Mechanochemical dynamics of collective cells and hierarchical topological defects in multicellular lumens

Abstract

Collective cell dynamics is essential for tissue morphogenesis and various biological functions. However, it remains incompletely understood how mechanical forces and chemical signaling are integrated to direct collective cell behaviors underlying tissue morphogenesis. Here, we propose a three-dimensional (3D) mechanochemical theory accounting for biochemical reaction-diffusion and cellular mechanotransduction to investigate the dynamics of multicellular lumens. We show that the interplay between biochemical signaling and mechanics can trigger either pitchfork or Hopf bifurcation to induce diverse static mechanochemical patterns or generate oscillations with multiple modes both involving marked mechanical deformations in lumens. We uncover the crucial role of mechanochemical feedback in emerging morphodynamics and identify the evolution and morphogenetic functions of hierarchical topological defects including cell-level hexatic defects and tissue-level orientational defects. Our theory captures the common mechanochemical traits of collective dynamics observed in experiments and could provide a mechanistic context for understanding morphological symmetry breaking in 3D lumen–like tissues.

The mechanochemical interplay causes diverse patterns and hierarchical topological defects in multicellular lumens.

INTRODUCTION

Collective cell dynamics refers to the coordinated behaviors of multiple cells, which occur across various natural processes, including embryonic development (1), tissue morphogenesis (2, 3), wound healing (4, 5), and cancer progression (6). One of the most fascinating features during these processes is the spatiotemporal coordination of the biochemical signaling and mechanical forces. On the one hand, precise and timely biochemical signals guide mechanical cues to reliably control tissue morphodynamics (7, 8). Collective cells can sense extracellular chemical gradients and undergo directed movements following the gradients, known as chemotaxis, which underlies critical physiological functions or pathological processes (9, 10). On the other hand, rich collective behaviors, such as cell deformations, migrations (11), rearrangements (12), and oscillations (13), often generate mechanochemical patterns (14–16) and may also influence biochemical signaling through mechanosensitive pathways (17). Because of the intricate interconnections involved, current understanding of these dynamic behaviors is limited.

A wide spectrum of intercellular wave patterns, both temporal and spatial, have been identified in multicellular systems spanning various biochemical intricacies (13, 18–23). In morphogenesis of murine cochlear duct, extracellular signal–regulated kinase (ERK) activation waves have been found to control cell migration and result in the duct elongation (Fig. 1A) (19). During Drosophila oogenesis, directional actomyosin and cell shape oscillations occur at the basal domain of ovarian epithelial follicle cells (Fig. 1B) (13), displaying rapid global pulse (Fig. 1C), which contributes to the tissue elongation and is regulated by an upstream biochemical guanosine triphosphatase Rho regulatory network. Collective spot-like ERK signaling waves with high frequency have been observed in the outer layer of mammary acini, which encode the cell fate for the later lumen formation (Fig. 1D) (20). Furthermore, numerous other signaling pathways, such as Wnt (23), Notch (16), and Ca²⁺ signals (21, 24), behave as system-wide wave-shaped regulators that play crucial roles in tissue morphogenesis. While the upstream biochemical messengers may differ between different systems, a common hallmark in these systems is the occurrence of downstream mechanical responses. The most predominant characteristic of these mechanical responses is the active contraction of actomyosin assemblies, which occurs either through the phosphorylation of the myosin light chain activated by calcium/calmodulin complexes (24, 25) or via the G protein–mediated Rho pathway (13, 26) that could be triggered by ERK activity (18, 20, 27).

Fig. 1. Examples of intercellular wave patterns and collective cell dynamics observed in lumen-like structures during development.

(A) Representative time-lapse snapshots of ERK activity maps in the floor plane of the cochlear duct [adapted from (19) with permission]. (B) Global change in basal myosin during Drosophila egg chamber development. Myosin accumulation is monitored by using the red fluorescent protein mCherry fused to the myosin regulatory light chain named Spaghetti Squash (Sqh-mCherry) [adapted from (13) with permission]. (C) Time-lapse series of one representative oscillating cell labeled with Sqh-mCherry, which shows that stage 9 follicle cells undergo rapid periodic contractions and myosin accumulation [adapted from (13) with permission]. (D) Representative time-series micrographs of the collective ERK wave mainly in outer cell layers in stage 2 during mammary acinar morphogenesis. In stage 3, the acinus consists of the outer cell layer and apoptosis of inner cells, allowing further formation of a hollow lumen (dashed white box) [adapted from (20) with permission].

Turing’s reaction-diffusion theory (28) provides a chemical basis for stationary pattern formation through pitchfork bifurcation, which involves the breaking of spatial symmetry. Moreover, under specific conditions, the reaction-diffusion system can also exhibit periodic oscillations or traveling waves through a Hopf bifurcation, involving the breaking of time symmetry (29). However, these chemical oscillatory mechanisms necessitate specific bifurcation conditions (30) and do not involve mechanical forces per se and, thereby, cannot capture the geometrical deformations in living systems. Recent experimental evidences (18, 20, 31) and theories (32–34) suggest that mechanochemical feedback is essential for chemical patterning coupled with shape deformation. Although some theoretical works on the interplay between biochemical and mechanical components based on continuous descriptions (35–41) or discrete models (34, 42–46) have been developed in two-dimensional (2D) monolayers or single-cell systems, how mechanical forces and chemical signaling at the cellular level are integrated and transmitted to collective cell dynamics that enables the establishment of 3D morphogenetic patterns of tissues remain to be explored.

