Activation and synchronization of the oscillatory morphodynamics in multicellular monolayer

Significance

Oscillatory morphodynamics of collective cells is of fundamental importance for concerting cellular events and tissue-level developments in many living systems. We demonstrate that the collective cell oscillations in an epithelium-like monolayer are attributed to a chemomechanical Hopf bifurcation tailored by external forces and boundary physics and geometry. Our findings not only offer mechanistic insights into the synchronization and activation of supracellular oscillations in Drosophila embryogenesis but also help uncover collective events in other scenarios, including wound healing and cancer invasion.

Abstract

Oscillatory morphodynamics provides necessary mechanical cues for many multicellular processes. Owing to their collective nature, these processes require robustly coordinated dynamics of individual cells, which are often separated too distantly to communicate with each other through biomaterial transportation. Although it is known that the mechanical balance generally plays a significant role in the systems’ morphologies, it remains elusive whether and how the mechanical components may contribute to the systems’ collective morphodynamics. Here, we study the collective oscillations in the Drosophila amnioserosa tissue to elucidate the regulatory roles of the mechanical components. We identify that the tensile stress is the key activator that switches the collective oscillations on and off. This regulatory role is shown analytically using the Hopf bifurcation theory. We find that the physical properties of the tissue boundary are directly responsible for synchronizing the oscillatory intensity and polarity of all inner cells and for orchestrating the spatial oscillation patterns inthe tissue.

Morphodynamics describes how a subject’s form changes over time. In living systems, the morphodynamic changes are both the effect and the cause of coordinated biochemical and biophysical processes. On the one hand, a system’s morphological changes result from intracellular force generation and intercellular force transmission through sequences of biological events. On the other hand, the morphodynamic changes provide various mechanical and physical cues that are critical for the morphogenesis of multicellular tissues (1, 2) and the development of organisms (3–5). Owing to the collective nature of many biological processes, it is of profound interest to understand the principle underlying the morphodynamics of a living thing that is built from individual yet coherent cells (6–8).

Oscillatory morphodynamics is an important category of collective morphodynamic phenomena that exists in many biological systems, including vertebrate segmentation (9), mesoderm invagination (10), and germband extension (11). These morphological oscillations are rooted in the active contraction of the actomyosin cytoskeleton in individual cells (12–15) and are often coupled with other intracellular biochemical signaling pathways (9, 13, 14). During the oscillatory process, the actomyosin cytoskeleton gets activated and forms an apical network beneath the membrane, which further facilitates the formation of cell–cell junctions that allow force transmission from a cell to its neighbors (16). These observations have motivated many modeling efforts that generally fall into two categories: The coupling-based models attempt to couple membrane tension with the actomyosin regulation pathway to reproduce the oscillation of single or multiple cells (14, 17); the input-based ratchet models treat the cortical actin–myosin cytoskeleton or the supracellular actin cable as a programed machine that drives the cells’ and/or the tissue’s oscillatory behaviors in the contraction–relaxation cycles (10, 13, 18). These progresses have successfully connected the mechanical concepts to the cell’s biochemical pathways. However, it is still elusive what the explicit roles of mechanics are in regulating the morphodynamics of tissues.

In this work, we use the Drosophila amnioserosa as a model system to address the above question. We propose and implement a chemomechanically coupled dynamic vertex model for an ensemble of cells confined within a 2D elliptical space. We show that a time-delayed negative feedback embedded in the chemomechanical coupling is capable of generating the autonomous oscillations without the need of material exchanging, which has been experimentally observed in Drosophila embryogenesis (19, 20). We computationally show and analytically derive that the tensile stress exerting on the boundary triggers a Hopf bifurcation in amnioserosa’s morphodynamics, which provides a critical and robust gating mechanism to switch the collective cell oscillations on and off. Furthermore, we discover that the mechanical and morphological properties of the amnioserosa boundary not only are important for maintaining the integrity of the tissue’s shape, but also are essential for orchestrating and synchronizing the oscillatory patterns across the length scale of hundreds of cells. These findings unveil the multifaceted roles of mechanics both as an activator and as a synchronizer in regulating the oscillatory morphodynamics at the tissue level.

Chemomechanical Model

The amnioserosa in the Drosophila embryo is an eye-shaped epithelium (Fig. 1A) containing several hundred cells confined by the lateral epidermis (21). We model the amnioserosa as a cell monolayer with an elliptic boundary, within which cells are approximated by polygons (Fig. 1B). Based on this vertex description, the mechanical energy of the monolayer can be described as (22, 23)

$$U=\sum _J\frac{1}{2}{k}_{\text{m}}{m}_J{L}_J^2+\sum _J\frac{1}{2}{K}_{\text{a}}{(A_J-A_0)}^2+\sum _{⟨i,j⟩}\mathrm{\Lambda }{l}_{ij},$$ [1]

where $L_J$ and $A_J$ are the perimeter and area of the $J$th cell, respectively; $m_J$ measures the myosin activity in cell $J$ and $k_{\text{m}}$ is the elastic coefficient of actin cortex; $A_0$ is the preferred area and $K_{\text{a}}$ denotes the areal stiffness of a cell; $l_{ij}$ is the edge length between vertices $i$ and $j$; and $\mathrm{\Lambda }$ represents the interfacial tension between two cells, such as cell–cell adhesion and cortical tension (24). The morphological changes of the monolayer can be simulated using the mechanical balance at each vertex ${𝐫}_i$ (Fig. 1C): $\eta \mathrm{d}{𝐫}_i/\mathrm{d}t=-\partial U/\partial {𝐫}_i$, where $\eta$ is the friction coefficient (23, 25). Based on the existing experimental observations (18, 21), we do not consider cell rearrangements.

Fig. 1.

Vertex-based chemomechanical coupling model. (A) Schematic of Drosophila dorsal closure. The central eye-shaped region is amnioserosa, surrounded by the lateral epidermis. (B) Geometric description of amnioserosa cells by connected polygons within an elliptic domain. (C) Forces acting at a vertex in the cell monolayer. (D) Interactive regulation between myosin activity and cell deformation. (E) Oscillations of area and myosin activity of a typical cell in the amnioserosa. (F) Spectrogram of area of the cells in the amnioserosa. (G) Scatter diagram of the oscillation phases ${\phi }_{\text{M}}$ and ${\phi }_{\text{A}}$ of all cells in the amnioserosa.

The first term on the right-hand side of Eq. 1 accounts for the cell contractility that is often attributed to the activity of myosin II. Through mechanosensitive proteins (15, 26–28), myosin molecules can be activated and/or recruited upon mechanical perturbations (e.g., stretching) with several minutes delay (29–31). Meanwhile, the increased myosin activity promotes cell contraction that can hinder the stretch. Therefore, the cell stretch and the myosin activity constitute a time-delayed negative feedback (Fig. 1D). We use a Hill function (32, 33) to quantify this chemomechanical negative feedback,

$$\frac{\mathrm{d}{m}_J}{\mathrm{d}t}=\alpha \frac{{[{{\lambda }_{\text{L}}^{(J)}|}_{(t-\tau )}]}^n}{K_{\text{L}}+{[{{\lambda }_{\text{L}}^{(J)}|}_{(t-\tau )}]}^n}-\beta m_J,$$ [2]

where $\alpha$ is the activation coefficient and $\beta$ is the deactivation rate; $K_{\text{L}}$ and $n$ are the apparent dissociation constant and the Hill coefficient, respectively; ${\lambda }_{\text{L}}^{(J)}=L_J/L_{\text{s}}$ characterizes the $J$th cell’s peripheral deformation with $L_{\text{s}}$ being the perimeter of a cell at rest; and $\tau$ denotes the time delay between sensing morphological changes and activating myosin II. The Hill function in Eq. 2 summarizes the nonlinear process of myosin activation. In fact, Hill functions have been widely used in describing switch-like biochemical reactions (34).

In this study, we apply the above model to investigate the monolayer’s morphodynamics under fixed, moving, and elastic boundary conditions. For the sake of generality, we normalize the equations and introduce two key parameters (Simulation of the Collective Cell Oscillations in a Monolayer): the dimensionless areal stiffness ${\stackrel{\sim }{K}}_{\text{a}}=K_{\text{a}}{A}_0\tau /\eta$ and the dimensionless contractile modulus ${\stackrel{\sim }{K}}_{\text{c}}=K_{\text{c}}\tau /\eta$, where $K_{\text{c}}=k_{\text{m}}\alpha /[(K_{\text{L}}+1)\beta ]$ defines the contractility of a cell at rest. In our calculations, we take ${\stackrel{\sim }{K}}_{\text{a}}=10.0$, ${\stackrel{\sim }{K}}_{\text{c}}=1.0$, $\mathrm{\Lambda }=0$, $\beta \tau =1.0$, $K_{\text{L}}=1.0$, and $n=8$, unless stated otherwise. Here, we set the interfacial tension $\mathrm{\Lambda }=0$ for simplicity. In the case of $\mathrm{\Lambda }\ne 0$, similar conclusions can be drawn (Fig. S1).

Fig. S1.

Simulation of collective cell oscillations in Drosophila amnioserosa. (A) Spatial distribution of the oscillation phases ${\phi }_{\text{A}}$ of cell area. (B) Histogram of the alignment orientation ${\theta }_{\text{align}}^{(J)}$ and the oscillatory orientation ${\theta }_{\text{aniso}}^{(J)}$. (C–G) Collective cell oscillations in a confined elliptic monolayer when the interfacial tension $\mathrm{\Lambda }\ne 0$. Parameters: ${\stackrel{\sim }{K}}_{\text{a}}=10$, ${\stackrel{\sim }{K}}_{\text{c}}=1.0$, $\stackrel{\sim }{\mathrm{\Lambda }}=-0.5$, $\stackrel{\sim }{\beta }=1.0$, $K_{\text{L}}=1.0$, and $n=8$. (C) Evolution of the area and the myosin activity of an arbitrary cell in the monolayer. (D) Scatter diagram of the oscillation phases of myosin activity ${\phi }_{\text{M}}$ and cell area ${\phi }_{\text{A}}$ of the cells. (E) Spectrogram of area of the cells in the amnioserosa. (F) Spatial distribution of the oscillation phases of cell area ${\phi }_{\text{A}}$. (G) Histogram of the alignment orientation ${\theta }_{\text{align}}^{(J)}$ and the oscillatory orientation ${\theta }_{\text{aniso}}^{(J)}$.

In this Supporting Information, we provide details of the theoretical derivations and numerical simulations to supplement the main text. Additional data are also included to support the results and conclusions given in the main text.

Simulation of the Collective Cell Oscillations in a Monolayer

Normalization of the Dynamic Equations.

