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

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.

MAC-MIGS develops computational modelling and its application to a range of economic sectors, including high-value manufacturing, energy, finance and healthcare. These fields contribute over £500 billion to the UK economy. The CDT involves collaborations with more than a dozen companies and organisations, including large corporations (AkzoNobel, IBM, Dassault, P&G, Aberdeen Standard Investments, Intel), mid-size firms, particularly in the engineering and power sectors (NM Group, which provides monitoring services to power grid operators in 30 countries, Artemis Intelligent Power, the world leader in digital displacement hydraulics, Leonardo, a provider of defense, security and aerospace services, and Oliver Wymans, a management consultancy firm) and startups such as Brainnwave, which develops data-modelling solutions, and Opengosim which designs state-of-the-art and massively parallel software for subsurface reservoir simulation. Government and other agencies involved will include the British Geological Survey, Forestry Commission, James Hutton Institute, and Scottish National Heritage. Engagement will be via internships, short projects and PhD projects. BIS has stated that "Organisations using computer generated modelling and simulations and Big Data analytics create better products, get greater insights, and gain competitive advantage over traditional development processes". Our partners share this vision and are keen to develop deeper collaborations with us over the duration of the CDT.

Our CDT will achieve the following:

- Produce 76 highly skilled mathematical scientists and professionals, ready to take up positions in academia or in companies such as our partners. The students will have exposure to projects, modelling camps and high-level international collaborations.

- Deliver economic and societal benefits through student research projects developed in close collaboration with our partners in industry, business and government and other agencies.

- Create pathways for impact on computer science, chemistry, physics and engineering by involving interdisciplinary partners from Heriot-Watt and Edinburgh Universities in the supervision and training of our students.

- Organise a large number of lectures and seminars which will be open to staff and students of the two universities. Such lectures will inform the wide university communities about the state-of-the-art in computational and mathematical modelling.

- Work with other CDTs both in Edinburgh and beyond to organise a series of workshops for undergraduates, intended to foster an increased uptake of PhD studentship places in technical areas by female students and those from ethnic minorities, with potential impact on the broader UK CDT landscape.

- Organise industrial sandpits and modelling camps which offer the possibility for our partners to present a challenge arising in their work, and to explore innovative ways to tackle that challenge, fully involving the CDT students. This will kick-start a change in the corporate mindset by exposing the relevant staff to new approaches.

- Develop a new course, "Entrepreneurship for Doctoral Students in the Mathematical Sciences" in conjunction with Converge Challenge (Scotland's largest entrepreneurial training programme) and UoE's School of Business. This and other support measures will develop an innovation culture and facilitate the translation of our students' ideas into commercial activities.