Here, we strive for a theoretically simple, 3D framework to capture the common mechanisms underlying diverse collective cell dynamics and mechanochemical pattern formation in tissues. We develop a mechanochemical theory based on a 3D vertex model to investigate the collective dynamics emerging in multicellular lumens or lumen-like structures. We demonstrate that the mechanochemical interplay can generate both static patterns and dynamic oscillations and waves through distinct bifurcations that are mediated by mechanochemical feedback. Growing experiments indicate that topological defects at different length scales in epithelial layers could serve as a key in many cellular processes (47–49). Thus, the functions of hierarchical topological defects, including hexatic and orientational defects at different scales, in mechanochemical patterning are identified. Our results underscore the inseparable convolution of biochemical signaling and mechanics in sculpting tissue morphogenesis.

RESULTS

3D mechanochemical model

During development, many tissues or organs display a lumen-like structure, where the outer multicellular epithelial sheet envelop fluids or a fluid-like interior that provides mechanical supporting for the outer sheets through hydrostatic pressure (2). Propagation of biochemical signals together with collective cell dynamics and deformation has been often observed in these lumen-like structures (Fig. 1) (13, 19, 20). Motivated by these experimental observations, we construct a 3D vertex model that consists of a single cell–thick epithelial shell and a fluid interior to examine the mechanochemical dynamics in multicellular systems, where all cells are represented by 3D prisms or frusta whose number of sides in apical and basal polygonal faces are the same, but the area may be different (Fig. 2A). Vertex models have been widely used to study the dynamics of 2D monolayers (42, 45, 50–53) and 3D multicellular structures (54–58). Different from the traditional approaches, the Schnakenberg model (59), which is one of the simplest reaction-diffusion theory and describes an autocatalytic reaction, is integrated into our 3D vertex model to account for the upstream biochemical signaling network. In this model, the concentration fields of signal protein, c_A, and its chemical reactants, c_R, are dispersed in cells. The signal protein represents a class of signals such as ERK and calcium, which can regulate force-generating molecules to activate the downstream contraction of actomyosin assemblies of cells (24, 27). The evolution of the signal protein, mediated by mechanochemical feedback, is governed by

$$\frac{\mathrm{d}{c}_{\mathrm{A}}^{\left(J\right)}}{\mathrm{d}t}=D_{\mathrm{A}}\sum _{I_{\left(J\right)}}\left[c_{\mathrm{A}}^{\left(I\right)}-c_{\mathrm{A}}^{\left(J\right)}\right]+k_{\mathrm{A},\text{on}}{c}_{\mathrm{A}}^{\left(J\right)2}{c}_{\mathrm{R}}^{\left(J\right)}+a-k_{\mathrm{A},\text{off}}{c}_{\mathrm{A}}^{\left(J\right)2}-\mathrm{\rho }f\left(\mathrm{\lambda }\right)c_{\mathrm{A}}^{\left(J\right)}$$ (1)

$$\frac{\mathrm{d}{c}_{\mathrm{R}}^{\left(J\right)}}{\mathrm{d}t}=D_{\mathrm{R}}\sum _{I_{\left(J\right)}}\left[c_{\mathrm{R}}^{\left(I\right)}-c_{\mathrm{R}}^{\left(J\right)}\right]-k_{\mathrm{R},\text{off}}{c}_{\mathrm{A}}^{{\left(J\right)}^2}{c}_{\mathrm{R}}^{\left(J\right)}+r$$ (2)

where $c_{\mathrm{A}}^{(J)}$ and $c_{\mathrm{R}}^{(J)}$ are the concentrations of activating signal protein and its chemical reactants in the Jth cell, respectively. The first terms in Eqs. 1 and 2 represent the diffusion due to intercellular signal propagation from the I_(J)th to the Jth cell, with diffusion rates D_A and D_R, respectively, where I_(J) denotes the cell adjacent to the Jth cell. The second terms describe the effects of chemical reactions, including autocatalysis of $c_{\mathrm{A}}^{(J)}$ and consumption of $c_{\mathrm{R}}^{(J)}$ , where $k_{\mathrm{A},\text{on}}$ is the self-activation rate of signal protein and $k_{\mathrm{R},\text{off}}$ the consumption rate of its chemical reactants. The positive constants a and r are the production rates of signal protein and its chemical reactants, respectively. $k_{\mathrm{A},\text{off}}$ is the constant deactivation rate of signal protein. The last term in Eq. 1 represents the deactivation caused by mechanochemical feedback, where ρ denotes the feedback intensity. This term characterizes the negative feedback of cellular deformation on biochemical signaling that activates cell contraction, as observed in previous experiments (18, 24, 34). The mechanochemical feedback is quantified by a Hill function (32)

