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* Department of Biomedical Engineering, Duke University, Durham, North Carolina 27708; and Department of Mathematics, University of California, Berkeley, California 94720
Correspondence: Address reprint requests to Wanda Krassowska, Tel.: 919-660-5105; Fax: 919-660-5405; E-mail: wanda.krassowska{at}duke.edu.
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ABSTRACT |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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; Potter, 1988; Weaver and Chizmadzhev, 1996). An emerging application of electroporation is gene delivery (Aihara and Miyazaki, 1998; Matthews et al., 1995; Nishi et al., 1996; Sukharev et al., 1992; Yoshizato et al., 2000; Zewert et al., 1995), which uses electric pulses to promote uptake of DNA. Although electroporation is of interest as an alternative to viral delivery methods, it has not yet been developed to the point that it can be applied routinely in clinical practice. One of the obstacles to further progress is the lack of a good theoretical model of the processes taking place during electroporation-mediated DNA delivery. Such a model must satisfy the following requirements:). Thus, one would expect that pores that admit DNA have radii of at least 10 nm, and possibly greater if the interactions of the DNA chain with the pore are of importance. ; Rols and Teissié, 1998).
The existing theory of electroporation fails to meet these requirements. Specifically, the Smoluchowski equation (SE), which has been used to describe the biophysical mechanism of the creation and evolution of pores since the late 1970s, cannot model pores large enough to admit plasmid DNA. When the applied transmembrane potential is low (e.g., below 0.4 V), the SE predicts that very few pores will be created, and that their radii will be below 1 nm (Joshi and Schoenbach, 2000). For larger transmembrane potentials, more pores are created but they expand so rapidly that in a matter of microseconds they exceed the radius beyond which electroporation becomes irreversible even if the transmembrane potential is removed (Freeman et al., 1994; Joshi and Schoenbach, 2000). This feature of the SE is well known and has been exploited in theoretical studies of irreversible breakdown and rupture of artificial lipid bilayers (Abidor et al., 1979) and biological cells (Joshi and Schoenbach, 2000). However, experiments involving uptake of plasmid DNA by cells do not support such a catastrophic scenario. Even though pulse durations used in such studies are on the order of milliseconds, a large percentage of cells survive, and they are healthy enough to express proteins coded by DNA (Klenchin et al., 1991; Rols and Teissié, 1998; Tekle et al., 1991; Wolf et al., 1994; Xie and Tsong, 1992; Yoshizato et al., 2000; Zhang et al., 1996). To account for this discrepancy, some researchers put forward a hypothesis that DNA entry into cells relies on the DNA-membrane interactions, which may be facilitated by a collection of small, 1 nm pores (Neumann et al., 1996; Rols and Teissié, 1998; Rols et al., 1998b; Sukharev et al., 1992, 1994).
An alternative way of resolving this discrepancy is presented here: a model that may be considered a nonlinear extension of the SE and that can predict the creation of stable macropores. The next section presents the governing equations of the model, outlines their numerical implementation, and describes the DNA uptake model of Neumann et al. (1996), which was added to our electroporation model to bring simulations closer to experiments. The Results section presents a detailed example of the process of creation and evolution of macropores during and after the pulse. The following sections describe how the number and size of pores, as well as DNA uptake, depend on the magnitude and duration of the electric pulse. Finally, the model is used to predict the outcome of a two-pulse protocol, which was proposed by Sukharev et al. (1992) specifically for DNA delivery.
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MODEL |
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; Glaser et al., 1988; Weaver, 1994; Weaver and Chizmadzhev, 1996), all pores are initially created hydrophobic at a rate determined by their energy (Fig. 1). Most of them are quickly destroyed by lipid fluctuations, but if hydrophobic pores of radius r 
r* are created, they convert spontaneously to long-lived hydrophilic pores, which are of interest in this study. Thus, most hydrophilic pores are created within a small range of radii just above r* and immediately expand to the minimum-energy radius rm (Neu and Krassowska, 1999). For the purpose of this study, we assume that pores are created with the initial radius rm at a rate determined by an ordinary differential equation (ODE) (DeBruin and Krassowska, 1999; Neu and Krassowska, 1999)
 | (1) |
is the creation rate coefficient. Neq is the equilibrium pore density for a given voltage Vm:
 | (2) |
), in which Eq. 1 is derived from the SE by a rigorous asymptotic analysis. Values of all parameters are given in Table 1.
