K-leap method for accelerating stochastic simulation of coupled chemical reactions

Leap methods are very promising for accelerating stochastic simulation of a well stirred chemically reacting system, while providing acceptable simulation accuracy. In Gillespie's tau-leap method [D. Gillespie, J. Phys. Chem. 115, 1716 (2001)], the number of firings of each reaction channel during a leap is a Poisson random variable, whose sample values are unbounded. This may cause large changes in the populations of certain molecular species during a leap, thereby violating the leap condition. In this paper, we develop an alternative leap method called the K-leap method, in which we constrain the total number of reactions occurring during a leap to be a number K calculated from the leap condition. As the number of firings of each reaction channel during a leap is upper bounded by a properly chosen number, our K-leap method can better satisfy the leap condition, thereby improving simulation accuracy. Since the exact stochastic simulation algorithm (SSA) is a special case of our K-leap method when K=1, our K-leap method can naturally change from the exact SSA to an approximate leap method during simulation, whenever the leap condition allows to do so.     

Accelerated stochastic simulation algorithm for coupled chemical reactions with delays

Some biochemical processes do not occur instantaneously but have considerably delays associated with them. In the existed methods which solve these chemically reacting systems with delays, averaging over a great deal of simulations is needed for reliable statistical characters. Here we present an accelerating approach, called the "Delay Final All Possible Steps" (DFAPS) approach, which does not alter the course of stochastic simulation, but reduces the required running times. Numerical simulation results indicate that the proposed method can be applied to a wide range of chemically reacting systems with delays and obtain a significant improvement on efficiency and accuracy over the existed methods.     

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.     

Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions

The dynamical solution of a well-mixed, nonlinear stochastic chemical kinetic system, described by the Master equation, may be exactly computed using the stochastic simulation algorithm. However, because the computational cost scales with the number of reaction occurrences, systems with one or more "fast" reactions become costly to simulate. This paper describes a hybrid stochastic method that partitions the system into subsets of fast and slow reactions, approximates the fast reactions as a continuous Markov process, using a chemical Langevin equation, and accurately describes the slow dynamics using the integral form of the "Next Reaction" variant of the stochastic simulation algorithm. The key innovation of this method is its mechanism of efficiently monitoring the occurrences of slow, discrete events while simultaneously simulating the dynamics of a continuous, stochastic or deterministic process. In addition, by introducing an approximation in which multiple slow reactions may occur within a time step of the numerical integration of the chemical Langevin equation, the hybrid stochastic method performs much faster with only a marginal decrease in accuracy. Multiple examples, including a biological pulse generator and a large-scale system benchmark, are simulated using the exact and proposed hybrid methods as well as, for comparison, a previous hybrid stochastic method. Probability distributions of the solutions are compared and the weak errors of the first two moments are computed. In general, these hybrid methods may be applied to the simulation of the dynamics of a system described by stochastic differential, ordinary differential, and Master equations.     

A new approximate method for the stochastic simulation of chemical systems: the representative reaction approach

We have developed two new approximate methods for stochastically simulating chemical systems. The methods are based on the idea of representing all the reactions in the chemical system by a single reaction, i.e., by the “representative reaction approach” (RRA). Discussed in the article are the concepts underlying the new methods along with flowchart with all the steps required for their implementation. It is shown that the two RRA methods {with the reaction as the representative reaction (RR)} perform creditably with regard to accuracy and computational efficiency, in comparison to the exact stochastic simulation algorithm (SSA) developed by Gillespie and are able to successfully reproduce at least the first two moments of the probability distribution of each species in the systems studied. As such, the RRA methods represent a promising new approach for stochastically simulating chemical systems. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2012     

A topological stochastic approach to the study of multidimensional potential energy surfaces of chemical reactions

In this article we propose a new approach for investigating the properties of multidimensional potential energy surfaces in chemical reactions, based on relations of each multidimensional surface to its one-dimensional image which is the chemical reaction tree. This approach makes it possible to find a common number of independent channels in chemical reactions for complex systems and to construct the probable models.     

