Fast Diffusion of Bound Ca ²⁺ : Analytical and Experimental Characterization of One- and Two-Dimensional Traveling Waves

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 Ca ²⁺ nanodomains generated by individual membrane channels, a widely used paradigm for intracellular Ca ²⁺ signaling. Previous models have primarily focused on stationary Ca ²⁺ nanodomains while neglecting diffusion and saturation of intracellular Ca ²⁺ buffers and sensors. In contrast, we derive analytical solutions without these simplifying assumptions.

Our analysis demonstrates that sustained Ca ²⁺ influx generates continuously expanding distributions of free Ca ²⁺ , whereas Ca ²⁺ -bound buffers and sensors propagate as traveling waves. 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 Ca ²⁺ currents. The analytical framework developed here readily 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.