Elongational flow model for transmission of supercoiled plasmid DNA during membrane ultrafiltration

Journal of Membrane Science 329 (2009) 201–208

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Elongational flow model for transmission of supercoiled plasmid DNA during membrane ultrafiltration D.R. Latulippe, A.L. Zydney ∗ Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802, United States

a r t i c l e

i n f o

Article history: Received 3 September 2008 Received in revised form 1 December 2008 Accepted 21 December 2008 Available online 30 December 2008 Keywords: Ultrafiltration Plasmid DNA DNA elongation Sieving coefficient Viscosity

a b s t r a c t The transmission of a 3.0 kbp supercoiled plasmid DNA through ultrafiltration membranes has recently been shown to be strongly dependent on the filtrate flux due to the elongation of the large DNA molecule in the converging flow field above the smaller membrane pore. The objective of this study was to extend these investigations to different size plasmids and to develop an improved theoretical model describing the effects of plasmid size, pore size, operating temperature, and solution viscosity on the critical filtrate flux for plasmid transmission. Experiments were performed in a stirred ultrafiltration cell using composite regenerated cellulose membranes. The critical filtrate flux, evaluated as the lowest filtrate flux at which plasmid transmission becomes significant, decreased with increasing membrane pore size but was essentially independent of the plasmid size (from 3.0 to 17 kbp). The critical filtrate flux was also a function of the operating temperature and glucose concentration through their effect on the viscosity of the buffer solution. The experimental results were in good agreement with predictions of a new model accounting for the effects of the DNA persistence length on the elongation of the supercoiled plasmid in the converging flow field into the membrane pore. © 2008 Elsevier B.V. All rights reserved.

1. Introduction There is considerable interest in using plasmid DNA as an advanced biotherapeutic for gene therapy applications and DNA-based vaccines. Recently, an important milestone in the development of this field was reached when the U.S. Department of Agriculture issued a conditional license for a plasmid DNA vaccine to treat canine melanoma [1,2]. Chromatographic processes are currently used for the large-scale purification of plasmid DNA, although the large mass transfer limitations and very low binding capacities associated with these very large bio-molecules pose significant technical challenges [3]. Sagar et al. [4] indicated that the chromatography resin accounted for more than 50% of the total raw material cost in manufacturing of plasmid DNA. Membrane processes are a potentially attractive alternative to chromatographic separations since they are only weakly affected by diffusional limitations. Several previous studies have shown the potential of using ultrafiltration for purification of plasmid DNA [5–7]. Latulippe et al. [8] recently demonstrated that transmission of plasmid DNA through ultrafiltration membranes is controlled by the elongation/deformation of the plasmid in the converging flow field into the membrane pores. Plasmid transmission was negligible

∗ 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 © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2008.12.045

(less than 1%) below a critical value of the filtrate flux, with this critical flux depending strongly on the membrane pore size [8] and solution conditions [9]. Latulippe et al. [8] also developed a modified version of an elongational flow model to describe the critical flux that was based directly on the original scaling analysis presented by Daoudi and Brochard [10]. The degree of polymer deformation was characterized by the Deborah number (De), which is equal to the time scale for polymer relaxation () 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. The time-scale for the flow was set equal to the inverse shear rate ( −1 ), which is a function of the polymer location within the detailed flow field. In contrast to the work by Daoudi and Brochard [10], Latulippe et al. [8] evaluated the relaxation time using the Rouse free-draining model [11] instead of the Zimm model, and the shear rate was evaluated when the plasmid was a distance of one superhelix radius (RS ) away from the pore entrance instead of a distance equal to the radius of gyration. The critical flux was thus given as:



Jcrit =

2 D 2 RG



εRS3 rp2



(1)

where D is the plasmid diffusion coefficient, RG is the plasmid radius of gyration, ε is the membrane porosity, and rp is the membrane pore radius. The model accurately described the experimental

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results for a 3.0 kilo base pair (kbp) supercoiled plasmid over a range of membrane pore size and solution ionic strength, with the dependence on ionic strength arising from changes in the plasmid diffusion coefficient, radius of gyration, and superhelix radius [9]. All of the experimental studies performed by Latulippe et al. [8] and Latulippe and Zydney [9] employed a single 3.0 kbp plasmid. Hirasaki et al. [5] examined the effect of plasmid size (3, 45, and 151 kbp) on plasmid transmission through hollow fiber virus filtration membranes and found that DNA transmission decreased only slightly over this very large range of DNA size. In contrast, Kong et al. [12] reported a significant reduction in transmission of supercoiled plasmids (from 6 to 29 kbp) and bacterial artificial chromosomes (33 to 116 kbp) with increasing DNA size, although these data were obtained with large 0.22 ␮m pore size microfiltration membranes. These results are inconsistent with the predicted decrease in the critical flux with increasing RG given by Eq. (1), suggesting that this model may be insufficient to fully describe plasmid transmission during membrane filtration. The objective of this study was to extend these previous investigations by evaluating the transmission of different sized supercoiled plasmids (3.0, 9.8, and 17 kbp) through a series of composite regenerated cellulose ultrafiltration membranes with different pore size. Data were also obtained over a range of solution temperature and viscosity to evaluate the effects of these parameters on plasmid transmission. The experimental results were analyzed using a new model that accounts for the effects of the DNA persistence length on the elongational properties of the supercoiled plasmid.

