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Autocatalytic set

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Collection of chemical entities

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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, ecology, and economics.

Autocatalytic sets also have the 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.

Before 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 molecular biology 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. The first empirical support came from Lincoln and Joyce, who obtained autocatalytic sets in which "two [RNA] enzymes catalyze each other's synthesis from a total of four component substrates."[1] Furthermore, an evolutionary process that began with a population of these self-replicators yielded a population dominated by recombinant replicators.

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[2] and others.

Formal definition<br>[edit]

Definition<br>[edit]

Given a set M of molecules, chemical reactions can be roughly defined as pairs r = (A, B) of subsets from M:[3]

a1 + a2 + ... + ak → b1 + b2 + ... + bk

Let R be the set of allowable reactions. A pair (M, R) is a reaction system (RS).

Let C be the set of molecule-reaction pairs specifying which molecules can catalyze which reactions:

C = {(m, r) | m ∈ M, r ∈ R}

Let F ⊆ M be a set of food (small numbers of molecules freely available from the environment) and R' ⊆ R be some subset of reactions. We define a closure of the food set relative to this subset of reactions ClR'(F) as the set of molecules that contains the food set plus all molecules that can be produced starting from the food set and using only reactions from this subset of reactions. Formally ClR'(F) is a minimal subset of M such that F ⊆ ClR'(F) and for each reaction r'(A, B) ⊆ R':

A ⊆ ClR'(F) ⇒ B ⊆ ClR'(F)

A reaction system (ClR'(F), R') is autocatalytic, if and only if for each reaction r'(A, B) ⊆ R':

there exists a molecule c ⊆ ClR'(F) such that (c, r') ⊆ C,

A ⊆ ClR'(F).

Example<br>[edit]

Let M = {a, b, c, d, f, g} and F = {a, b}. Let the set R contain the following reactions:

a + b → c + d, catalyzed by g<br>a + f → c + b, catalyzed by d<br>c + b → g + a, catalyzed by d or f

From the F = {a, b} we can produce {c, d} and then from {c, b} we can produce {g, a} so the closure is equal to:

ClR'(F) = {a, b, c, d, g}

According to the definition the maximal autocatalytic subset R' will consist of two reactions:

a + b → c + d, catalyzed by g<br>c + b → g + a, catalyzed by d

The reaction for (a + f) does not belong to R' because f does not belong to closure. Similarly the reaction for (c + b) in the autocatalytic set can only be catalyzed by d and not by f.

Probability that a random set is autocatalytic<br>[edit]

Studies...