The effect of domain growth on spatial correlations

Abstract

Mathematical models describing cell movement and proliferation are important research tools for the understanding of many biological processes. In this work we present methods to include the effects of domain growth on the evolution of spatial correlations between agent locations in a continuum approximation of a one-dimensional lattice-based model of cell motility and proliferation. This is important as the inclusion of spatial correlations in continuum models of cell motility and proliferation without domain growth has previously been shown to be essential for their accuracy in certain scenarios. We include the effect of spatial correlations by deriving a system of ordinary differential equations that describe the expected evolution of individual and pair density functions for agents on a growing domain.We then demonstrate how to simplify this system of ordinary differential equations by using an appropriate approximation. This simplification allows domain growth to be included in models describing the evolution of spatial correlations between agents in a tractable manner.

Variable species densities are induced by volume exclusion interactions upon domain growth

“…This approximation has been previously published [24], and reasonably implies that domain growth 'dilutes' the agent density [16]. Finally, we simplify Eq.…”

“…To do so we employ an agent-based, discrete random-walk model on a two-dimensional square lattice with volume-exclusion. We have previously shown that the way in which domain growth is implemented in an IBM can alter the population behaviour of agents [16,17]. Therefore, we hypothesised that the way in which domain growth is implemented in a simulation with multiple agent species could change the dominant agent species.…”

Modelling collective cell migration: neural crest as a model paradigm

“…Yates et al (2012) extended these studies and demonstrated that different ways to split the lattice sites ensures that individual-based stochastic simulations provide equivalent results to PDE models on any non-uniformly growing domain. Ross et al (2016Ross et al ( , 2017 investigated different ways to implement domain growth in lattice-based models. They simulated a one-dimensional random walk with volume exclusion, cell proliferation and death.…”

A power series method for solving ordinary and partial differentials equations motivated by domain growth

“…( 4) and (5) as a 'shape' parameter, however, its inclusion in Eqs. ( 4) and ( 5) is not necessary and β can be set to zero 1 A derivation of Eq. ( 1) can be found in the Supplementary material (SM1).…”

“…The method we present relies on a separation of variables in a system of equations we construct, and generates a power series weighted by coefficients written in terms of the initial condition. This method was motivated by work conducted on the mathematical modeling of domain growth [1].…”