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Turing bifurcation, wave-pinning or localised patterns for cell polarity formation; three sides of the same coin?

Presented by: 
Alan Champneys University of Bristol
Date: 
Tuesday 8th December 2015 - 09:00 to 10:00
Venue: 
INI Seminar Room 1
Abstract: 
In this talk I shall present recent work in collaboration with students Nicolas Verschuren and with Victor Brena motivated by problems of cellular level polarity formation motivated by a range of problems in plant biology. After reviewing some existing theories based on reaction-diffusion modelling, I will present some work on plant root hair formation in Arabidopsis in collaboration with Claire Grierson. By modelling the kinetics of the plant rho proteins, or ROPs, it will be argued that the key mechanism can be explained by the formation of a localised patch, which arises due to the presence of a subcritical Turing bifurcation and the recent theory of so-called homoclinic snaking. To see what happens in wild type, one needs to include spatial gradients, such that the dynamics of the patch can be explained asymptotically with the help of Michael Ward's semi-strong analysis technique. The mechanism is contrasted with that of the recent theory of wave pinning in mass-conservative reaction-diffusion equations. It is argued that small source and loss terms are biologically motivated by actions of the nucleus controlling the process and by proteins being recycled as symmetry-breaking takes hold. A new study is then undertaken of what happens under introduction of small source and loss terms to a canonical wave-pinning model. It is shown that localised patterns develop into snakes in one limit and in other limit develop into pinned fronts. A new asymptotic analysis shows how front selection occurs in the limit that the source and loss terms tend to zero.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons