Collective chemotaxis: how cells work together to migrate more efficiently

集体细胞迁移是一种复杂的生物学现象，例如在癌症和胚胎发育中观察到的。一个简化的建模假设是考虑一个同质的群体，其中一个群体的个体成员行为相同。本论文的目的是阐明异质群体的集体细胞迁移。集体细胞迁移是由不同的细胞-细胞相互作用，如共同吸引和接触抑制运动促进。这些机制通过调节小GTP酶（如Rac 1）的细胞内动力学作用于细胞极性，对定向细胞迁移至关重要。我们提出了一个有偏随机游走模型，其中的偏差取决于\textit{Rac 1}的内部状态，\textit{Rac 1}状态受到细胞与细胞和细胞与环境相互作用的影响。我们展示了广泛的模拟研究中的各种情况下的模型的范围和适用性。进而导出了相应的偏微分方程组.我们引入了一个特质结构的Keller-Segel模型来解释迁移细胞群体的异质性。细胞特性由配体占据的膜受体的比例给出，并且细胞通过将配体附着到其受体或从其受体分离来改变其特性。我们假设该性状与细胞的表型相关，并与其执行趋化性或增殖的能力相关。我们使用Hopf-Cole变换正式推导了行波解的性质，并将我们的分析结果与数值模拟的结果进行了比较。对于一个改进的特征结构Keller-Segel模型，我们使用线性稳定性分析，调查（在）稳定条件的Keller-Segel模型，源于离散化的特征变量在原始模型的系统。对于最简单的两态模型，我们推导出不稳定条件。我们推导出相应的标准的情况下，两个以上的状态，并支持这些数值模拟。

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 \textit{Rac1}. We propose a biased random walk model, where the bias depends on the internal state of \textit{Rac1}, and the \textit{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.\\ 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.\\ 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.