Using multiple Dirac delta points to describe inhomogeneous flux density over a cell boundary in a single-cell diffusion model

TL;DR

This paper introduces a method using multiple Dirac delta points to accurately model inhomogeneous flux density over cell boundaries in a single-cell diffusion model, improving computational efficiency.

Contribution

It demonstrates how to represent inhomogeneous flux distributions with clustered point sources, maintaining accuracy while simplifying the model.

Findings

Multiple Dirac delta points can replicate inhomogeneous flux profiles.

The method reduces computational complexity in cell diffusion simulations.

Accuracy is validated against spatial exclusion models.

Abstract

Biological cells can release compounds into their direct environment, generally inhomogeneously over their cell membrane, after which the compounds spread by diffusion. In mathematical modelling and simulation of a collective of such cells, it is theoretically and numerically advantageous to replace spatial extended cells with point sources, in particular when cell numbers are large, but still so small that a continuum density description cannot be justified, or when cells are moving. We show that inhomogeneous flux density over the cell boundary may be realized in a point source approach, thus maintaining computational efficiency, by utilizing multiple, clustered point sources (and sinks). In this report, we limit ourselves to a sinusoidal function as flux density in the spatial exclusion model, and we show how to determine the amplitudes of the Dirac delta points in the point source model, such that the deviation between the point source model and the spatial exclusion model is small.