Anomalous percolation features in molecular evolution

Self-replication underlies every species of living beings and simple physical intuition dictates that some sort of autocatalysis invariably constitutes a necessary ingredient for the emergence of molecular life. This led Worst et al. [E. G. Worst, P. Zimmer, E. Wollrab, K. Kruse, and A. Ott, New J. Phys. 18, 103003 (2016)] to study a model of molecular evolution of self-replicating molecules where spontaneous ligation and simple autocatalysis are in competition for their building blocks. We revisit this model, where irreversible aggregation leads to a transition from a regime of small molecules to macromolecules, and find an array of anomalous percolation features, some of them predicted for very specific percolation processes [R. M. D'Souza and J. Nagler, Nat. Phys. 11, 531 (2015)].

Figure 1

Illustration of spontaneous ligation (a) and autocatalysis (b). The rate of the autocatalytic reaction depends on the template molecule concentration [see Eq. (1)].

Figure 2

Relative size of the largest cluster $G$ as a function of time. Displayed are single realizations for the limiting cases $\alpha =0$ (a) and $\alpha \gg \beta$, specifically $\alpha ={10}^7$ (b). Independently of $\alpha$, the model shows a strong increase at the end of the process with jumps of order 1 close to $t=1$ (as $t\approx p_c=1$, see text). Jumps of size $1/2,1/4,\cdots$ in panel (b), for large system sizes $N$. Other parameters: $\beta =1$ and $N=2^{20}$.

Figure 3

Time course of molecule concentrations. Single realization for $\alpha ={10}^{-9}$ (a), $\alpha =0.1$ (b), and $\alpha ={10}^6$ (c). Other parameters: $\beta =1$ and $N=2^{17}$. Note that only the concentrations for molecules of size of powers of 2 are displayed as otherwise the curves would be too close together. (d) Critical value ${\alpha }_{c,n}$ of the autocatalytic rate for selection up to species $A_n$ (for $n=4,8,16$) as a function of the initial monomer concentration $N$.

Figure 4

Entropy as a function of time. Single realization for $\alpha =10, \beta =1$, and $N=2^{20}$. The entropy necessarily vanishes in the beginning ($N$ “clusters” of size $1/N$) and in the end of the process (one cluster of size $N$).

Figure 5

Relative size of the largest molecule as a function of the reaction time $t$ (a) and of the reaction number $p$ (b). Shown are the maximum, minimum, average (curve in between in green/gray), and three realizations [black, displayed only in panel (a)]. Parameters: $\alpha ={10}^7, \beta =1$, and $N=2^{20}$. The maximum, minimum, and average have been obtained from 10 000 realizations.

Figure 6

Non-self-averaging and discrete scale invariance. Relative variance $R_v$ of the order parameter $G$ as a function of the distance to the phase transition in the relative reaction number $p-p_c$ for $\alpha ={10}^{-6}$ (a), $\alpha ={10}^5$ (b), and $\alpha ={10}^7$ (c). Other parameters $\beta =1$ and system sizes $N=2^{15}, 2^{17}$, and $2^{20}$ in comparison.