. Kepka et al. [11] used hollow fiber polysulfone membranes to concentrate and buffer exchange plasmids with sizes from 2.6 to 11.5 kbp. Recovery of the 6.1 kbp plasmid after the ultrafiltration/diafiltration process was only 78%, which was apparently due to plasmid adsorption to the polysulfone membrane. Interestingly, the yields obtained with the 2.6 and 11.5 kbp plasmids were greater than that for the 6.1 kbp plasmid, although this might have been due to the different isoforms, and thus, the different extent of adsorption, for the 6.1 kbp plasmid. In addition to these studies of DNA ultrafiltration, several investigators have examined the use of microfiltration for clarification of DNA solutions. For example, Kendall et al. [12] reported complete transmission of both 6.9 and 20 kbp plasmids through 0.45 m nitrocellulose membranes, with no indication of any shear damage up to a TMP of 400 Pa. More recently, Kong et al. [13] examined the effect of solution conditions on sterile filtration of both plasmid DNA (6–29 kbp) and bacterial artificial chromosomes (33–116 kbp). DNA transmission was greatest for the smallest DNA molecules, a result that was attributed to the increase in deformation of the smaller DNA associated with the elongational flow field near the membrane pores. No quantitative analysis of the data was provided. Although these studies have demonstrated the potential of using membrane-based processes for plasmid purification, there is still considerable discrepancy regarding the effects of membrane pore size, filtrate flux rate, and sieving mechanisms on DNA transmission. The objective of this work was to obtain quantitative data on the transmission of a well-defined supercoiled circular plasmid through a series of ultrafiltration membranes with different nominal molecular weight cut-offs over a range of filtrate flux. The data were analyzed in terms of available filtration models to identify the contributions of DNA elongation, shear deformation, and concentration polarization on the observed plasmid transmission. 2. Materials and methods Ultrafiltration experiments were performed using a 3.0 kbp pBluescript II KS+ plasmid commercially available from Stratagene (La Jolla, CA). A stock solution of the plasmid in 10 mM
Tris-base–1 mM ethylenediaminetetraacetic acid (EDTA) was prepared by Aldevron (Fargo, ND) as follows. The plasmid was transfected into a DH5␣ bacterial strain, with a single colony used to inoculate a small starter culture. The starter culture was allowed to grow for approximately 7 h at 37 ◦ C and was then used to inoculate 2 l of a nutrient-rich media, which was grown overnight in a shaker flask. The resulting bacterial mass was harvested and lysed, with the lysate processed by anion exchange. Agarose gel electrophoresis verified that >95% of the circular plasmid was in the desired supercoiled (SC) topology. The stock solution was divided into 110 l aliquots and stored at −20 ◦ C. Plasmid solutions were prepared by diluting thawed stock solution with Tris–EDTA (TE) buffer made by mixing a TE concentrate (Sigma–Aldrich, 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 M cm. The ionic strength of the buffer solution was adjusted with sodium chloride (VWR, West Chester, PA) to obtain a final NaCl concentration of 10 mM. The solution pH (7.7 ± 0.1) was measured using a 420APlus pH meter (Thermo Orion, Beverly, MA), and the solution conductivity (1920 ± 25 S/cm) was measured using a 105APlus conductivity meter (Thermo Orion). Plasmid concentrations were determined by fluorescence detection using the nucleic acid stain PicoGreen (Invitrogen, Carlsbad, CA). Samples were analyzed in duplicate using a 96well GENios FL microplate reader (TECAN, Research Triangle Park, NC). Each well was initially filled with a 100 l sample plus 100 l 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. Filtration experiments were conducted using Ultracel composite regenerated cellulose membranes (Millipore Corp., Billerica, MA) having nominal molecular weight cut-offs of 100 kDa (Lot 112204BCH-1), 300 kDa (Lot K021805ACM4), and 1000 kDa (Lot 080304ACX-3). Limited data were also obtained using Durapore polyvinylidene fluoride membranes (Millipore) with pore size of 0.22 m. 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 insure thorough wetting of the pore structure. 2.1. Plasmid adsorption The extent of non-specific adsorption of the plasmid to the Ultracel and Durapore membranes was examined by immersing a membrane in a sealed plastic jar filled with 30 ml of a 250 ng/ml plasmid solution. The jar was placed on a Gyrotory Model G2 shaker table (New Brunswick Scientific Co., Edison, NJ) and agitated for 24 h at room temperature. A second jar containing
