Autocatalytic Cores in Reaction Networks

• Autocatalytic cores are minimal self-sustaining units within larger systems, defined across frameworks like RAF theory, stoichiometric networks, and technological cycles.
• They are rigorously characterized using positivity conditions, irreducibility, cyclic graph representations, and criteria specific to each formalism.
• Their study provides practical insights into network dynamics, evolutionary trajectories, and challenges in computational enumeration of core structures.

Autocatalytic cores are minimal or core-like self-amplifying structures inside larger reaction, catalytic, or innovation systems, but the exact object depends on the formalism. In RAF theory, the closest counterpart is the irreducible RAF (irrRAF), a minimal food-generated and reflexively autocatalytic reaction subset inside a larger maxRAF (Hordijk et al., 2012). In stoichiometric chemical-reaction-network theory, an autocatalytic core is a minimal square submatrix or subnetwork satisfying a positivity condition such as $Sc>0$, often with additional irreducibility and Metzler-sign constraints (Unterberger et al., 2021). In reaction networks with explicit catalysis, the same notion persists but must be reconstructed from child-selections and the bipartite Kőnig graph rather than from the stoichiometric matrix alone (Golnik et al., 3 Mar 2026). In technology networks, the “core” is the cyclic nucleus of an autocatalytic set of technological fields (Napolitano et al., 2017). In surface-mediated branching processes, the analogous object is a localized catalytic interface whose persistence is controlled by return statistics and a principal eigenvalue condition (Grebenkov, 23 Apr 2026).

1. Definitions and scope

Across the cited literatures, “autocatalytic core” does not denote a single universal mathematical object. It denotes a minimal or privileged self-sustaining unit relative to the modeling language being used: reaction subsets in RAF theory, stoichiometric submatrices in CRN theory, cyclic nuclei in directed innovation graphs, or localized replication zones in spatial branching models.

Framework	Core notion	Formal hallmark
RAF theory	irrRAF; sometimes closed RAF as a saturated module	Minimal RAF under reaction deletion; or RAF closed under all enabled reactions
Diluted stoichiometric CRNs	autocatalytic core	Minimal autonomous subnetwork with $Sc>0$
Explicit-catalysis CRNs	autocatalytic core via child-selection	Irreducible semipositive Metzler CS-matrix; induced fluffle
Technology networks	core of an ACS	Nodes connected through one or more cycles
Surface-mediated dynamics	localized replication zone	Self-sustaining surface patch determined by $\lambda_0=0$ threshold

A common thread is the distinction between a large ambient autocatalytic organization and a smaller subset that is either minimal, structurally indispensable, or dynamically privileged. This suggests a family resemblance rather than a single definition: a core is the smallest or most central self-amplifying organization recognized by the formalism at hand.

2. RAF-theoretic cores: irrRAFs, closed RAFs, and hierarchical substructure

In RAF theory the underlying object is a chemical reaction system $Q=\{X,R,C\}$ together with a food set $F\subset X$. A subset $R'\subseteq R$ is a RAF when it is both reflexively autocatalytic and food-generated: every reaction is catalyzed by at least one molecule generated within the set or present in food, and all reactants and catalysts needed for the set can be built from the food set using reactions in the set alone (Hordijk et al., 2012). The standard RAF algorithm returns the maxRAF, the largest RAF in the CRS. The key structural point is that the maxRAF is often internally decomposable into many subRAFs, and the most core-like objects are the irrRAFs, which are minimal under reaction deletion.

That minimality does not imply uniqueness. A single RAF can contain exponentially many irrRAFs. The canonical construction in the appendix of (Hordijk et al., 2012) yields $2^k$ irrRAFs using only $2k$ reactions and $5k$ molecules, so there is “no hope” for a polynomial-time algorithm that lists all irrRAFs in an arbitrary RAF. At the same time, the local inclusion structure is controlled: the number of maximal proper subRAFs of a RAF $R'$ never exceeds $Sc>0$0, and there are polynomial-time procedures to list them, test whether $Sc>0$1 is the union of two proper subRAFs, and determine whether a given subset is contained in every subRAF (Hordijk et al., 2012). The resulting partially ordered set of subRAFs can be represented by a Hasse diagram whose top node is the full RAF and whose bottom nodes are the irrRAFs.

