Using Multiple Dirac Delta Points to Describe Inhomogeneous Flux Density over a Cell Boundary in a Single-Cell Diffusion Model

Peng, Qiyao and Hille, Sander C. (2025) Using Multiple Dirac Delta Points to Describe Inhomogeneous Flux Density over a Cell Boundary in a Single-Cell Diffusion Model. In: Numerical Mathematics and Advanced Applications ENUMATH 2023, Volume 2 - European Conference, 2023 :. Lecture Notes in Computational Science and Engineering, 2 . Springer, Cham, pp. 243-251. ISBN 9783031861680

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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.