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 | (3) |
 | (4) |
There are two differences between our model and those published previously. First, the third term in Eq. 4 contains the effective tension of the membrane, 
eff, in place of a constant parameter 0, which represents the tension of a membrane without pores. eff is a function of the combined area of pores, 
:
 | (5) |
' is the energy per area of the hydrocarbon-water interface (Israelachvili, 1992) and A is the total area of the lipid bilayer. The use of eff instead of 0 introduces tension coupling between pores: as pores are created and expand, their total area Ap increases, decreasing the membrane tension eff "felt" by each pore. Eventually, the decrease in eff halts further expansion of pores. Thus, it is the presence of eff in Eq. 4 and its dependence on Ap that allows our model to predict stable pores with radii of tens of nanometers. Further details can be found in our previous publication (Neu and Krassowska, 2003), in which the coupling of pores through membrane tension seen in Eq. 4 is related to a nonlinear generalization of the SE, and the equation governing the evolution of pores (Eq. 3) is derived from this nonlinear SE by a rigorous asymptotic analysis.
The second difference is the revision of the contribution of the transmembrane potential Vm to the bilayer energy (last term in Eq. 4). Assuming that the inner surface of a pore is toroidal (Kandu
er et al., 2003), as illustrated in the inset to Fig. 2, F(r, Vm), the electric force acting on a pore is given by a formula
 | (6) |
) or decreased to zero (Barnett and Weaver, 1991; Joshi et al., 2002; Pastushenko and Chizmadzhev, 1982) with the increase in the pore radius. Previous formulas apply only to small pores with the cylindrical inner surface, whereas Eq. 6 applies to toroidal pores of arbitrary size. Further details can be found in our previous publication (Neu et al., 2003), in which the electrical force F acting on a pore is derived from first principles and computed numerically, and Eq. 6 arises as a heuristic approximation of the numerical solution.
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 | (7) |
is the combined current through all pores. The current-voltage relationship for an individual pore,
 | (8) |
gr2), and the input resistance, Ri = 1/(2gr) (Newman, 1966), where h is the membrane thickness and g is the conductivity of the solution. Because this study focuses on large pores, the pore resistance Rp is assumed ohmic and the interactions between ions and the pore wall (Barnett and Weaver, 1991; Pastushenko and Chizmadzhev, 1982) are ignored (they are of importance only for pores smaller than 
5 nm). Equation 7 is used with the initial condition Vm = 0, i.e., the rest potential of the cell is ignored. An abrupt increase of the current Ip when V exceeds threshold is due to creation of pores (in the model represented by the increase of the number of pores K) and the expansion of their radii (which in the model decreases resistances Rp and Ri).
Resealing
When the electric pulse is turned off, the membrane discharges very rapidly through the existing pores, Vm drops to zero, and the pores shrink to near the minimum-energy radius rm. Once pores become that small, they can reseal by converting to a hydrophilic configuration and being destroyed by lipid fluctuations. The resealing process is already included in Eq. 1: after the pulse has created a certain number of pores, the pore density N is larger than N0, the equilibrium pore density for Vm = 0. Hence, the right-hand side of Eq. 1 is negative and the pore density N decreases. With the parameters from Table 1, the time constant of resealing is 
3 s (Glaser et al., 1988).
Note that only pores that have shrunk to a radius near rm participate in resealing. If there exist any macropores with 
they cannot reseal by the mechanism represented in Eq. 1. The resealing of these giant pores is beyond the scope of this model because it involves such processes as a change in cell volume (Sandre et al., 1999) or active, exocytotic rebuilding of the lipid bilayer (McNeil and Steinhardt, 1997).
