A modified Next Reaction Method for simulating chemical systems with time dependent propensities and delays

Chemical reaction systems with a low to moderate number of molecules are typically modeled as discrete jump Markov processes. These systems are oftentimes simulated with methods that produce statistically exact sample paths such as the Gillespie Algorithm or the Next Reaction Method. In this paper we make explicit use of the fact that the initiation times of the reactions can be represented as the firing times of independent, unit rate Poisson processes with internal times given by integrated propensity functions. Using this representation we derive a modified Next Reaction Method and, in a way that achieves efficiency over existing approaches for exact simulation, extend it to systems with time dependent propensities as well as to systems with delays.Comment: 25 pages, 1 figure. Some minor changes made to add clarit

An asymptotic relationship between coupling methods for stochastically modeled population processes

This paper is concerned with elucidating a relationship between two common coupling methods for the continuous time Markov chain models utilized in the cell biology literature. The couplings considered here are primarily used in a computational framework by providing reductions in variance for different Monte Carlo estimators, thereby allowing for significantly more accurate results for a fixed amount of computational time. Common applications of the couplings include the estimation of parametric sensitivities via finite difference methods and the estimation of expectations via multi-level Monte Carlo algorithms. While a number of coupling strategies have been proposed for the models considered here, and a number of articles have experimentally compared the different strategies, to date there has been no mathematical analysis describing the connections between them. Such analyses are critical in order to determine the best use for each. In the current paper, we show a connection between the common reaction path (CRP) method and the split coupling (SC) method, which is termed coupled finite differences (CFD) in the parametric sensitivities literature. In particular, we show that the two couplings are both limits of a third coupling strategy we call the "local-CRP" coupling, with the split coupling method arising as a key parameter goes to infinity, and the common reaction path coupling arising as the same parameter goes to zero. The analysis helps explain why the split coupling method often provides a lower variance than does the common reaction path method, a fact previously shown experimentally.Comment: Edited Section 4.

Incorporating postleap checks in tau-leaping

By explicitly representing the reaction times of discrete chemical systems as the firing times of independent, unit rate Poisson processes, we develop a new adaptive tau-leaping procedure. The procedure developed is novel in that accuracy is guaranteed by performing postleap checks. Because the representation we use separates the randomness of the model from the state of the system, we are able to perform the postleap checks in such a way that the statistics of the sample paths generated will not be biased by the rejections of leaps. Further, since any leap condition is ensured with a probability of one, the simulation method naturally avoids negative population valuesComment: Final version. Minor change

Hybrid Pathwise Sensitivity Methods for Discrete Stochastic Models of Chemical Reaction Systems

Stochastic models are often used to help understand the behavior of intracellular biochemical processes. The most common such models are continuous time Markov chains (CTMCs). Parametric sensitivities, which are derivatives of expectations of model output quantities with respect to model parameters, are useful in this setting for a variety of applications. In this paper, we introduce a class of hybrid pathwise differentiation methods for the numerical estimation of parametric sensitivities. The new hybrid methods combine elements from the three main classes of procedures for sensitivity estimation, and have a number of desirable qualities. First, the new methods are unbiased for a broad class of problems. Second, the methods are applicable to nearly any physically relevant biochemical CTMC model. Third, and as we demonstrate on several numerical examples, the new methods are quite efficient, particularly if one wishes to estimate the full gradient of parametric sensitivities. The methods are rather intuitive and utilize the multilevel Monte Carlo philosophy of splitting an expectation into separate parts and handling each in an efficient manner.Comment: 30 pages. The numerical example section has been extensively rewritte