Theoretical Calculation of the Electric Field in the Vicinity of a Pore Formed in a Cell Membrane

Electroporation of biological cell membranes is a phenomena that occurs almost universally at a transmembrane potential of approximately 1000 mV. Theories of pore formation have been proposed based on electrocompression (Crowley 1972, Malderelli and Steebe 1992 ), local membrane defects ( Abidor and Chizdhmazdev 1979, Weaver and Mintzer 1981, Sugar 1981), and Statistical models were proposed by (Sowers and Lieber 1986). However, these theories do not in general derive the form of the electric field within the pore. In this paper, we present a view that is useful in describing the biological membrane. We also derive an approximate functional form for the electric field in the region of a pore formed in the membrane. Introduction Biological cells exist in a wide variety of shapes and sizes, which is dependent on their function in the organism. Although, in general, the shape of a cell is determined by its special function in the organism, a great many of the cells are possessed of an amorphous shape. A consequence of the fluid-like nature of the cell membrane is its ability to adopt a different shape. One of the most generally useful geometries that describe a wide range of cellular properties is that of a spherical shell. These cells may be partially embedded into a small pore in a material and made to assume an asymmetric dumbbell shape. Pores may be formed in the membrane by the application of an electric field of large enough magnitude across the membrane. The cell membrane, which separates the interior of the cell from its surrounding, is composed of a lipid bilayer which is approximately 10 nm thick. This lipid bilayer is composed of long chain fatty acids with their hydrophobic ends apposed to form an interior region of the membrane, which possesses hydrophobic characteristics. The faces of the membrane are composed of hydrophilic polar groups. The pores formed in the membrane of the cells may be transient, or permanent in nature, depending on the characteristics of the applied field. For relatively low field strengths, the pores formed may be very small and short-lived. For intermediate field strengths, the pores may be larger and remain for longer times. For large applied fields the pores may be permanent. A simple model of this arrangement is that of concentric spherical shells giving three regions. Region 1, has radius < R1, region 2 which has inner radius R1, and outer radius R2, and region 3 which has radius > R2. The three different regions of the shell namely region 1 with r < R1, region 2 with R1 < r < R2 and region 3 with r > R2, have conductivities given by σ σ σ 3 2 1 , , respectively. The potential distribution when these concentric shells are placed in a uniform electric field, can be readily found in spherical coordinates by solving Laplaces equation with appropriately chosen boundary conditions. In order to determine the potential that exists in the region of a pore formed in the membrane, it is necessary to examine the cell at a closer level. This view corresponds to a point of view such that |r-R2| << R2, the radius of the cell. To a first approximation the cell membrane may be viewed as a flat sheet, lipid bi-layer. The poration of the membrane produces an opening in this sheet. We seek to determine the form of the electric field that exists in the region of the pore formed in the membrane by an application of a uniform external field. In this treatment, time dependent behaviors will not be considered. The external field is taken to be uniform in the z direction. In spherical coordinates, the solution to Laplaces equation for the potential V, is ( ) ( ) [ ] ( ) θ θ P r B + r A = r, V l 1 + l l l l l cos − Σ where, due to the azimuthal symmetry and the requirement of periodicity of the -(Q) solution, we have set it to be a constant. The known boundary conditions may be expressed as For θ r E = z E V r 0 0 cos − − ⇒ ∞ → For finite is V 0 r → The potential in the three different regions may thus be written as ( ) ( ) θ θ P r A = r, V l l l l I cos Σ ( ) ( ) [ ] ( ) θ θ P r D + r C = r, V l 1 + l l l l l II cos − Σ ( ) ( ) [ ] ( ) θ θ P r G + r F = r, V l 1 + l l l l l III cos − Σ Inside the sphere includes 0 = B 0 = r ⇒ in region 1; Outside the sphere, region 3, for which r > R2, the potential must asymptotically approach the value, θ r E 0 cos − , thus the only values of l allowed is l=1. At the interfaces between the regions, the fields must satisfy the two boundary conditions, namely (I) Tangential component of E is continuous, (ii) Normal J is continuous. Taking σ σ σ 2 3 1 >> ≈ and writing t R = R 2 1 − where t is the membrane thickness, with 1 << R t 2 . Application of these conditions, ignoring terms that are second order in small parameters, yields the coefficients as