Long-range diffusion of bound Ca²⁺: time-dependent WKBJ solutions in one- and two-dimensional imaging

Reaction-diffusion (RD) systems play a fundamental role in numerous biochemical and biophysical processes. Here, we present a novel analytical framework for solving RD equations by applying the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) formalism to Ca²⁺ buffering. Previous models have primarily focused on stationary Ca²⁺ nanodomains while neglecting diffusion and saturation of intracellular Ca²⁺ buffers and sensors.

We derive analytical solutions using WKBJ ansatz and demonstrate that sustained Ca²⁺ source produces continuously expanding distributions of free Ca²⁺, whereas the concentration of Ca²⁺-bound buffers develops at much slower timescale. These predictions are supported experimentally by measurements of one-dimensional fluorescence profiles produced by single-channel activity and two-dimensional profiles generated by whole-cell Ca²⁺ currents.

The analytical framework developed here extends Michaelis-Menten-type kinetics to reaction-diffusion systems and may therefore be broadly applicable to biochemical and biophysical processes in which diffusion cannot be neglected.