Rare event statistics in reaction-diffusion systems

Abstract

We develop an efficient method to calculate probabilities of large deviations from the typical behavior (rare events) in reaction-diffusion systems. The method is based on a semiclassical treatment of underlying "quantum" Hamiltonian, encoding the system's evolution. To this end we formulate corresponding canonical dynamical system and investigate its phase portrait. The method is presented for a number of pedagogical examples.

Modeling the effect of transcriptional noise on switching in gene networks in a genetic bistable switch

“…(25) captures the effect of the promoter architecture. The same form of noise is also obtained for mRNA distributions, μ mRNA ¼ 1 m h i þ Δ m h i , where both Δ and Δ′ have been defined earlier.…”

“…Such a path integral representation has been derived for the case of a Michaelis-Menten enzyme attached to the membrane of a eukaryotic cell and later generalized to a network of reactions [23]. This theory is valid in the limit of short-lived mRNA [16][17][18][19][20][21][22][23][24][25][26][27][28]. Our first step is to consider the process of mRNA generation.…”

“…Here we do not provide the derivation of the semi-classical path integral approach and refer the reader to Text S1 in [24]. By considering the trajectories that dominate the dynamics, it was found that S(χ,t) is given by [25]:…”

“…A precise estimation of the prefactor σ is a hard task, although it can be obtained for one-dimensional systems; however it is not important because Eq. (21) is dominated by the exponent, when the system is not too close to the bifurcation point [25,30]. Strong fluctuations drive the system from its equilibrium metastable state to the on/off states along an optimal path that minimizes the action S. The optimal Hamiltonian trajectory that describes S starts from the metastable state and ends either at the on or off state.…”

“…This semi-classical method has been used earlier to calculate rare event statistics in reaction diffusion systems [25] and it has been applied to various epidemiological stochastic models [24,26,27]. In this method, we start with the eikonal ansatz:…”