Fast reactions with non-interacting species in stochastic reaction networks

<abstract><p>We consider stochastic reaction networks modeled by continuous-time Markov chains. Such reaction networks often contain many reactions, potentially occurring at different time scales, and have unknown parameters (kinetic rates, total amounts). This makes their analysis complex. We examine stochastic reaction networks with non-interacting species that often appear in examples of interest (e.g. in the two-substrate Michaelis Menten mechanism). Non-interacting species typically appear as intermediate (or transient) chemical complexes that are depleted at a fast rate. We embed the Markov process of the reaction network into a one-parameter family under a two time-scale approach, such that molecules of non-interacting species are degraded fast. We derive simplified reaction networks where the non-interacting species are eliminated and that approximate the scaled Markov process in the limit as the parameter becomes small. Then, we derive sufficient conditions for such reductions based on the reaction network structure for both homogeneous and time-varying stochastic settings, and study examples and properties of the reduction.</p></abstract>

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Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers

Entropy ◽

10.3390/e20090700 ◽

2018 ◽

Vol 20 (9) ◽

pp. 700 ◽

Cited By ~ 1

Author(s):

Michail Vlysidis ◽

Yiannis Kaznessis

Keyword(s):

Time Evolution ◽

Lagrange Multipliers ◽

Reaction Network ◽

Exact Algorithm ◽

Reaction Networks ◽

Moment Equations ◽

Interacting Species ◽

Oscillatory Systems ◽

Stochastic Reaction ◽

Stochastic Reaction Networks

The time evolution of stochastic reaction networks can be modeled with the chemical master equation of the probability distribution. Alternatively, the numerical problem can be reformulated in terms of probability moment equations. Herein we present a new alternative method for numerically solving the time evolution of stochastic reaction networks. Based on the assumption that the entropy of the reaction network is maximum, Lagrange multipliers are introduced. The proposed method derives equations that model the time derivatives of these Lagrange multipliers. We present detailed steps to transform moment equations to Lagrange multiplier equations. In order to demonstrate the method, we present examples of non-linear stochastic reaction networks of varying degrees of complexity, including multistable and oscillatory systems. We find that the new approach is as accurate and significantly more efficient than Gillespie’s original exact algorithm for systems with small number of interacting species. This work is a step towards solving stochastic reaction networks accurately and efficiently.

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Parameter Estimation of Kinetic Rates in Stochastic Reaction Networks by the EM Method

2008 International Conference on BioMedical Engineering and Informatics ◽

10.1109/bmei.2008.237 ◽

2008 ◽

Cited By ~ 5

Author(s):

Andr Horv ◽

Daniele Manini

Keyword(s):

Parameter Estimation ◽

Reaction Networks ◽

Kinetic Rates ◽

Em Method ◽

Stochastic Reaction ◽

Stochastic Reaction Networks

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Ergodicity analysis and antithetic integral control of a class of stochastic reaction networks with delays

10.1101/481085 ◽

2018 ◽

Author(s):

Corentin Briat ◽

Mustafa Khammash

Keyword(s):

State Space ◽

Reaction Network ◽

Reaction Networks ◽

Delayed Reaction ◽

Finite Dimensional ◽

Phase Type ◽

Free Network ◽

Stochastic Reaction ◽

Stochastic Reaction Networks ◽

Stochastic Reaction Network

AbstractDelays are important phenomena arising in a wide variety of real world systems, including biological ones, because of diffusion/propagation effects or as simplifying modeling elements. We propose here to consider delayed stochastic reaction networks, a class of networks that has been relatively few studied until now. The difficulty in analyzing them resides in the fact that their state-space is infinite-dimensional. We demonstrate here that by restricting the delays to be phase-type distributed, one can represent the associated delayed reaction network as a reaction network with finite-dimensional state-space. This can be achieved by suitably adding chemical species and reactions to the delay-free network following a simple algorithm which is fully characterized. Since phase-type distributions are dense in the set of probability distributions, they can approximate any distribution arbitrarily closely and this makes their consideration only a bit restrictive. As the state-space remains finite-dimensional, usual tools developed for non-delayed reaction network directly apply. In particular, we prove, for unimolecular mass-action reaction networks, that the delayed stochastic reaction network is ergodic if and only if the delay-free network is ergodic as well. Bimolecular reactions are more difficult to consider but slightly stronger analogous results are nevertheless obtained. These results demonstrate that delays have little to no harm to the ergodicity property of reaction networks as long as the delays are phase-type distributed, and this holds regardless the complexity of their distribution. We also prove that the presence of those delays adds convolution terms in the moment equation but does not change the value of the stationary means compared to the delay-free case. The covariance, however, is influenced by the presence of the delays. Finally, the control of a certain class of delayed stochastic reaction network using a delayed antithetic integral controller is considered. It is proven that this controller achieves its goal provided that the delay-free network satisfy the conditions of ergodicity and output-controllability.

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