Autocatalytic set
From Academic Kids

An autocatalytic set is a collection of entities, each of which is able to catalyze the creation of others within the set, such that as a whole, the set is able to catalyze its own replication. In this way the set as a whole is said to be autocatalytic. Autocatalytic sets were originally and most concretely defined in terms of entities which are replicating molecules, but have more recently been metaphorically extended to the study of systems in sociology and economics.
Prior to Watson and Crick, biologists considered autocatalytic sets the way metabolism functions in principle, i.e. one protein helps to synthesize another protein and so on. After the discovery of the double helix, the central dogma of genetics was formulated, which is that DNA is transcribed to RNA which is translated to protein. But this highly differentiated structure is clearly too complicated to explain the origin of life, which must have started from something less organized.
Therefore, several models of the origin of life are based on the notion that life may have arisen through the development of a molecular autocatalytic set. Most of these models which have emerged from the studies of complex systems predict that life arose not from a molecule with any particular trait (such as selfreplicating RNA) but from an an autocatalytic set. Modern life has the traits of an autocatalytic set, since no particular molecule, nor any class of molecules, is able to replicate itself. There are several models based on autocatalytic sets, including those of Stuart Kauffman and others.
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Formal definition
Given as set M of molecules, chemical reactions can be roughly defined as pairs r=(A,B) of subsets from M.
a_{1} + a_{2} + ... + a_{n} → b_{1} + b_{2} + ... + b_{m}
Let R be the set of allowable reactions. A pair (M,R) is a reaction system (RS).
A molecule m ∈ A ∩ B of a reaction r is a catalyst of this reaction.
A RS is autocatalytic, if all the catalysts for all its reactions are in M.
The above definition is not sufficient to describe dependency on external resources or nutrients. This can be formulated by a closure over a generating subset of M.
Formally, cl(S) denotes the smallest subset Y of M that contains S such that for each reaction (A,B)
A ⊆ S ∪ Y ⇒ B ⊆ Y
A RS is generated (over some resources S), if all reactants A in its reactions are in cl(S) and none of the resources is a catalyst.
A generated autocatalytic set is a RS that is both autocatalytic and generated.
Probability that a random set is autocalytic
Studies of the above model show that random RS can be autocatalytic with high probability under some assumptions. This comes from the fact that with growing number of molecules, the number of possible reactions and catalysations grows even stronger if the molecules grow in complexity, producing stochastically enough reactions and catalysations to make a part of the RS selfsupported. An autocatalytic set then extends very quickly with growing number of molecules for the same reason.
Such studies make autocatalytic sets candidates for a theoretical explanation of the very early origin of life, but are empirically unsupported in real chemistry.
Formal limitations
Formally, it is difficult to treat molecules as anything but unstructured entities, since the set of possible reactions (and molecules) would become infinite. Therefore, a derivation of arbitrarily long polymers as needed to model DNA, RNA or proteins is not possible, yet. Studies of the RNA World suffer from the same problem.