Modeling epithelial tissues as active-elastic sheets reproduce contraction pulses and predict rip resistance

Epithelial tissues are confluent cell sheets made of active, mechanically coupled cells that operate far from equilibrium and may exhibit unusual material properties, including solid-fluid transitions, super-elasticity, and exotic viscoelasticity. These tissues function as barriers and provide mechanical protection, requiring stability and integrity despite variable mechanical stresses. Observations across systems report dynamic contraction patterns such as contraction fronts during Drosophila development and density waves in migrating monolayers, with time scales of minutes to hours. Recent measurements in the early-divergent animal Trichoplax adhaerens revealed ultrafast traveling contraction pulses propagating at 10–30 micrometers per second over 1–3 cell lengths, with large, rapid cellular contractions and no cell rearrangements on short time scales, suggesting a mechanical propagation mechanism absent of neural or gap-junction signaling. Extension-induced contraction (EIC) is a common cellular response across epithelia, mediated by diverse molecular pathways, and cells may also soften or yield under stretch. The interplay of these antagonistic responses remains unclear. The research question is whether a minimal, purely mechanical model based on EIC alone can reproduce propagating contraction pulses and what emergent mechanical functions such dynamics confer, particularly regarding tissue cohesion and resistance to rupture. The study proposes that EIC is sufficient to generate propagating contraction pulses and that these dynamics enhance rip resistance by actively redistributing stresses.

Prior work has documented mechanically governed contraction waves and fronts in epithelia, including Drosophila embryonic tissues and MDCK monolayers under various conditions (confinement, expansion, substrate shear). Theoretical models have invoked couplings among mechanical, chemical, and geometric fields, including diffusion, active transport, and polarity, with some suggesting that contraction-under-tension alone can generate mechanically propagating waves. Experimentally, EIC is widespread in epithelial cells and involves multiple mechanisms (e.g., mechanosensitive calcium channels, ERK activation, actin alignment, myosin recruitment, junctional conformational changes). Cells can also respond to stretch by softening or yielding, and these responses can coexist. However, the precise mechanistic triggers (strain, stress, strain rate) and their integration at tissue scale are unresolved. Ultrafast contraction dynamics in T. adhaerens lack neural or gap-junction communication and occur too quickly for transcriptional processes, pointing to a mechanical trigger. This literature motivates a minimal mechanical model to test if EIC suffices for contraction wave propagation and to probe its physiological role.

The authors develop a minimal, purely mechanical model inspired by in vivo measurements in T. adhaerens. Single-cell model: a cell is modeled as an overdamped linear elastic spring (rest length l0, stiffness k) in a viscous medium (drag gamma) connected in parallel to an active contractile unit implementing extension-induced contraction (EIC). The active unit activates when the cell length exceeds a critical threshold lc, applying a fixed contractile force fc for a fixed duration tc (boxcar activation), then turns off. Parameter regime considered includes fc > k*lc and tc smaller than the passive relaxation time (tc < k/gamma). This circuit yields two single-cell regimes: excitable (l0 < lc), contracting only upon stretch, and oscillatory (lc < l0), functioning as a relaxation oscillator.

1D tissue model: a finite chain of N cells is assembled with cell-cell junctions as nodes; each cell follows the single-cell dynamics and experiences viscous drag. Free boundaries are used. In the excitable regime, a rim-cell is initially stretched by Delta l to trigger dynamics; in the oscillatory regime, all cells start at l0. Simulations track pulse propagation, with measurements of pulse speed V, width W, amplitude (amp), and conditions for pulse existence. Scaling relations are derived by analyzing the interface between active and passive cells, approximating two-cell dynamics with fixed boundaries to compute the excitation time of a neighbor and thus the pulse speed; width is W = V*tc; amplitude amp = 2(lc - l0). Non-dimensional parameters are defined to map regimes and thresholds for propagation.

2D tissue model: a vertex model on a hexagonal lattice represents a confluent epithelial sheet with free boundaries. Each cell has area elasticity (ka) and perimeter elasticity (kp). An active unit controls the perimeter: upon surpassing a critical area ac or critical perimeter pc, a contractile force fc acts for duration tc to shorten the perimeter. Parameter regimes relevant to T. adhaerens are explored (low area stiffness, high fc enabling ~50% area reduction, strain thresholds ~10%). The model produces radially, azimuthally, and spiral propagating pulses. A continuum approximation is formulated in the spirit of reaction-diffusion dynamics: time evolution of displacement u(r,t) follows du/dt = D∇²u + R(u,t), with D ~ k/gamma and R set by active parameters (fc, tc, thresholds).

Active cohesion tests: a numerical pulling experiment applies constant uniaxial forces at two opposite rim regions of a 2D sheet to compare passive versus active tissues. Strain and stress distributions are measured over time, comparing passive elasticity, EIC-only, and EIC plus yielding. Yielding is modeled as a threshold on junction stress sigma_c: when exceeded, the neighboring cells immediately soften by reducing stiffness k by a factor of two; stiffness recovers once stress drops below threshold. Metrics include time series of pulling distance L/L0, spatial maps of peak perimeter strain p/p0, area strain a/a0, and junction stress sigma, and histograms of their distributions.

