Percolation mechanism drives actin gels to the critically connected state

Cell motility and tissue morphogenesis depend crucially on the dynamic remodeling of actomyosin networks. An actomyosin network consists of an actin polymer network connected by cross-linker proteins and motor protein myosins that generate internal stresses on the network. A recent discovery shows that for a range of experimental parameters, actomyosin networks contract to clusters with a power-law size distribution [J. Alvarado, Nat. Phys. 9, 591 (2013)]. Here, we argue that actomyosin networks can exhibit a robust critical signature without fine-tuning because the dynamics of the system can be mapped onto a modified version of percolation with trapping (PT), which is known to show critical behavior belonging to the static percolation universality class without the need for fine-tuning of a control parameter. We further employ our PT model to generate experimentally testable predictions.

Figure 1

Schematics of the actin meshwork studied in the experiment by Alvarado et al. [10] [see also their Fig. 1]. Some actin filaments (black lines) connect to the top and the bottom boundaries. Interconnects are provided by cross-linking fascins (blue stars). Molecular motor myosins (red dumbbells) pull the filament together. The experiment was on a quasi-two-dimensional plane of size $2.5\times 2.5{\mathrm{mm}}^2$, and was observed over the course of about 2 h.

Figure 2

(a) The nodes adhered to the top and bottom boundaries are red and with adhesion strength $s_A$. The interior nodes (grey) have connection strengths drawn from a Poisson distribution and the higher the strength, the darker the color of the node. Each node is supposed to have a force $f$ pulling on it on all sides and the weakest node will disappear first (b).

Figure 3

(a) As random deletion of nodes start, the network starts to shrink. (b) As soon as a connected cluster detached from either top or bottom boundaries, it quickly shrinks to a focal point or to the boundary, and as a result, node deletion is no longer possible (c). (d) Retracing the path of each node, the areas at the initial time corresponding to the three disjoint, fragmented clusters can be obtained, which are schematically shown in (d).

Figure 4

Cluster density distribution. Simulation of our PT model on a $200\times 200$ square lattice produces a power-law distribution of cluster density ($n_s$) with an exponent close to 2 (black curve). The red curve corresponds to the power-law distribution with exponent $-187/91\simeq -2.05$ which corresponds to the exponent of static percolation on a 2D square lattice [17]. The inset figure shows the clusters generated from our model.

Figure 5

Finite size scaling. Simulations of our PT model on different sizes of square lattice. The inset shows that cluster density distribution collapse onto a single curve as predicted by static percolation theory, which dictates that $s^{\tau }{n}_s=\mathrm{\Phi }(s/L^{d_f})$ where $\mathrm{\Phi }$ is a scaling function and the exponents are $\tau =187/91$ and $d_f=91/48$ [17].

Figure 6

Predictions. (a) Size of clusters as a function of the time of their free contractions. The inset shows an example of the temporal ordering (light to dark) of cluster contractions. (b) Size of clusters as a function of the distance between their centers of mass and the closest adhesion boundary. Results are from simulations performed on a $200\times 200$ square lattice.