2.1. Plasmid sieving experiments

2. Materials and methods

LP =

Stirred cell ultrafiltration experiments were conducted using UltracelTM composite regenerated cellulose membranes (Millipore) with nominal molecular weight cutoffs of 100, 300, and 1000 kDa as previously described [8]. All membranes were initially soaked in isopropyl alcohol and then flushed with water to remove residual storage agents and to ensure thorough wetting of the pore structure. The membranes were soaked in a 250 ng/mL plasmid solution for 24 h at room temperature prior to being used in the filtration experiments. Filtration experiments were performed using three supercoiled plasmids of varying length. A 3.0 kbp pBluescript® II KS+ plasmid (Stratagene) was used as obtained from the supplier (i.e., no DNA fragments inserted). A 9.8 kbp plasmid, generated by insertion of a 6.8 kbp fragment into the SalI site of the pBluescript vector, was provided by Dr. Jeffrey Chamberlain at the University of Washington. A 17 kbp plasmid, generated by insertion of a 14 kbp fragment into the NotI site of the pBluescript vector, was provided by Dr. Paula Clemens from the University of Pittsburgh. In each case, a stock solution of the plasmid in Tris–base buffer was prepared by Aldevron as previously described [8]. Agarose gel electrophoresis (described subsequently) was used to verify that >95% of each plasmid was in the desired supercoiled form. The stock solutions were divided into 110 ␮L 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 10 mM Tris–HCl, 1 mM Na2 ·EDTA (TE) buffer (pH = 7.7 ± 0.1) made by mixing a TE concentrate with deionized distilled water. Sodium chloride was added to the buffer solution to obtain a final NaCl concentration of 10 mM and solution conductivity of 1920 ± 25 ␮S/cm. For select experiments, the buffer solution viscosity was adjusted by the addition of d-glucose (Sigma–Aldrich) at concentrations of 0, 7.5, 15, and 22.5% (mass glucose/mass buffer solution). Plasmid concentrations were determined by fluorescence detection using the nucleic acid stain PicoGreen® as previously described [8].

The filtrate flux (Jv ) was determined by timed collection, the transmembrane pressure difference (P) was evaluated using the digital pressure gauge, and the solution viscosity () was evaluated as a function of temperature and glucose concentration using literature data [13]. The majority of experiments were performed at room temperature (22–25 ◦ C). For those experiments conducted at non-ambient temperatures, the experimental apparatus was relocated into a temperature-controlled environmental chamber (Lab-Line Instruments) that maintained the temperature at the desired setpoint value ±0.5 ◦ C.

Sieving experiments were conducted using 10 mL (effective membrane area = 4.1 cm2 ) stirred ultrafiltration cells (Millipore) placed on a magnetic stir plate. The fluid flow was normal to the membrane surface, with good bulk mass transfer achieved by stirring the solution with a Teflon-coated stir bar at 730 rpm. The stirred cell was connected to a polycarbonate feed reservoir that was pressurized with compressed air. Sieving data were obtained over a range of transmembrane pressure from 0 to 62 kPa (corresponding to 0–9 psig) as measured by a digital differential pressure gauge. Both the stirred cell and feed reservoir were filled with the same plasmid DNA solution and a ‘pre-sieving’ 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 was set to the appropriate value to achieve the desired filtrate flux. Two filtrate samples were collected after a minimum of 2 min of filtration 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. Plasmid sieving coefficients were evaluated from the ratio of the filtrate concentration (mean from the two measurements) to the concentration in the stirred cell (determined from the average of the concentrations in the pre- and post-sieving samples). The stirred cell was then reassembled for a second experiment or emptied and filled with buffer solution to measure the membrane hydraulic permeability (LP ): Jv  P

(2)

2.2. Agarose gel electrophoresis The topology of the plasmid was examined by agarose gel electrophoresis. Horizontal gels were prepared by dissolving agarose in Tris–acetate-EDTA buffer. Gel loading samples were prepared by mixing 20 ␮L of the plasmid solution with 4 ␮L of TrackItTM loading buffer (Invitrogen). TrackItTM 1 kbp DNA ladder (Invitrogen) was loaded into the first well of the gel as a reference marker. The 3.0 kbp plasmid was analyzed using 0.8% agarose gels run at 60 V for 60 min in a Mini-Sub Cell GT (BioRad). The 9.8 and 17 kbp plasmids were analyzed using 0.7% agarose run at 50 V for 180 min. The gels were subsequently stained with 120 mL of SyBr® Gold working solution (Invitrogen) for 18 h on a rocker platform. Gel images were captured using a UV trans-illuminator and analyzed using AlphaImager software (Alpha Innotech). 3. Results and discussion Fig. 1 presents data for the ultrafiltration of 250 ng/mL solutions of the 3.0, 9.8, and 17 kbp plasmids in 10 mM NaCl TE buffer through the 1000 kDa UltracelTM membrane. Agarose gel electrophoresis of the filtrate and ‘post-sieving’ feed samples for all three plasmids

D.R. Latulippe, A.L. Zydney / Journal of Membrane Science 329 (2009) 201–208

Fig. 1. Observed sieving coefficients of the 3.0 (), 9.8 (), and 17 ( ) kbp plasmids through the 1000 kDa UltracelTM membrane in 10 mM NaCl TE buffer.