We integrate the time-delayed negative feedback between myosin activity and cell deformation into the dynamic vertex model to investigate collective cell oscillations in a monolayer. From $\eta \mathrm{d}{𝐫}_i/\mathrm{d}t=-\partial U/\partial {𝐫}_i$ given in the main text, we can derive the dynamic evolution of vertices as

$$\begin{array}{cc}\eta \frac{\mathrm{d}{𝐫}_i}{\mathrm{d}t}=\hfill & -\sum _{J\in C_i}{k}_{\text{m}}{m}_J{L}_J(\frac{{𝐫}_i-{𝐫}_{j1}}{|{𝐫}_i-{𝐫}_{j1}|}+\frac{{𝐫}_i-{𝐫}_{j2}}{|{𝐫}_i-{𝐫}_{j2}|})\hfill \\ & -\sum _{J\in C_i}\frac{1}{2}{K}_{\text{a}}(A_J-A_0)[𝐤\times ({𝐫}_{j2}-{𝐫}_{j1})]\hfill \\ & -\sum _{j\in V_i}\mathrm{\Lambda }\frac{{𝐫}_i-{𝐫}_j}{|{𝐫}_i-{𝐫}_j|},\hfill \end{array}$$ [S1]

where $𝐤$ denotes a unit vector normal to the cell monolayer. The summation ${\sum }_{J\in C_i}$ computes over all cells sharing vertex $i$, and $j_1$ and $j_2$ are the neighboring vertices of vertex $i$ in cell $J$, whereas the summation ${\sum }_{j\in V_i}$ is made over all neighboring vertices of vertex $i$.

We first normalize the dynamic equations for the vertices ${𝐫}_i$ (i.e., Eq. S1) and the myosin activity $m_J$ (i.e., Eq. 2 in the main text), using $\sqrt{A_0}$ as the length scale and $\tau$ as the timescale, respectively. The normalized coupled chemomechanical equations can be written as

$$\begin{array}{cc}\frac{\mathrm{d}{\stackrel{\sim }{𝐫}}_i}{\mathrm{d}\stackrel{\sim }{t}}=\hfill & -\sum _{J\in C_i}{\stackrel{\sim }{k}}_{\text{m}}{\stackrel{\sim }{m}}_J{\stackrel{\sim }{L}}_J(\frac{{\stackrel{\sim }{𝐫}}_i-{\stackrel{\sim }{𝐫}}_{j1}}{|{\stackrel{\sim }{𝐫}}_i-{\stackrel{\sim }{𝐫}}_{j1}|}+\frac{{\stackrel{\sim }{𝐫}}_i-{\stackrel{\sim }{𝐫}}_{j2}}{|{\stackrel{\sim }{𝐫}}_i-{\stackrel{\sim }{𝐫}}_{j2}|})\hfill \\ & -\sum _{J\in C_i}\frac{1}{2}{\stackrel{\sim }{K}}_{\text{a}}({\stackrel{\sim }{A}}_J-1)[𝐤\times ({\stackrel{\sim }{𝐫}}_{j2}-{\stackrel{\sim }{𝐫}}_{j1})]\hfill \\ & -\sum _{j\in V_i}\stackrel{\sim }{\mathrm{\Lambda }}\frac{{\stackrel{\sim }{𝐫}}_i-{\stackrel{\sim }{𝐫}}_j}{|{\stackrel{\sim }{𝐫}}_i-{\stackrel{\sim }{𝐫}}_j|}\hfill \end{array}$$ [S2]

and

$$\frac{\mathrm{d}{\stackrel{\sim }{m}}_J}{\mathrm{d}\stackrel{\sim }{t}}=\stackrel{\sim }{\alpha }\frac{{[{{\stackrel{\sim }{L}}_J|}_{(\stackrel{\sim }{t}-1)}]}^n}{{\stackrel{\sim }{K}}_{\text{L}}+{[{{\stackrel{\sim }{L}}_J|}_{(\stackrel{\sim }{t}-1)}]}^n}-\stackrel{\sim }{\beta }{\stackrel{\sim }{m}}_J.$$ [S3]

The dimensionless parameters are defined as

$$\stackrel{\sim }{t}=\frac{t}{\tau },\hspace{1em}{\stackrel{\sim }{m}}_J=\frac{m_J}{m_{\text{s}}},\hspace{1em}\stackrel{\sim }{\alpha }=\frac{\alpha \tau }{m_{\text{s}}},\hspace{1em}\stackrel{\sim }{\beta }=\beta \tau ,{\stackrel{\sim }{K}}_{\text{L}}=K_{\text{L}}{\left(\frac{L_{\text{s}}}{\sqrt{A_0}}\right)}^n=K_L{\stackrel{\sim }{L}}_{\text{s}}^n,\hspace{1em}{\stackrel{\sim }{L}}_J=\frac{L_J}{\sqrt{A_0}},{\stackrel{\sim }{A}}_J=\frac{A_J}{A_0},\hspace{1em}{\stackrel{\sim }{𝐫}}_i=\frac{{𝐫}_i}{\sqrt{A_0}},\hspace{1em}{\stackrel{\sim }{k}}_{\text{m}}=\frac{k_{\text{m}}{m}_{\text{s}}\tau }{\eta },{\stackrel{\sim }{K}}_{\text{a}}=\frac{K_{\text{a}}{A}_0\tau }{\eta },\hspace{1em}\stackrel{\sim }{\mathrm{\Lambda }}=\frac{\mathrm{\Lambda }\tau }{\eta \sqrt{A_0}},$$ [S4]

where $m_{\text{s}}=\alpha /\left[\beta \left(K_{\text{L}}+1\right)\right]$ and $L_{\text{s}}$ are the myosin activity and the perimeter of a cell at rest, respectively. Based on the reference myosin activity $m_{\text{s}}$, we define the contractile modulus of a cell as $K_{\text{c}}\stackrel{\mathrm{def}}{=}{k}_{\text{m}}{m}_{\text{s}}=k_{\text{m}}\alpha /\left[\beta \left(K_{\text{L}}+1\right)\right]$, with its dimensionless form given by ${\stackrel{\sim }{K}}_{\text{c}}=K_{\text{c}}\tau /\eta ={\stackrel{\sim }{k}}_{\text{m}}$. Note that $\stackrel{\sim }{\alpha }$, $\stackrel{\sim }{\beta }$, and $K_{\text{L}}$ are coupled as $\stackrel{\sim }{\alpha }=\stackrel{\sim }{\beta }(K_{\text{L}}+1)$, and thus only two of them are independent in this normalization scheme. Consequently, we list the independent dimensionless parameters as ${\stackrel{\sim }{K}}_{\text{a}}$, ${\stackrel{\sim }{K}}_{\text{c}}$, $\stackrel{\sim }{\mathrm{\Lambda }}$, $\stackrel{\sim }{\beta }$, $K_{\text{L}}$, and $n$. Eqs. S2 and S3 describe the dynamic evolution of a cell monolayer with chemomechanical coupling mechanism and can be numerically solved via the finite difference method.

Calculation of the Reference Cell Perimeter ${\text{𝑳}}_{\text{𝒔}}$.

According to Euler’s formula, the most probable cell shape is hexagonal in a 2D dynamic vertex model of a cell monolayer (22). Therefore, $L_{\text{s}}$ can be obtained analytically by solving the equilibrium state of a regular hexagonal cell. Let us consider a regular hexagonal cell unit undergoing isotropic contraction/stretching. In this scenario, the dynamic evolution of cell vertices, Eq. S1, reduces to

$$\eta \frac{\mathrm{d}R}{\mathrm{d}t}{𝐞}_r=-k_{\text{m}}m(6R){𝐞}_r-\frac{1}{2}{K}_{\mathrm{a}}(A-A_0)\sqrt{3}R{𝐞}_r-\frac{1}{2}\mathrm{\Lambda }{𝐞}_r,$$ [S5]

where $R$ is the radius (i.e., the edge length) of the regular hexagonal cell, and ${𝐞}_r$ is the unit vector pointing from the center of the hexagon to any of its vertices. It is worth noting that there is a factor $1/2$ in the interfacial tension term in Eq. S5 (i.e., $-1/2\mathrm{\Lambda }{𝐞}_r$). This is because each edge in the monolayer is shared by two neighboring cells and, therefore, the force induced by the interfacial tension $-\mathrm{\Lambda }{𝐞}_r$ is divided by $2$ when considering the force balance of a cell unit.

Consequently, the regular hexagonal cell’s coupled chemomechanical dynamics read

$$\eta \frac{\mathrm{d}R}{\mathrm{d}t}=-6k_{\text{m}}mR-\frac{\sqrt{3}}{2}{K}_{\mathrm{a}}(A-A_0)R-\frac{1}{2}\mathrm{\Lambda }$$ [S6]

and

$$\frac{\mathrm{d}m}{\mathrm{d}t}=\alpha \frac{{\left[{{\lambda }_{\text{L}}|}_{(t-\tau )}\right]}^n}{K_{\text{L}}+{\left[{{\lambda }_{\text{L}}|}_{(t-\tau )}\right]}^n}-\beta m,$$ [S7]

with $m$ being the myosin activity of the regular hexagonal cell. Applying a similar normalization process to that in Eqs. S2 and S3, we have the normalized equations

$$\frac{\mathrm{d}\widehat{R}}{\mathrm{d}\widehat{t}}=-{\widehat{k}}_{\text{m}}\widehat{m}\widehat{R}-{\widehat{K}}_{\text{a}}({\widehat{R}}^2-1)\widehat{R}-\widehat{\mathrm{\Lambda }}$$ [S8]

and

$$\frac{\mathrm{d}\widehat{m}}{\mathrm{d}\widehat{t}}=\widehat{\alpha }\frac{{\left[{\widehat{R}|}_{(\widehat{t}-1)}\right]}^n}{{\widehat{K}}_{\text{L}}+{\left[{\widehat{R}|}_{(\widehat{t}-1)}\right]}^n}-\widehat{\beta }\widehat{m},$$ [S9]

where the dimensionless parameters here are defined as

$$\widehat{t}=\frac{t}{\tau },\hspace{1em}\widehat{m}=\frac{m}{m_{\text{s}}},\hspace{1em}\widehat{\alpha }=\frac{\alpha \tau }{m_{\text{s}}},\hspace{1em}\widehat{\beta }=\beta \tau ,\hspace{1em}\widehat{R}=\frac{R}{R_{\text{a}}},\hspace{1em}{\widehat{K}}_{\text{L}}=K_{\text{L}}{(\frac{R_{\text{s}}}{R_{\text{a}}})}^n=K_{\text{L}}{\widehat{R}}_{\text{s}}^n,\hspace{1em}{\widehat{k}}_{\text{m}}=6\frac{k_{\text{m}}{m}_{\text{s}}\tau }{\eta }\stackrel{\text{def}}{=}{\widehat{K}}_{\text{c}},\hspace{1em}{\widehat{K}}_{\text{a}}=\frac{\sqrt{3}}{2}\frac{K_{\text{a}}{A}_0\tau }{\eta },\hspace{1em}\widehat{\mathrm{\Lambda }}=\frac{1}{2}\sqrt{\frac{3\sqrt{3}}{2}}\frac{\mathrm{\Lambda }\tau }{\sqrt{A_0}\eta },$$ [S10]

with $R_{\text{a}}=\sqrt{2A_0}/\sqrt[4]{27}$ being the cell radius corresponding to the preferred area $A_0$. Consequently, we have the following relationships between ${\widehat{K}}_{\text{c}}$, ${\widehat{K}}_{\text{a}}$, $\widehat{\mathrm{\Lambda }}$ and ${\stackrel{\sim }{K}}_{\text{c}}$, ${\stackrel{\sim }{K}}_{\text{a}}$, $\stackrel{\sim }{\mathrm{\Lambda }}$:

$${\widehat{K}}_{\text{c}}=6{\stackrel{\sim }{K}}_{\text{c}},\hspace{1em}{\widehat{K}}_{\text{a}}=\frac{\sqrt{3}}{2}{\stackrel{\sim }{K}}_{\text{a}},\hspace{1em}\widehat{\mathrm{\Lambda }}=\frac{1}{2}\sqrt{\frac{3\sqrt{3}}{2}}\stackrel{\sim }{\mathrm{\Lambda }}.$$ [S11]

