Section: New Results

Mechanics of tissue morphogenesis

Participants : Olivier Ali, Arezki Boudaoud [External Collaborator] , Guillaume Cerutti, Ibrahim Cheddadi [External Collaborator] , Florian Gacon, Christophe Godin, Bruno Leggio, Jonathan Legrand, Hadrien Oliveri, Jan Traas [External Collaborator] .

• Related Research Works: RW2 (Data-driven models) & RW3 (Plasticity & robustness of forms)

• Related Key Modeling Challenges: KMC2 (Efficient computational mechanical models of growing tissues) & KMC3 (Realistic integrated digital models)

As deformations supporting morphogenesis require the production of mechanical work within tissues, the ability to simulate accurately the mechanical behavior of growing living tissues is a critical issue of the MOSAIC project. From a macroscopic perspective, tissues mechanics can be formalized through the framework of continuum mechanics. However, the fact that they are composed, at the microscopic level, by active building blocks out of equilibrium (namely cells) offers genuine modeling challenges and opportunities. Integrating cellular behaviors such as mechano-sensitivity, intercellular fluxes of materials and cell division into a macroscopic mechanical picture of morphogenesis is the topic of this section.

Flattening mechanism during organogenesis in plants. Many plant species have thin leaf blades and axisymmetric elongating organs, such as stems and roots. From a morphoelastic perspective, such complex shapes are currently believed to emerge from the coordination between strain-based growth and stress-based stiffening at the cellular level.

To study the plausibility of such an hypothesis, we conducted numerical simulations where both a stress-based stiffening mechanism of cell walls [29] and a strain-based growth mechanism [24] have been implemented. We performed such simulations on multicellular and multilayered ellipsoidal structures and track their aspect ratio as they developed under various parametrization sets. One key aspect we wanted to investigate was the effect of an heterogeneous stress-based stiffening mechanism on the overall dynamics: Starting from a given initial shape, can we get significantly different shapes by assuming the stress-based stiffening mechanism active only in specific parts of the structures?

Our results, in accordance with experimental measurements conducted simultaneously by biologist colleagues, showed that: (i) Stress-based stiffening was mandatory to grow flat and axisymmetric organs; (ii) in order to grow flat structures, stress-based stiffening should only be active on anticlinal inner walls.

This work was part of Jan Traas's ERC grant Morphodynamics. This work is currently under review, see preprint version [23].

Influence of cell division during flat organogenesis in plants. One key limitation of our 3D modeling approach of leaf-like organogenesis is the lack of cell division implementation. This can be seen as a major flaw in the mechanical understanding of flattening since cell divisions, by increasing the number of load bearing walls, impact significantly the redistribution of mechanical stresses within the tissue.

To alleviate this limitation, we developed a 2D modeling approach to complement the 3D one. This 2D model encompasses the same biophysical processes as the 3D one (described in the previous subsection): a stress-based stiffening and a strain-based growth mechanisms of cell walls; augmented with a cell division module. We used this 2D framework to investigate the flattening dynamics of structures mimicking ellipsoid cross sections of growing organs. Such cross section were described as vertex-based, multicellular and multilayered structures.

We first reproduced the results obtained with the 3D approach to ensure that both models agreed on similar situations, where no cell division was implemented. We tested then several rules of cell division orientation and check which one(s) produced the most efficient flattening process. We were able to show that heterogeneity in the division rule between the epidermis and the inner tissues led to the more efficient flattening process and that a stress-based division rule was the most efficient to produce flat structure.

This analysis is part of the manuscript currently under review and available online in a preprint version [23].

Influence of mechanical stress anisotropy on the orientation of cell divisions in animal tissues. Tight regulation of cell division orientation is fundamental for tissue development. Recently, a great effort has been put into biophysical understanding of the long-axis division rules (Hertwig's rule for animal cells, Errera's rule for plant cells) and the systematic deviations from these rules observed in vivo. In both plants and animals, such deviations often correlate with anisotropic tensions within the tissue. To what extend these deviations are regulated or simply the result of stochasticity?

To address these questions in animal cells, we modeled theoretically and numerically cell division as an active process in a many-body system. We showed that under isotropic tension a cell's long axis emerges as the energetically optimal division orientation and that anisotropic stresses biased the energetics, leading to systematic deviations form Hertwig's rule. These deviations, as reported experimentally, are correlated to the main direction of stress anisotropy.

Our model successfully predicted division orientation distributions within two experimental systems: epidermis of the ascidian Phallusia mammillata (where deviations from Hertwig's rule have been so far eluding explanation) and of the pupal epithelium of the dorsal thorax of D. melanogaster.

This work was part of the Digem project and was presented in two international conferences: Mechanobiology and Physics of Life (Lyon) and Developmental and Cell Biology of the Future (Paris); and at the yearly InriaBio meeting in Lyon. A paper is currently under review and a preprint is available on bioRxiv [22].

Influence of water fluxes on plant morphogenesis. Since pressure appears as the “engine” behind growth-related deformation in Plants, its regulation by cells is a major control mechanism of morphogenesis. We developed 2D computational models to investigate the morphological consequences of the interplay between cell expansion, water fluxes between cells and tissue mechanics. This interdisciplinary work, between experiments and modeling, address the influence of turgor pressure heterogeneities on relative growth rate between cells. We showed that the coupling between fluxes and mechanics allows to predict observed morphological heterogeneities without any ad hoc assumption.

This work was part of the Agropolis fundation project MecaFruit3D and Arezki Boudaoud's ERC PhyMorph. It resulted in a publication in PLoS Computational Biology [7] that introduces the theoretical model and studies some of its properties. Another paper [27] presents the comparisons with experiments and is currently under review.

Development of de novo finite element (F.E.) library dedicated to mechanical simulations performed on complex cellularized structures. In order to compute accurately the mechanical stress field borne by multicellular pressurized 3D structures (such as plant tissues), we needed to update our existing library (tissueMeca, see [24]). Three key aspects had to be upgraded (i) the control over the F.E. solver, (ii) tracking of its precision and (iii) integration of the F.E. framework with the rest of our pipeline.

To that end, we decided to switch from Sofa to FEniCS (https://fenicsproject.org/) as the core F.E. framework used within our simulation pipeline. We started to develop a dedicated library, called CellFem, to solve F.E. problems on PropertyTopomesh instances (the data structure we developed within the team to describe multicellular plant tissues). CellFem provides a high level API to define and resolve variational problems to solve linear as well as non-linear elastic and elasto-plastic problems related to plant tissue morphogenesis.

In parallel, we also started the development of a meshing library (based on the GMSH library (http://gmsh.info/)) called CellMesh and dedicated to the triangulation of simplicial complexes. This work is currently under development.