confirmed that were no structural changes due to either passage through the membrane or stirring in the ultrafiltration cell. This was true even for the 17 kbp plasmid despite the greater susceptibility to shear for larger DNA [14]. The absence of any structural changes is consistent with previous results for the transmission of 3.0 kbp [8,9] and 9.5 kbp [15] plasmids through ultrafiltration membranes. In addition, there was no evidence of any membrane fouling in these experiments; the hydraulic permeability of the UltracelTM membrane after filtration was statistically indistinguishable (p-value >0.5) from the permeability of the fresh (un-used) membrane. The flux obtained with the plasmid solution was always within 10% of the value determined using the plasmid-free buffer. The y-axis in Fig. 1 shows the observed sieving coefficient (So ), which was evaluated as the ratio of the plasmid concentration in the filtrate solution to that in the feed. The error bars represent plus/minus one standard deviation as determined from standard propagation of error analysis based on the accuracy of the plasmid concentration measurements. For each experimental condition investigated, a minimum of two different sets of experiments were conducted on different days. Each set typically involved four to five individual measurements at different filtrate flux such that the entire flux profile could be evaluated. The sieving coefficients for each plasmid were a strong function of the filtrate flux, with the DNA transmission increasing significantly with increasing flux. The sieving coefficient data for the three different plasmids are very similar over the entire range of filtrate flux. For example, at a filtrate flux of 22 ␮m/s (corresponding to 79 L/m2 /h) the sieving coefficients of the 3.0, 9.8, and 17 kbp plasmids were 0.19 ± 0.01, 0.14 ± 0.02, and 0.16 ± 0.03, respectively. The sieving coefficients for the 3.0 and 17 kbp plasmids are not statistically different (pvalue = 0.10) while the value for the 9.8 kbp plasmid is slightly smaller than the results for the other plasmids. This is discussed in more detail subsequently. The data in Fig. 1 were used to evaluate the critical filtrate flux (Jcrit ) by linear extrapolation of the sieving coefficient versus filtrate flux data (for 0.02 ≤ So ≤ 0.50) to determine the x-intercept (i.e., So = 0). Fig. 2 shows the calculated values of the critical flux as an explicit function of the radius of gyration (RG ) of the plasmid DNA which was calculated using Eqs. (11) through (13) as described in further detail below. The error bars correspond to the range of x-intercepts determined from the 90% confidence band for the linear regression fit to the sieving coefficient data. For the 1000 kDa membrane, the critical filtrate flux is independent of plasmid DNA size; the slope of the best-fit line for Jcrit versus RG is not statis-

203

Fig. 2. Critical flux values for the 100 (), 300 (), and 1000 () kDa membranes as a function of the radius of gyration of the plasmid DNA. Data were obtained using 250 ng/mL solutions in 10 mM NaCl TE buffer. Error bars were determined from the 90% confidence band for a linear regression fit to the sieving coefficient versus filtrate flux data as described in the text.

tically different than zero (p-value > 0.32). Fig. 2 also shows data for the critical flux obtained with the 100 and 300 kDa UltracelTM membranes. The critical flux for any given plasmid increases as the nominal molecular weight cut-off (i.e., pore size) of the membrane decreases. For example, the critical flux for the 17 kbp plasmid goes from 11 ␮m/s for the 1000 kDa membrane to more than 50 ␮m/s for the 100 kDa membrane. For each membrane, the critical flux values for the 3.0 and 17 kbp plasmids were nearly identical (within 15%). Interestingly, the critical flux for the 9.8 kbp plasmid was larger than that for the 3.0 or 17 kbp plasmids for both the 300 and 100 kDa membranes, with this effect being most pronounced for the smallest pore size 100 kDa UltracelTM membrane. This result was confirmed for multiple membrane discs and for different plasmid samples. For example, the critical filtrate flux values for three separate experiments performed with the 9.8 kbp plasmid and two different 100 kDa membranes were 89 ± 6, 95 ± 8, and 97 ± 7 ␮m/s, with the error limits calculated from the 90% confidence band for the linear regression fit to each set of experimental data. The lack of any clear dependence of the critical flux on the plasmid size is inconsistent with the predictions of the modified elongational flow model developed by Latulippe et al. [8]. In particular, Eq. (1) predicts that the critical flux for the 3.0 kbp plasmid (RG = 69 nm) should be more than 6 times larger than that for the 17 kbp plasmid (RG = 177 nm) since the superhelix radius is essentially independent of the plasmid size [16,17]. However, the elongational flow model does, at least conceptually, explain how even very large plasmids are able to pass through ultrafiltration membranes with mean pore size that are as much as 20 times smaller than the size of the plasmid. 3.1. Critical flux model In contrast to the predictions of Eq. (1), the original elongational flow analysis presented by Daoudi and Brochard [10] does predict that the critical flux is independent of the plasmid size. This behav3 dependence of both the Zimm polymer ior arises because of the RG relaxation time: ≈

3 RG

kB T

(3)

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and the characteristic time for the fluid flow: ≈

Qp

=

x3

Jv rp2

(4)

εx3

where kB is the Boltzmann constant, T is the absolute temperature, and Qp is the volumetric flow rate through the membrane pore. Daoudi and Brochard [10] evaluated the critical distance from the membrane pore where elongation effects become significant (x) as 3 being equal to the radius of gyration of the polymer (giving an RG dependence in the denominator of Eq. (4)). The critical flux for polymer transmission was then calculated by assuming that elongation becomes significant when De =  = 1 giving: Jcrit =

εkB T

(5)

rp2 

Although Eq. (5) does predict that Jcrit is independent of the plasmid size, the predicted values of the critical flux are about three orders of magnitude larger than those observed experimentally [8], and this model is unable to explain the observed dependence of the critical flux on the salt concentration [9]. Our hypothesis is that the inconsistencies in Eqs. (1) and (5) are due to shortcomings in the evaluation of both the relaxation time and the characteristic flow time. In particular, there is significant experimental evidence that the relaxation time for plasmid DNA is more appropriately evaluated from the Rouse free-draining model using the expression given by Berne and Pecora [11]: =

2 RG

(6)