The equilibrium state of the cell unit $({\widehat{R}}_{\text{s}},{\widehat{m}}_{\text{s}})$ satisfies

$$\begin{array}{c}\hfill -{\widehat{k}}_{\text{m}}{\widehat{m}}_{\text{s}}{\widehat{R}}_{\text{s}}-{\widehat{K}}_{\text{a}}({\widehat{R}}_{\text{s}}^2-1){\widehat{R}}_{\text{s}}-\widehat{\mathrm{\Lambda }}=0,\\ \hfill \widehat{\alpha }\frac{{\widehat{R}}_{\text{s}}^n}{{\widehat{K}}_{\text{L}}+{\widehat{R}}_{\text{s}}^n}-\widehat{\beta }{\widehat{m}}_{\text{s}}=0;\end{array}$$

that is,

$${\widehat{K}}_{\text{a}}{\widehat{R}}_{\text{s}}^3+({\widehat{K}}_{\text{c}}-{\widehat{K}}_{\text{a}}){\widehat{R}}_{\text{s}}+\widehat{\mathrm{\Lambda }}=0,$$ [S12]

$${\widehat{m}}_{\text{s}}=1.$$ [S13]

Solving Eq. S12, we can obtain the cell radius at equilibrium ${\widehat{R}}_{\text{s}}$, and the cell perimeter at this state is $L_{\text{s}}=6R_{\text{a}}{\widehat{R}}_{\text{s}}$. Substituting it into Eqs. S2–S4, one can simulate the cellular dynamics in the monolayer with the chemomechanical coupling mechanism proposed in this work.

Estimation of the Dimensionless Parameters.

According to experimental measurements, the viscosity is in the order of ${10}^4-{10}^5\text{N}\cdot \text{s}\cdot {\text{m}}^{-2}$ for embryonic tissues (50); and the cell contraction force is in the range of $1-10\text{nN}$ for epithelial cell sheets (51). Therefore, we have the estimation of the friction coefficient $\eta \sim 0.01-0.1\text{N}\cdot \text{s}\cdot {\text{m}}^{-1}$ and the cell contraction modulus $K_{\text{c}}\sim {10}^{-4}-{10}^{-3}\text{N}\cdot {\text{m}}^{-1}$. Taking the cell areal stiffness $K_{\text{a}}\sim {10}^5-{10}^7\text{N}\cdot {\text{m}}^{-3}$ for epithelial tissues (18, 52), the delayed timescale $\tau \sim 1\text{min}$ for myosin activation (17, 29–31), and the preferred area $A_0=400\mathrm{\mu }{\text{m}}^2$, we have the scale of the dimensionless areal stiffness ${\stackrel{\sim }{K}}_{\text{a}}=K_{\text{a}}{A}_0\tau /\eta \sim 1-100$ and the dimensionless contractile modulus ${\stackrel{\sim }{K}}_{\text{c}}=K_{\text{c}}\tau /\eta \sim 0.1-1$. Furthermore, for simplicity, we set $\mathrm{\Lambda }=0$, which in fact does not affect our conclusions, as is shown in the following section.

Preparation of the Initial Monolayer.

We use the total stresses in the cell monolayer, ${\sigma }_{\mu \nu }$, to measure stress level of the monolayer, and also define the stress-free state of the monolayer. For an individual cell $J$, its stress ${\sigma }_{\mu \nu }^{(J)}$ is calculated by integrating the contributions from the areal deformation and the edge tension (53),

$${\sigma }_{\mu \nu }^{(J)}=S_J{\delta }_{\mu \nu }+\frac{1}{2}\frac{1}{A_J}\sum _{⟨i,j⟩\in E_J}{T}_{ij}\frac{l_{ij}^{\mu }{l}_{ij}^{\nu }}{l_{ij}},$$ [S14]

where the subscripts $\mu$ and $\nu$ take the values $x$ or $y$; ${\delta }_{\mu \nu }$ is the Kronecker delta; and $l_{ij}^x$ and $l_{ij}^y$ represent the projected components of edge $ij$ on the $x$ and $y$ axes, respectively. The summation ${\sum }_{⟨i,j⟩\in E_J}$ computes over all edges $E_J$ of cell $J$. $S_J=K_{\text{a}}(A_J-A_0)$ denotes the stress arising from the area variation, and $T_{ij}=\mathrm{\Lambda }+K_{\text{c}}(L_I+L_J)$ is the tension along the common edge $l_{ij}$ between cells $I$ and $J$. Thus, the total stresses in the whole cell monolayer are computed as

$${\sigma }_{\mu \nu }=\frac{{\sum }_J{\sigma }_{\mu \nu }^{(J)}{A}_J}{{\sum }_J{A}_J}.$$ [S15]

Based on the total stress in the whole monolayer, we can distinguish three states of the cell monolayer, i.e., stress-free state (or relaxed state), stretched state, and compressed state. In the main text, we have shown that the collective oscillatory morphodynamics can be tailored by mechanical stresses, and thus the simulations are based on the stress-free state except those for exploring the role of mechanical stretches. The details of generating the initial cell monolayer for simulations without or with stresses are given as follows.

For the cases without mechanical stresses, we start the simulations from a homogeneous and relaxed configuration, which is obtained in three steps: (i) We randomly seed points in the elliptic domain such that the average cell area is unity; (ii) we apply the Voronoi tessellation algorithm to mesh the elliptic domain into a cellular monolayer; and (iii) we move the vertices via the standard vertex model to minimize the stress within the monolayer, during which topological transitions (e.g., T1 transition and T2 transition) are allowed (23). No biochemical components are considered in the initialization steps.

Once the initial configuration has been obtained, we include the chemomechanical negative feedback by solving the coupled differential equations on the vertex positions ${𝐫}_i$ and the myosin activities $m_J$. The differential equations are solved using the forward-Euler difference method with time step $\mathrm{\Delta }t=0.002\tau$. All data are collected after the system reaches the dynamic equilibrium state, which typically takes about $\mathrm{20,000}$ simulation steps.

In the case with mechanical stresses, the initial configuration is constructed by applying an equiaxial stretch $\lambda$ to the homogeneous and relaxed configuration obtained above. The stretch of the cell monolayer is defined with respect to the relaxed state; e.g., ${\lambda }_x=L_x/L_{0x}$, where $L_{0x}$ and $L_x$ are the length of the cell monolayer along the $x$ axis before and after the stretching, respectively. Using the stretched configuration as the initial condition, we conduct simulations following the same procedure as above.

Influence of the Interfacial Tension on the Dynamics of Collective Cell Oscillations.

We have set the interfacial tension $\mathrm{\Lambda }=0$ to focus on the effects of active contraction and elastic deformation of cells in the main text for simplicity. When $\mathrm{\Lambda }\ne 0$, actually, collective cell oscillations could take place spontaneously due to the chemomechanical feedback, and similar oscillatory dynamics are observed. For example, it shows that the cell area and the myosin activity exhibit antiphase oscillations (Fig. S1 C and D). The oscillatory frequencies of cell area distribute within a concentrated regime around $f\sim 0.3{\tau }^{-1}$ (Fig. S1E), revealing collective coordinated and synchronized oscillatory dynamics. In addition, the oscillating cells tend to align along the short axis of the monolayer (Fig. S1 F and G), whereas they oscillate along the long axis (Fig. S1G).

Results

Spatiotemporal Structures of the Amnioserosa Oscillation.

We first show that the proposed coupling mechanism is able to reproduce the autonomous amnioserosa oscillation during the early stage of Drosophila dorsal closure (Fig. 1E). The spectrogram from spectral analysis indicates that the oscillatory frequencies of the cell area distribute within a concentrated regime around $f\sim 0.3{\tau }^{-1}$ (Fig. 1F), leading to a globally consistent period $T\approx 3.3\text{min}$, in agreement with experimental measurements (18, 35). The temporal correlation between the biomechanical and biochemical components is illustrated in the scatter diagram of the cell area (${\phi }_{\text{A}}$) and the myosin activity (${\phi }_{\text{M}}$) oscillatory phases (Fig. 1G). As expected for a negative-feedback–facilitated oscillator, (${\phi }_{\text{A}}$, ${\phi }_{\text{M}}$) mostly locate along two lines ${\phi }_{\text{A}}-{\phi }_{\text{M}}=\pm {180}^{\circ }$. Such an antiphase characteristic has also been observed in experiments (17, 35).

We extend our analysis to the spatial arrangement of the oscillation and find that cells with similar oscillation phases are aligned to form string-like patterns with a length scale spanning over the entire amnioserosa tissue (Fig. S1A). To quantify this observation, we define the alignment angle ${\theta }_{\text{align}}^{(J)}=\arccos [({𝐫}_J-{𝐫}_{I|J})\cdot {𝐞}_x/|{𝐫}_J-{𝐫}_{I|J}|]$ for every cell in the monolayer, where ${𝐫}_J$ is the center of cell $J$, ${𝐫}_{I|J}$ is the center of the neighboring cell $I$ whose oscillatory phase is the closest to cell $J$ among all neighbors, and ${𝐞}_x$ is the unit vector along the $x$ axis (Fig. 1B). The distribution of ${\theta }_{\text{align}}^{(J)}$ is peaked at $0^{\circ }$ with the majority below ${30}^{\circ }$ (Fig. S1B), confirming that the cells with similar oscillatory phases are spatially aligned along the short axis of the elliptic amnioserosa.

The amnioserosa oscillation also exhibits global polarity along the long axis of the elliptic tissue. We define a cell’s oscillatory orientation as ${\theta }_{\text{aniso}}^{(J)}=\arctan [{\mathrm{\Gamma }}_y^{(J)}/{\mathrm{\Gamma }}_x^{(J)}]$ with ${\mathrm{\Gamma }}_x^{(J)}$ and ${\mathrm{\Gamma }}_y^{(J)}$ being the projected components of the cell’s areal oscillation amplitude onto the $x$ (short) and $y$ (long) axes. We find that ${\theta }_{\text{aniso}}^{(J)}$ is close to ${90}^{\circ }$ (Fig. S1B). These results indicate that the intracellular chemomechanical coupling can propagate via the mechanical interfaces between cells and lead to spatiotemporally organized, aligned, and polarized cell oscillations at the tissue level.

