Statistics of percolating clusters in a model of photosynthetic bacteria

In photosynthetic organisms the energy of the illuminating light is absorbed by the antenna complexes and transmitted by the excitons to the reaction centers (RCs). The energy of light is either absorbed by the RCs, leading to their “closing” or is emitted through fluorescence. The dynamics of the light absorption is described by a simple model developed for exciton migration that involves the exciton hopping probability and the exciton lifetime. During continuous illumination the fraction of closed RCs $x$ continuously increases, and at a critical threshold $x_c$, a percolation transition takes place. Performing extensive Monte Carlo simulations, we study the properties of the transition in this correlated percolation model. We measure the spanning probability in the vicinity of $x_c$, as well as the fractal properties of the critical percolating cluster, both in the bulk and at the surface.

Figure 1

Illustration of the processes between an exciton (represented by a wavy line) and an RC. (a) If the RC is open (represented by an open circle), it will become closed (represented by a solid circle). (b) If the RC is closed, the exciton is redirected with probability $p$, or its energy is dissipated by emission of a fluorescence quantum (represented by pF) with probability $(1-p)$.

Figure 2

Illustration of the processes yielding fluorescence. First line: one-step process with the contribution $(1-p)G_1$. Second line: two-step process with the contribution $(1-p)p{G}_2$. Third line: $n$-step process with the contribution $p^{n-1}{G}_n$ since the energy of the exciton is radiated with probability 1. $G_k$ are multisite correlation functions of closed RCs, which are defined in the text. Note that for $k\ge 3$ step walks the same closed RC can be visited several times.

Figure 3

Illustration of the fractal structure of the giant cluster for $n=3$ and $p=0.5$ at an occupancy $x=0.589$, slightly above the percolation threshold (see Table 1) for $L=256$. Sites in the bulk are shown in red (dark gray), sites at the perimeter are indicated in green (light gray), and the hull is composed of the black and green (light gray) sites.

Figure 4

Spanning probability of the EM model with $p=0.5$ and $n=2$ on the square lattice with open boundary conditions having the rule ${\mathcal{R}}_1$ for different linear sizes $L=64,128,\cdots ,16384$, top to bottom. The crossing point of the curves defines the critical point, $x_c=0.5843$, where $R_1(x_c,\infty )\approx 0.5$. In the inset the scaling collapse of the points averaged through duality (see text) is plotted in terms of the variable in Eq. (7).

Figure 5

Spanning probability of the EM model with $p=0.9$ and $n=10$ on the square lattice with open boundary conditions having the rule ${\mathcal{R}}_2$ for different linear sizes $L=64,128,\cdots ,16384$, top to bottom. The crossing point of the curves defines the critical point, $x_c=.5817$, where $R_2(x_c,\infty )\approx 0.32$. In the inset the scaling collapse of the points is shown in terms of the variable in Eq. (7).

Figure 6

Fractal properties of the critical clusters in the EM model with $p=0.5$ and $n=2$ on the square lattice. Average mass $m\left(L\right)$, hull $h\left(L\right)$, and perimeter $s\left(L\right)$ as a function of linear size $L$ in log-log scale. The slope of the straight lines through the points have the fractal dimensions of traditional percolation in Eqs. (6) and (10).