Introduction

Multicellular systems exhibit a wide range of pulsatile and wave-like patterns during collective migration, development, and morphogenesis1,2,3. The appearance of these patterns can be attributed to various biochemical factors, depending on the specific phenomenon. These include waves of extracellular signal-related kinase (ERK)4,5, calcium waves6, periodic assembly and disassembly of myosin motors7,8, and the periodic release of chemoattractants9. Reaction–diffusion models10,11,12,13 and cellular automaton models14,15,16 have been widely used to study the mechanisms underlying biochemical pattern formation in multicellular systems. Mechanochemical patterns, on the other hand, have necessitated the development of new classes of models that integrate mechanical forces with chemical reactions17,18,19,20,21,22. For instance, the coupling of mechanical and chemical processes is particularly relevant in understanding the spatial propagation of contraction waves in Trichoplax adhaerens23, oscillatory morphodynamics in Drosophila amnioserosa tissue24, collective migration patterns20 and mechanical stress waves in expanding epithelial monolayers19,25,26. However, the role of cellular mechanics and geometry in the propagation of pulsatory signals remains poorly understood.

From the perspective of dynamical systems, pulsatory activity patterns can be regarded as manifestation of excitable dynamics, as seen in phenomena such as action potentials in nerve cells27 or pulses of actomyosin contractions observed in many cells28,29. Excitable systems share a common feedback motif: a rapid positive feedback that amplifies activity, followed by a delayed negative feedback that ultimately curtails the activity 30. While these positive and negative feedback loops have been identified in many biological systems, how such feedback motifs can be realized in mechanical systems remains largely unexplored.

One commonly observed mechanical feedback motif in cells is stretch-induced contraction, wherein a local stretching deformation triggers the recruitment of active components that induce contraction4,31,32,33,34. Recent studies have utilized the concept of stretch-induced contraction to explain phenomena such as wave propagation in active elastic media19,26, contraction pulses in epithelial tissues35, cell migration patterns in vitro20,36, cell and tissue morphogenesis24,37. Specifically, all these studies focused on dynamics in active elastic media, without considering the effects of geometric disorder and viscous dissipation on mechanochemical signal propagation.

In this study, we ask how cellular viscoelasticity and packing geometry regulate the propagation of active stresses at multicellular scales. To this end, we extended the framework of the cellular vertex models38,39,40,41 to incorporate feedback between cell junction strain and contractility. In addition, viscous dissipation is implemented by continuous strain relaxation in cell junctions. Inspired by the mechanical stretch-triggered activation of actomyosin contractility mediated by ERK4, we implement a simple feedback rule in which contractility in cell junctions is activated above a threshold junctional stretch. The junction remains active for a duration commensurate with the turnover rate of active elements. Subsequently, a refractory period ensues during which junctional contractility remains inactive, potentially due to the presence of ERK inhibitors. As a result of these rules, each cell junction behaves as a mechanically excitable unit that can exist in one of three states: active, inactive, and refractory.

Our proposed model elucidates the emergence of long-range propagation of contractile pulses and different patterns of self-sustained traveling waves, such as circular, elliptic, and spiral waves. We show that these tissue-level propagation patterns are controlled by the competition between the timescales associated with active and refractory states of the junction, and the characteristic timescale of junction strain relaxation. To explain these observations, we develop an effective theory of coupled excitable junctions, capable of explaining the emergence of the quiescent, wave-like and pulse-like patterns observed in vertex model simulations. Our theoretical framework predicts that shorter junctions promote re-activation of contractility, while larger junctions facilitate the propagation of activity over a broader region of the parameter space. We validate these predictions through simulations of disordered tissues in two dimensions. We find that geometrical disorder promotes sustained wave propagation at the tissue-level, and that the ability of junctions to propagate activity increases with junction length.