RAF theory also distinguishes minimal kernels from saturated modules. In a partitioned biochemical CRS $Sc>0$2, a closed RAF is a RAF $Sc>0$3 such that every reaction whose reactants and at least one catalyst are already available in $Sc>0$4 must itself belong to $Sc>0$5 (Smith et al., 2013). The closure $Sc>0$6 of a RAF is obtained by iteratively adding all such enabled reactions. This makes an irrRAF the natural minimal core, while a closed RAF is a self-complete autocatalytic module.

The distinction matters dynamically. In stochastic-Petri-net models of composed RAF systems, the full maxRAF may exist structurally while only some embedded closed RAF subsets are dynamically realized. Composition operations involving food entities or food complexes can suppress full-maxRAF realization and select only certain closed RAFs, with different runs realizing different closed subsets (Ravoni, 2020). A plausible implication is that minimality and dynamical relevance are not identical: a chemically minimal core can be latent, while a larger closed subset acts as the effective core under resource-constrained dynamics.

3. Stoichiometric and dynamical cores in chemical reaction networks

A second major line of work defines autocatalytic cores directly from stoichiometry. In the diluted-regime formulation, a network is stoichiometrically autocatalytic when there exists a strictly positive reaction vector $Sc>0$7 such that

$Sc>0$8

meaning every internal species has strictly positive net balance under that reaction combination (Unterberger et al., 2021). In the diluted regime, reactions with more than one low-concentration reactant are neglected, and the split graph $Sc>0$9 built from one-reactant reactions becomes the key object. The central criterion is (Top): each minimal class of $\lambda_0=0$0 must contain at least one internal one-to-several irreversible reaction. This condition is necessary and sufficient for stoichiometric autocatalysis in diluted networks, and with degradation small enough it also implies dynamical autocatalysis, characterized by a strictly positive Lyapunov exponent (Unterberger et al., 2021).

Within this formalism, the minimal autocatalytic subnetworks are the five core types of Blokhuis et al., and the diluted-regime dynamical theory preserves that classification. Minimal dynamical autocatalytic networks in the diluted regime are exactly Types I–V, the same five families previously obtained in the stoichiometric theory; moreover, the reversible extensions of these isolated cores have a unique positive stationary state without degradation, and with degradation they have at most one positive stationary state for all types except Type II with three catalytic loops or more (Nandan et al., 21 Jul 2025). This suggests that isolated cores are structurally robust elementary amplification motifs, while richer behaviors require additional nonlinear couplings between cores.

Allowing explicit catalysis changes the algebraic problem. If an entity appears as both reactant and product in the same reaction, the stoichiometric matrix $\lambda_0=0$1 records only net change and no longer determines whether explicit catalytic participation occurred. In this setting, autocatalytic cores remain minimal autocatalytic subnetworks, but their identification requires child-selections, associated CS-matrices, and the bipartite Kőnig graph. The generalized theory preserves both the matrix and graph pictures: cores still correspond to irreducible semipositive Metzler CS-matrices and to induced fluffles, but the diagonal of the CS-matrix is no longer necessarily strictly negative, and a single-reaction single-entity core with $\lambda_0=0$2 becomes possible (Golnik et al., 3 Mar 2026). The same paper introduces hard autocatalytic cores, defined as cores whose reversible extensions do not contain another autocatalytic core, and proves that for unit stoichiometries each autocatalytic core can be constructed as the superposition of at most two elementary circuits (Golnik et al., 3 Mar 2026).

A related instability-based framework places autocatalytic cores inside a larger class of unstable cores. In that formulation, unstable cores are minimal unstable Child-Selection matrices, and the determinant sign separates unstable-positive from unstable-negative feedbacks. Autocatalytic cores turn out to be exactly the unstable-positive feedbacks that are Metzler matrices (Vassena et al., 2023). This is conceptually important because it shows that stoichiometric sources of dynamical instability are broader than autocatalysis: there are unstable cores that yield multistationarity, superlinear growth, or oscillations without being autocatalytic.

