scholarly journals Beyond the chemical master equation: Stochastic chemical kinetics coupled with auxiliary processes

2021 ◽  
Vol 17 (7) ◽  
pp. e1009214
Author(s):  
Davin Lunz ◽  
Gregory Batt ◽  
Jakob Ruess ◽  
J. Frédéric Bonnans
Keyword(s):  
Master Equation ◽  
Population Level ◽  
Phenotypic Selection ◽  
Chemical Master Equation ◽  
Cellular Growth ◽  
Nonlinear Phenomena ◽  
Stochastic Chemical Kinetics ◽  
Transient Events ◽  
Numerical Solver ◽  
Passage Times

The chemical master equation and its continuum approximations are indispensable tools in the modeling of chemical reaction networks. These are routinely used to capture complex nonlinear phenomena such as multimodality as well as transient events such as first-passage times, that accurately characterise a plethora of biological and chemical processes. However, some mechanisms, such as heterogeneous cellular growth or phenotypic selection at the population level, cannot be represented by the master equation and thus have been tackled separately. In this work, we propose a unifying framework that augments the chemical master equation to capture such auxiliary dynamics, and we develop and analyse a numerical solver that accurately simulates the system dynamics. We showcase these contributions by casting a diverse array of examples from the literature within this framework and applying the solver to both match and extend previous studies. Analytical calculations performed for each example validate our numerical results and benchmark the solver implementation.

scholarly journals APPROXIMATE EXPONENTIAL ALGORITHMS TO SOLVE THE CHEMICAL MASTER EQUATION

2015 ◽  
Vol 20 (3) ◽  
pp. 382-395
Author(s):  
Azam Mooasvi ◽  
Adrian Sandu
Keyword(s):  
Master Equation ◽  
Stochastic Simulation ◽  
Stochastic Simulation Algorithm ◽  
Approximate Solutions ◽  
Chemical Master Equation ◽  
Matrix Exponential ◽  
Stochastic Chemical Kinetics ◽  
Stochastic Simulation Algorithms ◽  
The Matrix ◽  
Simulation Algorithms

This paper discusses new simulation algorithms for stochastic chemical kinetics that exploit the linearity of the chemical master equation and its matrix exponential exact solution. These algorithms make use of various approximations of the matrix exponential to evolve probability densities in time. A sampling of the approximate solutions of the chemical master equation is used to derive accelerated stochastic simulation algorithms. Numerical experiments compare the new methods with the established stochastic simulation algorithm and the tau-leaping method.

scholarly journals Novel domain expansion methods to improve the computational efficiency of the Chemical Master Equation solution for large biological networks

2020 ◽  
Vol 21 (1) ◽  
Author(s):  
Rahul Kosarwal ◽  
Don Kulasiri ◽  
Sandhya Samarasinghe
Keyword(s):  
Master Equation ◽  
State Space ◽  
Performance Management ◽  
Numerical Solutions ◽  
De Novo ◽  
Exact Analytical Solution ◽  
Chemical Master Equation ◽  
The State ◽  
Biochemical System ◽  
Biochemical Systems

Abstract Background Numerical solutions of the chemical master equation (CME) are important for understanding the stochasticity of biochemical systems. However, solving CMEs is a formidable task. This task is complicated due to the nonlinear nature of the reactions and the size of the networks which result in different realizations. Most importantly, the exponential growth of the size of the state-space, with respect to the number of different species in the system makes this a challenging assignment. When the biochemical system has a large number of variables, the CME solution becomes intractable. We introduce the intelligent state projection (ISP) method to use in the stochastic analysis of these systems. For any biochemical reaction network, it is important to capture more than one moment: this allows one to describe the system’s dynamic behaviour. ISP is based on a state-space search and the data structure standards of artificial intelligence (AI). It can be used to explore and update the states of a biochemical system. To support the expansion in ISP, we also develop a Bayesian likelihood node projection (BLNP) function to predict the likelihood of the states. Results To demonstrate the acceptability and effectiveness of our method, we apply the ISP method to several biological models discussed in prior literature. The results of our computational experiments reveal that the ISP method is effective both in terms of the speed and accuracy of the expansion, and the accuracy of the solution. This method also provides a better understanding of the state-space of the system in terms of blueprint patterns. Conclusions The ISP is the de-novo method which addresses both accuracy and performance problems for CME solutions. It systematically expands the projection space based on predefined inputs. This ensures accuracy in the approximation and an exact analytical solution for the time of interest. The ISP was more effective both in predicting the behavior of the state-space of the system and in performance management, which is a vital step towards modeling large biochemical systems.

scholarly journals Accurate Stochastic Simulation Methods for Homogeneous Biochemical Networks

2021 ◽  
Author(s):  
Farida Ansari
Keyword(s):  
Master Equation ◽  
Stochastic Simulation ◽  
Growth Factor Receptor ◽  
Stochastic Simulation Algorithm ◽  
Chemical Master Equation ◽  
Biochemical Networks ◽  
Random Time Change ◽  
Simulation Strategy ◽  
Epidermal Growth ◽  
Exact Stochastic Simulation

Stochastic models of intracellular processes are subject of intense research today. For homogeneous systems, these models are based on the Chemical Master Equation, which is a discrete stochastic model. The Chemical Master Equation is often solved numerically using Gillespie’s exact stochastic simulation algorithm. This thesis studies the performance of another exact stochastic simulation strategy, which is based on the Random Time Change representation, and is more efficient for sensitivity analysis, compared to Gillespie’s algorithm. This method is tested on several models of biological interest, including an epidermal growth factor receptor model.

scholarly journals Solving the chemical master equation using sliding windows

2010 ◽  
Vol 4 (1) ◽  
Author(s):  
Verena Wolf ◽  
Rushil Goel ◽  
Maria Mateescu ◽  
Thomas A Henzinger
Keyword(s):  
Master Equation ◽  
Chemical Master Equation ◽  
Sliding Windows

The role of non-equilibrium fluxes in the relaxation processes of the linear chemical master equation

2014 ◽  
Vol 141 (6) ◽  
pp. 065102 ◽  
Author(s):  
Luciana Renata de Oliveira ◽  
Armando Bazzani ◽  
Enrico Giampieri ◽  
Gastone C. Castellani
Keyword(s):  
Master Equation ◽  
Chemical Master Equation ◽  
Relaxation Processes ◽  
Non Equilibrium

scholarly journals Delay chemical master equation: direct and closed-form solutions

2015 ◽  
Vol 471 (2179) ◽  
pp. 20150049 ◽  
Author(s):  
Andre Leier ◽  
Tatiana T. Marquez-Lago
Keyword(s):  
Master Equation ◽  
Closed Form ◽  
Stochastic Simulation Algorithm ◽  
Chemical Master Equation ◽  
Kolmogorov Equation ◽  
Closed Form Solutions ◽  
Mrna Maturation ◽  
First Time ◽  
Nonlinear Markov Process ◽  
Reaction Schemes

The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.

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