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

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.

scholarly journals Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 700 ◽  
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.

scholarly journals Fast reactions with non-interacting species in stochastic reaction networks

2022 ◽  
Vol 19 (3) ◽  
pp. 2720-2749
Author(s):  
Linard Hoessly ◽  
◽  
Carsten Wiuf
Keyword(s):  
Markov Process ◽  
Sufficient Conditions ◽  
Reaction Network ◽  
Reaction Networks ◽  
Unknown Parameters ◽  
Interacting Species ◽  
Fast Reactions ◽  
Kinetic Rates ◽  
Stochastic Reaction ◽  
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>

scholarly journals Accelerated Sensitivity Analysis in High-Dimensional Stochastic Reaction Networks

PLoS ONE ◽  
2015 ◽  
Vol 10 (7) ◽  
pp. e0130825 ◽  
Author(s):  
Georgios Arampatzis ◽  
Markos A. Katsoulakis ◽  
Yannis Pantazis
Keyword(s):  
Sensitivity Analysis ◽  
Reaction Networks ◽  
High Dimensional ◽  
Stochastic Reaction ◽  
Stochastic Reaction Networks

scholarly journals Uncertainty quantification for quantum chemical models of complex reaction networks

2016 ◽  
Vol 195 ◽  
pp. 497-520 ◽  
Author(s):  
Jonny Proppe ◽  
Tamara Husch ◽  
Gregor N. Simm ◽  
Markus Reiher
Keyword(s):  
Free Energy ◽  
Electronic Structure ◽  
State Space ◽  
Time Scales ◽  
Reaction Network ◽  
Reaction Networks ◽  
Multiple Time Scales ◽  
Complex Reaction ◽  
Discrete State ◽  
Continuous State

For the quantitative understanding of complex chemical reaction mechanisms, it is, in general, necessary to accurately determine the corresponding free energy surface and to solve the resulting continuous-time reaction rate equations for a continuous state space. For a general (complex) reaction network, it is computationally hard to fulfill these two requirements. However, it is possible to approximately address these challenges in a physically consistent way. On the one hand, it may be sufficient to consider approximate free energies if a reliable uncertainty measure can be provided. On the other hand, a highly resolved time evolution may not be necessary to still determine quantitative fluxes in a reaction network if one is interested in specific time scales. In this paper, we present discrete-time kinetic simulations in discrete state space taking free energy uncertainties into account. The method builds upon thermo-chemical data obtained from electronic structure calculations in a condensed-phase model. Our kinetic approach supports the analysis of general reaction networks spanning multiple time scales, which is here demonstrated for the example of the formose reaction. An important application of our approach is the detection of regions in a reaction network which require further investigation, given the uncertainties introduced by both approximate electronic structure methods and kinetic models. Such cases can then be studied in greater detail with more sophisticated first-principles calculations and kinetic simulations.

scholarly journals Improved estimations of stochastic chemical kinetics by finite-state expansion

2021 ◽  
Vol 477 (2251) ◽  
pp. 20200964
Author(s):  
Tabea Waizmann ◽  
Luca Bortolussi ◽  
Andrea Vandin ◽  
Mirco Tribastone
Keyword(s):  
Master Equation ◽  
State Space ◽  
Stochastic Dynamics ◽  
Reaction Network ◽  
Discrete State ◽  
State Expansion ◽  
Analytical Approximations ◽  
Finite State ◽  
Stochastic Reaction ◽  
Discrete State Space

Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant. The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive, several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics. Here we propose finite-state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations and multi-stability.

scholarly journals Adaptive hybrid simulations for multiscale stochastic reaction networks

2015 ◽  
Vol 142 (3) ◽  
pp. 034118 ◽  
Author(s):  
Benjamin Hepp ◽  
Ankit Gupta ◽  
Mustafa Khammash
Keyword(s):  
Reaction Networks ◽  
Stochastic Reaction ◽  
Stochastic Reaction Networks ◽  
Hybrid Simulations
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