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Phenotype-structuring of non-local kinetic models of cell migration driven by environmental sensing
Tommaso Lorenzi, Nadia Loy, Chiara Villa
The capability of cells to form surface extensions to non-locally probe the surrounding environment plays a key role in cell migration. The existing mathematical models for migration of cell populations driven by this non-local form of environmental sensing rely on the simplifying assumption that cells in the population share the same cytoskeletal properties, and thus form surface extensions of the same size. To overcome this simplification, we develop a kinetic modelling framework wherein a population of migrating cells is structured by a continuous phenotypic variable that captures variability in structural properties of the cytoskeleton. This framework provides a multiscale representation of cell migration, from single-cell dynamics to population-level behaviours, as we start with a microscopic model that describes the dynamics of single cells in terms of stochastic processes. Next, we formally derive the mesoscopic counterpart of this model, which consists of a phenotype-structured kinetic equation that features a phenotype-dependent non-locality. Then, considering an appropriately rescaled version of this kinetic equation, we formally derive the corresponding macroscopic model, which takes the form of a partial differential equation for the cell number density. To validate the formal procedures employed to derive the macroscopic model from the microscopic model, through the mesoscopic one, we first compare the results of numerical simulations of the two models. We then compare numerical solutions of the macroscopic model with the results of cell locomotion assays, to test the ability of the model to recapitulate qualitative features of experimental observations.
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