Development of a Theoretical Model of Electroporation

Specific Aim: A membrane-field interaction model will be developed to study the changes in membrane electrical properties and the dynamics of membrane pores in the presence of an electric field. This model will couple the spatial distribution of membrane potential to that of membrane pores. The principles of cable theory will be used to express the transmembrane potential distribution. The pore population dynamics will be determined using the aqueous pore theory (Barnett and Weaver, 1991).

Preliminary Studies: The specific aim of this project is to construct a model that will encompass changes in membrane properties and distribution of membrane pores. Such a model was developed based on the principles of cable theory and of membrane pore theory of Barnett and Weaver (1991).

Figure 1: An experimental configuration for studying membrane-field interaction of an isolated cell (top left). The magnified view (top right) shows the membrane as an insulating shell. The cell is placed in a conducting medium between two parallel electrodes. The cable circuit model of a small patch of cell membrane is also shown (bottom). The intracellular and extracellular media are represented by resistances, and , respectively. The cell membrane is represented by a network of parallel resistors () and capacitors (), with a transmembrane potential across the membrane. The long axis of the cylindrical cell is assumed to be along the z-direction.

The system modeled in this project parallels the experimental arrangement shown in Fig. 1. A long skeletal muscle cell was assumed to be placed in a conducting medium between two parallel electrodes. The effects of an applied electric field on the properties of the cell membrane was modeled as a coupled process involving aqueous pore formation and generation of elevated transmembrane potentials. Membrane poration was described in terms of the presence of transient aqueous pores on the membrane. Changes in transmembrane potential distribution was determined using a non-uniform cable model.

Non-Uniform Cable Model

Using the cable theory, the electrical properties of a cell membrane were represented by a series of parallel resistors and capacitors (Rall, 1977), as illustrated in Fig. 1. In this model, the cell membrane was treated as a highly insulating cylindrical boundary separating two electrical conductors that represent the intracellular and extracellular fluids. The potential across the membrane can be expressed as the sum of resting and induced membrane potentials. Assuming that the interior of the cell is at isopotential, the induced membrane potential is a function of the applied electric field. When the resting membrane potential is zero, the induced membrane potential is linearly proportional to the applied voltage assuming that the membrane is uniformly of very low conductivity. For the case of a conducting membrane with spatially non-uniform properties, the membrane potential distribution can be described by the cable theory.

Using circuit theory, the induced transmembrane potential under an applied field along the z-direction at time t was represented by the partial differential equation:

                                          (1)

where and are the dimensionless space constant and time constant, respectively, given by

                          (2)

In Eq. 2, and are the resistances of the intracellular and extracellular fluids, respectively, and and (=1/) are the capacitance and conductance of the membrane, respectively. The membrane conductance, reciprocal of the membrane resistance, is a function of the spatial distribution of aqueous pores on the membrane and is given by

                                                              (3)

where n(r,z,t) is the pore density at time t in a unit membrane area around z (described in the next section) and and are the minimum and maximum pore radii. and are the spreading and pore resistances given by

                                                                          (4)

where h is the membrane thickness and and denote the conductivity of bulk medium and the conductivity of a pore of radius r, respectively. Equation 1 does not take into account the effect of resting membrane potential that arises in addition to the pore conductance considered in this model. A modification to the above equation will be incorporated into the model.

In the traditional uniform cable theory, space and time constants are spatially invariant. Such a treatment is valid only under steady-state conditions. In this model, these parameters are dependent on membrane conductance which, in turn, depends on the pore density. Thus, space and time constants are treated as spatially and temporally varying parameters.

Initial and Boundary Conditions: In order to determine an initial condition for solving the above differential equation, it was assumed that a spatial change in the induced transmembrane potential was due solely to the applied electric field. For the isolated cell case, it was assumed that the extracellular current was far greater than the intracellular current, i.e., , because the extracellular space is much larger than the intracellular space. Therefore, the product was assumed to be constant and equal to the applied field strength, , i.e., . This led to

                                                                          (5)

where is the resistance of each of the two hemispherical regions that constitute the endcaps (ec) of the cell (Gaylor et al. 1988).

The boundary condition constraining is written, using Kirchhoff's voltage law and charge conservation, as:

                                                                          (6)

where 2L is the length of the cell. Symmetry in transmembrane potential distribution gives the second boundary condition, , where z=0 refers to the center of the cell.