2 D

where D is the plasmid diffusion coefficient which can be evaluated using the Stokes–Einstein equation as: D=

kB T 6RH

(7)

where RH is the hydrodynamic radius. For example, Lewis and Pecora [18] evaluated the relaxation time of a 2.3 kbp linear DNA fragment by birefringence as  = 680 ␮s, which is in much better agreement with the predicted value of  = 580 ␮s given by Eq. (6) compared to  = 1780 ␮s as given by Eq. (3). Although Eq. (4), with x = RG , properly describes the shear rate for flow into an isolated pore [10], this equation cannot be used for the highly porous UltracelTM membranes since the fluid streamlines from adjacent pores will begin to interact at a distance above the surface of the membrane that is much less than the radius of gyration for the plasmids examined in this study. Latulippe et al. [8] set x equal to the superhelix radius of the plasmid, which is nearly independent of the plasmid size [16,17]. In contrast, we assume that the critical distance above the pore is directly proportional to the radius of gyration, i.e., x = ˇRG , with ˇ < 1. The critical flux is then evaluated using Eqs. (4), (6), and (7) with De =  yielding: Jcrit =

 6



ε rp2



ˇ3 Decrit 



kB T 



(8)

where Decrit is the critical Deborah number for plasmid elongation and  is the ratio of the hydrodynamic radius (RH ) to the radius of gyration (RG ) of the plasmid DNA. The critical flux is thus predicted to be independent of the radius of gyration (in good agreement with Figs. 1 and 2), but it is a function of the membrane properties (porosity and pore size), the plasmid conformation and flexibility (as given by  and Decrit ), and the solution properties (temperature and viscosity). 3.2. Temperature and viscosity effects Eq. (8) predicts a significant dependence of the critical flux on solution temperature and viscosity. Since there are no exper-

Fig. 3. Observed sieving coefficient for the 3.0 kbp plasmid through the UltracelTM 300 kDa membrane as a function of the filtrate flux for experiments performed at 10 ◦ C (), 20 ◦ C (), and 30 ◦ C (䊉) with 10 mM NaCl TE buffer.

imental data in the literature examining the effects of either of these parameters on plasmid DNA transmission during ultrafiltration, a series of experiments were performed in which the entire experimental apparatus (solution reservoir and stirred cell) was placed in a temperature-controlled environmental chamber that maintained the temperature within 0.5 ◦ C of the desired temperature set point. In each experiment, the apparatus was allowed to equilibrate for a minimum of 12 h to ensure that there were no temperature gradients within the system. Data were also obtained by independently varying the solution viscosity by addition of dglucose. Fig. 3 shows the observed sieving coefficient for the 3.0 kbp plasmid DNA through the 300 kDa UltracelTM membrane at temperatures of 10, 20, and 30 ◦ C. There was no evidence of any changes in DNA conformation, as determined by agarose gel electrophoresis, at any temperature. At a fixed value of the filtrate flux, the plasmid sieving coefficient increased with decreasing temperature. For example, at a filtrate flux of 39 ␮m/s (140 L/m2 /h) the sieving coefficient increased from less than 0.01 to So = 0.14 as the operating temperature was reduced from 30 to 10 ◦ C. This change in sieving coefficient was not due to any change in the membrane properties; experimental data for the membrane permeability (given by Eq. (2)) varied by less than 5% over the entire range of temperatures examined in this study. The data in Fig. 3 were used to evaluate the critical filtrate flux by extrapolation of the observed sieving coefficient data to So = 0. The results are plotted as the open symbols in Fig. 4 with the xaxis equal to the temperature-viscosity ratio, kB T/, which is the functional dependence predicted by Eq. (8). The data are highly linear when plotted in this manner with R2 > 0.99. The critical flux increased by a factor of 1.7 (from 25 to 43 ␮m/s) as the solution temperature increased from 10 to 30 ◦ C, which is in excellent agreement with the 1.75-fold increase in Jcrit predicted by Eq. (8) over this temperature range. Similar experiments were also conducted with the UltracelTM 1000 kDa membrane but using a much lower ionic-strength buffer. The critical filtrate flux again showed a very strong linear dependence (R2 > 0.99) on the temperatureviscosity ratio with the critical filtrate flux increasing by a factor of 2.0 as the temperature increased from 10 to 30 ◦ C (results not shown). Fig. 4 also shows data for the critical filtrate flux determined with the 3.0 kbp plasmid in buffer solutions containing high con-

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205

3.3. Comparison of model and data In order to provide a more quantitative comparison of the model calculations and experimental data, it is necessary to evaluate the membrane properties (pore radius and porosity) and the plasmid DNA parameter , which is equal to the ratio of the hydrodynamic radius to the radius of gyration. The membrane porosity was estimated as ε = 0.5 as discussed by Rao and Zydney [22]. The mean pore radius (rp ) was then calculated using the Hagen–Poiseuille equation as:



rp =

Fig. 4. Critical filtrate flux for the 3.0 kbp plasmid through the UltracelTM 300 kDa membrane as a function of temperature-viscosity ratio for experiments conducted at different temperature (♦) and different glucose concentrations () in 10 mM NaCl TE buffer. Solid line is linear regression fit for temperature experiments; dashed line is linear regression fit for glucose experiments. Error bars were calculated as in Fig. 2.

