# Mass action kinetics

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Mass action kinetics is a kinetic scheme for chemical reaction networks which says that the rate of a chemical reaction is proportional to the product of the concentrations of the reacting chemical species. It was first formulated by Cato Maximilian Guldberg and Peter Waage in the 1860's [1]. It remains one of the most common kinetic assumptions used by chemists, biologists, and mathematicians.

## Deterministic modeling

In the ordinary differential equation modeling of chemical reaction networks, the assumption of mass-action kinetics produces polynomial vector fields. For example, consider the reaction given by

$A + B \rightarrow C.$

If we let $[A]$ and $[B]$ denote the concentrations of A and B respectively, then the reaction occurs at a rate proportional to $[A][B]$. That is to say, we have

$[\mbox{rate of reaction}] \propto [A][B] \; \; \; \; \; \mbox{ or } \; \; \; \; \; [\mbox{rate of reaction}] = k [A][B].$

where $k>0$ is the (fixed) proportionality rate (or rate constant) associated with the reaction. Since each instance of the reaction produces a net decrease of one molecule of A and B each, and an increase of one molecule of C, we can model the reaction as

$\begin{array}{rcl} \frac{d[A]}{dt} & = & -k[A][B] \\ \frac{d[B]}{dt} & = & -k[A][B] \\ \frac{d[C]}{dt} & = & k[A][B]. \end{array}$

## References

1. C.M. Guldberg and P. Waage, Studies Concerning Affinity, C. M. Forhandlinger: Videnskabs-Selskabet i Christiana (1864), 35