We first describe the experimental observations, which serve as the basis for carrying out the agent-based simulations. Figure 1A shows the bright-field images of distinct stages during zebrafish morphogenesis. A 2D section of zebrafish blastoderm (Figure 1B) shows that there is considerable dispersion in cell sizes. The statistical properties of the cell sizes are shown in Appendix 1—figure 1D. Figure 1C shows that ${\displaystyle \eta }$ increases sharply over a narrow ${\displaystyle \varphi }$ range and saturates when ${\displaystyle \varphi }$ exceeds ${\displaystyle {\varphi }_S\approx 0.90}$.

Figure 1

Download asset Open asset

To account for the results in Figure 1C, we first simulated a mono-disperse system in which all the cells have identical radius (${\displaystyle R=8.5\mu m}$). Because the system crystallizes (Appendix 1—figure 1A and B), we concluded that the dynamics observed in experiments cannot be explained using this model. A 1:1 binary mixture of cells with different radii gives glass-like behavior for all ${\displaystyle \varphi }$, with the relaxation time ${\displaystyle {\tau }_{\alpha }}$ as well as the effective viscosity ${\displaystyle \overline{\eta }}$ (defined in Equation 1) following the VFT behavior (see Appendix 2).

In typical cell tissues, and zebrafish in particular, there is a great dispersion in the cell sizes, which vary in a single tissue by a factor of ∼5–6 (Petridou et al., 2021; Figure 1B, Appendix 1—figure 1D). In addition, the elastic forces characterizing cell–cell interactions are soft, which implies that the cells can overlap, with ${\displaystyle {\displaystyle r_{ij}-(R_i+R_j)<0}}$ when they are jammed (Figure 1B, D). Thus, both polydispersity (PD) and soft interactions between the cells must control the relaxation dynamics. To test this proposition, we simulated a highly polydisperse system (PDs) in which the cell sizes vary by a factor of ∼8 (Figure 1D , Appendix 1—figure 1E).

A simulation snapshot (Figure 1D) for ${\displaystyle \varphi =0.93}$ shows that different sized cells are well mixed. In other words, the cells do not phase separate. The structure of the tissue can be described using the pair correlation function, ${\displaystyle g(r)=\frac{1}{\rho }⟨\frac{1}{N}\sum _i^N\sum _{j\ne i}^N\delta (r-|{\overrightarrow{r}}_i-{\overrightarrow{r}}_j|)⟩}$, where ${\displaystyle \rho =\frac{N}{L^2}}$ is the number density, ${\displaystyle \delta }$ is the Dirac delta function, ${\displaystyle {\overrightarrow{r}}_i}$ is the position of the ith cell, and the angular bracket ${\displaystyle ⟨⟩}$ denotes an average over different ensembles. The ${\displaystyle g(r)}$ function (Figure 1E) has a peak around ${\displaystyle r\sim 17\mu m}$, which is approximately the average diameter of the cells. The absence of peaks in ${\displaystyle g(r)}$ beyond the second one suggests there is no long-range order. Thus, the polydisperse cell system exhibits liquid-like structure even at the high ${\displaystyle \varphi }$.

A fit of the experimental data for ${\displaystyle \eta }$ using the VFT (Tammann and Hesse, 1926; Fulcher, 1925) relation in the range ${\displaystyle \varphi \le 0.87}$ (Figure 1C) yields ${\displaystyle {\varphi }_0\approx 0.95}$ and ${\displaystyle D\approx 0.51}$ (Sinha and Thirumalai, 2021). The VFT equation for cells, which is related to the Doolittle equation (White and Lipson, 2016) for fluidity (${\displaystyle \frac{1}{\eta }}$) that is based on free space available for motion in an amorphous system (Doolittle and Doolittle, 1957; Cohen and Turnbull, 1959), is ${\displaystyle \eta ={\eta }_0\exp \left[\frac{D}{{\varphi }_0/\varphi -1}\right]}$, where ${\displaystyle D}$ is the apparent activation energy. In order to compare with experiments, we calculated an effective shear viscosity (${\displaystyle \overline{\eta }}$) for the polydisperse system using a Green–Kubo-type relation (Hansen and McDonald, 2013)

The stress tensor ${\displaystyle P_{\mu \nu }(t)}$ in the above equation is

where ${\displaystyle \mu ,\nu \in (x,y)}$ are the Cartesian components of coordinates, ${\displaystyle {\overrightarrow{r}}_{ij}={\overrightarrow{r}}_i-{\overrightarrow{r}}_j}$, ${\displaystyle {\overrightarrow{f}}_{ij}}$ is the force between ith and jth cells, and ${\displaystyle A}$ is the area of the simulation box. Note that ${\displaystyle \overline{\eta }}$ should be viewed as a proxy for shear viscosity because it does not contain the kinetic term and the factor ${\displaystyle \frac{A}{k_{B}T}}$ is not included in Equation (1) because temperature is not a relevant variable in the highly over-damped model for cells considered here.

Plot of ${\displaystyle \overline{\eta }}$ as a function of ${\displaystyle \varphi }$ in Figure 2A shows qualitatively the same behavior as the estimate of viscosity (using dimensional arguments) made in experiments. Two features about Figures 1C and 2A are worth noting. (i) Both simulations and experiments show that up to ${\displaystyle \varphi \approx 0.90}$, ${\displaystyle \overline{\eta }(\varphi )}$ follows the VFT relation with ${\displaystyle {\varphi }_0\sim 0.94}$ and ${\displaystyle D\sim 0.5}$. More importantly, ${\displaystyle \overline{\eta }}$ is independent of ${\displaystyle \varphi }$ when ${\displaystyle {\displaystyle \varphi >0.90}}$. (ii) The values of ${\displaystyle {\varphi }_0}$ and ${\displaystyle D}$ obtained by fitting the experimental estimate of ${\displaystyle \eta }$ to the VFT equation and simulation results are almost identical. Moreover, the onset of the plateau packing fraction in simulations and experiments occurs at the same value (${\displaystyle {\varphi }_S\sim 0.90}$). The overall agreement with experiments is remarkable given that the model was not created to mimic the zebrafish tissue.

