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Chemical master equation

Garrett R Dowdy, Paul I Barton
The method of moments has been proposed as a potential means to reduce the dimensionality of the chemical master equation (CME) appearing in stochastic chemical kinetics. However, attempts to apply the method of moments to the CME usually result in the so-called closure problem. Several authors have proposed moment closure schemes, which allow them to obtain approximations of quantities of interest, such as the mean molecular count for each species. However, these approximations have the dissatisfying feature that they come with no error bounds...
February 28, 2018: Journal of Chemical Physics
Daniil A Andrienko, Iain D Boyd
Simulation of vibrational energy transfer and dissociation in O2 -N2 collisions is conducted using the quasi-classical trajectory method on an ab initio potential energy surface. Vibrationally resolved rate coefficients are obtained in a high-temperature region between 8000 and 20 000 K by means of the cost-efficient classical trajectory propagation method. A system of master equations is constructed using the new dataset in order to simulate thermal and chemical nonequilibrium observed in shock flows. The O2 relaxation time derived from a solution of the master equations is in good agreement with the Millikan and White correlation at lower temperatures with an increasing discrepancy toward the translational temperature of 20 000 K...
February 28, 2018: Journal of Chemical Physics
Adil A Gangat, Ian P McCulloch, Ying-Jer Kao
The driven-dissipative Bose-Hubbard model can be experimentally realized with either negative or positive onsite detunings, inter-site hopping energies, and onsite interaction energies. Here we use one-dimensional matrix product density operators to perform a fully quantum investigation of the dependence of the non-equilibrium steady states of this model on the signs of these parameters. Due to a symmetry in the Lindblad master equation, we find that simultaneously changing the sign of the interaction energies, hopping energies, and chemical potentials leaves the local boson number distribution and inter-site number correlations invariant, and the steady-state complex conjugated...
February 27, 2018: Scientific Reports
Guan-Rong Huang, David B Saakian, Chin-Kun Hu
Studying gene regulation networks in a single cell is an important, interesting, and hot research topic of molecular biology. Such process can be described by chemical master equations (CMEs). We propose a Hamilton-Jacobi equation method with finite-size corrections to solve such CMEs accurately at the intermediate region of switching, where switching rate is comparable to fast protein production rate. We applied this approach to a model of self-regulating proteins [H. Ge et al., Phys. Rev. Lett. 114, 078101 (2015)PRLTAO0031-900710...
January 2018: Physical Review. E
Divya Singh, Srabanti Chaudhury
In this work, we study the catalytic activity of a single enzyme in the presence of multiple substrates. We broadly discuss three different mechanisms of bisubstrate binding, namely ordered sequential, random sequential and ping-pong non-sequential pathway. Using the chemical master equation approach we obtain exact expressions for the waiting time distributions the mean turnover time and the randomness parameter as a function of the substrate concentration, such that both concentrations are fixed but one of them is changed quasistatically...
February 2, 2018: Chembiochem: a European Journal of Chemical Biology
Yen Ting Lin, Nicolas E Buchler
Single-cell experiments show that gene expression is stochastic and bursty, a feature that can emerge from slow switching between promoter states with different activities. In addition to slow chromatin and/or DNA looping dynamics, one source of long-lived promoter states is the slow binding and unbinding kinetics of transcription factors to promoters, i.e. the non-adiabatic binding regime. Here, we introduce a simple analytical framework, known as a piecewise deterministic Markov process (PDMP), that accurately describes the stochastic dynamics of gene expression in the non-adiabatic regime...
January 2018: Journal of the Royal Society, Interface
Matthias Reis, Justus A Kromer, Edda Klipp
Multimodality is a phenomenon which complicates the analysis of statistical data based exclusively on mean and variance. Here, we present criteria for multimodality in hierarchic first-order reaction networks, consisting of catalytic and splitting reactions. Those networks are characterized by independent and dependent subnetworks. First, we prove the general solvability of the Chemical Master Equation (CME) for this type of reaction network and thereby extend the class of solvable CME's. Our general solution is analytical in the sense that it allows for a detailed analysis of its statistical properties...
January 20, 2018: Journal of Mathematical Biology
Chen Jia, Michael Q Zhang, Hong Qian
Single-cell gene expression is inherently stochastic; its emergent behavior can be defined in terms of the chemical master equation describing the evolution of the mRNA and protein copy numbers as the latter tends to infinity. We establish two types of "macroscopic limits": the Kurtz limit is consistent with the classical chemical kinetics, while the Lévy limit provides a theoretical foundation for an empirical equation proposed in N. Friedman et al., Phys. Rev. Lett. 97, 168302 (2006)PRLTAO0031-900710.1103/PhysRevLett...
October 2017: Physical Review. E
Robin Shannon, David R Glowacki
The chemical master equation is a powerful theoretical tool for analysing the kinetics of complex multi-well potential energy surfaces in a wide range of different domains of chemical kinetics spanning combustion, atmospheric chemistry, gas-surface chemistry, solution phase chemistry, and biochemistry. There are two well-established methodologies for solving the chemical master equation: a stochastic "kinetic Monte Carlo" approach and a matrix-based approach. In principle, the results yielded by both approaches are identical; the decision of which approach is better suited to a particular study depends on the details of the specific system under investigation...
January 12, 2018: Journal of Physical Chemistry. A
Leo Bronstein, Heinz Koeppl
Approximate solutions of the chemical master equation and the chemical Fokker-Planck equation are an important tool in the analysis of biomolecular reaction networks. Previous studies have highlighted a number of problems with the moment-closure approach used to obtain such approximations, calling it an ad hoc method. In this article, we give a new variational derivation of moment-closure equations which provides us with an intuitive understanding of their properties and failure modes and allows us to correct some of these problems...
