Embryonic Tissues as Active Foams

Abstract

The physical state of embryonic tissues emerges from non-equilibrium, collective interactions among constituent cells. Cellular jamming, rigidity transitions and characteristics of glassy dynamics have all been observed in multicellular systems, but it is unclear how cells control these emergent tissue states and transitions, including tissue fluidization. Combining computational and experimental methods, here we show that tissue fluidization in posterior zebrafish tissues is controlled by the stochastic dynamics of tensions at cell-cell contacts. We develop a computational framework that connects cell behavior to embryonic tissue dynamics, accounting for the presence of extracellular spaces, complex cell shapes and cortical tension dynamics. We predict that tissues are maximally rigid at the structural transition between confluent and non-confluent states, with actively-generated tension fluctuations controlling stress relaxation and tissue fluidization. By directly measuring strain and stress relaxation, as well as the dynamics of cell rearrangements, in elongating posterior zebrafish tissues, we show that tension fluctuations drive active cell rearrangements that fluidize the tissue. These results highlight a key role of non-equilibrium tension dynamics in developmental processes.

Extended Data Fig. 1. Power law relation between NE rate and MSD at long timescales

Power law relation between long time MSD values and NE rate when the systems are close to confluence for high adhesion levels. NE rate and longtime MSD show a power law relation with an exponent of 0.75.

Extended Data Fig. 2. Comparison of solid/fluid phase diagrams obtained from stress relaxation and from cell movements

Solid/fluid phase diagrams determined by mechanical measurement of stress relaxation (left) and cell movements, MSD=1/2 (middle) and MSD=1/4 (right). Green region indicates confluent states.

Figure 1:. Characteristics of multicellular systems and simulation framework.

a, Confocal section of embryonic tissue in a zebrafish embryo (membranes labeled; yellow). b, Schematics of cortical tension and adhesion at cell-cell contacts in a multicellular system with spaces (red) between cells (gray). c, Confocal section through zebrafish embryonic tissues showing the intensity (I_s) of fluorescent reporter protein secreted extracellularly as well as cell membranes (green), indicating the presence of extracellular spaces (red range). d, Schematics defining the cell size L₀, cell number N and simulation box area A_T, which specify cell density. e, Kymograph of membrane signal intensity (I) along a tissue region (pink; left panel) containing a cell-cell contact, showing contact length ℓ fluctuations (bottom). f, Simulated tension fluctuations causing cell-cell contact length fluctuations. Increasing (decreasing) tension shortens (lengthens) cell-cell junctions (bottom). g, Schematics of the dynamic vertex model formulation. Triple vertices (physical vertices; red) and non-physical intermediate vertices (blue) are shown.

Figure 2:. Equilibrium configurations and structural transitions.

a-c, Representative equilibrium configurations (a), volume fraction ϕ (b) and mean number of neighbors z (c) for varying values of relative adhesion W/T₀ and cell density ρ. d, number N_E (top) and average area Ā_E (bottom) of extracellular spaces for varying relative cell adhesion and different cell densities, showing a sharp structural transition at W/T₀ ≈ 0.23 (gray line) leading to the opening of large extracellular spaces. e, Lowering (dashed line) and increasing (dash-dotted line) relative adhesion quasi-statically shows bistable states and strong hysteresis in equilibrium configurations. Equilibrium quenched states are also shown (solid line). f, Average neighbor number (cell contacts) z as the system volume fraction changes at vanishing cell adhesion, showing the existence of a jamming transition at ϕ_c ≃ 0.83 (configuration shown in inset). Power law fits z – z_c = z₀(ϕ − ϕ_c)^1/2 + z₁(ϕ − ϕ_c) with non-zero z₀ and z₁ (red line) and with z₁ = 0 (gray line) are shown. g, Average shape factor for varying relative adhesion showing a sharp increase at W/T₀ = 2 (vanishing tensions), leading to anisotropic cell shapes (inset), recapitulating density-independent transitions. h, Schematics of a simple shear deformation imposing a large strain step (ϵ_xy = 1.5), with associated temporal evolution of both strain and shear stress. i, Temporal relaxation of shear stress σ_xy (normalized to ${\sigma }_0\equiv \sqrt{\rho }{T}_0∕L_0$) after the imposed strain step for varying adhesion levels. j-k, Dependence of the yield stress σ_Y on the relative adhesion strength (j) and on both cell density and relative adhesion (k), showing a maximum at the structural transition between confluent and non-confluent states (green line). Error bands = SD. N= 20 (b,c,d,f,g) and N=10 (e,i,j,k) independent simulations for each parameter set.