Results

Cellular vertex model with mechanical feedback

To elucidate the emergent dynamic patterns in an excitable tissue, we use the framework of the vertex model38,39,40,41, where a monolayer tissue is modeled as a two-dimensional polygonal tiling. The polygons represent the cells, and the edges represent the cell-cell junctions. Each vertex i, with position ri, is subject to friction with coefficient μ, and elastic forces and inter-cellular tensions arising from a Hamiltonian H. The Hamiltonian governing tissue mechanical energy is given by

$$H=\frac{K}{2}\mathop{\sum}_{\alpha }{({A}_{\alpha }-{A}_{0})}^{2}+\mathop{\sum}_{\langle i,j\rangle }\Lambda {l}_{ij}\,,$$
(1)

where the first energy term is a sum over all cells α, and the second term is a sum over the cell-cell junctions defined by the adjacent vertices i and j. The first term in Eq. (1) is the elastic energy that penalizes changes in cell area, where K is the bulk elastic modulus, Aα and A0 are the actual and preferred cell areas, respectively. The second term represents an interfacial energy, with tension Λ along each cell junction of length lij.

Active contractile forces arise at each junction from the actomyosin cortex, generating an active force per unit length, Γij(t). Consequently, the active force at each vertex can be written as a length-dependent tension: \({{{{{{{{\bf{F}}}}}}}}}_{i}^{{{{{{{{\rm{act}}}}}}}}}=-{\sum }_{\langle i,j\rangle }{\Gamma }_{ij}(t){l}_{ij}\left(\partial {l}_{ij}/\partial {{{{{{{{\bf{r}}}}}}}}}_{i}\right)\). Increase in active tension with edge length results in strain that is independent of the initial edge length, as observed in optogenetic experiments upon activation of cell-cell junction contractility42. We note that the length-dependent active tension is qualitatively similar to the perimeter-dependent contractility term in the classical vertex model energy functional40. However, our formulation treats each junction as a separate mechanical unit, rather than assuming homogeneous junction property across the cell.

we consider a time-dependent contractility Γij(t), whose dynamics depend on junctional strain and memory of mechanical state. To compute the time-evolution of each vertex, we assumed an overdamped limit, such that the equations of motion are given by:

$$\mu \frac{{{{{{{{\rm{d}}}}}}}}{{{{{{{{\bf{r}}}}}}}}}_{i}}{{{{{{{{\rm{d}}}}}}}}t}=-\frac{\partial H}{\partial {{{{{{{{\bf{r}}}}}}}}}_{i}}+{{{{{{{{\bf{F}}}}}}}}}_{i}^{{{{{{{{\rm{act}}}}}}}}}\,.$$
(2)

The above equation of motion is coupled to the dynamics of junctional strain and contractility, as described below.

Stretch-induced contractility is a commonly observed regulatory mechanism for controlling the level of active contractile stress in cells4,31,32,33,43,44. A local stretch in cell junctions could trigger actin fiber alignment34,45,46, myosin recruitment33, and also the activation of the ERK signaling4 that would promote contractility. We therefore implement a simple model of cellular junctions as viscoelastic materials subject to a strain-tension feedback (Fig. 1a). Here, a local stretch triggers the activation of contractility, which in turn reduces stretch via contractile forces. Additionally, junction strain continuously relaxes over time due to viscous dissipation and contractility undergoes turnover as part of a self-regulatory mechanism.

Fig. 1: Model for stretch-induced contraction and activity dynamics in cell junctions.
figure 1

a Local junction stretch induces the activation of ERK signaling, which in turn promotes contractility. Subsequently, myosin-based contractile forces act to reduce the stretch. The triggering quantity, strain, and the resulting induced contractility are influenced by memory mechanisms, as depicted by the strain relaxation rate kL and the active contraction lifetime τ. b Junction-level rules to change its state. Red denotes active state, gray denotes inactive state, and blue denotes a refractory state. c Representative section of a simulated tissue with hexagonal cells, using kL = 0.5 and τ = 2. Red junction color denotes active state, which spreads to neighboring junctions as they are stretched. d Representative dynamics of junction length, junction rest length, junction strain, and active contractility for an initially inactive junction (gray), indicated by the arrow in panel (c), with εon = 0.1 (indicated by the red dashed line).