$$f\left(\mathrm{\lambda }\right)=\frac{{\left[{\mathrm{\lambda }}_{\mathrm{r}}-{\mathrm{\lambda }}^{\left(J\right)}\right]}^n}{K_{\mathrm{A}}+{\left[{\mathrm{\lambda }}_{\mathrm{r}}-{\mathrm{\lambda }}^{\left(J\right)}\right]}^n}$$ (3)

where λ^(J) = V_J/V₀ is the volume fraction of the Jth cell to the preferred volume and λ_r is the reference deformation constant. K_A represents the apparent dissociation constant of the activating signal protein, and n is the Hill coefficient (see the Supplementary Materials for details). In this way, we bridge the field of signaling and deformation of cells (Fig. 2B). Because of the osmotic equilibrium, alterations in cortical tension typically lead to cell volume changes less than 10% (60). Meanwhile, experimental observations have suggested that individual cell volume decrease can generate tissue contraction (61). To focus on mechanochemical interplay, we disregard the role of osmotic pressure in maintaining volume while allow for volume deformations of cells and tissue contraction.The potential energy of the multicellular lumen can be described as (42, 54)

$$U={\displaystyle \sum _J}\frac{1}{2}{K}_{\mathrm{v}}{(V_J-V_0)}^2+{\displaystyle \sum _{<I,J>}}{K}_{\mathrm{s}}{A}_{I,J}+{\displaystyle \sum _J}\frac{1}{2}{k}_{\mathrm{m}}{m}_J{A}_J^2+\frac{1}{2}{K}_{\mathrm{v}}^{\mathrm{l}}{(V^{\mathrm{l}}-V_0^{\mathrm{l}})}^2$$ (4)

where the four terms denote, respectively, cell volume energy, interfacial energy, active contraction, and lumen volume energy. K_v is the volume stiffness of the cell, K_s denotes the interfacial tension, A_I,J represents the area of the surface shared by the Ith and Jth cell, and ∑_<I,J> stands for the summation over all neighboring cells. Actomyosin is assumed to be recruited to the surface of the cell with area A_J to generate an active surface tension and cell contraction. $m_J=\mathrm{\mu }{c}_{\mathrm{A}}^{(J)}$ measures the actomyosin activity of the Jth cell, where μ is the conversion coefficient from signal protein concentration to actomyosin activity. $k_{\mathrm{m}}$ describes the elastic coefficient for the actin cortex. $K_{\mathrm{v}}^{\mathrm{l}}$ is the volume stiffness of the lumen whose current volume and preferred volume are V^l and $V_0^{\mathrm{l}}$ , respectively.

Fig. 2. 3D mechanochemical vertex model and linear stability analysis.

(A) Schematics of a multicellular lumen, a representative cell J, and a seven-cell system. The lumen is surrounded by confluent cells, in which the actomyosin cortex consisting of actin filaments and myosin motors is considered. The deformations and movement of cells are described by the motion of vertices. ${\mathbf{v}}_i$ is the velocity of vertex i, ${\mathbf{F}}_i^{\mathrm{r}}=-\mathrm{\eta d}{\mathbf{r}}_i/\mathrm{d}t$ , and ${\mathbf{F}}_i^{\mathrm{e}}=-\mathrm{\partial }U/\mathrm{\partial }{\mathbf{r}}_i$ are the dissipative force and potential force acting at the vertex, respectively. In the seven-cell system, the boundary vertices of cells are marked red, and the central vertices are marked orange. (B) Mechanochemical pathway that involves the chemical reaction, cell deformation, and mechanochemical negative feedback. (C) 3D morphological diagram of the seven-cell system as a function of k_m, D_A, and ρ (movie S1). (D) 2D cross sections of morphological diagram by fixing one parameter. The orange and blue lines correspond to the pitchfork and Hopf bifurcation, respectively.

For an overdamped multicellular system, the motion of vertex i obeys the Langevin equation:

$$\mathrm{\eta }\frac{\mathrm{d}{\mathbf{r}}_i}{\mathrm{d}t}=-\frac{\mathrm{\partial }U}{\mathrm{\partial }{\mathbf{r}}_i}$$ (5)

where η denotes the damping coefficient and r_i is the position of vertex i (Fig. 2A). We first theoretically analyze a single cell (Fig. 2A) to preliminarily examine the effects of mechanical, chemical, and geometric properties on the mechanochemical equilibrium of a cellular unit (see the Supplementary Materials for details). As expected, with low stiffness, high contractility, or small thickness, the cells are likely to experience large deformations, while the strong mechanochemical feedback hinders deformation by inhibiting the reactions of signal protein (fig. S2).

