Stochastic Pattern Formation in Mathematical Biology

Many varied biological systems exhibit visually striking patterns both static and dynamic, from the pigmentations of animal skins to the spatial distribution of vegetation growth. Deterministic continuum (partial differential equation) models have been widely used to analyse and successfully predict various characteristics of these patterns via the diffusion-driven `Turing instability' mechanism. Initially proposed as an explanatory model for embryogenesis, the Turing instability describes the spontaneous emergence of spatially inhomogeneous structure from simple homogeneous initial conditions through the coupled dynamics of localised `reactions' and spatial diffusion.

Such PDE models often suffer from issues of robustness and parameter tuning. Sometimes a continuum of pattern instabilities is possible, and the model cannot predict which pattern will persist over time. Further, for realistic reaction rates, the predicted instability thresholds typically require an order of magnitude (or greater) disparity in species diffusivities, which is rarely observed in real-world systems.

Stochastic models can avoid these problems while simultaneously allowing analysis of the impacts of noise and (on microscopic scales) discreteness on the pattern formation. Such models have also expanded the Turing mechanism to include random fluctuation-driven `stochastic Turing patterns'. Compartment-based (mesoscopic) stochastic models offer an analytically tractable approach to analysing the spatiotemporal characteristics and statistics of patterning instabilities, both diffusion-driven and fluctuation-driven.

The aims of this project are first and foremost to develop a cohesive analytical framework for the identification, classification, and prediction of spontaneous patterning instabilities; starting with compartment-based models, but eventually spanning a range of discrete stochastic reaction-diffusion set-ups, encompassing both deterministic and stochastic Turing patterns. Secondly, this project will apply this framework to analysing patterning instabilities in more recently developed stochastic reaction-diffusion models, such as volume exclusion processes. With this, we aim to further the analysis of patterning instabilities by enabling comparison between different models and types of instability, to better understand the underlying physical mechanisms driving pattern formation.

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.