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an equal volume of the plasmid solution without a membrane was used as a control. Samples were taken from both jars at 0, 12, 18, and 24 h to evaluate the extent of adsorption from the change in plasmid concentration in the bulk solution. The membrane hydraulic permeability (Lp ): Lp =
JV P
(1)
was also evaluated before and after exposure to the plasmid solution using the apparatus described below. The filtrate flux (Jv ) was determined by timed collection, and the transmembrane pressure difference (P) was determined by differential pressure measurement using a digital pressure gauge (Omega, Stamford, CT). 2.2. Plasmid sieving experiments Sieving experiments were conducted 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 1531AB phototachometer (General Radio Co., Concord, MA). The stirred cell was connected to a polycarbonate feed reservoir that was pressurized with compressed air. The reservoir was fitted with a ball valve for pressure relief and a DPG1203 digital differential pressure gauge (Omega, Stamford, CT). Sieving data were obtained over a range of transmembrane pressure from 0 to 62 kPa (corresponding to 0–9 psig). The stirred cell was filled with the appropriate plasmid solution and a sample was taken directly from the stirred cell (before sealing the assembly). The stirred cell was then connected to the feed reservoir and the entire system was pressurized. The stirrer was turned on, the filtrate line was unclamped, and the feed pressure set to the appropriate value to achieve the desired filtrate flux. Samples were collected after a minimum of 2 min and after collection of at least 1 ml of filtrate. 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 samples were stored at 4 ◦ C until analysis. The stirred cell was then reassembled for a second experiment or emptied and filled with buffer solution to measure the membrane hydraulic permeability. All experiments were performed at room temperature.
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3. Results and discussion The extent of plasmid adsorption to the Ultracel and Durapore membranes was evaluated by solution depletion over a period of 24 h. Data obtained with a 250 ng/ml solution of the 3.0 kbp plasmid showed no measurable change in the solution concentration (compared to that for the control) over a 24-h static adsorption experiment at room temperature, even at the very low plasmid concentrations used in these studies. In addition, the hydraulic permeability of the membrane evaluated after adsorption was statistically indistinguishable from the value obtained for the fresh membrane (at a significance level of 0.01). These data indicate that there was minimal plasmid adsorption to either the Ultracel or Durapore membranes over the range of conditions examined in this study. Typical data for the observed sieving coefficient (So ) of the 3.0 kbp plasmid through the 300 kDa Ultracel membrane are shown in Fig. 1 as a function of the filtrate flux. Experiments were performed using a feed concentration of 250 ng/ml and a stirrer speed of 730 rpm. The observed sieving coefficient was evaluated as the ratio of the plasmid concentration in the filtrate solution to that in the feed, with the feed concentration determined as the arithmetic average of the concentrations evaluated from samples obtained from the stirred cell immediately before and after the sieving measurements. The error bars in the figure represent plus/minus one standard deviation as determined from the accuracy of the concentration measurements. The sieving coefficient was a strong function of filtrate flux, with So increasing by over 70-fold as the flux increased from approximately 20 to 125 m/s (corresponding to transmembrane pressures from 3.4 to 28 kPa). The sieving coefficient was negligible (So < 0.02) for filtrate flux below 20 m/s, with this value of the flux defined as the critical flux (Jcrit ) for plasmid transmission through the 300 kDa membrane under these experimental conditions. There was no evidence of any fouling in these experiments; the hydraulic permeability of the membrane after filtration was statistically indistinguishable from the per-
2.3. Agarose gel electrophoresis The topology of the plasmid was examined by agarose gel electrophoresis. Horizontal gels were made with 0.8% agarose in Tris–acetate–EDTA buffer containing ethidium bromide. Samples were prepared by mixing 20 l of the plasmid solution with 4 l of TrackIt buffer (Invitrogen). TrackIt 1 kbp DNA ladder (Invitrogen) was loaded into the first well of the gel as a reference. Gels were run at 60 V for 40–60 min in a Mini-Sub Cell GT (BioRad, Hercules, CA) with a PowerPac Basic power supply (BioRad). Gel images were captured using a UV transilluminator and AlphaImager software (Alpha Innotech Corp., San Leandro, CA).