Numerical implementation
The model described by Eqs. 1–8 has been implemented using MATLAB. An important feature of this implementation is that it represents two populations of pores in two different ways. Small pores, whose radii congregate near the minimum-energy radius rm, are accounted for by a pore density N(t). All pores in this population are assumed to have the same radius rm, and N(t) increases or decreases according to Eq. 1. Large pores, whose radii are larger than rm, are represented individually: the radius of each pore evolves according to Eq. 3. The program accounts for the exchange of pores between these two populations.
For a typical simulation, initial conditions assume an intact membrane and no transmembrane potential. Thus, Vm(0) = 0, N(0) = N0 (i.e., there is an equilibrium pore density at r = rm), and K = 0 (i.e., there are no large pores). The time loop contains the following steps:
eff. The numerical integration of Eqs. 1, 3, and 7 is performed using MATLAB function ode23t. The initial time step is 10 ps, but an adaptive time-stepping algorithm increases the time step as the simulation progresses to decrease the total run time. The run time depends on the number of large pores created by the pulse. Stronger pulses, which create more than 104 pores, result in unacceptably long runs. To decrease run times in such cases, the program launches groups of pores rather than individual pores. All pores in the group are assumed to have the same radius, and the program keeps track of the number of pores in each launched group. For example, the simulation presented in Fig. 4, in which a 1.25 V, 10 ms pulse created 40,057 large pores, took 3 min to run on a Sun Blade 1000 (750 MHz) work station with pores launched in groups.
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for the uptake of the YEp 351 plasmid DNA by yeast cells. The uptake of the DNA is described by the Nernst-Planck equation, which accounts for both the diffusive and electrophoretic transport:
 | (9) |
Equation 9 differs from Eq. 12 in Neumann et al. (1996) in that it follows a slightly different notation and incorporates both influx and efflux of DNA from the cell. In addition, Neumann et al. used their Eq. 12 for the entire cell membrane, so that S(t) was the combined area of all pores and Vp(t) was equal to the transmembrane potential Vm(t), measured away from the pore. In contrast, we apply Eq. 9 to each individual pore (provided its radius r is above that of the supercoiled DNA, 10 nm (Rybenkov et al., 1997)), and hence S(t) is the area of one pore, evaluated at each time step by our electroporation model. Vp(t) is also evaluated at each time step, but in our model it represents only the fraction of Vm(t) that corresponds to an actual voltage drop across the pore:
 | (10) |
Other parameters of Eq. 9 were chosen as follows:
is too large for the 5.6 kb pair YEp 351 plasmid, even if measured in water, and it does not reflect that the movement of DNA slows down inside the pore. Evaluated from the molecular weight (Fournier, 1998), D0 for YEp 351 DNA in water is 9.8 x 10–12 m2/s. Assuming a sevenfold decrease inside the pore (Ambjörnsson et al., 2002), we used D0 = 1.3 x 10–12 m2/s. )
 | (11) |
), D0 = 9.8 x 10–12 m2/s (see above), and therefore zeff = –27. ).
To compute DNA uptake, the program has been extended by including in each time step the computation of DNA transport through each pore by integrating numerically Eq. 9. Only pores of radii r > 10 nm are assumed to be permeable to DNA. The uptakes for all pores are summed and divided by two to account for the fact that negatively charged DNA molecules enter predominantly through the depolarized half of the cell (Klenchin et al., 1991). The resulting intracellular DNA concentration, [DNA]i, is reported relative to the threshold concentration required for the cell transformation given by Neumann et al., [DNA]th = 1.7 x 10–7 mol/m3.
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RESULTS |
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(Eq. 1). The nascent pores grow rapidly (Fig. 4 C; see also the left panel of Fig. 5 A, which shows the first 10 µs of this pulse) because of the high effective membrane tension and the large electric force on the pores, which depends on (Eq. 6). The creation and growth of pores cause the fractional pore area (FPA 
Ap/A) to increase (Fig. 4 D).
