Approximation of a compound-exchanging cell by a Dirac point

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.