Environmental influence on the pattern formation of collectively moving cells

Research Area: Mathematical Biology

The thesis will be devoted to deriving, studying and analysing models of collectively moving particles, such as sperm or other cells. We will assume that the particles can self-propel and that they interact with a complex, heterogeneous environment. Collective dynamics studies what happens when a large number of agents, such as animals, people, micro-organisms, crystals, etc., interact with each other. A particular focus is the emergence of large scale order or patterns. Famous examples include global alignment in crystals, lane formation for people, waves and aggregation in bacteria, milling in schools of fish or swarming in birds.

We consider systems where the environment plays an important role for the dynamics. One example are cells, such as fibroblasts or invading cancer cells, moving through a tissue made up of fibres and other components. Another example are sperm cell moving through the female reproductive tract and interacting with the present mucus. We will assume the environment is modified by the movement and vice versa.

Mathematically we will start with an individual based model for the cells and the fibres. This will form the basis to derive mean-field and macroscopic equations for the densities and directions of the cells and the fibres. We will examine appropriate scaling limits. We will simulate both the individual-based and continuum models and compare the behaviour. Additionally we will use tools such as linear stability analysis to gain insight into the system. If possible we will compare the model prediction with data such as cell tracks or sperm data.