Figure 2

Download asset Open asset

To provide additional insights into the dynamics, we calculated the isotropic self-intermediate scattering function, ${\displaystyle F_s(q,t)}$,

where ${\displaystyle \overrightarrow{q}}$ is the wave vector, and ${\displaystyle {\overrightarrow{r}}_j(t)}$ is the position of a cell at time ${\displaystyle t}$. The degree of dynamic correlation between two cells can be inferred from the decay of ${\displaystyle F_s(q,t)}$. The angle bracket ${\displaystyle ⟨...⟩}$ is an average over different time origins and different trajectories. We chose ${\displaystyle q=\frac{2\pi }{r_{\text{max}}}}$, where ${\displaystyle r_{\text{max}}}$ is the position of the first peak in ${\displaystyle g(r)}$ between all cells (see Figure 1E). The relaxation time ${\displaystyle {\tau }_{\alpha }}$ is calculated using ${\displaystyle F_s(q,t={\tau }_{\alpha })=\frac{1}{e}}$.

From Figure 2B and C, which show ${\displaystyle F_s(q,t)}$ as a function of ${\displaystyle t}$ for various ${\displaystyle \varphi }$, it is clear that the dynamics become sluggish as ${\displaystyle \varphi }$ increases. The relaxation profiles exhibit a two-step decay with a plateau in the intermediate time scales. The dynamics continues to slow down dramatically until ${\displaystyle \varphi \le 0.90}$. Surprisingly, the increase in the duration of the plateau in ${\displaystyle F_s(q,t)}$ ceases when ${\displaystyle \varphi }$ exceeds ${\displaystyle \approx 0.90}$ (Figure 2C), a puzzling finding that is also reflected in the dependence of ${\displaystyle {\tau }_{\alpha }}$ on ${\displaystyle \varphi }$ in Figure 2D. The relaxation time increases dramatically, following the VFT relation, till ${\displaystyle \varphi \approx 0.90}$, and subsequently reaches a plateau (see the inset in Figure 2D).

If the VFT relation continued to hold for all ${\displaystyle \varphi }$, as in glasses or in binary mixture of 2D cells (see Appendix 2), then the fit yields ${\displaystyle {\varphi }_0\approx 0.95}$ and ${\displaystyle D\approx 0.50}$. However, the simulations show that ${\displaystyle {\tau }_{\alpha }}$ is nearly a constant when ${\displaystyle \varphi }$ exceeds ${\displaystyle 0.90}$. We should note that the behavior in Figure 2D differs from the dependence of ${\displaystyle {\tau }_{\alpha }}$ on ${\displaystyle \varphi }$ for 2D monodisperse polymer rings, used as a model for soft colloids. Simulations (Gnan and Zaccarelli, 2019) showed ${\displaystyle {\tau }_{\alpha }}$ increases till a critical ${\displaystyle {\varphi }_S}$ but it decreases substantially beyond ${\displaystyle {\varphi }_S}$ with no saturation.

Plot of ${\displaystyle {\tau }_{\alpha }}$ as a function of the radius of cells ${\displaystyle R_i}$ (Figure 3A) shows nearly eight orders of magnitude change. The size dependence of ${\displaystyle {\tau }_{\alpha }}$ on ${\displaystyle \varphi }$ is striking. That ${\displaystyle {\tau }_{\alpha }}$ should increase for large-sized cells (see the data beyond the vertical dashed line in Figure 3A) is not unexpected. However, even when cell sizes increase beyond ${\displaystyle R_i=4.25\mu m}$, the dispersion in ${\displaystyle {\tau }_{\alpha }}$ is substantial, especially when ${\displaystyle \varphi }$ exceeds ${\displaystyle {\varphi }_S}$. The relaxation times for cells with ${\displaystyle R_i<4.25\mu m}$ are relatively short even though the system as a whole is jammed. For ${\displaystyle \varphi \ge 0.90}$, ${\displaystyle {\tau }_{\alpha }}$ for small-sized cells have a weak dependence on ${\displaystyle \varphi }$. Although ${\displaystyle {\tau }_{\alpha }}$ for cells with radius <4 µm is short, it is clear that for a given ${\displaystyle \varphi }$ (e.g., ${\displaystyle \varphi =0.93}$) the variations in ${\displaystyle {\tau }_{\alpha }}$ are substantial. In contrast, ${\displaystyle {\displaystyle {\tau }_{\alpha }\mathrm{s}}}$ for larger cells (${\displaystyle R\ge 7\mu m}$) are substantially large, possibly exceeding the typical cell division time in experiments. In what follows, we interpret these results in terms of available free area ${\displaystyle ⟨A_{\text{free}}⟩}$ for cells. The smaller-sized cells have the largest ${\displaystyle ⟨A_{\text{free}}⟩\approx 50\mu m^2\approx \pi R_S^2(R_S\approx 4\mu m)}$ (${\displaystyle R_S}$ is the radius of the small cell).

Figure 3

Download asset Open asset

The effect of jamming on the dramatic increase in ${\displaystyle {\tau }_{\alpha }}$ occurs near ${\displaystyle R_i\approx 4.5\mu m}$, which is comparable to the length scale of short-range interactions. For ${\displaystyle \varphi \le 0.90}$, ${\displaystyle {\tau }_{\alpha }}$ increases as the cell size increases. However, at higher packing fractions, even cells of similar sizes show substantial variations in ${\displaystyle {\tau }_{\alpha }}$, which change by almost 3–4 orders of magnitude (see the data around the vertical dashed line for ${\displaystyle \varphi \ge 0.915}$ in Figure 3A).

This is a consequence of large variations in the local density (Appendix 6—figure 1). Some of the similar-sized cells are trapped in the jammed environment, whereas others are in less crowded regions (see Appendix 6—figure 1). The spread in ${\displaystyle {\tau }_{\alpha }}$ increases dramatically for ${\displaystyle {\displaystyle \varphi >{\varphi }_S}}$ (${\displaystyle \approx 0.90}$) and effectively overlap with each other. This is vividly illustrated in the histogram, ${\displaystyle P(\log ({\tau }_{\alpha }))}$, shown in Figure 3B. For ${\displaystyle {\displaystyle \varphi <{\varphi }_s}}$, the peak in ${\displaystyle P(\log ({\tau }_{\alpha }))}$ monotonically shifts to higher ${\displaystyle \log ({\tau }_{\alpha })}$ values. In contrast, when ${\displaystyle \varphi }$ exceeds ${\displaystyle {\varphi }_S}$ there is overlap in ${\displaystyle P(\log ({\tau }_{\alpha }))}$, which is reflected in the saturation of ${\displaystyle \overline{\eta }}$ and ${\displaystyle {\tau }_{\alpha }}$.