Figure 3:. Tissue dynamics with finite tension fluctuations.

a, MSD for varying magnitudes of tension fluctuations ΔT/T₀, showing sub-diffusive (0 < α < 1) and diffusive (α = 1) behaviors as tension fluctuation increase (inset). b, Snapshots of dynamic configurations with examples of cell trajectories over t/τ_T = 10². c, MSD at long timescales (t = 10²τ_T ⪢ τ_T > τ_R), showing non-monotonous behavior for varying relative adhesion strength and minimal values at the structural transition. d, Cellular neighbor exchange (NE) rate for varying relative adhesion. Distinct types of neighbor exchange events: gain/loss of cell contacts (left) and T1 transitions (right). Error bands = SD. 10 independent simulations for each set of parameters.

Figure 4:. Stress relaxation and structure in active multicellular systems.

a-b, Stress relaxation (a) and temporal changes in cellular NE rate (b) after the imposed strain step for varying magnitudes of tension fluctuations. The long timescale stress relaxation follows stretched exponentials (black dashed lines) and eventually reaches the average value σ_A of active shear stress fluctuations (horizontal lines in inset). The cellular NE rate quickly decays to zero in the absence of activity (t ~ τ_R), but remains finite in the presence of activity. c, Sketch showing the dynamics NE induced by the externally applied shear strain (passive) and by tension fluctuations in cells (active). Active NE enables further stress relaxation and tissue fluidization after the initial passively induced NE. d, Stress relaxation timescale τ_SR for varying magnitude of tension fluctuations and relative adhesion, showing a sharp increase as the structural transition between non-confluent and confluent (green background) states. e, Phase diagram showing the transition between fluid and solid tissue states for different activity values and relative adhesion strength. Solid states surround the structural transition and are found both in confluent and non-confluent configurations for low enough activity. f-h, Dependence of the system volume fraction ϕ (f) and average shape factor $\stackrel{‒}{s}$ (g; with red/blue squares indicating fluid/solid states, respectively) on relative adhesion for different magnitudes of tension fluctuations. Representative snapshots of dynamic configurations for fixed relative adhesion (W/T₀ = 1; vertical dashed line in g) and increasing tension fluctuations are shown (h), with red/blue contours indicating fluid/solid states, respectively. Error bands = SD. 10 independent simulations for each set of parameters.

Figure 5:. Stress relaxation and tissue fluidization in posterior tissues during body axis elongation.

a, Lateral view of a zebrafish embryo and sketches showing lateral and dorsal views of posterior tissues, indicating the fluid-like MPZ and solid-like PSM. A confocal section showing a portion of the MPZ (cell membranes, yellow) with a magnetic droplet (magenta) is highlighted. b, Schematic sketches and confocal snapshots of magnetic droplet actuation in the MPZ tissue. c, Time evolution of the strain, (b – b₀)/R, before, during and after magnetic actuation. d, Relaxation of anisotropic stress σ_A(t) after magnetic actuation in both linear and log-linear scales, with ${\sigma }_A^0={\sigma }_A(t=0)$. The fit (black line) corresponds to a stretched-exponential function. e, Example of NE events. f, Normalized frequency of amplitude of junctional length fluctuations for the MPZ, both in the absence and presence of blebbistatin (average amplitudes in inset). g, Cumulative NE events in the MPZ (away from the droplet, cyan) and in the close neighborhood of the droplet (around droplet, red). h, Temporal evolution of NE rate in the MPZ (cyan) and also in the region around droplet (red). i, Average NE rates in the region around droplet (A.D., both during initial droplet relaxation (green) and at its final stages (pink)) and in the MPZ, both in the absence (cyan) and presence (orange) of blebbistatin. j, Experimentally measured and simulated dynamics of cell shapes in both MPZ and PSM, showing faster dynamics in the MPZ and largely static cell boundaries in the PSM. Experimental data is an average intensity projection of a confocal section timelapse. k, MSRD shows uncaging behavior for the MPZ but caged for the PSM of both wild type and cdh2^−/− embryos. Error Band=SD (d,g), SEM (i,k). N=10 (d), N=298 from 4 embryos (g,h,i; A.D), N=396 from 3 embryos (g,h,i; MPZ). Data in f, MPZ bleb in i and MRSD for MPZ and PSM in wild type embryos in k were reanalyzed from Ref. .