Modeling ionic flow between small targets: insights from diffusion and electro-diffusion theory

TL;DR

This work addresses how ion flux through a membrane channel perturbs local concentration and voltage in neuronal nanodomains. It employs the Poisson-Nernst-Planck ($PNP$) electro-diffusion framework to derive asymptotic expressions for steady-state concentration differences and voltage under diffusion-only and two-species electro-diffusion, focusing on small channel windows on the boundary. For a single-species diffusion problem, the leading concentration difference scales as $c(oldsymbol{x}_1) - c(oldsymbol{x}_2) \sim \frac{2I}{\mathcal{F}\pi A D}$, with curvature and geometrical corrections captured by higher-order terms; in spherical domains a three-term expansion incorporates inter-window distance $L$. In the two-species case, the density differences for positive and negative ions match and yield a voltage drop given by $v(\boldsymbol{x}_1) - v(\boldsymbol{x}_2) = \frac{k_B\mathcal{T}}{e}\log(...)$, with curvature and geometry again modulating the response; nonlinear I–V relations emerge, predicting nanodomain voltages on the order of 10–25 nm and highlighting curvature’s regulatory role. Together, these results illuminate how localized fluxes and microdomain geometry shape ionic regulation near channels, with implications for voltage gating and channel interactions in dendritic spines and similar structures.

Abstract

The flow of ions through permeable channels causes voltage drop in physiological nanodomains such as synapses, dendrites and dendritic spines, and other protrusions. How the voltage changes around channels in these nanodomains has remained poorly studied. We focus this book chapter on summarizing recent efforts in computing the steady-state current, voltage and ionic concentration distributions based on the Poisson-Nernst-Planck equations as a model of electro-diffusion. We first consider the spatial distribution of an uncharged particle density and derive asymptotic formulas for the concentration difference by solving the Laplace's equation with mixed boundary conditions. We study a constant particles injection rate modeled by a Neumann flux condition at a channel represented by a small boundary target, while the injected particles can exit at one or several narrow patches. We then discuss the case of two species (positive and negative charges) and take into account motions due to both concentration and electrochemical gradients. The voltage resulting from charge interactions is calculated by solving the Poisson's equation. We show how deep an influx diffusion propagates inside a nanodomain, for populations of both uncharged and charged particles. We estimate the concentration and voltage changes in relations with geometrical parameters and quantify the impact of membrane curvature.

Figures (8)

• Figure 1: Channels dynamics and voltage in neuronal nano-domains.A) Dendritic spine receiving a calcium ions influx. B) Inner and outer membranes of a mitochondrion, with the inner membrane curvature thought to regulate voltage and calcium ions intake emboJ. C) Formation of a voltage nano-domain in the vicinity of two ion channels (cf. zamponi2011).
• Figure 2: 3-D geometrical domain types.A) An arbitrary domain with general mean mean curvature function $H(\hbox{\boldmath$x$})$. B) A ball domain of radius $R$. C) For two nearby narrow windows we can approximate the boundary by a tangent plane.
• Figure 3: Spherical domain case. Density difference between the influx and outflux as a function of the current amplitude $I$ (A), their distance $L$ from each other (B) and the narrow window radius $A$ (C). The green dashed curves indicate the asymptotic solution \ref{['eq:threeTerm_neumann']}. Model parameters are $R = 500$ nm, $A = 10$ nm, $C_0 = 200$ mM, $D = 200\,\mu{\rm m}^2/s$. For the last case (C) the two windows are antipodes located, thus $L = 1\,\mu{\rm m}$. Values for other physical parameters are given in Table \ref{['table:electrical_param']}.
• Figure 4: Projection of spherical to planar boundaries. Using the coordinates \ref{['eq:cart_coord']} we can map \ref{['eq:model_neumann']} to a diffusion problem in the infinite half-space (eq. \ref{['eq:inf_half']}).
• Figure 5: Field lines trajectories and penetration length. (A) For a distance $L = 60$ nm and a current $I = 100$ pA, examples of trajectories where the gradient is computed using COMSOL comsol (red curve) and from the asymptotic solution \ref{['eq:inner_sol']} via the 2-D dynamical system \ref{['eq:dyna_neu']} (black dashed curve). The green line indicates the penetration length $L_{\rm pe}$. (B) Penetration length $L_{\rm pe}$ versus the current amplitude $I$ (C) Penetration length $L_{\rm pe}$ versus the distance $L$ between the narrow windows. Parameter values are the same as in Fig. \ref{['fig:fig2']}.