Automatic identification of model reductions for discrete stochastic simulation

Multiple time scales in cellular chemical reaction systems present a challenge for the efficiency of stochastic simulation. Numerous model reductions have been proposed to accelerate the simulation of chemically reacting systems by exploiting time scale separation. However, these are often identified and deployed manually, requiring expert knowledge. This is time-consuming, prone to error, and opportunities for model reduction may be missed, particularly for large models. We propose an automatic model analysis algorithm using an adaptively weighted Petri net to dynamically identify opportunities for model reductions for both the stochastic simulation algorithm and tau-leaping simulation, with no requirement of expert knowledge input. Results are presented to demonstrate the utility and effectiveness of this approach.     

The stochastic quasi-steady-state assumption: reducing the model but not the noise

Highly reactive species at small copy numbers play an important role in many biological reaction networks. We have described previously how these species can be removed from reaction networks using stochastic quasi-steady-state singular perturbation analysis (sQSPA). In this paper we apply sQSPA to three published biological models: the pap operon regulation, a biochemical oscillator, and an intracellular viral infection. These examples demonstrate three different potential benefits of sQSPA. First, rare state probabilities can be accurately estimated from simulation. Second, the method typically results in fewer and better scaled parameters that can be more readily estimated from experiments. Finally, the simulation time can be significantly reduced without sacrificing the accuracy of the solution.     

Single-substrate enzyme kinetics: the quasi-steady-state approximation and beyond

We analyze the standard model of enzyme-catalyzed reactions at various substrate-enzyme ratios by adopting a different scaling scheme and computational procedure. The regions of validity of the quasi-steady-state approximation are noted. Certain prevalent conditions are checked and compared against the actual findings. Efficacies of a few other measures, obtained from the present work, are highlighted. Some recent observations are rationalized, particularly at moderate and high enzyme concentrations.     

Aluminosilicate dissolution kinetics: a general stochastic model

We apply a kinetic model developed for understanding the behavior of crystal dissolution to aluminosilicate dissolution kinetics. Without making any assumptions about specific dissolution mechanisms, the model is a vigorous stochastic exploration of all of the elementary reactions and basic processes involved in dissolution: bond breakage, bond formation, surface diffusion, and departure and arrival of Si- and Al- units. In the stochastic model, the interdependence of these elementary reactions and basic processes is strictly determined by the complicated three-dimensional surface structure in which interconnected Si- and Al- atoms share oxygen atoms. The modeling results are consistent with experimental data in various aspects, such as saturation state dependence of the dissolution rate, aluminum inhibition effects, surface chemistry evolution, anisotropic dissolution, and alteration product. The stochastic model integrates all microscopic information at the atomic scale and elucidates the reasons for the observed kinetic results in experimental studies, improving our fundamental understanding of aluminosilicate dissolution.     

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.     

Identifiability classes in reaction kinetics

Some difficulties of eliciting reliable rate constants from experimental data are considered. Different classes of ill-posed estimation problems are given and demonstrated by simple examples.

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Binomial leap methods for simulating stochastic chemical kinetics

This paper discusses efficient simulation methods for stochastic chemical kinetics. Based on the tau-leap and midpoint tau-leap methods of Gillespie [D. T. Gillespie, J. Chem. Phys. 115, 1716 (2001)], binomial random variables are used in these leap methods rather than Poisson random variables. The motivation for this approach is to improve the efficiency of the Poisson leap methods by using larger stepsizes. Unlike Poisson random variables whose range of sample values is from zero to infinity, binomial random variables have a finite range of sample values. This probabilistic property has been used to restrict possible reaction numbers and to avoid negative molecular numbers in stochastic simulations when larger stepsize is used. In this approach a binomial random variable is defined for a single reaction channel in order to keep the reaction number of this channel below the numbers of molecules that undergo this reaction channel. A sampling technique is also designed for the total reaction number of a reactant species that undergoes two or more reaction channels. Samples for the total reaction number are not greater than the molecular number of this species. In addition, probability properties of the binomial random variables provide stepsize conditions for restricting reaction numbers in a chosen time interval. These stepsize conditions are important properties of robust leap control strategies. Numerical results indicate that the proposed binomial leap methods can be applied to a wide range of chemical reaction systems with very good accuracy and significant improvement on efficiency over existing approaches.     

