Pushing coarse-grained models beyond the continuum limit using equation learning

Abstract

Mathematical modelling of biological population dynamics often involves proposing high-fidelity discrete agent-based models that capture stochasticity and individual-level processes. These models are often considered in conjunction with an approximate coarse-grained differential equation that captures population-level features only. These coarse-grained models are only accurate in certain asymptotic parameter regimes, such as enforcing that the time scale of individual motility far exceeds the time scale of birth/death processes. When these coarse-grained models are accurate, the discrete model still abides by conservation laws at the microscopic level, which implies that there is some macroscopic conservation law that can describe the macroscopic dynamics. In this work, we introduce an equation learning framework to find accurate coarse-grained models when standard continuum limit approaches are inaccurate. We demonstrate our approach using a discrete mechanical model of epithelial tissues, considering a series of four case studies that consider problems with and without free boundaries, and with and without proliferation, illustrating how we can learn macroscopic equations describing mechanical relaxation, cell proliferation, and the equation governing the dynamics of the free boundary of the tissue. While our presentation focuses on this biological application, our approach is more broadly applicable across a range of scenarios where discrete models are approximated by approximate continuum-limit descriptions.

Random walk models for the propagation of signalling molecules in one-dimensional spatial networks and their continuum limit

“…The extension of our model to higher dimensions and to dynamic, adaptive networks [18] is of strong interest as it would enable more comprehensive modelling of several network systems. Deriving continuum limits for arbitrary networks in higher dimensions is a non-trivial extension but it can be anticipated that average transport properties of signals may also be governed by the reaction-diffusion-advection equations since these types of equations express conservation laws that hold at the discrete level [60,61]. In bone, such models may be able to further our understanding of how signals propagating through the osteocyte network contribute to bone formation and resorption processes, and how osteocytes set the mechanical memory of bone [62].…”

Discrete and continuous mathematical models of sharp-fronted collective cell migration and invasion

“…Discrete models can be implemented to visualize snapshots of the spreading population in a way that is directly analogous to performing and imaging an experiment to reveal the positions of individual cells within the population. Another advantage of working with discrete stochastic models is that the discrete mechanism can be coarse-grained into an approximate continuum model, which means that we can encode different individual-level rules into a simulation-based model, and then convert these rules into approximate continuum PDE models, and the solution of these coarse-grained models can be compared with averaged discrete data obtained by repeated simulation [48,[58][59][60][61][62][63][64][65][66]. As described previously, there has been a great deal of effort devoted to understanding how different forms of continuum PDE models predict smooth or sharp-fronted solution profiles; however, far less attention has been devoted to understanding what individual-level mechanisms lead to smooth or sharp fronts in discrete models of cell migration.…”