Highlights - New lattice-free stochastic model of collective cell migration with an arbitrary number of distinct interacting subpopulations

 - Infinite hierarchy of moment equations derived to describe the stochastic model

 - Mean field and moment dynamics approximations are used to mimic different kinds of monoculture and coculture cell biology experiments

 - Moment dynamics description is necessary when adhesive forces are sufficiently strong

 - Mean field description is sufficient when adhesive forces are sufficiently weak

 Abstract 

Cell migration within tissues involves the interaction of many cells from distinct subpopulations. In this work, we present a discrete model of collective cell migration where the motion of individual cells is driven by random forces, short range repulsion forces to mimic crowding, and longer range attraction forces to mimic adhesion. This discrete model can be used to simulate a population of cells that is composed of K ≥ 1 distinct subpopulations. To analyse the discrete model we formulate a hierarchy of moment equations that describe the spatial evolution of the density of agents, pairs of agents, triplets of agents, and so forth. To solve the hierarchy of moment equations we introduce two forms of closure: (i) the mean field approximation, which effectively assumes that the distributions of individual agents are independent ; and (ii) a moment dynamics description that is based on the Kirkwood superposition approximation . The moment dynamics description provides an approximate way of incorporating spatial patterns, such as agent clustering, into the continuum description. Comparing the performance of the two continuum descriptions confirms that both perform well when adhesive forces are sufficiently weak. In contrast, the moment dynamics description outperforms the mean field model when adhesive forces are sufficiently large. This is a first attempt to provide an accurate continuum description of a lattice-free, multi-species model of collective cell migration.

Continuum approximations for lattice-free multi-species models of collective cell migration

Continuum descriptions of spatial spreading for heterogeneous cell populations: Theory and experiment.

Variability in cell populations is frequently observed in both in vitro and in vivo settings. Intrinsic differences within populations of cells, such as differences in cell sizes or differences in rates of cell motility, can be present even within a population of cells from the same cell line. We refer to this variability as cell heterogeneity. Mathematical models of cell migration, for example, in the context of tumour growth and metastatic invasion, often account for both undirected (random) migration and directed migration that is mediated by cell-to-cell contacts and cell-to-cell adhesion. A key feature of standard models is that they often assume that the population is composed of identical cells with constant properties. This leads to relatively simple single-species homogeneous models that neglect the role of heterogeneity. In this work, we use a continuum modelling approach to explore the role of heterogeneity in spatial spreading of cell populations. We employ a three-species heterogeneous model of cell motility that explicitly incorporates different types of experimentally-motivated heterogeneity in cell sizes: (i) monotonically decreasing; (ii) uniform; (iii) non-monotonic; and (iv) monotonically increasing distributions of cell size. Comparing the density profiles generated by the three-species heterogeneous model with density profiles predicted by a more standard single-species homogeneous model reveals that when we are dealing with some forms of heterogeneity a simple and computationally efficient single-species homogeneous model can be remarkably accurate in describing the evolution of a heterogeneous cell population. In contrast, we also identify other forms of heterogeneity for which the simpler single-species homogeneous model performs relatively poorly. Additional results for heterogeneity in parameters describing both undirected and directed cell migration are also considered, and we find that similar results apply.

https://biorxiv.org/content/biorxiv/early/2019/07/17/705434.full-text.pdf

Continuum descriptions of spatial spreading for heterogeneous cell populations: theory and experiment

Using Experimental Data and Information Criteria to Guide Model Selection for Reaction–Diffusion Problems in Mathematical Biology

Reaction--diffusion models describing the movement, reproduction and death of individuals within a population are key mathematical modelling tools with widespread applications in mathematical biology. A diverse range of such continuum models have been applied in various biological contexts by choosing different flux and source terms in the reaction--diffusion framework. For example, to describe collective spreading of cell populations, the flux term may be chosen to reflect various movement mechanisms, such as random motion (diffusion), adhesion, haptotaxis, chemokinesis and chemotaxis. The choice of flux terms in specific applications, such as wound healing, is usually made heuristically, and rarely is it tested quantitatively against detailed cell density data. More generally, in mathematical biology, the questions of model validation and model selection have not received the same attention as the questions of model development and model analysis. Many studies do not consider model validation or model selection, and those that do often base the selection of the model on residual error criteria after model calibration is performed using nonlinear regression techniques. In this work, we present a model selection case study, in the context of cell invasion, with a very detailed experimental data set. Using Bayesian analysis and information criteria, we demonstrate that model selection and model validation should account for both residual errors and model complexity. These considerations are often overlooked in the mathematical biology literature. The results we present here provide a clear methodology that can be used to guide model selection across a range of applications. Furthermore, the case study we present provides a clear example where neglecting the role of model complexity can give rise to misleading outcomes.

/pdf/using-experimental-data-and-information-criteria-to-guide-548zmj0xj7.pdf

Using experimental data and information criteria to guide model selection for reaction–diffusion problems in mathematical biology

Moving fronts of cells are essential for development, repair and disease progression. Therefore, understanding and quantifying the details of the mechanisms that drive the movement of cell fronts is of wide interest. Quantitatively identifying the role of intercellular interactions, and in particular the role of cell pushing, remains an open question. Indeed, perhaps the most common continuum mathematical idealization of moving cell fronts is to treat the population of cells, either implicitly or explicitly, as a population of point particles undergoing a random walk that neglects intercellular interactions. In this work, we report a combined experimental-modelling approach showing that intercellular interactions contribute significantly to the spatial spreading of a population of cells. We use a novel experimental data set with PC-3 prostate cancer cells that have been pretreated with Mitomycin-C to suppress proliferation. This allows us to experimentally separate the effects of cell migration from cell proliferation, thereby enabling us to focus on the migration process in detail as the population of cells recolonizes an initially-vacant region in a series of two-dimensional experiments. We quantitatively model the experiments using a stochastic modelling framework, based on Langevin dynamics, which explicitly incorporates random motility and various intercellular forces including: (i) long range attraction (adhesion); and (ii) finite size effects that drive short range repulsion (pushing). Quantitatively comparing the ability of this model to describe the experimentally observed population-level behaviour provides us with quantitative insight into the roles of random motility and intercellular interactions. To quantify the mechanisms at play, we calibrate the stochastic model to match experimental cell density profiles to obtain estimates of cell diffusivity, D, and the amplitude of intercellular forces, f_0. Our analysis shows that taking a standard modelling approach which ignores intercellular forces provides a poor match to the experimental data whereas incorporating intercellular forces, including short-range pushing and longer range attraction, leads to a faithful representation of the experimental observations. These results demonstrate a significant role for intercellular interactions in cell invasion.

/pdf/mechanistic-and-experimental-models-of-cell-migration-reveal-49z0ua9psy.pdf

Mechanistic and experimental models of cell migration reveal the importance of intercellular interactions in cell invasion

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Mechanistic and experimental models of cell migration reveal the importance of intercellular interactions in cell invasion