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For the 1.25 V pulse, Vm remains sufficiently large (near 0.9 V) for the creation of pores to continue, causing the transmembrane potential and accumulation radius to decrease slowly in time (Fig. 4, A and C) and the number of pores to increase (Fig. 4 B). FPA also increases slightly as the combined area of new pores is greater than the pore area lost by the decrease in the accumulation radius (Fig. 4 D). If the applied voltage is larger (e.g., 1.35 V in Fig. 4), more pores are created, Vm decreases well below threshold, and the creation rate decreases to a very small value. Thus, for sufficiently large pulses, the accumulation radius, transmembrane potential, fractional pore area, and number of pores remain effectively constant after the initial transient.
Postshock evolution of macropores and cell survival
Once the pulse is turned off and the membrane discharges, so that Vm 0, the population of pores can evolve according to two different scenarios: shrinkage or coarsening (Neu and Krassowska, 2003). These are illustrated in Fig. 5 for 10 µs pulses of strengths 1.25 V (Fig. 5 A) and 1.15 V (Fig. 5 B). In shrinkage, all pores decrease their radii to the minimum energy radius rm (Fig. 5 A). Shrinkage occurs when the pulse is sufficiently strong and long to create a large number of pores, i.e., K must exceed a critical value 
(for all pores accumulated at the same radius (Neu and Krassowska, 2003)). If a weaker or shorter pulse creates a number of pores below Kc, coarsening occurs. In coarsening, one pore expands to a radius of 2.23 µm (for a cell of 10 µm radius), whereas all other pores shrink to rm (Fig. 5 B). Coarsening and shrinkage may correspond to irreversible and reversible breakdown, respectively, since the former has been observed to occur at lower pulse strengths than the latter (Benz et al., 1979; Weaver and Chizmadzhev, 1996).
From the viewpoint of cell survival, shrinkage is preferable, as pores of radii near rm readily reseal (Eq. 1). Coarsening creates a giant, 2.23 µm pore, leaving the cell vulnerable to leakage and death. Thus, it is important to use pulses sufficiently strong and long enough to avoid the postshock coarsening.
Effects of pulse magnitude and duration
The pore population depends on the magnitude and duration of the electric pulse applied to the membrane (Tekle et al., 1991; Wolf et al., 1994) and the model is used to explore this dependence. Table 2 lists simulation results for 10 ms pulses of seven magnitudes, ranging from V0 = 0.8 V to 1.4 V.
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; Sukharev et al., 1994): as expected from Eq. 1, each of the pulses listed in Table 2 creates some pores. However, not all these pulses will be recognized as electroporating in experiments, which typically identify electroporation by the uptake of small molecules (Gabriel and Teissié, 1997) or the decrease in Vm during the pulse (Benz and Zimmermann, 1980; Hibino et al., 1991; Powell et al., 1989). Using the latter criterion and assuming that a 5% decrease is experimentally detectable, 0.8 and 0.9 V pulses will be considered subthreshold, and 1 V and larger pulses, suprathreshold.
K appears to be still increasing at the end of pulses with V0 
1.3 V but not for stronger pulses (1.35 V pulse in Fig. 4 B, millisecond timescale). Strictly speaking, K is increasing for all pulses, but if the number of pores created within 1 µs is <10–6 times the number of existing pores, we consider the creation process terminated. This is why K is considered still increasing for the 0.8 V pulse but not for the 1.4 V pulse, even though final Vm is nearly identical for these two pulses.
The distribution of pore radii is characterized by the maximum radius rmax and the mean radius 
These both decrease with the pulse magnitude, indicating that stronger pulses create more but smaller pores. The decrease in pore radii does not quite compensate for the increase in the number of pores, and thus the fractional pore area, FPA, increases approximately linearly with the pulse strength. The difference between rmax and 
quantifies how wide the pore population is. For weak pulses, there is a large difference between rmax and during the initial transient (Table 2 and Fig. 4 C, microsecond scale) but on the millisecond scale, rmax and are equal, indicating that all pores have accumulated at the same radius. For strong pulses, above 1.3 V, rmax and are very close even during the initial transient, indicating that the pore population is very compact from the beginning (Table 2 and Fig. 4 C).