Hopf Bifurcation of the Chemomechanical System.

To gain analytical insights into the dynamic mechanism of cell oscillations, we study a minimal seven-cell system that contains a regular hexagonal cell surrounded by and linked to six others (Fig. 2A). This hexagonal approximation is based on Euler’s formula that the most probable cell shape in a 2D cell monolayer is a hexagon. To examine the role of mechanical forces, we apply an equiaxial stretch $\lambda$ to the system and then fix the boundary vertices. From the linear stability analysis, the characteristic equation is (Hopf Bifurcation Analysis of the Minimal System)

$$(\zeta +\overline{\beta })(\zeta +\frac{7}{6}{\overline{K}}_{\text{c}}+2{\overline{K}}_{\text{a}}{\lambda }^2)+\frac{7}{6}\frac{n{\overline{K}}_{\text{c}}\overline{\beta }{K}_{\text{L}}}{K_{\text{L}}+{\lambda }^n}{e}^{-\zeta }=0,$$ [3]

where $\zeta$ denotes the eigenvalue of the Jacobian matrix for the seven-cell system with chemomechanical coupling; $\overline{\beta }=\beta \tau$, ${\overline{K}}_{\text{c}}=6{\stackrel{\sim }{K}}_{\text{c}}(K_{\text{L}}+1){\lambda }^n/(K_{\text{L}}+{\lambda }^n)$ and ${\overline{K}}_{\text{a}}=21K_{\text{a}}\tau R_{\text{s}}^2/(8\eta )$ are the dimensionless deactivation rate, cell contraction modulus, and cell area stiffness for the seven-cell system, respectively, with $R_{\text{s}}$ being the edge length of a hexagonal cell at rest. It should be noted that a slightly different normalization scheme is used for the seven-cell system to simplify the analytical derivation (see Hopf Bifurcation Analysis of the Minimal System for details). The mathematical definitions and physical meanings of the parameters used in this study are included in Table S1.

Fig. 2.

Stability analysis using the Hopf bifurcation theory. (A) Schematic of a minimal system with seven cells. The red solid circles and blue solid circles mark the boundary vertices and inner vertices, respectively. (B) Phase diagram of the seven-cell system’s dynamics, obtained by examining the eigenvalue $\zeta$: stable state for $\text{Re}(\zeta )<0$ and oscillatory state for $\text{Re}(\zeta )>0$ and $\text{Im}(\zeta )\ne 0$; the collapsed state refers to collapse of the model. Parameters: ${\stackrel{\sim }{K}}_{\text{a}}=10.0$, ${\stackrel{\sim }{K}}_{\text{c}}=1.0$, $\mathrm{\Lambda }=0$, and $\lambda =1$. (C) Effects of external stretch $\lambda$ on the seven-cell system’s dynamics. The magenta circles correspond to the critical stretches for the Hopf bifurcation. Parameters: ${\stackrel{\sim }{K}}_{\text{a}}=10.0$, ${\stackrel{\sim }{K}}_{\text{c}}=1.2$, $\mathrm{\Lambda }=0$, $\beta \tau =1.0$, $K_{\text{L}}=1.0$, and $n=8$. (D) Phase diagram of the amnioserosa monolayer’s dynamics, obtained by examining the average areal oscillation amplitude $A_{\text{D}}$: stable state for $A_{\text{D}}=0$, oscillatory state for $A_{\text{D}}>0$, and collapsed state for the collapse of model. Parameters: $\mathrm{\Lambda }=0$, $\beta \tau =1.0$, $K_{\text{L}}=1.0$, and $n=8$.

Table S1.

Summarization of the key physical parameters and their dimensionless forms

	Normalization scheme
Symbol	Cell monolayer	Seven-cell system	Physical meaning
$\alpha$	$\stackrel{\sim }{\alpha }={\displaystyle \frac{\alpha \tau }{m_{\text{s}}}}=\stackrel{\sim }{\beta }(K_{\text{L}}+1)$	$\overline{\alpha }={\displaystyle \frac{\alpha \tau }{m_0}}=\overline{\beta }{\displaystyle \frac{K_{\text{L}}+{\lambda }^n}{{\lambda }^n}}$	Myosin activation coefficient
$\beta$	$\stackrel{\sim }{\beta }=\beta \tau$	$\overline{\beta }=\beta \tau =\stackrel{\sim }{\beta }$	Myosin deactivation rate
$K_{\text{L}}$	${\stackrel{\sim }{K}}_{\text{L}}=K_{\text{L}}{\stackrel{\sim }{L}}_{\text{s}}^n$	$K_{\text{L}}$	Apparent dissociation constant
$n$	$n$	$n$	Hill coefficient
$K_{\text{a}}$	${\stackrel{\sim }{K}}_{\text{a}}={\displaystyle \frac{K_{\text{a}}{A}_0\tau }{\eta }}$	${\overline{K}}_{\text{a}}={\displaystyle \frac{21}{8}}{\displaystyle \frac{K_{\text{a}}\tau R_{\text{s}}^2}{\eta }}={\displaystyle \frac{21}{8}}{\stackrel{\sim }{K}}_{\text{a}}{\stackrel{\sim }{R}}_{\text{s}}^2$	Cell area stiffness
$K_{\text{c}}\stackrel{def}{=}{k}_{\text{m}}{m}_{\text{s}}$	${\stackrel{\sim }{K}}_{\text{c}}={\displaystyle \frac{K_{\text{c}}\tau }{\eta }}$	${\overline{K}}_{\text{c}}=6{\displaystyle \frac{(K_{\text{L}}+1){\lambda }^n}{K_{\text{L}}+{\lambda }^n}}{\displaystyle \frac{K_{\text{c}}\tau }{\eta }}=6{\displaystyle \frac{(K_{\text{L}}+1){\lambda }^n}{K_{\text{L}}+{\lambda }^n}}{\stackrel{\sim }{K}}_{\text{c}}$	Cell contraction modulus
$\mathrm{\Lambda }$	$\stackrel{\sim }{\mathrm{\Lambda }}={\displaystyle \frac{\mathrm{\Lambda }\tau }{\eta \sqrt{A_0}}}$	—	Intercellular interfacial tension
$m_{\text{s}}={\displaystyle \frac{\alpha }{\beta (K_{\text{L}}+1)}}$	—	—	Reference myosin activity of a cell at rest
$m_0={\displaystyle \frac{\alpha }{\beta }}{\displaystyle \frac{{\lambda }^n}{K_{\text{L}}+{\lambda }^n}}$	—	—	Reference myosin activity of a cell under stretch
$L_{\text{s}}=6R_{\text{s}}$	${\stackrel{\sim }{L}}_{\text{s}}={\displaystyle \frac{L_{\text{s}}}{\sqrt{A_0}}}$	—	Perimeter of a cell at rest
$R_{\text{s}}$	${\stackrel{\sim }{R}}_{\text{s}}={\displaystyle \frac{R_{\text{s}}}{\sqrt{A_0}}}$	—	Radius of a cell at rest
$A_0$	—	—	Preferred cell area
$\tau$	—	—	Time delay
$\eta$	—	—	Friction coefficient
$\lambda$	—	—	Mechanical stretch

$\zeta$ determines the seven-cell system’s dynamics. As visualized in the phase diagram (Fig. 2B), two requirements for collective cell oscillations emerge: First, it is necessary to have a strong nonlinearity in the chemomechanical coupling (i.e., a large $n$) to destabilize any balance between the biochemical and biomechanical components in the negative feedback (34); second, it is also necessary to have a long enough delay in the negative feedback between the myosin activation and the cell stretch (i.e., a large $\tau$) such that the system’s biochemical component can maintain a working memory about the previous state of the system’s biomechanical component, providing an effective inertia for oscillation to occur.

The role of mechanical forces in cell oscillations is clarified by varying stretch $\lambda$. In fact, Eq. 3 leads to the necessary condition for cell oscillation (Hopf Bifurcation Analysis of the Minimal System):

$$\{\begin{array}{c}1+\frac{12{\overline{K}}_{\text{a}}{\lambda }^2}{7{\overline{K}}_{\text{c}}}<\frac{n{K}_{\text{L}}}{K_{\text{L}}+{\lambda }^n},\hfill \\ 1+\frac{6\overline{\beta }}{7{\overline{K}}_{\text{c}}}+\frac{12{\overline{K}}_{\text{a}}{\lambda }^2}{7{\overline{K}}_{\text{c}}}<\overline{\beta }\frac{n{K}_{\text{L}}}{K_{\text{L}}+{\lambda }^n}.\hfill \end{array}$$ [4]

It is clear that for the two limit cases, $\lambda \to 0^{+}$ and $\lambda \to +\mathrm{\infty }$, the above inequalities can never be satisfied, indicating that collective cell oscillations can take place only if the stretching or compressing falls in the range that triggers the Hopf bifurcation (${\lambda }_{\text{cr}1}<\lambda <{\lambda }_{\text{cr}2}$, Fig. 2C). Outside of this range, mechanical stretches or forces may hinder cell oscillation.

We next extend our analysis to the full monolayer and plot the phase diagram of the collective cell dynamics in Fig. 2D. It is clear that the amnioserosa oscillation relies on both the cell areal stiffness ${\stackrel{\sim }{K}}_{\text{a}}$ and the cell contractility ${\stackrel{\sim }{K}}_{\text{c}}$. Interestingly, for a given ${\stackrel{\sim }{K}}_{\text{a}}$, a weak ${\stackrel{\sim }{K}}_{\text{c}}$ results in a stable resting phase of the collective cell dynamics, whereas strong contraction may trigger Hopf bifurcation (Fig. S2), leading to sustained collective shape oscillations.

Fig. S2.

Bifurcation diagram of the cell dynamics in the amnioserosa monolayer, regulated by cell contractile modulus ${\stackrel{\sim }{K}}_{\text{c}}$. Here, $A_{\max }={⟨{\max }_t{A}_J(t)⟩}_J$ and $A_{\min }={⟨{\min }_t{A}_J(t)⟩}_J$. Parameters: ${\stackrel{\sim }{K}}_{\text{a}}=10.0$, $\mathrm{\Lambda }=0$, $\beta \tau =1.0$, $K_{\text{L}}=1.0$, and $n=8$.

The above analysis suggests that the observed Drosophila amnioserosa oscillation is a biological demonstration of the Hopf bifurcation, and mechanical forces could play a key regulating role, as is further demonstrated in the following.

Mechanical Activation and Deactivation of the Amnioserosa Oscillation.