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The mechanical strain in cell junctions is defined as \({\varepsilon }_{ij}=({l}_{ij}-{l}_{ij}^{0})/{l}_{ij}^{0}\), where \({l}_{ij}^{0}\) is the rest length of the junction shared by the vertices i and j. The viscoelastic nature of the junctions is modeled through rest length remodeling at a rate kL, leading to continuous strain relaxation42,

$$\frac{{{{{{{{\rm{d}}}}}}}}{l}_{ij}^{0}}{{{{{{{{\rm{d}}}}}}}}t}=-{k}_{{{{{{{{\rm{L}}}}}}}}}({l}_{ij}^{0}-{l}_{ij}).$$
(3)

Rest length remodeling is a natural consequence of actomyosin networks with turnover, where strained elements are replaced by unstrained ones31. The feedback between junction strain and contractility is implemented as follows. Each cell junction can exist in one of three states: inactive (Γij = 0), active (Γij = Γ0), and refractory (Γij = 0). While both inactive and refractory states lack contractility, refractory junctions are those that cannot be active for a duration τref, representing the timescale associated with the presence of inhibitors of contractility. The rules describing junction state changes are given below (Fig. 1b):

  • Inactive junctions become active if their strain εij exceeds a threshold value εon.

  • Active junctions become refractory after being active for a time period τact.

  • Refractory junctions become inactive after a duration τref.

As an example, we’ll examine the scenario depicted in Fig. 1c, d, where gray, red, and blue junctions represent the inactive, active, and refractory states, respectively. It is important to note that Fig. 1c represents a section of a larger tissue. Initially, the junction marked by the black arrow in Fig. 1c is set to an inactive state, while the middle red junction is manually activated. The contraction of the middle junction induces a moderate length increase in the inactive junction, leading to an increase in junction strain (Fig. 1d). When the strain reaches the critical value ϵon, ERK signaling is activated4. This activation, in turn, leads to the production of ERK inhibitors13. The threshold value εon ensures immunity to small perturbations, allowing junctions to be excitable units. ERK activation induces contractility, changing the junction state from inactive to active, increasing contractility to Γij = Γ0, resulting in junction contraction (Fig. 1d). The active state persists for a time period τact (red phase in Fig. 1b–d), which represents an effective timescale arising from the turnover time of actomyosin, and the inactivation of ERK. During this phase, the next inactive junction becomes activated, causing the observed length increase in Fig. 1d at t ~ 0.7t0, where t0 = (μ/[0.636KA0]). Subsequently, the initially manually activated junction returns to Γij = 0 resulting in the observed length decrease in Fig. 1d at t ~ 2t0. The remaining levels of ERK inhibitors keep the junction in a refractory state, in which it can not be re-activated (blue phase in Fig. 1b–d). The initial increase in length observed in this phase is due to the sudden drop in active contractility, while the subsequent decrease in length is a result of neighbor dynamics. Finally, after a time period τref, the inhibitors reach low enough levels to take the junction back to the inactive state (final state in Fig. 1d).

Although the activation time period (τact) and the refractory time period (τref) are distinct parameters, a recent study13 has demonstrated that, to adequately describe the observed ERK activity waves, the characteristic timescales for ERK activation and inactivation by inhibitors tend to be similar, typically of the order of a few minutes. For simplicity, we will first focus on the case where τact = τref = τ, and we will refer to this timescale simply as the activation period unless otherwise specified. The strain relaxation rate (kL) and the activation period (τ) will jointly impact the dynamics at both the junction and tissue scales, as discussed later.

Traveling pulse and waves in ordered tissues

To characterize the emergent dynamic states arising from junction-level mechanical feedback, we first simulated an ordered tissue, composed of 260 hexagonal cells, in a box of sides \({L}_{x} \sim 14\sqrt{{A}_{0}}\) and \({L}_{y} \sim 18.6\sqrt{{A}_{0}}\), under periodic boundary conditions. In simulations, we non-dimensionalized force scales by \(K{({A}_{0})}^{3/2}\), length scales by \(\sqrt{{A}_{0}}\), and timescales by t0 = μ/(0.636KA0), setting A0 = 1, K = 1, and μ = 0.636.

