Model and configurations.

(A) A single cell modeled as a closed loop of beads and springs. Each bead experiences an outward-normal pressure and tangential spring forces (Top). Additionally, in an active cell, all beads move with a self-propulsion speed vo along a noisy polarity direction, (Bottom). (B) Steady-state tissue configurations. Increasing adhesion and noise strength can lead to cell jamming with reduced intercellular space.

Fluid to solid transition with increasing intercellular adhesion.

(A) MSD of cell centers for different values of (top to bottom: = 8.3 × 10−5, 8.3 × 10−3, 0.025, 0.042, 0.083, 0.25), showing diffusive to subdiffusive and caged behavior as increases. The dashed line indicates a slope of 1 on the log-log plot. (B-C) Zoomed-in snapshots of cell collectives and the cell center trajectories at low (B) and high (C) adhesion strengths (corresponding to extreme values of , denoted by (B) and (C) in Panel A). Regular polygonal shapes arise in C, but mostly rounded shapes emerge in B with noticeable intercellular gaps. Cell center trajectories look diffusive in B, while they appear caged in C. In B (left), red and green cells show spontaneous neighbor exchange (T1 transition) in the liquid phase. (D) The effective diffusivity, quantified as an order parameter, is shown against . (E) MSD exponent, as a function of . Parameters: = 16.6 × 10−3, = 0.2, = 5.2 × 10−4. Other parameters are from Table S1.

Tissue phase diagrams.

(A-C) Phase diagrams are shown in the 2D parameter space of 1/ vs. 1/ (A), 1/ vs. 1/ (B), and 1/ vs. 1/ (C). Blue dots represent the fluid phase with Deff > 0.001, and red triangles represent the solid phase with Deff < 0.001. Arrowheads in panel A correspond to extreme opposite phases that are discussed in Fig. 4. (D) The 2D phase planes are organized into a schematic 3D phase diagram. Letters ‘F’ and ‘S’ denote the fluid and solid phases, respectively. Parameters: = 16.6 × 10−3, = 0.2, = 5.2 × 10−4, = 8.3 × 10−4 (Note that two of these parameters are varied in panels A-C). Other parameters are from Table S1.

Tension and shape fluctuations in tissue monolayers.

(A-B) Heat maps showing cellular edge tensions in the tissue kept in fluid (A) and solid (B) phases, as shown by arrowheads in the phase diagram of Fig. 3A. Parameters: = 8.3 × 10−6, = 0.8 × 10−4 for A and = 0.42, = 0.001 for B. Other parameter values are as mentioned in Fig. 3 caption. (C-E) For a single tissue, the average tension (C), area fraction (D), and shape index (E) are plotted over time in the steady state. The grey and black lines denote two opposite phases (fluid and solid, respectively) corresponding to panels A and B. (F) The shape index, averaged over time and many ensembles, is shown with for various values. Inset: The dynamical order parameter, Deff, versus 〈S〉 for × 104 = 5. (G) Standard deviation of the shape index showed a peak when plotted with . Blue circles and red triangles represent fluid and solid phases, respectively, in F and G. (H-I) Instantaneous zoomed configurations (H) and probability distributions of shape indices (I), corresponding to the marked points, (i), (ii), and (iii) in the panels F, G. In H(ii), regular polygonal cell shapes (within the dashed red circle) coexist with elliptical shapes, suggesting shape fluctuations. In I(ii), the shape index distribution is much broader than the other regions (I(i) & I(ii)). (J) Each cell is color mapped with the amount of distance it traversed within a time window (from 2 × 105 to 2 × 106 iterations) in the three specified regions, (i), (ii), and (iii) of Panel G. In J(ii), cells with large and small displacements coexist together, forming connected clusters. Parameters: = 8.3 × 10−4 for region-i, = 0.25 for region-ii, and = 0.46 for region-iii. Other parameters are from Fig. 3 and Table S1.

Dynamic heterogeneity and glassy dynamics.

