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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." Furthermore, an evolutionary process that began with a population of these self-replicators yielded a population dominated by recombinant replicators.

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." Furthermore, an evolutionary process that began with a population of these self-replicators yielded a population dominated by recombinant replicators.

生命起源的几个模型都是基于这样一个理念，即生命可能是通过一个随着时间演化的原初分子自催化集合的发展而产生的。从复杂系统研究中产生的大多数模型都这样预测，生命并非起源于具有任何特定特征的分子(如能自复制的RNA) ，而是起源于一个自催化的集合。第一个经验性的支持来自林肯（Lincoln）和乔伊斯（Joyce），他们构造出了自催化集合（两种[RNA]酶通过四种底物组件互相催化对方的合成）。此外，一个始于这些自复制因子的群体通过进化产生了一个重组复制因子占主导的群体。

生命起源的几个模型都是基于这样一个理念，即生命可能是通过一个随着时间演化的原初分子自催化集合的发展而产生的。从复杂系统研究中产生的大多数模型都这样预测，生命并非起源于具有任何特定特征的分子(如能自复制的RNA) ，而是起源于一个自催化的集合。第一个经验性的支持来自林肯（Lincoln）和乔伊斯（Joyce），他们构造出了自催化集合（两种[RNA]酶通过四种底物组件互相催化对方的合成）。此外，一个始于这些自复制因子的群体通过进化产生了一个以重组复制因子占主导的群体。

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]]<ref>Kauffman, Stuart A. (2008) ''Reinventing the Sacred: A New View of Science, Reason, and Religion''.  [Basic Books],  {{ISBN|0-465-00300-1}}, chapter 5, especially pp. 59–71</ref> and others.

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]]<ref>Kauffman, Stuart A. (2008) ''Reinventing the Sacred: A New View of Science, Reason, and Religion''.  [Basic Books],  {{ISBN|0-465-00300-1}}, chapter 5, especially pp. 59–71</ref> and others.

==Formal definition==

==Formal definition==

= = 正式定义 = =  

= = 形式化定义 = =  

===Definition===

===Definition===

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

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

给定一组 m 分子，化学反应可以粗略地定义为 m 中子集的对 r = (a，b) :

给定一组分子的集合M，化学反应可以粗略地定义为M的子集对r = (A, B):

   a₁ + a₂ + ... + a_k → b₁ + b₂ + ... + b_k

   a₁ + a₂ + ... + a_k → b₁ + b₂ + ... + b_k

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

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

设 r 是允许反应的集合。一对(m，r)是一个反应体系(RS)。

设R是可发生的反应的集合。一个(M, R)对是一个反应系统(RS)。

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

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

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

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

设 c 为一组分子反应对，指定哪些分子可以催化哪些反应:

设C为一组分子反应对的集合来指定哪些分子可以催化哪些反应:

   C = {(m, r) | m ∈ M, r ∈ R}

   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':

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':

设 f something m 是一组食物(从环境中自由获得的少量分子) ，r ′ r 是反应的子集。我们定义了一个食物集合的闭合相对于这个反应子集 ClR’(f)作为包含食物集合的分子集合加上所有可以从食物集合中产生的分子，并且只使用这个反应子集合的反应。形式上，ClR’(f)是 m 的极小子集，使得 f something ClR’(f)和每个反应 r’(a，b) something r’:

设F（F ⊆ M）是一组食物（可从环境中自由获得的少量分子) 的集合，并设R'（R' ⊆ R）是一些反应的子集。我们定义了一个食物集合相对于该反应子集 ClR'(F)的闭包，作为一个包含了食物集合加上所有可以从食物集合中产生的分子的集合，并且，只使用该反应子集中的反应。形式上， ClR'(F)是 M的最小子集，使得F ⊆ ClR'(F)以及每个反应r'(A, B) ⊆ R':

   A ⊆ Cl_R'(F) ⇒ B ⊆ Cl_R'(F)

   A ⊆ Cl_R'(F) ⇒ B ⊆ Cl_R'(F)

# A ⊆ ClR'(F).

# A ⊆ ClR'(F).

反应体系(ClR’(f) ，r’)是自催化的，当且仅当对于每个反应 r’(a，b) something r’: # 存在分子 c something ClR’(f)使得(c，r’) something c，# a something ClR’(f)。

反应系统(Cl_R'(F), R')是自催化的，当且仅当对于每个反应r'(A, B) ⊆ R':  

 

