Kim and Sönmez show self-sustaining networks can emerge from few catalysts

Jinsu Kim and Ercan Sönmez posted a mathematical model of self-sustaining chemical networks to arXiv on Nov. 4, 2026, describing how reflexively autocatalytic food-generated sets, self-sustaining chemical reaction networks, can arise from simple molecules through a few catalysts.

Reflexively autocatalytic food-generated sets give mathematicians a way to study how self-sustaining chemical reaction networks can emerge from simple molecules, a question at the center of chemical origin-of-life research. The sources describe the origin of life as the problem of how organized chemistry could arise from available simple molecules, and they present these sets as the mathematical tool for that investigation.

Research on these sets begins with Kauffman’s early work on autocatalytic networks, followed by the Mossel–Steel probabilistic framework. Hordijk, Steel, and collaborators later broadened RAF theory toward heterogeneous catalysis, including reaction-dependent probabilities or power-law distributed catalytic activity.

Mossel and Steel established fundamental conditions making RAFs likely in random polymer frameworks, placing a bound on the catalysis required as maximum polymer length grows. Hordijk, Wills and Steel extended the framework to reaction-dependent catalysis probabilities inside the binary polymer model.

Hordijk and Steel found that the distribution of catalytic activity strongly influences RAF probability and size, while linear bounds on average catalysis needed for RAF formation still apply. Varanasi and Korenaga also studied phase transitions in RAF probability as catalytic probability changes.

Kim and Sönmez offer a stochastic model that blends uneven molecular catalytic activity with compatibility that depends on both the molecule and the reaction. The model aims to be concrete and flexible while capturing two forms of heterogeneity and remaining mathematically tractable.

Each molecule receives a random catalytic weight, and the authors pair this with an explicit probability rule for whether a molecule catalyzes a given reaction, creating a setting that is concrete, flexible and tractable. Random weights stand for intrinsic differences in catalytic activity among molecules, and the authors place no moment assumptions on the weights at the outset.

The system starts from a finite alphabet with at least two symbols and treats molecules as non-empty strings up to length n. Reactions include ligation, in which two strings join into one, and the reverse cleavage, with each treated as a distinct directed reaction. The full set contains every non-empty string up to length n over that alphabet, with total count given by the sum of kappa to the j for j from 1 to n.

The model fixes a food set of molecules up to length t, independent of n, with n greater than t, so the food set does not grow with the maximum polymer length. A reaction set counts as food-generated when every reactant it needs can be produced from the food set.

A RAF set is non-empty and food-generated, and each reaction in it is catalyzed by at least one molecule available from the food set or produced by the set. The two conditions are food-generation and reflexive autocatalysis.

Each molecule receives a random weight for intrinsic catalytic activity, and conditional on those weights, catalysis indicators are independent. The probability that a reaction lacks a catalyst equals exp(-lambda_n M_n(r)) based on its weighted catalytic mass M_n(r).

Weights are independent and identically distributed with probability strictly between zero and infinity, and the authors impose no moment assumptions on them at the outset. Catalysis probability is determined by global intensity lambda_n, the molecular weight W_x, and compatibility K_n(x,r); when compatibility is zero, that molecule cannot catalyze that reaction.

When W_x K_n(x,r) is larger, the conditional catalysis probability rises, while indicator independence holds only after conditioning on the weights. Relative to Mossel and Steel, the model drops the requirements of mutually independent catalysis and catalysis probabilities depending only on the molecule, retaining indicator independence only conditional on the weights.

The paper derives new general criteria governing RAF formation or failure in the weighted catalysis model on forward ligation reactions. The authors show that classical phase-transition criteria need not remain valid when catalysis depends on the reaction.

Earlier Mossel–Steel criteria were built under reaction-independent assumptions, which the new model relaxes. Reaction dependence by itself produces counterexamples to efforts to extend those criteria beyond their original assumptions.

