Neu and Krassowska32 have shown that under scaling of the radius r by r* = 0.5nm, PDE (8) reduces to an ODE, dN = « (i_ N \ (9)
where N(t) is the density of hydrophilic pores defined as
and Neq(Vm) is the equilibrium pore density for a given voltage Vm,
Constants a, Vep, No, q are defined in Appendix 1. ODE (9) is usually solved with the initial condition N(0) = 0 (no pores).
Equation (9) shows that hydrophilic pores appear at a rate that is exponentially dependent on the square of the transmembrane potential Vm. Vep is the characteristic voltage of electroporation. Note that, just like the creation rate density (4), ODE (9) does not have a distinct threshold for pore creation. In this ODE, a sharp increase in pore creation occurs at approximately 4 Vep. The value of Vep in Appendix 1 was chosen so that Vm < 1 V can be considered subthreshold for shocks up to 1 ms duration.
Equation (9) describes not only creation of pores but also their resealing: after the shock has created a certain number of pores, the pore density N becomes larger than N0, the equilibrium pore density for Vm = 0. Hence, if the shock is turned off and Vm drops near zero, the right-hand side of (9) becomes negative and the pore density N starts decreasing. With the parameters from Appendix 1, the time constant of resealing is approximately 3 s.31 Note that the only pores that participate in resealing are those that have shrunk to a radius near rm. If there exist any pores with r ^ rm, they cannot reseal by the mechanism represented in (9). The resealing of these macropores is beyond the scope of the present model because it involves such processes as a change in cell volume52 or active, exocytotic rebuilding of the lipid bilayer.53
Neu and Krassowski43 have shown that the diffusion term in the PDE (8) is at least two orders of magnitude smaller than the drift term and can be eliminated. Hence, (8) reduces to a first-order advection PDE, which can be further transformed using the method of characteristics.54 This procedure leads to ODEs governing the time evolution of individual pore radii.
As discussed in sections "Pore Creation" and "Pore Evolution," hydrophilic pores are created with the initial radius of r* and they subsequently change size to minimize the energy of the lipid bilayer. For a membrane with a total number of K pores, the rate of change of their radii, rj, is determined by a set of ODEs:
where u is the drift velocity given by (7).
Compared to the PDE (8), the ODEs (9) and (12) of the asymptotic model contain a smaller number of parameters and most of them are related in a straightforward way to experimental measurements.31 The connection between the parameters of the ODE (9) and the molecular-level constants appearing in the PDE can be found in our previous publications.32'55
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