Inherently unstable networks collapse to a critical point

Nonequilibrium systems that are driven or drive themselves towards a critical point have been studied for almost three decades. Here we present a minimalist example of such a system, motivated by experiments on collapsing active elastic networks. Our model of an unstable elastic network exhibits a collapse towards a critical point from any macroscopically connected initial configuration. Taking into account steric interactions within the network, the model qualitatively and quantitatively reproduces results of the experiments on collapsing active gels.

Figure 1

An example of the collapse process of a triangular network from the initial to final, collapsed state. Only a quarter part of the $100\times 112$ nodes network is shown. Note, that during the process the total mass (total number of nodes) is conserved. In the final state of the network all the mass is concentrated in foci. The movie of the collapse of the full network can be seen in the Supplemental Material [10].

Figure 2

An illustration of the influence of steric interactions on the collapsed state properties. The initial state presented in (a) without steric interactions collapses to two foci with masses 3 and 9, as shown in (b). Due to steric interactions the enclave is trapped by its surroundings, leading to a collapsed state with one focus of mass 12, as shown in (c).

Figure 3

Cluster mass (in nodes) distributions. (a) Critical point of the random percolation model with (circles) and without (squares) absorption of the enclaves into their surroundings (NEP model [11]). Insets illustrate the enclaves—clusters fully surrounded by another cluster—and their absorption in the NEP model. In the bottom inset the enclaves are marked by dashed boundaries and nonenclaves by solid boundaries. In the top inset the result of absorption of enclaves from the configuration of the bottom inset is shown. (b) Collapsed state of the collapse model without (squares) and with (circles) steric interactions. The fits to squares in both panels are with the power law of the random percolation model $\tau =187/91>2$, while for the circles the fit is with the NEP model's Fisher exponent from Eq. (3). Insets demonstrate the effective absorption of enclaves during the collapse with steric interactions. The result of collapse of the configuration from Fig. 2 is shown for the case of collapse without (bottom inset) and with (top inset) steric interactions.

Figure 4

(a), (b) Clusters' structure at the critical point of the random percolation model without (a) and with (b) absorbed enclaves—NEP model. Dashed circle in (a) indicates an enclave, absorbed in (b). (c), (d) Modeling of the motor-driven collapse with and without steric interactions. Configuration of clusters modeled without (c) and with (d) steric interactions. Each cluster is indicated by a different gray level (color in the online version) on each plot.

Figure 5

Experimental results of a fascin-crosslinked actin network, collapsed by myosin motors [11]. The concentrations of crosslinks, actin and molecular motors are given by $0.24$, 12, and $0.12\mu \text{M}$, respectively. The dimensions are $3.18\text{mm}\times 2\text{mm}$, while the sample thickness is $80\mu \text{m}$. (a) The network after 2 min. (b) The network after 104 min. (c) Configuration of the clusters. Colors indicate the largest (blue) and the second-largest (pink) clusters. (d) Histogram (circles) of cluster areas, averaged over 26 samples. For the critically connected regime, the data is statistically more consistent ($1.4$ standard errors from the Hill estimator of $\tau =1.91\pm 0.06$ [36]) with a power-law distribution with a NEP model's (or the collapse model with steric interactions in two dimensions) Fisher exponent from Eq. (3) (solid line). The agreement of the data with the random percolation model Fisher exponent $\tau =187/91$, indicated by the dashed line, is significantly worse ($2.4$ standard errors from the Hill estimator).