Amnioserosa oscillation emerges in the early stage of the Drosophila dorsal closure and gradually decays until it completely disappears around the middle stage of the dorsal closure. During the dorsal closure process, the amnioserosa cells actively self-contract to sew the lateral epidermis, thus yielding a tensile force in the tissue (36, 37). Based on this fact and our theoretical study on the seven-cell system, we argue that this tensile force may contribute to the robust gating of the amnioserosa oscillation. To test this hypothesis, we first mimic the mechanical environment, using an external equiaxial stretch $\lambda$ applied on an elliptic monolayer ($\lambda <1$ for compressing and $\lambda >1$ for stretching, Fig. 3A, Inset). It becomes immediately evident that sufficiently high compression (e.g., $\lambda <0.7$) and stretch (e.g., $\lambda >1.2$) can both hinder collective cell oscillations (Fig. 3A), further confirming the role of mechanical forces on cell oscillation.

Fig. 3.

Mechanically gated collective cell oscillations in an elliptic monolayer. (A) The equiaxial stretch $\lambda$ regulates the oscillation intensity $A_D$ of the monolayer. Inset shows schematic of an elliptic monolayer under equiaxial stretch $\lambda$. (B) Simulation of Drosophila dorsal closure: evolution of cell area and myosin activity of a typical amnioserosa cell.

We then explicitly simulate the early-to-middle stage of dorsal closure. It has been reported that the myosin activity in amnioserosa cells increases as dorsal closure proceeds (35), implying an enhanced amnioserosa contraction due to the increase in myosin activation coefficient $\alpha$ in Eq. 2 (38). To account for this biochemical feature, we gradually increase $\alpha$ while keeping the myosin deactivation rate $\beta$ fixed. Meanwhile, the boundary of the monolayer is allowed to move inward progressively, with a speed slower than the active contraction of the amnioserosa (Simulation of the Drosophila Dorsal Closure Process). Such mismatch in speed is adopted to generate tensile stress within the monolayer (Fig. S3). We find that the areal oscillatory amplitude decays gradually, whereas the myosin activity increases in an oscillatory manner due to the chemomechanical coupling (Fig. 3B). Our results agree well with experimental observations (18, 35) and suggest that the gradual disappearance of the amnioserosa oscillation results from the increasing tensile stress built up by the force balance between the active amnioserosa contraction and the passive epidermis resistance. Therefore, the tensile stress provides a robust mechanical gating mechanism that switches collective cell oscillations in Drosophila amnioserosa on and off.

Fig. S3.

Mechanically gated collective cell oscillations in an elliptic monolayer. (A and B) Phase diagrams of (A) the areal oscillation intensity and (B) the myosin activity regulated by the myosin activation coefficient $\alpha$ and the elastic strain ${\epsilon }_{\text{e}}$. Here, $m_{\text{a}}={⟨m_J(t)⟩}_{J,t}$ is the mean myosin activity of all amnioserosa cells over time. (C) Evolution of myosin activation coefficient $\alpha$, elastic strain ${\epsilon }_{\text{e}}$, contractile strain ${\epsilon }_{\text{c}}$, and total strain $\epsilon$. (D) Evolution of mean normal stress ${\sigma }_{\text{m}}^{(\text{as})}$ and area $A_{\text{as}}$ of the amnioserosa.

Regulatory Roles of the Amnioserosa Boundary Stiffness.

Recent experiments showed that weakening the leading edge of the lateral epidermis results in an irregular amnioserosa morphology (Fig. 4A, adapted from ref. 39), indicating the influence of the boundary constraint on embryonic development (37, 39). To identify the specific regulatory roles of this mechanical constraint, we attach the amnioserosa’s boundary vertices to an array of elastic linear springs (Fig. 4B). The stiffness of the boundary springs $K_{\text{b}}$ characterizes the strength of mechanical constraints from the boundary and we set its normalized form, ${\stackrel{\sim }{K}}_{\text{b}}=K_{\text{b}}\tau /\eta$, as the boundary stiffness.

Fig. 4.

Effects of boundary stiffness on the oscillatory structure. (A) Experimentally observed amnioserosa morphology in the absence of the boundary supracellular actin cable. Adapted from ref. 39 with permission from Macmillan Publishers Ltd: Nature Cell Biology, copyright (2016). (B) Schematic of an elliptic cell monolayer with elastic boundary constraints. Results are obtained using a monolayer with aspect ratio $\xi =a/b=2.0$, where $a$ and $b$ are the semilong and semishort axes of the ellipse domain, respectively. (C) Tissue morphology and areal oscillatory phases ${\phi }_{\text{A}}$ under different boundary stiffness ${\stackrel{\sim }{K}}_{\text{b}}$. (D) The tissue’s morphological integrity vs. ${\stackrel{\sim }{K}}_{\text{b}}$. The red line is the mean displacement $D_{\text{b}}={⟨|D_j(t)|⟩}_{j,t}$ with $D_j(t)$ being the displacement of the boundary vertex $j$ at time $t$, and the blue line is the mean curvature fluctuation $C_{\text{b}}={⟨C_b(t)⟩}_t$ with $C_b(t)$ being the SD of the curvature along the boundary at time $t$. (E) The areal oscillation amplitudes $A_{\text{D}}$ of the inner and boundary cells vs. the ${\stackrel{\sim }{K}}_{\text{b}}$. Here, $A_{\text{D}}={⟨A_{\text{D}}^{(J)}⟩}_J$, where $A_{\text{D}}^{(J)}$ is the areal oscillation amplitude of cell $J$ and ${⟨\cdot ⟩}_J$ computes an average over the inner or boundary cells. (F) The mean oscillatory orientation ${\theta }_{\text{aniso}}={⟨{\theta }_{\text{aniso}}^{(J)}⟩}_J$ and the mean oscillatory alignment ${\theta }_{\text{align}}={⟨{\theta }_{\text{align}}^{(J)}⟩}_J$ vs. ${\stackrel{\sim }{K}}_{\text{b}}$. (G) Spatial distributions of the areal oscillation amplitudes under different ${\stackrel{\sim }{K}}_{\text{b}}$. $r$ is the minimum distance of a cell to boundary cells and $d_{\text{c}}$ is the mean distance between neighboring cells. (H) The SD of areal oscillation amplitudes vs. ${\stackrel{\sim }{K}}_{\text{b}}$ for inner and total cells.

Intriguingly, we find that higher boundary stiffness not only smoothens the overall amnioserosa morphology, but also restores the organized spatial pattern of the amnioserosa oscillation as seen in the fixed-boundary simulation (Fig. 4C and Figs. S1A and S4A). On the one hand, the boundary displacement and roughness are significantly reduced by increasing ${\stackrel{\sim }{K}}_{\text{b}}$ (Fig. 4D): The boundary cells oscillate with high amplitudes when ${\stackrel{\sim }{K}}_{\text{b}}$ is weak, and such boundary oscillations are largely suppressed once ${\stackrel{\sim }{K}}_{\text{b}}$ is beyond a critical level ${\stackrel{\sim }{K}}_{\text{b}}^{(\text{cr})}\sim (\mathrm{1,10})$ (Fig. 4E and Fig. S4B). This finding, together with the recent experiments (37, 39), demonstrates that a stiff boundary is essential for maintaining the mechanical integrity of the amnioserosa tissue both statically and dynamically.

Fig. S4.

Effect of boundary constraints on collective cell oscillations. (A) Collective oscillatory phases of myosin activity ${\phi }_{\text{M}}$ under different boundary stiffness ${\stackrel{\sim }{K}}_{\text{b}}$. (B) The myosin oscillation amplitudes $m_{\text{D}}$ of cells within inner and boundary regions vs. the boundary stiffness ${\stackrel{\sim }{K}}_{\text{b}}$. (C) Boundary stiffness ${\stackrel{\sim }{K}}_{\text{b}}$ regulates the distribution of oscillatory alignment ${\theta }_{\text{align}}^{(J)}$. (D) Spatial distributions of the myosin oscillation amplitudes under different ${\stackrel{\sim }{K}}_{\text{b}}$. (E) Boundary stiffness regulates the distribution of oscillatory orientation ${\theta }_{\text{aniso}}^{(J)}$. (F) The SD of myosin oscillation amplitudes vs. boundary stiffness for inner and total cells.

On the other hand, the oscillation dynamics of inner cells are orchestrated by increasing ${\stackrel{\sim }{K}}_{\text{b}}$ as well. This is first evidenced by the emergence of the string-like oscillatory patterns while tuning up ${\stackrel{\sim }{K}}_{\text{b}}$ (Fig. 4C and Fig. S4A). Quantitatively, the distribution of the alignment angle ${\theta }_{\text{align}}^{(J)}$ evolves from a flat shape to one with a peak at $0^{\circ }$ (Fig. S4C), resulting in a significant decrease in ${\theta }_{\text{align}}={⟨{\theta }_{\text{align}}^{(J)}⟩}_J$ (Fig. 4F). Furthermore, the spatial distribution of the oscillatory amplitudes exhibits critical dependence on ${\stackrel{\sim }{K}}_{\text{b}}$: For soft boundaries, the oscillatory amplitude decreases while moving away from the boundary; whereas when the boundary stiffness is higher than the critical value, the oscillatory amplitude of inner cells becomes consistently higher than that of the boundary cells (Fig. 4 E and G and Fig. S4 B and D). In fact, experiments have observed that the oscillations of peripheral-most cells are weaker than those of inner cells during Drosophila dorsal closure (18). From a mechanical perspective, our analysis reveals the regulatory role of the mechanical constraint as an organizer that orchestrates the oscillations of individual cells across the entire tissue.

A surprising result in Fig. 4G (also Fig. S4D) is that with a stiff boundary (${\stackrel{\sim }{K}}_{\text{b}}=5,10,100$), all inner cells oscillate with nearly the same amplitude, independent of their locations. This globally synchronized behavior does not exist when the boundary is soft (${\stackrel{\sim }{K}}_{\text{b}}=0.1,1$). Meanwhile, larger ${\stackrel{\sim }{K}}_{\text{b}}$ also leads to the globally polarized oscillation (increasing ${\theta }_{\text{aniso}}={⟨{\theta }_{\text{aniso}}^{(J)}⟩}_J$ in Fig. 4F and Fig. S4E). To quantify this global synchronization phenomenon, we calculate the standard deviations of the oscillation amplitudes for all cells and for the inner cells. The standard deviations dramatically drop around ${\stackrel{\sim }{K}}_{\text{b}}^{(\text{cr})}$ (Fig. 4H and Fig. S4F). These results indicate that it is the boundary stiffening that synchronizes the cells’ oscillation within the amnioserosa tissue.

Regulatory Role of the Amnioserosa Boundary Geometry.

Because the oscillation exhibits preferential alignments along the short axis as well as predominate polarity along the long axis, we hypothesize that the elliptic boundary geometry may also influence the tissue’s morphodynamics. We test this hypothesis by investigating the evolvement of the oscillation patterns upon changing the aspect ratio ($\xi$) of the fixed elliptic boundary.

