Branching process approach for Boolean bipartite networks of metabolic reactions

The branching process (BP) approach has been successful in explaining the avalanche dynamics in complex networks. However, its applications are mainly focused on unipartite networks, in which all nodes are of the same type. Here, motivated by a need to understand avalanche dynamics in metabolic networks, we extend the BP approach to a particular bipartite network composed of Boolean AND and OR logic gates. We reduce the bipartite network into a unipartite network by integrating out OR gates and obtain the effective branching ratio for the remaining AND gates. Then the standard BP approach is applied to the reduced network, and the avalanche-size distribution is obtained. We test the BP results with simulations on the model networks and two microbial metabolic networks, demonstrating the usefulness of the BP approach.

Figure 1

Schematic illustration of avalanche dynamics in a Boolean bipartite network. When a reaction (rectangle) is deactivated, a metabolite (ellipse) would be deactivated if all the reactions have that metabolite as a product. A reaction cannot take place if any of the substrate metabolites is absent. Thus, metabolites and reactions correspond to logic OR and AND gates, respectively. When a metabolite is generated from a single reaction (with probability $p_1$), it can block $\ell$ reactions [with probability $p_{o,m}\left(\ell \right)$] and each of the metabolite branches $k_i$ ($i=1,\cdots ,\ell$) [with probability $p_{o,r}\left(k_i\right)$]. The resulting value $K=k_1+\cdots +k_{\ell }$ corresponds to the branching number, which has probability $q_K$, as derived in Eq. (1).

Figure 2

The avalanche-size distribution $p_a\left(s\right)$ for the Boolean cascade model on the directed bipartite scale-free network with the in- and out-degree distribution $p_d\left(k\right)\sim k^{-\gamma }$ and $\gamma =2.5$ (red, $\square$), $3.5$ (green, $◯$), and $4.5$ (blue, $▵$). The curves are a good fit to the theoretical function $p_a\left(s\right)=A{s}^{-\tau }\exp (-s/s_c)$ with $\tau$ values of $1.71$ (red, $\square$), $1.56$ (green, $◯$), and $1.47$ (blue, $▵$). Note that we are in the supercritical regime so that isolated peaks at large $s$ appear. The system size is $N={10}^5$.

Figure 3

The avalanche-size distributions on real-world metabolic networks. Panel (a) is for E. coli and panel (b) is for S. cerevisiae. Circles are for simulation results, and squares are for BP predictions with empirical degree distributions. Solid lines are guidelines with the slope of $-1.83$, which satisfies the relation $\tau =\gamma /(\gamma -1)$ with $\gamma =2.2$. Insets are out-degree distributions of metabolites (triangles) and reactions (diamonds). Solid guidelines in the insets have a slope of $-2.2$.