Electrical Field Action on Lipid Bilayer - Mathematical Model

ELECTRICAL FIELD ACTION ON LIPID BILAYER - MATHEMATICAL MODEL

ELECTRICAL FIELD ACTION ON LIPID BILAYER - MATHEMATICAL MODEL

D.E.Creanga

Univ. Al. I. Cuza, Fac. of Physics, 11 A Bd. Carol I, Iasi

Lipid bilayer behavior was studied starting from the dependence of the rate of hydrophilic pore formation on the pore activation energy, that is depending on the area of a single lipid molecule as well as on the temporal parameters o the electrical field application and is increased under the action of an electrical field. The mathematical model proposed inhere describes the permeability of the lipid bilayer for different values of the electrical constraint. The main differential equation intended for the mathematical model development led finally to a cubic solution that takes various graphical forms for different values of the parameter d, related to the presence of a cation channel controlled by other cation species. The negative slope of the hysteresis type curve that was noticed for certain d values, may be taken as an indication of the self-adjusting phenomena underlying the charged species transport phenomena through membrane pores under the electrical filed influence. Though not fitted yet with experimental data, the model may be useful in the study of therapeutical protocols where drug substances are ionized molecules.

Phenomenological background. Artificial membranes consistent with lipid bilayers are convenient biophysical models used for the study of charged species transport. Permeability of lipid bilayers can be controlled using electrical fields, the rate of hydrophilic pore formation being calculated by Glasser et al. (1) as an exponential function of the activation energy,,in dependence also of the single lipid area, s, the temperature T and the frequency of lateral fluctuations of the lipid molecule . The pore activation energy was assumed proportional to the voltage drop u, pore radius and the relative permittivity of water inside pore and of the lipid molecule . So, the rate of pore activation was found as being equal to:

(1)

where: N is the pore density, h is the layer thickness, is the dielectric constant, k is Boltzmann’s constant, is the activation energy in the lack of the electrical field,, and being the electrical pulse duration, the interval between pulses and the pore resealing time. It is obvious that the electroporation rate is lower when a certain number of pores are already formed.

We propose the study o the lipid bilayer with protein incorporated so that ion channels permeable for a species of cations (C1) are present in the artificial membrane.

Mathematical model. The mathematical model that was developed inhere assumes that ion channels are not voltage dependent but they are activated by a second species of cations C2 while in the two compartments separated by the lipid bilayer both cation species can be found.

(i)The electrical field is applied in the direction of C2 ions gradient (for instance from the left to the right) so that electroporation facilitates C2 transport through the membrane.

(ii)Once they pass through the membrane they are able to activate C1 channels so that the rate of cation concentration increase in the right compartment is enhanced significantly.

(iii)As a consequence, a local field opposite to the external one is generated and the electroporation rate is slowed down rapidly as well as that of cation accumulation in the right side compartment.

(iv)The attachment of C2 cations to the C1 ion channels diminishes cation concentration in the right side compartment.

The differential equation describing the above presumptions proposed in this article has the form:

(2)

where the first term in the right side corresponds to the above (i) hypothesis, the nominator of the second term corresponds to the hypothesis (ii), the denominator of the second term expresses the presumption (iii) while the third term describes the (iv) assumption; a, b, c and d are the parameters describing the rates of the four processes invoked above. It is evident that instead of the exponential dependence the second order polynomial dependence was used.

Result and discussion. The investigations aiming the describing of the stationary states, the left side of the equation (2) was taken as equal to zero so that, left side of the equation (2) vanishes and for calculation simplicity one can take all coefficients equal to the unit except that we foccused on for the interpretation of the cation dynamics:

(2’)

The equation (3) will be further discussed for different values of d:

(3)

The interesting cases were found for certain d values and for relatively small values

of the independent variable (u - up to 1 kV). The initial value of cation concentration in the right compartment is taken formally as equal to zero. In all below graphs the independent variable, u, is noted with x while n is noted with y(x). Monotone curve corresponds to the case of two complex solutions and was obtained for d higher than 0.65 (fig. 1). For d values lower than 0.65 the curve shape changes so that three distinct zones can be seen: between two zones with monotone shapes a negative slope curve zone appears, very narrow at the beginning but covering whole range between 0.25 and 0.0 (fig. 2-3) for d approximately equal to 0.6. The negative slope of the hysteresis type curve may be usually taken as an indication of the self-adjusting phenomena underlying the charged species transport phenomena through membrane pores under the electrical filed influence.

In fig. 4 a turning point is visible having the coordinates (0;1). For lower d values the loop resulted in the small x values becomes smaller and smaller while the distance between the loop and the monotone branch situated at high y values becomes larger. For further decreasing d values two distinct branches of y(x) are visible suggesting that the cation system can pass suddently from one low-energy state to another state with higher energy and lower stability. The stable energetical states correspond to relatively low values of the electrical field, u, while it is enhancing from zero to about 0.25 kV. When the total electrical field is diminished following the generation of a local field of opposite direction, then the cation system can pass to an unstable state, its concentration remaining non-zeroed. After the activation of ion channels assuring membrane permeability for the second cation species the system may pass to a higher energetical state (for the same value of the external electrical field) non-stable because of the complexity of phenomena involved in the ion transport through the artificial membrane.

Conclusion. Non-linear behavior of ion concentration under the action of an electrical field could be revealed for certain values of the parameter corresponding to cation concentration diminution after attaching to ion channel cation-controlled. Electrical measurements with adequate devices, able to recorded time real data, could validate or not the proposed model.

References

1. Chizmadzhev, Y.A., Zarnitsin, V.G., Weaver, J.C., Potts, R.O., Mechanism of electroinduced ionic species transport through a multilamellar lipid system, Biophysical Journal, 1995, 68 (3), 749-765
2. Clarke, R.J., Zouni, A., Holzwarth, J.F., Voltage sensitivity of the fluorescent probe RH421 in a model membrane system, Biophysical journal, 1995, 68 (4), 1406-1416