Fig. 1. Effect of filtrate flux on the sieving coefficient of the 3.0 kbp plasmid through the Ultracel 300 kDa membrane at a plasmid concentration of 250 ng/ml and a stirring speed of 730 rpm.
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meability of the fresh (un-used) membrane. In addition, the flux obtained with the plasmid solution was always within 10% of the value determined using the buffer. Agarose gel electrophoresis confirmed that there were no topological changes in the plasmid, with the DNA in the feed and filtrate solutions both migrating as supercoiled DNA. The increase in plasmid transmission with increasing filtrate flux seen in Fig. 1 is qualitatively consistent with the build-up of a high concentration of retained solute near the membrane surface due to the effects of concentration polarization [14]: So =
Sa (1 − Sa ) exp(−Jv /km ) + Sa
(2)
where Sa is the actual sieving coefficient, equal to the ratio of the solute concentration in the filtrate solution to that in the solution immediately upstream of the membrane surface, and km is the bulk mass transfer coefficient. Eq. (2) can be re-arranged as: 1 Jv 1 − 1 = ln −1 − (3) ln So Sa km indicating that a plot of ln(1/So − 1) should be linear in the flux assuming that the membrane Peclet number (Pem ) is large enough (typically greater than about 4) so that Sa is independent of the filtrate flux. It is difficult to obtain quantitative estimates for the membrane Peclet number for the plasmid DNA due to the absence of theoretical results for the hindrance factors describing the convective and diffusive transport of large (flexible) polymers through narrow pores. The most conservative estimate of Pem is obtained assuming that the hindrance factors for diffusion and convection are both negligible, in which case Pem is greater than 4 for filtrate flux above 32 m/s. A more realistic value for Pem can be obtained using expressions for the hindrance factors for a hard-sphere in a cylindrical pore [14] assuming that the characteristic size of the plasmid DNA in the membrane pore is one-half the pore radius. This gives Pem > 4 over the entire range of filtrate flux examined in this study. The experimental data in Fig. 1 were re-plotted in the form suggested by Eq. (3) with the results shown in Fig. 2. The data show a strong non-linear response over this range of filtrate flux, with the slope (equal to 1/km ) varying by more than a factor of 10 for data obtained at low and high filtrate flux. These results strongly suggest that the increase in So seen in Fig. 1 is not due to traditional concentration polarization effects, which is also consistent with the similarity in the filtrate flux obtained with solutions of the plasmid DNA and the buffer. Note that the observed dependence on the flux cannot be explained by any flux-dependence of the actual sieving coefficient; this phenomenon would cause the data to be concave down in contrast to the strong upward concavity seen in Fig. 2. Instead, the increase in sieving coefficient with increasing filtrate flux is more likely due to some type of flow-induced elongation or deformation of the DNA. This is discussed in more detail subsequently. In order to obtain additional insights into the possible effects of concentration polarization on the plasmid sieving coefficient, a series of experiments were performed over a range of stirring speeds corresponding to a range of bulk mass transfer
Fig. 2. Concentration polarization analysis of the sieving coefficient data in Fig. 1. Slope is equal to the reciprocal of the mass transfer coefficient if the flux dependence is due to concentration polarization.
coefficients in the stirred cell. Results are shown in Fig. 3 for the 300 kDa Ultracel membrane at three different values of the filtrate flux. The plasmid transmission was completely independent of the stirring speed, with values varying by less than 10% between 300 and 970 rpm even at the highest filtrate flux. The observed sieving coefficients at the lowest filtrate flux were all less than 0.004, with no measurable variation over the range of stirring speed examined in these experiments. These data provide additional confirmation that the flux dependence seen in Fig. 1 is not governed by concentration polarization. The experimental data in Fig. 1 indicate that the plasmid is able to pass relatively easily through the pores of the 300 kDa membrane at high values of the filtrate flux, with greater than 60% transmission at the highest flux. This high degree of transmission is surprising given the large size of the plasmid DNA. The radius of gyration (RG ) of the supercoiled plasmid DNA used in these experiments can be approximated using the ‘wormlike chain’ model originally developed by Kratky and Porod. The
Fig. 3. Effect of stirring speed on the plasmid sieving coefficients through the Ultracel 300 kDa membrane at a filtrate flux of 13 m/s (♦), 44 m/s (), and 120 m/s () with a feed concentration of 250 ng/ml.