Linear stability analysis

To gain theoretical insights into the mechanochemical states, we perform stability analysis of a seven-cell unit, which is a simplified minimal system containing a regular hexagonal frustum surrounded by and linked to six others (Fig. 2A). The mechanochemical dynamics of the minimal seven-cell system can be fully described by five independent variables (see the Supplementary Materials for details). By introducing infinitesimal perturbations to the stationary state of the seven-cell system, we deduce the incremental dynamics of the mechanochemical system and then analyze the eigenvalues of the Jacobian matrix to gain different states. As an example, fig. S3 illustrates the emergence of Hopf bifurcation, where the maximum real part of the eigenvalue shifts from negative to positive as the mechanochemical feedback is enhanced.

To comprehensively capture the morphodynamics in the system, we construct a 3D phase diagram as a function of chemical diffusivity D_A, mechanical contractility k_m, and mechanochemical feedback ρ (Fig. 2C and movie S1). The 3D phase diagram and its 2D cross section (Fig. 2D) reveal that there exist three possible states: stable, Turing instability, and periodic oscillation. Specifically, when the intensity of ρ is low, the system primarily displays the characteristics of a biochemical reaction-diffusion system, in which a low D_A triggers pitchfork bifurcation, resulting in Turing patterns with nonuniform chemical concentrations. These chemical signals can induce actomyosin contraction, giving rise to nonuniform mechanical deformations. In contrast, a high D_A favors a stable state. When ρ increases up to exceeding a certain threshold, the system undergoes Hopf bifurcation, yielding periodic oscillations that involve the contributions from both chemical signaling and mechanical forces. The strong contractility of cells is conducive to mechanical deformation and, thereby, favors collective oscillations. These results suggest that two types of bifurcations mediated by mechanochemical feedback break spatial and temporal translation symmetry, respectively, giving rise to Turing patterns and periodic oscillations concomitantly with mechanical deformations.

Morphodynamics induced by mechanochemical instability

We next perform numerical simulations to examine the 3D morphodynamics of the whole multicellular lumen based on the mechanochemical model described above. Multicellular dynamics in simulations involves three processes: (i) intra- and intercellular chemical reactions and diffusion, (ii) vertex movements and cell deformations along the energy gradient, and (iii) cell rearrangements based on a reversible network reconnection (55). In the initial state, a configuration of 400 cells that interconnect to form an isotropic confluent epithelial shell is set up by cell growth and division randomly from a small multicellular lumen. All cells are in a stationary state with a uniform chemical concentration of signal protein and reactants. Time integration is performed using the explicit Euler method (Materials and Methods). Our simulations produce rich mechanochemical patterns in the multicellular lumen, both static and dynamic (Fig. 3A). These patterns are collected in a 3D phase diagram, as a function of D_A, k_m, and ρ (Fig. 3B and movie S6). The in silico results are qualitatively consistent with those predicted by the linear stability analysis in the simplified seven-cell system (Fig. 2, C and D).

Fig. 3. Mechanochemical instability–induced static patterns and dynamic oscillations.

(A) Mechanical deformations in different static or dynamic states obtained from simulations. Three static states include (I) spot pattern, (II) stripe pattern, and (III) uniform pattern. Four dynamic oscillatory modes include (IV) spot wave (movie S2), (V) spin wave (movie S3), (VI) traveling wave (movie S4), and (VII) global fluctuation (movie S5). Parameters used in simulations are: (I) (k_m, ρ) = (0.04,0.3), D_A = 0.3 for six spots, D_A = 0.4 for five spots, and D_A = 0.5 for four spots; (II) (k_m, D_A, ρ) = (0.04,0.8,0.3); (III) (k_m, D_A, ρ) = (0.04,1.2,0.3); (IV) (k_m, D_A, ρ) = (0.04,0.4,3.0); (V) (k_m, D_A, ρ) = (0.04,1.0,3.0); (VI) (k_m, D_A, ρ) = (0.08,1.0,3.0); (VII) (k_m, D_A, ρ) = (0.05,1.0,3.0). (B) 3D morphological diagram obtained from simulations (movie S6).

Static patterns

It has been shown that myosin-II patterns can induce curvature variation in intestinal organoids (14). Our results demonstrate that static patterns of mechanical deformations and actomyosin activity can be engendered (Fig. 3A and fig. S4) when the mechanochemical feedback is weak (see ρ = 0.3 in Fig. 3B for example). By decreasing D_A, the pitchfork bifurcation takes place, and the uniform stationary state (III) gives way to spatially periodic states with either stripe pattern (II) or spot pattern (I) (fig. S4). This Turing-type instability is caused by an increase in the relative diffusion coefficient D_R/D_A, known as “diffusion-induced instability” in classical Turing theory. The wavelength of the emerging pattern is controlled by the dynamic diffusion process. Specifically, six-, five-, and four-spot and stripe patterns can be generated sequentially with the decrease of D_R/D_A. However, in stark contrast to pure chemical process, Turing-type instability in our system not only induces spot or stripe actomyosin patterns but also causes nonuniform deformations and symmetry breaking in the geometrical configuration due to the mechanochemical interplay (Fig. 3A), which can be considered as mechanical instability in the multicellular lumen. Therefore, the mechanochemical instability can render the formation of static bud-shaped and disk-shaped patterns, which hold potential in driving 3D morphogenetic process from spherical to complex branching shapes.