centrations of glucose to directly vary the solution viscosity. At room temperature, the solution viscosity increases from 0.9 × 10−3 to 1.6 × 10−3 Pa s as the glucose concentration (mass glucose/mass buffer solution) increases from 0 to 22.5% [13]. At a fixed value of the filtrate flux, the plasmid sieving coefficient increased with increasing glucose concentration, consistent with the behavior seen in Fig. 3 for decreasing temperature, both of which cause an increase in solution viscosity. For example, at a filtrate flux of 34 ␮m/s the plasmid sieving coefficient increased from 0.03 ± 0.01 to 0.19 ± 0.03 as the glucose concentration increased from 0 to 22.5%. The critical filtrate flux values, again evaluated by extrapolating the sieving coefficient data to So = 0, are highly linear when plotted as a function of kB T/ with R2 > 0.99 (filled symbols in Fig. 4). The critical flux increased by a factor of 1.6, from 18 to 28 ␮m/s, as the glucose concentration decreased from 22.5 to 0%, which is only slightly less than the 1.8-fold increase in Jcrit predicted by Eq. (8) over this viscosity range. Note that it was not possible to evaluate the critical filtrate flux over a larger range of viscosity because higher glucose concentrations caused considerable interference with the sensitivity of the PicoGreen® reagent. The slope of the critical flux data obtained at different glucose concentrations was about 30% smaller than that obtained at different temperatures, which may simply be due to small differences in the membrane properties (i.e. pore radius, porosity) of the different membrane discs used for each set of experiments. The hydraulic permeability of the two membrane discs used in these experiments were within 5% which would correspond to less than a 3% difference in the mean pore radius assuming equivalent porosity. Alternatively, there is experimental and theoretical evidence that the DNA persistence length, a measure of the DNA flexibility, is a function of temperature [19] and this could influence the temperature dependence seen in Fig. 4. It is also possible that the highly concentrated glucose altered the DNA structure/flexibility; previous studies have shown that the addition of glucose reduces the dielectric constant [20], which could alter the intramolecular electrostatic interactions between the charge groups along the DNA backbone. There is also experimental evidence that highly concentrated sucrose solutions (with similar viscosities to the glucose solutions examined in Fig. 4) increase DNA flexibility [21].

8ım Lp ε

1/2

(9)

where the membrane hydraulic permeability (LP ) was evaluated using Eq. (2) from the slope of experimental data for the filtrate flux (Jv ) as a function of the transmembrane pressure (P) obtained with plasmid-free TE buffer. The membrane thickness (ım ) was estimated as 1.0 ␮m [23] giving rp = 6.6, 8.3, and 11.3 nm for the 100, 300, and 1000 kDa UltracelTM membranes, respectively. The evaluation of  = RH /RG was more involved. The hydrodynamic radius, RH , was evaluated from the Stokes–Einstein equation (Eq. (7)) with the diffusion coefficient for the supercoiled plasmid calculated using the empirical expression developed by Prazeres [24]: D = A2

T −2/3 (bp) 

(10)

where A2 = 3.31 × 10−15 and bp is the number of DNA base pairs. The diffusion coefficient, and hence RH , was assumed to be independent of solution ionic strength based on experimental data obtained by Gebe et al. [25] and Hammermann et al. [26]. The radius of gyration of the supercoiled plasmid is related to that of a linear plasmid with the same number of base pairs as:

 1

RG = 0.715(RG-OC ) = 0.715

2

2 RG-Linear

 (11)

where RG-OC is the radius of gyration of the open-circular (i.e., nicked) form of the plasmid and RG-Linear is the radius of gyration of the linear isoform. The second equality in Eq. (11) which relates RG-OC and RG-Linear , is based on a random coil model for flexible polymers [27], while the first equality is developed from a linear fit of RG data from static light scattering experiments [26,28–30]. The radius of gyration of the linear form of the plasmid was calculated using the ‘worm-like chain’ model [31]:

 RG-Linear = a

 2

2a L a −1+ −2 3a L L

1/2 (1 − e−L/a )

(12)

where L, the contour length of the DNA, is calculated by multiplying the number of DNA base pairs by the axial rise per base pair (0.34 nm/bp). The persistence length (a) is a measure of the DNA stiffness and is function of the solution ionic strength due to intramolecular electrostatic interactions involving the phosphate groups in the DNA helix [32]. Manning [32] recently presented an equation for the persistence length as a function of the solution ionic strength: a=

 a∗ 2/3 R4/3  2

Z 2 B

(2Z − 1)

be−b 1 − e−b



− 1 − ln(1 − e−b )

(13)

where a* = 7.5 nm is the persistence length of a hypothetically uncharged DNA, R = 1 nm is the radius of the DNA double-helix, Z is the valence of the salt cation, B = 0.71 nm at room temperature is the Bjerrum length for pure water,  is the inverse Debye length, and is the DNA charge density = B /b where b is the spacing between DNA charge sites. Eq. (13) predicts a dependence on ionic

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Fig. 5. Critical filtrate flux as a function of the bracketed terms in Eq. (8). Open, top-filled, and filled symbols represent results for the 3.0, 9.8, and 17 kbp plasmids, respectively, using the UltracelTM 100 kDa (diamonds), 300 kDa (circles), and 1000 kDa (squares) membranes. Solid line is the zero-intercept linear regression fit; dashed line is zero-intercept linear regression fit to the data for the 3.0 and 17 kbp plasmids only. Error bars were calculated as in Fig. 2.