Table 3 lists simulation results for pulses of different durations, from 10 µs to 10 ms. Two pulse strengths were examined: a just subthreshold pulse (1.15 V) and the largest pulse still resulting in DNA uptake (1.35 V). The results show that the number of pores, K, radii rmax and 
and the fractional pore area, FPA, depend on pulse duration only for weak pulses. For strong pulses, the initial transient is below 10 µs, so there is little dependence of K, rmax and and FPA on the duration.
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Uptake of DNA predicted by the model
Tables 2 and 3 also list the intracellular concentration of DNA (relative to [DNA]th) at the end of the pulse. Sufficient DNA uptake can occur even with a slightly subthreshold pulse (0.9 V) and increases with pulse strength. For any pulse strength, the DNA uptake increases with the pulse duration (Table 3). However, DNA uptake for short pulses (10 and 100 µs) is insufficient to result in cell transformation and thus would not be detected experimentally. Pulses of millisecond duration are needed for cell transformation, a result that agrees with experimental evidence (Klenchin et al., 1991; Rols and Teissié, 1998). The uptake stops abruptly at pulses above 1.4 V because the pores created by such strong pulses are smaller than the assumed radius of the DNA, 10 nm (Table 2).
More comprehensive results on DNA uptake are presented in Fig. 6, which shows the level curves (solid lines) of relative DNA concentration for durations up to 10 ms and pulse strengths up to 1.4 V. The model predicts that cell transformation will be observed only for pulse parameters inside the region enclosed by bold lines. The upper boundary of this region is a horizontal line corresponding to 1.4 V: no DNA uptake occurs above this line because stronger pulses create pores too small to be permeable to DNA (see also Table 2). The lower boundary is the strength-duration relationship for cell transformation (Fig. 6, line labeled 1).
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Simulations of the two-pulse protocol
Sukharev et al. (1992) proposed a two-pulse protocol specifically for gene delivery applications. The authors argued that the first pulse, large in magnitude and short in duration, will quickly produce a large number of pores, whereas the second pulse, small in magnitude and longer in duration, will maintain large pore radii and will facilitate electrophoretic movement of DNA into the cell. There was a break of variable duration between the two pulses. Their experiments, as well as the study by Satkauskas et al. (2002), have confirmed that the uptake of DNA for the two-pulse protocol exceeded that for a single pulse.
The model was used to investigate the mechanism behind the greater efficiency of the two-pulse protocol. Fig. 7 A shows the distribution of pore radii during the 1.25 V, 10 µs first pulse, during the first 30 µs of the 100 ms break, and during the first 70 µs of the 0.5 V, 100 ms second pulse. By the end of the first pulse, 18,025 pores were created and started to accumulate at a radius of 34 nm. During the break, all of these pores shrank to radius rm and the resealing process started, which decreased the number of pores to 17,906 by the end of the break. During the second pulse, all of the remaining pores expanded to a radius of 18.2 nm and remained stable for the duration of the pulse. Thus, the second pulse facilitated entry of DNA into the cell by keeping the pore radii sufficiently large and by exerting an electric force on the DNA in a direction perpendicular to the membrane.
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The simulations of Fig. 7 suggest that the two-pulse protocol allows control of the number and size of pores during the second pulse. The number of pores is determined by the magnitude of the first pulse but it can be decreased to a desired number by changing the duration of the break between pulses. The pore size can be increased in three different ways: by decreasing the magnitude of the first pulse (fewer pores will be created and they will expand to larger radii), by increasing the break between pulses (more pores will reseal), and by increasing the magnitude of the second pulse. Although there are limits on how much the pulse strength can be manipulated, the two-pulse protocol gives better control of pore size and, by allowing very long durations of the second pulse, increases the DNA uptake with less of an effect on cell viability. This combination of properties may be responsible for the effectiveness of the two-pulse protocol in achieving high transfection efficiencies.
