On the origins of approximations for stochastic chemical kinetics

Abstract

This paper considers the derivation of approximations for stochastic chemical kinetics governed by the discrete master equation. Here, the concepts of ͑1͒ partitioning on the basis of fast and slow reactions as opposed to fast and slow species and ͑2͒ conditional probability densities are used to derive approximate, partitioned master equations, which are Markovian in nature, from the original master equation. Under different conditions dictated by relaxation time arguments, such approximations give rise to both the equilibrium and hybrid ͑deterministic or Langevin equations coupled with discrete stochastic simulation͒ approximations previously reported. In addition, the derivation points out several weaknesses in previous justifications of both the hybrid and equilibrium systems and demonstrates the connection between the original and approximate master equations. Two simple examples illustrate situations in which these two approximate methods are applicable and demonstrate the two methods' efficiencies.

Stochastic thermodynamics under coarse graining

“…For convenience, we are going to call the true system states "microstates," and states which lump together multiple true system states will be called "mesostates." While recent studies have investigated various aspects of such coarse-graining at the level of the stochastic description [60][61][62][63][64], as well as some of its implications at the level of the thermodynamic description [65][66][67][68][69], the present paper analyzes the effect of coarse-graining on stochastic thermodynamics as a whole. A rewarding outcome of this study is that the coarse-graining procedure a posteriori unambiguously clarifies the implicit assumptions made when formulating stochastic thermodynamics at the microscopic level.…”

(Hybrid) automata and (stochastic) programs * The hybrid automata lattice of a stochastic program

“…In general, we will choose a reference model and compare different hybrid semantics with this one. The choice falls on the original CTMC-based semantics, which is also in line with previous work on hybrid simulation strategies for biochemical systems [31]. In order to study the goodness of the approximation, we consider three different strategies: compare different simulation runs, prove analytical theorems or use a logic-based framework.…”

“…A crucial problem is clearly how to define such a partitioning scheme. Such a question is still open and has received a lot of attention in the field of biochemical reactions, see for instance [26,31,36]. We are currently investigating it in our setting.…”

“…The drawback is that rates depend on variables, hence it may be difficult to asses a priori whether an action is fast or slow. Dynamic partitioning schemes may improve such an approach, see Section 8 and [26,31].…”

The stochastic quasi-steady-state assumption: Reducing the model but not the noise

“…Such stiffness is analogous to the deterministic concept of reaction equilibrium and has been the subject of several recent studies. [15][16][17][18][19][20][21] A separate but equally important source of stiffness results from highly reactive species. This phenomenon results when reaction intermediates, known as quasi-steady-state (QSS) species, react so rapidly that their average number throughout the simulation is nearly zero or their average number is much smaller than the other species (the reactants and products).…”