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Chemical master equation

Youfang Cao, Anna Terebus, Jie Liang
The discrete chemical master equation (dCME) provides a fundamental framework for studying stochasticity in mesoscopic networks. Because of the multi-scale nature of many networks where reaction rates have large disparity, directly solving dCMEs is intractable due to the exploding size of the state space. It is important to truncate the state space effectively with quantified errors, so accurate solutions can be computed. It is also important to know if all major probabilistic peaks have been computed. Here we introduce the Accurate CME (ACME) algorithm for obtaining direct solutions to dCMEs...
2016: Multiscale Modeling & Simulation: a SIAM Interdisciplinary Journal
Neil U M Howes, James P A Lockhart, Mark A Blitz, Scott A Carr, Maria Teresa Baeza-Romero, Dwayne E Heard, Robin J Shannon, Paul W Seakins, T Varga
Using laser flash photolysis coupled to photo-ionization time-of-flight mass spectrometry (PIMS), methyl radicals (CH3) have been detected as primary products from the reaction of OH radicals with acetaldehyde (ethanal, CH3CHO) with a yield of ∼15% at 1-2 Torr of helium bath gas. Supporting measurements based on laser induced fluorescence studies of OH recycling in the OH/CH3CHO/O2 system are consistent with the PIMS study. Master equation calculations suggest that the origin of the methyl radicals is from prompt dissociation of chemically activated acetyl products and hence is consistent with previous studies which have shown that abstraction, rather than addition/elimination, is the sole route for the OH + acetaldehyde reaction...
September 29, 2016: Physical Chemistry Chemical Physics: PCCP
Cyrille Costentin, Jean-Michel Savéant
Among the many virtues ascribed to catalytic nanoparticles, the prospect that the passage from the macro- to the nanoscale may change product selectivity attracts increasing attention. To date, why such effects may exist lacks explanation. Guided by recent experimental reports, we propose that the effects may result from the coupling between the chemical steps in which the reactant, intermediates, and products are involved and transport of these species toward the catalytic surface. Considering as a thought experiment the competitive formation of hydrogen and formate upon reduction of hydrogenocarbonate ions on metals like palladium or platinum, a model is developed that allows one to identify the governing parameters and predict the effect of nanoscaling on selectivity...
September 29, 2016: Proceedings of the National Academy of Sciences of the United States of America
Colin S Gillespie, Andrew Golightly
Solving the chemical master equation exactly is typically not possible, so instead we must rely on simulation based methods. Unfortunately, drawing exact realisations, results in simulating every reaction that occurs. This will preclude the use of exact simulators for models of any realistic size and so approximate algorithms become important. In this paper we describe a general framework for assessing the accuracy of the linear noise and two moment approximations. By constructing an efficient space filling design over the parameter region of interest, we present a number of useful diagnostic tools that aids modellers in assessing whether the approximation is suitable...
October 1, 2016: Statistical Applications in Genetics and Molecular Biology
Sayuri K Hahl, Andreas Kremling
In the mathematical modeling of biochemical reactions, a convenient standard approach is to use ordinary differential equations (ODEs) that follow the law of mass action. However, this deterministic ansatz is based on simplifications; in particular, it neglects noise, which is inherent to biological processes. In contrast, the stochasticity of reactions is captured in detail by the discrete chemical master equation (CME). Therefore, the CME is frequently applied to mesoscopic systems, where copy numbers of involved components are small and random fluctuations are thus significant...
2016: Frontiers in Genetics
Zachary Fox, Gregor Neuert, Brian Munsky
Emerging techniques now allow for precise quantification of distributions of biological molecules in single cells. These rapidly advancing experimental methods have created a need for more rigorous and efficient modeling tools. Here, we derive new bounds on the likelihood that observations of single-cell, single-molecule responses come from a discrete stochastic model, posed in the form of the chemical master equation. These strict upper and lower bounds are based on a finite state projection approach, and they converge monotonically to the exact likelihood value...
