14,891 research outputs found
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
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