Mathematical modeling of collective cell migration : cell trait structures and intracellular variables

Abstract

Collective cell migration is a complex biological phenomenon observed, for example, in cancer and embryonic development. A simplifying modeling assumption is to consider a homogeneous population, where the individual members of a group behave identically. The aim of this thesis is to shed some light onto the collective cell migration of heterogeneous populations. Collective cell migration is promoted by different cell-cell interactions, such as co-attraction and contact inhibition of locomotion. These mechanisms act on cell polarity, crucial for directed cell migration, through modulating the intracellular dynamics of small GTPases such as Rac1. We propose a biased random walk model, where the bias depends on the internal state of Rac1, and the Rac1 state is influenced by cell-cell and cell-environment interactions. We demonstrate the scope and applicability of the model in various scenarios in an extensive simulation study. Furthermore, we derive a corresponding system of partial differential equations. Using this model, we successfully replicated key observations from biological experiments. Consistent with these observations, contact inhibition of locomotion seemed crucial for successful collective migration. Additionally, we established a link between the natural deactivation rate of the intracellular state and the persistence of directional movement. We introduce a trait-structured Keller-Segel model to account for heterogeneity in migrating cell populations. The cell trait is given by the proportion of membrane receptors occupied by ligands, and cells change their trait by attaching or detaching ligands to or from their receptors. We assume that the trait is linked to the phenotype of a cell and, with that, to its ability to perform chemotaxis or proliferate. We formally derive properties of traveling wave solutions using the Hopf-Cole transformation and compare our analytical findings to results from numerical simulations. The derivation of this novel model is a key accomplishment of this thesis. A significant finding was the explicit expression for the dominant trait within invading waves of heterogeneous cell populations under specific parameter regimes. Additionally, we identified a theoretical minimal wave speed for traveling waves. Under trade-of assumptions between chemotactic ability and proliferation, we discovered a distinct structure within the traveling waves, with proliferative cells located at the back and migratory cells at the front. For a modified trait-structured Keller-Segel model, we use a linear stability analysis to investigate (in-)stability conditions for a system of Keller-Segel models that stems from discretising the trait variable in the original model. For the simplest, two-state model, we derive instability conditions. We deduce corresponding criteria for cases with more than two states, and support these by numerical simulations. The main result is a novel criterion for Turing instabilities in specific parameter regimes, stemming from our model’s explicit consideration of ligand-receptor bindings.