A Reaction-Based Model of the State Space of Chemical Reaction Systems Enables Efficient Simulations

The choice of the state space representation of a system can turn into a prominent advantage or burden in any endeavour to mathematically model dynamical systems since it entails the analytical tractability of the related modelling formalism and the efficiency of the numerical computation. The Reaction-Based Model (RBM) of the state space, which is presented in this article, is a novel formalization of the kinetics of a system of interacting molecules. According to our representation, the state
$S_\mu$
of a system of
$M$
reactions and
$N$
molecular species, is identified with the occurrence of the reaction
$R_\mu$
(
$\mu = 1, \dots, M$
). The transition between any two states
$S_\mu$
and
$S_\nu$
is modelled as a first-order reaction
$S_\mu \rightarrow S_\nu$
and described by mass action-like equation for the partial time derivative of the variables
$P(S_\mu, t)$
and
$P(S_\nu, t)$
, which denote the probabilities that the system lies in the two states, respectively. The rate equations for the state probabilities are coupled with those for the abundance of molecular species. Altogether, the rate equations along with the specification of the initial conditions define the Cauchy problem whose solution describes the time-evolution of the system. The RBM has been applied to a typical severely stiff biological case study. The numerical solutions of the system's dynamics turned out to be computationally more efficient and in agreement with the results of the stochastic and hybrid stochastic/deterministic simulation algorithms.