Past exam of the mathematical physiology course of the University of Oxford / 2023 / 1 / a

Write $a=[\mathrm{A}]$, $b=[\mathrm{B}]$ and $c=[\mathrm{C}]$ for the three concentrations. The letters denote chemical species; the forward arrow represents association of one molecule of each reactant into one complex, and the reverse arrow represents dissociation. The positive constants $k_{+}$ and $k_{-}$ are the respective reaction rate constants. For an elementary chemical reaction, the law of mass action gives forward and reverse reaction ratesEvery forward event removes one molecule of each reactant and adds one complex; a reverse event reverses those changes. Thus the reaction-rate equations areThe units of $k_{+}$ are inverse concentration per unit time, whereas $k_{-}$ has units of inverse time. Adding the equations gives the independent conservation lawsIn particular $a-b$ is constant, but this is a consequence of those two conservation laws. The total $a+b+c$ need not be constant: association changes the number of separate molecules.

For the general stoichiometry, take $p,q$ to be positive integers. The net reaction rate and stoichiometric vector areConsequentlyThe factors $p,q$ outside $R$ count molecules consumed per event; the powers inside $R$ express the law of mass action. The corresponding conservation laws are $a+pc=\text{constant}$ and $b+qc=\text{constant}$. No factorial is required in this deterministic convention for the reaction rate constant.