strength even at very high salt concentrations (above 1 M), while the Odijk–Skolman–Fixman (OSF) model predicts that the DNA persistence length becomes independent of ionic strength above approximately 50 mM [33,34]. Both the OSF and Manning theories give similar results at low to moderate ionic strength, although Eq. (13), in combination with Eq. (12), is in better agreement with experimental data for the radius of gyration of linear DNA reported by Borochov et al. [35] over a range of NaCl concentrations. Fig. 5 shows a comparison of the model calculations and experimental results for the critical filtrate flux for three different plasmid sizes (3.0, 9.8, and 17 kbp, corresponding to radius of gyration of 69, 133, and 177 nm), three different pore size membranes (UltracelTM 100, 300, and 1000 kDa, corresponding to pore radii of 6.6, 8.3, and 11.3 nm), three different temperatures (10, 20, and 30 ◦ C), and four different glucose concentrations (0, 7.5, 15, and 22.5%). The x-axis in Fig. 5 is of the form given by Eq. (8) with  = RH /RG evaluated using Eqs. (7) and (10) to (13). Although there is clearly some scatter in the data, the results for the different membranes, plasmids, temperatures, and viscosities all tend to collapse to a single line when plotted in this manner. A linear regression fit to the data for the 3.0 and 17 kbp plasmids gave R2 = 0.84; the R2 value dropped to 0.70 when data for all 3 plasmids were included in the analysis. The data for the 9.8 kbp plasmid tend to fall above the data for the 100 and 300 kDa membranes, consistent with the results in Fig. 2. This is discussed in more detail subsequently. The best fit value of the slope in Fig. 5, using the results for all three plasmids with the intercept fixed at the origin, is ˇ3 Decrit /6 = 8.6 ± 0.6 × 10−4 . This corresponds to ˇ = 0.12 assuming that Decrit = 1, which is consistent with direct measurements of the elongation of linear DNA obtained by video fluorescence microscopy [36,37]. This implies that the critical distance from the pore entrance where plasmid elongation becomes significant is approximately 12% of the radius of gyration of the plasmid. The calculated value of x = ˇRG for the 3.0 kbp plasmid is 8.3 nm which is in fairly good agreement with the value of the superhelix radius (Rs = 10 nm) used by Latulippe et al. [8] in developing Eq. (1). Thus, even though Eq. (1) is unable to explain the observed dependence on the plasmid size, the calculated values of the critical flux are in fairly good agreement with the experimental results for the 3.0 kbp plasmid (but not for the other plasmids). It is worth noting that

the critical flux values reported by Latulippe et al. [8] and Latulippe and Zydney [9] are slightly different than the data obtained in this study. This is probably due to small discrepancies in “room temperature” between these studies; the plasmid solutions in [8] and [9] were stored at 4 ◦ C to minimize endonuclease activity, with no direct measurements made of the actual solution temperature during the ultrafiltration experiment. Fairly small differences in temperature, e.g., between 18 and 23 ◦ C, would be expected to cause more than a 15% change in the critical flux due to the kB T/ term in Eq. (8). Although the experiments reported in this study were performed using buffer solutions containing 10 mM NaCl, Latulippe and Zydney [9] clearly showed that the critical flux increases significantly with decreasing salt concentration. This effect is captured in Eq. (8) by the parameter , which accounts for the change in plasmid conformation arising from the intramolecular electrostatic interactions between the negatively charged phosphate groups in the DNA backbone. For example, Latulippe and Zydney [9] found that the critical flux for the 3.0 kbp plasmid through the 300 kDa membrane decreased by 48% as the salt concentration was increased from 10 to 150 mM. Eq. (13) yields a 20% reduction in the DNA persistence over this range of salt concentrations, which would correspond to a 23% reduction in the critical flux as given by Eq. (8). This is quite a bit smaller than what was observed experimentally, although this discrepancy may have been at least partially due to small differences in solution temperature between the experiments. 4. Conclusions The experimental data obtained in this study provide the first quantitative results for the effects of plasmid DNA size, solution temperature, and viscosity on the transmission of supercoiled plasmids through different pore size ultrafiltration membranes. In all cases, plasmid transmission increased from essentially zero at low flux to greater than 50% at high filtrate flux. The plasmid sieving coefficients were essentially independent of the plasmid size (from 3.0 to 17 kbp) but increased with increasing solution viscosity. The critical filtrate flux for plasmid transmission, defined as the flux at which plasmid transmission first becomes significant, was essentially independent of the plasmid size (from 3.0 to 17 kbp) but increased with decreasing membrane pore size and decreasing solution viscosity, with the latter effect arising from either an increase in temperature or a reduction in glucose concentration. The experimental results for the critical filtrate flux were analyzed using a new form of the flow-induced polymer elongation model originally developed by Daoudi and Brochard [10]. Two significant modifications were made in the analysis. First, the plasmid relaxation time was evaluated using the Rouse free-draining model with the intrinsic structure of the plasmid included through the parameter , the ratio of the hydrodynamic radius to the radius of gyration, which is in turn a function of the DNA persistence length and thus the salt concentration. Second, the characteristic time for the fluid flow was evaluated at a critical distance above the membrane pore that is a small fraction of the radius of gyration, x = ˇRG , to account for the interaction between the flow streamlines above adjacent pores in the high pore density UltracelTM membranes used in this study. The model properly predicts that the critical flux is independent of the radius of gyration of the plasmid DNA, and it also captures the observed dependence on solution temperature and viscosity. The experimental data for the different pore size membranes, plasmid sizes, operating temperatures, and solution viscosities effectively collapse to a single straight line when plotted in terms of the form given by the model, with the best-fit value of ˇ = 0.12. The elongational flow model for plasmid transmission developed in this study can be used to estimate the critical flux for