Domain catalyzed chemical reactions: a molecular dynamics simulation of isomerization kinetics

The model for domain catalyzed isomerization kinetics in condensed fluids is applied for a diluted mixture of a chiral solute with a consolute temperature. The solution is quench to phase separation at temperatures below the consolute temperature. The droplet coalescence enhances the isomerization kinetics due to the substantial excess pressure inside the small droplets given by the Laplace equation. The domain catalyzed isomerization kinetics breaks the symmetry, and the droplets end with only one dominating species. We argue that D-glyceraldehyde which is only moderately solvable in water and which has played a crucial role in the evolution is a candidate for the stereo specific ordering in bio-organic matter.     

Deterministic modeling and stochastic simulation of poly-alkoxylation reactions

The main scope of this work is to show the feasibility and the advantage of using a stochastic approach to describe the poly-alkoxylation kinetics of different substrates. For this purpose, the reactions of ethylene and propylene oxides with respectively ethylene glycol, 1-octanol, and 2-octanol were considered. Two kinetic models were used for interpreting all the kinetic runs available in the literature, one deterministic and another one stochastic, for a useful comparison between the two different approaches. As the adopted reaction mechanism, rate laws, and related kinetic parameters were the same for both the kinetic models, the obtained results for what concerns the substrate consumption, and the oligomers distribution profiles were the same in both cases. In the case of the stochastic kinetic approach, the calculations must be made on a small volume of the reacting mixture containing a sufficiently high number of molecules that is suitable for the statistical analysis but as small as possible for reducing the calculation time. The calculations made have allowed individuating this optimal volume. This study is propaedeutic to the application of a stochastic kinetic approach to the study of ethylene-propylene oxides copolymerization that cannot be faced with a deterministic model for the extremely long or impracticable calculation time due to the great number of material balance differential equations that must be integrated.     

Implementation of a molecular dynamics approach with rigid fragments to simulation of chemical reactions in biomolecular systems

We describe a new implementation of the molecular dynamics method aimed at simulation of the properties of biomolecular systems in which chemical reactions are possible. The quantum mechanical/molecular mechanical method based on the effective fragment potential theory is used for calculating the energies and forces along trajectories. Due to specific features of the effective fragment theory, the behavior of the molecular mechanical subsystem is described by rigid body dynamics. The method has been applied to simulation of proton transfer along the chain of water molecules inside the gramicidin channel.     

Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates

An efficient simulation algorithm for chemical kinetic systems with disparate rates is proposed. This new algorithm is quite general, and it amounts to a simple and seamless modification of the classical stochastic simulation algorithm (SSA), also known as the Gillespie [J. Comput. Phys. 22, 403 (1976); J. Phys. Chem. 81, 2340 (1977)] algorithm. The basic idea is to use an outer SSA to simulate the slow processes with rates computed from an inner SSA which simulates the fast reactions. Averaging theorems for Markov processes can be used to identify the fast and slow variables in the system as well as the effective dynamics over the slow time scale, even though the algorithm itself does not rely on such information. This nested SSA can be easily generalized to systems with more than two separated time scales. Convergence and efficiency of the algorithm are discussed using the established error estimates and illustrated through examples.     

Digital simulation of electrochemical processes involving very fast chemical reactions : Part 1. A simple general approach

A simple approach based on the steady-state assumption and linear concentration profiles in the reaction layer is proposed to simulate electrochemical systems involving very fast homogeneous chemical reactions. The flux equations are derived in detail and the method is shown to be suitable for catalytic e.c.e and dimerization reaction mechanism. The accuracy of the proposed method is checked by comparing the results obtained with those already available in the literature.     

An analytical approach to solutions of master equations for stochastic nonlinear reactions

In this paper we consider a class of nonlinear reactions which are important in stochastic reaction networks. We find the exact solution of the chemical master equation for a class of irreversible and reversible nonlinear reactions. We also present the explicit form of the equilibrium probability solution of the reactions. The results can be used for analyzing stochastic dynamics of important reactions such as binding/unbinding reaction and protein dimerization.     