We initiate our simulations with a mechanically equilibrated tissue, where all junctions are initially in the inactive state. We then perturb the equilibrium state by manually activating a single junction positioned near the center of the simulation window (Fig. 2a). When the rate of strain relaxation is sufficiently slow, corresponding to a small value of kL, we observe the emergence of two distinct activity patterns depending on the activation period τ: traveling waves (Fig. 2b) and traveling pulse (Fig. 2c). For small τ, we find waves of activity traveling radially outwards, as shown in Fig. 2b (Supplementary Movie 1). These self-sustaining waves are characterized by alternating rings and regions of red (indicating activity) and blue (indicating refractory) junctions. The tissue activity reaches a steady-state when the wavefront traverses the entire tissue (around t ~ 5t0 in Fig. 2d). Conversely, for larger values of τ, we do not observe self-sustaining waves due to the lack of junction reactivation events. Instead, a single transient activity pulse travels across the tissue (Fig. 2c, e, Supplementary Movie 2). Over an extended time period, the tissue eventually becomes entirely inactive.

Fig. 2: Emergent activity patterns in ordered tissues with a local junction activation.
figure 2

a Initial condition of every simulation, with a single chosen junction manually activated (red). b Snapshots of persistent activity waves in an ordered tissue, in a simulation using the values of (τ, kL) = (0.8, 0.7) (left white dot in panel (g)). c Snapshots of an activity pulse traveling across an ordered tissue, in a simulation using the values of (τ, kL) = (1.6, 0.7) (right white dot in panel (g)). Junction color in panels (a–c) denotes the state of activity: Red—active, blue—refractory, gray—inactive. d Fraction of junctions in inactive state (gray line), active state (red line), and refractory state (blue line), as a function of time, for activity waves shown in (b). Data show that at long times only active and refractory states persist, with almost no junctions in the inactive state. e Fraction of junctions in each state, as a function of time, for activity pulse shown in (c). Data show a transient pulse of activity before the entire tissue becomes inactive. Color scheme is the same as in panel (d). f Total junction strain as a function of time, for the wave (b) and the pulse (c). g Phase diagram showing the three distinct emergent states: traveling pulse, traveling waves, and a quiescent state with no propagation of activity. The emergent states are controlled by the rest length remodeling rate kL (in units of \({t}_{0}^{-1}\)), and the activation period τ (in units of t0). Colormap shows the value of the maximum active junction fraction. Cyan dashed line is the boundary between “No propagation” and “Traveling waves” phase, whereas white dashed line is the boundary between “Traveling waves” and “Traveling pulse” phase.

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To quantify the mechanical deformation due to these traveling activity patterns, we calculated the total junction strain as ∑〈i, j〉εij. Fig. 2f shows the dynamics of the total junction strain for both wave (Fig. 2b) and pulse-like (Fig. 2c) patterns. The pulse causes a positive peak in strain, followed by a negative peak, ultimately returning to zero strain due to mechanical relaxation. Conversely, in the traveling wave pattern, while there is a peak in strain, it eventually stabilizes as a result of activity-induced mechanical fluctuations and the relaxation of strain at the junction level.

These propagating activity states are only observed when the value of kL is sufficiently small. A large kL causes the strain in the neighboring junctions of an active junction to relax before activation can occur, resulting in a quiescent state without any propagation. To quantify the extent of tissue-scale activity, we calculated the maximum fraction of active junctions throughout the simulation. This measurement enables us to identify the phase boundary, determined by the critical value of kL, that separates the regimes with activity propagation (either wave or pulse) from those without propagation (cyan-dashed boundary in Fig. 2g). Moreover, by quantifying the active junction fraction at the final steady state, we can differentiate between the propagating modes, leading to the delineation of the wave-to-pulse phase boundary (white-dashed boundary in Fig. 2g).

Effective theory predicts emergent dynamic states

To predict the emergence of excitable pulses, quiescent states, and oscillatory patterns as functions of the strain relaxation rate kL and activation period τ, we developed an effective one-dimensional theory of coupled excitable junctions. Our minimal model consists of three interconnected junctions with fixed boundaries, as shown in Fig. 3a. Each unit comprises an elastic component with a spring constant k and natural length L (representing the one-dimensional version of cell elasticity), connected in parallel with a dashpot of friction coefficient μ, and an active element with contractility Γ1,2. If the junction is inactive or refractory then Γ1,2 = 0, and Γ1,2 = Γ0 if the junction is active. These active and elastic elements are connected in parallel with a tensile element with line tension Λ1,2. The central junction has a length l1(t), while the outer junctions have lengths l2(t). The fixed boundary conditions ensure that l1(t) + 2l2(t) = 3L.

