The transport porperties of a dilute gas are determined by binary collision integrals. In order to predict the first density corrections to the transport properties, it is necessary to consider collision integrals that account for the effects of collisions among three molecules. In this technical report we derive and formulate such three-particle collision integrals for the coefficients of thermal conductivity, viscosity and self-diffus ion of a gas of hard spherical molecules. An evaluation of these three-particle collision integrals will be presented in a subsequent technical report.

Simulating the evolution of a chemically reacting system using the bimolecular propensity function, as is done by the stochastic simulation algorithm and its reaction-diffusion extension, entails making statistically inspired guesses as to where the reactant molecules are at any given time. Those guesses will be physically justified if the system is dilute and well-mixed in the reactant molecules. Otherwise, an accurate simulation will require the extra effort and expense of keeping track of the positions of the reactant molecules as the system evolves. One molecule-tracking algorithm that pays careful attention to the physics of molecular diffusion is the enhanced Green's function reaction dynamics (eGFRD) of Takahashi, Tănase-Nicola, and ten Wolde [Proc. Natl. Acad. Sci. U.S.A. 107, 2473 (2010)]. We introduce here a molecule-tracking algorithm that has the same theoretical underpinnings and strategic aims as eGFRD, but a different implementation procedure. Called the small-voxel tracking algorithm (SVTA), it combines the well known voxel-hopping method for simulating molecular diffusion with a novel procedure for rectifying the unphysical predictions of the diffusion equation on the small spatiotemporal scale of molecular collisions. Indications are that the SVTA might be more computationally efficient than eGFRD for the problematic class of non-dilute systems. A widely applicable, user-friendly software implementation of the SVTA has yet to be developed, but we exhibit some simple examples which show that the algorithm is computationally feasible and gives plausible results.

The chemical master equation (CME) and the mathematically equivalent stochastic simulation algorithm (SSA) assume that the reactant molecules in a chemically reacting system are “dilute” and “well-mixed” throughout the containing volume. Here we clarify what those two conditions mean, and we show why their satisfaction is necessary in order for bimolecular reactions to physically occur in the manner assumed by the CME and the SSA. We prove that these conditions are closely connected, in that a system will stay well-mixed if and only if it is dilute. We explore the implications of these validity conditions for the reaction-diffusion (or spatially inhomogeneous) extensions of the CME and the SSA to systems whose containing volumes are not necessarily well-mixed, but can be partitioned into cubical subvolumes (voxels) that are. We show that the validity conditions, together with an additional condition that is needed to ensure the physical validity of the diffusion-induced jump probability rates of molecules between voxels, require the voxel edge length to have a strictly positive lower bound. We prove that if the voxel edge length is steadily decreased in a way that respects that lower bound, the average rate at which bimolecular reactions occur in the reaction-diffusion CME and SSA will remain constant, while the average rate of diffusive transfer reactions will increase as the inverse square of the voxel edge length. We conclude that even though the reaction-diffusion CME and SSA are inherently approximate, and cannot be made exact by shrinking the voxel size to zero, they should nevertheless be useful in many practical situations.

1. The Fickian theory of diffusion 2. A review of random variable theory 3. Einstein's theory of diffusion 4. Implications and limitations of the Einstein theory of diffusion 5. The discrete-stochastic approach 6. Master equations and simulation algorithms for the discrete-stochastic approach 7. Continuous Markov process theory 8. Langevin's theory of diffusion 9. Implications of Langevin's theory 10. Diffusion in an external force field 11. The first-passage time approach

We outline our perspective on stochastic chemical kinetics, paying particular attention to numerical simulation algorithms. We first focus on dilute, well-mixed systems, whose description using ordinary differential equations has served as the basis for traditional chemical kinetics for the past 150 years. For such systems, we review the physical and mathematical rationale for a discrete-stochastic approach, and for the approximations that need to be made in order to regain the traditional continuous-deterministic description. We next take note of some of the more promising strategies for dealing stochastically with stiff systems, rare events, and sensitivity analysis. Finally, we review some recent efforts to adapt and extend the discrete-stochastic approach to systems that are not well-mixed. In that currently developing area, we focus mainly on the strategy of subdividing the system into well-mixed subvolumes, and then simulating diffusional transfers of reactant molecules between adjacent subvolumes together with chemical reactions inside the subvolumes.

Previous chapters have focused mainly on the problem of computing where a diffusing solute molecule will be at a specified future time. In this concluding chapter the chapter takes a look at the subtly different problem of computing when a diffusing solute molecule will first reach a specified location. This is known as the first-passage time problem. Within the framework of the Einstein model of diffusion, the first-passage time problem has been solved analytically in only a few relatively simple cases. In the context of the Langevin model the problem is even more challenging, and apparently no exact results are known at present. The chapter considers the first-passage time problem only within the framework of the Einstein model, and the chapter examines only its two simplest forms. The aim is to illuminate the concepts and issues involved, and to demonstrate some solution strategies. In the course of doing that, the chapter will uncover some results that will deepen the understanding of the cell-jumping hypothesis that underlies the discrete- stochastic model of diffusion.