There are cells (typically with small sizes) that move faster even in a highly jammed environment (see Appendix 5—figures 1C and 2). The motions of the fast-moving cells change the local environment, which effectively facilitates the bigger cells to move in a crowded environment (see Appendix 5—figures 1D and 2, Video 1 (${\displaystyle {\displaystyle \varphi =0.92>{\varphi }_S}}$) and Video 2 (${\displaystyle \varphi =0.90={\varphi }_S}$)). In contrast, for ${\displaystyle {\displaystyle \varphi =0.85<{\varphi }_S}}$, small- and large-sized cells move without hindrance because of adequate availability of free area (Video 3). The videos vividly illustrate the large-scale facilitated rearrangements that enable the large-sized cells to move.

Video 1

Download asset

Video 2

Download asset

Video 3

Download asset

The dependence of ${\displaystyle {\tau }_{\alpha }}$ on ${\displaystyle \varphi }$ for ${\displaystyle \varphi \le {\varphi }_S}$ (Figure 2D) implies that the polydisperse cell systems behave as a soft glass in this regime. On theoretical grounds, it was predicted that ${\displaystyle P(\ln ({\tau }_{\alpha }))\sim \exp [-c(\ln (\frac{{\tau }_{\alpha }}{{\tau }_0}){)}^2]}$ in glass-forming systems (Kirkpatrick and Thirumalai, 2015). Remarkably, we found that this prediction is valid in the polydisperse cell system (Figure 3C). However, above ${\displaystyle {\varphi }_S}$ the predicted relation is not satisfied (see Figure 3D).

We explain the saturation in the viscosity by calculating the available free area per cell, as ${\displaystyle \varphi }$ increases. In a hard disk system, one would expect that the free area would decrease monotonically with ${\displaystyle \varphi }$ until it is fully jammed at the close packing fraction (∼0.84; Drocco et al., 2005; Reichhardt and Reichhardt, 2014). Because the cells are modeled as soft deformable disks, they could overlap with each other even when fully jammed. Therefore, the region where cells overlap creates free area in the immediate neighborhood.

The extent of overlap (${\displaystyle h_{ij}}$) is reflected in distribution ${\displaystyle P(h_{ij})}$. The width in ${\displaystyle P(h_{ij})}$ increases with ${\displaystyle \varphi }$, and the peak shifts to higher values of ${\displaystyle h_{ij}}$ (Figure 4A). The mean, ${\displaystyle ⟨h_{ij}⟩}$, increases with ${\displaystyle \varphi }$ (Figure 4B). Thus, even if the cells are highly jammed at ${\displaystyle \varphi \approx {\varphi }_S}$, free area is available because of an increase in the overlap between cells (see Figure 5).

Figure 4

Download asset Open asset

Figure 5

Download asset Open asset

When ${\displaystyle \varphi }$ exceeds ${\displaystyle {\varphi }_S}$, the mobility of small-sized cells facilitates the larger cells to move, as is assumed in the free volume theory of polymer glasses (Cohen and Turnbull, 1959; Turnbull and Cohen, 1961; Turnbull and Cohen, 1970; Falk et al., 2020). As a result of the motion of small cells, a void is temporarily created, which allows other (possibly large) cells to move. In addition to the release of space, the cells can also interpenetrate (Figure 4A and B). If ${\displaystyle h_{ij}}$ increases, as is the case when the extent of compression increases (Figure 4A), the available space for nearby cells would also increase. This effect is expected to occur with high probability at ${\displaystyle {\varphi }_S}$ and beyond, resulting in high overlap between the cells. These arguments suggest that the combined effect of PD and cell–cell overlap creates, via the self-propulsion of cells, additional free area that drives larger cells to move even under jammed conditions.

In order to quantify the physical picture given above, we calculated an effective area for each cell by first calculating Voronoi cell area ${\displaystyle A}$. A plot for Voronoi tessellation is presented in Figure 5A for ${\displaystyle \varphi =0.93}$, and the histogram of ${\displaystyle A}$ is shown in Figure 5B. As ${\displaystyle \varphi }$ increases, the distribution shifts toward lower Voronoi cell size ${\displaystyle ⟨A⟩}$. The mean Voronoi cell size ${\displaystyle ⟨A⟩}$ as a function of ${\displaystyle \varphi }$ in Figure 5C shows ${\displaystyle ⟨A⟩}$ decreases as ${\displaystyle \varphi }$ is increased. As cells interpenetrate, the Voronoi cell size will be smaller than the actual cell size (${\displaystyle \pi R_i^2}$) in many instances (Figure 5A). To demonstrate this quantitatively, we calculated ${\displaystyle A_{\text{free,i}}=A_i-\pi R_i^2}$. The value of ${\displaystyle A_{\text{free}}}$ could be negative if the overlap between neighboring cells is substantial; ${\displaystyle A_{\text{free}}}$ is positive only when the Voronoi cell size is greater than the actual cell size. Positive ${\displaystyle A_{\text{free}}}$ is an estimate of the available free area. The histograms of ${\displaystyle A_{\text{free}}}$ for all the packing fractions in Figure 5D show that the distributions saturate beyond ${\displaystyle \varphi =0.90}$. All the distributions have a substantial region in which ${\displaystyle A_{\text{free}}}$ is negative. The negative value of ${\displaystyle A_{\text{free}}}$ increases with increasing ${\displaystyle \varphi }$, which implies that the amount of interpenetration between cells increases.

Because of the overlap between the cells, the available free area fraction ${\displaystyle {\varphi }_{\text{free}}}$ is higher than the expected free area fraction (${\displaystyle 1.0-\varphi }$) for all ${\displaystyle \varphi }$. We define an effective free area fraction ${\displaystyle {\varphi }_{\text{free}}}$ as

where ${\displaystyle N_p}$ is the number of positive free area in jth snapshots, ${\displaystyle N_t}$ is the total number of snapshots, ${\displaystyle A_{\text{box}}}$ is the simulation box area, and ${\displaystyle A_{{\text{free}}_{+,i}}^j}$ is the positive free area of ith cell in jth snapshot.

The calculated ${\displaystyle {\varphi }_{\text{free}}}$, plotted as a function of ${\displaystyle \varphi }$ in Figure 5E, shows that ${\displaystyle {\displaystyle {\varphi }_{\text{free}}}}$ decreases with ${\displaystyle \varphi }$ until ${\displaystyle \varphi =0.90}$, and then it saturates near a value ${\displaystyle {\varphi }_{\text{free}}\approx 0.22}$ (see the right panel in Figure 5E). Thus, the saturation in ${\displaystyle \overline{\eta }}$ as a function of ${\displaystyle \varphi }$ is explained by the free area picture, which arises due to combined effect of the size variations and the ability of cells to overlap.

