Self-organized criticality of a catalytic reaction network under flow

Self-organized critical behavior in a catalytic reaction network system induced by smallness in the molecule number is reported. The system under a flow of chemicals is shown to undergo a transition from a stationary to an intermittent reaction phase when the flow rate is decreased. In the intermittent reaction phase, two temporal regimes with active and halted reactions alternate. The number frequency of reaction events at each active regime and its duration time are shown to obey a universal power law with the exponents 4/3 and 3/2, respectively, independently of the parameters and network structure. These power laws are explained by a one-dimensional random-walk representation of the number of catalytically active chemicals. Possible relevance of the result to reaction dynamics in artificial and biological cells is briefly discussed.

Figure 1

(a) An illustration of a catalytic reaction $B+C\to A+C$. Arrows indicate the reaction, their origin and end-point (“B” and “A”) indicate the substrate and product, and the species on the arrow (“C”) indicates the catalyst. (b) An example of a catalytic reaction network.

Figure 2

(Color online) Typical temporal evolutions of $N\left(t\right)$ (black) and $RN\left(t\right)$ [gray (red online)] for $M=300$ and $K=12$ with (a) $Q=0.3$ and (b) $Q=0.001$.

Figure 3

(Color online) $\rho$, $\kappa$, and ${\kappa }^{\prime }$ as a function of $Q$ for several $M$, $K$, $M_{in}$, and $M_{out}$. $\rho$, $\kappa$, and ${\kappa }^{\prime }$ are computed by sampling data over ${10}^4\le t\le {10}^7$.

Figure 4

(Color online) Frequency of $S$ for $M=1000$ and some values of $K$ and $Q$, with $M_{in}=M$ and $M_{out}=1$ for (a) and $M_{in}\sim M/K$ and $M_{out}=3$ for (b). These histograms are computed by sampling data over ${10}^4\le t\le {10}^9$.

Figure 5

(Color online) (a) Frequencies of $D$ for $M=1000$, $Q=0.0003$, $M_{in}=M$, and $M_{out}=1$ with $K=18$ and $K=36$, and (b) $D\text{−}S$ characteristics for the same conditions. The histograms are computed by sampling data over (a) ${10}^4\le t\le {10}^9$ and (b) ${10}^4\le t\le 2\times {10}^6$.