Dynamic oscillations

When the mechanochemical feedback is strong, the system is likely to undergo Hopf bifurcation, which leads to dynamic oscillations with multiple modes, including spot waves (IV), spin waves (V), traveling waves (VI), and global fluctuations (VII) (Fig. 3A and fig. S4). When the mechanochemical feedback ρ = 3.0, for example, oscillations caused by mechanochemical instability are dominant (Fig. 3B). The emergence of different modes in simulations can be attributed to the variations in mechanochemical properties. With a low D_A, a strong ρenables the signal of active contraction to propagate regularly among collective cells in the form of spots (mode IV in Figs. 3A and 4A and movie S2). As D_A increases and k_m is weak, the stripe-shaped pattern emerges and rotates periodically (mode V in Fig. 3A and movie S3), yielding spin-wave oscillations (Fig. 4B). With further increasing D_A and k_m, global fluctuations induced by Hopf bifurcation happens (mode VII in Figs. 3A and 4D and movie S5). This dynamic mode is spatially homogeneous but temporally oscillatory. When k_m, D_A, and ρ are large enough, the system favors traveling waves (mode VI in Fig. 3A and movie S4), characterized by random changes in both wavelength and direction (Fig. 4C). These dynamic modes can be found in experiments, such as spot-like ERK wave in epithelial cells of mammary acinar (20), rapid traveling calcium waves during gastrulation of Oikopleura dioica (21), and the global fluctuations of Drosophila follicle cells (13). Phase diagram in Fig. 3B also indicates that when ρ is relatively strong but D_A or k_m is weak, the patterns remain static, which suggests that both chemical diffusion and mechanical contraction are indispensable for the collective oscillations.

Fig. 4. Oscillatory dynamics in collective cells.

(A to D) Kymographs of different actomyosin patterns in the spherical coordinate system (Materials and Methods), where t denotes the dimensionless time. (θ, φ) in (D) is the spherical coordinate system of the lumen. (E) Maximum activity of actomyosin in collective cells under different D_A and k_m, where ρ = 3.0. (F) Concentration of actomyosin in a representative cell over time. (G) Mechanochemical oscillations in a representative cell J.

To evaluate the oscillation intensity in different dynamic modes, we calculate the maximum intensity of actomyosin activity in collective cells (Fig. 4E). It shows that the lower D_A and k_m will give rise to stronger oscillations after bifurcation. Furthermore, we examine the characteristics of oscillatory dynamics at the cellular scale. The actomyosin activity intensity inside and the deformation of a representative cell over time are plotted in Fig. 4F. Markedly, spot and spin waves exhibit robust oscillations with large amplitudes and low frequencies; global fluctuations display high frequency and small amplitude; traveling waves, however, display random oscillations with varying amplitudes and rapid propagation velocity as a result of the varying oscillatory phases coupling across different spatial regions. From the temporal correlation between the mechanical deformation and biochemical signaling, it can be seen that there is a negative phase shift between c_A and λ (Fig. 4G), which results from the negative feedback–facilitated oscillator (32). Meanwhile, the two chemical concentrations, c_A and c_R, are also out of phase, and the concentration of reactants is higher than that of signal protein. This arises from the oscillatory characteristic of the self-activator–depleted reaction kinetics. These in silico results are consistent with our linear stability analysis and reveal the mechanochemical mechanisms of diverse collective cell dynamics in the multicellular lumens.

Oscillation-facilitated reduction of hexatic defects

We next explore the cellular organization during oscillations. Topologically, the hexagonal crystal structure is considered to be stable and ideal in a planar monolayer system. On a spherical surface, however, nonhexagonal units are unavoidable and considered as hexatic disclinations or defects, which constrain the crystal order to conform to the geometric shape (62). In the in silico multicellular lumen, because of the randomness of initial cell growth and division, these nonhexagonal hexatic defects at the cellular level are engendered in both the apical and basal surfaces. Distinct from the defect singularities that are hard to be captured by continuum theory, the hexatic defects at the cellular and supracellular levels can be exactly described by cell shape and arrangement in our cell-based vertex model. Therefore, their energy is described by the potential energy in Eq. 4 without additionally introduced. The presence of nonhexagonal cells can induce local nonuniform stress that affects the homeostatic state of tissues. We track the development and evolution of hexatic defects during oscillations of mode IV whose oscillatory intensity is high (Fig. 3A). In the beginning, there are some nonhexagonal cells such as pentagons and heptagons in the multicellular lumen. As oscillations start, the proportion of hexagons increases, while the numbers of pentagons and heptagons decrease, eventually reaching a stable state (Fig. 5A). Meanwhile, all squares and octagons disappear.

Fig. 5. Scutoid-based topological transitions during oscillations.

(A) Distribution of the shapes of cell basal surfaces during oscillations. (B) 2D phase diagram of (left) oscillation intensity and (right) the number of reduced crystallography defects as a function of D_A and ρ. (C) Evolution of four-cell energy E₄ during the scutoid-based topological transition. The yellow region represents the state when the scutoids appear, while the blue regions represent the state when the prisms or frusta exist only. (D) A complete process of a representative scutoid-based topological transition during oscillations (movie S7). The slant lateral surface of scutoid is marked in red.

