| Abstract: |
| Several cellular activities, such as directed migration, are coordinated by an intricate network of biochemical reactions which lead to a polarised state of the cell, in which cellular symmetry is broken, causing the cell to have a well defined front and back. Balancing biological complexity with mathematical tractability is one of the aims of the works by Mori {\it et al} (2008), Holmes and Edelstein-Keshet (2016), in which they propose a famous minimal model for polarisation, known as {\it wave pinning} model.
We will present an extension of these works in a bulk-surface setting for 3-dimensional domains. We will show how a local perturbation of the surface component can trigger a propagative activation over the membrane. The interplay with the bulk component generates a stable profile with only a portion of the membrane strongly active. Depending on the type of proteins, these might represent the front and the back of the cell, or {\it viceversa}.
Furthermore, we will present the {\it Coupled Bulk Surface Finite Element Method}, used to solve the model, together with a predictor-corrector method for the temporal discretisation. Numerical results will be presented that validate the mathematical predictions. \
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{\bf \Large References}
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Mori, Y., Jilkine, A., and Edelstein-Keshet, L. (2008)
{\it Wave-pinning and cell polarity from a bistable reaction-diffusion system}.
Biophysical journal, 94(9):3684-3697.\vspace{0.1cm}
Holmes, W. R. and Edelstein-Keshet, L. (2016)
{\it Analysis of a minimal Rho-GTPase circuit regulating cell shape}
Physical biology, 13(4):046001. |
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