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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.
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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).
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Fig. 3. Growth rate as a function of the mode number l for different stationary RhoA activities (A) c∗R=0.15
and (B) c∗R=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 c∗R
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) c∗R=0.15, k˜M=0.1; (II) c∗R=0.5, k˜M=0; (III) c∗R=0.15, k˜M=0.2 (traveling wave) and c∗R=0.5, k˜M=5
(standing wave); (IV) c∗R=0.8, k˜M=0. (D) RhoA and actomyosin concentrations change with time when global oscillation occurs. Parameters are c∗R=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.
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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.
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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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Fig. S1. Typical refined FitzHugh-Nagumo equation for purely biochemical reaction system. (A–C) The cR − cA phase diagrams and (D–F) cR time course of excitable, oscillatory and stable state. Fixed parameters are given as k˜R = 13, k˜A = 1.5, α1 = 0.25, α2 = 0.85, β = 1, β0 = 0.1, ε˜ = 0.02.
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Fig. S2. Four typical dispersion relations. Parameters are given as: (A) c∗R = 0.1, k˜M = 2; (B) c∗R = 0.1, k˜M = 6; (C) c∗R = 0.5, k˜M = 2; (D) c∗R = 0.5, k˜M = 6.
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Fig. S3. Standing waves. (A) RhoA activity and (B) normal displacement field. Parameters are c∗R = 0.5 and k˜M = 5.
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Fig. S4. (Color online) With initial RhoA and actomyosin acitivities concentrated on two caps near the poles, the active shell oscillates between dumbbell shape and spheroidicity. (A) RhoA activity and (B) normal displacement field. Mechanical feedback k˜M = 0.2.
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Fig. S5. (Color online) Mechanical feedback rectifies the nematic axis with different initial imperfections of RhoA activity. (A) Mechanical feedback-driven transitions from spiral waves to traveling waves with ε˜ = 0.01. Different initial imperfections of RhoA activity are discussed with θA = π/3, π/3 + 0.1π, π/3 + 0.2π. (B) Mechanical feedback-driven transitions from crescent-shape patterns to traveling waves with ε˜ = 0.05. Crescent-shape patterns emerge and disappear with k˜M = 0 (upper row) and k˜M = 0.1 (middle row). Large enough mechanical feedback can trigger traveling waves with k˜M = 1 (bottom row) and then the traveling waves disappear because of intrinsic features of the
biochemical reaction equation. Other parameters are fixed as c∗R = 0.15, D˜ = 0.1, α˜1 = 0.25, k˜R = 8, k˜A = 1.5, h˜ = 0.15, Ac = 1, Ai = 2, k˜H = 20.
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Fig. S6. 3D dynamical morphological evolution of viscoelastic active shells. The viscous timescales τv = 1e3, 1e4, and 1e6 with different mechanical feedback kM = 0.1 and kM = 0.2 (or kM = 1) are compared. Other parameters are fixed as c∗R = 0.15, D˜ = 0.1, ε˜ = 0.01, α˜1 = 0.25, k˜R = 8, k˜A = 1.5, h˜ = 0.15, Ac = 1, Ai = 2,
and k˜H = 20
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