Heavy-tailed weights mean catalytic activity can be highly uneven across the molecular population. With heavy-tailed catalytic weights, the authors identify and measure a further phase transition. For finite-mean weights, the catalytic intensity parameter is what determines whether a RAF forms or fails to form.

If the mean catalytic weight is infinite, a RAF can still arise with probability approaching one even as the catalytic intensity parameter drops toward zero. “For infinite-mean weights, however, we show that RAFs may emerge with probability tending to one even when the catalytic intensity parameter tends to zero,” Kim and Sönmez wrote. It implies that autocatalysis can become likely as the system grows without strong catalysis everywhere.

|X_t|^2 forward reactions have food-set reactants, and these remain independent of n once n is at least 2t; every non-empty food-generated forward set includes at least one such reaction. Because t is fixed, these food-set reactions control a key obstruction to RAF formation.

The weighted catalytic mass criterion states phase transitions for RAF probability based on each reaction’s total weighted catalytic mass, independent of any specific weight distribution. The criterion applies to forward ligation reactions with arbitrary positive finite iid molecular weights.

The weighted catalytic masses of reactions that begin at the food set form a barrier to RAF formation, while enough mass spread across all forward reactions makes RAF emergence probable. This mechanism shows how the criterion separates regimes in which a RAF is unlikely from regimes in which it becomes probable as the system grows.

The authors identify conditions under which the probability that a RAF exists tends to one or zero as the maximum polymer length n grows without bound. Theorem 1 describes these phase-transition regimes, which represent behavior differing qualitatively from earlier RAF models.

The paper finds that a RAF can become nearly certain even when catalysis rests with a single molecule, according to Proposition 10. This result supports the broader conclusion that broadly distributed catalysis is not necessary for collectively autocatalytic systems to emerge.

Heavy-tailed weights give a few molecules much greater catalytic influence while total intensity may fall with system size; strong, well-distributed interactions can still make RAF emergence likely as size grows. The authors conclude that broadly distributed catalytic activity is not required for collectively autocatalytic systems, since a few highly active molecules can be decisive.

The authors describe this as a mathematical perspective rather than a direct description of early-life chemical dynamics. They warn that the model does not directly account for early-life chemistry, but see it as a possible mathematical angle for thinking about organized autocatalysis in mixed molecular populations.

The authors say their result about RAFs emerging from a limited number of catalysts matches numerical simulations reported earlier by Hordijk and Steel. A collectively autocatalytic system can become likely when just a few molecules drive catalysis, including a case with a single catalytic molecule.

The authors base their proofs on a constructive method from Mossel and Steel that Hordijk, Wills and Steel later adapted to reaction-dependent catalysis. They apply that strategy to a weighted model where catalysis probabilities hinge on the catalyst molecule together with the particular reaction.

The authors compare their extension to the change from Erdös–Rényi random graphs to graphs where edge chances depend on endpoint features. The analogy shows why catalysis probabilities in the new model can depend on the catalyst and the reaction together instead of only the catalyst.

Lab experiments show that RNA systems can self-sustain replication and form cooperative catalytic networks. Recent work has explored connections among autocatalysis, molecular replication, catalytic regulation, spatial structure, and prebiotic chemistry’s arrangement.

Hordijk, Wills and Steel focused on four particular catalysis mechanisms in extending earlier frameworks to reaction-dependent catalysis probabilities inside the binary polymer model.

Hordijk and Steel studied how catalytic activity varies unevenly across alphabets of any finite size by assigning each molecule a random number of reactions to catalyze through uniform choices. The authors contrast their weight-based approach with this earlier random-number approach.

Jinsu Kim and Ercan Sönmez authored the paper. The arXiv submission is version 1, a 1,019 KB file uploaded on Nov. 4, 2026 at 06:38:47 UTC.

Keywords are origin of life, autocatalytic reaction networks, chemical self-organization, RAF sets, heavy-tailed distributions, and phase transitions. The keyword list signals the paper’s placement in origin-of-life and statistical-physics literature.