Approximation of a compound-exchanging cell by a Dirac point

TL;DR

This paper introduces a simplified point source model for cell communication via diffusive compounds, replacing complex boundary conditions with Dirac delta functions, and demonstrates its mathematical well-posedness and comparability to traditional models.

Contribution

It develops a novel point source approximation for cell diffusion models, proving its mathematical validity and showing its potential to simplify complex biological simulations.

Findings

The point source model is well-posed in suitable Sobolev spaces.

Solutions exhibit non-$H^1$-smoothness at the Dirac point.

Numerical comparisons suggest high similarity between the models.

Abstract

Communication between single cells or higher organisms by means of diffusive compounds is an important phenomenon in biological systems. Modelling therefore often occurs, most straightforwardly by a diffusion equation with suitable flux boundary conditions at the cell boundaries. Such a model will become computationally inefficient and analytically complex when there are many cells, even more so when they are moving. We propose to consider instead a point source model. Each cell is virtually reduced to a point and appears in the diffusion equation for the compound on the full spatial domain as a singular reaction term in the form of a Dirac delta `function' (measure) located at the cell's centre. In this model, it has an amplitude that is a non-local function of the concentration of compound on the (now virtual) cell boundary. We prove the well-posedness of this particular parabolic problem with non-local and singular reaction term in suitable Sobolev spaces. We show for a square bounded domain and for the plane that the solution cannot be $H^1$-smooth at the Dirac point. Further, we show a preliminary numerical comparison between the solutions to the two models that suggests that the two models are highly comparable to each other.