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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A + B \rightarrow C. }

If we let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [A]} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [B]} denote the concentrations of A and B respectively, then the reaction occurs at a rate proportional to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [A][B]} . That is to say, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\mbox{rate of reaction}] \propto [A][B] \; \; \; \; \; \mbox{ or } \; \; \; \; \; [\mbox{rate of reaction}] = k [A][B]. }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} }