Additional simulations were performed to examine qualitative agreement between the model and the experiment. Quantitative comparison is not possible because the exact relationship between DNA uptake (predicted by the model) and transfection efficiency (measured experimentally) is not known. Fig. 8 A shows the intracellular DNA concentration as a function of the second pulse duration. The monotonic, nearly linear increase of DNA concentration is qualitatively similar to the results reported in Fig. 5 of Sukharev et al. (1992), although the initial nonlinear part is more pronounced in the experiment. This difference may come from the fact that the experiment used decaying exponential rather than rectangular pulses.
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), DNA uptake and transformation efficiency decrease in a sigmoidal fashion. However, only a part of the sigmoidal curve is seen in the experiment for the same range of break durations. The difference is due to the model parameters not corresponding to the cell type used in the experiment, but the agreement can be improved by choosing a longer time constant for resealing. |
DISCUSSION |
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; Joshi and Schoenbach, 2000). It is in these regions of the cell that the transmembrane voltage is the largest and most pores are created (Hibino et al., 1991). Thus, these regions are also most important to gene delivery. Relating the circuit model of Fig. 3 to a spherical cell of radius a, V0 can be interpreted as the maximum transmembrane potential induced in a cell by the external electric field E: V0 = 1.5 E a.
Even though electroporation in cells is fundamentally the same as in membranes (Chernomordik et al., 1987), some differences are expected. In contrast to a uniformly polarized membrane, Vm in a cell varies with the position (Hibino et al., 1991; Pavlin and Miklav
i, 2003), and the number of pores and their sizes are expected to vary as well. This leads to an important difference in the behavior of a patch versus a cell. In an electroporated patch, Vm decreases considerably below threshold, even to almost zero if the pulse is sufficiently strong (see Benz and Zimmermann, 1980, for experimental evidence and Barnett and Weaver, 1991, for modeling results). Such a drastic decrease does not occur in a cell, where the adjacent nonelectroporated regions maintain Vm at an approximately threshold level, even for very strong shocks (see Hibino et al., 1991, for experimental evidence and DeBruin and Krassowska, 1999, for modeling results).
This drop in Vm occurring in a membrane patch is the reason why our study could not reproduce the exact electrical conditions of the two-pulse experiment of Sukharev et al. (1992). In both the experiment and the model, the second pulse was chosen to be approximately half of the threshold, but the ratio of the first and second pulse strengths was 30 in the Sukharev study and only 2.5 in the model. If the model uses an appropriately strong first pulse, then the number of pores is very large and Vm during the second pulse drops so low that the pores cannot expand and no uptake takes place. To trigger pore expansion, the second pulse should exceed threshold, which would defeat the purpose of the two-pulse protocol. Thus, we have chosen to use a smaller first pulse, which limits the degree of electroporation and prevents a large drop in Vm. Of course, this adjustment would not have been necessary had the model represented a true cell, which would maintain a decent Vm even after a strong pulse.
Fortunately, our model of electroporation (Eqs. 1 and 3) can be used with spatially extended systems. For a single cell in an external field, the cell membrane can be divided into equipotential slices and Eqs. 1 and 3 will apply to each slice. All slices will be coupled by common membrane tension eff and by Vm, which in a spatially extended system will be governed by a partial differential equation (PDE) instead of an ODE (Eq. 7). Such a model will be computationally more expensive, mostly because of the need to solve a PDE at each time step. Nevertheless, this approach has been used successfully in our previous studies that explored electroporation in a cell, a one-dimensional fiber, and a two-dimensional sheet of tissue, although with a much simpler electroporation model (Aguel et al., 1999; DeBruin and Krassowska, 1998, 1999).
In deriving Eq. 5 for effective membrane tension, the tension was assumed to depend linearly on the change in membrane area. This assumption breaks down in the low-tension regime, in which thermal fluctuations affect the shape of the membrane (Rawicz et al., 2000). The nonlinear area-tension relation may influence of the later stages of pore evolution. Another simplification is that the cell volume is assumed constant. In reality, the intracellular fluid will leak out through the macropores, resulting in reduced membrane tension. The reduced membrane tension will halt the tendency of pores to expand and will contribute to the eventual resealing of pores. Thus, another extension of this model should be the addition of changes in cell volume. The coupling of pore evolution with a change in cell volume has been proposed before, although for the case of only one pore present (Brochard-Wyart et al., 2000; Sandre et al., 1999).