Our main result, which we explain by adopting the free volume theory developed in the context of glasses (Cohen and Turnbull, 1959; Turnbull and Cohen, 1961; Turnbull and Cohen, 1970; Falk et al., 2020), is that above a critical packing fraction ${\displaystyle {\varphi }_S\sim 0.90}$ the viscosity saturates. Relaxation time, ${\displaystyle {\tau }_{\alpha }}$, measured using dynamic light scattering, in nearly monodisperse microgel poly(N- isopropylacrylamide) (PNiPAM) (Philippe et al., 2018) was found to depend only weakly on the volume fraction (3D), if ${\displaystyle {\varphi }_V}$ exceeds a critical value. It was suggested that the near saturation of ${\displaystyle {\tau }_{\alpha }}$ at high ${\displaystyle {\varphi }_V}$ is due to aging, which is a non-equilibrium effect. If saturation in viscosity and relaxation time in the embryonic tissue at high ${\displaystyle \varphi }$ is due to aging, then ${\displaystyle {\tau }_{\alpha }}$ should increase sharply as the waiting time, ${\displaystyle {\tau }_{\omega }}$, is lengthened. We wondered if aging could explain the observed saturation of ${\displaystyle \eta }$ in the embryonic tissue above ${\displaystyle {\varphi }_S}$. If aging causes the plateau in the tissue dynamics, then ${\displaystyle \eta }$ or ${\displaystyle {\tau }_{\alpha }}$ should be an increasing function of the waiting time, ${\displaystyle {\tau }_{\omega }}$. To test the effect of ${\displaystyle {\tau }_{\omega }}$ on ${\displaystyle {\tau }_{\alpha }}$, we calculated the self-intermediate scattering function ${\displaystyle F_s(q,t+{\tau }_{\omega })}$ as a function of ${\displaystyle t}$ by varying ${\displaystyle {\tau }_{\omega }}$ over three orders of magnitude at ${\displaystyle \varphi =0.92}$ (Figure 6A). There is literally no change in ${\displaystyle F_s(q,t+{\tau }_{\omega })}$ over the entire range of ${\displaystyle {\tau }_{\omega }}$. We conclude that, ${\displaystyle {\tau }_{\alpha }}$, extracted from ${\displaystyle F_s(q,t+{\tau }_{\omega })}$ is independent of ${\displaystyle {\tau }_{\omega }}$. The variations in ${\displaystyle {\tau }_{\alpha }}$ (Figure 6B), with respect to ${\displaystyle {\tau }_{\omega }}$, are significantly smaller than the errors in the simulation. Thus, the saturation in ${\displaystyle \eta }$ or ${\displaystyle {\tau }_{\alpha }}$ when ${\displaystyle {\displaystyle \varphi >{\varphi }_S}}$ is not a consequence of aging.

Figure 6

Download asset Open asset

There are two implications related to the absence of aging in the dynamics of the non-confluent embryonic tissues. (i) Although active forces drive the dynamics of the cells, as they presumably do in reality, the cell collectives can be treated as being near equilibrium, justifying the use of Green–Kubo relation to calculate ${\displaystyle \eta }$. (ii) Parenthetically, we note that the absence of significant non-equilibrium effects, even though zebrafish is a living system, further justifies the use of equilibrium rigidity percolation theory to analyze the experimental data (Petridou et al., 2021).

Following our earlier studies (Malmi-Kakkada et al., 2018; Sinha et al., 2020), we simulated a 2D version of a particle-based cell model. We did not explicitly include cell division in the simulations. This is physically reasonable because in the experiments (Petridou et al., 2021) the time scales over which cell division induced local stresses relax are short compared to cell division time. Thus, local equilibrium is established in between random cell division events. We performed simulations in 2D because experiments reported the dependence of viscosity as a function of area fraction.

In our model, cells are modeled as soft deformable disks (Matoz-Fernandez et al., 2017; Drasdo and Höhme, 2005; Schaller and Meyer-Hermann, 2005; Malmi-Kakkada et al., 2018) interacting via short-ranged forces. The elastic (repulsive) force between two cells with radii ${\displaystyle R_i}$ and ${\displaystyle R_j}$ is Hertzian, which is given by

where ${\displaystyle h_{ij}=\text{max}[0,R_i+R_j-|{\overrightarrow{r}}_i-{\overrightarrow{r}}_j|]}$. The repulsive force acts along the unit vector ${\displaystyle {\overrightarrow{n}}_{ij}}$, which points from the center of the jth cell to the center of the ith cell. The total force on the ith cell is

where ${\displaystyle NN(i)}$ is the number of near-neighbor cells that are in contact with the ith cell. The jth cell is the nearest neighbor of the ith cell, if ${\displaystyle {\displaystyle h_{ij}>0}}$. The near-neighbor condition ensures that the cells interpenetrate each other to some extent, thus mimicking the cell softness. For simplicity, we assume that the elastic moduli (${\displaystyle E}$) and the Poisson ratios (${\displaystyle \nu }$) for all the cells are identical. PD in the cell sizes is important in recovering the plateau in the viscosity as a function of packing fraction. Thus, the distribution of cell areas (${\displaystyle A_i=\pi R_i^2}$) is assumed to have a distribution that mimics the broad area distribution discovered in experiments.

In addition to the repulsive Hertz force, we include an active force arising from self-propulsion mobility (µ), which is a proxy for the intrinsically generated forces within a cell. For illustration purposes, we take µ to independent of the cells, although this can be relaxed readily. We assume that the dynamics of each cell obeys the phenomenological equation

where ${\displaystyle {\gamma }_i}$ is the friction coefficient of ith cell, and ${\displaystyle {\mathcal{W}}_i(t)}$ is a noise term. The friction coefficient ${\displaystyle {\gamma }_i}$ is taken to be ${\displaystyle {\gamma }_0{R}_i}$ (Sinha et al., 2022). By scaling ${\displaystyle t}$ by the characteristic time scale, ${\displaystyle \tau =\frac{{⟨R⟩}^2}{{\mu }^2}}$ in Equation (6), one can show that the results should be insensitive to the exact value of µ. The noise term ${\displaystyle {\mathcal{W}}_i(t)}$ is chosen such that ${\displaystyle ⟨{\mathcal{W}}_i(t)⟩=0}$ and ${\displaystyle ⟨{\mathcal{W}}_i^{\alpha }(t){\mathcal{W}}_j^{\beta }(t^{\mathrm{\prime }})⟩=\delta (t-t^{\mathrm{\prime }}){\delta }_{i,j}{\delta }^{\alpha ,\beta }}$. In our model, there is no dynamics with only systematic forces because the temperature is zero. The observed dynamics arises solely due to the self-propulsion (Equation 6).