(A-B) Cell center displacements over a time window (from 105 to 2 × 106 iterations) for low and high adhesion strengths (marked by (i) and (ii), respectively, corresponding to Fig. 4G). For lower adhesion (region-i, A), instantaneous displacements are random and uncorrelated, whereas the displacement field shows swirling patterns for higher adhesion (region-ii, B). (C) The non-Gaussian parameter, α2(Δt), shows a peak for higher adhesion (region-ii) around the lag time Δt*. The shaded region spanning At* indicates the time window where the displacement fields (panels A, B) and trajectories (panel F) were observed. (D-E) Probability distributions of cell center displacements at the lag time Δt*. Black dashed lines indicate the best fit Gaussian. The Blue dashed line (in E) shows an exponential fit. (F) Sample cell center trajectories within a time window (corresponding to the shaded region in C). The trajectory is diffusive for lower adhesion (region-i), but a cage rearrangement event (hopping trajectory) was seen for higher adhesion (region-ii). (G) Probability distribution of diffusion coefficients (D) measured from time-averaged MSD curves of individual cells for the region-ii. The red line is an exponential fit. (H-I) Probability distributions of cell center displacements (H) and scaled displacements (I) for different values of and . Note that the tails of all distributions of normalized displacements (defined by ) follow a single master curve with exponent 1 (indicated by the dashed line in I). Insets of H and I show zoomed tail parts of corresponding distributions. The parameters are the same as in Fig. 4.

Bead-spring model of a single cell.

The force components due to the tangential spring tension are shown in black arrows. The blue arrows denote the normal components of the pressure force. The neighbors of the i-th bead are the (i − 1)-th and (i + 1)-th beads.

List of system parameters.

A single cell (at equilibrium) idealized as an n-sided regular polygon (here n = 8).

An isotropic pressure force Pl0 inflates the cell, pushing each vertex point outward and increasing the radius of the circumcircle from r0 to r. Correspondingly, the surface springs at the edges extend from l0 to (l0 + 2Δl). The diagram is not drawn to scale and is magnified for visual clarity.

Fluid to solid transition with respect to three dimensionless parameters.

The mean-squared displacements of cell centers show transitions from the fluid-like diffusive to the solid-like sub-diffusive behavior (A) with increasing intracellular pressure (top to bottom: = 0.20, 0.22, 0.24, 0.33), (B) with decreasing motility (top to bottom: = 16.6 × 10−3, 15.0 × 10−3, 13.2 × 10−3, 10.8 × 10−3), and (C) with increasing rotational noise strength (top to bottom: = 5.2 × 10−4, 7.5 × 10−4, 13.0 × 10−4, 47 × 10−4). Parameters: = 16.6 × 10−3, = 0.2, = 5.2 × 10−4, = 8.3 × 10−4 (Note that one of these parameters are varied in panels A-C). Other parameters are from Table S1.

Phase diagrams in the parameter space of (A)

vs. 1/, (B) vs. 1/, and (C) vs. 1/. Dashed lines correspond to the phase boundaries. Parameters: = 0.2, = 5.2 × 10−4, = 8.3 × 10−4 (Note that one of these parameters are varied along with in panels A-C). Other parameters are from Table S1.

Edge-tension distributions.

Probability distributions of the edge elongation (li − l0) in the tissue kept in fluid (blue) and solid (red) phases, corresponding to Fig. 4C (main paper). Parameters: = 8.3 × 10−6, = 0.8 × 10−4 (blue) and = 0.42, = 0.001 for B. Other parameter values are as mentioned in Fig. 3 caption.

Cell-area distributions.

Probability distributions of the scaled cell-area, Ã = A/〈A〉, for region-i (red), ii (green), and iii (blue), corresponding to Fig. 4F. Solid lines are the fits with the generalized k-Gamma distribution, . Note that the k value increases with an increase in as the tissue progressively solidifies. Here, 〈A〉 denotes the mean of cell-area. Parameters: Same as Fig. 4F.

Spontaneous formation of a muticellular rosette.

