Percolation Model for Enzyme Gel Degradation

We study a model for the gel degradation by an enzyme, where the gel is schematized as a cubic lattice, and the enzyme as a random walker, that cuts the bonds over which it passes. The model undergoes a (reverse) percolation transition, which for low density of enzymes falls in a universality class different from random percolation. In particular, we have measured a gel fraction critical exponent $\beta =1.0\pm 0.1$, in excellent agreement with experiments made on the real system.

Figure 1

(a) Pacman percolation model on a square lattice of size ${64}^2$, with a single enzyme and after 12 000 steps, when the density of nonhydrolyzed bonds is $p=0.42$. (b) The random percolation model with the same bond density.

Figure 2

(a) Percolation probability $\Pi (p,L)$ and (b) density $\rho (p,L)$ of the percolating cluster as a function of the bond density $p$, with a single enzyme and for cubic lattices of size $L=30$, 40, 50, 60. Insets: data collapses obtained plotting $\Pi (p,L)$ and $\rho (p,L)L^{\beta /\nu }$ $(p-p_c)L^{1/\nu }$, with $\nu =1.8$ and $\beta =1.0$.

Figure 3

Spatial correlation $G(r)$ in the occupation of the bonds, with a single enzyme on a cubic lattice of size $L=100$, near the percolation threshold $p\simeq 0.14$.

Figure 4

Conductivity $\Sigma (p,L)$ of (a) the random-resistor network and (b) the conductor-superconductor network as a function of the bond density $p$, with a single enzyme and for the same lattice sizes of Fig. 2. Insets: data collapses obtained plotting $\Sigma (p,L)L^{\mu /\nu }$ versus $(p-p_c)L^{1/\nu }$, with $\nu =1.8$, $\mu =3.5$, and $s=1.1$.