To identify the influence of dynamic oscillations on the hexatic defects, we record the oscillation intensity and the number of defects under varying D_A and ρ. It shows that only when the oscillations happen with high intensity can the defects be reduced efficiently (Fig. 5B). In the nonoscillating state with low D_A and ρ, the proportion of polygons barely changes despite the high actomyosin activity in cells. While in weak oscillations of mode V or VII, the number of defects remains almost unchanged. The numerical results also suggest that, despite the high oscillation intensity in traveling wave (mode VI), the defects rarely reduce. This phenomenon may be attributed to the considerably faster intercellular wave propagation compared with the cellular deformations, which implies a potential requirement for the match of time scales to reduce defects.

This evolution of hexatic topological defects at the cellular level is achieved through spontaneous topological transitions that enable the rearrangement of neighboring cells. Intriguingly, the topological transition involves the generation of scutoids, which, in contrast to prisms or frusta, indicate the disparity of side number between the apical and basal surface and present nonplanar lateral faces (Fig. 5D) absent in 2D and quasi-3D systems (63). These scutoids have been observed experimentally in 3D packing of nonplanar epithelia (64). Our simulations show that as the spot waves propagate, some scutoids emerge spontaneously and vanish subsequently through T1 topological transitions (55). A representative scutoid-based topological transition during oscillations is illustrated in Fig. 5D. Because of the nonuniform contraction in the tissue, the two cells with heptagon-shaped apical and basal surfaces are stretched. Then, the apical surface undergoes T1 topological transition firstly, while the basal surface does not, giving rise to a 3D scutoid, where the apical and basal surfaces are not coordinated. The scutoid-shaped cells undergo continuous deformation with a slant lateral surface (marked by red) that extends progressively. After the slant surface extends across the lumen wall entirely, the basal surface completes T1 transition as well, resulting in prism-shaped cells with the same topology between the apical and basal surfaces (movie S7). Therefore, during the scutoid-based transition, both neighbor cell exchange and apico-basal intercalation take place. We calculate the energy of the four cells involved in the topological transition (Materials and Methods), as depicted in Fig. 5C. Different from the energy characteristics of T1 transition in 2D tissues (65), there exists a sharp energy decrease when the scutoids emerge. These results demonstrate that the scutoid-based topological transitions could act as a geometrical solution for cell migration and rearrangement in the curved lumen, which enables the cross-surface coordination and reduce hexatic defects in the 3D-curved tissue efficiently.

Activation and motion of aster-like topological defects

Cells in epithelial tissues can self-organize into nematic patterns that exhibit a long-range order, which may give rise to supracellular scale topological defects (66, 67), an orientational defect that is different from hexatic defects. We next investigate the spatiotemporal dynamics of the large-scale supracellular nematic texture in the lumen structures. The orientation field of the multicellular lumens is evaluated by calculating the long axes of cells (Materials and Methods).

Dynamic defect motion

Because of nonuniform cell contraction and deformations induced by mechanochemical interplay, aster-shaped +1 topological defects are activated within the regions that localize high actomyosin activity (Fig. 6A). As the actomyosin waves propagate, +1 topological defects follow them with a short time delay. The 3D kymograph shows spatiotemporally oscillations of the chemical patterns, which illustrate the trajectory of +1 topological defects (Fig. 6C and movie S8). The deformation field demonstrates that there exists a chain-like cell deformation wave ahead of the moving defects (Fig. 6B) and also propagates periodically with the defects (Fig. 6D and movie S9). The motion of defects is regulated by a synergistic interplay between biochemical diffusion and intercellular mechanotransduction (Fig. 6, C and D). Because the mechanochemical interplay and cellular activity may be modulated via chemical drugs, our results could provide a promising strategy to program defect dynamics in living multicellular lumens, complementary to the traditional control of topological defects (68).

Fig. 6. Spatiotemporal dynamics of topological defects resulting from cell orientations.

(A) Typical trajectory of a topological defect (+1) along the movements of actomyosin pattern in state (IV). The red solid circle marks the +1 defect, and the dashed line with an arrow represents the trajectory and its moving direction. (B) Displacement field of vertex, ∣dr_i∣, which characterizes the cell motion and deformation during oscillations. The white dashed line denotes a chain-like wavefront of deformation wave. (C and D) Kymographs of defect motion and deformation wave propagation during oscillations (Materials and Methods). The red and blue dashed lines describe the defect trajectory (movie S8). The yellow and green dashed lines describe the propagation of deformation wave (movie S9). (E) Experimental observation on the formation of topological defects during Hydra regeneration [adapted from (47) with permission]. (F and G) Evolution of the defect-induced morphogenesis in the growing multicellular lumen (movie S10). The actomyosin patterns and mean curvature (Materials and Methods) at different time are displayed, respectively. N is the total number of cells during lumen morphogenesis.

