Material for “ Information Processing by Biochemical Networks : A Dynamic Approach ”

A Stochastic Kinetic Model (SKM) is a continuous-time pure jump process describing the dynamic evolution of the state of the molecular network, x(t). Any change in x(t) is the result of the occurrence of some biochemical reaction, m. The stoichiometric matrix is given by S := [S1, S2, ..., SM ], where the column vector Sm consists of the changes in the number of molecules of each species caused by reaction m. The stochastic dynamic evolution of x(t) is governed by the conditional reaction intensities, [λm(t); m = 1, ...,M ], which can be thought of as the instantaneous reaction rates at time t conditional on the trajectory (or ‘history’), X(t) = (x(s); 0 ≤ s ≤ t). Conditional intensities, λA(t), for the levels of any group of species A are defined and constructed by Bowsher (2010). The term SKM implies that each reaction intensity, λm(t), ‘depends only on’ (is measurable with respect to) the history of the reactants of reaction m, XR[m](t) = (xR[m](s); 0 ≤ s ≤ t). Stochastic chemical kinetic theory (Gillespie, 1992) implies that these reaction intensities take the form