August 21, 2016: Journal of Chemical Physics
Mohamad Akbar Ali, Jason A Sonk, John R Barker
The reaction of methanimine (CH2NH) with the hydroperoxy (HO2) radical has been investigated by using a combination of ab initio and density functional theory (CCSD(T)/CBSB7//B3LYP+Dispersion/CBSB7) and master equation calculations based on transition state theory (TST). Variational TST was used to compute both canonical (CVTST) and microcanonical (μVTST) rate constants for barrierless reactions. The title reaction starts with the reversible formation of a cyclic prereactive complex (PRC) that is bound by ∼11 kcal/mol and contains hydrogen bonds to both nitrogen and oxygen...
September 15, 2016: Journal of Physical Chemistry. A
Neil L Wesch, Laura J Burlock, Robert J Gooding
The lengths of the telomere regions of chromosomes in a population of cells are modelled using a chemical master equation formalism, from which the evolution of the average number of cells of each telomere length is extracted. In particular, the role of the telomere-elongating enzyme telomerase on these dynamics is investigated. We show that for biologically relevant rates of cell birth and death, one finds a critical rate, R crit, of telomerase activity such that the total number of cells diverges. Further, R crit is similar in magnitude to the rates of mitosis and cell death...
2016: Physical Biology
Bryan Lau, Ofer Kedem, Mark A Ratner, Emily A Weiss
Ratchets rectify the motion of randomly moving particles, which are driven by isotropic sources of energy such as thermal and chemical energy, without applying a net, time-averaged force between source and drain. This paper describes the behavior of a damped electron, modeled by a quantum Lindblad master equation, within a flashing ratchet (a one-dimensional potential that oscillates between a flat surface and a periodic asymmetric surface). By examining the complete space of all biharmonic potential shapes and a large range of oscillation frequencies, two modes of ratchet operation, differentiated by their oscillation frequencies (relative to the rate of electron relaxation), are identified...
June 2016: Physical Review. E
Daniil A Andrienko, Iain D Boyd
Investigation of O2-N collisions is performed by means of the quasi-classical trajectory method on the two lowest ab initio potential energy surfaces at temperatures relevant to hypersonic flows. A complete set of bound-bound and bound-free transition rates is obtained for each precollisional rovibrational state. Special attention is paid to the vibrational and rotational relaxations of oxygen as a result of chemically non-reactive interaction with nitrogen atoms. The vibrational relaxation of oxygen partially occurs via the formation of an intermediate NO2 complex...
July 7, 2016: Journal of Chemical Physics
Stephen Smith, Ramon Grima
The chemical master equation (CME) is the exact mathematical formulation of chemical reactions occurring in a dilute and well-mixed volume. The reaction-diffusion master equation (RDME) is a stochastic description of reaction-diffusion processes on a spatial lattice, assuming well mixing only on the length scale of the lattice. It is clear that, for the sake of consistency, the solution of the RDME of a chemical system should converge to the solution of the CME of the same system in the limit of fast diffusion: Indeed, this has been tacitly assumed in most literature concerning the RDME...
May 2016: Physical Review. E
Tyler M Earnest, John A Cole, Joseph R Peterson, Michael J Hallock, Thomas E Kuhlman, Zaida Luthey-Schulten
Ribosomes-the primary macromolecular machines responsible for translating the genetic code into proteins-are complexes of precisely folded RNA and proteins. The ways in which their production and assembly are managed by the living cell is of deep biological importance. Here we extend a recent spatially resolved whole-cell model of ribosome biogenesis in a fixed volume [Earnest et al., Biophys J 2015, 109, 1117-1135] to include the effects of growth, DNA replication, and cell division. All biological processes are described in terms of reaction-diffusion master equations and solved stochastically using the Lattice Microbes simulation software...
October 2016: Biopolymers
Junwei Lucas Bao, Xin Zhang, Donald G Truhlar
Understanding the falloff in rate constants of gas-phase unimolecular reaction rate constants as the pressure is lowered is a fundamental problem in chemical kinetics, with practical importance for combustion, atmospheric chemistry, and essentially all gas-phase reaction mechanisms. In the present work, we use our recently developed system-specific quantum RRK theory, calibrated by canonical variational transition state theory with small-curvature tunneling, combined with the Lindemann-Hinshelwood mechanism, to model the dissociation reaction of fluoroform (CHF3), which provides a definitive test for falloff modeling...
