Euler’s equation, the ‘most beautiful equation in mathematics’, startlingly connects the five most important constants in the subject: 1, 0, π, e and i. Central to both mathematics and physics, it has also featured in a criminal court case and on a postage stamp, and has appeared twice in The Simpsons. So what is this equation – and why is it pioneering?

Robin Wilson is an Emeritus Professor of Pure Mathematics at the Open University, Emeritus Professor of Geometry at Gresham College, London, and a former Fellow of Keble College, Oxford.

28 February 2018, 5pm-6pm, Mathematical Institute, Oxford

Please email @email to register

Bacteria swim by rotating semi-rigid helical flagellar filaments, using an ion driven rotary motor embedded in the membrane. Bacteria are too small to sense a spatial gradient and therefore sense changes in time, and use the signals to bias their direction changing pattern to bias overall swimming towards a favourable environment. I will discuss how interdisciplinary research has helped us understand both the mechanism of motor function and its control by chemosensory signals.

Please see https://www.eventbrite.co.uk/e/qbiox-colloquium-dunn-school-seminar-hil…

for details.

We describe a locally finite graph naturally associated to each knot type K, called the Reidemeister graph. We determine several local and global properties of this graph and prove that the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly (time permitting), we introduce another object, relating the Reidemeister and Gordian graphs, and briefly present an application to the study of DNA.

Many symptoms of Parkinson’s disease are connected with abnormally high levels of synchrony in neural activity. A successful and established treatment for a drug-resistant form of the disease involves electrical stimulation of brain areas affected by the disease, which has been shown to desynchronize neural activity. Recently, a closed-loop deep brain stimulation has been developed, in which the provided stimulation depends on the amplitude or phase of oscillations that are monitored in patient’s brain. The aim of this work was to develop a mathematical model that can capture experimentally observed effects of closed-loop deep brain stimulation, and suggest how the stimulation should be delivered on the basis of the ongoing activity to best desynchronize the neurons. We studied a simple model, in which individual neurons were described as coupled oscillators. Analysis of the model reveals how the therapeutic effect of the stimulation should depend on the current level of synchrony in the network. Predictions of the model are compared with experimental data.

The derivation of so-called `effective descriptions' that explicitly incorporate microscale physics into a macroscopic model has garnered much attention, with popular applications in poroelasticity, and models of the subsurface in particular. More recently, such approaches have been applied to describe the physics of biological tissue. In such applications, a key feature is that the material is active, undergoing both elastic deformation and growth in response to local biophysical/chemical cues.

Here, two new macroscale descriptions of drug/nutrient-limited tissue growth are introduced, obtained by means of two-scale asymptotics. First, a multiphase viscous fluid model is employed to describe the dynamics of a growing tissue within a porous scaffold (of the kind employed in tissue engineering applications) at the microscale. Secondly, the coupling between growth and elastic deformation is considered, employing a morpho-elastic description of a growing poroelastic medium. Importantly, in this work, the restrictive assumptions typically made on the underlying model to permit a more straightforward multiscale analysis are relaxed, by considering finite growth and deformation at the pore scale.

In each case, a multiple scales analysis provides an effective macroscale description, which incorporates dependence on the microscale structure and dynamics provided by prototypical `unit cell-problems'. Importantly, due to the complexity that we accommodate, and in contrast to many other similar studies, these microscale unit cell problems are themselves parameterised by the macroscale dynamics.

In the first case, the resulting model comprises a Darcy flow, and differential equations for the volume fraction of cells within the scaffold and the concentration of nutrient, required for growth. Stokes-type cell problems retain multiscale dependence, incorporating active cell motion [1]. Example numerical simulations indicate the influence of microstructure and cell dynamics on predicted macroscale tissue evolution. In the morpho-elastic model, the effective macroscale dynamics are described by a Biot-type system, augmented with additional terms pertaining to growth, coupled to an advection--reaction--diffusion equation [2].

[1] HOLDEN, COLLIS, BROOK and O'DEA. (2018). A multiphase multiscale model for nutrient limited tissue growth, ANZIAM (In press)

[2] COLLIS, BROWN, HUBBARD and O'DEA. (2017). Effective Equations Governing an Active Poroelastic Medium, Proceedings of the Royal Society A. 473, 20160755

Dr Nicola Beer heads up the Department of Stem Cell Engineering at the new Novo Nordisk Research Centre Oxford. Her team will use human stem cells to derive metabolically-relevant cells and tissues such as islets, hepatocytes, and adipocytes todiscover novel secreted factors and corresponding signalling pathways which modify cell function, health, and viability. Bycombining in vitro-differentiated human stem cell-derived models with CRISPR and other genomic targeting techniques, the teamassay cell function from changes in a single gene up to a genome-wide scale. Understanding the genes and pathways underlying cell function (and dysfunction) highlights potential targets for new Type 2 Diabetes therapeutics. Dr Beer will talk about the work ongoing in her team, as well as more broadly about the role of human stem cells in drug discovery and patient treatment.

Please note that this seminar will take place at the Physical and Theoretical Chemistry Laboratory within the
Department of Chemistry, room, PTCL lecture theatre.