Thu, 24 Oct 2024

12:00 - 13:00
L3

Effective elasticity and dynamics of helical filaments under distributed loads

Michael Gomez
(Kings College London)
Abstract

Slender elastic filaments with intrinsic helical geometry are encountered in a wide range of physical and biological settings, ranging from coil springs in engineering to bacteria flagellar filaments. The equilibrium configurations of helical filaments under a variety of loading types have been well studied in the framework of the Kirchhoff rod equations. These equations are geometrically nonlinear and so can account for large, global displacements of the rod. This geometric nonlinearity also makes a mathematical analysis of the rod equations extremely difficult, so that much is still unknown about the dynamic behaviour of helical rods under external loading.

An important class of simplified models consists of 'equivalent-column' theories. These model the helical filament as a naturally-straight beam (aligned with the helix axis) for which the extensional and torsional deformations are coupled. Such theories have long been used in engineering to describe the free vibrations of helical coil springs, though their validity remains unclear, particularly when distributed forces and moments are present. In this talk, we show how such an effective theory can be derived systematically from the Kirchhoff rod equations using the method of multiple scales. Importantly, our analysis is asymptotically exact in the small-wavelength limit and can account for large, unsteady displacements. We then illustrate our theory with two loading scenarios: (i) a heavy helical rod deforming under its own weight; and (ii) axial rotation (twirling) in viscous fluid, which may be considered as a simple model for a bacteria flagellar filament. More broadly, our analysis provides a framework to develop reduced models of helical rods in a wide variety of physical and biological settings, as well as yielding analytical insight into their tensile instabilities.

Thu, 17 Oct 2024

12:00 - 13:00
L3

Microswimmer motility and natural robustness in pattern formation: the emergence and explanation of non-standard multiscale phenomena

Mohit Dalwadi
(Mathematical Institute)
Abstract
In this talk I use applied mathematics to understand emergent multiscale phenomena arising in two fundamental problems in fluids and biology.
 
In the first part, I discuss an overarching question in developmental biology: how is it that cells are able to decode spatio-temporally varying signals into functionally robust patterns in the presence of confounding effects caused by unpredictable or heterogeneous environments? This is linked to the general idea first explored by Alan Turing in the 1950s. I present a general theory of pattern formation in the presence of spatio-temporal input variations, and use multiscale mathematics to show how biological systems can generate non-standard dynamic robustness for 'free' over physiologically relevant timescales. This work also has applications in pattern formation more generally.
 
In the second part, I investigate how the rapid motion of 3D microswimmers affects their emergent trajectories in shear flow. This is an active version of the classic fluid mechanics result of Jeffery's orbits for inert spheroids, first explored by George Jeffery in the 1920s. I show that the rapid short-scale motion exhibited by many microswimmers can have a significant effect on longer-scale trajectories, despite the common neglect of this motion in some mathematical models, and how to systematically incorporate this effect into modified versions of Jeffery's original equations.

The European Congress of Mathematics took place in Seville, Spain this week (weren't they fab in the Euros), and two ex-Oxford Mathematicians won prizes. Cristiana De Filippis, now Associate Professor at the University of Parma, and Freddie Manners, now Associate Professor at University of California, San Diego.

Current Oxford mathematicians  Martin Bridson, Heather Harrington and James Newton were plenary, special and invited speakers respectively.

Photo of five students

Our final year students have now left us, in some cases never to return to mathematics, but in others to pursue the subject as researchers, in Oxford and across the globe.

However, before they left we asked some of them them to reflect on what they'd liked and disliked about the mathematics course, which parts of mathematics had really grabbed them, and whether they had any regrets - a fun and instructive exit interview, if you like.

Minimal activation with maximal reach: reachability clouds of bio-inspired slender manipulators
Kaczmarski, B Moulton, D Goriely, A Kuhl, E Extreme Mechanics Letters
A geometric dual of F-maximization in massive type IIA
Couzens, C Lüscher, A (21 Jun 2024)
Free field realizations for rank-one SCFTs
Beem, C Deb, A Martone, M Meneghelli, C Rastelli, L (01 Jul 2024)
Fri, 06 Dec 2024

11:00 - 12:00
L5

Spatial mechano-transcriptomics of mouse embryogenesis

Prof Adrien Hallou
(Dept of Physics University of Oxford)
Abstract

Advances in spatial profiling technologies are providing insights into how molecular programs are influenced by local signalling and environmental cues. However, cell fate specification and tissue patterning involve the interplay of biochemical and mechanical feedback. Here, we propose a new computational framework that enables the joint statistical analysis of transcriptional and mechanical signals in the context of spatial transcriptomics. To illustrate the application and utility of the approach, we use spatial transcriptomics data from the developing mouse embryo to infer the forces acting on individual cells, and use these results to identify mechanical, morphometric, and gene expression signatures that are predictive of tissue compartment boundaries. In addition, we use geoadditive structural equation modelling to identify gene modules that predict the mechanical behaviour of cells in an unbiased manner. This computational framework is easily generalized to other spatial profiling contexts, providing a generic scheme for exploring the interplay of biomolecular and mechanical cues in tissues.

Fri, 29 Nov 2024

11:00 - 12:00
L5

Algebraic approaches in the study of chemical reaction networks

Dr Murad Banaji
(Mathematical Institute University of Oxford)
Abstract

Underlying many biological models are chemical reaction networks (CRNs), and identifying allowed and forbidden dynamics in reaction networks may 
give insight into biological mechanisms. Algebraic approaches have been important in several recent developments. For example, computational 
algebra has helped us characterise all small mass action CRNs admitting certain bifurcations; allowed us to find interesting and surprising 
examples and counterexamples; and suggested a number of conjectures.   Progress generally involves an interaction between analysis and 
computation: on the one hand, theorems which recast apparently difficult questions about dynamics as (relatively tractable) algebraic problems; 
and on the other, computations which provide insight into deeper theoretical questions. I'll present some results, examples, and open 
questions, focussing on two domains of CRN theory: the study of local bifurcations, and the study of multistationarity.

Fri, 22 Nov 2024

11:00 - 12:00
L5

Bifurcations, pattern formation and multi-stability in non-local models of interacting species

Dr Valeria Giunta
( Dept of Maths Swansea University)
Abstract

Understanding the mechanisms behind the spatial distribution, self-organisation and aggregation of organisms is a central issue in both ecology and cell biology. Since self-organisation at the population level is the cumulative effect of behaviours at the individual level, it requires a mathematical approach to be elucidated.
In nature, every individual, be it a cell or an animal, inspects its territory before moving. The process of acquiring information from the environment is typically non-local, i.e. individuals have the ability to inspect a portion of their territory. In recent years, a growing body of empirical research has shown that non-locality is a key aspect of movement processes, while mathematical models incorporating non-local interactions have received increasing attention for their ability to accurately describe how interactions between individuals and their environment can affect their movement, reproduction rate and well-being. In this talk, I will present a study of a class of advection-diffusion equations that model population movements generated by non-local species interactions. Using a combination of analytical and numerical tools, I will show that these models support a wide variety of spatio-temporal patterns that are able to reproduce segregation, aggregation and time-periodic behaviours commonly observed in real systems. I will also show the existence of parameter regions where multiple stable solutions coexist and hysteresis phenomena.
Overall, I will describe various methods for analysing bifurcations and pattern formation properties of these models, which represent an essential mathematical tool for addressing fundamental questions about the many aggregation phenomena observed in nature.
 

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