Coupling Trace and Aggregated Finite Element Methods to learn how Surface Polarity patterns Animal Embryos

  • Neiva, Eric (Center for Interdisciplinary Research in Biology (CIRB), Collège de France, CNRS, INSERM, Université PSL)
  • Turlier, Hervé (Center for Interdisciplinary Research in Biology (CIRB), Collège de France, CNRS, INSERM, Université PSL)

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All living organisms emerge from a single cell. In early embryo development of animals, the interplay of surface dynamics and polarity proteins forms segregated polarity domains on the cell surface, which the cell uses to generate different daughter cells upon division. A classical biological model to learn how animal cells polarise and pattern is the 1-mm long C. elegans roundworm. Establishment of cell polarity in the C. elegans zygote is understood to a great extent. Yet, beyond the one-cell stage, the biophysical mechanisms controlling cell polarity remain largely unknown [1]. In this talk, we will describe our efforts to shed light onto this matter with unfitted finite element methods. The problem is characterised by the coupling of the dynamics of the cell surface (viscous actomyosin flows) with the transport of polarity proteins on both the surface and interior of the cell. In order to deal with this, we have coupled trace FE formulations [2] for the surface physics with aggregated FE formulations [3] for the bulk physics. We will provide details on our coupled formulations and their stability, performance aspects of the computer implementation, and some numerical examples, including qualitative experimental validation. [1] J. Nance. Getting to know your neighbour: cell polarization in early embryos. The Journal of Cell Biology, Vol. 206(7), pp. 823–832, 2014. [2] T. Jankuhn, M. A. Olshanskii, A. Reusken and A. Zhiliakov. Error analysis of higher order Trace Finite Element Methods for the surface Stokes equation. Journal of Numerical Mathematics, Vol. 29(3), pp. 245–267, 2021. [3] S. Badia, F. Verdugo, and A. F. Mart ́ın. The aggregated unfitted finite element method for elliptic problems. Computer Methods in Applied Mechanics and Engineering, Vol. 336, pp. 533–553, 2018.