A configuration showing a five-cell junction (highlighted in red) known as a rosette that can be formed when five or more cells share a vertex. Parameters: = 8.3 × 10−5, = 0.2, = 16.6 × 10−3, = 5.2 × 10−4, l0 = 0.14. Other parameters are from Table S1.

Jamming onset in the ‘passive limit’.

We obtained the ‘passive limit’ of our model by switching off the self-propulsion speed (υ0 = 0) in the absence of cell-cell adhesion (Kadh = 0). Jammed states are formed by systematically increasing the intracellular pressure (consequently increasing the packing fraction) in a box with rigid boundaries. For more details, see the SI text, Section V. (A-C) Steady state simulation snapshots with increasing packing fractions: ϕ/ϕmax — 0.84 (A), 0.95 (B), and 1.0 (C). (D) Normalized area fraction (ϕ/ϕmax) and (E) the average contact number per cell (Z), as functions of the averaged shape index (〈S〉). The tissue reaches the confluence when the normalized packing fraction becomes unity (ϕ/ϕmax − 1) at the measured shape index 〈S〈 ≈ 3.81 (shown by the vertical dashed line in D). The maximal packing fraction achieved is ϕmax = 0.883, as shown by the horizontal dotted line in panel D. The average coordination number Z ~ 5.7 indicates a jammed confluent state at 〈S〉 ∔ 3.81 (indicated by the vertical dashed line in E). Parameters: = 0, υ0 = 0, l0 − 0.1 and varying from 0.0875 to 0.75. Other parameters are from Table S1.

Calculation of shape indices by Voronoi tessellation (corresponding to Fig. 4F).

(A) A configuration of polygonal cells (red solid lines), encompassing the original cells (grey), obtained from the Voronoi tesselation of the space in the liquid phase (corresponding to the blue star-marked point in panel B). (B) The effective diffusivity from long time MSD slope, Deff, is shown against for different values of (rotational noise strength). Data points with Deff > 0.001 are marked as liquid (blue circles), and Deff ≤ 0.001 are marked as solid (red triangles). (C) The average shape index measured from the Voronoi polygons(〈S〉V or) is plotted with adhesion strength () for different values of . The horizontal dashed line corresponds to 〈S〉V or − 3.81. The black, green, and brown curves in panel B correspond to Fig. 4F of the main paper. In Panels B and C, blue circles and red triangles represent fluid and solid phases, respectively. Parameters are from Fig. 4 of our manuscript and Table S1.

Solidification at highly confluent tissues.

(A-B) Tissue configurations for = 0 (A) and = 0.25 (B). Note that high confluence was achieved by increasing l0. (C) MSD of cell centers (corresponding to values as in panels A and B), showing a transition from sub-diffusive to caged behavior. The dashed line shows the MSD exponent of 0.67 for = 0. (D) MSD exponent (β) shows transition with at a lower (red curve) and at a higher (blue curve). The blue and red arrowheads indicate the transition points (and corresponding values) based on a cut-off, β = 0.2, as shown by the dotted horizontal line. (E) The area fraction (both ϕ and ϕ′, as defined in Sec. IVD, SI) and (F) the average contact number per cell (Z), are plotted against for the highly confluent tissues at two different values (red and blue ones). Note that in panels D, E, and F, the black dashed curve is for the parameters corresponding to Fig. 2E (main text), given here for comparison with a near-confluent tissue. Notably, ϕ and Z are almost constant for the fully confluent tissue compared to the near-confluent tissue denoted by the dashed reference curves. (G) The average shape index measured from the Voronoi tessellated polygons (〈S〉V or), and (H) the average shape index (〈S〉) obtained from the actual cell contours, are plotted against . In G, the blue and red arrowheads indicate the values corresponding to 〉S〉V or = 3.81, which coincide with the liquid-solid transition points located in panel D. For all panels, l0 = 0.14 except the dashed reference curve (in D-F), for which l0 = 0.1. Other parameters are from Fig. 3 (main text) and Table S1.