Anomalous Discontinuity at the Percolation Critical Point of Active Gels

Abstract

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. DOI: 10.1103/PhysRevLett.114.098104 PACS numbers: 87.16.Ka, 64.60.ah, 64.60.Bd, 64.60.De Percolation theory has become pervasive in a number of fields ranging from physics to mathematics and even computer science [1]. In particular, it successfully describes connectivity and elastic properties of polymer networks [2,3]. The simplest percolation model is the random percolation (RP) model, consisting of a collection of nodes with controlled connectivity, p, representing the fraction of occupied bonds between the nodes. As a function of p, the order parameter-the mass fraction of the largest clusterbecomes finite above the percolation threshold p c . The nature of the transition is of special interest because the system properties are highly tunable at this point, especially if the transition is discontinuous; in that case, just a few bonds can have a significant impact, even for very large systems [4]. Usually, however, percolation transitions are second order, with a continuous variation of the order parameter and various critical signatures. More specialized percolation models can exhibit different phase behavior, including discontinuous transitions between the two phases (see discussion below).Here, we present a simple model based on random percolation that develops a discontinuous jump in the order parameter in the thermodynamic limit, while exhibiting other features of criticality in such quantities as the correlation length and susceptibility. Interestingly, the transition we observe occurs for the same p c < 1 as for random percolation. Moreover, our model can account for recent experimental results on active biopolymer gels that have been shown to self-organize towards a critical connectivity point [5]. The experimentally observed cluster properties at this point were found to be inconsistent with the ordinary random percolation model.In these experiments, we studied a model cytoskeletal system, composed of actin filaments, fascin cross-links and myosin motors in a quasi-2D chamber of dimensions 3 mm × 2 mm × 80 μm [5] (see Supplemental Material [6]). We observed a motor-driven collapse of the network into disjointed clusters (see Fig. 1(a) and movie of the collapse in the Supplemental Material [6]). The configuration of the clusters prior to the collapse is obtained by analyzing the timereversed movie [see Fig. 1(b)] and their masses were estimated from their initial areas. We found that, over a wide range of the experimental parameters, the number n s of clusters of mass s exhibits a power-l...

Inherently unstable networks collapse to a critical point

“…In two-dimensional (2D) systems these interactions trap enclavic clusters in their surrounding clusters during the collapse (see Fig. 2 for illustration) [11]. As we show below, this qualitatively and quantitatively impacts the collapse process and its final state in the experiment.…”

“…Motor activity can lead to network failure and its subsequent collapse to isolated foci [7,31,32] and, as reported in Refs. [5,11], can robustly lead to scale-free structures.…”

“…[5,11], motor-driven collapse of a model cytoskeletal system, composed of actin filaments, fascin cross links, and myosin motors was studied in chambers with dimensions 3.18 mm × 2 mm. The sample thickness is 80 μm (not much larger than a typical length of an actin filament, 20 μm), such that the chambers can be considered as quasi-2D.…”

“…Since the density of the largest cluster vanishes in the thermodynamic limit, the percolation transition is continuous [14]. However, the properties of the NEP model are different [11]. The configuration of the clusters possesses no enclaves, as shown in Fig.…”

Generic Long-Range Interactions Between Passive Bodies in an Active Fluid

“…They have attracted much attention [2,3] due to a host of interesting physical phenomena [4][5][6][7][8][9], their relevance to many biological systems [10][11][12][13], and their potential use for self-assembly applications [14]. They have also been suggested as tools for novel engineering applications -for example, active fluids have been used to power microscopic gears [15][16][17][18][19][20][21].…”