Fig. 5A shows that a circular boundary results in random and isotropic oscillation within the monolayer where no oscillatory pattern can be identified; whereas an increased aspect ratio helps individual cells differentiate the directions of polarity (long axis) and alignment (short axis). As summarized in Fig. 5B, increasing the aspect ratio of the boundary enhances the oscillatory polarity and the phase alignment simultaneously. We emphasize that it is of general importance for individual cells to “recognize” the tissue’s geometry and to arrange their morphodynamics accordingly. Similar to Drosophila dorsal closure, cells in Drosophila germband extension also undergo polarized collective oscillations where the oscillation amplitudes along the long anterior–posterior axis are larger than those along the short dorsal–ventral axis (19). Our results suggest that in addition to genetic regulation, boundary geometry may also modulate morphodynamic patterns to provide necessary physical cues for embryogenesis.

Fig. 5.

Effect of boundary geometry on collective cell oscillations. (A) Breakdown of the isotropy in the boundary geometry leads to organized oscillatory patterns. (B) Dependence of the mean alignment orientation ${\theta }_{\text{align}}$ and the mean oscillatory orientation ${\theta }_{\text{aniso}}$ on the aspect ratio $\xi$.

Cell Population and the Spatiotemporal Oscillatory Structures.

The spatiotemporal oscillatory structures require precise transmission of the regulatory information within the amnioserosa tissue. Recent experiments showed that Madin–Darby canine kidney epithelial cells exhibit a highly coordinated motility structure on small circular plates, whereas such structure disappears when the plate becomes too large even if the cell density remains the same (40). This result would lead to a conclusion that the fidelity of the information transmission would reduce as the monolayer’s size increases. However, our results show otherwise.

We analyze the morphodynamics of the elliptic cell monolayer with different cell populations $N$. As summarized in Fig. S5A, when the cell population is very low ($N=50$), the cell oscillations within the monolayer exhibit only very weak alignment and polarity. In other words, the oscillatory structure is noisy. When the cell population increases, more organized and polarized oscillatory patterns start emerging ($N=100\sim 500$) and become more distinguishable. These observations can be further confirmed by the smaller variances in oscillatory amplitudes (Fig. S5 B–E), the higher polarity (Fig. S5F), and the stronger alignment (Fig. S5G). Interestingly, at $N=200$, which is approximately the cell number in the Drosophila amnioserosa, both the polarity and alignment reach high levels close to their maximum values (Fig. S5H).

Fig. S5.

Cell population analysis. (A) Collective oscillatory patterns in a confined elliptic cell monolayer with different cell populations $N$. The emergence of the oscillatory patterns is characterized using ${\theta }_{\text{aniso}}^{(J)}$ and ${\theta }_{\text{align}}^{(J)}$. (B–E) Synchronization of cell oscillation for (B and C) inner cells and (D and E) total cells, characterized using the SD of the oscillatory amplitudes of (B and D) cell area and (C and E) myosin activity. (F and G) The mean oscillatory orientation ${\theta }_{\text{aniso}}$ (F) and the mean oscillatory alignment ${\theta }_{\text{align}}$ (G) vs. ${\stackrel{\sim }{K}}_{\text{b}}$, with different cell populations. (H) Dependence of the mean alignment orientation ${\theta }_{\text{align}}$ and the mean oscillatory orientation ${\theta }_{\text{aniso}}$ on the cell population $N$, where ${\stackrel{\sim }{K}}_{\text{b}}=100$.

The idea of phase transition has recently been proposed to explain the collective motility of many-cell systems such that increasing cell density would lead to glass transition that globally jams the systems’ motion (41–44). Our results may suggest a similar “disorder-to-order” transition in collective cell oscillations such that some critical cell population is required to ensure spatiotemporally organized, aligned, and polarized cell oscillations in the amnioserosa tissue.

Hopf Bifurcation Analysis of the Minimal System

To unravel how mechanical forces and boundary confinement modulate collective cell oscillations in the monolayer, we consider a minimal system with seven cells, as shown in Fig. 2A in the main text, where a regular hexagonal cell is surrounded by and linked to six others. We first apply an equiaxial stretch $\lambda$ to the system and then fix all of the boundary vertices. The vertices of the central cell move according to the dynamic functions introduced in the main text. Owing to the symmetry of the minimal system, there are three independent variables, i.e., the radius of the central cell $R$ and the myosin activities of the central cell $m_{\text{C}}$ and of the boundary cells $m_{\text{B}}$. Therefore, the dynamics of the minimal system are fully described by

$$\begin{array}{c}\eta \frac{\mathrm{d}R}{\mathrm{d}t}=-k_{\text{m}}(m_{\text{C}}{L}_{\text{C}}-m_{\text{B}}{L}_{\text{B}})-\frac{\sqrt{3}}{2}{K}_{\text{a}}(A_{\text{C}}-A_{\text{B}})R,\hfill \\ \frac{\mathrm{d}{m}_{\text{C}}}{\mathrm{d}t}=\alpha \frac{{[{{\lambda }_{\text{L}}^{(\text{C})}|}_{(t-\tau )}]}^n}{K_{\text{L}}+{[{{\lambda }_{\text{L}}^{(\text{C})}|}_{(t-\tau )}]}^n}-\beta m_{\text{C}},\hfill \\ \frac{\mathrm{d}{m}_{\text{B}}}{\mathrm{d}t}=\alpha \frac{{[{{\lambda }_{\text{L}}^{(\text{B})}|}_{(t-\tau )}]}^n}{K_{\text{L}}+{[{{\lambda }_{\text{L}}^{(\text{B})}|}_{(t-\tau )}]}^n}-\beta m_{\text{B}},\hfill \end{array}$$ [S16]

where

$$\begin{array}{c}{L}_{\text{C}}=6R,\hspace{1em}{L}_{\text{B}}=7\lambda R_{\text{s}}-R,\hspace{1em}\hfill \\ A_{\text{C}}=\frac{3\sqrt{3}}{2}{R}^2,\hspace{1em}{A}_{\text{B}}=\frac{\sqrt{3}}{4}(7{\lambda }^2{R}_{\text{s}}^2-R^2),\hfill \end{array}$$ [S17]

and thus

$${\lambda }_{\text{L}}^{(\text{C})}=\frac{R}{R_{\text{s}}},\hspace{1em}{\lambda }_{\text{L}}^{(\text{B})}=\frac{7\lambda R_{\text{s}}-R}{6R_{\text{s}}}.$$ [S18]

Substituting Eqs. S17 and S18 into Eq. S16, we obtain

$$\begin{array}{c}\eta \frac{\mathrm{d}R}{\mathrm{d}t}=-k_{\text{m}}\left[6m_{\text{C}}R-m_{\text{B}}(7\lambda R_{\text{s}}-R)\right]-\frac{21}{8}{K}_{\text{a}}(R^2-{\lambda }^2{R}_{\text{s}}^2)R,\hfill \\ \frac{\mathrm{d}{m}_{\text{C}}}{\mathrm{d}t}=\alpha \frac{{\left[{R|}_{(t-\tau )}\right]}^n}{K_{\text{L}}{R}_{\text{s}}^n+{\left[{R|}_{(t-\tau )}\right]}^n}-\beta m_{\text{C}},\hfill \\ \frac{\mathrm{d}{m}_{\text{B}}}{\mathrm{d}t}=\alpha \frac{{\left[\frac{7}{6}\lambda R_{\text{s}}-{\frac{1}{6}R|}_{(t-\tau )}\right]}^n}{K_{\text{L}}{R}_{\text{s}}^n+{\left[\frac{7}{6}\lambda R_{\text{s}}-{\frac{1}{6}R|}_{(t-\tau )}\right]}^n}-\beta m_{\text{B}}.\hfill \end{array}$$ [S19]

These equations are normalized as

$$\begin{array}{c}\frac{\mathrm{d}\overline{R}}{\mathrm{d}\overline{t}}=-{\overline{k}}_{\text{m}}[{\overline{m}}_{\text{C}}\overline{R}-{\overline{m}}_{\text{B}}(\frac{7}{6}\lambda -\frac{1}{6}\overline{R})]-{\overline{K}}_{\text{a}}({\overline{R}}^2-{\lambda }^2)\overline{R},\hfill \\ \frac{\mathrm{d}{\overline{m}}_{\text{C}}}{\mathrm{d}\overline{t}}=\overline{\alpha }\frac{{[{\overline{R}|}_{(\overline{t}-1)}]}^n}{K_{\text{L}}+{[{\overline{R}|}_{(\overline{t}-1)}]}^n}-\overline{\beta }{\overline{m}}_{\text{C}},\hfill \\ \frac{\mathrm{d}{\overline{m}}_{\text{B}}}{\mathrm{d}\overline{t}}=\overline{\alpha }\frac{{[\frac{7}{6}\lambda -{\frac{1}{6}\overline{R}|}_{(\overline{t}-1)}]}^n}{K_{\text{L}}+{[\frac{7}{6}\lambda -{\frac{1}{6}\overline{R}|}_{(\overline{t}-1)}]}^n}-\overline{\beta }{\overline{m}}_{\text{B}},\hfill \end{array}$$ [S20]

where the dimensionless parameters are defined as

$$\overline{t}=\frac{t}{\tau },\hspace{1em}\overline{R}=\frac{R}{R_{\text{s}}},\hspace{1em}{\overline{m}}_{\text{C}}=\frac{m_{\text{C}}}{m_0},\hspace{1em}{\overline{m}}_{\text{B}}=\frac{m_{\text{B}}}{m_0},\hspace{1em}{m}_0=\frac{\alpha }{\beta }\frac{{\lambda }^n}{K_{\text{L}}+{\lambda }^n}=\frac{(K_{\text{L}}+1){\lambda }^n}{K_{\text{L}}+{\lambda }^n}{m}_{\text{s}},\hspace{1em}\overline{\alpha }=\frac{\alpha \tau }{m_0}=\beta \tau \frac{K_{\text{L}}+{\lambda }^n}{{\lambda }^n}=\overline{\beta }\frac{K_{\text{L}}+{\lambda }^n}{{\lambda }^n},\hspace{1em}\overline{\beta }=\beta \tau ,\hspace{1em}{\overline{k}}_{\text{m}}=6\frac{k_{\text{m}}{m}_0\tau }{\eta }=6\frac{(K_{\text{L}}+1){\lambda }^n}{K_{\text{L}}+{\lambda }^n}{\stackrel{\sim }{k}}_{\text{m}}\stackrel{\text{def}}{=}{\overline{K}}_{\text{c}},\hspace{1em}{\overline{K}}_{\text{a}}=\frac{21}{8}\frac{K_{\text{a}}\tau R_{\text{s}}^2}{\eta }=\frac{21}{8}{\stackrel{\sim }{K}}_{\text{a}}\frac{R_{\text{s}}^2}{A_0}=\frac{21}{8}{\stackrel{\sim }{K}}_{\text{a}}{\stackrel{\sim }{R}}_{\text{s}}^2,$$ [S21]

with $R_{\text{s}}$ determined by Eq. S12.

