Autocatalytic set

An autocatalytic set is a collection of entities, each of which can be created catalytically by other entities within the set, such that as a whole, the set is able to catalyze its own production. In this way the set as a whole is said to be autocatalytic. Autocatalytic sets were originally and most concretely defined in terms of molecular entities, but have more recently been metaphorically extended to the study of systems in sociology and economics.

Autocatalytic sets also have the uncanny ability to replicate themselves if they are split apart into two physically separated spaces. Computer models illustrate that split autocatalytic sets will reproduce all of the reactions of the original set in each half, much like cellular mitosis. In effect, using the principles of autocatalysis, a small metabolism can replicate itself with very little high level organization. This property is why autocatalysis is a contender as the foundational mechanism for complex evolution.

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. The molecular structure of DNA and RNA, as well as the metabolism that maintains their reproduction, are believed to be too complex to have arisen spontaneously in one step from a soup of chemistry.

Several models of the origin of life are based on the notion that life may have arisen through the development of an initial molecular autocatalytic set which evolved over time. 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 self-replicating RNA) but from 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 a set M of molecules, chemical reactions can be roughly defined as pairs r=(A, B) of subsets from M.

 a1 + a2 + ... + an → b1 + b2 + ... + bm

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 autocatalytic

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 self-supported. 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.

Linguistic aspects

Contrary to the above definition, which applies to the field of Artificial chemistry, no agreed-upon notion of autocatalytic sets exists today.

While above, the notion of catalyst is secondary insofar that only the set as a whole has to catalyse its own production, it is primary in other definitions, giving the term "Autocatalytic Set" a different emphasis. There, every reaction (or function, transformation) has to be mediated by a catalyst. As a consequence, while mediating its respective reaction, every catalyst denotes its reaction, too, resulting in a self denoting system, which is interesting for two reasons. First, real metabolism is structured in this manner. Second, self denoting systems can be considered as an intermediate step towards self describing systems.

From both a structural and a natural historical point of view, one can identify the ACS as seized in the formal definition the more original concept, while in the second, the reflection of the system in itself is already brought to an explicit presentation, since catalysts represent the reaction induced by them. In ACS literature, both concept are present, but differently emphasised.

To complete the classification from the other side, generalised self reproducing systems move beyond self-denotation. There, no unstructured entities carry the transformations anymore, but structured, described ones. Formally, a generalised self reproducing system consists of two function, u and c, together with their descriptions Desc(u) and Desc(c) along following definition:

    u : Desc(X) -> X
    c : Desc(X) -> Desc(X)

where the function 'u' is the "universal" constructor, that constructs everything in its domain from appropriate descriptions, while 'c' is a copy function for any description. Practically, 'u' and 'c' can fall apart into many subfunctions or catalysts.

Note that the (trivial) copy function 'c' is necessary because though the universal constructor 'u' would be able to construct any description, too, the description it would base on, would in general be longer than the result, rendering full self replication impossible.

This last concept can be attributed to von Neumann's work on self reproducing automata, where he holds a self description necessary for any non trivial (generalised) self reproducing system to avoid interferences. Von Neumann planned to design such a system for a model chemistry, too.

Non-autonomous autocatalytic sets

Virtually all articles on autocatalytic sets leave open whether the sets are to be considered autonomous or not. Often, autonomy of the sets is silently assumed.

Likely, the above context has a strong emphasis on autonomous self replication and early origin of life. But the concept of autocatalytic sets is really more general and in practical use in various technical areas, e.g. where self sustaining tool chains are handled. Clearly, such sets are not autonomous and object of human agency.

Examples of practical importance of non-autonomous autocatalytic sets can be found e.g. in the field of compiler construction and in operating systems, where the self referential nature of the respective constructions is explicitly discussed, very often in terms of the chicken and egg problem.