The second-order partial differential equation (Eq. 1) describes the spatial distribution of transmembrane potential under an applied field. This is coupled to the pore distribution through the spatially and temporally varying space and time constants. These parameters are determined from pore distribution using the membrane pore model.

Membrane Pore Model

Transmembrane potential distribution obtained from linear cable theory was used to evaluate a corresponding pore distribution. Creation, destruction, and evolution of membrane pores were modeled using the aqueous pore theory.

Following the approach developed by Barnett and Weaver (1991), the evolution of aqueous pores on the cell membrane was modeled as a transmembrane potential-driven process. According to this model, the cell membrane was assumed to contain random local thermal defects or pores. The presence of a energy gradient causes the pressure inside a pore to exceed that of the surrounding bath. The pressure difference drives the water molecules out of the pore and causes the pore to contract. On the other hand, the expansion of a pore occurs as a result of the movement of water molecules into the pore. The energy gradient is very large under quiescent conditions which prevents the pores from expanding to a larger radius. However, the application of an external electric field elevates the transmembrane potential thus lowering the energy gradient. Thus, an increased transmembrane potential induces the expansion of a pore.

A change in pore density occurs due to the difference between expanding and restoring forces. Diffusion of water contributes to both expansion and contraction of a pore. This can be expressed in the form of a convection-diffusion equation. Let n(r,z,t)dr be the number of pores in the membrane with radius between r and r+dr at time t within a unit area of the membrane around z. The total change in n(r,z,t) is described by Smoluchowski equation (Gardiner, 1985)

                                                            (7)

where represents the effective diffusion coefficient of a membrane pore in radius space, is the pore energy, kT is the product of Boltzmann's constant and temperature (in ).

Pore energy, , denotes the energy required to form a pore of a given radius, r. It can be described as the sum of a mechanical () and an electrical () component. The mechanical energy describes the balance between expanding and restoring surface tension forces on the pore wall. The electrical energy relates to the transmembrane potential-induced expansive force. The pore energy, , is a function of pore radius r and transmembrane potential and is given by

                                (8)

where is the surface tension of the membrane-water interface, and is the edge energy of the pore. In the above equation, and denote the dielectric constant of water and lipid, respectively and and are the conductivities of a pore of radius r and of the bulk electrolyte medium.

Initial and Boundary Conditions: In order to solve Smoluchowski equation (Eq. 7), the following constraints were introduced. It was assumed that no pores of size exist and that the number of pores changed within a given time interval due to creation and destruction of pores of radius giving the boundary condition

                                                         (9)

where and are the creation and destruction rates of pores of radius and J is the flux of pores in radius space, given by

                                         (10)

Pore formation was thought of as an excitation of the membrane, wherein the membrane goes through configurations which are energetically favorable. This was characterized using an energy barrier, , a sum of two terms, one dependent on transmembrane potential and the other independent of the potential, such that . Here, is the voltage independent component of the energy barrier to pore creation, and a is the capacitance associated with a hydrophobic pore that is subsequently converted to a hydrophilic pore. The pore creation rate can be written as

                                                                         (11)

where is the attempt rate of thermal fluctuations and is a constant with dimensions of inverse time. Assuming that the probability that a pore of radius is destroyed is independent of , pore destruction rate can be expressed as

                                                                                  (12)

where is the destruction rate prefactor with dimensions of velocity and is the voltage independent pore destruction rate. Thus, the boundary condition at was

    (13)

The probability that a large pore will decrease in size is very small. Thus, an "absorbing wall" condition, n(r,z,t)=0 was used at . Here, was assumed to equal , greater than the radius corresponding to the radius of energy maximum at .

At t=0, the membrane was assumed to be in equilibrium with . The quasi-steady state solution of Smoluchowski equation under this constraint, found analytically was

                      (14)

where is the critical radius of a pore.

The coupling between membrane properties and field-induced pore population is two-fold. First, the space and time constants depend on membrane conductance which, in turn, is a function of pore density. Second, the pore energy, which governs the changes in pore population, is a function of transmembrane potential. Thus, Eq. 1 and Eq. 7 together describe a strongly coupled system. Solving these coupled differential equations permits the determination of pore population in terms of measurable quantities such as transmembrane current and potential.