• EIC suffices to generate propagating contraction pulses in overdamped tissues. In 1D excitable chains, a single initial stretch triggers a contractile pulse that propagates at constant speed, with a symmetric profile consisting of an extension front followed by a contraction front. Cells within the pulse remain at rest length while actively contracting.
• Scaling laws: Pulse speed V is set by the neighbor activation time determined by fc, k, and gamma; pulse width W scales as V*tc; pulse amplitude amp equals 2(lc - l0). Numerical simulations across parameter ranges collapse onto these predictions. The model accounts for both fast pulses (~100 micrometers per second in T. adhaerens) and much slower waves (~2–3 micrometers per minute in MDCK monolayers).
• Threshold for propagation: A sufficient active contraction impulse relative to the excitation threshold is required; if subthreshold, perturbations decay. For fixed activation parameters, sufficiently prolonged or strong stimulation produces spike trains: series of identical pulses carrying fixed strain quanta (amplitude determined solely by lc - l0).
• Oscillatory tissues (lc < l0) organize into ordered pulsations: pulses repeatedly initiate at the rims and annihilate in the center, after a transient global shrinkage to a steady average tissue length L. Dynamics display limit cycles with characteristic frequency and amplitude; other regimes include flickering no-deactivation, non-physical collapse when forces are too strong, long transients at high viscosity, and irregular (chaotic) pulsations at low viscosity.
• 2D sheets exhibit analogous behavior with circular and spiral pulses; as in 1D, pulses feature an expansion front followed by a shrinkage front. Geometrical frustration in 2D (especially at rims and for concave cell shapes) leads to quiescent intervals interspersed with bursts of activity as fc increases.
• Active cohesion and rip resistance: Under constant tensile pulling, passive sheets focus strain and stress near pulling points or along the axis. Active EIC sheets enter a dynamic steady state with spatiotemporal oscillations that homogenize strains across the tissue, reducing high peak strains and engaging most cells. Adding a yielding response caps high junction stresses by trading them for localized strains, thereby mitigating both failure modes (cell rupture from high strain and junction failure from high stress) when thresholds are set below rupture values. The combined EIC+yielding maintains integrity without parameter fitting.
• The continuum description recasts the dynamics as a reaction-diffusion-like system for displacement/strain, with diffusion set by viscoelasticity and reaction by cellular activity parameters.

The findings address whether purely mechanical EIC can drive contraction wave propagation and confer functional advantages to epithelial tissues. The model shows that active, thresholded contractions with a finite memory (duration) produce robust, non-dissipative trigger-wave pulses in overdamped, confluent tissues. These pulses can transmit mechanical information rapidly over long distances, providing a potential mechanism for intercellular communication in tissues lacking neural or gap-junction signaling, such as T. adhaerens.

Functionally, the dynamics enhance rip resistance: rather than allowing strain focusing under continuous tension (as in passive materials), EIC-driven activity redistributes loads across time and space, homogenizing strain and thereby avoiding high local deformations that risk rupture. However, contraction-induced reductions in cell strain can increase junctional stress. Introducing a complementary yielding response that softens cells upon high junction stress caps these stresses, striking a balance that prevents both cell overstrain and junction failure. Thus, active cohesion emerges from the interplay of contraction and softening feedbacks.

These results generalize across parameter regimes and dimensionality (1D chains and 2D sheets), with predictable scaling of pulse properties and identifiable transitions to irregular dynamics. The minimalist model, supported by experimental observations in T. adhaerens and compatible with slower waves in other epithelia, suggests a broad relevance to tissues experiencing mechanical challenges and offers principles for designing synthetic active materials.

The study introduces a minimal, experiment-inspired mechanical model in which extension-induced contraction at the single-cell level is sufficient to generate propagating contraction pulses in epithelial tissues. The authors derive scaling laws for pulse speed, width, and amplitude, map dynamic regimes (excitable and oscillatory), and extend the framework to 2D vertex models and a reaction-diffusion-like continuum description. A key emergent prediction is active cohesion: EIC-driven dynamics under tension homogenize strain and, when combined with a yielding response to junction stress, simultaneously limit both cell strain and junction stress, enhancing resistance to rupture.

Future work should experimentally quantify EIC triggers and parameters (e.g., whether strain, stress, or strain rate governs activation), validate pulse properties at the tissue scale, and test predictions across diverse epithelia, including embryonic tissues and mechanically challenged organs (heart, lung, gut, bladder, vasculature). Investigations in early-divergent animals may illuminate the evolution of excitation-contraction coupling, mechanical information processing in living materials, and the origins of neuromuscular systems.

• The model assumes activation by cell strain and yielding by junction stress; the true biological triggers (strain, stress, strain rate) remain uncertain and hard to distinguish experimentally.
• Active contraction is modeled as a fixed force applied for a fixed duration (boxcar), and yielding as an immediate, reversible halving of stiffness; these are simplifying assumptions due to limited quantitative data.
• Linear elasticity is assumed for cells; non-linearities (e.g., volume conservation or compressibility limits) are largely neglected, leading to non-physical collapse in extreme parameter regimes.
• Parameter values are inspired by T. adhaerens measurements and qualitative observations; comprehensive experimental validation across tissues is pending.
• The continuum reduction is phenomenological, capturing reaction-diffusion-like behavior without detailed molecular specificity.