Ubiquitous “glassy” relaxation in catalytic reaction networks

Study of reversible catalytic reaction networks is important not only as an issue for chemical thermodynamics but also for protocells. From extensive numerical simulations and theoretical analysis, slow relaxation dynamics to sustain nonequlibrium states are commonly observed. These dynamics show two types of salient behaviors that are reminiscent of glassy behavior: slow relaxation along with the logarithmic time dependence of the correlation function and the emergence of plateaus in the relaxation-time course. The former behavior is explained by the eigenvalue distribution of a Jacobian matrix around the equilibrium state that depends on the distribution of kinetic coefficients of reactions. The latter behavior is associated with kinetic constraints rather than metastable states and is due to the absence of catalysts for chemicals in excess and the negative correlation between two chemical species. Examples are given and generality is discussed with relevance to bottleneck-type dynamics in biochemical reactions as well.

Figure 1

(Color online) Relaxation time course for four sets of networks $\left(M=24,S=24,K=8\right)$ for several $\beta$.

Figure 2

(Color online) Relaxation time as a function of $\beta$ for the sample reaction networks shown in Fig. 1.

Figure 3

(Color online) (a) Time course of $C\left(t\right)$ for two sample networks, I and II, given in (b), with $S=5$ and $\beta =16/\epsilon$. In (b), the chemicals attached to the arrows showing reactions indicate the catalysts. Time courses of $x_i^{dev}\left(t\right)$ of (c) network I and (d) network II corresponding to (a). In (a), (c), and (d), all chemicals have equal number initially (i.e., $\beta =0$). (e) Time courses of $x_i^{dev}\left(t\right)$ of network I from the initial condition $x_i=1$ for $i=0,2,3$, $x_1=0.4$, and $x_4=1.6$.

Figure 4

(Color) (a) Reaction network with $M=12$ and $K=4$ $\left(S=12\right)$, where the color of each arrow indicates the catalyst for the reaction, (b) “major relaxation” for each chemical (see text), (c) “major relaxation” path for $\left\{X_3,X_6\right\}$ and $\left\{X_5,X_7\right\}$ clusters (see text), (d) time course of $C\left(t\right)$ for network, and (e) time courses of $x_i^{dev}\left(t\right)$ for $\beta =20/\epsilon$.