Mathematical modelling of mammalian pigmentation patterns: Stochastic modelling of melanoblast neural crest cells.

Embryogenesis defines the early stages of embryonic development. Many such developments are attained via a
family of cells called the neural crest cells. Neural crest cells play a vital role in many biological developments in the
early stages of the growing embryo, e.g. formation of bones, cartilage & pigmentation of hair & skin.
Epidermal pigmentation is a product of melanogenesis which is achieved via melanocytes. Melanoblasts, the early
precursors of melanocytes, are the pigment-producing cells responsible for producing melanin. Melanoblasts
originate in the trunk region of the neural crest from which they delaminate & migrate dorsoventrally along their
migratory pathway to colonise the developing epidermis. They then differentiate into melanocytes & start producing
melanin. Survival of melanoblasts is dependent on signalling between the receptor Kit & its ligand Kitl. Mutations in
the Kit gene can alter the signalling mechanism causing melanoblasts to behave erroneously which ultimately leads
to incomplete colonisation of the epidermis. Recent research suggests that the melanoblasts of Kit mutant mice
exhibit longer cell-cycle times, leaving parts of the epidermis deprived of these pigment-producing cells. The resulting
neurocristopathy is called piebaldism. A piebald mouse shows a white spot on the belly. Erroneous behaviour of
other neural crest cells leads to more serious conditions such as Neurofibromatosis & Hirschsprung's disease.
Larger mammals also show piebaldism, e.g. patches of unpigmented skin in cows.
My project will concern modelling melanoblast behaviour using mathematical models. So far, using experimentally
parameterised stochastic agent-based models I have been able to replicate the white belly spots in mice on a
growing domain. We also showed that varying the parameter values our model can produce patch-like patterns of
larger mammals such as those seen in some cow species. We would now like to expand on this work along the follo
wing key lines
Pattern formation in larger mammals: Although the on-lattice agent-based model could seemingly successfully
replicate patches in cows, our work lacks mathematical analysis to accompany these results. We will be developing a
more mathematically rigorous continuum model to accompany the simulation algorithm which will quantify the patch
formation in cows. It has been observed that cow patches have sharper & well-defined boundaries in contrast to
the belly spots in mice. We will develop hybrid deterministic-continuum models which exploit Turing's theory of
diffusion-driven pattern formation in order to investigate this phenomenon.
Localisation of mesenchyme dermal cells to hair follicles: Mesenchymal dermal cells play a key role in hair follicle
morphogenesis. Responding to certain signalling pathways the mesenchymal cells aggregate & form a periodic
pattern of dermal condensates. The locations of these condensates mark the positions of future hair follicles. In this
strand of the project, we are interested in the mechanisms which drive this periodic pattern. We will use simulation
models to understand this patterning.
Realistic cell-cycle time distribution: Much of the stochastic modelling during this PhD will employ the Gillespie
Stochastic Simulation Algorithm. The Gillespie algorithm assumes that cell-cycle times are exponentially distributed
& exhibit the memoryless property. However, it is well-known that cells-cycle times are not drawn from this
distribution in reality. It has been shown previously that cell-cycle times of certain cells in mice are more accurately
modelled using an Erlang distribution. We will be developing models to realistically model cell-cycle time data for
melanoblasts. Evidence from existing work suggests correlations between cell cycle times of daughter cells & their
more distant relatives. We would like to expand on the existing models to incorporate the effects of correlation
between generations of

Combining specialised modelling techniques with complex data analysis in order to deliver prediction with quantified uncertainties lies at the heart of many of the major challenges facing UK industry and society over the next decades. Indeed, the recent Government Office for Science report "Computational Modelling, Technological Futures, 2018" specifies putting the UK at the forefront of the data revolution as one of their Grand Challenges.

The beneficiaries of our research portfolio will include a wide range of UK industrial sectors such as the pharmaceutical industry, risk consultancy, telecommunications and advanced materials, as well as government bodies, including the NHS, the Met Office and the Environment Agency.

Examples of current impactful projects pursued by students and in collaboration with stake-holders include:

- Using machine learning techniques to develop automated assessment of psoriatic arthritis from hand X-Rays, freeing up consultants' time (with the NHS).

- Uncertainty quantification for the Neutron Transport Equation improving nuclear reactor safety (co-funded by Wood).

- Optimising the resilience and self-configuration of communication networks with the help of random graph colouring problems (co-funded by BT).

- Risk quantification of failure cascades on oil platforms by using Bayesian networks to improve safety assessment for certification (co-funded by DNV-GL).

- Krylov regularisation in a Bayesian framework for low-resolution Nuclear Magnetic Resonance to assess properties of porous media for real-time exploration (co-funded by Schlumberger).

- Machine learning methods to untangle oceanographic sound data for a variety of goals in including the protection of wildlife in shipping lanes (with the Department of Physics).

Future committed partners for SAMBa 2.0 are: BT, Syngenta, Schlumberger, DNV GL, Wood, ONS, AstraZeneca, Roche, Diamond Light Source, GKN, NHS, NPL, Environment Agency, Novartis, Cytel, Mango, Moogsoft, Willis Towers Watson.

SAMBa's core mission is to train the next generation of academic and industrial researchers with the breadth and depth of skills necessary to address these challenges. SAMBa's most sustained impact will be through the contributions these researchers make over the longer term of their careers. To set the students up with the skills needed to maximise this impact, SAMBa has developed a bespoke training experience in collaboration with industry, at the heart of its activities. Integrative Think Tanks (ITTs) are week-long workshops in which industrial partners present high-level research challenges to students and academics. All participants work collaboratively to formulate mathematical
models and questions that address the challenges. These outputs are meaningful both to the non-academic partner, and as a mechanism for identifying mathematical topics which are suitable for PhD research. Through the co-ownership of collaboratively developed projects, SAMBa has the capacity to lead industry in capitalising on recent advances in mathematics. ITTs occur twice a year and excel in the process of problem distillation and formulation, resulting in an exemplary environment for developing impactful projects.

SAMBa's impact on the student experience will be profound, with training in a broad range of mathematical areas, in team working, in academic-industrial collaborations, and in developing skills in communicating with specialist and generalist audiences about their research. Experience with current SAMBa students has proven that these skills are highly prized: "The SAMBa approach was a great template for setting up a productive, creative and collaborative atmosphere. The commitment of the students in getting involved with unfamiliar areas of research and applying their experience towards producing solutions was very impressive." - Dr Mike Marsh, Space weather researcher, Met Office.