Modeling electroporation in a single cell

Abstract

Electroporation uses electric pulses to promote delivery of DNA and drugs into cells. This study presents a model of electroporation in a spherical cell exposed to an electric field. The model determines transmembrane potential, number of pores, and distribution of pore radii as functions of time and position on the cell surface. For a 1-ms, 40 kV/m pulse, electroporation consists of three stages: charging of the cell membrane (0-0.51 micros), creation of pores (0.51-1.43 micros), and evolution of pore radii (1.43 micros to 1 ms). This pulse creates approximately 341,000 pores, of which 97.8% are small ( approximately 1 nm radius) and 2.2% are large. The average radius of large pores is 22.8 +/- 18.7 nm, although some pores grow to 419 nm. The highest pore density occurs on the depolarized and hyperpolarized poles but the largest pores are on the border of the electroporated regions of the cell. Despite their much smaller number, large pores comprise 95.3% of the total pore area and contribute 66% to the increased cell conductance. For stronger pulses, pore area and cell conductance increase, but these increases are due to the creation of small pores; the number and size of large pores do not increase.

FIGURE 1

Schematic of a spherical cell considered in this study. Locations of particular interest are indicated by labels: D (H), depolarized (hyperpolarized) pole, D_b (H_b), border between electroporated and nonelectroporated region in the depolarized (hyperpolarized) part of the cell, E, cell equator. An arrow indicates the direction of the electric field E, polar angle θ measures the position along the cell circumference.

FIGURE 2

Time evolution of electroporation in a cell. (A) Transmembrane potential at the locations indicated by labels. (B) Number of all pores K^θ at locations D, H, and H_b. There is only one pore at H_b, no pores at D_b or E. K^θ of 10⁴ corresponds to a pore density of 4.07 × 10¹³ pores/m². (C) Radii of 12 selected pores at the depolarized pole. (D) Maximum radii at locations indicated by labels. The pore at location D_b was created at 0.32 ms. Note the different timescales in panels A and B versus C and D. Dotted vertical lines in panels A and B indicate the start and the end of the pore nucleation stage.

FIGURE 3

Spatial distribution of electroporation in a cell. (A) Transmembrane potential at times indicated in the legend. The dotted line is the steady-state potential that would have been achieved without electroporation. The times 0.51 and 1.43 μs are the start and the end of the pore nucleation stage. (B) Number of all pores K^θ and number of large pores at 1 ms. Note the 10-fold difference in scales for K^θ and . (C) Maximum radius at the end of the nucleation stage (1.43 μs) and at 1 ms. (D) Membrane conductance G^θ at the times indicated in the legend. In panels A–D, solid circles on the 1-ms plots indicate locations D, D_b, E, H_b, and H (left to right).

FIGURE 4

Distribution of pore radii at 30 μs, 100 μs, and 1 ms. (A) Pores at the depolarized pole D. (B) Pores on the entire cell. Only large pores are included in the distributions and their number K_lg is given on each panel. The total number of pores, K, is reported in the upper panels. Two lower panels of panel B cover only part of the range of pore radii; maximum radii r_max are reported in each panel. The bin width = 2 nm.

FIGURE 5

Postpulse evolution of the pores in a cell. The electric field has been turned off at 1 ms. (A) Radii of 12 selected pores (seven large and five small) at the depolarized pole D. This panel is a continuation of Fig. 2 C. (B) Membrane conductance G^θ at the end of the pulse (1 ms, thin line) and after the pulse (1.5 ms, thick line). Solid circles on the 1.5-ms plots indicate locations D, D_b, E, H_b, and H (left to right).

FIGURE 6

Asymmetry of the electroporation process. (A) Transmembrane potential over the depolarized and hyperpolarized hemispheres (overlaid). (B) Membrane conductance, G^θ, over the depolarized and hyperpolarized hemispheres (overlaid). In both panels, solid circles indicate the position of the pole D and the border D_b; open circles indicate the pole H and the border H_b.

FIGURE 7

Transmembrane potential at the equator of the cell. The dashed line indicates V_m = 0.

FIGURE 8

Dependence of cell electroporation on field strength. All results were collected at the end of a 1-ms pulse and apply to the whole cell. (A) Number of all pores (K) and number of small (K_sm) and large (K_lg) pores. The lines corresponding to K and K_sm overlap. (B) Number of pores in large pore population shown on an expanded vertical scale. (C) Average radius () of the large pore population. The vertical bars indicate standard deviation. For clarity, the range of the field strengths was truncated to 50 kV/m; continues at the same level beyond 50 kV/m. (D) Total area of pores reported as a fraction of the cell surface area. (E) Total conductance of the cell membrane. (F) Extent of electroporation in the cell measured by the polar radii θ_d and θ_h. To facilitate comparison with θ_d, the figure plots π–θ_h. The lines labeled an-dep and an-hyp show the extent of electroporation estimated from Eq. 12 for the depolarized and hyperpolarized hemisphere, respectively. The legend in panel A applies also to panels B, D, and E. Dotted vertical line indicates the electric field of 40 kV/m, i.e., the default field-strength used throughout this article.