Robustness of force and stress inference in an epithelial tissue

During morphogenesis, the shape of a tissue emerges from collective cellular behaviors, which are in part regulated by mechanical and biochemical interactions between cells. Quantification of force and stress is therefore necessary to analyze the mechanisms controlling tissue morphogenesis. Recently, a mechanical measurement method based on force inference from cell shapes and connectivity has been developed. It is non-invasive, and can provide space-time maps of force and stress within an epithelial tissue, up to prefactors. We previously performed a comparative study of three force-inference methods, which differ in their approach of treating indefiniteness in an inverse problem between cell shapes and forces. In the present study, to further validate and compare the three force inference methods, we tested their robustness by measuring temporal fluctuation of estimated forces. Quantitative data of cell-level dynamics in a developing tissue suggests that variation of forces and stress will remain small within a short period of time ($\sim$minutes). Here, we showed that cell-junction tensions and global stress inferred by the Bayesian force inference method varied less with time than those inferred by the method that estimates only tension. In contrast, the amplitude of temporal fluctuations of estimated cell pressures differs less between different methods. Altogether, the present study strengthens the validity and robustness of the Bayesian force-inference method.Comment: 4 pages, 4 figure

A migrating epithelial monolayer flows like a Maxwell viscoelastic liquid

We perform a bidimensional Stokes experiment in an active cellular material: an autonomously migrating monolayer of Madin-Darby Canine Kidney (MDCK) epithelial cells flows around a circular obstacle within a long and narrow channel, involving an interplay between cell shape changes and neighbour rearrangements. Based on image analysis of tissue flow and coarse-grained cell anisotropy, we determine the tissue strain rate, cell deformation and rearrangement rate fields, which are spatially heterogeneous. We find that the cell deformation and rearrangement rate fields correlate strongly, which is compatible with a Maxwell viscoelastic liquid behaviour (and not with a Kelvin-Voigt viscoelastic solid behaviour). The value of the associated relaxation time is measured as $\tau = 70 \pm 15$~min, is observed to be independent of obstacle size and division rate, and is increased by inhibiting myosin activity. In this experiment, the monolayer behaves as a flowing material with a Weissenberg number close to one which shows that both elastic and viscous effects can have comparable contributions in the process of collective cell migration.Comment: 17 pages, 15 figure

Fast determination of coarse grained cell anisotropy and size in epithelial tissue images using Fourier transform

Mechanical strain and stress play a major role in biological processes such as wound healing or morphogenesis. To assess this role quantitatively, fixed or live images of tissues are acquired at a cellular precision in large fields of views. To exploit these data, large numbers of cells have to be analyzed to extract cell shape anisotropy and cell size. Most frequently, this is performed through detailed individual cell contour determination, using so-called segmentation computer programs, complemented if necessary by manual detection and error corrections. However, a coarse grained and faster technique can be recommended in at least three situations. First, when detailed information on individual cell contours is not required, for instance in studies which require only coarse-grained average information on cell anisotropy. Second, as an exploratory step to determine whether full segmentation can be potentially useful. Third, when segmentation is too difficult, for instance due to poor image quality or too large a cell number. We developed a user-friendly, Fourier transform-based image analysis pipeline. It is fast (typically $10^4$ cells per minute with a current laptop computer) and suitable for time, space or ensemble averages. We validate it on one set of artificial images and on two sets of fully segmented images, one from a Drosophila pupa and the other from a chicken embryo; the pipeline results are robust. Perspectives include \textit{in vitro} tissues, non-biological cellular patterns such as foams, and $xyz$ stacks.Comment: 13 pages; 9 figure

Are large perimeter-minimizing two-dimensional clusters of equal-area bubbles hexagonal or circular?

A computer study of clusters of up to 200,000 equal-area bubbles shows for the first time that rounding conjectured optimal hexagonal planar soap bubble clusters reduces perimeter.Comment: 10 pages, 9 figure

Screening in two-dimensional foams

Using the Surface Evolver software, we perform numerical simulations of point-like deformations in a two-dimensional foam. We study perturbations which are infinitesimal or finite, isotropic or anisotropic, and we either conserve or do not conserve the number of bubbles. We measure the displacement fields around the perturbation. Changes in pressure decrease exponentially with the distance to perturbation, indicating a screening over a few bubble diameters

Quasicrystalline three-dimensional foams

We present a numerical study of quasiperiodic foams, in which the bubbles are generated as duals of quasiperiodic Frank-Kasper phases. These foams are investigated as potential candidates to the celebrated Kelvin problem for the partition of three-dimensional space with equal volume bubbles and minimal surface area. Interestingly, one of the computed structures falls close (but still slightly above) the best known Weaire-Phelan periodic candidate. This gives additional clues to understanding the main geometrical ingredients driving the Kelvin problem

Cell adhesion and cortex contractility determine cell patterning in the Drosophila retina

Hayashi and Carthew (Nature 431 [2004], 647) have shown that the packing of cone cells in the Drosophila retina resembles soap bubble packing, and that changing E- and N-cadherin expression can change this packing, as well as cell shape. The analogy with bubbles suggests that cell packing is driven by surface minimization. We find that this assumption is insufficient to model the experimentally observed shapes and packing of the cells based on their cadherin expression. We then consider a model in which adhesion leads to a surface increase, balanced by cell cortex contraction. Using the experimentally observed distributions of E- and N-cadherin, we simulate the packing and cell shapes in the wildtype eye. Furthermore, by changing only the corresponding parameters, this model can describe the mutants with different numbers of cells, or changes in cadherin expression.Comment: revised manuscript; 8 pages, 6 figures; supplementary information not include

Simulations of viscous shape relaxation in shuffled foam clusters

We simulate the shape relaxation of foam clusters and compare them with the time exponential expected for Newtonian fluid. Using two-dimensional Potts Model simulations, we artificially create holes in a foam cluster and shuffle it by applying shear strain cycles. We reproduce the experimentally observed time exponential relaxation of cavity shapes in the foam as a function of the number of strain steps. The cavity rounding up results from local rearrangement of bubbles, due to the conjunction of both a large applied strain and local bubble wall fluctuations

Deformation of grain boundaries in polar ice

The ice microstructure (grain boundaries) is a key feature used to study ice evolution and to investigate past climatic changes. We studied a deep ice core, in Dome Concordia, Antarctica, which records past mechanical deformations. We measured a "texture tensor" which characterizes the pattern geometry and reveals local heterogeneities of deformation along the core. These results question key assumptions of the current models used for dating