4. Graph representations, decomposition, and enumeration

The graph-theoretic representation that underlies much of the modern theory is the directed bipartite Kőnig graph $\lambda_0=0$3, with species-to-reaction edges $\lambda_0=0$4 whenever $\lambda_0=0$5 is a reactant of $\lambda_0=0$6, and reaction-to-species edges $\lambda_0=0$7 whenever $\lambda_0=0$8 is a product of $\lambda_0=0$9. In explicit-catalysis theory, a child-selection becomes a perfect matching on the reactant-to-reaction side, and irreducible Metzler candidates correspond to fluffles: bipartite strong blocks with equal numbers of species and reaction vertices, out-degree $Q=\{X,R,C\}$0 at each species, and in-degree $Q=\{X,R,C\}$1 at each reaction (Golnik et al., 3 Mar 2026). For true cores, the relevant objects are induced fluffles.

This structure supports enumeration. The 2025 enumeration paper derives sufficient graph conditions for subgraphs supporting irreducible autocatalytic systems in the bipartite Kőnig representation, and on that basis develops an efficient algorithm to enumerate autocatalytic subnetworks and, as a special case, autocatalytic cores in full-size metabolic networks. The same approach can be restricted to core enumeration alone, and it is demonstrated on the core metabolism of E. coli as well as on larger networks from E. coli, human erythrocytes, and Methanosarcina barkeri (Golnik et al., 24 Nov 2025). A plausible implication is that the child-selection/fluffle formalism has moved autocatalytic-core analysis from small stylized networks to full-scale metabolic reconstructions.

The computational landscape remains uneven. In RAF theory, finding the maxRAF is polynomial-time, listing maximal proper subRAFs is polynomial-time, testing whether a RAF is a union of two proper subRAFs is polynomial-time, and testing whether a reaction or subset is contained in every subRAF is polynomial-time (Hordijk et al., 2012). By contrast, listing all irrRAFs is not expected to be polynomial-time in general because there may be exponentially many of them (Hordijk et al., 2012). In kinetic RAF theory, deciding whether a CRN contains a kinetic RAF with $Q=\{X,R,C\}$2 is NP-complete (Smith et al., 2013). With inhibition, deciding whether a $Q=\{X,R,C\}$3-RAF exists is NP-hard, though fixed-parameter tractable in the total number of inhibitors; a polynomial-time iterative sufficient-condition algorithm can nevertheless detect substantial $Q=\{X,R,C\}$4-RAFs in practice (Hordijk et al., 2016). These results together show that minimality, dynamical viability, and inhibition-resilience are progressively harder properties than bare autocatalytic existence.

5. Beyond molecular chemistry: technology, economy, surfaces, and spatial localization

The same language has been exported beyond chemistry, but with important shifts in meaning. In technology networks built from patent data, an autocatalytic set is defined as a “subgraph, each of whose nodes has at least one incoming link from a node belonging to the same subgraph,” and the core is “any set of nodes connected through one or more cycles.” The periphery consists of nodes catalyzed by the core but not feeding back into the closed path (Napolitano et al., 2017). Empirically, the technology network exhibited a unique ACS from 1980 to 2010, its size increased over time, by 2010 it spanned approximately half of the network, and in the last decade it concentrated almost $Q=\{X,R,C\}$5 of total fitness while containing only a bit more than half the nodes (Napolitano et al., 2017). Here the core is not a minimal chemical subnetwork but the cyclic nucleus of mutually reinforcing knowledge spillovers.

Economic RAF models push the analogy back toward standard RAF theory. In the TAP-based production-network model, goods map to molecule types, production functions to reactions, catalysts are goods, and the initial goods form the food set. RAFs emerge with high probability once catalysis is strong enough, but the resulting maxRAFs are typically very large—often almost the whole network. In one run where food-set goods were disallowed as catalysts, a maxRAF of size $Q=\{X,R,C\}$6 contained sampled irrRAFs of sizes about $Q=\{X,R,C\}$7 to $Q=\{X,R,C\}$8, with average size $Q=\{X,R,C\}$9, and the average pairwise overlap among $F\subset X$0 sampled irrRAFs was close to $F\subset X$1 (Hordijk et al., 2022). This suggests a substantial shared backbone without implying a unique canonical core.