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size of a linear 3.0 kbp DNA can be estimated as [15]: 1/2 a 2 2a L −L/a −1+ −2 (1 − e ) RG-Linear = a 3a L L
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ments for the shear-induced elongation of highly flexible chains approaching a single cylindrical pore as: (4)
The contour length of the DNA (L) is calculated by multiplying the number of base pairs by the axial rise per base pair (0.34 nm/bp). The persistence length (a) is proportional to the DNA stiffness and has a relatively constant value of 50 nm over a range of solution ionic strength [16]. The size of the supercoiled form is reduced compared to that of the linear DNA since the double-helix is covalently closed resulting in a self-coiling ‘plectonemic’ structure. Previous studies indicate that the size of the supercoiled DNA is slightly less than that for the opencircular (OC) form [17,18], with the latter estimated from the radius of gyration of the linear form as [19]: 1/2 1 2 RG-OC = (5) RG-Linear 2 Eq. (5) predicts a radius of gyration of 86 nm for a 3.0 kbp plasmid in the open circular form, with the radius of the supercoiled form estimated as 73 nm based on the data from Voordouw et al. [17] and Fishman and Patterson [18]. The radius of gyration for the supercoiled plasmid examined in this study is more than 8 times larger than the effective pore radius of the 300 kDa membrane, which was estimated as 8.6 nm using the Hagen–Poiseuille equation [20]: 8μδm Lp 1/2 rp = (6) ε assuming that the membrane is composed of a parallel array of uniform cylindrical pores. The hydraulic permeability of the 300 kDa membrane, determined from the buffer flow measurements, was 4.6 × 10−6 m/(s kPa). The membrane thickness, δm , was estimated as 0.5 m for the skin layer in the asymmetric composite regenerated cellulose membrane, and the membrane porosity, ε, was estimated as 50% [21]. Calculations using a range of membrane thickness (from 0.2 to 1 m) and porosity (from 20 to 80%) gave a membrane pore radius from about 4 to 20 nm, which is still significantly smaller than the radius of gyration of the plasmid DNA. Thus, the high degree of plasmid transmission obtained in these experiments can only occur if there is significant elongation or deformation of the plasmid during ultrafiltration. There is extensive previous work on the flow-induced deformation of flexible polymer chains. The extent of polymer deformation is typically analyzed in terms of the Deborah number (De), which is the ratio of the time-scale for polymer relaxation (τ) to the characteristic time for the fluid flow (γ −1 ) where γ is the shear-rate of the fluid flow [22]. Significant polymer deformation typically begins to occur when De = τγ ≥ 1. One of the earliest attempts to model the effects of chain deformation on the transport of large linear polymers into cylindrical pores was due to Daoudi and Brochard [22]. For very large polymers, the analysis predicts a sharp transition in polymer transmission as the filtrate flux increases, with the critical flux characterizing this transmission evaluated using scaling argu-
Jcrit =
εkB T rp2 μ
(7)
where kB is the Boltzmann constant, T the absolute temperature, μ the fluid viscosity, and rp is the pore radius. Long and Anderson [23] used Eq. (7) to analyze data for the transmission of linear polystyrene through track-etched mica membranes, with the results in good qualitative agreement with the model under conditions where the radius of gyration of the polystyrene was larger than the membrane pore radius. Eq. (7) gives a critical flux for plasmid transmission through the 300 kDa membrane of approximately 2.8 × 104 m/s, assuming a membrane porosity of 0.5 and a pore radius of 8.6 nm, which is more than three orders of magnitude larger than the critical flux seen in Fig. 1. This discrepancy is much larger than any uncertainty in the calculated value of the membrane pore radius. Eq. (7) also predicts that the critical flux varies with the reciprocal of the pore radius squared; thus, membranes with larger pore size should have a smaller critical flux. This effect is examined in more detail in Fig. 4, which shows data for plasmid transmission through three different nominal molecular weight cut-off Ultracel membranes. The hydraulic permeabilities of the 100 and 1000 kDa membrane were 2.5 × 10−6 and 8.7 × 10−6 m/(s kPa), which correspond to pore radii of 6.4 and 11.8 nm, respectively. The plasmid sieving coefficient at any given flux increases significantly with increasing membrane pore size. For example, the sieving coefficient through the 1000 kDa membrane at a filtrate flux of 50 m/s was about 100 times larger than that through the 100 kDa membrane at a comparable flux. Also shown for comparison is the plasmid transmission through a 0.2 m pore size Durapore membrane. The plasmid sieving coefficients for the Durapore membrane are above 0.9 over the entire range of filtrate flux examined in Fig. 4, indicating that there is nearly complete plasmid transmission through this membrane, consistent with the large pore size (200 nm) relative to the size of the plasmid DNA (73 nm).