On the origins of approximations for stochastic chemical kinetics

This paper considers the derivation of approximations for stochastic chemical kinetics governed by the discrete master equation. Here, the concepts of (1) partitioning on the basis of fast and slow reactions as opposed to fast and slow species and (2) conditional probability densities are used to derive approximate, partitioned master equations, which are Markovian in nature, from the original master equation. Under different conditions dictated by relaxation time arguments, such approximations give rise to both the equilibrium and hybrid (deterministic or Langevin equations coupled with discrete stochastic simulation) approximations previously reported. In addition, the derivation points out several weaknesses in previous justifications of both the hybrid and equilibrium systems and demonstrates the connection between the original and approximate master equations. Two simple examples illustrate situations in which these two approximate methods are applicable and demonstrate the two methods' efficiencies.     

The total quasi-steady-state for multiple alternative substrate reactions

Journal of Mathematical Chemistry - The Michaelis–Menten–Briggs–Haldane approximation and its extension, the total quasi-steady-state approximation (tQSSA) are famous assumptions...     

Logistic-exponential model for chemiluminescence kinetics

A logistic-exponential (LE) model for chemiluminescence (ChL) kinetics was constructed as a superposition of a logistic function, representing the ascending sigmoidal-in-shape phase, and an exponential function representing the descending phase of the ChL time course. The logistic component of the LE model expresses a non-linear autocatalytic reversible reaction counteracting a rise in the ChL which is not considered by a classical two-exponential model of the time course of ChL, whereas the exponential component of the LE model represents a first-order reaction of a ChL decay. The proposed reactions, that underlie the time course of ChL, were shown to form a second-order dynamic system. Main characteristics of the LE model such as the ChL peak value (CL(m)), the peak time (t(m)) and the inflexion points' times (t(i)) were determined as well as the error calculus for the LE model. Moreover, several applications of the LE model to ChL processes generated by both native and perturbed polymorphonuclear granulocytes (PMNs) and red blood cells (RBCs), intoxicated yeast, ferrous ion-treated bull spermatozoa, autoxidising l-3,4-dihydroxyphenylalanine (l-DOPA), or luminol oxidised in the Fenton reaction were made.     

Asymmetric stochastic localization in geometry controlled kinetics

We consider the motion of Brownian particles confined in a two-dimensional symmetric bilobal enclosure with uneven cross section. Varying cross section of the confinement results in an effective entropic potential in reduced dimension. By employing two external noise forces, one additive and another multiplicative along x direction, we demonstrate that a correlation between them causes a symmetry breaking of entropic stability, i.e., a difference in relative stability of two lobes. This leads to an asymmetric localization of population in the stationary state. A two-state model is proposed to explain the asymmetric localization of population due to entropic diffusion.     

A study of the accuracy of moment-closure approximations for stochastic chemical kinetics

Moment-closure approximations have in recent years become a popular means to estimate the mean concentrations and the variances and covariances of the concentration fluctuations of species involved in stochastic chemical reactions, such as those inside cells. The typical assumption behind these methods is that all cumulants of the probability distribution function solution of the chemical master equation which are higher than a certain order are negligibly small and hence can be set to zero. These approximations are ad hoc and hence the reliability of the predictions of these class of methods is presently unclear. In this article, we study the accuracy of the two moment approximation (2MA) (third and higher order cumulants are zero) and of the three moment approximation (3MA) (fourth and higher order cumulants are zero) for chemical systems which are monostable and composed of unimolecular and bimolecular reactions. We use the system-size expansion, a systematic method of solving the chemical master equation for monostable reaction systems, to calculate in the limit of large reaction volumes, the first- and second-order corrections to the mean concentration prediction of the rate equations and the first-order correction to the variance and covariance predictions of the linear-noise approximation. We also compute these corrections using the 2MA and the 3MA. Comparison of the latter results with those of the system-size expansion shows that: (i) the 2MA accurately captures the first-order correction to the rate equations but its first-order correction to the linear-noise approximation exhibits the wrong dependence on the rate constants. (ii) the 3MA accurately captures the first- and second-order corrections to the rate equation predictions and the first-order correction to the linear-noise approximation. Hence while both the 2MA and the 3MA are more accurate than the rate equations, only the 3MA is more accurate than the linear-noise approximation across all of parameter space. The analytical results are numerically validated for dimerization and enzyme-catalyzed reactions.     

Binomial tau-leap spatial stochastic simulation algorithm for applications in chemical kinetics

In cell biology, cell signaling pathway problems are often tackled with deterministic temporal models, well mixed stochastic simulators, and/or hybrid methods. But, in fact, three dimensional stochastic spatial modeling of reactions happening inside the cell is needed in order to fully understand these cell signaling pathways. This is because noise effects, low molecular concentrations, and spatial heterogeneity can all affect the cellular dynamics. However, there are ways in which important effects can be accounted without going to the extent of using highly resolved spatial simulators (such as single-particle software), hence reducing the overall computation time significantly. We present a new coarse grained modified version of the next subvolume method that allows the user to consider both diffusion and reaction events in relatively long simulation time spans as compared with the original method and other commonly used fully stochastic computational methods. Benchmarking of the simulation algorithm was performed through comparison with the next subvolume method and well mixed models (MATLAB), as well as stochastic particle reaction and transport simulations (CHEMCELL, Sandia National Laboratories). Additionally, we construct a model based on a set of chemical reactions in the epidermal growth factor receptor pathway. For this particular application and a bistable chemical system example, we analyze and outline the advantages of our presented binomial tau-leap spatial stochastic simulation algorithm, in terms of efficiency and accuracy, in scenarios of both molecular homogeneity and heterogeneity.     

