Autocatalysis

Acid-catalyzed hydrolysis of esters produces carboxylic acids that also catalyze the same reaction. Indeed, the observation of an accelerating hydrolysis of gamma valerolactone to gamma-hydroxyvaleric acid led to the introduction of the concept of autocatalysis in 1890.[4]

The oxidation of hydrocarbons by air or oxygen is the basis of autoxidation. Like many radical reactions, the rate vs time plot shows a sigmoidal behavior, characteristic of autocatalysis.[5] Many reactions of organic compounds with halogen involve autocatalytic radical mechanisms. For example the reaction of acetophenone with bromine to give phenacyl bromide.

Oscillating reactions such as the Belousov–Zhabotinsky reaction are more complicated examples that involve autocatalysis.[2] In such reactions the concentrations of some intermediates oscillate, as does the rate of formation of products. Other notable examples are the Lotka–Volterra equations for the predator-prey model, and the Brusselator model.

Autocatalysis applies also to reactions involving solids. Crystal growth provide dramatic examples of autocatalysis: the growth rate depends on the surface area of the growing crystal. The growth of metal films from solution using the technique of electroless plating is autocatalytic. The rate of plating accelerates after some deposition has occurred, i.e., nucleation.[6]

Autocatalytic reactions are those in which at least one of the products is also a reactant. A simple autocatalytic reaction can be written[3]

${\displaystyle A+B\rightleftharpoons 2B}$

with the rate equations (for an elementary reaction)

${\displaystyle {d \over dt}[A]=-k_{+}[A][B]+k_{-}[B]^{2}\,}$

${\displaystyle {d \over dt}[B]=+k_{+}[A][B]-k_{-}[B]^{2}\,}$.

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; 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

${\displaystyle [B]={\frac {[A]_{0}+[B]_{0}}{({\frac {[A]_{0}}{[B]_{0}}}-{\frac {k_{-}}{k_{+}}})e^{-k_{+}([A]_{0}+[B]_{0})t}+1+{\frac {k_{-}}{k_{+}}}}}}$

and

${\displaystyle [A]={\frac {([A]_{0}+[B]_{0})(({\frac {[A]_{0}}{[B]_{0}}}-{\frac {k_{-}}{k_{+}}})e^{-k_{+}([A]_{0}+[B]_{0})t}+{\frac {k_{-}}{k_{+}}})}{({\frac {[A]_{0}}{[B]_{0}}}-{\frac {k_{-}}{k_{+}}})e^{-k_{+}([A]_{0}+[B]_{0})t}+1+{\frac {k_{-}}{k_{+}}}}}}$.

For an irreversible reaction (i.e. ${\displaystyle k_{-}=0}$)[3][7]

${\displaystyle [A]={\frac {[A]_{0}+[B]_{0}}{1+{\frac {[B]_{0}}{[A]_{0}}}e^{([A]_{0}+[B]_{0})kt}}}}$

and

${\displaystyle [B]={\frac {[A]_{0}+[B]_{0}}{1+{\frac {[A]_{0}}{[B]_{0}}}e^{-([A]_{0}+[B]_{0})kt}}}}$.

The graph for these equations is a 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 esters to carboxylic acids and alcohols.[7] 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.[7]

• Assembly theory
• Catalytic cycle
• Reaction–diffusion system
• Morphogenesis

• Some Remarks on Autocatalysis and Autopoiesis (Barry McMullin)
• Jain, Sanjay; Krishna, Sandeep (21 December 1998). "Autocatalytic Sets and the Growth of Complexity in an Evolutionary Model". Physical Review Letters. 81 (25): 5684–5687. arXiv:adap-org/9809003. Bibcode:1998PhRvL..81.5684J. doi:10.1103/PhysRevLett.81.5684. S2CID 14471886.