Communication: limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks

It is commonly believed that, whenever timescale separation holds, the predictions of reduced chemical master equations obtained using the stochastic quasi-steady-state approximation are in very good agreement with the predictions of the full master equations. We use the linear noise approximation to obtain a simple formula for the relative error between the predictions of the two master equations for the Michaelis-Menten reaction with substrate input. The reduced approach is predicted to overestimate the variance of the substrate concentration fluctuations by as much as 30%. The theoretical results are validated by stochastic simulations using experimental parameter values for enzymes involved in proteolysis, gluconeogenesis, and fermentation.     

A data-integrated method for analyzing stochastic biochemical networks

Variability and fluctuations among genetically identical cells under uniform experimental conditions stem from the stochastic nature of biochemical reactions. Understanding network function for endogenous biological systems or designing robust synthetic genetic circuits requires accounting for and analyzing this variability. Stochasticity in biological networks is usually represented using a continuous-time discrete-state Markov formalism, where the chemical master equation (CME) and its kinetic Monte Carlo equivalent, the stochastic simulation algorithm (SSA), are used. These two representations are computationally intractable for many realistic biological problems. Fitting parameters in the context of these stochastic models is particularly challenging and has not been accomplished for any but very simple systems. In this work, we propose that moment equations derived from the CME, when treated appropriately in terms of higher order moment contributions, represent a computationally efficient framework for estimating the kinetic rate constants of stochastic network models and subsequent analysis of their dynamics. To do so, we present a practical data-derived moment closure method for these equations. In contrast to previous work, this method does not rely on any assumptions about the shape of the stochastic distributions or a functional relationship among their moments. We use this method to analyze a stochastic model of a biological oscillator and demonstrate its accuracy through excellent agreement with CME/SSA calculations. By coupling this moment-closure method with a parameter search procedure, we further demonstrate how a model's kinetic parameters can be iteratively determined in order to fit measured distribution data.     

Bistable stochastic biochemical networks: large chemical networks and systems with many molecules

In this paper we continue the program started in Hwang and Velázquez (J Math Chem, to appear). We describe some chemical systems exhibiting bistable behavior with reaction constants of order one, but where bistability is due to the presence of a large number of chemical species or a large number of molecules of some of the species. We derive generalizations of the classical Kramers’ formula that gives the switching times for some particular systems exhibiting a large number of species.     

Grand canonical monte carlo modeling of hydrogen adsorption on phosphorus-doped open carbon framework

Mechanism of hydrogen adsorption in high surface area carbon-based porous materials has been studied. Influence of chemical modification of the adsorbing surface has been simulated using grand canonical Monte Carlo method. Special attention has been paid to the competition between increasing the surface of open carbon frameworks and heterogeneous distribution of the energy of adsorption. Additionally, it has been shown that the molecular mass of atoms which substitute carbon atoms is an important factor determining the final hydrogen uptake.     

Discrete-time stochastic modeling and simulation of biochemical networks

Since inherent randomness in chemically reacting systems is evident, stochastic modeling and simulation are exceedingly important for investigating complex biological networks. Within the most common stochastic approach a network is modeled by a continuous-time Markov chain governed by the chemical master equation. We show how the continuous-time Markov chain can be converted to a stochastically identical discrete-time Markov chain and obtain a discrete-time version of the chemical master equation. Simulating the discrete-time Markov chain is equivalent to the Gillespie algorithm but requires less effort in that it eliminates the generation of exponential random variables. Thus, exactness as possessed by the Gillespie algorithm is preserved while the simulation can be performed more efficiently.     

Eliminating fast reactions in stochastic simulations of biochemical networks: a bistable genetic switch

In many stochastic simulations of biochemical reaction networks, it is desirable to "coarse grain" the reaction set, removing fast reactions while retaining the correct system dynamics. Various coarse-graining methods have been proposed, but it remains unclear which methods are reliable and which reactions can safely be eliminated. We address these issues for a model gene regulatory network that is particularly sensitive to dynamical fluctuations: a bistable genetic switch. We remove protein-DNA and/or protein-protein association-dissociation reactions from the reaction set using various coarse-graining strategies. We determine the effects on the steady-state probability distribution function and on the rate of fluctuation-driven switch flipping transitions. We find that protein-protein interactions may be safely eliminated from the reaction set, but protein-DNA interactions may not. We also find that it is important to use the chemical master equation rather than macroscopic rate equations to compute effective propensity functions for the coarse-grained reactions.     