The system is initialized in a mechanical equilibrium state, and we perturb it by activating the central junction (Γ1 = Γ0, Γ2 = 0). We then let the system to evolve following the equations of motion: μdli/dt = − ∂Heff/∂li, Eq. (3), and the rules governing the junction states. The effective Hamiltonian governing the system is defined as:

$${H}_{{{{{{{{\rm{eff}}}}}}}}}=\frac{k}{2}{({l}_{1}-L)}^{2}+k{({l}_{2}-L)}^{2}+{\Lambda }_{1}{l}_{1}+2{\Lambda }_{2}{l}_{2}+\frac{{\Gamma }_{1}}{2}{l}_{1}^{2}+{\Gamma }_{2}{l}_{2}^{2}.$$
(4)
Fig. 3: Effective model of coupled excitable junctions.
figure 3

a Schematic of a minimal three-junction model. Each junction is composed of a linear elastic element with a spring constant k and natural length L, connected in parallel with a viscous element of friction coefficient μ, and an active element with contractility Γ1 (middle junction) and Γ2 (end junctions). Tension in the middle junction is denoted by Λ1 and those in the end junctions are given by Λ2. b Phase diagram of the three-junction model as a function of activation period τ and rest lenght remodeling rate kL, showing three different phases: no propagation (yellow shading), pulse propagation (blue shading) and pulse reactivation (orange shading). Here the junctions are assumed to have equal tension. c, d Phase diagrams for pulse propagation and reactivation for asymmetric junction tensions, with (Λ1, Λ2 = 0.1, 0.2) (c), and (Λ1, Λ2 = 0.4, 0.1) (d). All cases consider Γ0 = 1 and ϵon = 0.1.

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We initially considered the scenario of symmetric junctions, wherein Λ1 = Λ2. This corresponds to an ordered tissue where junction tensions and lengths are uniform. To explore the behavior of the system, we non-dimensionalized force scales by kL, length scales by L, and timescales by μ/k (setting L = 1, k = 1, μ = 1), and numerically solved the system of equations for different values of τ and kL, from t = 0 to t = 2τ. The simulation outcomes can be categorized as follows: i) If the outer junctions remain inactive throughout the simulation, it is classified as a case of No propagation; ii) If the outer junctions become active but the central junction does not re-activate, we observe a single Pulse; and iii) finally, if the outer junctions become active and the central junction re-activates, it falls into the category of Re-activation. Fig. 3b shows the phase diagram of the model in τ-kL phase space showing the emergence of the three outcomes described above. A comparison with the phase diagram for the ordered tissue (Fig. 2g) reveals that the effective model successfully captures both key features of the vertex model: a critical value of kL for propagation of activity, which diminishes for small τ, and a small region of reactivation corresponding to wave-like states.

For the simple case of kL = 0, we can analytically determine the condition for propagation as τ > τp, with

$${\tau }_{{{{{{{{\rm{p}}}}}}}}}=\frac{1}{2\left(3+2{\Gamma }_{0}\right)}\ln \left(\frac{{\Gamma }_{0}}{{\Gamma }_{0}-(3+2{\Gamma }_{0}){\epsilon }_{{{{{{{{\rm{on}}}}}}}}}}\right).$$
(5)

The τ-reactivation range is determined by solving an inequality of the form \(A{e}^{-6\tau }-B{e}^{-6(2+{\Gamma }_{0})\tau }-{\epsilon }_{{{{{{{{\rm{on}}}}}}}}} \, > \, 0\), where

$$A=\frac{2{\Gamma }_{0}}{3+{\Gamma }_{0}}\left(1-{e}^{-2\left(3+{\Gamma }_{0}\right){\tau }_{{{{{{{{\rm{p}}}}}}}}}}\right){e}^{6{\tau }_{{{{{{{{\rm{p}}}}}}}}}},$$
(6)
$$B=\frac{2{\Gamma }_{0}}{3+2{\Gamma }_{0}}\left(1-{e}^{-2\left(3+2{\Gamma }_{0}\right){\tau }_{{{{{{{{\rm{p}}}}}}}}}}\right){e}^{2\left(3+2{\Gamma }_{0}\right){\tau }_{{{{{{{{\rm{p}}}}}}}}}}.$$
(7)

