Solving reaction-diffusion equations on evolving surfaces defined by biological image data

TL;DR

This paper introduces a computational method for solving reaction-diffusion equations on dynamically changing biological surfaces derived from 3D cell images, highlighting the importance of geometry in experimental analysis.

Contribution

It develops a finite element approach on evolving surface triangulations from image data and models material transport with a normal velocity, applied to biological cell simulations.

Findings

Geometry changes significantly affect FRAP recovery times.

Membrane movement influences pattern evolution in reaction-diffusion systems.

Evolving surface modeling improves interpretation of biological experiments.

Abstract

We present a computational approach for solving reaction-diffusion equations on evolving surfaces which have been obtained from cell image data. It is based on finite element spaces defined on surface triangulations extracted from time series of 3D images. A model for the transport of material between the subsequent surfaces is required where we postulate a velocity in normal direction. We apply the technique to image data obtained from a spreading neutrophil cell. By simulating FRAP experiments we investigate the impact of the evolving geometry on the recovery. We find that for idealised FRAP conditions, changes in membrane geometry, easily account for differences of $\times 10$ in recovery half-times, which shows that experimentalists must take great care when interpreting membrane photobleaching results. We also numerically solve an activator -- depleted substrate system and report on the effect of the membrane movement on the pattern evolution.