A nonlocal-to-local approach to aggregation-diffusion equations

TL;DR

The paper addresses how to model cell–cell adhesion–driven aggregation by deriving a local, fourth-order aggregation-diffusion model as a short-range limit of nonlocal interactions. It first analyzes a one-species local model, yielding the gradient-flow form $\partial_t\rho = -\nabla\cdot\left(\rho\nabla\left(\Delta\rho + \mu^2\rho\right)\right)$, and then extends to a two-species thin-film system with $\partial_t\rho = -\nabla\cdot\left(\rho\nabla\left(\kappa\Delta\rho + \alpha\Delta\eta + \mu\rho + \omega\eta\right)\right)$ and a coupled equation for $\eta$, all derived from short-range interaction potentials. The authors establish the gradient-flow structure with an energy functional $\mathcal{F}_2[\rho,\eta]$, analyze linear stability, and perform extensive numerical simulations in 1D and 2D to reproduce Steinberg’s four pattern configurations (sorting, partial engulfment, engulfment, mixing), linking cross-adhesion parameters $\alpha$ and $\omega$ to observed patterns. The local model offers analytical tractability, energy-based insights, and easier calibration to data compared to nonlocal models, while remaining faithful to the same qualitative adhesion-driven phenomena. The work suggests that local, high-order aggregation-diffusion equations provide a practical and theoretically principled framework for understanding tissue patterning and for future inference from experimental data.

Abstract

Over the past decades, nonlocal models have been widely used to describe aggregation phenomena in biology, physics, engineering, and the social sciences. These are often derived as mean-field limits of attraction-repulsion agent-based models, and consist of systems of nonlocal partial differential equations. Using differential adhesion between cells as a biological case study, we introduce a novel local model of aggregation-diffusion phenomena. This system of local aggregation-diffusion equations is fourth-order, resembling thin-film or Cahn-Hilliard type equations. In this framework, cell sorting phenomena are explained through relative surface tensions between distinct cell types. The local model emerges as a limiting case of short-range interactions, providing a significant simplification of earlier nonlocal models, while preserving the same phenomenology. This simplification makes the model easier to implement numerically and more amenable to calibration to quantitative data. Additionally, we discuss recent analytical results based on the gradient-flow structure of the model, along with open problems and future research directions.

Figures (7)

• Figure 1: Aggregation is possible in the local model as long as $\mu^2>0$. Numerical simulations with periodic boundary conditions and parameters: $\mu^2 = 1,\,L = 40,\,\Delta x = 0.2,\,\Delta t = 0.01$. Initial data corresponds to the spatially homogeneous steady state $\rho(x,0) = 1$ for $x\in[-L,L]$ plus a small perturbation.
• Figure 2: (left) Aggregation in the two-dimensional local model. Initial data is $\rho(x,0) = 1$ plus a small perturbation. Density configuration at $t = 250$. (right) Convergence to steady state in the two-dimensional model. Radial density profiles at different time points and stationary solution given by $\rho_{j_{1,1}/\mu}(r)$ (see falco2024local for details on steady states). Simulation parameters are $\Delta t = 0.01,\,\Delta x=0.1,\,\Delta y =0.1,\,\mu = 1$ and domain specifications: $L = 15$ for (a) and $L=5$ for (b).
• Figure 3: Possible configurations for Steinberg experiments in terms of the cross-adhesion and the self-adhesion of a system of two species (adapted from MurakawaTogashi). In the weak cross-adhesion regime we might have two patterns depending on whether the cross-adhesion strength is strictly zero or positive. Sorting is observed when there is no cross-adhesion between the two species, and partial engulfment when cross-adhesion is small compared to the self-adhesion of each population. When the cross-adhesion is stronger, the system might evolve to an engulfment pattern, where the more cohesive species is surrounded by the less cohesive one; or to complete mixing of the cell populations. The first corresponds to the case in which the cross-adhesion is stronger than the self-adhesion of one species but weaker than the self-adhesion of the other one. The latter occurs when the cross-adhesion strength is comparable to both self-adhesion forces.
• Figure 4: Understanding the impact of changing model parameters. Imposing that $\eta$ is the less cohesive population implies $\mu>\kappa>1$ as discussed in the text. We focus then on the cross-interactions. Parameter ranges for $\alpha$ and $\omega$ shown above: the blue-shaded region represents the weak cross-adhesion regime, while the red-shaded region corresponds to the case of strong cross-adhesion. Below we plot the numerically found steady states for the parameter values given by the square points: sorting, $\omega = -2.38,\,\alpha =0.03$; partial engulfment, $\omega = -0.04,\,\alpha =0.52$; engulfment, $\omega = 1.69,\,\alpha = 1.21$; mixing, $\omega = 5.51,\,\alpha =1.40$. In every case $\kappa = 2$ and $\mu = 4$. We observe the different patterns seen in the Steinberg experiments and the transition from sorting to mixing as we increase the cross-adhesion, agreeing with the model interpretation. Numerical simulations performed on a domain of length $L = 5$ and $\Delta x = 0.2,\,\Delta t = 0.01$ with periodic boundary conditions and initial condition $\rho(x,0) = \eta(x,0) = \mathds{1}_{|x|<1.5}/2$. See figshare for an animated movie with the stationary states corresponding to each point in the dashed line.
• Figure 5: Solutions of the local model using model parameters related to the Steinberg experiments. Each column represents the solution with the same set of parameters and at different times. Sorting, $\alpha = 0 ,\,\omega =-1$; partial engulfment, $\alpha = 0.8 ,\,\omega = 0.2$; engulfment, $\alpha = 1.3 ,\,\omega =2$; mixing, $\alpha = 1.4 ,\,\omega =6$. In every case $\kappa = 2$ and $\mu = 4$ and also $L = 25,\,\Delta x = 0.2,\, \Delta t =0.01$. See figshare for animated movies.