Topological defects as a mechanical morphogen

Tissue morphogenesis is characterized by a series of spatial and temporal modulations that occur in distinct stages. The role of topological defects in epithelial tissues has been proved to be crucial for development (47, 48, 69). For instance, in Hydra regeneration, the +1 defects are found to break the original spherical symmetry to drive the formation of tentacles (Fig. 6E) (47). Motivated by this, we probe the function of topological defects in the mechanochemical scenario by allowing cell division in simulations (Materials and Methods). Our theory provides a robust approach to examine topological defect dynamics and tissue morphogenesis simultaneously at single-cell resolution (Fig. 6, F and G, and movie S10). As the tissue grows, the mechanochemical interplay leads to actomyosin localization with spatial patterns, which cause localized deformations and induce +1 topological defects. This innate coupling between biochemical and mechanical fields is able to drive buckling around the defects, leading to out-of-plane tissue deformation and collective cell migration, which underlie the variation of local curvature and, thereby, tissue protrusion, recapitulating the defect-induced morphogenesis observed in experiments (47, 48). These results suggest that the mechanochemical interplay–induced bifurcation provides a blueprint to determine the locations where de novo symmetry breaking takes place, while topological defects serve as a mechanical morphogen that directly drives branching and subsequent tentacle elongation.

Together, mechanochemical feedback determines two types of defect motion: surface propagation (Fig. 6A) and out-of-plane protrusion (Fig. 6F), which mirror different morphological features and transport properties of topological defect. On the one hand, topological defects can move dynamically in the direction down the chemical concentration gradient and also along the orientation field (Fig. 6A). On the other hand, mechanical forces at the defects can result in the elongation and rearrangement of cells, steering collective cells to align with the tissue shape. These observed phenomena highlight the potential significance of mechanochemical interplay in the defect dynamics.

DISCUSSION

From system-wide spatiotemporal pattern to shape formation, cell and tissue mechanics can play crucial roles concurrently (31, 70). In this work, we have proposed a mechanochemical framework based on the 3D vertex model involving biochemical signaling and mechanical deformations to describe the self-organized collective cell dynamics and patterns in the multicellular lumens. Our results show that under the regulation of mechanochemical feedback, abundant static patterns and collective oscillations including spot waves, spin waves, traveling waves, and global fluctuations will emerge in the lumens. These findings highlight the essential role of mechanochemical interplay in tissue morphogenesis.

Our simulations not only capture deformations and rearrangements of cells but also quantify large-scale supracellular nematic texture during patterning of the multicellular lumens. Accompanied by periodic cell oscillations, paired nonhexagonal disclinations (e.g., pentagons and heptagons) gradually neutralize through the formation and elimination of scutoids, by which the lumen structures attain cross-surface coordination and reduce hexatic defects. As might be expected from the emergence of scutoids in 3D packing of epithelia (64), it is intriguing to find that scutoids, as a transition state, enable the multicellular systems to gain an energetically favorable configuration via the efficient path during oscillations. We expect that more experiments could validate this theoretical prediction. Furthermore, the mechanochemical instability–induced dynamic behaviors of topological orientational defects underlying a strategy of “defect spirograph” (71) in active systems of collective cells, while by weakening mechanochemical feedback, topological defects could act as a “mechanical morphogen” (47) to drive 3D complex morphogenesis in the multicellular lumens.

In this work, the simplified Schnakenberg model is used to describe the upstream biochemical signaling network. It is imperative to recognize that the intracellular signaling pathways [e.g., multitiered reaction networks (43, 72)] and their intercellular propagation [e.g., directional distributions influenced by tissue polarity (23, 73)] exhibit a level of intricacy beyond the scope of the Schnakenberg model. For the mechanical components, the way how cells contract or are stretched varies across different systems, encompassing the behaviors such as overall volume change (61), the maintenance of nearly constant volume (18), or deformations of the basal surface (74). In addition, the differences in time scales (75) and time delays (42) between chemical reactions and mechanotransduction may affect the mechanochemical feedback and thus collective cell dynamics. Last, previous studies have shown a crucial fact that luminal hydraulic pressure, as a long-range mechanical signal, can drive the hydraulically gated oscillations and coordinate tissue-level self-organization (76, 77), during which multiple cross-talk between mechanical and biochemical signaling happens (78). While our theory has not yet addressed these issues, the proposed theoretical framework can be specifically and easily extended to involve these factors and unveil mysterious morphodynamics in other multicellular structures. Notwithstanding the limitations, we believe that this work could deepen our understanding of collective dynamics of cells and topological defects underlying 3D tissue morphogenesis.

