Multiple-scale stochastic processes: Decimation, averaging and beyond

Abstract

The recent experimental progresses in handling microscopic systems have allowed to probe them at levels where fluctuations are prominent, calling for stochastic modeling in a large number of physical, chemical and biological phenomena. This has provided fruitful applications for established stochastic methods and motivated further developments. These systems often involve processes taking place on widely separated time scales. For an efficient modeling one usually focuses on the slower degrees of freedom and it is of great importance to accurately eliminate the fast variables in a controlled fashion, carefully accounting for their net effect on the slower dynamics. This procedure in general requires to perform two different operations: decimation and coarse-graining. We introduce the asymptotic methods that form the basis of this procedure and discuss their application to a series of physical, biological and chemical examples. We then turn our attention to functionals of the stochastic trajectories such as residence times, counting statistics, fluxes, entropy production, etc. which have been increasingly studied in recent years. For such functionals, the elimination of the fast degrees of freedom can present additional difficulties and naive procedures can lead to blatantly inconsistent results. Homogenization techniques for functionals are less covered in the literature and we will pedagogically present them here, as natural extensions of the ones employed for the trajectories. We will also discuss recent applications of these techniques to the thermodynamics of small systems and their interpretation in terms of information-theoretic concepts.

From conformational spread to allosteric and cooperative models of E. coli flagellar motor

“…However, as experiments show [13], in the flagellar motor regulation mechanism of E. coli it is possible to identify a hierarchy of widely separated time scales. This opens up the possibility of operating a reduction of the set of states by gradually integrating out/decimating fast degrees of freedom, operating a quasi-stationary approximation: the time scale of the slow degrees of freedom is much longer than the time needed for the fast variables to relax to a stationary distribution; hence, the fast degrees of freedom enter the slow dynamics only through quantities averaged over such stationary distribution (conditioned to the state of the slow variables) [16][17][18]. The application of such techniques to the study of the allosteric regulation of the motor of E. coli will be the subject of the following sections.…”

Thermodynamically consistent coarse graining of biocatalysts beyond Michaelis–Menten

“…In these cases, the dissipation in a nonequililibrium process is typically underestimated-although also overestimations may occur [23]. For a general overview of coarse-graining in Markov processes, see [24] and references therein.…”

Dynamical phase diagram of an auto-regulating gene in fast switching conditions

“…In this case, the Markovian model illustrated in Fig. 1(b) can be reduced to a much simpler one by using a classical simplification method of multiscale Markov jump processes called averaging [24][25][26]. Since σ b and σ u are large, for each n ≥ m, the two microstates (0, n) and (1, n − m) are in rapid equilibrium and thus can be aggregated into a group that is labelled by group n, as depicted in Fig.…”