A new mathematical model for cell motility with nonlocal repulsion from saturated areas

TL;DR

This work develops a new nonlocal saturation mechanism for cell motility models by embedding repulsion from saturated regions into the nonlocal drift term. It presents two complementary frameworks—Eulerian density-based and Lagrangian particle-based formulations—and shows that both converge to the same macroscopic PDEs under corresponding limits. Three drift variants are analyzed: the APS-like strong aggregation with $K(u)=\nabla_{NL}u$, the CMSTT local saturation with $K(u)=(1-u)\nabla_{NL}u$, and a nonlocal saturation with $K(u)=(1-\frac{1}{K}\int_{x-R}^{x+R}u(y,t)dy)\nabla_{NL}u$. Numerical simulations in a bounded domain illustrate how saturation and repulsion shape aggregation, boundary behavior, and dispersion, highlighting the robustness of the dual perspectives and the ability to capture a spectrum from strong clustering to widespread distribution. These insights advance mathematically rigorous modeling of cell–cell adhesion with saturation effects.

Abstract

The main purpose of this work is the mathematical modelling of large populations of cells under different deterministic interactions among themselves, in balance with naturally random diffusion. We focus on cell-cell adhesion mechanisms for a single population confined to an isolated domain. Our most relevant contribution is to derive a mathematical model including a nonlocal saturation coefficient as part of an appropriate nonlocal drift term, including repulsion effects, depending on the level of saturation of the area. For this purpose, we use two discrete approaches taking into account different perspectives: Eulerian and Lagrangian reference systems.

Figures (27)

• Figure 1: APS model. First row trajectories with $N=300$ cells, second row densities of the PDE model (§ 3.5.1) and third one histograms of SDEs (§ 4.1). Both at intermediate time on the left and at final time on the right.
• Figure 2: Drifts for APS model. First row PDE model (§ 3.5.1) and second row SDEs (§ 4.1). Both at intermediate time on the left and at final time on the right.
• Figure 3: Local saturation model, $K=1$. First row trajectories with $N=3500$ cells, second row densities of the PDE model (§ 3.5.2) and third one histograms of SDEs (§ 4.3). Both at intermediate time on the left and at final time on the right.
• Figure 4: Local saturation model, $K=1$. First row trajectories with $N=3500$ cells, second row densities of the PDE model (§ 3.5.2) and third one histograms of SDEs (§ 4.3). Both at intermediate time on the left and at final time on the right.
• Figure 5: Local saturation coefficients for SDEs (§ 4.3), $K=1$. Initial time on the left and final time on the right.