化学反应的内禀反应坐标法I:甲硫醛脱氢反应的IRC解析

本文用内禀反应坐标法讨论了甲硫醛分子的脱氢反应机理. 在4-31G基组上对此反应做了量子化学从头计算和反应路解析, 得出过渡态结构, 反应势能曲线, 活化能, 反应热以及沿反应坐标反应系的一些物理量的变化, 并对过渡态做了振动分析. 所得结果与甲醛,甲硫醇和甲醇等分子的脱氢反应结果做了对比. 还给出沿反应坐标分子间弹性碰撞阶段和最佳碰撞角等信息. 同时, 计算了反应的频率因子A,讨论了反应速率常数k值与温度的关系.     

Analysis of electronically nonadiabatic chemical reactions: An information theoretic approach

A practical procedure for the determination of branching ratios for reactions which lead to either excited or electronically ground state products is outlined. The method is applied to four reactions which could (on energetic grounds) produce an electronically excited iodine atom. No case of a complete inversion is found, but one reaction (F + HI) is predicted to yield a statistical, (one half), I^*(²P_1/2) to I(²P_3/2) ratio.     

A partial-propensity formulation of the stochastic simulation algorithm for chemical reaction networks with delays

Several real-world systems, such as gene expression networks in biological cells, contain coupled chemical reactions with a time delay between reaction initiation and completion. The non-Markovian kinetics of such reaction networks can be exactly simulated using the delay stochastic simulation algorithm (dSSA). The computational cost of dSSA scales with the total number of reactions in the network. We reduce this cost to scale at most with the smaller number of species by using the concept of partial reaction propensities. The resulting delay partial-propensity direct method (dPDM) is an exact dSSA formulation for well-stirred systems of coupled chemical reactions with delays. We detail dPDM and present a theoretical analysis of its computational cost. Furthermore, we demonstrate the implications of the theoretical cost analysis in two prototypical benchmark applications. The dPDM formulation is shown to be particularly efficient for strongly coupled reaction networks, where the number of reactions is much larger than the number of species.     

Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas

A kinetic approach is presented for the analysis of a gas mixture with two kinds of nonconservative interactions. In a bimolecular chemical reaction, mass transfer and energy of chemical link arise, and in inelastic mechanical encounters, molecules get excited or de‐excited due to their quantized structure. Molecules undergo transitions between energy levels also by absorption and emission of photons of the self‐consistent radiation field. From the kinetic Boltzmann‐type equations, the problem of equilibria and of their stability is addressed. A detailed balance principle is proved and a Lyapunov functional is constructed; mass action law and Planck's law of radiation are recovered. This revised version was published online in July 2006 with corrections to the Cover Date.     

Digital simulation of electrochemical processes involving very fast chemical reactions : Part 2. A new criterion for determining rate constants of consecutive second-order irreversible chemical reactions

Electrode reactions followed by very fast chemical reactions are considered. A simple approach, in which steady state and linear concentration profiles in the reaction layer are assumed, is proposed for the simulation of these processes when the substrate is not present in large excess. When the substrate/depolarizer mole ratio is less than one, two well-separated peaks are detected; under such conditions, working curves that enable the corresponding second-order homogeneous rate constant to be evaluated can be derived by the finite-difference simulation technique. The method is applied to elucidation of the reduction of the tetrakis(triphenylphosphine)nickel(I) complex in the presence of hydrogen ions in acetonitrile at ?30°C.     

Mapping tertiary interactions in protein folding reactions: a novel mass spectrometry- and chemical synthesis-based approach

A novel mass spectrometry- and chemical synthesis-based approach for studying protein folding reactions is described, and its initial application to study the folding/unfolding reaction of a homo-hexameric enzyme 4-oxalocrotonate (4OT) is reported. This new approach involves the application of total chemical synthesis to prepare protein analogues that contain a photoreactive amino acid site-specifically incorporated into their primary amino acid sequence. To this end, a photoreactive amino acid-containing analogue of 4OT in which Pro-1 was replaced with p-benzoyl-l-phenylalanine (Bpa) was prepared. This analogue can be used to map structurally specific protein-protein interactions in 4OT's native folded state. These photocrosslinking studies and peptide mapping results with (PlBpa)4OT indicate that this construct is potentially useful for probing the structural properties of equilibrium and kinetic intermediates in 4OT's folding reaction.