The model of DNA permeation is the least developed part of this work. As a first step, we used the simplest available formulation proposed by Neumann et al. (1996), which is based on the Nernst-Planck equation. It is not clear to what extent this equation applies to long DNA chains (Puc et al., 2003). For example, it accounts for the DNA size only indirectly, through its effective charge zeff. Also, the relative radii of the DNA and the pores affect permeation in a very simplistic way: a critical pore size is chosen (e.g., 10 nm) and pores below this size do not admit DNA. Clearly, there is room for improvement here. In the future, we plan to incorporate a more sophisticated model of DNA uptake, which will be based on the recent theoretical and experimental results related to the translocation of the DNA chain through the pore (Ambjörnsson et al., 2002; de Gennes, 1999; Han et al., 1999; Kasianowicz et al., 1996; Sung and Park, 1996).
An additional source of discrepancy between the model's predictions and experimental results is that the model is deterministic, whereas experiments contain numerous random components: cells differ in diameter and shape, which exposes them to different Vm in the same electric field, and the processes of pore creation, evolution, and DNA uptake are stochastic in nature (Powell and Weaver, 1986; Sung and Park, 1996). Were these random elements included in the model, as was done by Puc et al. (2003), certain results of this study would be modified. For example, the sharp drop in the DNA uptake for V0 above 1.4 V is clearly unphysiological and results from the lack of random factors in the model.
Finally, this study could not even attempt to reproduce quantitatively the results of any particular experiment. The reason is twofold. First, the results reported in experiments, such as transfection efficiency (TE), cannot be related in a rigorous way to the intracellular DNA concentration predicted by the model. One study reported that TE, assessed by the activity of ß-galactosidase, depended linearly on the external DNA concentration (Klenchin et al., 1991), and thus, also on the internal DNA concentration. However, no such relation was found in another study (Rols et al., 1998a), which indicates that TE, when measured hours or days after the experiment using bulk colorimetric methods, depends on a number of biological factors (Chang et al., 1991; Rols et al., 1998b; Sukharev et al., 1994). Second, it is not possible to find in the literature all parameters required by the model for a single cell type. The most comprehensive parameter set is available for artificial lipid bilayers, but experiments on lipid bilayers do not yield data on DNA uptake. On the other hand, cell lines used in the DNA uptake experiments have not been used in experiments that determine basic parameters of electroporation. Typically, only electroporation threshold is measured (Sukharev et al., 1992; Wolf et al., 1994), and sometimes the resealing time constant as well (Bier et al., 1999; Golzio et al., 1998; Hama-Inaba et al., 1987).
Relevance for electroporation-mediated DNA delivery
Even with these limitations, the predictions of the model are in qualitative and sometimes even quantitative agreement with experiments. Most important, the model confirms numerous experimental findings that pulses of millisecond duration, slightly above the threshold for electroporation, are most effective in delivering DNA to cells, whereas the short and strong pulses used in drug delivery are not effective (Gehl and Mir, 1999; Hama-Inaba et al., 1987; Rols et al., 1998a; Yoshizato et al., 2000). The model explains these results by demonstrating that just suprathreshold pulses create pores large enough to admit DNA and of sufficient number to assure the postshock shrinkage of all pores to rm, which facilitates resealing and cell survival (Tables 2 and 3). Larger pulses (>1.35 V) create many pores but they are too small for significant DNA uptake because they accumulate at a radius <10 nm (Tables 2 and 3). Thus, the predicted voltage range for successful uptake is relatively narrow (Fig. 6), which is consistent with some experimental results (e.g., 1.2–1.6 kV/cm in Tekle et al., 1991). However, other studies observed a somewhat wider voltage range (e.g., maximum uptake observed at 2.5 times threshold in Hama-Inaba et al., 1987).
