The Effect of the Nonlinearity of the Response of Lipid Membranes to Voltage Perturbations on the Interpretation of Their Electrical Properties. A New Theoretical Description

Abstract

Our understanding of the electrical properties of cell membranes is derived from experiments where the membrane is exposed to a perturbation (in the form of a time-dependent voltage or current change) and information is extracted from the measured output. The interpretation of such electrical recordings consists in finding an electronic equivalent that would show the same or similar response as the biological system. In general, however, there is no unique circuit configuration, which can explain a single electrical recording and the choice of an electric model for a biological system is based on complementary information (most commonly structural information) of the system investigated. Most of the electrophysiological data on cell membranes address the functional role of protein channels while assuming that the lipid matrix is an insulator with constant capacitance. However, close to their melting transition the lipid bilayers are no inert insulators. Their conductivity and their capacitance are nonlinear functions of both voltage, area and volume density. This has to be considered when interpreting electrical data. Here we show how electric data commonly interpreted as gating currents of proteins and inductance can be explained by the nonlinear dynamics of the lipid matrix itself.

Keywords: capacitance; conduction; electrophysiology; impedance; inductance; lipid membrane.

Figure 1

(A).The equivalent circuit of the lipid membrane, containing a resistor (Rm) and a capacitor (Cm) in parallel. (B). An approximate equivalent circuit for the membrane of the squid giant axon with additional inductance, L.

Figure 2

Longitudinal impedance data shown in a complex plot (Nyquist plot) showing the negative of the imaginary part (y axis) and the real part (x axis) of the impedance for different values of the frequency (open circles). (A). Calculated impedance spectrum of ideal capacitance resistance membrane like the one in Figure 1A. Frequencies are given in multiples of the characteristic frequency. (B). Measured impedance data for squid giant axon membrane in a frequency range from 10 kHz down to 30 Hz indicating an inductive element in the electrical circuit. Figures adapted from [18].

Figure 3

Calculated Nyquist plot of the impedance of a biological membrane (Equation (17)) shown for different degrees of nonlinearity: (A)–(B): blue: $\Delta G_0=0$ and $\Delta C_0=0$, green, $\Delta G_0=0$ and $\Delta C_0=0.5·C_0$, red, $\Delta G_0=2·G_0$ and $\Delta C_0=0$, black, $\Delta G_0=2·G_0$ and $\Delta C_0=0.5·C_0$. Membrane background conductance: $G_0=1mS/c{m}^2$ [1,9] (A), $G_0=10mS/c{m}^2$ (B). Membrane capacitance is $C_0=1\mu F/c{m}^2$ and the characteristic relaxation time is $\tau =1$ ms. (C): different values of $\Delta G_0$ . (D): different values of membrane conductance $G_0$.

Figure 4

(A). Voltage jump at time t = 0 from a holding voltage of $V_h=-100$mV to an end voltage of $V_e=-60$mV ($\Delta V=40$mV, red) and $V_e=+60$ mV ($\Delta V=160$mV, black). (B)–(C). The capacitive current response to the voltage jump. (B). shown for membrane with no offset polarization assumed. (C). shown for a polar membrane with spontaneous polarization $P_{0,f}=1$mC/m${}^2$ in the fluid phase and $P_{0,g}=0$mC/m${}^2$ in the gel phase. Values used are from LUV of DPPC (see Appendix), the temperature is $T=314.5$K and $\tau =1$ms is assumed.

Figure 5

(A). Voltage jumps at t= 0 from a holding voltage of $V_h=0$ to an end voltage of $V_e=100$ mV (positive jump, black) and $V_e=-100$ mV (negative jump, red). (B). The resistive current response to positive and negative voltage jump according to Equation (21). The assumed conductance is $G_0=1mS/c{m}^2$ and $\Delta G_m=10$mS/cm${}^2$, and the characteristic relaxation time is $\tau =1$ ms. No polarization or holding voltage is assumed.