We place ${\displaystyle N}$ cells in a square box that is periodically replicated. The size of the box is ${\displaystyle L}$ so that the packing fraction (in our 2D system it is the area fraction) is ${\displaystyle \varphi =\frac{\sum _{i=1}^N\pi R_i^2}{L^2}}$. We performed extensive simulations by varying ${\displaystyle \varphi }$ in the range ${\displaystyle 0.700\le \varphi \le 0.950}$. The results reported in main text are obtained with ${\displaystyle N=500}$. Finite size effects are discussed in Appendix 7.

To mimic the variations in the area of cells in a tissue (Petridou et al., 2021), we use a broad distribution of cell radii (see Appendix 1 for details). The parameters for the model are given in Table 1. In this study, we do not consider the growth and division of cells. Thus, our simulations describe steady-state dynamics of the tissue. For each ${\displaystyle \varphi }$, we performed simulations for at least (5–10)${\displaystyle {\tau }_{\alpha }}$ before storing the data. For each ${\displaystyle \varphi }$, we performed 24 independent simulations. The calculation of viscosity was performed by averaging over 40 independent simulations at each ${\displaystyle \varphi }$.

We calculated the effective viscosity (${\displaystyle \overline{\eta }}$) for various values of ${\displaystyle \varphi }$ by integrating the off-diagonal part of the stress–stress correlation function ${\displaystyle ⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}$ using the Green–Kubo relation (Hansen and McDonald, 2013) (without the pre-factor ${\displaystyle \frac{A}{k_{B}T}}$)

where µ and ${\displaystyle \nu }$ denote Cartesian components (${\displaystyle x}$ and ${\displaystyle y}$) of the stress tensor ${\displaystyle P_{\mu \nu }(t)}$ (see main text for the definition of ${\displaystyle P_{\mu \nu }(t)}$). The definition of ${\displaystyle \overline{\eta }}$, which relates the decay of stresses as a function of times in the non-confluent tissue, is akin to the methods used to calculate viscosity in simple fluids (Equation 7). The time dependence of ${\displaystyle ⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}$, normalized by ${\displaystyle ⟨P_{\mu \nu }(0{)}^2⟩}$, for different values of ${\displaystyle \varphi }$ (Figure 7A and B) shows that the stress relaxation is clearly non-exponential, decaying to zero in two steps. After an initial rapid decay followed by a plateau at intermediate times (clearly visible for ${\displaystyle \varphi \ge 0.91}$), the normalized ${\displaystyle ⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}$ decays to zero as a stretched exponential. The black dashed lines in Figure 7C show that a stretched exponential function, ${\displaystyle C_s\exp [-{\left(\frac{t}{{\tau }_{\eta }}\right)}^{\beta }]}$, where ${\displaystyle {\tau }_{\eta }}$ is the characteristic time in which stress relax and ${\displaystyle \beta }$ is the stretching exponent, provides an excellent fit to the long time decay of ${\displaystyle {\displaystyle ⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}}$ (from the plateau region to zero) as a function of ${\displaystyle t}$. Therefore, we utilized the fit function, ${\displaystyle C_s\exp [-{\left(\frac{t}{{\tau }_{\eta }}\right)}^{\beta }]}$, to replace the noisy long time part of ${\displaystyle ⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}$ by a smooth fit data before evaluating the integral in Equation (7). The details of the procedure to compute ${\displaystyle \overline{\eta }}$ are described below.

Figure 7

Download asset Open asset

We divided ${\displaystyle ⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}$ in two parts. (i) The short time part (${\displaystyle {⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}_{\text{short}}}$) – the smooth initial rapid decay until the plateau is reached (e.g., see the blue circles in Figure 7D for ${\displaystyle \varphi =0.93}$). For the ${\displaystyle n}$ data points at short times, ${\displaystyle (t_1,{⟨P_{\mu \nu }(t_1)P_{\mu \nu }(0)⟩}_{\text{\% short}})}$,…, ${\displaystyle (t_n,{⟨P_{\mu \nu }(t_n)P_{\mu \nu }(0)⟩}_{\text{\% short}})}$, we constructed a spline ${\displaystyle S(t)}$ using a set of cubic polynomials:

The polynomials satisfy the following properties. (a) ${\displaystyle S_i(t_i)={⟨P_{\mu \nu }(t_i)P_{\mu \nu }(0)⟩}_{\text{\% short}}}$ and ${\displaystyle S_i(t_{i+1})={⟨P_{\mu \nu }(t_{i+1})P_{\mu \nu }(0)⟩}_{\text{short}}}$ for ${\displaystyle {\displaystyle i=1,...,n-1}}$ which guarantees that the spline function ${\displaystyle S(t)}$ interpolates between the data points. (b) ${\displaystyle S_{i-1}^{\mathrm{\prime }}(t)=S_i^{\mathrm{\prime }}(t)}$ for ${\displaystyle {\displaystyle i=2,...,n-1}}$ so that ${\displaystyle S^{\mathrm{\prime }}(t)}$ is continuous in the interval ${\displaystyle [t_1,t_n]}$. (c) ${\displaystyle S_{i-1}^{\mathrm{\prime }\mathrm{\prime }}(t)=S_i^{\mathrm{\prime }\mathrm{\prime }}(t)}$ for ${\displaystyle {\displaystyle i=2,...,n-1}}$ so that ${\displaystyle S^{\mathrm{\prime }\mathrm{\prime }}(t)}$ is continuous in the interval ${\displaystyle [t_1,t_n]}$. By solving for the unknown parameters, ${\displaystyle b_i,c_i}$, and ${\displaystyle d_i}$, using the above-mentioned properties, we constructed the function S(t). We used ${\displaystyle S(t)}$ to fit ${\displaystyle {⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}_{\text{short}}}$ to get an evenly spaced (${\displaystyle \delta t=10s}$) smooth data (solid blue line in Figure 7D). The fitting was done using the software ‘Xmgrace’.

(ii) The long time part (${\displaystyle {⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}_{\text{long}}}$) – from the plateau until it decays to zero – is shown by the red circles in Figure 7D. The long time part was fit using the analytical function ${\displaystyle C_s\exp [-{\left(\frac{t}{{\tau }_{\eta }}\right)}^{\beta }]}$ (black dashed line in Figure 7D). We refer to the fit data (${\displaystyle \delta t=10s}$) as ${\displaystyle {⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}_{\text{long}}^{\text{fit}}}$.

