Voltage Laws in Nanodomains Revealed by Asymptotics and Numerical Simulations of Electrodiffusion Equations
Frédéric Paquin-Lefebvre, A. Barea Moreno, David Holcman · Multiscale Modeling and Simulation · 2026
Abstract. Characterizing the local voltage distribution within nanophysiological domains, driven by ionic currents through membrane channels, is crucial for studying cellular activity in modern biophysics, yet it presents significant experimental and theoretical challenges. Theoretically, the complexity arises from the difficulty of solving electrodiffusion equations in three-dimensional domains. Currently, there are no general methods available for obtaining asymptotic computations or approximate solutions of these nonlinear equations, and, numerically, it is challenging to explore solutions across both small and large spatial scales. In this work, we develop a method for solving the Poisson–Nernst–Planck equations with ionic currents entering and exiting through two narrow circular window channels located on the boundary. The inflow through the first window is composed of a single cation species, while the outflow maintains a constant ionic density satisfying local electroneutrality conditions. Employing regular expansions and Green’s function representations, we derive steady-state solutions of ionic profiles and voltage drops in both small and large ionic charge regimes. We explore how the local surface curvature and size of windows influence voltage dynamics and validate our theoretical predictions through numerical simulations, assessing the accuracy of our asymptotic computations. These novel relationships between current, voltage, concentrations, and geometry can enhance the characterization of physiological behaviors of nanodomains.