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MiR@W day: Cell Motility and PDEs in Evolving and Complex Domains

9th November 2015

Organisers: Charlie Elliott, Bjorn Stinner

Draft Schedule

12.00-13:30 Lunch in the Mathematics Institute Common Room

13.30-13:45 Bjorn Stinner (Warwick) Introductory remarks

13.45-14:30 P Pozzi

Title: On the numerical analysis for geometric PDEs: the curve shortening flow

Abstract: In this talk I will discuss some important ideas and challenges that arise in developing and analyzing numerical schemes for geometric problems. After introducing a model in which a geometric evolution equation for a curve is coupled to a parabolic equation on the evolving curve (a problem that to some extent is motivated by applications in cell biology), I will then concentrate on the purely geometric motion of curve, the so-called curve shortening flow. A discretization scheme by finite elements will be presented and discussed.

14.30-15:15 J Etienne

Title: Cells and embryos as flowing shells or Stokes problem on a curved surface

Abstract: The mechanical properties of cells and of early embryos are mostly due to a thin layer of 'active' material, actomyosin, sitting at the periphery of the object. In the long-time limit, actomyosin dynamics are governed by a compressible version of Stokes equations with a source term for 'activity'. A fair mechanical model is thus a surface on which Stokes flow has to be solved. After introducing the problem and showing analytical resolution in a simple geometry, I will present a finite element technique which allows to solve this problem when the surface is fixed.

15.15-16:00 Tea & Coffee in the Mathematics Institute Common Room

16.00-16:45 A Straube

Title: How human cells move

Abstract: I will present a cell biologist's view of how human cells crawl over a surface, in particular discuss factors that determine their directionality, decisions to turn and collective behaviour. I will then describe ongoing work on analysing morphological transitions in migrating cells using similarity-based shape space mapping. Our algorithm analyses cell shape from image timelapse sequences and learns the intrinsic low-dimensional structure of cell shape space. We use the resultant shape space map to visualise differences in cell shape distribution following perturbation experiments and to analyse the quantitative relationships between shape and migration behaviour. Core of our algorithm is a new, rapid, landmark-free shape difference measure that allows unbiased analysis of the widely varying morphologies exhibited by migrating epithelial cells. We used our method to predict cell turning from dynamic cell shape information.

16.45-17:30 C Venkataraman

Title: An Optimal Control Approach to Whole Cell Tracking

Abstract: Whole cell tracking refers to the process of reconstructing dynamic whole cell morphologies from static imaging data. Such techniques are needed for the estimation of the dynamics of morphological features from static data, e.g., circularity, number of pseudopodia, maximal and minimal normal velocities, etc. In this talk, I will discuss an approach to whole cell tracking in which we formulate the problem as an inverse problem for fitting a (simplified) mathematical model for cell motility to experimental data. The resulting mathematical problem consists of the optimal control of a geometric evolution law. One advantage of the proposed approach is that physical features included in the model are reflected in the computed morphologies, we illustrate this with the exemplar case of volume conservation. Time permitting efficient solution methods for the approximation of the computationally intensive optimal control problem will also be discussed. Based on joint work with Blazakis, Madzvamuse, Styles, Yang (all Sussex) and Reyes-Aldasoro (City).

17.30-18:15 Charlie Elliott (Warwick) Concluding remarks

18.15 Dinner in the Mathematics Institute Common Room