Electron-transfer kinetics and activation barriers for the reductions of nitrosonium and nitronium ions in aprotic solvents

Using the cyclic voltammetry (CV), the electron-transfer kinetics for the reductions of NO⁺ and NO₂⁺ cations have been studied at the Pt electrode in nitromethane, sulfolane, and propylene carbonate. The heterogeneous rate constants have been determined by two independent procedures from the transfer coefficient α, the diffusion coefficient D, from a detailed examination of the CV-peak separations, and from an inspection of the values of the cathodic peak potentials at different scan rates. The results have been compared to those reported in the literature, and discussed. In the classical model, outer-sphere electron-transfer reactions are considered subject to an activation energy arising from solvent reorganization and bond reorganization processes. The solvent and molecular reorganizational barriers for these electroreductions have been assessed in aprotic media. The Marcus-Hush theory has been applied to the self-exchange reactions of the NO₂⁺/NO₂ and NO⁺/NO couples in an attempt to predict the rate of electron transfer. The findings indicate some improvement between theory and experiment. However, it should be noted that the experimental values of k_s found for the NO₂⁺ reduction in the solvents used are still too high in comparison with those determined theoretically. In view of the fairly strong coordination of the solvent molecule(s) as ligand(s) to NO₂⁺ and NO⁺ cations, we believe that such discrepancies should stem, to some extent, from the involvement of an inner-sphere pathway by generation of an activated complex on the surface of the Pt electrode. © 1993 John Wiley & Sons, Inc.     

On the method of quasi-steady-state approximation

Sufficient conditions for the validity of the quasi-steady-state approximation widely used in chemical kinetics are considered by means of the qualitative geometric theory of differential equations with small parameters.     

Cluster kinetics model for mixtures of glassformers

For glassformers we propose a binary mixture relation for parameters in a cluster kinetics model previously shown to represent pure compound data for viscosity and dielectric relaxation as functions of either temperature or pressure. The model parameters are based on activation energies and activation volumes for cluster association-dissociation processes. With the mixture parameters, we calculated dielectric relaxation times and compared the results to experimental values for binary mixtures. Mixtures of sorbitol and glycerol (seven compositions), sorbitol and xylitol (three compositions), and polychloroepihydrin and polyvinylmethylether (three compositions) were studied.     

Effect of reactant size on discrete stochastic chemical kinetics

This paper is aimed at understanding what happens to the propensity functions (rates) of bimolecular chemical reactions when the volume occupied by the reactant molecules is not negligible compared to the containing volume of the system. For simplicity our analysis focuses on a one-dimensional gas of N hard-rod molecules, each of length l. Assuming these molecules are distributed randomly and uniformly inside the real interval [0,L] in a nonoverlapping way, and that they have Maxwellian distributed velocities, the authors derive an expression for the probability that two rods will collide in the next infinitesimal time dt. This probability controls the rate of any chemical reaction whose occurrence is initiated by such a collision. The result turns out to be a simple generalization of the well-known result for the point molecule case l=0: the system volume L in the formula for the propensity function in the point molecule case gets replaced by the "free volume" L-Nl. They confirm the result in a series of one-dimensional molecular dynamics simulations. Some possible wider implications of this result are discussed.     

Diffusion-collision model algorithms for protein folding kinetics

The diffusion-collision model (DCM) of protein folding is described qualitatively and quantitatively. The input parameters required to perform a calculation are explained, and the output data are outlined. Three examples are given of calculating DCM folding kinetics: the Engrailed Homeodomain (a three-helix bundle with three helical microdomains, pdb code 1ENH), protein G (with three microdomains having a beta-hairpin-alpha-helix-beta-hairpin motif, pdb code 1PGA), and apomyoglobin (with eight helices and seven strong microdomain-microdomain pairings).     

超微盘电极线性扫描伏安法准稳态准可逆波理论

本文提出超微盘电极上线性扫描准稳态准可逆伏安曲线表达式, 并用铂超微盘电极、(NH~4)~2Fe(SO~4)~2-1mol·dm^-^3H~2SO~4体系进行了验证, 理论与实验相符。