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Chemical master equation

Stephen Smith, Ramon Grima
Models of chemical kinetics that incorporate both stochasticity and diffusion are an increasingly common tool for studying biology. The variety of competing models is vast, but two stand out by virtue of their popularity: the reaction-diffusion master equation and Brownian dynamics. In this review, we critically address a number of open questions surrounding these models: How can they be justified physically? How do they relate to each other? How do they fit into the wider landscape of chemical models, ranging from the rate equations to molecular dynamics? This review assumes no prior knowledge of modelling chemical kinetics and should be accessible to a wide range of readers...
May 21, 2018: Bulletin of Mathematical Biology
Andreas Petrides, Glenn Vinnicombe
This paper is concerned with the potential multistability of protein concentrations in the cell. That is, situations where one, or a family of, proteins may sit at one of two or more different steady state concentrations in otherwise identical cells, and in spite of them being in the same environment. For models of multisite protein phosphorylation for example, in the presence of excess substrate, it has been shown that the achievable number of stable steady states can increase linearly with the number of phosphosites available...
May 17, 2018: PLoS Computational Biology
Mark Pfeifle, Yong-Tao Ma, Ahren W Jasper, Lawrence B Harding, William L Hase, Stephen J Klippenstein
Ozonolysis produces chemically activated carbonyl oxides (Criegee intermediates, CIs) that are either stabilized or decompose directly. This branching has an important impact on atmospheric chemistry. Prior theoretical studies have employed statistical models for energy partitioning to the CI arising from dissociation of the initially formed primary ozonide (POZ). Here, we used direct dynamics simulations to explore this partitioning for decomposition of c-C2 H4 O3 , the POZ in ethylene ozonolysis. A priori estimates for the overall stabilization probability were then obtained by coupling the direct dynamics results with master equation simulations...
May 7, 2018: Journal of Chemical Physics
Thanh Lam Nguyen, James H Thorpe, David H Bross, Branko Ruscic, John F Stanton
A combination of high-level coupled-cluster calculations and two-dimensional master equation approaches based on semi-classical transition state theory is used to reinvestigate the classic prototype unimolecular isomerization of methyl isocyanide (CH3NC) to acetonitrile (CH3CN). The activation energy, reaction enthalpy, and fundamental vibrational frequencies calculated from first principles agree well with experimental results. In addition, the calculated thermal rate constants adequately reproduce those of experiment over a large range of temperature and pressure in the fall-off region, where experimental results are available, and are generally consistent with statistical chemical kinetics theory (such as Rice-Ramsperger-Kassel-Marcus (RRKM) and transition state theory (TST))...
April 26, 2018: Journal of Physical Chemistry Letters
Per Lötstedt
An algorithm for computing the linear noise approximation (LNA) of the reaction-diffusion master equation (RDME) is developed and tested. The RDME is often used as a model for biochemical reaction networks. The LNA is derived for a general discretization of the spatial domain of the problem. If M is the number of chemical species in the network and N is the number of nodes in the discretization in space, then the computational work to determine approximations of the mean and the covariances of the probability distributions is proportional to [Formula: see text] in a straightforward implementation...
April 11, 2018: Bulletin of Mathematical Biology
Lisa Weber, William Raymond, Brian Munsky
In quantitative analyses of biological processes, one may use many different scales of models (e.g., spatial or non-spatial, deterministic or stochastic, time-varying or at steady-state) or many different approaches to match models to experimental data (e.g., model fitting or parameter uncertainty/sloppiness quantification with different experiment designs). These different analyses can lead to surprisingly different results, even when applied to the same data and the same model. We use a simplified gene regulation model to illustrate many of these concerns, especially for ODE analyses of deterministic processes, chemical master equation and finite state projection analyses of heterogeneous processes, and stochastic simulations...
April 6, 2018: Physical Biology
Marsha I Lester, Stephen J Klippenstein
Alkene ozonolysis is a primary oxidation pathway for anthropogenic and biogenic alkenes emitted into the troposphere. It is also an important source of atmospheric hydroxyl (OH) radicals, often called the atmosphere's detergent. Alkene ozonolysis takes place through a highly exothermic reaction pathway with multiple intermediates and barriers prior to releasing the OH radical products. This Account focuses on a key reaction intermediate with a carbonyl oxide functional group (-COO), known as the Criegee intermediate, which is formed along with a carbonyl coproduct in alkene ozonolysis reactions...
April 3, 2018: Accounts of Chemical Research
Nima Ghaderi
The analytic solutions, that is, populations, are derived for the K-adiabatic and K-active bimolecular master equations, separately, for a single and multiple potential wells and reaction channels, where K is the component of the total angular momentum J along the axis of least moment of inertia of the recombination products at a given energy E. The analytic approach provides the functional dependence of the population of molecules on its K-active or K-adiabatic dissociation, association rate constants and the intermolecular energy transfer, where the approach may complement the usual numerical approaches for reactions of interest...
March 27, 2018: Journal of Physical Chemistry. A
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
Herein, the catalytic activity of a single enzyme in the presence of multiple substrates is studied. Three different mechanisms of bisubstrate binding, namely, ordered sequential, random sequential and ping-pong nonsequential pathway, are broadly discussed. By means of the chemical master equation approach, 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 quasi-statically are obtained...
April 16, 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...
October 2017: Physical Review. E
Robin Shannon, David R Glowacki
The chemical master equation is a powerful theoretical tool for analyzing the kinetics of complex multiwell 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...
February 15, 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
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