The equilibrium state of this system, $({\overline{R}}_{\text{h}},{\overline{m}}_{\text{C}}^{(\text{h})},{\overline{m}}_{\text{B}}^{(\text{h})})$, satisfies

$$\begin{array}{c}\hfill -{\overline{k}}_{\text{m}}\left[{\overline{m}}_{\text{C}}^{(\text{h})}{\overline{R}}_{\text{h}}-{\overline{m}}_{\text{B}}^{(\text{h})}\left(\frac{7}{6}\lambda -\frac{1}{6}{\overline{R}}_{\text{h}}\right)\right]-{\overline{K}}_{\text{a}}({\overline{R}}_{\text{h}}^2-{\lambda }^2){\overline{R}}_{\text{h}}=0,\\ \hfill \overline{\alpha }\frac{{\overline{R}}_{\text{h}}^n}{K_{\text{L}}+{\overline{R}}_{\text{h}}^n}-\overline{\beta }{\overline{m}}_{\text{C}}^{(\text{h})}=0,\\ \hfill \overline{\alpha }\frac{{\left[\frac{7}{6}\lambda -\frac{1}{6}{\overline{R}}_{\text{h}}\right]}^n}{K_{\text{L}}+{\left[\frac{7}{6}\lambda -\frac{1}{6}{\overline{R}}_{\text{h}}\right]}^n}-\overline{\beta }{\overline{m}}_{\text{B}}^{(\text{h})}=0,\end{array}$$

which lead to

$${\overline{R}}_{\text{h}}=\lambda ,\hspace{1em}{\overline{m}}_{\text{C}}^{(\text{h})}=1,\hspace{1em}{\overline{m}}_{\text{B}}^{(\text{h})}=1.$$ [S22]

Applying small perturbations to the differential system Eq. S20, we can obtain the perturbed equations around the equilibrium state $({\overline{R}}_{\text{h}},{\overline{m}}_{\text{C}}^{(\text{h})},{\overline{m}}_{\text{B}}^{(\text{h})})$,

$$\begin{array}{c}\frac{\mathrm{d}(\delta \overline{R})}{\mathrm{d}\overline{t}}=-{\overline{k}}_{\text{m}}\lambda \delta {\overline{m}}_{\text{C}}+{\overline{k}}_{\text{m}}\lambda \delta {\overline{m}}_{\text{B}}-(\frac{7}{6}{\overline{k}}_{\text{m}}+2{\overline{K}}_{\text{a}}{\lambda }^2)\delta \overline{R},\hfill \\ \frac{\mathrm{d}(\delta {\overline{m}}_{\text{C}})}{\mathrm{d}\overline{t}}=-\overline{\beta }\delta {\overline{m}}_{\text{C}}+{\overline{\alpha }{K}_{\text{L}}\frac{n{\lambda }^{n-1}}{{\left(K_{\text{L}}+{\lambda }^n\right)}^2}\delta \overline{R}|}_{(\overline{t}-1)},\hfill \\ \frac{\mathrm{d}(\delta {\overline{m}}_{\text{B}})}{\mathrm{d}\overline{t}}=-\overline{\beta }\delta {\overline{m}}_{\text{B}}-{\frac{1}{6}\overline{\alpha }{K}_{\text{L}}\frac{n{\lambda }^{n-1}}{{\left(K_{\text{L}}+{\lambda }^n\right)}^2}\delta \overline{R}|}_{(\overline{t}-1)},\hfill \end{array}$$

where $\delta \overline{R}$, $\delta {\overline{m}}_{\text{C}}$, and $\delta {\overline{m}}_{\text{B}}$ are the corresponding increments away from equilibrium. Setting $(\delta \overline{R},\delta {\overline{m}}_{\text{C}},\delta {\overline{m}}_{\text{B}})=({\delta }_1,{\delta }_2,{\delta }_3)e^{\zeta \overline{t}}$, we have

$$\begin{array}{c}\hfill -(\frac{7}{6}{\overline{k}}_{\text{m}}+2{\overline{K}}_{\text{a}}{\lambda }^2+\zeta ){\delta }_1-{\overline{k}}_{\text{m}}\lambda {\delta }_2+{\overline{k}}_{\text{m}}\lambda {\delta }_3=0,\\ \hfill \overline{\alpha }{K}_{\text{L}}\frac{n{\lambda }^{n-1}}{{\left(K_{\text{L}}+{\lambda }^n\right)}^2}{e}^{-\zeta }{\delta }_1-(\overline{\beta }+\zeta ){\delta }_2=0,\\ \hfill -\frac{1}{6}\overline{\alpha }{K}_{\text{L}}\frac{n{\lambda }^{n-1}}{{\left(K_{\text{L}}+{\lambda }^n\right)}^2}{e}^{-\zeta }{\delta }_1-(\overline{\beta }+\zeta ){\delta }_3=0.\end{array}$$ [S23]

The existence of nontrivial solution $({\delta }_1,{\delta }_2,{\delta }_3)$ requires the corresponding Jacobian matrix satisfying

$$\text{det}\left(\begin{array}{ccc}\hfill -\left(\frac{7}{6}{\overline{k}}_{\text{m}}+2{\overline{K}}_{\text{a}}{\lambda }^2+\zeta \right)\hfill & \hfill -{\overline{k}}_{\text{m}}\lambda \hfill & \hfill {\overline{k}}_{\text{m}}\lambda \hfill \\ \hfill \overline{\alpha }{K}_{\text{L}}\frac{n{\lambda }^{n-1}}{{\left(K_{\text{L}}+{\lambda }^n\right)}^2}{e}^{-\zeta }\hfill & \hfill -(\overline{\beta }+\zeta )\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{6}\overline{\alpha }{K}_{\text{L}}\frac{n{\lambda }^{n-1}}{{\left(K_{\text{L}}+{\lambda }^n\right)}^2}{e}^{-\zeta }\hfill & \hfill 0\hfill & \hfill -(\overline{\beta }+\zeta )\hfill \end{array}\right)=0.$$

That is,

$$\begin{array}{c}{(\zeta +\overline{\beta })}^2(\zeta +\frac{7}{6}{\overline{k}}_{\text{m}}+2{\overline{K}}_{\text{a}}{\lambda }^2)\hfill \\ +\frac{7}{6}{\overline{k}}_{\text{m}}\overline{\alpha }{K}_{\text{L}}\frac{n{\lambda }^n}{{\left(K_{\text{L}}+{\lambda }^n\right)}^2}(\zeta +\overline{\beta })e^{-\zeta }=0.\hfill \end{array}$$ [S24]

Eq. S24 has a real and two complex eigenvalues. The real eigenvalue is negative, i.e., $\zeta =-\overline{\beta }$, which corresponds to a stable state. Whereas the other two eigenvalues satisfy

$$\begin{array}{c}(\zeta +\overline{\beta })(\zeta +\frac{7}{6}{\overline{k}}_{\text{m}}+2{\overline{K}}_{\text{a}}{\lambda }^2)\hfill \\ +\frac{7}{6}{\overline{k}}_{\text{m}}\overline{\alpha }{K}_{\text{L}}\frac{n{\lambda }^n}{{\left(K_{\text{L}}+{\lambda }^n\right)}^2}{e}^{-\zeta }=0.\hfill \end{array}$$ [S25]

Substituting $\overline{\alpha }=\overline{\beta }(K_{\text{L}}+{\lambda }^n)/{\lambda }^n$ into Eq. S25, we obtain

$$(\zeta +\overline{\beta })(\zeta +\frac{7}{6}{\overline{k}}_{\text{m}}+2{\overline{K}}_{\text{a}}{\lambda }^2)+\frac{7}{6}{\overline{k}}_{\text{m}}\overline{\beta }\frac{n{K}_{\text{L}}}{K_{\text{L}}+{\lambda }^n}{e}^{-\zeta }=0.$$ [S26]

For brevity, setting

$$a_{\text{H}}=\frac{7}{6}{\overline{k}}_{\text{m}}\overline{\beta }\frac{n{K}_{\text{L}}}{K_{\text{L}}+{\lambda }^n},\hspace{1em}{b}_{\text{H}}=\overline{\beta },\hspace{1em}{c}_{\text{H}}=\frac{7}{6}{\overline{k}}_{\text{m}}+2{\overline{K}}_{\text{a}}{\lambda }^2,$$ [S27]

then Eq. S26 can be rewritten as

$$(\zeta +b_{\text{H}})(\zeta +c_{\text{H}})+a_{\text{H}}{e}^{-\zeta }=0.$$ [S28]

Denoting $\zeta =r+\text{i}\omega$, Eq. S28 becomes

$$\begin{array}{c}\hfill -{\omega }^2+(r+b_{\text{H}})(r+c_{\text{H}})+a_{\text{H}}{e}^{-r}\cos \omega =0,\\ \hfill \omega (2r+b_{\text{H}}+c_{\text{H}})-a_{\text{H}}{e}^{-r}\sin \omega =0.\end{array}$$ [S29]

These two equations can be numerically solved to obtain the eigenvalues, which determine the dynamic features of cells in the seven-cell system. By setting $r=0$, one obtains the critical parameters for the seven-cell system undergoing a Hopf bifurcation

$$\begin{array}{c}\hfill -{\omega }^2+b_{\text{H}}{c}_{\text{H}}+a_{\text{H}}\cos \omega =0,\\ \hfill \omega (b_{\text{H}}+c_{\text{H}})-a_{\text{H}}\sin \omega =0;\end{array}$$ [S30]

that is,

$$\begin{array}{c}\hfill -{\omega }^2+\overline{\beta }({\displaystyle \frac{7}{6}}{\overline{k}}_{\text{m}}+2{\overline{K}}_{\text{a}}{\lambda }^2)+{\displaystyle \frac{7}{6}}{\overline{k}}_{\text{m}}\overline{\beta }{\displaystyle \frac{n{K}_{\text{L}}}{K_{\text{L}}+{\lambda }^n}}\cos \omega =0,\\ \hfill \omega (\overline{\beta }+{\displaystyle \frac{7}{6}}{\overline{k}}_{\text{m}}+2{\overline{K}}_{\text{a}}{\lambda }^2)-{\displaystyle \frac{7}{6}}{\overline{k}}_{\text{m}}\overline{\beta }{\displaystyle \frac{n{K}_{\text{L}}}{K_{\text{L}}+{\lambda }^n}}\sin \omega =0,\end{array}$$ [S31]

where $\omega >0$ is the angular frequency of the dynamic system at the Hopf bifurcation. Beyond the Hopf bifurcation, the central cell, as well as the boundary cells, undergoes spontaneous sustained oscillation. The characteristic angular frequency ${\omega }_{\text{H}}$ at such a critical state is given by

$${\omega }_{\text{H}}^2=-\frac{1}{2}(b_{\text{H}}^2+c_{\text{H}}^2)+\frac{1}{2}\sqrt{{(b_{\text{H}}^2-c_{\text{H}}^2)}^2+4a_{\text{H}}^2}.$$ [S32]

