Biochemical Reactions as Renewal Processes: the case of mRNA Degradation

In the last decade, the improvements of wet-lab techniques is making available novel experimental data about biochemical reactions occurring in single living cells. This information provides evidences that random events arising at the molecular level play important roles in determining the overall behaviour of biological organisms [2, 10]. As a consequence, new interest is rising towards stochastic models of biochemical systems. In this context, the Continuous Time Markov Chain (CTMC) is the most popular modeling approach and it is associated with the well-known Gillespie’s Chemical Master Equation (CME) [8, 1]. CTMCs are characterised by two properties: (i) The probability of transition events (chemical reactions) towards the future states depend only on the current state (Markov property); (ii) Transition probabilities are time-independent and, thus, the process is time-invariant. Importantly, the first property was proven to entail negative exponential distributions of the inter-event times or Waiting Times (WTs). The aforementioned biological data show stochastic temporal dynamics which significantly deviates from those expected according to a CTMC process. In particular, in many cases, the WTs elapsing between subsequent chemical reactions are found to be distributed according to non-exponential distributions [6, 3–5], leading to the loss of both the Markov property and time-invariance. Therefore, modelling approaches grounding on CTMCs (and the CME) can result inadequate.