Mass action kinetics

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 . 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

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