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

TL;DR

This work tackles approximating an inhomogeneous boundary flux from a single cell in diffusion problems by replacing the spatially extended cell with a cluster of Dirac delta sources. It develops a symmetry-based localization of point sources, deriving explicit intensities $\\widetilde{\\Phi}_C(t)$ and $\\widetilde{\\Phi}_D(t)$ to reproduce the target flux form $\\phi(\\boldsymbol{x})=\\phi_0+ A\\sin(n\\theta)$, and analyzes convergence as $t\\rightarrow\\infty$ and $r\\rightarrow0^+$. The authors formulate the spatial-exclusion model (BVP_S) and the point-source model (BVP_P), establish flux-matching conditions, and demonstrate via numerical experiments that a multi-Dirac configuration reduces flux and solution discrepancies compared to a single Dirac source, especially for small localization radius $r$. The approach offers a computationally efficient route for simulating diffusive signaling in scenarios with many or moving cells, with planned extensions to multi-cell interactions and time-varying flux patterns in future work.

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.

Figures (4)

• Figure 1: A schematic presentation of a single cell ($\Omega_C$ in orange) within domain ($\Omega$ as a square). $\Omega\setminus\Omega_C$ (blue domain) is the extracellular environment, into which the cell releases diffusive compounds that cannot escape through the boundary $\partial\Omega$. A typical direction of the normal vector $\boldsymbol{n}$ on $\partial\Omega_C$ is indicated.
• Figure 2: The locations of Dirac delta points for $n=1$ and $n=2$ are shown in Panel (a) and (b), respectively. The distance between the non-centred Dirac delta point and the center is denoted as $r$ and all the non-centred Dirac delta points are identical.
• Figure 3: Expression $\tilde{\phi}(\boldsymbol{x}_\theta,t)$ with symmetric configuration of the point sources and intensities $\widetilde{\Phi}_D(t)$ and $\widetilde{\Phi}_C(t)$ in comparison with $\phi(\boldsymbol{x}_\theta)$ given by \ref{['Eq_phi']} with $n=1$. (a) The value of $r$ is fixed and $\tilde{\phi}(\boldsymbol{x}_\theta,t)$ is plotted at different $t$ for $n = 1$; (b) We take $t\rightarrow+\infty$ and the dashed curves show $\tilde{\phi}(\boldsymbol{x}_\theta,t)$ for large $t$ at different values of $r$ for $n = 1$. Panel (c) and (d) are similar plots as (a) and (b), respectively, with $n = 2$.
• Figure 4: Multi-Dirac and single-Dirac approaches in the point source model with 'symmetric configuration' of points and $\varepsilon$-truncated intensities $\widetilde{\Phi}_D(t)$ and $\widetilde{\Phi}_c(t)$ with $n=1$ in comparison with the spatial exclusion model. The four quantities are plotted against the time iterations. They are computed by numerical integration after firstly interpolating the solutions to both models to the common mesh. $D=5$, timestep $\Delta t = 0.04$, terminal time $T = 40$, average mesh size $h = 0.0875$.