Regularity and approximability of the solutions to the chemical master equation

Gauckler, Ludwig J.; Yserentant, Harry

doi:10.1051/m2an/2014018

Cited by 15 publications

(10 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A related existence result for the Markov semigroup of the CME was established in [ 22 ], which however does not apply to the case of reversible RRE, because of the restrictions on the growth of the transition rates.…”

Section: The Chemical Master Equationmentioning

confidence: 99%

Modeling of Chemical Reaction Systems with Detailed Balance Using Gradient Structures

Maas

Mielke

2020

J Stat Phys

View full text Add to dashboard Cite

We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary $$\Gamma $$ Γ -convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels.

show abstract

Section: The Chemical Master Equationmentioning

confidence: 99%

Modeling of Chemical Reaction Systems with Detailed Balance Using Gradient Structures

Maas

Mielke

2020

J Stat Phys

View full text Add to dashboard Cite

show abstract

“…In this approach all the outgoing transitions from the truncated state-space are simply eliminated by setting their propensities to zero. It has been shown that this reflected version of FSP yields accurate estimates of the stationary distribution for some reaction network examples [17,12]. However there is no theoretical guarantee that this approach will work well in general.…”

Section: Introductionmentioning

confidence: 99%

A finite state projection algorithm for the stationary solution of the chemical master equation

Gupta

Mikelson

Khammash

2017

The Journal of Chemical Physics

View full text Add to dashboard Cite

The chemical master equation (CME) is frequently used in systems biology to quantify the effects of stochastic fluctuations that arise due to biomolecular species with low copy numbers. The CME is a system of ordinary differential equations that describes the evolution of probability density for each population vector in the state-space of the stochastic reaction dynamics. For many examples of interest, this state-space is infinite, making it difficult to obtain exact solutions of the CME. To deal with this problem, the Finite State Projection (FSP) algorithm was developed by Munsky and Khammash [J. Chem. Phys. 124(4), 044104 (2006)], to provide approximate solutions to the CME by truncating the state-space. The FSP works well for finite time-periods but it cannot be used for estimating the stationary solutions of CMEs, which are often of interest in systems biology. The aim of this paper is to develop a version of FSP which we refer to as the stationary FSP (sFSP) that allows one to obtain accurate approximations of the stationary solutions of a CME by solving a finite linear-algebraic system that yields the stationary distribution of a continuous-time Markov chain over the truncated state-space. We derive bounds for the approximation error incurred by sFSP and we establish that under certain stability conditions, these errors can be made arbitrarily small by appropriately expanding the truncated state-space. We provide several examples to illustrate our sFSP method and demonstrate its efficiency in estimating the stationary distributions. In particular, we show that using a quantized tensor-train implementation of our sFSP method, problems admitting more than 100 × 10 states can be efficiently solved.

show abstract

“…For all models encountered in practice, the error bound g(t) decreases monotonically as Ω expands [42]. Under some regularity conditions on the propensity functions, it could be shown that p(t) −p Ω (t) 1 → 0 as Ω → S [22]. 3.1.…”

Section: Where (26)mentioning

confidence: 97%

A Parallel Implementation of the Finite State Projection Algorithm for the Solution of the Chemical Master Equation

Munsky

2020

Preprint

View full text Add to dashboard Cite

AbstractStochastic reaction networks are a popular modeling framework for biochemical processes that treat the molecular copy numbers within a single cell as a continuous time Markov chain, whose forward Chapman-Kolmogorov equation is known in biochemistry literature as the chemical master equation (CME). The solution of the CME contains extremely useful information that can be compared to experimental data in order to improve the quantitative understanding of biochemical reaction networks within the cell. However, this solution is costly to compute as it requires integrating an enormous system of differential equations that grows exponentially with the number of chemical species. To address this issue, we introduce a novel multiple-sinks Finite State Projection algorithm that approximates the CME with an adaptive sequence of reduced-order models with an effecient parallelization based on MPI. The implementation is tested on models of sizable state spaces using a high-performance computing node on Amazon Web Services, showing favorable scalability.

show abstract

Regularity and approximability of the solutions to the chemical master equation

Cited by 15 publications

References 21 publications

Modeling of Chemical Reaction Systems with Detailed Balance Using Gradient Structures

Modeling of Chemical Reaction Systems with Detailed Balance Using Gradient Structures

A finite state projection algorithm for the stationary solution of the chemical master equation

A Parallel Implementation of the Finite State Projection Algorithm for the Solution of the Chemical Master Equation

Contact Info

Product

Resources

About