The linear noise approximation (LNA) is a way of approximating the stochastic time evolution of a well-stirred chemically reacting system. It can be obtained either as the lowest order correction to the deterministic chemical reaction rate equation (RRE) in van Kampen's system-size expansion of the chemical master equation (CME), or by linearising the two-term-truncated chemical Kramers-Moyal equation. However, neither of those derivations sheds much light on the validity of the LNA. The problematic character of the system-size expansion of the CME for some chemical systems, the arbitrariness of truncating the chemical Kramers-Moyal equation at two terms, and the sometimes poor agreement of the LNA with the solution of the CME, have all raised concerns about the validity and usefulness of the LNA. Here, the authors argue that these concerns can be resolved by viewing the LNA as an approximation of the chemical Langevin equation (CLE). This view is already implicit in Gardiner's derivation of the LNA from the truncated Kramers-Moyal equation, as that equation is mathematically equivalent to the CLE. However, the CLE can be more convincingly derived in a way that does not involve either the truncated Kramers-Moyal equation or the system-size expansion. This derivation shows that the CLE will be valid, at least for a limited span of time, for any system that is sufficiently close to the thermodynamic (large-system) limit. The relatively easy derivation of the LNA from the CLE shows that the LNA shares the CLE's conditions of validity, and it also suggests that what the LNA really gives us is a description of the initial departure of the CLE from the RRE as we back away from the thermodynamic limit to a large but finite system. The authors show that this approach to the LNA simplifies its derivation, clarifies its limitations, and affords an easier path to its solution.

Chapter 3 showed how Einstein's analysis of Brownian motion shifted the focus of the classical diffusion equation from the average behavior of many solute molecules to the probabilistic behavior of an individual solute molecule. This chapter deduces some further implications of the Einstein theory of diffusion, using not only analytical reasoning but also numerical simulation. This chapter shows how simulation can be used to construct plots of a single diffusing molecule's position probability density function for some simple boundary conditions. The chapter also shows how simulation can be used to construct “snapshots” of an unrestricted solute molecule's position at successive instants of time. But the chapter will discovers that such snapshots expose a serious physical limitation of the Einstein theory. The chapter rationalizes a quick fix that allows us to complete the derivation which has begun in Chapter 3 of a formula for the stochastic rate of a diffusion-controlled bimolecular chemical reaction. But from a broader view, it will become apparent that, while the Einstein theory of diffusion has a wide range of practical utility, a physically more accurate theory is needed.

Michaelis-Menten kinetics are commonly used to represent enzyme-catalysed reactions in biochemical models. The Michaelis-Menten approximation has been thoroughly studied in the context of traditional differential equation models. The presence of small concentrations in biochemical systems, however, encourages the conversion to a discrete stochastic representation. It is shown that the Michaelis-Menten approximation is applicable in discrete stochastic models and that the validity conditions are the same as in the deterministic regime. The authors then compare the Michaelis-Menten approximation to a procedure called the slow-scale stochastic simulation algorithm (ssSSA). The theory underlying the ssSSA implies a formula that seems in some cases to be different from the well-known Michaelis-Menten formula. Here those differences are examined, and some special cases of the stochastic formulas are confirmed using a first-passage time analysis. This exercise serves to place the conventional Michaelis-Menten formula in a broader rigorous theoretical framework.

We derive an explicit formula for the propensity function (stochastic reaction rate) of a generic bimolecular chemical reaction in which the reactant molecules move about by diffusion, as solute molecules in a bath of much smaller and more numerous solvent molecules. Our derivation assumes that the solution is macroscopically well stirred and dilute in the solute molecules. It effectively extends the physical rationale for the chemical master equation and the stochastic simulation algorithm from well-stirred dilute gases to well-stirred dilute solutions, with the former becoming a limiting case of the latter. This extension is important for cellular systems, where the solvent molecules are typically water and the solute (reactant) molecules are much larger organic structures, whose relatively low populations often require a discrete-stochastic formalism. In the course of our derivation, we illuminate some limitations on the ability of the classical diffusion equation to accurately describe how a diffusing molecule moves on spatial and temporal scales that are relevant to collision-induced chemical reactions.

An analysis is presented of the approximating assumptions that underlie a recently proposed derivation of the traditional deterministic reaction rate equation from a discrete-stochastic formulation of chemical kinetics. It is shown that if the system is close enough to the thermodynamic limit, in which the molecular populations and the containing volume all approach infinity in such a way that the molecular concentrations remain finite, then the required approximating assumptions will be justified for practically all spatially homogeneous systems that one is likely to encounter.

This chapter reviews the theory of stochastic chemical kinetics and several simulation methods that are based on that theory. An effort is made to delineate the logical connections among the major elements of the theory, such as the chemical master equation, the stochastic simulation algorithm, tau-leaping, the chemical Langevin equation, the chemical Fokker-Planck equation, and the deterministic reaction rate equation. Focused presentations are given of two approximate simulation strategies that aim to improve simulation efficiency for systems with “multiscale” complications of the kind that are often encountered in cellular systems: The first, explicit tau-leaping, deals with systems that have a wide range of molecular populations. The second, the slow-scale stochastic simulation algorithm, is designed for systems that have a wide range of reaction rates. The latter procedure is shown to provide a stochastic generalization of the Michaelis-Menten analysis of the enzyme-substrate reaction set.