The coefficients in Eqs. S30 and S31 should satisfy the following constraints for the existence of ${\omega }_{\text{H}}$, which leads to the necessary conditions for the occurrence of Hopf bifurcation,

$$\{\begin{array}{c}{\omega }_{\text{H}}^2>0,\hfill \\ \frac{\sin {\omega }_{\text{H}}}{{\omega }_{\text{H}}}<1.\hfill \end{array}$$ [S33]

Substituting Eqs. S27, S30, and S32 into Eq. S33, the necessary conditions are recast as

$$\{\begin{array}{c}1+\frac{12}{7}\frac{{\overline{K}}_{\text{a}}{\lambda }^2}{{\overline{k}}_{\text{m}}}<\frac{n{K}_{\text{L}}}{K_{\text{L}}+{\lambda }^n},\hfill \\ 1+\frac{6\overline{\beta }}{7{\overline{k}}_{\text{m}}}+\frac{12}{7}\frac{{\overline{K}}_{\text{a}}{\lambda }^2}{{\overline{k}}_{\text{m}}}<\overline{\beta }\frac{n{K}_{\text{L}}}{K_{\text{L}}+{\lambda }^n}.\hfill \end{array}$$ [S34]

Inserting ${\overline{k}}_{\text{m}}=6{\stackrel{\sim }{k}}_{\text{m}}(K_{\text{L}}+1){\lambda }^n/(K_{\text{L}}+{\lambda }^n)$ into Eq. S34 yields

$$\{\begin{array}{c}1+\frac{2}{7}\frac{(K_{\text{L}}+{\lambda }^n){\overline{K}}_{\text{a}}}{{\stackrel{\sim }{k}}_{\text{m}}(K_{\text{L}}+1){\lambda }^{n-2}}<\frac{n{K}_{\text{L}}}{K_{\text{L}}+{\lambda }^n},\hfill \\ 1+\frac{(K_{\text{L}}+{\lambda }^n)[\overline{\beta }+2{\overline{K}}_{\text{a}}{\lambda }^2]}{7{\stackrel{\sim }{k}}_{\text{m}}(K_{\text{L}}+1){\lambda }^n}<\overline{\beta }\frac{n{K}_{\text{L}}}{K_{\text{L}}+{\lambda }^n}.\hfill \end{array}$$ [S35]

The two inequations in Eq. S35 designate the necessary conditions for the onset of sustained oscillations in the minimal system. For a preliminary estimate, inserting ${\stackrel{\sim }{K}}_{\text{a}}=10.0$, ${\stackrel{\sim }{K}}_{\text{c}}=1.0$, and $\stackrel{\sim }{\mathrm{\Lambda }}=0$ into Eqs. S11 and S12, we have $R_{\text{s}}/R_{\text{a}}\approx 0.5542$, and thus ${\overline{K}}_{\text{a}}=(21/8){\stackrel{\sim }{K}}_{\text{a}}(R_{\text{s}}^2/A_0)=(21/8){\stackrel{\sim }{K}}_{\text{a}}[(2/3\sqrt{3}){(R_{\text{s}}/R_{\text{a}})}^2]\approx 3.1036$. Further taking $K_{\text{L}}=1$ and $\lambda =1$, we obtain $n>3.77$ from the first line of Eq. S35. In addition, inserting these parameter values into the second line of Eq. S35, we get $n>1/7+3.77/\overline{\beta }$, which indicates that $\beta$ tailors the dynamics in the seven-cell system. In the present study, we take $n=8$.

Simulation of the Drosophila Dorsal Closure Process

During Drosophila embryogenesis, the amnioserosa tissue (the eye-shaped cells highlighted using green color in Fig. 1A in the main text) exhibits distinct and sustained collective cell oscillations from the end of germband retraction to the early stage of dorsal closure. As dorsal closure proceeds, however, the oscillation in the whole amnioserosa decays until no shape oscillations can be detected (18, 35). During dorsal closure, the amnioserosa monolayer contracts actively, with the assistance of a supracellular actin cable along the monolayer boundary. In addition, it was revealed that the myosin activity in amnioserosa cells increases as dorsal closure proceeds (35). This could stem from the increase in the myosin activation coefficient $\alpha$ (38), which leads to an increase in myosin activation (Eq. 2 in the main text) and thus elevates the myosin activity in amnioserosa cells. In turn, such an increase of myosin activity enhances the contractility of the amnioserosa cells and promotes the contraction of amnioserosa. The increase in amnioserosa contraction then promotes the sewing up of lateral epidermis (cells surrounding the amnioserosa in Fig. 1A in the main text), which incurs a mechanical resistance against the dorsal closure. Such competition between the active contraction of the amnioserosa and the resistance from the lateral epidermis gives rise to an elevated tensile stress in the amnioserosa tissue as dorsal closure proceeds (36). Based on the above biological facts and also on our theoretical studies of the minimal system, we argue that the gradual disappearance of cell oscillations in the amnioserosa could be attributed to the increasing tensile stress, built up by the force balance between the active amnioserosa contraction and the passive epidermis resistance. To verify our hypothesis, we simulate the process of dorsal closure.

First, we investigate how the active contraction of the amnioserosa (increasing myosin activation coefficient $\alpha$) and the resistance from the lateral epidermis regulate the collective oscillatory dynamics in the amnioserosa. To quantify the resistance from the lateral epidermis, we introduce the total strain to describe the shrinkage of the amnioserosa

$$\epsilon =(L^{\text{as}}-L_0^{\text{as}})/L_0^{\text{as}},$$ [S36]

where $L_0^{\text{as}}$ is the initial perimeter of the monolayer and $L^{\text{as}}$ is the perimeter after contraction. Then we additively decompose the total strain as $\epsilon ={\epsilon }_{\text{c}}+{\epsilon }_{\text{e}}$, where ${\epsilon }_{\text{c}}$ is the contractile strain induced by active contraction of amnioserosa in the absence of lateral resistance, and ${\epsilon }_{\text{e}}$ is the elastic strain describing the deformation due to resistance from the lateral epidermis and thus quantifies such resistance. When there is no boundary resistance, $\epsilon ={\epsilon }_{\text{c}}$ and thus ${\epsilon }_{\text{e}}$ vanishes, leading to a stress-free state in the monolayer. Fig. S3A shows a phase diagram that illustrates how the myosin activation coefficient $\alpha$ (quantification of the active contraction of the amnioserosa) and the elastic strain ${\epsilon }_{\text{e}}$ (quantification of the resistance from the lateral epidermis) tailor the cellular dynamics in the amnioserosa. It is seen that the rising tensile elastic strain (or, equivalently, tensile stress) gradually eliminates collective cell oscillations. Cell oscillations could be completely suppressed when the elastic strain is sufficiently large, similar to what happens in the minimal system and the cell monolayer under external stretches. When the lateral resistance is completely relaxed, the amnioserosa cells will undergo sustained oscillations. Although the elastic strain hinders collective cell oscillations in the amnioserosa, it tends to enhance the myosin activity (Fig. S3B) because of the myosin activation by cell stretch (Eq. 2 and Fig. 1D in the main text).

Second, we simulate the dynamic process of dorsal closure. We focus on the phase from early stage to middle stage, during which cell oscillations disappear gradually and no obvious cell rearrangements (e.g., cell intercalation and apoptosis) take place (18, 21). Therefore, we do not consider cell rearrangements in this study. In the simulations, the myosin activation coefficient $\alpha$ is gradually enhanced while the myosin deactivation rate $\beta$ is fixed. Simultaneously, the boundary of the monolayer is allowed to move inward progressively while keeping the elliptic shape of the monolayer, corresponding to an increasing total strain $|\epsilon |$. The evolution of total strain $\epsilon$ is selected according to the following criteria: (i) Choose an evolution route $(\alpha ,{\epsilon }_{\text{e}})$, both of which increase with time (corresponding to increasing active contraction of the amnioserosa and increasing resistance from the lateral epidermis, respectively), as shown in Fig. S3C; (ii) calculate the contractile strain ${\epsilon }_{\text{c}}$ according to the increasing $\alpha$, and compute the total strain $\epsilon ={\epsilon }_{\text{e}}+{\epsilon }_{\text{c}}$; and (iii) simulate the dorsal closure as mentioned earlier according to the evolution route $(\alpha ,\epsilon )$, as shown in Fig. S3C. It shows that the tensile stress ${\sigma }_{\text{m}}^{(\text{as})}$ increases in the amnioserosa as dorsal closure proceeds (Fig. S3D). Meanwhile, the oscillation amplitudes of both cell area and myosin activity decay gradually (Fig. 3B in the main text). Although its oscillation fades away, the myosin activity increases, because the enhanced tension can promote myosin activity through the chemomechanical feedback (Eq. 2 and Fig. 1D in the main text). These results agree well with experimental observations (18, 35), suggesting that the gradual disappearance of cell oscillations in the amnioserosa as dorsal closure proceeds could be the direct consequence of the increasing tensile stress built up due to the force balance between the active contraction of amnioserosa and the resistance from lateral epidermis.

Discussion

Living systems are generally built from a large quantity of individual elements, such as cells, whose biochemical and biophysical dynamics need to be coordinated for collective functions. Although active and passive chemical transportations play essential roles in biological regulation, the speed of material transport, which often takes hours (1, 45), can hardly satisfy the needs of coordinating many fast morphodynamic behaviors of individual cells over large length scales. Therefore, other ways of transmitting regulatory information have to be used. It has been experimentally shown that mechanical tension can serve as a global inhibitor to help an elongated cell quickly establish its polarity over a length scale of tens of micrometers (46). Here, we rigorously show that mechanical forces from a stiff anisotropic boundary can activate, organize, and synchronize individual cells’ oscillatory morphodynamics within the entire Drosophila amnioserosa tissue, even in the absence of chemical exchange and transportation.

It has been shown that the biomechanical environment can largely influence a cell’s gene expression, proliferation, and fate (47). We believe that the coordinated oscillatory morphodynamics, including the aligned oscillatory patterns, the polarized oscillation directions, and the synchronized oscillatory amplitudes, may help establish globally coordinated biomechanical environments for cells that are spatially separated, so that their cell cycles and fates can be coordinated as well. This is of extreme importance for embryonic development. In this way, the amnioserosa oscillation may serve as a pacemaker to control the process of embryogenesis.

Recently, critical progress has been made in understanding how the mechanical behavior of tissues depends on single-cell properties. It has demonstrated that higher cell–cell adhesion can weaken the overall rigidity of the entire tissue (48, 49). In our work, we pursue the reverse question of how single-cell behaviors are regulated by the physical properties of tissues. Importantly, the key concept of active mechanical contraction in our model has recently been experimentally observed (37, 39). We expect that this morphodynamics framework will also help experimentalists develop other testable hypotheses. For example, it will be interesting to experimentally test the predicted gating mechanism that the tensile stress determines the occurrence of Hopf bifurcation, which activates/deactivates the autonomous amnioserosa oscillation. We believe that the “tissue-to-cell” mor-phodynamic analysis presented in this paper provides a unique perspective to understand the precise regulation of large-scale living systems.