arXiv subjects

Thomas Gatter

Publications and source records attributed to Thomas Gatter.

6 recordsLinked to original sources

arXiv2026

Bridging two theoretical frameworks of autocatalysis: RAF sets and stoichiometric autocatalysis

Autocatalysis lies at the heart of many (bio)chemical processes and is key to processes leading up to the origin of life. Two seemingly very different formalisms have emerged that define autocatalysis. Kauffman introduced collective autocatalysis to describe systems of molecules that mutually catalyze each other's formation, emphasizing the self-sustaining character of autocatalytic systems. This view is mathematically formalized in the theory of Reflexively Autocatalytic and Food-generated sets (RAF). In parallel, stoichiometric autocatalysis emerged from the theory of Chemical Reaction Networks (CRN), focusing on the net-productive, self-amplifying character of autocatalytic subnetworks. These two frameworks have coexisted independently in the literature, since RAF theory considers each reaction as explicitly catalyzed, while the CRN approach often excludes explicitly catalyzed reactions altogether. Nevertheless, both frameworks describe reaction networks and thus admit a common mathematical representation in terms of stoichiometric matrices. We highlight this connection and show that the two formalisms are less disparate than they might appear. To illustrate this point we prove that, under mild and general conditions, any RAF is stoichiometrically autocatalytic.

q-bio.MN↗

arXiv2026

Autocatalytic Cores in Reaction Networks with Explicit Catalysis

Autocatalytic cores are minimal units in reaction networks (RNs) responsible for the emergence of autocatalysis. In the absence of explicit catalysis, i.e., when an entity appears both as reactant and product in the same reaction, they are known to be encoded by square submatrices of the stoichiometric matrix whose columns can be reordered as an irreducible child-selection (CS) matrix with negative diagonal and nonnegative off-diagonal (Metzler matrix). In the bipartite Koenig graph representing the RN, these CS matrices can be identified by fluffles, i.e., strong blocks with an identical number of entity and reaction vertices that have out- and in-degree 1, respectively. Here, we adapt the concepts derived for autocatalytic cores to RNs with explicitly catalyzed reactions, which emerge as digons, i.e., elementary circuits in the Koenig graph of length 2. In this setting, we confirm that an inspection of the stoichiometric matrix alone is inconclusive concerning the presence and number of autocatalytic cores, requiring a more delicate algebraic analysis. Nevertheless, this generalization preserves both the graph and the matrix representation as fluffles and irreducible Metzler CS matrices, respectively, although the diagonal is no longer necessarily strictly negative. We introduce the notion of hard autocatalytic cores, i.e. those that do not yield other autocatalytic cores upon inclusion of all reverse reactions. Finally, we consider the case of unit stoichiometries and show that each autocatalytic core can be constructed as the superposition of at most 2 elementary circuits. In particular, autocatalytic cores involving explicitly catalyzed reactions always contain a spanning subgraph consisting of a single elementary circuit together with a simple entity-to-reaction chord. Moreover, we identify the essentially unique example for which at least two circuits are required.

math.CO↗

arXiv2025

Enumeration of Autocatalytic Subsystems in Large Chemical Reaction Networks

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 K\H{o}nig 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.

q-bio.MN↗

arXiv2025

BiRNe: Symbolic bifurcation analysis of reaction networks with Python

Computer algebra methods for analyzing reaction networks often rely on the assumption of mass-action kinetics, which transform the governing ODEs into polynomial systems amenable to techniques such as Gr\"obner basis computation and related algebraic tools. However, these methods face significant computational complexity, limiting their applicability to relatively small networks involving only a handful of species. In contrast, building on recent theoretical advances, we introduce here \textsc{BiRNe} (BIfurcations in Reaction NEtworks) Python module, which relies on a symbolic approach designed to detect bifurcations in larger reaction networks (up to 10-20 species, depending on the network's connectivity) equipped with parameter-rich kinetics. This class includes enzymatic kinetics such as Michaelis--Menten, ligand-binding kinetics like Hill functions, and generalized mass-action kinetics. For a given network, the current algorithm identifies all minimal autocatalytic subnetworks and fully characterizes the presence of bifurcations associated with zero eigenvalues, thus determining whether the network admits multistationarity. It also detects oscillatory bifurcations arising from positive-feedback structures, capturing a significant class of possible oscillations.

q-bio.MN↗

arXiv2024

A Short Note on Relevant Cuts

The set of relevant cuts in a graph is the union of all minimum weight bases of the cut space. A cut is relevant if and only if it is the a minimum weight cut between two distinct vertices. Moreover, we give a characterization in terms of Picard-Queyranne Directed Acyclic Graphs that can be used to accelerate the enumeration of the relevant cuts. Finally, we perform an experimental evaluation by comparing with state-of-the-art algorithms.

math.CO↗

arXiv2020

Convexity deficit of benzenoids

In 2012, a family of benzenoids was introduced by Cruz, Gutman, and Rada, which they called convex benzenoids. In this paper we introduce the convexity deficit, a new topological index intended for benzenoids and, more generally, fusenes. This index measures by how much a given fusene departs from convexity. It is defined in terms of the boundary-edges code. In particular, convex benzenoids are exactly the benzenoids having convexity deficit equal to 0. Quasi-convex benzenoids form the family of non-convex benzenoids that are closest to convex, i.e., they have convexity deficit equal to 1. Finally, we investigate convexity deficit of several important families of benzenoids.

math.CO↗