The above inequality cannot be analytically solved in general, but we can solve several limiting cases to gain intuition on the dynamics. Initially, we examine the case where ϵon = 0. Here, τp = 0, indicating that activity propagates regardless of the activation period τ. Consequently, A = B = 0. Thus, in the absence of rest length remodeling and critical strain threshold, only one outcome prevails: propagation of activity pulse. Subsequently, when considering Γ0 = 1, as in the numeric solution shown in Fig. 3b, the left-hand side of the inequality assumes a cubic form for \(y=\exp \left(-6\tau \right)\). Within the range 0.077 < ϵon < 0.2, the inequality yields a range of activation timescales τ1(ϵon) < τ < τ2(ϵon) where reactivation occurs. The exact positioning of τ1 relative to τp varies depending on ϵon. Specifically for ϵon = 0.1 (as considered in Fig. 3b), we find that τp < τ1 < τ2, with τ1 − τp = 0.003. It is noteworthy that across all the plots shown in Fig. 3, τ1 − τp≤0.025, thus the contours of τ1 are not displayed.

We then used the one-dimensional effective model (Fig. 3a) to investigate the role of disorder in the propagation of activity. Disorder was introduced by removing the condition of homogeneous line tension, letting Λ1 ≠ Λ2. First, we analyzed the case Λ1 < Λ2. Due to identical mechanical properties of each junction before activation (other than tension values), the initial equilibrium state featured a central long junction (l1 > L) flanked by two shorter junctions (l2 < L) (Fig. 3c). By solving the system of equations numerically, we found that the larger junction (l1 > L) could propagate activity over a broader region in the (τ, kL) parameter space, while the re-activation region is substantially diminished. This is because larger junctions produce greater active contractile forces, while shorter neighboring junctions require a lower extension to achieve the strain threshold for activation εon. Conversely, when Λ1 > Λ2, the opposite behavior was observed (Fig. 3d). Our effective model thus reveals two main effects of the geometrical heterogeneity (or disorder) on cellular response to active contractility. Large junctions promote propagation of activity, while shorter junctions facilitated re-activation, leading to oscillatory patterns.

Tissue disorder promotes self-sustained wave propagation

Motivated by the predictions of the effective model on the impact of geometric heterogeneity, we now investigate the effect of disorder in cell packing geometry on activity propagation in two-dimensional tissue simulations. To this end, we constructed a tissue comprising 208 cells, within a rectangular box with dimensions approximately equal to \({L}_{x} \sim 14\sqrt{{A}_{0}}\) and \({L}_{y} \sim 15\sqrt{{A}_{0}}\), subject to periodic boundary conditions. In these simulations, all mechanical properties at cell and junction levels are the same, with disorder restricted to geometric heterogeneity only. The initial state of the tissue corresponded to a state of mechanical equilibrium, characterized by varying junction lengths and polygon sidedness, as depicted in Fig. 4a.

Fig. 4: Sustained propagation of activity waves in disordered tissues.
figure 4

a Snapshots of activity waves traveling across a disordered tissue, using (τ, kL) = (1.6, 0.7) (white dot in (b)). Colored segments represent inactive (gray), active (red), and refractory (blue) junctions. (b) Phase diagram for the emergence of waves, pulses and quiescent states, varying strain relaxation rate kL (in units of \({t}_{0}^{-1}\)) and activation period τ (in units of t0). The cyan boundary indicates a maximum active junction fraction equal to 0.1, whereas the white dashed line is the boundary between 'Traveling waves' and 'Traveling pulse'. c Scatter plot of the critical rest length remodeling rate kL for a fixed τ = 3, as a function of the initially activated junction length, for 100 different chosen junctions. The critical kL follows the same definition as the cyan boundary in (b). Red-vertical line represents the junction length in the ordered hexagonal tissue. Blue-horizontal line represents the critical kL in the ordered tissue.

