Characterization of Cell Membrane Electroporation

T. R. Gowrishankar, R. C. Lee
The University of Chicago
and
J. C. Weaver
Massachusetts Institute of Technology

Scientific Problem

The plasma membrane of a cell serves the vital function of partitioning the molecular contents of the cytoplasm from its external environment. These membranes are largely composed of amphiphilic lipids which self-assemble into highly insulating structures and thus present a large energy barrier to transmembrane ionic transport. However, the lipid matrix can be disrupted by a strong external electric field leading to an increase in transmembrane conductivity and diffusive permeability. These effects are the result of formation of aqueous pores in the membrane, which also alter the electrical potential across the membrane. These events are encountered in practice, both by design and by accident. Electroporation of cell membranes is used as a tool in injecting drugs and DNA into the cell (Tsong, 1991). Electroporation is also the basic mechanism of tissue injury in high-voltage electric shock (Lee, 1994).

The objective of this project is to develop a model to characterize the molecular events involved in electroporation. Electroporation occurs as a result of the reorientation of lipid molecules of the bilayer membrane to form hydrophilic pores in the membrane. The distribution of such pores, both in terms of size and number, determine the electrical properties of the cell membrane. Changes in pore radius are effected by surface tension forces on the pore wall, diffusion of water molecules into and out of the pore and an electric field induced force of expansion. Pore distribution in the presence of an external electric field can be described by Smoluchowski's equation (Barnett and Weaver, 1991). In order to estimate the changes in pore distribution and membrane electrical properties, the partial differential equation was solved subject to appropriate boundary conditions.

The Crank-Nicolson method of implicit finite difference scheme was used to solve the partial differential equation (Mü, 1990). The tridiagonal matrix was partitioned into p subsystems, each processor thus requiring to work on n/p equations where n is the order of the original matrix.

Results

The molecular events underlying electroporation determine the kinetics of opening and closing of membrane pores. The opening and closing of pores occur as a result of the rotation of lipid molecules that form the pore walls. The movement of lipid molecules in electroporation is characterized in the theoretical model by the diffusion coefficient of lipid molecules in pore radius space. The relaxation of transmembrane current following the removal of the external pulse occurs as a result of the reorientation of the lipid molecules to close the membrane pores or shrink them. The post-field relaxation time constant of transmembrane current for different transmembrane potentials was determined from voltage clamp measurements (Fig. 1).

Figure 1: Post-field relaxation time constant of transmembrane current for different transmembrane potentials. The relaxation time constants determined from voltage clamp measurements of transmembrane current at different transmembrane potentials were used to estimate the diffusion coefficient of lipid molecules in pore radius space. Theoretical estimations of relaxation time constants obtained from numerical simulation of electroporation on SP1 were used to fit the experimental data.

The decrease in relaxation time constant with transmembrane potential is related to the size and distribution of membrane pores prior to the termination of the pulse. Theoretical estimations of relaxation time constants were used to fit the experimental data. The diffusion coefficient was estimated from this fit to be around 5x10e-16 m2/sec.

Table 1: Range of diffusion coefficient of membrane components reported in the literature. The diffusion coefficient of pores in radius space is orders of magnitude smaller than most membrane components.

The diffusion coefficient of pores in radius space (D) was estimated from relaxation time constants determined from experimental measurements and model simulations to be 5x10e-16 m$2/sec.\; The\; estimated\; value\; of\; D\; is\; lower\; than\; the\; estimates\; for\; other\; membrane\; components\; (Table\; 1).\; This\; suggests\; that\; certain\; pores\; are\; stable\; over\; a\; few\; msec\; after\; the\; pulse\; is\; turned\; off\; owing\; to\; a\; slower\; rotation\; of\; lipid\; molecules\; following\; electroporation\; resulting\; in\; a\; rectification\; of\; transmembrane\; current.\; The\; two\; orders\; of\; magnitude\; reduction\; in\; diffusion\; coefficient\; following\; electroporation\; may\; be\; attributed\; to\; the\; decreased\; diffusivity\; of\; lipids\; in\; the\; presence\; of\; stable\; pore-protein\; complexes.$

The diffusion coefficient estimated from the experimental measurements was used in simulating electroporation at different transmembrane potentials. The evolution of pore distribution is shown (Fig. 2) for different transmembrane potentials.

Figure 2: Total number of pores at different transmembrane potentitals. The abrupt decrease in the number of pores immediately following the pulse is indicative of the spontaneous shrinking of small pores.

The abrupt decrease in the number of pores immediately following the pulse is indicative of the spontaneous shrinkage of small pores. Further decrease in the number of pores indicates the decrease in size of larger pores after the pulse is removed. The simulation results agree with experimental measurements which show that the transmembrane potential threshold for electroporation is around -240 mV. Below this threshold, pore creation and destruction occur at a comparable rate resulting in a negligible increase in transmembrane current during the pulse. The corresponding membrane conductance profiles for different transmembrane potentials are also shown (Fig. 3).

Figure 3: Membrane Conductance at different transmembrane potentials. The orders of magnitude increase in membrane conductance reflects the large increase in the total number of pores.

An order of magnitude increase in the total number of pores is reflected by a corresponding increase in membrane conductance. This large increase in membrane conductance for large magnitude pulses removes the ionic gradient across the membrane. Maintenance of this ionic gradient requires metabolic energy. A continuous loss of energy will lead to cell death. Thus, it is essential that the membrane pores be sealed in order to restore the ionic gradient and thus the metabolic energy of the cell in case of membrane injury by electric field.

References

1.

Tsong, T. Y. 1991. Electroporation of cell membranes. Biophys. J. 60:297-306.

2.

Lee, R. C. 1994. In Advances in Electromagnetic Fields in Living Systems. Vol. 1. Ed.: J. C. Lin. 81-127. Plenum Press, New York:NY.

3.

Barnett, A. and J. C. Weaver. 1991. Electroporation: a unified, quantitative theory of reversible electrical breakdown and mechanical rupture in artificial planar bilayer membranes. Bioelectrochem. Bioenerg. 25:163-182.

4.

Mü, S. M. 1990. A Method to Parallelize Tridiagonal Solvers. Proc. 5th Dist. Mem. Conf. South Carolina. Eds: T. L. Huntsberger and B. A. Huntsberger. IEEE Press.