Periodic Reporting for period 1 - FEM4Embryo (New implicit-surface Finite Element Methods to discover universal mechanisms of animal embryo development)
Reporting period: 2023-05-01 to 2025-04-30
All living organisms emerge from a single cell, the zygote. In early embryo development of animals, the interplay of cell surface dynamics, polarity proteins and intracellular flows forms distinct polarity domains, which the cell uses to specify the fate of its daughter cells. A classical biological model to learn how animal cells polarise is the 1-mm long C. elegans roundworm. Establishment of cell polarity in the C. elegans zygote is well studied, yet the biophysical mechanisms controlling cell polarity beyond that stage remain largely unexplored. Biophysical experiments and Finite Element (FE) models could together elucidate this matter. However, FE models of multicellular morphogenesis, other than rare, rely on explicit surface descriptions. Explicit methods are not ideal to simulate cell polarity at the multicellular scale, because they can hardly deal simultaneously with 3D foamlike geometries, coupled surface-bulk physics and topology changes. Conversely, FE methods based on implicit surfaces have recently shown convincing potential to bypass this weakness. In this project, I will launch a new implicit-surface FE paradigm to simulate early embryo development and I will apply it to gain new insights on the development of animal embryos. To this end, I will pioneer implicit-surface FE methods for coupled surface-bulk dynamics in foamlike geometries. The success of the project is ensured by joining leading expertise in implicit-surface FEs from myself, biophysics of morphogenesis from my supervisor, and cell biology at my host institution. I will be based at the Center for Interdisciplinary Research in Biology of the CNRS, located at Collège de France; a unique and ideal place to create new synergies between a FE researcher like me, biophysicists and cell biologists, as well as to grow and diversify my scientific and transferable skills to become one of the few independent leaders in FEs for multicellular morphogenesis in Europe.
We formulated and implemented a novel unfitted finite element framework to simulate surface or coupled surface-bulk problems in time-dependent domains, focusing on fluid-fluid interactions in single animal cells between the actomyosin cortex and the cytoplasm. We proposed a sharp-interface framework that uniquely combines the trace finite element method for surface flows with the aggregated finite element method for bulk flows. This approach enables accurate and stable simulations on fixed Cartesian grids without remeshing. The model also incorporates mechanochemical feedback through the surface transport of a molecular regulator of active tension. We solve the resulting mixed-dimensional system on a fixed Cartesian grid using a level-set-based method to track the evolving surface. Numerical experiments validated the accuracy and stability of the method, capturing phenomena such as self-organised pattern formation, curvature-driven relaxation, and cell cleavage. The novel framework offers a powerful and extendable tool for investigating increasingly complex morphogenetic processes in animal cells.
The new finite element framework is the first one that addresses models and simulations in animal cell morphogenesis with sharp-interface (level-set) methods. This will potentially enable new applications in the field that are difficult to tackle with current approaches because, e.g. they cannot deal with topological changes (body-fitted ALE) or are too costly in 3D (diffuse-interface/phase-field approaches).