Fig. 4. Plasmid sieving coefficients through the Ultracel 100 kDa (♦), Ultracel 300 kDa (), Ultracel 1000 kDa (), and Durapore 0.22 m () membranes at a feed concentration of 250 ng/ml and a stirring speed of 730 rpm.
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Fig. 5. Critical filtrate flux as a function of the membrane pore size. Solid line is prediction of the elongational flow model (Eq. (7)); dash-dot line is prediction of shear deformation model (Eq. (9)); dashed line is prediction of modified elongational flow model (Eq. (12)).
In order to examine the pore size dependence in more detail, the calculated values of the critical flux are shown as an explicit function of the membrane pore radius in Fig. 5. For each membrane, the critical flux was evaluated by linear regression of the sieving coefficient versus flux data in Fig. 4, using data obtained for sieving coefficients between 0.02 and 0.5, with Jcrit determined as the x-intercept (corresponding to So = 0) on the filtrate flux axis. The log–log plot in Fig. 5 yields a slope of −3.4, a significantly stronger dependence than that given by Eq. (7), and the predicted values of the critical flux using the elongation model (solid curve) are nearly three orders of magnitude larger than the experimental results. One possible reason for the large discrepancy between the values of the critical flux predicted by Eq. (7) and the experimental data is that the model neglects the effects of intermolecular interactions on the elongation of the plasmid DNA. Daoudi and Brochard [22] and Nguyen and Neel [24] extended this analysis to account for the reduction in polymer length caused by polymer–polymer interactions, with the critical flux now a function of the polymer concentration: Jcrit
εkB T = 2 rp μ
C∗ C
15/4 (8)
where C* is the polymer overlap concentration. Eq. (8) predicts that the critical flux is a strong inverse function of the polymer concentration. Experimental data for the plasmid sieving coefficient at feed concentrations of 250, 750, and 2500 ng/ml are shown in Fig. 6. In each case, the feed was prepared by diluting an appropriate amount of the 250 g/ml stock solution with 10 mM NaCl–TE buffer to obtain a final volume of 100 ml. The plasmid sieving coefficients were completely independent of the feed concentration over the entire range of filtrate flux, in sharp contrast to the nearly 4-order of magnitude variation in critical flux predicted by Eq. (8). The data clearly indicate that intermolecular interactions have no significant effect on plasmid transmission through the 300 kDa Ultracel membrane.
Fig. 6. Plasmid sieving coefficients through the Ultracel 300 kDa membrane at feed concentrations of 250 ng/ml (♦), 750 ng/ml (), and 2500 ng/ml () and a stirring speed of 730 rpm.
