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. 2018 Dec 10;145(24):dev167387.
doi: 10.1242/dev.167387.

A Simplified Mechanism for Anisotropic Constriction in Drosophila Mesoderm

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Free PMC article

A Simplified Mechanism for Anisotropic Constriction in Drosophila Mesoderm

Konstantin Doubrovinski et al. Development. .
Free PMC article

Abstract

Understanding how forces and material properties give rise to tissue shapes is a fundamental issue in developmental biology. Although Drosophila gastrulation is a well-used system for investigating tissue morphogenesis, a consensus mechanical model that explains all the key features of this process does not exist. One key feature of Drosophila gastrulation is its anisotropy: the mesoderm constricts much more along one axis than along the other. Previous explanations have involved graded stress, anisotropic stresses or material properties, or mechanosensitive feedback. Here, we show that these mechanisms are not required to explain the anisotropy of constriction. Instead, constriction can be anisotropic if only two conditions are met: the tissue is elastic, as was demonstrated in our recent study; and the contractile domain is asymmetric. This conclusion is general and does not depend on the values of model parameters. Our model can explain results from classical tissue-grafting experiments and from more-recent laser ablation studies. Furthermore, our model may provide alternative explanations for experiments in other developmental systems, including C. elegans and zebrafish.

Keywords: Anisotropy; Drosophila; Gastrulation.

Conflict of interest statement

Competing interestsThe authors declare no competing or financial interests.

Figures

Fig. 1.
Fig. 1.
Model for gastrulation. Schematic of tissue dynamics with illustrations of the model underneath. Left: cross-section through Drosophila embryo. The embryo consists of a single layer of epithelial cells surrounding a central unstructured yolk sack. Apical surfaces face outward, basal surfaces face inward. Middle: the contractile domain (mesoderm) comprises a rectangular patch of cells some 20 cells wide and 80 cells long. Elastic elements are illustrated as springs and active forces are represented by ‘motors’ (M). Both are attached to the two adjacent ‘beads’ in parallel. Beads are assumed to feel viscous drag from the ambient environment when moving. It is assumed that forces exerted by motors are constant in time and space. Right: the contractile region shrinks anisotropically, contracting much more strongly along the width than along the length.
Fig. 2.
Fig. 2.
Simulations of the model. (A) Schematic showing the geometry of the problem and the quantities describing the resulting deformation. After the deformation sets in, a point located at spatial position r displaces to a new spatial position r+u(r), where u has components ux and uy. (B) An example of a final mechanical equilibrium state. Parameters are E=1, σ=0.2 and φ0=0.5 (see main text for notation); entire domain size is 50×50; contractile domain is 10×2. For readability, only the noticeably deformed middle portion of the domain is shown. Simulation was carried out using finite differences. (C) Same simulation as in B showing the distribution of the deformation as a field of displacement vectors. (D) Asymptotic (equilibrium) aspect ratio of the contractile domain as a function of Poisson's ratio (x-axis) and contractile active stress (three different curves). All parameters (except the ones that are varied) are as in B,C. Black, red and green curves correspond to φ0=0.5, φ0=0.75 and φ0=1, respectively. (E) Equilibrium length-to-width ratio (Lx/Ly) of the contractile domain as a function of the initial ratio at time zero for three values of active contractile stress. All parameters, except those being varied are as in B. (F) Symmetric domains contract isotropically. All parameters except domain size are as in B. Contractile domain size is 4×4.
Fig. 3.
Fig. 3.
Simulation of an ablation experiment. (A) Final state following a simulated laser ablation. A set of vertices in the center of the contractile domain are removed (‘ablated’) after a finite deformation sets in. Parameters and the details of implementation are given in the Materials and Methods. (B) Schematic intuitively explaining the outcome of the simulation in A (see main text for the details).

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