13:30 Arrival and coffee
14:00 Dr. Andrew Krause (Durham University)
Title: A Mathematician's View of Challenges in Turing-Type Pattern Formation.
Abstract: Motivated by recent work with biologists, I will showcase some mathematical results on Turing instabilities in complex domains, and in particular discuss their limitations. This is scientiﬁcally related to understanding developmental tuning in a variety of settings such as mouse whiskers, human ﬁngerprints, bat teeth, and more generally pattern formation on multiple scales and evolving domains. Some of these problems are natural extensions of classical reaction-diffusion models, amenable to standard linear stability analysis, whereas others require the development of new tools and approaches. These approaches also help close the vast gap between the simple theory of diﬀusion-driven pattern formation, and the messy reality of biological development, though there is still much work to be done in validating even complex theories against the rich pattern dynamics observed in nature. I will emphasize throughout the role that Turing's 1952 paper had in these developments, and how much of our modern progress (and difficulties) were predicted in this paper. I will close by discussing a range of open questions, many of which fall well beyond the extensions I will discuss, but at least some of which were known to Turing.
15:00 Tea and coffee break
15:30 Georgia Brennan (University of Oxford)
Title: From Mice, to Machine, to Man: Modelling clearance and proteopathy in Alzheimer’s disease.
Abstract: Every day, over 28,000 people are diagnosed with dementia, making it a leading cause of death and economic burden worldwide. The most common form of dementia is Alzheimer's disease (AD). Our best defence against AD has, until recently, been insights gathered from experiments using mice and numerous such experiments have studied the fundamental importance of the brain's clearance mechanisms. Mathematical modelling has joined the fight against AD and allows for safe, ethical, and cost-effective in-silico experimentation in man. My research delivers the first theoretical model of coupled brain clearance and AD progression, and my data-driven computational approach simulates 40 years of AD progression in less than 14 seconds of computational time.
In this talk, I will describe a new model of AD and its coupled relationship with brain clearance. The resulting high-dimensional, network diffusion-reaction dynamical system yields theoretical insights into the neurodegeneration process and its relationship with clearance. Computational results, on high-resolution brain graphs constructed from the data of 426 patients, demonstrate the connection between clearance and AD progression. A key finding is that the coupling between proteopathic spreading and regional brain clearance may not only alter the trajectory of AD but also provide a potential window into understanding AD subtypes. AD research is changing, and mathematics is providing the critical bridge from mice to machine to man.
16:30 Dr. Alex Fletcher (University of Sheffield)
Title: Understanding self-organised tissue patterning across scales.
Abstract: Polarisation is one of the most basic levels of cell and tissue scale pattern formation. In developing epithelial tissues, planar polarisation is vital for coordinated cell behaviours during morphogenesis. Alongside experimental approaches, mathematical modelling offers a useful tool with which to unravel the underlying mechanisms. I will describe our recent and ongoing efforts to model the planar polarised behaviours of cells in developing epithelial tissues, how these models have given new mechanistic insights into various aspects of Drosophila development, and the mathematical and computational challenges associated with this work.
17:30 Poster session
09:30 Prof. Karen Page (UCL)
Title: Mathematical models of Notch-Delta signalling.
Abstract: The pattern of bristles on the Drosophila notum is controlled by Notch-Delta signalling. There is evidence that dynamic protrusions affect patterning. We analyse this using linear stability analysis, and attempt to define minimal requirements for longer range patterning.
10:30 Tea and coffee break
11:00 Dr. Nicolas Verschueren van Rees (University of Exeter)
Title: Patterns on biologically-relevant models: from arteries to reaction diffusion models on finite circular domains.
Abstract: In this talk, we will present the results of the study of two pattern-forming models solved on a finite domain. In the first, the dynamics of the real and complex cubic-quintic Swift-Hohenberg equation over a finite disk with no-flux boundary conditions are studied. We predict the unstable modes of the trivial state using a linear stability analysis. These modes are followed via numerical continuation, revealing a great variety of spatially extended and spatially localized behaviors. Notably, we find solutions localized in the interior as well as solutions localized along the boundary or part of the boundary. Bifurcation diagrams summarizing these results and their stability properties are presented, linking the different solutions. The findings of this study are likely relevant to nonlinear optics, combustion as well as convection.
In the second model, A simple equation modeling an inextensible elastic lining subject to an imposed pressure is derived from the idealised elastic properties of the lining and the pressure. The equation aims to capture the wrinkling response of arterial endothelium to blood pressure changes. A bifurcation diagram is computed via numerical continuation. Wrinkling, buckling, folding, and mixed-mode solutions are found and organised according to system-response measures including tension, in-plane compression, maximum curvature and energy. Approximate wrinkle solutions are constructed using weakly nonlinear theory, in excellent agreement with numerics. We explain how the wavelength of the wrinkles is selected as a function of the parameters in compressed wrinkling systems and show how localized folds and mixed-mode states form in secondary bifurcations from wrinkled states.
12:00 Prof. Jonathan Dawes (University of Bath)
Title: Pattern formation with nonlocal terms, and Alan Turing’s later work on morphogenesis.
Abstract: I will describe the influence of a nonlocal nonlinear term on the well-known dynamics of the model equation usually ascribed to Swift and Hohenberg (1977). Although a nonlocal term allows a continuous transition between purely local and completely global coupling there are interesting and perhaps unexpected aspects of the dynamics in intermediate cases.
It turns out that the analysis of this model problem is closely related to the problem Alan Turing worked on after the publication of his well-known 1952 paper in mathematical biology. Unfinished, unpublished archive material reveals fascinating insights into his attempt to tackle a much more complex mathematical pattern formation problem. I will survey this material and show how it goes far beyond the 1952 paper in both mathematical content and ambition.