Enumeration of Autocatalytic Subsystems in Large Chemical Reaction Networks

TL;DR

This paper develops a graph-theoretic framework for enumerating autocatalytic subsystems in large chemical reaction networks by leveraging the König representation. It introduces fluffles and circuitnets as structural motifs that capture all irreducible autocatalytic CS-matrices, and defines centralized autocatalysis to classify cores. The authors implement a scalable, modular algorithm (autogato) that enumerates elementary circuits, constructs CS-equivalence classes, tests for autocatalysis, and identifies autocatalytic cores, outperforming existing ILP approaches on large networks. Demonstrations on E. coli, human erythrocytes, and Methanosarcina barkeri reveal pervasive autocatalysis across life domains, with substantial numbers of cores and centralized motifs; the work provides a practical tool for analyzing self-sustaining substructures in metabolism and related CRNs.

Abstract

Autocatalysis is an important feature of metabolic networks, contributing crucially to the self-maintenance of organisms. Autocatalytic subsystems of chemical reaction networks (CRNs) are characterized in terms of algebraic conditions on submatrices of the stoichiometric matrix. Here, we derive sufficient conditions for subgraphs supporting irreducible autocatalytic systems in the bipartite König representation of the CRN. On this basis, we develop an efficient algorithm to enumerate autocatalytic subnetworks and, as a special case, autocatalytic cores, i.e., minimal autocatalytic subnetworks, in full-size metabolic networks. The same algorithmic approach can also be used to determine autocatalytic cores only. As a showcase application, we provide a complete analysis of autocatalysis in the core metabolism of E. coli and enumerate irreducible autocatalytic subsystems of limited size in full-fledged metabolic networks of E. coli, human erythrocytes, and Methanosarcina barkeri (Archea). The mathematical and algorithmic results are accompanied by software enabling the routine analysis of autocatalysis in large CRNs.

Figures (15)

• Figure 1:
• Figure 2: Left:Multiple CS may exist in a given induced subgraph of $\mathbf{K}$. In the example, there are indeed two perfect matchings and thus two CS $\bm{\kappa}_1=(X_\kappa,R_\kappa,\kappa_1)$ and $\bm{\kappa}_2(X_\kappa,R_\kappa,\kappa_2)$: $\kappa_1(x_1)=r_1$, $\kappa_1(x_2)=r_2$, $\kappa_1(x_3)=r_3$ and $\kappa_2(x_1)=r_2$, $\kappa_2(x_2)=r_3$, $\kappa_2(x_3)=r_1$. Right:A strongly connected child selective subgraph may not have a CS matrix with irreducible Metzler part. Here, choosing, $\kappa(x_1)=r_1$, $\kappa(x_2)=r_2$, $\kappa(x_3)=r_3$, and $\kappa(x_4)=r_4$ (edges depicted in red) yields a block diagonal $\mathfrak{M}(\mathbf{S}[\kappa])$ composed of two $2\times 2$ blocks.
• Figure 3: Left: Strongly-connected bipartite graph without a cut-vertex but not child-selective, as the four reaction vertices are more than the three species vertices. Right: Strongly-connected bipartite digraph with a cut-vertex and child-selective (red edges).
• Figure 4: Unions of two elementary circuits, depicted in green and blue, yielding different configurations: a is not connected and thereby violates the fluffle condition $(iii)$ in Prop. \ref{['prop:usefluffles']}, b-e possess more metabolites than reaction vertices or vice-versa and contradict fluffle condition $(i)$ in Prop. \ref{['prop:usefluffles']}. In addition, b and c are also not a strong block and thus violate $(iii)$ as well. The union f has a substrate vertex $x_1$ with out-degree and a reaction vertex $r_1$ with in-degree two, respectively, which contradicts fluffle condition $(ii)$ in Prop. \ref{['prop:usefluffles']}. The only combination consistent with the fluffle definition is depicted in g where indeed the intersection of the two elementary circuits is a path from a substrate to a reaction vertex, as prescribed by Thm. \ref{['thm:woffle-union']}. The two elementary circuits in d, e, f, moreover, also, consitute an example of circuitnets that are not associated to fluffles.
• Figure 5: The fluffle corresponding to an autocatalytic core of Type V, Eq. \ref{['eq:coretype5']}. The fluffle $F$ is depicted as a union of the elementary circuits in two different circuitnets. The circuitnet $\mathcal{C}_1=\{C_1,C_2,C_3\}$ (left) comprises the three elementary circuit $C_1=(x_1,r_1,x_3,r_3,x_1)$ (green), $C_2=(x_2,r_2,x_3,r_3,x_2)$ (blue), and $C_3=(x_1,r_1,x_2,r_2,x_1)$ (red); the circuitnet $\mathcal{C}_2=\{C_4,C_5\}$ (right) consists of the two elementary circuits $C_4=(x_1,r_1,x_3,r_3,x_2,r_2,x_1)$ (teal) and $C_5=(x_1,r_1,x_2,r_2,x_3,r_3,x_1)$ (purple). We have $F=\bigcup(\mathcal{C}_1)=\bigcup(\mathcal{C}_2)$.

Theorems & Definitions (93)

• Definition 1
• Definition 2
• Definition 3
• Proposition 4: Cor. 5.1 in vassena_unstable_2024
• Definition 5
• Definition 6
• Definition 7
• Proposition 8
• Proposition 9
• Corollary 10