D.R. Latulippe, A.L. Zydney / Journal of Membrane Science 329 (2009) 201–208

any given plasmid in terms of its underlying properties. In particular, the model accounts for the ionic strength dependence of the plasmid conformation through the parameter  = RH /RG , where (a) the hydrodynamic radius was evaluated from the Stokes–Einstein equation using an empirical expression for the diffusion coefficient and (b) the radius of gyration was evaluated using the ‘worm-like chain’ model using a DNA persistence length that is a function of solution ionic strength as given by the theoretical expression developed by Manning [32]. The model predicts a decrease in the critical filtrate flux with increasing solution ionic strength (i.e., increasing ) that is in good agreement with experimental results previously reported by Latulippe and Zydney [9]. As discussed previously, the critical flux for the 9.8 kbp plasmid was somewhat larger than that for either the 3.0 or 17 kbp plasmids, with this effect being most pronounced for the smallest pore size 100 kDa membrane. Although the origin of this behavior is unclear, our current hypothesis is that the lower transmission of the 9.8 kbp plasmid is due to the underlying base sequence of this particular plasmid. Recent theoretical calculations by Bomble and Case [38] showed that the persistence length of linear DNA sequences with different base pairs had persistence lengths varying from 47.5 to 74.3 nm due to the different stacking and hydrogen-bonding interactions between adjacent bases. DNA structures with long repeats of single base pairs had significantly less flexibility than DNA containing multiple repeats of CGG, consistent with experimental results for nucleosome formation [39,40]. These differences in persistence length and flexibility could easily explain the anomalous results for the 9.8 kbp plasmid, with these phenomena captured in the elongational flow model through the parameters  and possibly Decrit . Future studies using plasmids with well-defined base pair sequences will be needed to explore this phenomenon in more detail. Acknowledgements The authors would like to acknowledge Dr. Jeffrey Chamberlain from the University of Washington who provided the 9.8 kbp plasmid, Dr. Paula Clemens from the University of Pittsburgh who provided the 17 kbp plasmid, and Millipore Corporation for donation of the UltracelTM membranes. The authors would also like to thank Janelle Konietzko for assistance with some of the experiments.

Nomenclature a a* b D De Jcrit Jv kB B L LP Qp rp R RG RH RS So T

DNA persistence length (m) uncharged DNA persistence length (m) spacing between DNA charge sites (m) plasmid diffusion coefficient (m2 /s) Deborah number critical filtrate flux (␮m/s) filtrate flux (␮m/s) Boltzmann constant (J/K) Bjerrum length for pure water (m) DNA contour length (m) membrane hydraulic permeability (m) volumetric flow rate through a single pore (m3 /s) membrane pore radius (m) radius of DNA double-helix (m) radius of gyration (m) hydrodynamic radius (m) superhelix radius (m) observed sieving coefficient temperature (K)

x Z

207

critical distance from the membrane pore (m) unsigned valence of the salt solution cation

Greek symbols ˇ ratio of critical distance for elongation to radius of gyration  shear rate (1/s) ım membrane thickness (m) P transmembrane pressure difference (Pa) ε membrane porosity  solution viscosity (Pa s)  inverse Debye length (m−1 )  ratio of the hydrodynamic radius to the radius of gyration

DNA charge density  relaxation time (s)

References [1] P.J. Bergman, M.A. Camps-Palau, J.A. McKnight, N.F. Leibman, D.M. Craft, C. Leung, J. Liao, I. Riviere, M. Sadelain, A.E. Hohenhaus, P. Gregor, A.N. Houghton, M.A. Perales, J.D. Wolchok, Development of a xenogeneic DNA vaccine program for canine malignant melanoma at the animal medical center, Vaccine 24 (2006) 4582–4585. [2] Merial, USDA grants conditional approval for first therapeutic vaccine to treat cancer, 2007. [3] M.M. Diogo, J.A. Queiroz, D.M.F. Prazeres, Chromatography of plasmid DNA, J. Chromatogr. A 1069 (2005) 3–22. [4] S. Sagar, M.P. Watson, A. Lee, Y.G. Chau, Capacity challenges in chromatographybased purification of plasmid DNA, in: A.S. Rathore, A. Velayudhan (Eds.), Scaleup and Optimization in Preparative Chromatography, Marcel Dekker, Inc., New York, 2003, pp. 251–272. [5] T. Hirasaki, T. Sato, T. Tsuboi, H. Nakano, T. Noda, A. Kono, K. Yamaguchi, K. Imada, N. Yamamoto, H. Murakami, S.-I. Manabe, Permeation mechanism of DNA molecules in solution through cuprammonium regenerated cellulose hollow fiber, J. Membr. Sci. 106 (1995) 123–129. [6] D.W. Kahn, M.D. Butler, D.L. Cohen, M. Gordon, J.W. Kahn, M.E. Winkler, Purification of plasmid DNA by tangential flow filtration, Biotechnol. Bioeng. 69 (2000) 101–106. [7] C. Kepka, R. Lemmens, J. Vasi, T. Nyhammar, P.-E. Gustavsson, Integrated process for purification of plasmid DNA using aqueous two-phase systems combined with membrane filtration and lid-bead chromatography, J. Chromatogr. A 1057 (2004) 115–124. [8] D.R. Latulippe, K. Ager, A.L. Zydney, Flux-dependent transmission of supercoiled plasmid DNA through ultrafiltration membranes, J. Membr. Sci. 294 (2007) 169–177. [9] D.R. Latulippe, A.L. Zydney, Salt-induced changes in plasmid DNA transmission through ultrafiltration membranes, Biotechnol. Bioeng. 99 (2008) 390–398. [10] S. Daoudi, F. Brochard, Flows of flexible polymer solutions in pores, Macromolecules 11 (1978) 751–758. [11] B.J. Berne, R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology and Physics, Wiley, New York, NY, 1976. [12] S. Kong, N. Titchener-Hooker, M.S. Levy, Plasmid DNA processing for gene therapy and vaccination: studies on the membrane sterilisation filtration step, J. Membr. Sci. 280 (2006) 824–831. [13] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics, 88th edition, CRC Press/Taylor and Francis, Boca Raton, FL, 2008. [14] M.S. Levy, I.J. Collins, S.S. Yim, J.M. Ward, N. Titchener-Hooker, P. Ayazi Shamlou, P. Dunnill, Effect of shear on plasmid DNA in solution, Bioprocess. Eng. 20 (1999) 7–13. [15] E. Arkhangelsky, B. Steubing, E. Ben-Dov, A. Kushmaro, A. Gitis, Influence of pH and ionic strength on transmission of plasmid DNA through ultrafiltration membranes, Desalination 227 (2008) 111–119. [16] T.C. Boles, J. White, H.N.R. Cozzarelli, Structure of plectonemically supercoiled DNA, J. Mol. Biol. 213 (1990) 931–951. [17] M. Hammermann, N. Brun, K.V. Klenin, R. May, K. Toth, J. Langowski, Saltdependent DNA superhelix diameter studied by small angle neutron scattering measurements and Monte Carlo simulations, Biophys. J. 75 (1998) 3057– 3063. [18] R.J. Lewis, R. Pecora, Comparison of predicted Rouse-Zimm dynamics with observations for a 2311 base pair DNA fragment, Macromolecules 19 (1986) 2074–2075. [19] Y.J. Lu, B. Weers, N.C. Stellwagen, DNA persistence length revisited, Biopolymers 61 (2002) 261–275. [20] X.J. Liao, G.S.V. Raghavan, J.M. Dai, V.A. Yaylayan, Dielectric properties of ␣-dglucose aqueous solutions at 2450 MHz, Food Res. Int. 36 (2003) 485–490.