We then combined ${\displaystyle {⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}_{\text{short}}}$ and ${\displaystyle {⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}_{\text{long}}^{\text{\% fitted}}}$ to obtain ${\displaystyle {⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}_{\text{combined}}}$ (see inset of Figure 7D). Finally, we calculated ${\displaystyle \overline{\eta }}$ using the equation,

where ${\displaystyle t_1\delta t}$ is the end point of ${\displaystyle {⟨P_{\mu \nu }(t)P_{\mu \nu }(0)⟩}_{\text{short}}}$ and ${\displaystyle T\delta t}$ is the end point of ${\displaystyle {⟨P_{\mu \nu }(i\delta t)P_{\mu \nu }(0)⟩}_{\text{combined}}}$.

Appendix 1—figure 1

Download asset Open asset

To explain the observed saturation in viscosity (Figure 1C) when cell area fraction, ${\displaystyle \varphi }$, exceeds ${\displaystyle {\varphi }_S}$ (${\displaystyle \approx 0.90}$), we first simulated a monodisperse cell system using the model described in the ‘Materials and methods’ section. The monodisperse cell system (${\displaystyle R=8.5\mu m}$) crystallizes (see Appendix 1—figure 1A and B), which excludes it from being a viable model for explaining the experimental findings. We also find that a system consisting of 50:50 binary mixture of cells cannot account for the experimental data even though crystallization is avoided (see the next section). These findings forced us to take PD into account.

In order to develop an agent-based model that accounts for PD effects, we first extracted the distribution, ${\displaystyle P(A_i)}$, of the area (${\displaystyle A_i}$) in the zebrafish cells from the experimental data (Petridou et al., 2021). Appendix 1—figure 1D shows that ${\displaystyle P(A_i)}$ is broad, implying that cell sizes are highly heterogeneous. Based on this finding, we sought a model of the non-confluent tissue that approximately mimics PD found in experiments. In other words, ${\displaystyle \frac{\mathrm{\Delta }A}{⟨A⟩}}$ (${\displaystyle \mathrm{\Delta }A}$ is the dispersion in ${\displaystyle P(A_i)}$ and ${\displaystyle ⟨A⟩}$ is the mean value) should be similar to the data in Appendix 1—figure 1D. The radii (${\displaystyle R_i}$) of the cells in the simulations are sampled from a Gaussian distribution ${\displaystyle \sim \exp (-(R_i-⟨R⟩{)}^2/2\mathrm{\Delta }{R}^2)}$, with ${\displaystyle ⟨R⟩=8.5\mu m}$ and ${\displaystyle \mathrm{\Delta }R=4.5\mu m}$. The resulting ${\displaystyle P(A_i)}$ for one of the realizations (see Appendix 1—figure 1C) is shown in Appendix 1—figure 1E. The value of ${\displaystyle \frac{\mathrm{\Delta }A}{⟨A⟩}}$ is ${\displaystyle \sim 0.5}$, which compares favorably with the experimental estimate (∼0.4). The unusual dependence of the viscosity (${\displaystyle \eta }$) as a function of packing fraction (${\displaystyle \varphi }$) cannot be reproduced in the absence of PD.

We showed that the saturation in the relaxation time above a critical area fraction, ${\displaystyle {\varphi }_S}$, is related to two factors. One is that there ought to be dispersion in the cell sizes (Appendix 1—figure 1). The extent of dispersion is likely to less in three rather in two dimensions. The second criterion is that the cells should be soft, allowing them to overlap at high area fraction. In other words, the cell diameters are explicitly non-additive. The Hertz potential captures the squishy nature of the cells.

In order to reveal the importance of PD, we first investigated if a binary mixture of cells (see Appendix 2—figure 1A) would reproduce the observed dependence of ${\displaystyle {\tau }_{\alpha }}$ on ${\displaystyle \varphi }$. We created a 50:50 mixture with a cell size ratio ${\displaystyle \sim 1.4}$ (${\displaystyle R_B=8.5\mu m}$ and ${\displaystyle R_S=6.1\mu m}$). All other parameters are the same as in the polydisperse system (see Table 1). The radial distribution function, ${\displaystyle g(r)}$, calculated by considering both the cell types, exhibits intermediate- but not long-range translational order (Appendix 2—figure 1B). A peak in ${\displaystyle g(r)}$ appears at ${\displaystyle r_{max}=14.0\mu m}$.

In order to determine the dependence of the relaxation time, ${\displaystyle {\tau }_{\alpha }}$, on ${\displaystyle \varphi }$, we first calculated ${\displaystyle F_s(q,t)}$ at ${\displaystyle q=2\pi /r_{max}}$ (Appendix 2—figure 1C). The decay of ${\displaystyle F_s(q,t)}$ is similar to what one finds in typical glass-forming systems.

As ${\displaystyle \varphi }$ increases, the decay of ${\displaystyle F_s(q,t)}$ slows down dramatically. When ${\displaystyle \varphi }$ exceeds ${\displaystyle 0.93}$, there is a visible plateau at intermediate times followed by a slow decay. By fitting the long time decay of ${\displaystyle F_s(q,t)}$ to ${\displaystyle \exp (-(t/{\tau }_{\alpha }{)}^{\beta })}$ (${\displaystyle \beta }$ is the stretching exponent), we find that ${\displaystyle {\tau }_{\alpha }(\varphi )}$ as a function of ${\displaystyle \varphi }$ is well characterized by the VFT relation (Appendix 2—figure 1D). There is no evidence of saturation in ${\displaystyle {\tau }_{\alpha }(\varphi )}$ at high ${\displaystyle \varphi }$. The ${\displaystyle \varphi }$ dependence of the effective shear viscosity ${\displaystyle \overline{\eta }}$ (Appendix 2—figure 1E), calculated using Equation (8), also shows no sign of saturation. The VFT behavior, with ${\displaystyle {\varphi }_0\approx 1}$ and ${\displaystyle D\approx 1.2}$, shows that the two-component cell system behaves as a ‘fragile’ glass (Angell, 1991).

