The theory described here uses a continuum representation of pore energetics, which is appropriate for large pores but is unlikely to remain valid as the size of the pore approaches the size of a lipid molecule. Studies of electroporation that use molecular dynamics (MD) simulations give us a fascinating picture of molecular-level events occurring during electroporation.70,71 Unfortunately, MD simulations are so expensive computationally that at present these models are limited to very small pieces of the membrane (e.g., 256 lipid molecules) and very early stages of electroporation (up to 50 ns). Nevertheless, they confirm some of the assumptions of the theory presented here, such as that the pore creation occurs in two steps, starting with hydrophobic pores (called "water wires") that subsequently change conformation to hydrophilic.
Both experiments and MD simulations show that the electroporation process is probabilistic in nature, with the pore creation rate and the change in its radius subject to fluctuations. These fluctuations are of importance when one attempts to model a small number of pores, created very slowly in response to threshold-level shocks.18,72 With stronger shocks, which create a large number of pores, individual variations in creation and expansion rates are of less significance, since they are not readily visible in the total current Ip. Hence, the theory presented here is an averaged, deterministic description of the electroporation process and its intended use is for above-threshold shocks.
The third limitation is the representation of the flow of matter through the pores. The present model is concerned only with the flow of electric current, which allows one to determine the decrease in the membrane resistance during electroporation. Previous studies have separated that current into currents carried by Na+, K+, Ca2+, and Cl_ ions and examined the changes in intracellular ionic concentrations as a result of pore creation.58'73 However, to date our group did not attempt to model flow of water through pores and the resulting change of cell surface area and volume: the cell geometry was assumed constant. In reality, the intracellular fluid will leak through the macropores, decreasing cell volume and reducing membrane tension.52 This is an additional factor that can slow the growth and facilitate resealing of the pores, although cell swelling and growth of pores have also been observed.25 Thus, future extension of the present theory should involve changes in cell volume. The coupling of pore evolution with a change in cell volume has been proposed before, although only in the case when one pore is present.52'74 As a result of the constant cell volume assumption, the model presented here is valid for relatively short time intervals (milliseconds) before the flow of water through pores affects the cell volume. That is usually sufficient, as electroporation shocks are rarely longer than a few milliseconds.
The most significant limitation of all electroporation models is the lack of a consistent set of model parameters that would represent a specific tissue under study. This is because it is not possible to find in the literature all parameters required by the model for a single cell type. Typically, only the electroporation threshold is measured,14'64'75 and sometimes the resealing time constant as well.45'76'77 The most comprehensive parameter set is available for artificial lipid bilayers.31 Although electroporation in cells is fundamentally the same as in artificial bilayers,78 many parameters depend sensitively on the composition of the lipid bilayer.79'80 There exist parameter sets that approximate electroporation in cardiac muscle60 and in skeletal muscle,81 but only for the earlier version of the asymptotic model that does not include the growth of pores. In the example included here, a "default" parameter set was used, which was developed from the measurements of Glaser et al.31 on artificial lipid bilayers and then adjusted to match the study of Hibino et al.,82 who used potentiometric dyes to visualize the evolution of transmembrane potential during electroporation of sea urchin eggs. Therefore, in interpreting the results given in the section "Example of the Electroporation Process," one needs to keep in mind that urchin eggs have a higher electroporation threshold than cardiac muscle: 1 V versus 0.4-0.5 V.8'83
Finally, electroporation presents considerable challenges to numerical simulations. Under conditions corresponding to most practical applications, the governing equations are stiff because of strong exponential dependence of the pore creation rate on the square of the transmembrane voltage. Even with the asymptotic approximations for pore creation and evolution, simulations of electroporation are expensive. This is because during an early part of the creation transient, the number of pores and their radii have to be tracked very accurately, which requires small time steps. Any errors in the number and distribution of pore radii would propagate to the transmembrane voltage. The data in Fig. 4b show that errors in Vm as small as 0.02 V can result in a threefold increase in the pore creation rate, and consequently, the number of pores would be predicted incorrectly. Recent research attempts to bypass this difficulty by using singular perturbation to "peel away" the strong exponential dependence of pore creation rate upon the transmembrane voltage Vm.84 In particular, during the pore creation phase, the full system of ODEs (9) and (12) reduces to a single integrodifferential equation for the transmembrane voltage plus an expression for the pore density distribution. Hopefully, further progress in this direction will allow us to study both temporal and spatial aspects of electroporation in three-dimensional tissue without prohibitive computational costs.
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