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MASDOC - Mathematics and Statistics Doctoral Training Centre

Cell motility and PDEs in evolving and complex domains

Authors: Lewis Church, Matthew Collins, Simon Etter, Katharina Hopf, Kamil Kosiba
Supervisors: Charles Elliott, Bjorn Stinner

Cell motility refers to the way individual cells are able to move spontaneously around an organism. The specific mechanism that drives this motion is known as the actin treadmill. Actin polymers are proteins that are able to push out the front of the cell and then retract the back of it. This is what allows the cell to move forward. The model we shall use to describe the motion of the cell boundary will be a coupled system consisting of a geometric evolution equation and a PDE on the surface. The basis of the model is the force balance equation

(\mathcal{F}_p + \mathcal{F}_v + \mathcal{F}_{visc} + \mathcal{F}_s + \mathcal{F}_b ) \, \nu = 0.


Here, \nu denotes the outward pointing unit normal of the surface and each \mathcal{F} is the scalar value of a force acting on the surface in normal direction.

  • \mathcal{F}_p=f(c) is a protrusive force depending non-linearly on the density of a chemical species c governed by the conservation law
    \frac{\text{d}}{\text{d}t} \int c \, dA = \int_{\partial\Gamma'} \nabla c \cdot \mu \, dA,\hspace{1cm}\Gamma'\subset\Gamma,\;\; \mu\text{ conormal to } \partial\Gamma'.
  • \mathcal{F}_v=\lambda represents a volume constraint.
  • \mathcal{F}_{visc}=-\omega \,v\cdot\nu is the viscous force opposing the movement of the cell membrane.
  • \mathcal{F}_s=-k_s H is the mean curvature force arising from the surface energy E_s=\int_{\Gamma} k_s.
  • \mathcal{F}_{b}=k_b \left( \Delta_\Gamma H+H|\nabla_\Gamma\nu|^2-\tfrac{1}{2}H^3 \right) is the Willmore force related to the bending energy E_b=\int_{\Gamma} k_bH^2.

    Combining each of these terms we obtain the following system of equations:

    v=\left(- k_s H + k_b \left( \Delta_\Gamma H+H|\nabla_\Gamma\nu|^2-\tfrac{1}{2}H^3 \right) +\lambda +f(c)\right)\nu,
    \int_\Gamma v = 0,
    \partial^\bullet c = k_c \, \triangle_{\Gamma}c - c \nabla_{\Gamma} \cdot v.

    In order to solve the coupled system analytically or numerically, we need to give a description of our surface. There are several approaches we could take - parametric, graph, level set, phase field - each of which have their own advantages and disadvantages. We will consider the first two methods. For the parametric approach we will have the issue of mesh degeneration, caused by the fact that the mesh points are evolving with normal velocity. We aim to resolve this issue with the DeTurck trick, which is a reparametrisation that introduces tangential motion that will hopefully counter the mesh degeneration. For the graph approach, we have the disadvantage that it is quite restrictive in terms of modelling. However, working in this setting will enable us to prove local existence. Furthermore the ideas and techniques presented will be a useful tool in understanding the general situation.

    Acknowledgements

    We acknowledge the funding body EPSRC and the support from MASDOC CDT.

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