A comment regarding the binary system of cells is in order. The variables that characterize this system are ${\displaystyle \lambda =R_B/R_S}$ (${\displaystyle R_B}$ (${\displaystyle R_S}$) is the radius of the big (small) cells), ${\displaystyle {\mathrm{\Phi }}_B=N_B/(N_A+N_B)}$ (${\displaystyle N_B}$ (${\displaystyle N_A}$) is the number of big (small) cells) with ${\displaystyle {\mathrm{\Phi }}_A=1-{\mathrm{\Phi }}_B}$, and the packing fraction, ${\displaystyle \varphi =\pi (N_B{R}_B^2+N_A{R}_A^2)/A_S}$, where ${\displaystyle A_S}$ is the area of the sample. The results in Appendix 2—figure 1 were obtained using ${\displaystyle \lambda =1.4}$, ${\displaystyle {\mathrm{\Phi }}_B=0.5}$. With this choice, the value of PD is ${\displaystyle (\lambda -1)/(\lambda +1)\approx 0.17}$, which is smaller than the experimental value. It is possible that by thoroughly exploring the parameter space, ${\displaystyle \lambda }$ and ${\displaystyle \varphi }$, one could find regions in which the ${\displaystyle {\tau }_{\alpha }}$ in the two-component cell system would saturate beyond ${\displaystyle {\varphi }_S}$. However, in light of the results in Appendix 1—figure 1D, we choose to simulate systems that also have high degree of PD (Appendix 1—figure 1E).

Appendix 2—figure 1

Download asset Open asset

In the main text, we established that the effective viscosity, ${\displaystyle \overline{\eta }}$, which should be a proxy for the true ${\displaystyle \eta }$ and the relaxation time (${\displaystyle {\tau }_{\alpha }}$) in the polydisperse cell system, saturates beyond ${\displaystyle {\varphi }_S}$. The dynamics saturates beyond ${\displaystyle {\varphi }_S}$ because the available area per cells (quantified as ${\displaystyle {\varphi }_{\text{free}}}$) is roughly a constant in this region (see Figure 5E). In the binary system, however, we do not find any saturation in the ${\displaystyle {\tau }_{\alpha }}$. Therefore, if ${\displaystyle {\displaystyle {\varphi }_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}}}$ controls the dynamics, one would expect that ${\displaystyle {\varphi }_{\text{free}}}$ should decrease monotonically with ${\displaystyle \varphi }$ for the binary cells system. We calculated the average Vornoi cell size ${\displaystyle ⟨A⟩}$ (Appendix 3—figure 1A) and ${\displaystyle {\varphi }_{\text{free}}}$ (Appendix 3—figure 1B) for the binary system. Both ${\displaystyle ⟨A⟩}$ and ${\displaystyle {\varphi }_{\text{free}}}$ decrease with ${\displaystyle \varphi }$, which is consistent with the free volume picture proposed in the context of glasses.

Appendix 3—figure 1

Download asset Open asset

We have shown in the earlier section that the saturation of ${\displaystyle {\tau }_{\alpha }}$ above ${\displaystyle {\varphi }_S}$ is not a consequence of aging. In the context of glasses and supercooled liquids (Thirumalai et al., 1989), it has been shown that if ergodicity is broken, then a variety of observables would depend on initial conditions. In order to test if ergodicity is broken in our model of non-confluent tissues, we define a measure ${\displaystyle \mathrm{\Omega }(t)}$ given by

where ${\displaystyle k}$ and ${\displaystyle l}$ represent systems with different initial conditions, and ${\displaystyle {\omega }^k(t)=\frac{1}{t}{\int }_0^t{P}_{\mu \nu }^k(s)ds}$ with ${\displaystyle P_{\mu \nu }(s)}$ being the value of stress (see Equation 2 in the main text) at time ${\displaystyle s}$. If the system is ergodic, implying the system has explored the whole phase space on the simulation time scale, the values of ${\displaystyle {\omega }^k(t)}$ and ${\displaystyle {\omega }^l(t)}$ would be independent of ${\displaystyle k}$ and ${\displaystyle l}$. Therefore, ${\displaystyle \mathrm{\Omega }(t)}$ should vanish at very long time for ergodic systems. On the other hand, if ergodicity is broken, then ${\displaystyle \mathrm{\Omega }(t)}$ would be a constant whose value would depend on ${\displaystyle k}$ and ${\displaystyle l}$.

The long time values of ${\displaystyle \mathrm{\Omega }(t)}$, normalized by ${\displaystyle \mathrm{\Omega }(0)}$ (at ${\displaystyle t=0}$), are 0.01, 0.016, and 0.026 for ${\displaystyle \varphi =0.85,0.90}$, and 0.92, respectively (Appendix 4—figure 1A–C). Because these values are sufficiently small, we surmise that effectively ergodicity is established. Therefore, our conclusion in the earlier section that the polydisperse cell system is in near equilibrium is justified and also explains the absence of aging. Furthermore, it was also predicted previously (Thirumalai et al., 1989) that in long time the ergodic measure (${\displaystyle \mathrm{\Omega }(t)}$ in our case) should decay as ${\displaystyle \approx 1/t}$. Appendix 4—figure 1D shows that this is indeed the case – at long time ${\displaystyle \mathrm{\Omega }(t)/\mathrm{\Omega }(0)}$ decays approximately as ${\displaystyle 1/t}$.

Appendix 4—figure 1

Download asset Open asset

The structural and the dynamical behavior of the small (${\displaystyle R_S\le 4.5\mu m}$) and large (${\displaystyle R_B\ge 12.0\mu m}$) cells is dramatically different in the non-confluent tissue. The pair correlation functions between small cells (${\displaystyle g_{SS}(r)}$) and between large cells (${\displaystyle g_{BB}(r)}$) (Appendix 5—figure 1A and B) for ${\displaystyle \varphi =0.905}$ and ${\displaystyle \varphi =0.92}$ show that both small and large cells exhibit liquid-like disordered structures. However, it is important to note that ${\displaystyle g_{SS}(r)}$ has only one prominent peak and a modest second peak. In contrast, ${\displaystyle g_{BB}(r)}$ has three prominent peaks. Thus, the smaller-sized cells exhibit liquid-like behavior, whereas the large cells are jammed. This structural feature is reflected in the decay of ${\displaystyle F_s(q,t)}$ with ${\displaystyle q=\frac{2\pi }{r_{max}}}$, where ${\displaystyle r_{max}}$ is the position of the first peak in the ${\displaystyle g(r)}$ (Appendix 5—figure 1C and D).

There is a clear difference in the decay of ${\displaystyle F_s(q,t)}$, spanning nearly eight orders of magnitude, between small and large cells (compare Appendix 5—figure 1C and D). At the highly jammed packing fraction (${\displaystyle \varphi =0.92}$), small-sized cells have the characteristics of fluid-like behavior.

