Dependence of Electroporation Detection Threshold on Cell Radius: An Explanation to Observations Non Compatible with Schwan’s Equation Model

Abstract

It is widely accepted that electroporation occurs when the cell transmembrane voltage induced by an external applied electric field reaches a threshold. Under this assumption, in order to trigger electroporation in a spherical cell, Schwan’s equation leads to an inversely proportional relationship between the cell radius and the minimum magnitude of the applied electric field. And, indeed, several publications report experimental evidences of an inverse relationship between the cell size and the field required to achieve electroporation. However, this dependence is not always observed or is not as steep as predicted by Schwan’s equation. The present numerical study attempts to explain these observations that do not fit Schwan’s equation on the basis of the interplay between cell membrane conductivity, permeability, and transmembrane voltage. For that, a single cell in suspension was modeled and the electric field necessary to achieve electroporation with a single pulse was determined according to two effectiveness criteria: a specific permeabilization level, understood as the relative area occupied by the pores during the pulse, and a final intracellular concentration of a molecule due to uptake by diffusion after the pulse, during membrane resealing. The results indicate that plausible model parameters can lead to divergent dependencies of the electric field threshold on the cell radius. These divergent dependencies were obtained through both criteria and using two different permeabilization models. This suggests that the interplay between cell membrane conductivity, permeability, and transmembrane voltage might be the cause of results which are noncompatible with the Schwan’s equation model.

Appendix

Variation in Extracellular Concentration

Assuming that the initial intracellular concentration is zero and there is no change in cell volume during the experiments, initial and final concentrations can be written as:

$$\left\{ {\begin{array}{*{20}l} {c_{i,0} = 0} \hfill \\ {c_{e,0} = N_{0} /V\left( {1 - F} \right)} \hfill \\ {c_{i,f} = N/F \times V} \hfill \\ {c_{e,f = } \left( {N_{0} - N} \right)/V\left( {1 - F} \right)} \hfill \\ \end{array} } \right.$$

(18)

being N ₀ the total number of molecules, N the number of molecules that enter the cell, V the total volume of the suspension, and F the cell volume fraction. If we express the final intracellular concentration as a fraction of the initial extracellular concentration $c_{i,f} = a \times c_{e,i}$, then from (18):

$$a = \frac{{c_{i,f} }}{{c_{e,i} }} = \frac{{N\left( {1 - F} \right)}}{{N_{0} F}}$$

(19)

On the other hand, from (18), the relationship between final and initial extracellular concentration is:

$$\frac{{c_{e,f} }}{{c_{e,0} }} = \frac{{N_{0} - N}}{{N_{0} }}$$

(20)

Combining 19 and 20 we can obtain an expression of the term $c_{e,f} /c_{e,0}$ as a function of F and a:

$$\frac{{c_{e,f} }}{{c_{e,0} }} = \frac{{1 - F\left( {1 + a} \right)}}{1 - F}$$

(21)

For a cell volume fraction of 0.2 and a final intracellular concentration of 5 % the initial extracellular concentration, the variation of the extracellular concentration is 1.25 %.