Bistable stochastic biochemical networks: highly specific systems with few chemicals

In this paper we describe a class of stochastic biochemical systems exhibiting bistable behavior, in the sense that the invariant measure associated to the system is concentrated near two different classes of states of the system. We develop methods that allow us to describe the behavior of the invariant measure in some suitable asymptotic limits, as well as the switching times for the transitions between the states close to each of the states with high probability. Due to the discrete character of the problem, switching times cannot be computed using the classical Kramers’ formula, and alternative methods are required.     

Deficiency zero for random reaction networks under a stochastic block model framework

Journal of Mathematical Chemistry - Deficiency zero is an important network structure and has been the focus of many celebrated results within reaction network theory. In our previous paper...     

Grand canonical ensemble approach to electrochemical thermodynamics,kinetics, and model Hamiltonians

The unique feature of electrochemistry is the ability to control reaction thermodynamics and kinetics by the application of electrode potential. Recently, theoretical methods and computational approaches within the grand canonical ensemble (GCE) have enabled to explicitly include and control the electrode potential in first principles calculations. In this review, recent advances and future promises of GCE density functional theory and rate theory are discussed. Particular focus is devoted to considering how the GCE methods either by themselves or combined with model Hamiltonians can be used to address intricate phenomena such as solvent/electrolyte effects and nuclear quantum effects to provide a detailed understanding of electrochemical reactions and interfaces.     

On a new self-consistent-field theory for the canonical ensemble

A significant amount of many-body problems of quantum or classical equilibrium statistical mechanics are conveniently treated at fixed temperature and system size. In this paper, we present a new functional integral approach for solving canonical ensemble problems over the entire coupling range, relying on the method of Gaussian equivalent representation of Efimov and Ganbold. We demonstrate its suitability and competitiveness for performing approximate calculations of thermodynamic and structural quantities on the example of a repulsive potential model, widely used in soft matter theory.     

Grand canonical Monte Carlo simulation of a model colloid-polymer mixture: coexistence line, critical behavior, and interfacial tension

Grand canonical Monte Carlo simulations are used to study phase separation in a simple colloid-polymer model, the so-called Asakura-Oosawa model. To overcome the problem of small acceptance rates of the grand-canonical moves, cluster moves are introduced. Successive umbrella sampling, recently introduced by Virnau and Muller [J. Chem. Phys. 120, 10925 (2004)], is used to access the phase-separated regime. The unmixing binodal and the interfacial tension are measured and compared to theoretical predictions. By means of finite-size scaling, the behavior close to the critical point is also investigated. Close to criticality, we observe substantial deviations from mean-field behavior.     

The dynamics of the stochastic multi-molecule biochemical reaction model

The paper introduces the dynamics of a stochastic multi-molecule biochemical reaction model.First, we show that there is a unique positive solution of the stochastic model. Furthermore, we deduce the conditions when the reaction will end and when the reaction being proceed. At last, we derive that the solution of (1.5) oscillates around the endemic proportion equilibrium $P^*(x^*,y^*)$ , and the intensity of fluctuation is proportional to white noise. The key to the analysis in this paper is choosing appropriate Lyapunov function. The outcomes are illustrated by computer simulations throughout this paper.     

Quantum stochastic theory of exchange dephasing model for a vibrating molecule in dense medium

The exchange dephasing model is investigated theoretically. An equation of motion satisfied by a reduced density matrix for the observed high-frequency mode is derived by a cumulant perturbation expansion method up to fourth order. The calculation shows a temperature dependence of the width and shift of the spectral line.     

A stochastic theory of collision phenomena: one-dimensional atom-diatom collision model

A stochastic model of one-dimensional atom—diatom collisions is explicitly solved and studied in this paper. By treating the translational motion as an irrelevant variable, a second order partial differential equation is obtained for which the energy conservation is strictly satisfied. This equation is shown to be solvable in terms of Mathieu functions. It is shown that when certain further approximations are made in the equation, it leads to the equations studied by other authors in the stochastic model of collision phenomena. Various approximate but analytic distribution functions are obtained for the distribution of the observables and some aspects of the model are discussed in terms of the solutions obtained.     