In addition to solute deformation due to the elongational flow near the pore entrance, the local shear stress at the pore wall can cause an orientation and elongation of the polymer chain [25]. The shear rate in the pore can be evaluated assuming Hagen–Poiseuille flow with the critical flux again estimated as occurring at De = 1 yielding: kB T εrp Jcrit = (9) 4 μR3G The dash-dot curve in Fig. 5 shows the model predictions for the critical flux based on Eq. (9). This shear flow model predicts a critical flux of 11 m/s for the 300 kDa membrane (with rp = 8.6 nm), which is in good agreement with the experimental data. However, the shear-flow model predicts that the critical flux increases with increasing membrane pore size due to the reduction in the wall shear rate in the larger pores, in sharp contrast to the behavior observed experimentally. Thus, Eq. (9) would give a critical flux of 16 m/s for the 1000 kDa membrane and 260 m/s for the 0.2 m pore size Durapore membrane, while the experimental data in Fig. 4 are consistent with Jcrit values of 8 m/s and below 6 m/s, respectively. Note that the application of this model to the Durapore membrane should be done with caution since the radius of gyration of the plasmid is actually smaller than the membrane pore size, eliminating the requirement for plasmid elongation/deformation for entry into the pore. The development leading to Eqs. (7) and (9) both used the Zimm ‘non-free draining’ model to evaluate the time-scale for polymer relaxation, τ [26]. Rouse [27] developed an alternative model to describe the dynamics of polymer molecules in dilute solution that ignored the effects of intramolecular hydrodynamic interactions on polymer dynamics. Berne and Pecora [28] derived the following expression for the ‘free-draining’ relaxation time in terms of experimentally measurable quantities: τ=
R2G π2 D
(10)
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where D is the translational diffusion coefficient. Lewis and Pecora [29] evaluated the relaxation time of a 2.3 kbp DNA fragment by birefringence experiments as τ = 680 s. This value is in much better agreement with predictions of the ‘free-draining’ model by Eq. (10) compared to the ‘non-free draining’ model. Light scattering studies of a slightly larger 3.7 kbp supercoiled plasmid yielded a value of D = 4 × 10−12 m2 /s and this value was used for this work [18]. Incorporation of Eq. (10) into the elongational flow model gives: εR3G π2 D (11) Jcrit = rp2 R2G which predicts a critical flux of 1.9 × 104 m/s for the 300 kDa membrane. This value is still three orders of magnitude larger than the experimental data in Fig. 1; thus, the difference between the model and data is not simply related to the choice of a model for the deformation of a flexible polymer chain (either freedraining or non-free-draining) to evaluate the relaxation time for the plasmid DNA. The elongational flow model developed by Daoudi and Brochard [22] was based on flow through a single cylindrical pore located within an infinite flat (non-porous) plane. The critical flux was then evaluated by assuming that polymer deformation becomes significant when the polymer is located at a distance of one radius of gyration above the pore entrance. Although this physical picture might be appropriate for a nonporous plane with a single pore, this model will break down for a real membrane since the fluid streamlines from adjacent pores will begin to interact at a distance above the surface of the membrane approximately equal to the pore–pore spacing, which is much less than the 73 nm radius of gyration for the plasmid DNA examined in this study. Given the geometry of the supercoiled plasmid, it is likely that the plasmid would be able to approach the membrane surface to within a distance characterized by the radius of the plasmid superhelix, RS , which is the average distance between the superhelix axis and the axis of the DNA double helix [30]. Thus, it seems appropriate to evaluate the critical flux for DNA transmission using a modified form of the elongation flow model presented by Daoudi and Brochard, in which the critical distance from the membrane surface is evaluated as the radius of the superhelix and the relaxation time for the DNA is evaluated using the expression for a ‘free-draining’ polymer chain (Eq. (10)) giving: εR3S π2 D Jcrit = (12) rp2 R2G Boles et al. [30] estimated the superhelix radius of 3.5 and 7.0 kbp supercoiled plasmid DNA as approximately 10 nm based on image analysis of electron micrographs. The dashed curve in Fig. 5 shows the model predictions for the critical flux based on Eq. (12). For the 300 kDa membrane, the predicted critical flux of 50 m/s is a factor of 2.5 larger than the experimental data (Jcrit = 20 m/s), although this small discrepancy is not significant since the original theoretical development pre-
