Before replication, before selection, before anything we recognize as Darwinian, there is a more fundamental question: how does a chemical system become self-sustaining? Not self-replicating—that comes later. Just: how does a network of reactions maintain itself? How do you get a system whose outputs include all the catalysts it needs to keep running?
Stuart Kauffman asked this in the 1970s and called the answer autocatalytic sets. Mike Steel and Wim Hordijk gave it mathematical teeth in the 2000s with RAF theory—Reflexively Autocatalytic and F-generated sets. The result is one of the sharpest frameworks we have for understanding the transition from chemistry to proto-life.
Consider a set of molecules. Some of them can react to form others. But most biochemical reactions are impossibly slow without catalysis. So each reaction needs a catalyst—itself a molecule that must be produced by some other reaction in the system. This is the chicken-and-egg problem of origins: every reaction needs a catalyst, and every catalyst needs a reaction to produce it.
The formal framework is a chemical reaction system (CRS), defined as a tuple:
X is your universe of molecule types. R is the set of reactions, each consuming reactants and producing products. C tells you which molecules catalyze which reactions. And F—the food set—is the set of molecules assumed freely available from the environment. The food is what you start with. Everything else must be built.
A subset R′ ⊆ R is a RAF set if it satisfies two conditions simultaneously:
Together: the set makes everything it needs (F-generated) and catalyzes everything it does (reflexively autocatalytic). It pulls itself up by its own bootstraps.
Place molecules and reactions on the canvas below. Green circles are food molecules (click an existing molecule to toggle food status). Squares are reactions. Draw arrows by dragging from one node to another: solid arrows for reactant/product connections, dashed for catalysis. Then check whether your network forms a RAF.
Not all self-sustaining networks are created equal. There is a hierarchy of bootstrappability, and the distinctions matter for understanding what can actually start from chemistry.
Kauffman’s original intuition was that autocatalytic sets arise spontaneously once a chemical system is sufficiently complex. Steel and Hordijk proved this rigorously using the binary polymer model: molecules are binary strings of length up to N, and each molecule catalyzes each reaction independently with probability p. The expected number of reactions each molecule catalyzes is μ.
The result is a phase transition. Below a threshold, RAFs almost never exist. Above it, they almost always do. And the threshold is surprisingly low:
RAFs need only linear catalytic density. CAFs need exponential. This gap is the mathematical signature of the bootstrap advantage: tolerating circular dependencies (as RAFs do) is exponentially easier than requiring strict constructive buildup (as CAFs do).
Here is the surprising result that defies naive intuition: when RAFs first emerge at the phase transition, they are never small. You might expect that as you increase catalytic density, first a tiny two-reaction RAF appears, then it grows. But no. The first RAF to appear has size at least 2cn for some constant c > 0.
This is fundamentally different from directed cycles in random graphs (Bollobás-Rasmussen), which appear at all sizes near the threshold. The organizational closure demanded by a RAF—simultaneously satisfying F-generation and autocatalysis—imposes a minimum complexity. You cannot have a little bit of self-sustenance. When it appears, it appears at scale.
A maximal RAF (maxRAF) may contain many smaller RAFs within it. These sub-RAFs form a partially ordered set under inclusion, and at the bottom of this poset sit the irreducible RAFs—ur-RAFs that cannot be decomposed further. Every sub-RAF is a union of ur-RAFs.
This structure is the key to evolvability. Different sub-RAFs can be “expressed” or “suppressed” depending on environmental conditions. The maxRAF becomes a space of possibilities, and selection can act on which sub-RAFs dominate. This is how a RAF framework can support something like competition and evolution before template replication exists.
The RAF framework is more general than chemistry. Any system whose operations sustain their own conditions of operation is structurally analogous to a RAF. The food set is whatever is given freely from outside; the reactions are the system’s operations; the catalysis is whatever accelerates or enables those operations; and the closure condition says the system produces what it needs.
What the mathematics tells us is that these self-sustaining loops are not infinitely improbable. They emerge with a phase transition, they resist perturbation through their sub-RAF structure, and they demand a minimum complexity that ensures, once present, they are robust. The universe does not need a miracle to bootstrap self-sustenance. It needs only sufficient catalytic density and a food set. The rest is mathematics.
Kauffman called it “order for free.” Steel and Hordijk showed exactly how free it is: the cost is linear in system size, not exponential. The bootstrap problem is not as hard as it looks. The chicken and the egg can emerge together, as long as there are enough of them.