Anomalous Discontinuity at the Percolation Critical Point of Active Gels

We develop a percolation model motivated by recent experimental studies of gels with active network remodeling by molecular motors. This remodeling was found to lead to a critical state reminiscent of random percolation (RP), but with a cluster distribution inconsistent with RP. Our model not only can account for these experiments, but also exhibits an unusual type of mixed phase transition: We find that the transition is characterized by signatures of criticality, but with a discontinuity in the order parameter.

Figure 1

(a) Experimental results of a fascin-cross-linked actin network, collapsed by myosin motors (see Supplemental Material [6] for details and movie of the collapse). (b) Initial configuration of the collapsed clusters in (a). Colors indicate the largest (blue) and the second-largest (pink) clusters. (c) Histogram (squares) of cluster masses, averaged over 26 samples. For the critically connected regime, the data are statistically more consistent (1.4 standard errors from the Hill estimator of $\tau =1.91\pm 0.06$ [10]) with a power-law distribution with a NEP model’s Fisher exponent from Eq. (9) (solid line). The agreement of the data with the RP model Fisher exponent ${\tau }^{\prime }=187/91$, indicated by the dashed line, is significantly worse (2.4 standard errors from the Hill estimator). Lower insets demonstrate an enclave (solid line) and its surrounding cluster (dashed line). In the upper inset, the enclave is absorbed into its surrounding cluster. (d) Clusters’ structure at the critical point of the NEP model (RP model with absorbed enclaves). The absorption of enclaves is implemented by identifying the longest boundary of each cluster and absorbing all the nodes within this boundary in the cluster (see Supplemental Material [6] for more details).

Figure 2

(a) Cluster mass distribution at the percolation transition of the RP (diamonds) and NEP (circles) models, with $M={2000}^2$. The long lines indicate power laws with ${\tau }^{\prime }=187/91$ (dashed line) and $\tau =1.82$ (solid line) [Eq. (9)]. The short line indicates the experimental $\tau \simeq 1.91$ observed over 2 decades. Insets illustrate cluster structures of the RP (left inset) and NEP models (right inset) on a triangular lattice for $0<p<1$ (see Supplemental Material [6] for $p=0,1$ cases). Colors (online) represent different clusters. The cross in the left inset indicates an enclave that is absorbed in the NEP model (right inset). The absorption of enclaves in the NEP model changes the distribution of clusters sizes and can account for the experimental observation of $\tau <2$ in Ref. [5]. (b) Demonstration of the discontinuous transition in the NEP model. Strength of the largest cluster, $P$, as a function of bond density, $p$ for different system sizes, from $M=12$ to 400. Inset: collapse of the data from the main figure using Eq. (4) and $\gamma =8/3$. (c) Demonstration of the critical transition in the NEP model. Average cluster, $S$, as a function of bonds density, $p$ for the same system sizes as in (b). Inset: collapse of the data from the main figure using Eq. (6) and $\gamma =8/3$. (d) Probability of an $s$ cluster to be not an enclave of another cluster, $\mathcal{P}$, at the percolation transition of the RP model for different system sizes, from $M=6$ to 2000. Inset: collapse of the data from the main figure using Eq. (7). The dashed lines indicate power law $a=0.1706$ obtained from Eqs. (8) and (9).