Appendix 5—figure 1

Download asset Open asset

A typical snapshot from one of the simulations (${\displaystyle \varphi =0.91}$) and the trajectories for a few small- and big-sized cells are displayed in Appendix 5—figure 2A–D. The figures reflect the decay in ${\displaystyle F_s(q,t)}$. Not only are the mobilities heterogeneous, it is also clear that the displacements of the small cells are greater than the large cells.

Appendix 5—figure 2

Download asset Open asset

The mobilities of all the cells, even under highly jammed conditions (${\displaystyle \varphi \ge {\varphi }_S}$), are only possible if the local area fraction changes dynamically. This is a collective effect, which is difficult to quantify because it would involve multi-cell correlation function. As explained in the main text, the movement of a jammed cell can only occur if several neighboring cells move to create space. This picture is not that dissimilar to kinetic facilitation in glass-forming materials (Hedges et al., 2009; Biroli and Garrahan, 2013). However, in glass-like systems, facilitation is due to thermal excitation, but in the active system, it is self-propulsion that causes the cells to move.

The creation of free space may be visualized by tracking the positions of the nearest-neighbor cells. In Appendix 6—figure 1, we display the local free area of a black-colored cell at different times. The top panels show the configurations where the black cell is completely jammed by other cells. The cell, colored in black, can move if the neighboring cells rearrange (caging effect in glass-forming systems) in order to increase the available free space. The bottom panels show that upon rearrangement of the cells surrounding the black cell its mobility increases. Such rearrangement occurs continuously, which qualitatively explains the saturation in viscosity in the multicomponent cell system.

Appendix 6—figure 1

Download asset Open asset

In the main text, we report results for ${\displaystyle N=500}$. To asses if the unusual dynamics is not an effect of finite system size, we performed additional simulations with ${\displaystyle N=200}$ and ${\displaystyle N=750}$. As shown in Appendix 7—figure 1A and B, ${\displaystyle F_s(q,t)}$ saturates at ${\displaystyle \varphi \ge {\varphi }_S}$, which is reflected in the logarithm of ${\displaystyle {\tau }_{\alpha }}$ as a function of ${\displaystyle \varphi }$ (Appendix 7—figure 1C and D). The saturation value ${\displaystyle {\varphi }_S\sim 0.90}$ is independent of the system size. The value of ${\displaystyle {\varphi }_0}$ (${\displaystyle \approx 0.95}$) is also nearly independent of system size. Therefore, the observed dynamics, reflected in the plateau in the viscosity at high ${\displaystyle \varphi }$, is likely not an effect of finite system size.

Appendix 7—figure 1

Download asset Open asset

In the zebrafish blastoderm experiment (Petridou et al., 2021), the change in viscosity (${\displaystyle \eta }$) was shown as a function of mean connectivity ${\displaystyle ⟨C⟩}$. To test whether our 2D tissue simulations also exhibit similar behavior, we calculated viscosity as a function of mean coordination number ${\displaystyle ⟨N_c⟩}$ (defined below), which is equivalent to ${\displaystyle ⟨C⟩}$. We define the coordination number, ${\displaystyle N_c}$, as the number of cells that are in contact with a given cell. Two cells with indices ${\displaystyle i}$ and ${\displaystyle j}$ are in contact if ${\displaystyle {\displaystyle h_{ij}=R_i+R_j-r_{ij}>0}}$. We calculate ${\displaystyle N_c}$ for all the cells for each ${\displaystyle \varphi }$ and calculate the histogram ${\displaystyle P(N_c)}$. The distributions of ${\displaystyle {\displaystyle P(N_c)}}$ are well fit by a Gaussian distribution function ${\displaystyle A\exp [-(\frac{x-\mu }{\sigma }{)}^2]}$ (Appendix 8—figure 1A–C). The calculated mean, ${\displaystyle ⟨N_c⟩}$, from the fit is linearly related to the cell area fraction ${\displaystyle \varphi }$ (Appendix 8—figure 1D). We also calculate the average connectivity ${\displaystyle ⟨C⟩}$ defined in the following way. Each cell is defined as a node, and an edge is defined as the line connecting two nodes. If a snapshot has ${\displaystyle n}$ nodes and ${\displaystyle m}$ edges, then the connectivity is defined as ${\displaystyle C=\frac{2m}{n}}$ (Petridou et al., 2021). We calculate ${\displaystyle C}$ for all the snapshots for each ${\displaystyle \varphi }$ and estimated its mean value ${\displaystyle ⟨C⟩}$. We find that ${\displaystyle ⟨C⟩}$ and ${\displaystyle ⟨N_c⟩}$ are of similar values (Appendix 8—figure 2A), and the dependence of viscosity ${\displaystyle \overline{\eta }}$ on ${\displaystyle ⟨N_c⟩}$ and ${\displaystyle ⟨C⟩}$ is similar to experimental results (Appendix 8—figure 2B and C; Petridou et al., 2021).

Appendix 8—figure 1

Download asset Open asset

Appendix 8—figure 2

Download asset Open asset

To pictorially observe the percolation transition in the simulations, we plot the connectivity map for various values ${\displaystyle \varphi }$ in Appendix 9—figure 1. Note that for smaller ${\displaystyle \varphi \le 0.89}$ the map shows that cells are loosely connected, suggesting a fluid-like behavior. For ${\displaystyle \varphi \ge 0.89}$, the connectivity between the cells spans the entire system, which was noted and analyzed elsewhere thoroughly (Petridou et al., 2021). Cells at one side of simulation box are connected to cells at the side. The cell connectivity extends throughout the sample. The transition from a non-percolated state to a percolated state occurs over a very narrow range of ${\displaystyle \varphi }$, which corresponds to the onset of rigidity percolation transition (Petridou et al., 2021). It is gratifying that the simple simulations reproduce the experimental observations.

Appendix 9—figure 1

Download asset Open asset

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We acknowledge Anne D Bowen at the Visualization Laboratory (Vislab), Texas Advanced Computing Center, for help with video visualizations. We are grateful to Mauro Mugnai and Nicoletta Petridou for useful discussions. This work was supported by grants from the National Science Foundation (PHY 23-10639 and CHE 23-20256) and the Welch Foundation (F-0019).

1. Preprint posted: November 26, 2022 (view preprint)
2. Sent for peer review: April 17, 2023
3. Preprint posted: July 3, 2023 (view preprint)
4. Preprint posted: September 28, 2023 (view preprint)
5. Preprint posted: December 18, 2023 (view preprint)
6. Version of Record published: January 19, 2024 (version 1)

You can cite all versions using the DOI https://doi.org/10.7554/eLife.87966. This DOI represents all versions, and will always resolve to the latest one.

© 2023, Das et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.