Editing Autocatalysis (section)

==Mathematical description==
[[Image:Sigmoid curve for an autocatalytical reaction.jpg|256px|right|thumb|Sigmoid variation of product concentration in autocatalytic reactions]]
Autocatalytic reactions are those in which at least one of the products is also a reactant. A simple autocatalytic reaction can be written<ref name=Steinfeld/> 
:<math>  A + B \rightleftharpoons 2B</math>

with the rate equations (for an elementary reaction)
:<math>{d \over dt}[ A ] =-  k_+ [ A ] [B ]  + k_{-} [B ]^2 \,</math>
:<math>{d \over dt}[ B ] = + k_+ [ A ] [B ]  -k_{-} [B ]^2 \,</math>.

This reaction is one in which a molecule of species A interacts with a molecule of species B. The A molecule is converted into a B molecule. The final product consists of the original B molecule plus the B molecule created in the reaction.

The key feature of these rate equations is that they are [[Nonlinear system#Nonlinear differential equations|nonlinear]]; the second term on the right varies as the square of the concentration of B. This feature can lead to multiple fixed points of the system, much like a [[quadratic equation]] can have two roots. Multiple fixed points allow for multiple states of the system. A system existing in multiple [[macroscopic]] states is more orderly (has lower entropy) than a system in a single state.

The concentrations of A and B vary in time according to
: <math>[B]=\frac{[A]_0+[B]_0}{(\frac{[A]_0}{[B]_0}-\frac{k_-}{k_+})e^{-k_+([A]_0+[B]_0)t}+1+\frac{k_-}{k_+}}</math>
and
: <math>[A]=\frac{([A]_0+[B]_0)((\frac{[A]_0}{[B]_0}-\frac{k_-}{k_+})e^{-k_+([A]_0+[B]_0)t}+\frac{k_-}{k_+})}{(\frac{[A]_0}{[B]_0}-\frac{k_-}{k_+})e^{-k_+([A]_0+[B]_0)t}+1+\frac{k_-}{k_+}}</math>.

For an irreversible reaction (i.e. <math>k_- = 0</math>)<ref name=Steinfeld/><ref name=Moore>Moore J.W. and [[Ralph Pearson|Pearson R.G.]] ''Kinetics and Mechanism'' (John Wiley 1981) p.26  {{ISBN|0-471-03558-0}}</ref> 
:<math>[A]=\frac{[A]_0+[B]_0}{1+\frac{[B]_0}{[A]_0}e^{([A]_0+[B]_0)kt}}</math>
and 
:<math>[B]=\frac{[A]_0+[B]_0}{1+\frac{[A]_0}{[B]_0}e^{-([A]_0+[B]_0)kt}}</math>.
The graph for these equations is a [[sigmoid function|sigmoid curve]] (specifically a [[logistic function]]), which is typical for autocatalytic reactions: these chemical reactions proceed slowly at the start (the [[induction period]]) because there is little catalyst present, the rate of reaction increases progressively as the reaction proceeds as the amount of catalyst increases and then it again slows down as the reactant concentration decreases. If the concentration of a reactant or product in an experiment follows a sigmoid curve, the reaction may be autocatalytic.

These kinetic equations apply for example to the acid-catalyzed hydrolysis of some [[ester]]s to [[carboxylic acid]]s and [[Alcohol (chemistry)|alcohol]]s.<ref name=Moore/> There must be at least some acid present initially to start the catalyzed mechanism; if not the reaction must start by an alternate uncatalyzed path which is usually slower. The above equations (which do not consider the alternate pathway) for the catalyzed mechanism would imply that the concentration of acid product remains zero forever.<ref name=Moore/>

===Asymmetric autocatalysis===
Asymmetric autocatalysis occurs when the reaction product is [[chiral]] and thus serves as a catalyst for its own production. Reactions of this type, such as the [[Soai reaction]], have the property that they can amplify a very small [[enantiomeric excess]] into a large one.<ref>{{cite journal |doi=10.1021/acs.chemrev.9b00557 |title=Autocatalytic Models for the Origin of Biological Homochirality |year=2020 |last1=Blackmond |first1=Donna G. |journal=Chemical Reviews |volume=120 |issue=11 |pages=4831–4847 |pmid=31797671 |arxiv=1909.13015 }}</ref> In another example, [[sodium chlorate]] crystallizes as an equilibrium mixture of left- and right-handed crystals.  When seeded appropriated, saturated solutions of this salt (which is optically inactive), will produce batches of single enantiomeric crystals.<ref>{{cite journal |doi=10.1021/acs.chemrev.0c00819 |title=Spontaneous Deracemizations |year=2021 |last1=Buhse |first1=Thomas |last2=Cruz |first2=José-Manuel |last3=Noble-Terán |first3=María E. |last4=Hochberg |first4=David |last5=Ribó |first5=Josep M. |last6=Crusats |first6=Joaquim |last7=Micheau |first7=Jean-Claude |journal=Chemical Reviews |volume=121 |issue=4 |pages=2147–2229 |pmid=33464058 |s2cid=231640216 }}</ref>

=== Possible role in origin of life ===
{{main|Abiogenesis}}
[[File:FormoseRxn.svg|thumb|Autocatalytic cycle of formose reaction showing how glyceraldehyde can be both the catalyst and the product of one portion of this complex reaction type.|left|288px]]
An early example of autocatalysis is the [[formose reaction]], in which formaldehyde and base produce sugars and related polyols.  Characteristic of autocatalysis, this reaction rate is extremely slow initially but accelerates with time.  This kind of reaction has often been cited as being relevant to the origin of life.<ref name=ACIE/>

Autocatalysis is one explanation for [[abiogenesis]].<ref>{{cite book|author=Stuart Kauffman|title=At Home in the Universe: The Search for the Laws of Self-Organization and Complexity|isbn=978-0-19-509599-9|publisher=Oxford University Press|year=1995|url-access=registration|url=https://archive.org/details/athomeinuniverse00kauf_0}}</ref><ref>Ecology, the Ascendent Perspective", Robert Ulanowicz, Columbia Univ. Press 1997.</ref><ref>Investigations, Stuart Kauffman.</ref> Illustrative is the reaction amino adenosine and pentafluorophenyl ester in the presence of amino adenosine triacid ester (AATE).  This experiment demonstrated that autocatalysts could exhibit competition within a population of entities with heredity, which could be interpreted as a rudimentary form of [[natural selection]], and that certain environmental changes (such as irradiation) could alter the chemical structure of some of these self-replicating molecules (an analog for mutation) in such ways that could either boost or interfere with its ability to react, thus boosting or interfering with its ability to replicate and spread in the population.<ref>{{cite journal|last=Rebeck|first=Julius|title=Synthetic Self-Replicating Molecules|journal=Scientific American|date=July 1994|volume=271 |issue=1 |pages=48–55|doi=10.1038/scientificamerican0794-48 |bibcode=1994SciAm.271a..48R }}</ref>