MATERIALS AND METHODS

Simulation scheme

The governing equations of the mechanochemical system can be fully described by Eqs. 1, 2, and 5. In our simulations, the equations are discretized and numerically solved using the explicit Euler method with a time step, denoted as Δt, whose value is set to be 0.01. For all the results shown in Fig. 3, the initial geometric configurations of 400 cells that interconnect to form an isotropic confluent epithelial shell applied in the simulations are the same. Meanwhile, the initial distribution of two chemical concentrations is set to be uniform throughout all the cells, specifically, ${c_{\mathrm{A}}^{(J)}\mid }_{t=0}=0.2{V_J\mid }_{t=0}$ and ${c_{\mathrm{R}}^{(J)}\mid }_{t=0}=0.1{V_J\mid }_{t=0}$ , respectively. As the evolution progresses, chemical reactions occur within cells and diffuse between the neighboring cells, followed by the cellular deformation and movement at each time step, during which when the edge of cell becomes shorter than a threshold length of reconnection, the reconnection (55) of the vertices happens. The numerical parameters are shown in table S1. The simulations of this study are performed in MATLAB R2021a.

3D and 2D kymographs

The 3D kymographs in Fig. 6 are obtained by analyzing the actomyosin pattern and vertex displacement on the basal surfaces of cells at each time step. We focus on the region defined by φ ∈ (−0.9π,0.9π) and θ ∈ (0.1π,0.9π) (Fig. 4D), which consists of nonuniformly spaced basal vertices. Then, we define N_φ × N_θ bins along the azimuthal φ-direction with width dφ and the azimuthal θ-direction with width dθ, respectively. The initial surface is fitted and interpolated at the points specified under the uniform grid in spherical coordinate system. Here, we choose N_φ = 57, N_θ = 26, and dφ = dθ = 0.1. By keeping azimuth angle θ fixed at π/2, the 3D kymograph can undergo a degeneration process to 2D (Fig. 4, A to D).

Four-cell energy involved in topological transition

The energy of the four cells during the scutoid-based topological transition depicted in Fig. 5C is calculated as

$$E_4={\displaystyle \sum _{J=1}^4}\frac{1}{2}{K}_{\mathrm{v}}{(V_J-1)}^2+{\displaystyle \sum _{<I,J>}}{K}_{\mathrm{s}}{A}_{I,J}+{\displaystyle \sum _{J=1}^4}\frac{1}{2}{k}_{\mathrm{m}}{m}_J{A}_J^2$$ (6)

where ${\sum }_{J=1}^4$ and ∑_<I,J> stand for the summation over the four cells and all lateral surfaces shared by neighboring cells.

Orientation field analysis of the multicellular lumen

Considering that scutoids primarily emerge as an intermediate state facilitating apical-basal coordination, and the thickness of cells is thin, the orientation of an individual cell can be represented by the long axes of the basal surface. The cell aspect ratio is defined by

$$\mathrm{\xi }=g_1/g_2$$ (7)

where g₁ and g₂ represent the largest and second-largest eigenvalues of the gyration tensor calculated from the basal surface of each cell, respectively. The orientation of each cell is determined by the principal direction, i.e., unit eigenvector v₁, associated with the largest eigenvalue g₁ of the gyration tensor. Thus, the long axes of cell J are represented as v_J.

To determine the local mean orientation field, we partition the multicellular domain into a uniform grid consisting of 150 points. Each point falls within the range of an actual cell, which is referred to as the central cell associated with the point. The neighboring cell domain of each grid point is defined as the set of this central cell and its neighboring first-row cells. The mean orientation field at each point is determined by the long axes of the cell within the neighboring cell domain that has the largest aspect ratio.

Simulation of the defect-induced morphogenesis

On the basis of the static spot pattern (mode I in Fig. 3A), we introduce cell growth and division into simulations to explore the impacts of the mechanical morphogen. We use the static pattern at t = 300 as the initial configuration for the following calculations. To make cells grow, the reference volume V₀ in Eq. 4 is replaced by an equilibrium volume V_eq, V_eq = 2V₀. Directional cell division is integrated into the model, with direction of division along the long axes of the basal surface of the cell. Since previous studies have shown that the mechanical stretch can facilitate and orient the cell division (79), we simplify the criteria of directional division. A cell that is about to divide must meet the following conditions: (i) V_J > 1.5V₀, the cells become large enough; (ii) the aspect ratio ξ_J > 2, and the cells are stretched long enough. As a result, the cells grow and divide, driving the out-of-plane tissue protrusion, which is aligned with the orientation field (Fig. 6, F and G, and movie S10).

Calculation of the local mean curvature

The mean radius of curvature at each cell’s location of apical and basal surface can be estimated as $R_{\mathrm{a}}=\sqrt{{\mathit{NA}}_{\mathrm{a}}/4\mathrm{\pi }}$ and $R_{\mathrm{b}}=\sqrt{{\mathit{NA}}_{\mathrm{b}}/4\mathrm{\pi }}$ , respectively, where A_a and A_b are the area of the cell’s apical and basal surface and N is the total number of the cells in the lumen. The value of local mean curvature depicted in Fig. 6G is defined by ∣H∣ = 2/(R_a + R_b). H > 0 when R_a < R_b, H < 0 when R_a > R_b, while H = 0 when R_a = R_b.

Supplementary Materials

This PDF file includes:

Supplementary Text

Figs. S1 to S4

Table S1

Legends for movies S1 to S10

Other Supplementary Material for this manuscript includes the following:

Movies S1 to S10