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sented by Daoudi and Brochard was based on scaling arguments and would thus, be expected to give only order of magnitude agreement with the data. The predicted values of the critical flux for the 100 kDa (90 m/s) and 1000 kDa (26 m/s) membrane are also in good agreement with the experimental results of 64 and 8 m/s, respectively, particularly considering the approximations involved in both the model development and the evaluation of the membrane pore radius. In addition, the predicted value of the critical flux for the 0.2 m PVDF membrane (Jcrit = 0.1 m/s) is consistent with the very high degree of DNA transmission seen in Fig. 4 (although the plasmid DNA probably does not need to elongate to enter the pores of this microfiltration membrane). 4. Conclusions The experimental results obtained in this study provide some of the most quantitative data currently available for the transmission of plasmid DNA through semipermeable ultrafiltration membranes. The extent of plasmid transmission is a very strong function of the filtrate flux, with the sieving coefficient increasing from essentially zero to nearly one as the flux increases. This large increase in sieving coefficient with increasing flux was not due to concentration polarization effects; the functional dependence on the flux was inconsistent with predictions of the polarization model and the sieving coefficient showed no dependence on the stirring speed. Instead, the flux dependence was likely due to some type of deformation or elongation of the plasmid during filtration, consistent with the ability of the large plasmid (radius of gyration > 70 nm) to pass through the 8.6 nm pores of the 300 kDa Ultracel membrane examined in these experiments. The plasmid sieving coefficient was also a strong function of the membrane pore size, with the sieving coefficient through the 1000 kDa membrane being nearly two orders of magnitude larger than that through the 100 kDa membrane at a filtrate flux of 50 m/s. The critical flux decreased significantly with increasing membrane pore size, which is consistent with predictions of the elongation flow model developed by Daoudi and Brochard [22], but the magnitude of the critical flux was approximately three orders of magnitude smaller than the predicted values. In contrast, the shear flow deformation model gave a reasonably good prediction of the critical flux for the 300 kDa membrane, but this model predicts an increase in critical flux with increasing pore size, exactly opposite of the trend found experimentally. A modified form of the elongational flow model using the plasmid superhelix radius as the characteristic dimension and the freedraining model for the plasmid relaxation time was shown to be in significantly better agreement with the experimental data than any of the published models for polymer deformation in membrane systems. According to the elongation flow model, the plasmid transmission should increase rapidly from zero to essentially one when the flux exceeds the critical flux. However, the experimental results obtained with the Ultracel membranes show a much broader variation in plasmid transmission, with the sieving coefficient increasing from 0 to 1 over about an order of magnitude
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range in the filtrate flux. Similar behavior was seen by Long and Anderson [23] for the transmission of large linear polystyrene molecules through microporous track-etched mica membranes, which the authors attributed to possible discrepancies in the velocity flow field above the rhomboidal pores of the mica membrane. The gradual increase in the plasmid sieving coefficient observed in this study could be due to a number of phenomena. First, Perkins et al. [31] showed that there is a very large heterogeneity in the stretching of individual DNA molecules as determined from direct observations of fluorescently labeled DNA in an elongational flow field. This heterogeneity could lead to a gradual variation in DNA transmission with increasing filtrate flux. Second, the Ultracel membranes have a significant pore size distribution. Plasmid transmission would initially become significant when the filtrate flux exceeds the critical flux in the largest pores, with the gradual increase in transmission reflecting the variation in the critical flux due to the variation in the pore radius throughout the distribution. The presence of a broad pore size distribution might also explain the experimental observation that the critical flux decreases with increasing pore size with a power law dependence of approximately rp−3.4 , which is a much larger exponent than predicted by the elongational flow model. Since the critical flux was defined at the onset of significant plasmid transmission, Jcrit would be governed by the largest pores in the distribution, which may not be properly described using the mean pore size evaluated from the hydraulic permeability data. Future work will be required to fully characterize the nature of the flow/deformation that governs the sieving characteristics of plasmid DNA during filtration through semipermeable ultrafiltration membranes. Acknowledgements The authors would like to acknowledge Millipore Corporation for donation of the Ultracel membranes. The authors would also like to thank Aldevron for providing technical details on the process used to prepare the plasmid DNA.
Nomenclature a C C* D De Jcrit Jv kB km L Lp P Pem rp RG
DNA persistence length (m) polymer concentration (kg/m3 ) polymer overlap concentration (kg/m3 ) diffusion coefficient (m2 /s) Deborah number critical filtrate flux (m/s) filtrate flux (m/s) Boltzmann constant (J/K) bulk mass transfer coefficient (m/s) DNA contour length (m) membrane hydraulic permeability (m/(s kPa)) transmembrane pressure difference (kPa) membrane Peclet number membrane pore radius (m) radius of gyration (m)
RS Sa So T
superhelix radius (m) actual sieving coefficient observed sieving coefficient absolute temperature (K)
Greek symbols γ shear rate (s−1 ) membrane thickness (m) δm ε membrane porosity μ fluid viscosity (Pa s) τ relaxation time (s)
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