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Abstract
We outline our perspective on stochastic chemical kinetics, paying particular attention to numerical simulation algorithms. We first focus on dilute, wellmixed 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 discretestochastic approach, and for the approximations that need to be made in order to regain the traditional continuousdeterministic 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 discretestochastic approach to systems that are not wellmixed. In that currently developing area, we focus mainly on the strategy of subdividing the system into wellmixed subvolumes, and then simulating diffusional transfers of reactant molecules between adjacent subvolumes together with chemical reactions inside the subvolumes.
The work of D.T.G. was funded by the University of California, Santa Barbara under professional services Agreement No. 130401A40, pursuant to National Institutes of Health (NIH) Award No. R01EB01487701. The work of A.H. and L.R.P. was funded by National Science Foundation (NSF) Award No. DMS1001012, ICB Award No. W911NF090001 from the U.S. Army Research Office, NIBIB of the NIH under Award No. R01EB01487701, and (U.S.) Department of Energy (DOE) Award No. DESC0008975. The content of this paper is solely the responsibility of the authors and does not necessarily represent the official views of these agencies.
I. INTRODUCTION
II. DILUTE WELLMIXED CHEMICAL SYSTEMS
A. The chemical master equation and the propensity function
B. Physical justification for the propensity function
C. The stochastic simulation algorithm
D. Tauleaping
E. Connection to the traditional ODE approach
F. Stiff systems and the slowscale SSA
G. Rare events
H. Sensitivity analysis
III. BEYOND WELLMIXED SYSTEMS
A. The reactiondiffusion master equation and simulation algorithm
B. Algorithms for spatial stochastic simulation
C. The RDME on small length scales
IV. ACCOMPLISHMENTS AND CHALLENGES
Key Topics
 Diffusion
 26.0
 Chemical reactions
 21.0
 Poisson's equation
 18.0
 Chemical kinetics
 14.0
 Langevin equation
 9.0
Figures
Stochastic chemical kinetics is premised on the definition (2) of the propensity function in the top box, a definition which must look to molecular physics for its justification. The two solidoutlined boxes in yellow denote mathematically exact consequences of that definition: the chemical master equation (1) and the stochastic simulation algorithm (4) . Dashedoutlined boxes denote approximate consequences: tauleaping (7) , the chemical Langevin equation (9) , the chemical FokkerPlanck equation (not discussed here but see Ref. 35 ), and the reaction rate equation (10) . The bracketed condition by each dashed inference arrow is the condition enabling that approximation: reading from top to bottom, those conditions are the first leap condition, the second leap condition, and the thermodynamic limit. The rationale for viewing the linear noise approximation (LNA) 41 as an intermediate result between the CLE and the RRE is detailed in Ref. 42 . It has been shown 37,42 that for realistic propensity functions, getting “close enough” to the thermodynamic limit will ensure simultaneous satisfaction of the first and second leap conditions, at least for finite spans of time; therefore, the toptobottom progression indicated in the figure will inevitably occur as the molecular populations and the system volume become larger. But a given chemical system might be such that the largest value of τ that satisfies the first leap condition will not be large enough to satisfy the second leap condition; in that case, there will be no accurate description of the system below the discretestochastic level in the figure.
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Stochastic chemical kinetics is premised on the definition (2) of the propensity function in the top box, a definition which must look to molecular physics for its justification. The two solidoutlined boxes in yellow denote mathematically exact consequences of that definition: the chemical master equation (1) and the stochastic simulation algorithm (4) . Dashedoutlined boxes denote approximate consequences: tauleaping (7) , the chemical Langevin equation (9) , the chemical FokkerPlanck equation (not discussed here but see Ref. 35 ), and the reaction rate equation (10) . The bracketed condition by each dashed inference arrow is the condition enabling that approximation: reading from top to bottom, those conditions are the first leap condition, the second leap condition, and the thermodynamic limit. The rationale for viewing the linear noise approximation (LNA) 41 as an intermediate result between the CLE and the RRE is detailed in Ref. 42 . It has been shown 37,42 that for realistic propensity functions, getting “close enough” to the thermodynamic limit will ensure simultaneous satisfaction of the first and second leap conditions, at least for finite spans of time; therefore, the toptobottom progression indicated in the figure will inevitably occur as the molecular populations and the system volume become larger. But a given chemical system might be such that the largest value of τ that satisfies the first leap condition will not be large enough to satisfy the second leap condition; in that case, there will be no accurate description of the system below the discretestochastic level in the figure.
Parts of a Cartesian mesh (a) and an unstructured triangular mesh (b). Molecules are assumed to be wellmixed in the local volumes that make up the dual elements of the mesh (depicted in pink color). For the Cartesian grid (a), the dual is simply the staggered grid. The dual of the triangular mesh in (b) is obtained by connecting the midpoints of the edges and the centroids of the triangles. (c) shows how a model of a eukaryotic cell with a nucleus (green) can be discretized with a mesh made up of triangles and tetrahedra. The figure is adapted from Ref. 70 , where a model of nuclear import was simulated on this domain using the URDME software.
Click to view
Parts of a Cartesian mesh (a) and an unstructured triangular mesh (b). Molecules are assumed to be wellmixed in the local volumes that make up the dual elements of the mesh (depicted in pink color). For the Cartesian grid (a), the dual is simply the staggered grid. The dual of the triangular mesh in (b) is obtained by connecting the midpoints of the edges and the centroids of the triangles. (c) shows how a model of a eukaryotic cell with a nucleus (green) can be discretized with a mesh made up of triangles and tetrahedra. The figure is adapted from Ref. 70 , where a model of nuclear import was simulated on this domain using the URDME software.
Schematic representation of the RDME's behavior as a function of the voxel size h. For h < h*, no local correction to the conventional mesoscopic reaction rates exists that will make the RDME consistent with the Smoluchowski model for the simple problem of diffusion to a target. Figure adapted from Ref. 90 .
Click to view
Schematic representation of the RDME's behavior as a function of the voxel size h. For h < h*, no local correction to the conventional mesoscopic reaction rates exists that will make the RDME consistent with the Smoluchowski model for the simple problem of diffusion to a target. Figure adapted from Ref. 90 .
Tables
Simulation times for a spatial stochastic system simulated to a final time of 200 s with the nextsubvolume method, as implemented in URDME.
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Simulation times for a spatial stochastic system simulated to a final time of 200 s with the nextsubvolume method, as implemented in URDME.
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Abstract
We outline our perspective on stochastic chemical kinetics, paying particular attention to numerical simulation algorithms. We first focus on dilute, wellmixed 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 discretestochastic approach, and for the approximations that need to be made in order to regain the traditional continuousdeterministic 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 discretestochastic approach to systems that are not wellmixed. In that currently developing area, we focus mainly on the strategy of subdividing the system into wellmixed subvolumes, and then simulating diffusional transfers of reactant molecules between adjacent subvolumes together with chemical reactions inside the subvolumes.
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