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As previously, we activated a randomly chosen cell junction (see Fig. 4a, t = 0.0τ), and let the tissue evolve from t = 0 to t = 20, for different values of strain relaxation rate kL ∈ (0, 1.5) and activation period τ ∈ (0.2, 5.0). By measuring the maximum active junction fraction (Fig. 4b), we again observe that propagation occurs below a critical kL, for sufficiently large τ. Unlike in ordered tissues (Fig. 2g), wave states are now possible for a wide range of τ values, and propagating solitary pulse only occurs in particular cases with exceedingly large activation periods. Consistent with the predictions of the one-dimensional effective model, we find that the presence of short junctions in disordered tissues promotes junction reactivation, thereby facilitating the emergence of self-sustaining wave-like states. As an illustrative example, Fig. 4a (corresponding to the white dot in Fig. 4b) represents a wave-like state arising in a tissue with parameters (τ = 1.6, kL = 0.7) (Supplementary Movie 3), which led to pulse propagation in the ordered tissue (Fig. 2g). Interestingly, the junction that is reactivated by the end of an oscillatory cycle need not necessarily be the same one initially chosen for activation. This introduces a non-local effect of disorder in promoting sustained wave-like patterns. We find that the critical kL required for wave propagation increases with the length of the initially activated junction (Fig. 4c), as predicted by the one-dimensional effective model.

Interestingly, we observe that certain long junctions, surrounded by shorter ones, retain their inactive state throughout the entirety of the simulations. This phenomenon is rooted in the longer activation period necessary to propagate activity in such configurations, as predicted by the one-dimensional effective model (Fig. 3d). In certain scenarios, this effect merely delays the onset of activation, whereas in others, it renders activation unattainable.

Controlling the geometry of wavefronts

Our theory and simulations have elucidated that the propagation of activity at the tissue scale is governed by two distinct characteristic timescales of the system: the activation period (τ) (taken to be equal to the refractory time) and the rest length remodeling timescale \(({k}_{{{{{{{{\rm{L}}}}}}}}}^{-1})\). We now investigate the impact of varying the activation (τact) and refractory periods (τref) on the resulting dynamic patterns that emerge within the tissue. In particular, we show that the ratio of activation to refractory period controls the geometry of wave patterns.

We initiated the simulations by activating a single junction (Fig. 5a) or a partial row of junctions (Fig. 5b) within the tissue, while the remaining junctions remained in the inactive state. We find that the ratio of the refractory period to the activation period, Δ = τref/τact, controls the wavelength of the propagating waves (see Figs. 5c–e). Specifically, at higher values of Δ, propagating waves fail to materialize, and instead, we observe the presence of a solitary traveling pulse of activity (Fig. 5f).

Fig. 5: Emergent dynamical patterns arising in an ordered tissue.
figure 5

The tissue is composed of 2759 cells, varying Δ = τref/τact. Colored segments represent inactive (gray), active (red), and refractory junctions. a, b Zoomed in configuration of two different initial conditions for junction states: a single junction activation, and b partial row activation with neighbors underneath in refractory states. c Wavelength versus Δ, for the initial condition shown in (a). d–f Collective dynamics arising from the initial condition in (a), for different values of τref, with τact = 2. g–i Collective dynamical states arising from the initial condition (b), for different values of τref, with τact = 1. Panels d, e, g–i represent self-sustained waves, while panel f represents a solitary traveling pulse. All these simulations consider kL = 0.5, and integration time step Δt = 0.05.