January 7, 2018: Journal of Chemical Physics
Tomás Aquino, Marco Dentz
Kinetic Monte Carlo methods such as the Gillespie algorithm model chemical reactions as random walks in particle number space. The interreaction times are exponentially distributed under the assumption that the system is well mixed. We introduce an arbitrary interreaction time distribution, which may account for the impact of incomplete mixing on chemical reactions, and in general stochastic reaction delay, which may represent the impact of extrinsic noise. This process defines an inhomogeneous continuous time random walk in particle number space, from which we derive a generalized chemical master equation...
December 8, 2017: Physical Review Letters
Andrius Gelzinis, Edvardas Rybakovas, Leonas Valkunas
Accurate simulations of open quantum system dynamics is a long standing issue in the field of chemical physics. Exact methods exist, but are costly, while perturbative methods are limited in their applicability. Recently a new black-box type method, called transfer tensor method (TTM), was proposed [J. Cerrillo and J. Cao, Phys. Rev. Lett. 112, 110401 (2014)]. It allows one to accurately simulate long time dynamics with a numerical cost of solving a time-convolution master equation, provided many initial system evolution trajectories are obtained from some exact method beforehand...
December 21, 2017: Journal of Chemical Physics
Stefan Hellander, Andreas Hellander, Linda Petzold
The reaction-diffusion master equation (RDME) is a model that allows for efficient on-lattice simulation of spatially resolved stochastic chemical kinetics. Compared to off-lattice hard-sphere simulations with Brownian dynamics or Green's function reaction dynamics, the RDME can be orders of magnitude faster if the lattice spacing can be chosen coarse enough. However, strongly diffusion-controlled reactions mandate a very fine mesh resolution for acceptable accuracy. It is common that reactions in the same model differ in their degree of diffusion control and therefore require different degrees of mesh resolution...
December 21, 2017: Journal of Chemical Physics
Frits Veerman, Carsten Marr, Nikola Popović
The inherent stochasticity of gene expression in the context of regulatory networks profoundly influences the dynamics of the involved species. Mathematically speaking, the propagators which describe the evolution of such networks in time are typically defined as solutions of the corresponding chemical master equation (CME). However, it is not possible in general to obtain exact solutions to the CME in closed form, which is due largely to its high dimensionality. In the present article, we propose an analytical method for the efficient approximation of these propagators...
December 15, 2017: Journal of Mathematical Biology
Minh V Duong, Hieu T Nguyen, Tam V-T Mai, Lam K Huynh
Master equation/Rice-Ramsperger-Kassel-Marcus (ME/RRKM) has shown to be a powerful framework for modeling kinetic and dynamic behaviors of a complex gas-phase chemical system on a complicated multiple-species and multiple-channel potential energy surface (PES) for a wide range of temperatures and pressures. Derived from the ME time-resolved species profiles, the macroscopic or phenomenological rate coefficients are essential for many reaction engineering applications including those in combustion and atmospheric chemistry...
December 15, 2017: Physical Chemistry Chemical Physics: PCCP
Xinyue Zhang, Dong Zhang, Chenhan Zhao, Kai Tian, Ruicheng Shi, Xiao Du, Andrew J Burcke, Jing Wang, Shi-Jie Chen, Li-Qun Gu
The chemical properties and biological mechanisms of RNAs are determined by their tertiary structures. Exploring the tertiary structure folding processes of RNA enables us to understand and control its biological functions. Here, we report a nanopore snapshot approach combined with coarse-grained molecular dynamics simulation and master equation analysis to elucidate the folding of an RNA pseudoknot structure. In this approach, single RNA molecules captured by the nanopore can freely fold from the unstructured state without constraint and can be programmed to terminate their folding process at different intermediates...
November 13, 2017: Nature Communications
Khanh N Dinh, Roger B Sidje
Monte Carlo methods such as the stochastic simulation algorithm (SSA) have traditionally been employed in gene regulation problems. However, there has been increasing interest to directly obtain the probability distribution of the molecules involved by solving the chemical master equation (CME). This requires addressing the curse of dimensionality that is inherent in most gene regulation problems. The finite state projection (FSP) seeks to address the challenge and there have been variants that further reduce the size of the projection or that accelerate the resulting matrix exponential...
November 3, 2017: Physical Biology
Michael P Burke, Stephen J Klippenstein
Termolecular association reactions involve ephemeral collision complexes-formed from the collision of two molecules-that collide with a third and chemically inert 'bath gas' molecule that simply transfers energy to/from the complex. These collision complexes are generally not thought to react chemically on collision with a third molecule in the gas-phase systems of combustion and planetary atmospheres. Such 'chemically termolecular' reactions, in which all three molecules are involved in bond making and/or breaking, were hypothesized long ago in studies establishing radical chain branching mechanisms, but were later concluded to be unimportant...
November 2017: Nature Chemistry
L Vereecken
The reaction of CH2OO, the smallest carbonyl oxide (Criegee intermediate, CI), with several acids was investigated using the CCSD(T)/aug-cc-pVTZ//M06-2X/aug-cc-pVTZ quantum chemical method, as well as microvariational transition state theory and RRKM master equation theoretical kinetic methodologies. For oxoacids HNO3 and HCOOH, a 1,4-insertion mechanism allows for barrierless reactions with high rate coefficients, in agreement with literature experimental data. This mechanism relies on the presence of a double bond in the α-position to the acidic OH group...
November 1, 2017: Physical Chemistry Chemical Physics: PCCP
Ankit Gupta, Jan Mikelson, Mustafa Khammash
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...
October 21, 2017: Journal of Chemical Physics
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