A distinct generalization treats a core as a localized catalytic interface. In the surface-mediated branching-diffusion theory, particles diffuse in a bounded domain $F\subset X$2 and branch only when they hit specific boundary regions $F\subset X$3, with generating function

$F\subset X$4

The entire many-body dynamics reduces to a nonlinear renewal equation built from the single-particle first-hit kernel at the catalytic surface, and the transition between extinction and self-sustaining growth is controlled sharply by the principal eigenvalue $F\subset X$5: the critical condition is $F\subset X$6, subcritical behavior has $F\subset X$7, and supercritical self-sustaining growth has $F\subset X$8 (Grebenkov, 23 Apr 2026). In this literature the core is explicitly spatial rather than combinatorial.

Experimental surface science provides a material analogue. On partially hydroxylated $F\subset X$9, $R'\subseteq R$0 can spontaneously dissociate into two surface-bound $R'\subseteq R$1 fragments, and those sub-carbonyl motifs are proposed as the true precursors of subsequent autocatalytic cobalt deposition (Muthukumar et al., 2012). This suggests a physically localized seed-state notion of core: first surface-induced dissociation, then self-reinforcing growth centered on the resulting Co-containing fragments.

6. Evolvability, persistence, and recurrent controversies

One reason autocatalytic cores matter is that they turn existence claims into structural and evolutionary claims. In RAF theory, the poset of subRAFs and its Hasse diagram are interpreted as possible evolutionary or assembly trajectories: larger RAFs can arise by adding reactions to irrRAFs or by combining subRAFs, and one RAF can enable another by providing needed molecules (Hordijk et al., 2012). In composition-based stochastic RAF models, this becomes dynamically explicit: source-sharing and food-complex exchange can force only some closed RAF subsets to emerge, so different embedded core-like motifs become the realized units of long-term behavior (Ravoni, 2020).

Compartment models make the evolutionary interpretation sharper. In a protocell containing the ACS

$R'\subseteq R$2

with $R'\subseteq R$3 catalyzing the ligation reactions and $R'\subseteq R$4 contributing membrane material, the deterministic intracellular dynamics becomes bistable over the interval

$R'\subseteq R$5

The inactive state and active ACS state have distinct protocell growth rates, and stochastic reaction noise plus partitioning noise induces transitions between them. At the population level, the active lineage can arise in a population of inactive protocells and then dominate because its protocells divide faster (Singh et al., 2023). In this setting the autocatalytic core is not only self-amplifying but heritable at the compartment level.

Persistence theory provides another criterion for core-like sufficiency. For bimolecular autocatalytic systems of the form

$R'\subseteq R$6

relative populations $R'\subseteq R$7 satisfy an autonomous polynomial system after time rescaling, and for several families of autocatalytic recombination networks the induced relative-population network is strongly endotactic or dynamically equivalent to a weakly reversible single-linkage-class network. In repeated-reactant systems, strong connectivity of the production graph is enough to imply permanence of the relative-population dynamics (Craciun et al., 2020). This suggests a persistence-based notion of core: a strongly connected self-producing subnetwork that prevents extinction on the simplex.

A persistent controversy concerns whether cyclic topology alone is enough. It is not. In random catalytic polymer networks there is a broad region between $R'\subseteq R$8 and $R'\subseteq R$9 levels of catalysis where strongly connected components exist but RAFs do not, because food-generation and closure fail. Moreover, when catalytic food molecules are present, RAFs can be realized by linear catalyst-to-product structures rather than by SCCs (Filisetti et al., 2015). The misconception that “autocatalytic core” always means “cycle” is therefore too narrow. Another unresolved boundary concerns unification with epidemiology: one recent review argues that epidemiologic strains behave like critical minimal siphons and minimal autocatalytic cores, and proposes the disease-free face as the union of all minimal siphons or all cores, while also emphasizing that the equivalence remains heuristic rather than a general theorem (Avram et al., 30 Oct 2025).

Taken together, these literatures support a stable but plural conclusion. An autocatalytic core is the minimal or privileged self-amplifying organization relative to a chosen formalism: an irrRAF inside a maxRAF, a minimal stoichiometric subnetwork with $2^k$0, an induced fluffle, a cyclic spillover nucleus, or a localized replication interface. What varies is not the intuition of a core, but the mathematical language used to make that intuition exact.