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Inspired by self-sustained spiral patterns observed in excitable systems16,47,48,49, we inquired whether we could design an initial state that would break the circular symmetry of the emergent wavefronts. Previous theoretical work using a three-state (inactive, active, refractory) cellular automata model has shown that spiral waves can emerge from an initial state consisting of a layer of excited cells and an adjacent layer of refractory cells50,51,52. We therefore initialized our simulations by activating a partial row of junctions, followed by neighbors underneath in refractory states, while the remaining junctions remained in inactive states (Fig. 5b). This initial condition leads to elliptical wavefronts for the cases of Δ < 1 (Fig. 5g) and Δ = 1 (Fig. 5h, Supplementary Movie 4). Similar to the case of single junction activation, smaller values for Δ decrease the wavelength of the traveling wavefront (Fig. 5g). For higher values of Δ we observe the emergence of a pair of self-sustaining spirals (Fig. 5i, Supplementary Movie 5). This can be explained as follows. The initial condition of a partial row of active junctions followed by a layer of refractory junctions instigates two distinct patterns. Firstly, it leads to the emergence of a propagating wavefront. Secondly, it initiates the formation of two open ends within the wavefront, resulting initially in the addition of an excited element and subsequently in the development of a curved wave segment. This curved segment then propagates outward, adopting a spiral shape. Thus, each open end leads to the emergence of an spiral. Note that due to the periodic boundary conditions in our model, the least number of open ends that can be created is two. These results show that by designing appropriate initial states and disparate timescales for junction activation and refractory periods, the geometry and wavelength of the emergent wavefronts can be precisely controlled in our model.

Discussion

Excitable dynamics are widespread in nature, involved in diverse biological and chemical processes, including action potentials in nerve cells27, calcium wave propagation53, the Belousov-Zhabotinsky chemical reaction54, pulsatory Rho dynamics28,55 and Ras-GTP (guanosine triphosphate)56 signaling networks. Although excitable dynamics have been well-studied in the context of biochemical networks, there are a few examples of mechanical circuits that display excitable behavior57. In this work, we have elucidated the minimal design principles for mechanical excitability in living tissues using the framework of the cellular vertex model. Our extended vertex model incorporates agent-based rules at the level of cell junctions, representing coarse-grained biochemical reactions that connect junction deformations with the activation of contractility. Through this model, we have demonstrated how the timescales regulating excitability and cell-level geometrical disorder control the formation of different spatio-temporal patterns and facilitate long-range signal propagation in the tissue.

Prior work has examined pattern formation in excitable tissues, considering various triggering factors, including cell-level tension35, cell size36, and active ERK concentration4. However, none of these models have considered the effects of viscous dissipation or explored the potential roles of geometrical disorder in pattern formation dynamics. Here we investigated the role of viscous dissipation and geometrical disorder on tissue-level pattern formation. We showed that mechanical strain can trigger the emergence of three tissue-level states: quiescent (no propagation), traveling waves, and traveling pulses. These states arise from the interplay between the characteristic timescales associated with junction activation, inactivation, and refractory states.

To explain these emergent dynamics, we have developed an effective junction-scale theory that qualitatively captures the observed behaviors in the vertex model. Our model also provides insights into the impact of geometrical tissue disorder on tissue-level activity states, demonstrating that large junctions promote propagation, while small junctions facilitate re-activation. These predictions have been corroborated through two-dimensional vertex-like simulations, although experimental validation in epithelial tissues remains an avenue for future exploration.

Furthermore, we have demonstrated that the geometry of the emerging traveling wavefronts is influenced by the initial state of junctions and the ratio between the durations of junction active states and refractory states. This intricate interplay results in variations in wavelengths, transitions from waves to pulses, formation of elliptic wavefronts, and pairs of self-sustained spiral wavefronts. The predicted patterns arising from specific initial junction states could potentially be experimentally tested using optogenetic tools to spatially activate myosin contractility 58, ERK59, and Fluorescence resonance energy transfer (FRET) imaging to visualize the resulting patterns4.

Methods

The custom simulation code for the vertex model, implemented using Python 3, uses the Euler integration method to integrate the differential equations. All the simulation codes can be accessed on GitHub (https://github.com/BanerjeeLab/Excitable_Tissue). In implementing T1 transitions, a similar approach to that described in a previous work by some of us was adopted60. This involves enforcing the creation and instantaneous resolution of a 4-fold vertex whenever a junction’s length becomes smaller than lT1. A newly created junction is set to have l = l0 = 1.5lT1 and be in the inactive state. The simulations encompassed tissues of varying sizes, as specified in the respective figure captions. Default model parameters used in the simulations are listed in Table 1. The numerical analysis of the one-dimensional effective model was done in Mathematica 12.

Table 1 Default model parameters
Full size table