Yin S et al. (2022), Three-dimensional chiral morphodynamics of chem...

ECB-ART-51140

Proc Natl Acad Sci U S A 2022 Dec 06;11949:e2206159119. doi: 10.1073/pnas.2206159119.

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Three-dimensional chiral morphodynamics of chemomechanical active shells.

Yin S , Li B , Feng XQ .

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Morphogenesis of active shells such as cells is a fundamental chemomechanical process that often exhibits three-dimensional (3D) large deformations and chemical pattern dynamics simultaneously. Here, we establish a chemomechanical active shell theory accounting for mechanical feedback and biochemical regulation to investigate the symmetry-breaking and 3D chiral morphodynamics emerging in the cell cortex. The active bending and stretching of the elastic shells are regulated by biochemical signals like actomyosin and RhoA, which, in turn, exert mechanical feedback on the biochemical events via deformation-dependent diffusion and inhibition. We show that active deformations can trigger chemomechanical bifurcations, yielding pulse spiral waves and global oscillations, which, with increasing mechanical feedback, give way to traveling or standing waves subsequently. Mechanical feedback is also found to contribute to stabilizing the polarity of emerging patterns, thus ensuring robust morphogenesis. Our results reproduce and unravel the experimentally observed solitary and multiple spiral patterns, which initiate asymmetric cleavage in Xenopus and starfish embryogenesis. This study underscores the crucial roles of mechanical feedback in cell development and also suggests a chemomechanical framework allowing for 3D large deformation and chemical signaling to explore complex morphogenesis in living shell-like structures.

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	Fig. 1. Active shell model of a cell cortex. (A) A typical chemomechanical spiral pattern in the Xenopus embryo. Adapted from ref. 12 with permission. (B) Torsion deformation of the cell cortex induced by spiral chemical concentration. (C) The active contraction of the cell cortex stems from the motion of its actomyosin network, which is regulated by the activated RhoA concentration (i.e., activity). (D) A three-component feedback system composed of RhoA, actomyosin, and cortex contraction shows the chemomechanical interplay
	Fig. 2. Schematics of chemical regulations on mechanical active deformation. (A) In-membrane isotropic stretching and (B) curvature deviation, as a function of local actomyosin activity. (C) Schematic of the mechanical feedback φ(H) as a function of the relative mean curvature changes ΔΓ(H/H*−1).
	Fig. 3. Growth rate as a function of the mode number l for different stationary RhoA activities (A) cR=0.15 and (B) cR=0.5. Colored solid lines represent the real part of the largest growth rate, and dashed lines represent the complex part. (C) Phase diagram as a function of stationary RhoA activity cR* and dimensionless strength of negative mechanical feedback k˜M, which is obtained from linear stability analysis. Four regions including (I) pulsatory spiral wave (pink), (II) global relaxation oscillation (orange), (III) traveling and standing waves (blue), and (IV) stable region (white) can be distinguished. The red lines represent a chemical-induced pitchfork bifurcation, while the blue lines represent the mechanical feedback-induced pitchfork bifurcation. The insets show numerical simulations of the evolution of these patterns on deforming shells. Initial conditions are assumed as locally concentrated RhoA near the north pole. Parameters used in the simulations are (I) cR=0.15, k˜M=0.1 ; (II) cR=0.5, k˜M=0; (III) cR=0.15, k˜M=0.2 (traveling wave) and cR=0.5, k˜M=5 (standing wave); (IV) cR=0.8, k˜M=0. (D) RhoA and actomyosin concentrations change with time when global oscillation occurs. Parameters are cR=0.5, k˜M=0, and ε˜=0.02. (E) The oscillation period T as a function of the parameter ε˜, showing the scaling law T~ε˜−0.88.
	Fig. 4. 3D large deformation of the active elastic shell under the regulation of biochemical and mechanical interplay with negative mechanical feedback strength (A) k˜M=0.1 and (B) k˜M=0.2. In (B), solitary spiral waves can transit to traveling waves. The upper rows in each panel represent RhoA activity cR in the intact deforming shell, the middle rows show the front and back of spiral and traveling waves, and the bottom rows represent the normal displacement un.
	Fig. 5. Multiple spiral waves of RhoA signaling in active shells are accompanied by large deformations. (A) Experimental observation in the cellular cortex of the starfish oocytes. Adapted from ref. 55 with permission. (B) Numerical simulations.

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