Biochemical reactions as renewal processes: the case of mRNA degradation

Biochemical processes are typically described in terms of Continuous Time Markov Chains (CTMCs), which is the stochastic pro- cess associated with the well-known Gillespie's Chemical Master Equa- tion (CME). However, this approach is limited by the basic features of CTMC, that is, Markov property, time-invariance and, consequently, exponential decay of both correlation functions and distribution of Wait- ing Times (WTs) between successive reactions. Here we propose a model based on the theory of renewal point processes, i.e., stochastic processes defined as sequences of critical events occurring randomly in time and in- dependent from each other. Renewal theory allows to generalize CTMC modeling to the case of non-exponential behavior observed in many bio- chemical systems at the cell scale and is the natural framework for the study of intermittent time series. In particular, renewal modeling allows to include directly in a simple way non-exponential WT distribution such as slow power-law decay or stretched exponential. In the specific appli- cation of mRNA degradation, a renewal model can include whatever functional form of the degradation rate.