Two other key experimental findings are that the observable uptake of macromolecules requires a pulse duration on the order of milliseconds (Klenchin et al., 1991; Rols and Teissié, 1998) and that, for multipulse protocols, the uptake decreases with the break between pulses (Satkauskas et al., 2002; Sukharev et al., 1992; Wolf et al., 1994). These results are consistent with Table 3 and Fig. 7 B. The time constant of the decrease in uptake appears to be longer in experiments than in the model (e.g., minutes rather than seconds (Bier et al., 1999; Golzio et al., 1998; Satkauskas et al., 2002; Tekle et al., 1991)). This is because the resealing time constant in the model, 3 s, comes from the measurements on the lipid bilayers (Glaser et al., 1988) and may not be appropriate for cell lines used experimentally.
The model can also explain the experimental observation that the permeable state is long-lived for small, but not large, molecules (Rols and Teissié, 1998; Satkauskas et al., 2002; Wolf et al., 1994). As seen in Fig. 5, in less than a millisecond after the pulse pores shrink to the minimum energy radius rm 0.8 nm, where they persist until resealing takes place. This radius is large enough to admit small marker molecules such as propidium iodide or calcein (radius 0.6 nm, evaluated from molecular weight (Fournier, 1998)), which can enter driven by the concentration gradient, but not for the macromolecules with a radius much larger than rm.
It is not clear whether the model's prediction of the drop in the DNA uptake at very strong pulses (Table 2) agrees with experiments. The difficulty here is the decrease in the cell viability with the pulse strength, which may affect the measurement of the uptake. Some studies demonstrated a decrease in DNA uptake for strong pulses in several cell lines, but this decrease disappeared when the uptake per viable cell was reported (Chang et al., 1991; Hama-Inaba et al., 1987). On the other hand, other groups report a decrease in DNA uptake for strong pulses even if the cell viability decrease is taken into account (Tekle et al., 1991; Wolf et al., 1994).
Most important, our model greatly simplifies the intuitive picture of the DNA uptake by the cells, which is still debated in the literature. To date, theoretical models could predict stable pores of only a few nanometers in radius; larger pores were unstable (Freeman et al., 1994; Joshi and Schoenbach, 2000). These predictions were confirmed by some experiments, in which high-voltage, short pulses were used that must have created a large number of pores with radii not substantially larger than 1 nm (Glaser et al., 1988; Kakorin and Neumann, 2002; Schwister and Deuticke, 1985). To reconcile these results with the experimental evidence of the significant increase in DNA uptake after electric pulses, some researchers postulated that DNA entry into cells relies on the DNA-membrane interactions, which may be facilitated by a collection of small, 1 nm pores (Neumann et al., 1996; Rols and Teissié, 1998; Rols et al., 1998b; Sukharev et al., 1992, 1994). However, the physical mechanism behind such a permeation process remains unclear and direct experimental confirmation is still lacking.
Our model, together with the experimental and theoretical evidence of the existence of stable pores with radii on the order of tens to hundreds of nanometers (Chang and Reese, 1990; Fonari et al., 2003; Kanduer et al., 2003; Lieber and Steck, 1982; Sandre et al., 1999; Tieleman et al., 2003; Zhelev and Needham, 1993), supports a simpler mechanism, in which DNA enters the membrane through stable macropores. Our model predicts pores large enough to permit the uptake of DNA (Fig. 4; Tables 2 and 3), even in its circular or supercoiled conformation (Blackburn and Gait, 1996). These pores remain open for the entire duration of the electric pulse (Fig. 4; Tables 2 and 3) providing adequate time for the DNA chain to enter the cell (Rols and Teissié, 1998; Sukharev et al., 1994). Although this mechanism needs further experimental confirmation, the qualitative agreement between the modeling and experimental results, presented here, speaks in its favor. With further improvements, the model presented here may become a valuable tool in theoretical investigations of electroporation-mediated DNA delivery.
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This work was supported in part by National Science Foundation grant BES-0108408.
Submitted on October 14, 2003; accepted for publication January 21, 2004.
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