# 存在一个分子c ⊆ ClR'(F)使得(c, r') ⊆ C，

# A ⊆ ClR'(F).。

===Example===

===Example===

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

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

设 m = { a，b，c，d，f，g }和 f = { a，b }。设 r 包含以下反应:

设M = {a, b, c, d, f, g}和F = {a, b}。设集合R包含以下反应:

   a + b  → c + d, catalyzed by g

   a + b  → c + d, catalyzed by g

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

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

A + b → c + d，g a + f → c + b，d c + b → g + a，d 或 f 催化

a + b → c + d，由g催化

 

a + f → c + b， 由d催化

 

c + b → g + a，由d或f催化

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:

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:

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:

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:

从 f = { a，b }我们可以产生{ c，d } ，然后从{ c，b }我们可以产生{ g，a } ，所以闭包等于:

由F = {a, b}，我们可以产生{c, d}，然后从{c, d}中，我们可以产生{g, a}，所以闭包等于:

   Cl_R'(F) = {a, b, c, d, g}

   Cl_R'(F) = {a, b, c, d, g}

   ClR'(F) = {a, b, c, d, g}

   ClR'(F) = {a, b, c, d, g}

ClR’(f) = { a，b，c，d，g }

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

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

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

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

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

根据定义，最大自催化子集 r’包括两个反应:

根据定义，最大自催化子集R'包含两个反应:

   a + b  → c + d, catalyzed by g

   a + b  → c + d, catalyzed by g

   c + b  → g + a, catalyzed by d

   c + b  → g + a, catalyzed by d

A + b → c + d，g c + b → g + a，d 催化

a + b → c + d，由g催化

 

c + b → g + a，由d催化

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.

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.

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.

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.

(a + f)的反应不属于 r’，因为 f 不属于闭包。同样，自催化体系中(c + b)的反应只能用 d 催化，而不能用 f 催化。

(a + f)的反应不属于 R'，因为f不属于闭包。同样，自催化集合中(c + b)的反应只能用d催化，而不能用f催化。

==Probability that a random set is autocatalytic==

==Probability that a random set is autocatalytic==

==Probability that a random set is autocatalytic==

==Probability that a random set is autocatalytic==

= = 一个随机集是自动催化的概率 = =  

= = 一个随机集合是自催化的概率 = =  

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 a growing number of molecules, the number of possible reactions and catalysations grows even larger if the molecules grow in complexity, producing stochastically enough reactions and catalysations to make a part of the RS self-supported.<ref>{{cite journal | author = Mossel E, Steel M. | title = Random biochemical networks and the probability of self-sustaining autocatalysis | journal = Journal of Theoretical Biology | volume = 233 | issue = 3 | pages = 327–336 | year = 2005 | pmid =  15652142| doi = 10.1016/j.jtbi.2004.10.011| bibcode = 2005JThBi.233..327M | citeseerx = 10.1.1.133.9352 }}</ref> An autocatalytic set then extends very quickly with growing number of molecules

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 a growing number of molecules, the number of possible reactions and catalysations grows even larger if the molecules grow in complexity, producing stochastically enough reactions and catalysations to make a part of the RS self-supported.<ref>{{cite journal | author = Mossel E, Steel M. | title = Random biochemical networks and the probability of self-sustaining autocatalysis | journal = Journal of Theoretical Biology | volume = 233 | issue = 3 | pages = 327–336 | year = 2005 | pmid =  15652142| doi = 10.1016/j.jtbi.2004.10.011| bibcode = 2005JThBi.233..327M | citeseerx = 10.1.1.133.9352 }}</ref> An autocatalytic set then extends very quickly with growing number of molecules

for the same reason. These theoretical results make autocatalytic sets attractive for scientific explanation of the very early origin of life.

for the same reason. These theoretical results make autocatalytic sets attractive for scientific explanation of the very early origin of life.

==Formal limitations==

==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.

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.

从形式上讲，除了非结构化的实体以外，很难把分子看作任何东西，因为一系列可能的反应(和分子)将变得无穷无尽。因此，根据模拟 DNA、 RNA 或蛋白质所需的任意长度的聚合物是不可能的。对 RNA 世界的研究也遇到了同样的问题。

从形式上讲，很难将分子视为非结构化实体之外的任何东西，因为可能的反应（和分子）集合会变得无限大。因此，根据模拟 DNA、 RNA 或蛋白质所需的任意长度的聚合物是不可能的。对 RNA 世界的研究也遇到了同样的问题。

==Linguistic aspects==

==Linguistic aspects==