A Diffuse-Interface Method for Pore Dynamics in Lipid Membranes under Electric Fields

TL;DR

This work introduces a diffuse-interface framework for membrane electroporation that couples a phase-field pore evolution model to a quasi-static electrolyte potential and a spatially varying leaky-dielectric transmembrane voltage $V_m$. A stabilized semi-implicit scheme updates $V_m$ by treating the stiff leakage term implicitly while lagging the electrolyte-to-membrane current, and a semi-analytical spectral Laplace solver provides exact ionic currents for each transverse mode. The method reproduces the sharp-interface critical-radius bifurcation, captures electric-field focusing through conductive pores, and supports stochastic pore nucleation by including thermal noise in the phase-field dynamics. This yields a robust, grid-convergent predictive tool for electroporation, enabling design of electrically controlled release protocols and providing a basis for future extensions to curvature effects and heterogeneous membranes.

Abstract

We develop a diffuse-interface continuum model for membrane electroporation that couples a phase field for pore geometry to a quasi-static electrolyte potential and a spatially varying leaky-dielectric model for the transmembrane voltage. The main contribution is a stabilized time-integration strategy for transmembrane voltage $V_m$: the stiff leakage term is treated implicitly while the electrolyte-to-membrane ionic current is lagged, yielding a closed-form update that removes the restriction imposed by the fast dielectric relaxation time. The electrolyte potential is computed efficiently using a semi-analytical spectral Laplace solver: a 2D DCT in the membrane plane reduces the 3D problem to independent 1D ODEs in $z$, solved in closed form and reconstructed by an inverse transform. The coupled method is robust under grid refinement, reproduces the sharp-interface critical-radius bifurcation, and captures electric-field focusing through conductive pores. We also demonstrate stochastic pore nucleation by adding thermal noise to the phase-field dynamics, enabling fully emergent electroporation events without prescribing initial defects.

Figures (12)

• Figure 1: (a) The computational domain $\Gamma = \Gamma_{\text{lipid}} \cup \Gamma_{\text{pore}}$. (b) A cross section of the planar lipid membrane placed between parallel electrodes that impose a uniform potential difference $V$
• Figure 2: Schematic of the singular-interface electrostatic domain. The potential jump $V_m$ is enforced at $z=0$, while the current flux $J_{\text{elec}}$ remains continuous across the interface.
• Figure 3: (a) Model setup with a central pore of radius $R_0$ at the center (b) Pore-radius evolution: pores with $R_0<R_c$ close, whereas $R_0>R_c$ leads to expansion.
• Figure 4: (a) the membrane potential shows a clear depression in the vicinity of the pore; (b) the phase field delineates the pore geometry; (c) the area-averaged transmembrane voltage exhibits the expected charging dynamics; and (d) a cross-section reveals that the water-filled pore, being far more conductive than the lipid, acts as a low-resistance pathway-equipotential contours visibly converge and funnel through the pore.
• Figure 5: Membrane dynamics for applied voltages near the threshold. (a) Average transmembrane potential $\overline{V}_m$ (charging curve) over time. (b) Pore radius $R(t)$ showing expansion above critical voltage and closure below the critical.