June 22, 2016: Physical Chemistry Chemical Physics: PCCP
Margaritis Voliotis, Philipp Thomas, Ramon Grima, Clive G Bowsher
Simulation of biomolecular networks is now indispensable for studying biological systems, from small reaction networks to large ensembles of cells. Here we present a novel approach for stochastic simulation of networks embedded in the dynamic environment of the cell and its surroundings. We thus sample trajectories of the stochastic process described by the chemical master equation with time-varying propensities. A comparative analysis shows that existing approaches can either fail dramatically, or else can impose impractical computational burdens due to numerical integration of reaction propensities, especially when cell ensembles are studied...
June 2016: PLoS Computational Biology
Jiajun Zhang, Qing Nie, Tianshou Zhou
Traditional moment-closure methods need to assume that high-order cumulants of a probability distribution approximate to zero. However, this strong assumption is not satisfied for many biochemical reaction networks. Here, we introduce convergent moments (defined in mathematics as the coefficients in the Taylor expansion of the probability-generating function at some point) to overcome this drawback of the moment-closure methods. As such, we develop a new analysis method for stochastic chemical kinetics. This method provides an accurate approximation for the master probability equation (MPE)...
May 21, 2016: Journal of Chemical Physics
Chantal Sleiman, Sergio González, Stephen J Klippenstein, Dahbia Talbi, Gisèle El Dib, André Canosa
The gas phase reaction between the CN radical and acetonitrile CH3CN was investigated experimentally, at low temperatures, with the CRESU apparatus and a slow flow reactor to explore the temperature dependence of its rate coefficient from 354 K down to 23 K. Whereas a standard Arrhenius behavior was found at T > 200 K, indicating the presence of an activation barrier, a dramatic increase in the rate coefficient by a factor of 130 was observed when the temperature was decreased from 168 to 123 K. The reaction was found to be pressure independent at 297 K unlike the experiments carried out at 52 and 132 K...
June 1, 2016: Physical Chemistry Chemical Physics: PCCP
Aaron Kelly, Andrés Montoya-Castillo, Lu Wang, Thomas E Markland
Generalized quantum master equations (GQMEs) are an important tool in modeling chemical and physical processes. For a large number of problems, it has been shown that exact and approximate quantum dynamics methods can be made dramatically more efficient, and in the latter case more accurate, by proceeding via the GQME formalism. However, there are many situations where utilizing the GQME approach with an approximate method has been observed to return the same dynamics as using that method directly. Here, for systems both in and out of equilibrium, we provide a more detailed understanding of the conditions under which using an approximate method can yield benefits when combined with the GQME formalism...
May 14, 2016: Journal of Chemical Physics
Khanh N Dinh, Roger B Sidje
The finite state projection (FSP) method has enabled us to solve the chemical master equation of some biological models that were considered out of reach not long ago. Since the original FSP method, much effort has gone into transforming it into an adaptive time-stepping algorithm as well as studying its accuracy. Some of the improvements include the multiple time interval FSP, the sliding windows, and most notably the Krylov-FSP approach. Our goal in this tutorial is to give the reader an overview of the current methods that build on the FSP...
2016: Physical Biology
Fuke Wu, Tianhai Tian, James B Rawlings, George Yin
The frequently used reduction technique is based on the chemical master equation for stochastic chemical kinetics with two-time scales, which yields the modified stochastic simulation algorithm (SSA). For the chemical reaction processes involving a large number of molecular species and reactions, the collection of slow reactions may still include a large number of molecular species and reactions. Consequently, the SSA is still computationally expensive. Because the chemical Langevin equations (CLEs) can effectively work for a large number of molecular species and reactions, this paper develops a reduction method based on the CLE by the stochastic averaging principle developed in the work of Khasminskii and Yin [SIAM J...
May 7, 2016: Journal of Chemical Physics
Huy D Vo, Roger B Sidje
A stochastic model of cellular p53 regulation was established in Leenders, and Tuszynski (2013 Front. Oncol. 3 1-16) to study the interactions of p53 with MDM2 proteins, where the stochastic analysis was done using a Monte Carlo approach. We revisit that model here using an alternative scheme, which is to directly solve the chemical master equation (CME) by an adaptive Krylov-based finite state projection method that combines the stochastic simulation algorithm with other computational strategies, namely Krylov approximation techniques to the matrix exponential, divide and conquer, and aggregation...
2016: Physical Biology
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