208

D.R. Latulippe, A.L. Zydney / Journal of Membrane Science 329 (2009) 201–208

[21] D. Londos-Gagliardi, G. Serros, G. Aubel-Sadron, Comparison of the physicochemical properties of native DNA and sonicated DNA. II. Effect of sucrose solutions, J. Chim. Phys. Phys.-Chim. Biol. 68 (1971) 670–673. [22] S. Rao, A.L. Zydney, Controlling protein transport in ultrafiltration using small charged ligands, Biotechnol. Bioeng. 91 (2005) 733–742. [23] L.J. Zeman, A.L. Zydney, Microfiltration and Ultrafiltration Principles and Applications, Marcel Dekker, Inc., New York, 1996. [24] D.M.F. Prazeres, Prediction of diffusion coefficients of plasmids, Biotechnol. Bioeng. 99 (2008) 1040–1044. [25] J.A. Gebe, J.J. Delrow, P.J. Heath, B.S. Fujimoto, D.W. Stewart, J.M. Schurr, Effects of Na+ and Mg2+ on the structures of supercoiled DNAs: comparison of simulations with experiments, J. Mol. Biol. 262 (1996) 105–128. [26] M. Hammermann, C. Steinmaier, H. Merlitz, U. Kapp, W. Waldeck, G. Chirico, J. Langowski, Salt effects on the structure and internal dynamics of superhelical DNAs studied by light scattering and Brownian dynamics, Biophys. J. 73 (1997) 2674–2687. [27] V.A. Bloomfield, D.M. Crothers, I. Tinoco Jr., Nucleic Acids Structure, Properties, and Functions, University Science Books, Sausalito, California, 2000. [28] G. Voordouw, Z. Kam, N. Borochov, H. Eisenberg, Isolation and physical studies of the intact supercoiled, the open circular, and the linear forms of ColE1-plasmid DNA, Biophys. Chem. 8 (1978) 171–189. [29] D.M. Fishman, G.D. Patterson, Light scattering studies of supercoiled and nicked DNA, Biopolymers 38 (1996) 535–552.

[30] J. Langowski, M. Hammermann, K. Klenin, R. May, K. Toth, Superhelical DNA studied by solution scattering and computer models, Genetica 106 (1999) 49–55. [31] H. Benoit, P. Doty, Light scattering from non-Gaussian chains, J. Phys. Chem. 57 (1953) 958–963. [32] G.S. Manning, The persistence length of DNA is reached from the persistence length of its null isomer through an internal electrostatic stretching force, Biophys. J. 91 (2006) 3607–3616. [33] T. Odijk, Polyelectrolytes near rod limit, J. Polym. Sci. Pt. B-Polym. Phys. 15 (1977) 477–483. [34] J. Skolnick, M. Fixman, Electrostatic persistence length of a wormlike polyelectrolyte, Macromolecules 10 (1977) 944–948. [35] N. Borochov, H. Eisenberg, Z. Kam, Dependence of DNA conformation on the concentration of salt, Biopolymers 20 (1981) 231–235. [36] T.T. Perkins, D.E. Smith, S. Chu, Single polymer dynamics in an elongational flow, Science 276 (1997) 2016–2021. [37] P.K. Wong, Y.-K. Lee, C.-M. Ho, Deformation of DNA molecules by hydrodynamic focusing, J. Fluid Mech. 497 (2003) 55–65. [38] Y.J. Bomble, D.A. Case, Multiscale modeling of nucleic acids: insights into DNA flexibility, Biopolymers 89 (2008) 722–731. [39] Z.W. Zhu, D.J. Thiele, A specialized nucleosome modulates transcription factor access to a c-glabrata metal responsive promoter, Cell 87 (1996) 459–470. [40] H.C.M. Nelson, J.T. Finch, B.F. Luisi, A. Klug, The structure of an oligo(dA) oligo(dT) tract and its biological implications, Nature 330 (1987) 221–226.