# 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

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## Abstract

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## 1. Introduction

**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].

#### 1.1. Nonlinear Capacitance

#### 1.2. Nonlinear Conduction

## 2. Theory

#### 2.1. Impedance Spectroscopy

**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 $\Delta {C}_{0}=0.5\xb7{C}_{0}$,

**red**, $\Delta {G}_{0}=2\xb7{G}_{0}$ and $\Delta {C}_{0}=0$,

**black**, $\Delta {G}_{0}=2\xb7{G}_{0}$ and $\Delta {C}_{0}=0.5\xb7{C}_{0}$. Membrane background conductance: ${G}_{0}=1\phantom{\rule{0.277778em}{0ex}}mS/c{m}^{2}$ [1,9] (

**A**), ${G}_{0}=10\phantom{\rule{0.277778em}{0ex}}mS/c{m}^{2}$ (

**B**). Membrane capacitance is ${C}_{0}=1\phantom{\rule{0.277778em}{0ex}}\mu F/c{m}^{2}$ and the characteristic relaxation time is $\tau =1$ ms. (

**C**): different values of $\Delta {G}_{0}$ . (

**D**): different values of membrane conductance ${G}_{0}$.

#### 2.2. Voltage Jumps

**Figure 4.**(

**A**). Voltage jump at time t = 0 from a holding voltage of ${V}_{h}=-100\phantom{\rule{0.277778em}{0ex}}$mV to an end voltage of ${V}_{e}=-60\phantom{\rule{0.277778em}{0ex}}$mV ($\Delta V=40\phantom{\rule{0.277778em}{0ex}}$mV,

**red**) and ${V}_{e}=+60\phantom{\rule{0.277778em}{0ex}}$ mV ($\Delta V=160\phantom{\rule{0.277778em}{0ex}}$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 ${P}_{0,f}=1\phantom{\rule{0.277778em}{0ex}}$mC/m${}^{2}$ in the fluid phase and ${P}_{0,g}=0\phantom{\rule{0.277778em}{0ex}}$mC/m${}^{2}$ in the gel phase. Values used are from LUV of DPPC (see Appendix), the temperature is $T=314.5\phantom{\rule{0.277778em}{0ex}}$K and $\tau =1\phantom{\rule{0.277778em}{0ex}}$ms is assumed.

**Figure 5.**(

**A**). Voltage jumps at t= 0 from a holding voltage of ${V}_{h}=0$ to an end voltage of ${V}_{e}=100\phantom{\rule{0.277778em}{0ex}}$ mV (positive jump, black) and ${V}_{e}=-100\phantom{\rule{0.277778em}{0ex}}$ mV (negative jump, red). (

**B**). The resistive current response to positive and negative voltage jump according to Equation (21). The assumed conductance is ${G}_{0}=1\phantom{\rule{0.277778em}{0ex}}mS/c{m}^{2}$ and $\Delta {G}_{m}=10\phantom{\rule{0.277778em}{0ex}}$mS/cm${}^{2}$, and the characteristic relaxation time is $\tau =1\phantom{\rule{0.277778em}{0ex}}$ ms. No polarization or holding voltage is assumed.

## 3. Summary and Discussion

## 4. Conclusions

**;**the conduction through the membrane shows great similarities with the conducting properties commonly associated to protein ion channels.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

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**MDPI and ACS Style**

Mosgaard, L.D.; Zecchi, K.A.; Heimburg, T.; Budvytyte, R.
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. *Membranes* **2015**, *5*, 495-512.
https://doi.org/10.3390/membranes5040495

**AMA Style**

Mosgaard LD, Zecchi KA, Heimburg T, Budvytyte R.
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. *Membranes*. 2015; 5(4):495-512.
https://doi.org/10.3390/membranes5040495

**Chicago/Turabian Style**

Mosgaard, Lars D., Karis A. Zecchi, Thomas Heimburg, and Rima Budvytyte.
2015. "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" *Membranes* 5, no. 4: 495-512.
https://doi.org/10.3390/membranes5040495