On the Qualitative Behaviour of Oscillating Biochemical Systems: The Stochastic Approach

Some crucial enzymatic reaction systems, often showing feedbacks, give rise to oscillatory mechanisms. It occurs in well-known biochemical processes, such as, to give some examples, the PER-TIM circuit related to circadian rhythms, the mitogen-activated protein kinase (MAPK) cascade, the glycolysis and so on. In this work, we address the problem of modeling some complex biological processes by means of a stochastic approach, with a focus on the PER circadian clock model. As it is well-known, in such a scenario, a network of biochemical reactions is well described in terms of a Continuous-Time Markov Chain (CTMC), whose stochastic behaviour is fully characterized by the CTMC initial probability distribution and the probabilities per unit of time of each reaction (propensities). In order to cope with the burden of the system dimensionality, we resort to Quasi-Steady State Approximations (QSSAs), in particular to the standard QSSA and the total QSSA (sQSSA and tQSSA, respectively), which are widely used in enzyme kinetics to simplify the equations modeling the reactions, thus speeding up numerical simulations. It has been shown that, both in a deterministic and in a stochastic setting, the tQSSA is valid in a much wider range of parameters and initial conditions than the sQSSA. With the aim of investigating the asymptotic course of the process, we study the stationary distribution of the CTMC by applying the stochastic approach to the non-approximated original chemical networks (later referred to as full system), as well as to the standard and total QSSAs. Our simulation results fully confirm the superiority of the performances of tQSSA with respect to sQSSA in terms of qualitative and quantitative agreement of probability distributions with the stochastic modeling of the full system. Furthermore, the agreement (in spite of the large complexity reduction) between full system and tQSSA also in the stochastic framework allows to reveal non-trivial correlations among the species involved in the reactions, which cannot be investigated in a purely deterministic framework.