An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks

Stochastic chemical kinetics more accurately describes the dynamics of "small" chemical systems, such as biological cells. Many real systems contain dynamical stiffness, which causes the exact stochastic simulation algorithm or other kinetic Monte Carlo methods to spend the majority of their time executing frequently occurring reaction events. Previous methods have successfully applied a type of probabilistic steady-state approximation by deriving an evolution equation, such as the chemical master equation, for the relaxed fast dynamics and using the solution of that equation to determine the slow dynamics. However, because the solution of the chemical master equation is limited to small, carefully selected, or linear reaction networks, an alternate equation-free method would be highly useful. We present a probabilistic steady-state approximation that separates the time scales of an arbitrary reaction network, detects the convergence of a marginal distribution to a quasi-steady-state, directly samples the underlying distribution, and uses those samples to accurately predict the state of the system, including the effects of the slow dynamics, at future times. The numerical method produces an accurate solution of both the fast and slow reaction dynamics while, for stiff systems, reducing the computational time by orders of magnitude. The developed theory makes no approximations on the shape or form of the underlying steady-state distribution and only assumes that it is ergodic. We demonstrate the accuracy and efficiency of the method using multiple interesting examples, including a highly nonlinear protein-protein interaction network. The developed theory may be applied to any type of kinetic Monte Carlo simulation to more efficiently simulate dynamically stiff systems, including existing exact, approximate, or hybrid stochastic simulation techniques.     

Grand canonical Monte Carlo simulation of the adsorption isotherms of water molecules on model soot particles

The grand canonical Monte Carlo method is used to simulate the adsorption isotherms of water molecules on different types of model soot particles. The soot particles are modeled by graphite-type layers arranged in an onionlike structure that contains randomly distributed hydrophilic sites, such as OH and COOH groups. The calculated water adsorption isotherm at 298 K exhibits different characteristic shapes depending both on the type and the location of the hydrophilic sites and also on the size of the pores inside the soot particle. The different shapes of the adsorption isotherms result from different ways of water aggregation in or/and around the soot particle. The present results show the very weak influence of the OH sites on the water adsorption process when compared to the COOH sites. The results of these simulations can help in interpreting the experimental isotherms of water adsorbed on aircraft soot.     

Continuous medium theory for nonequilibrium solvation: I. How to correctly evaluate solvation free energy of nonequilibrium

Considering the influences of electrostatic potential Phi upon the change of solute charge distribution deltarho and rho upon the change deltaPhi at the same time, a more reasonable integral formula of dG = (1/2) integral (V) (rhodeltaPhi + Phideltarho)dV is used to calculate the change of the electrostatic free energy in charging the solute-solvent system to a nonequilibrium state, instead of the one of dG = integral (V) PhideltarhodV used before. This modification improves the expressions of electrostatic free energy and solvation free energy, in which no quantity of the intermediate equilibrium state is explicitly involved. Detailed investigation reveals that the solvation free energy of nonequilibrium only contains the interaction energy between the field due to the solute charge in vacuum, and the dielectric polarization at the nonequilibrium state. The solvent reorganization energies of forward and backward electron transfer reactions have been redefined because the derivations lead to a remarkable feature that these quantities are direction-dependent, unlike the theoretical models developed before. The deductions are given in the electric field-displacement form. Relevant discussions on the reliability of theoretical models suggested in this work have also been presented.     

A three-phase nonequilibrium model for catalytic distillation

Nonequilibrium model for steady state simulation of catalytic distillation is presented. Mathematical model takes into account both mass and heat transfers across the gas liquid interface and through the liquid-solid (catalyst) interface. Equations describing the mentioned phenomena are based on the effective diffusivity approach. The resulting system of nonlinear algebraic equations was implemented in the FORTRAN programming language and solved by the BUNLSI (Ferraris & Tronconi, 1986) solver. The described model was verified using the experimental data obtained from a continuous distillation column equipped with catalytic packing. As an experimental model system, synthesis of propyl propionate from propan-1-ol and propionic acid was chosen. Comparison of experimental and simulation data is presented, and appropriateness of the developed model for other types of catalytic distillation processes is discussed. Presented at the 35th International Conference of the Slovak Society of Chemical Engineering, Tatranské Matliare, 26–30 May 2008.     

Communication: Decoherence in a nonequilibrium environment: an analytically solvable model

We describe an analytically solvable model of quantum decoherence in a nonequilibrium environment. The model considers the effect of a bath driven from equilibrium by, for example, an ultrafast excitation of a quantum chromophore. The nonequilibrium response of the environment is represented by a nonstationary random function corresponding to the fluctuating transition frequency between two quantum states coupled to the surroundings. The nonstationary random function is characterized by a Fourier series with the phase of each term starting initially with a definite value across the ensemble but undergoing random diffusion with time. The decay of the off-diagonal density matrix element is shown to depend significantly on the particular pattern of initial phases of the terms in the Fourier series, or equivalently, the initial phases of bath modes coupled to the quantum subsystem. This suggests the possibility of control of quantum decoherence by the detailed properties of an environment that is driven from thermal equilibrium.