Hybrid pathwise sensitivity methods for discrete stochastic models of chemical reaction systems

Abstract

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.

Variance decomposition in stochastic simulators

“…In the context of reaction networks presenting an inherent stochastic dynamics, analyses have been restricted to averaged functionals of the stochastic model solution: the analyses characterize the sensitivity of such averages with respect to parameters, for instance coefficients in the propensity functions defining the stochastic network. [18][19][20][21][22] Parametric sensitivity analyses have to be contrasted with the use of the Sobol decomposition we are proposing in the present work. Here, we assume no variability in any model parameter.…”

Stochastic averaging and sensitivity analysis for two scale reaction networks

“…This representation is tremendously useful in conducting analysis of the trajectories. In particular, it leads to formulations of the Next-Reaction Method 27,28 and interpreting simulated trajectories in the path-space to allow for coupling paths 19,29,30 as well as path-wise differentiation 16,17 . When simulating exact trajectories (using any exact method; Direct SSA, Next-Reaction, etc), the propensity functions λ r (x; θ) probabilistically determine both the time between reactions ∆t as well as the next reaction r * to fire.…”

“…The sensitivities give important insight into the system, indicating directions for gradient-descent type optimization as well as determining bounds for quantifying the uncertainty 15 . Current techniques for estimating the sensitivities have large variance, requiring many more samples than those for estimating E θ {f (X)} alone [16][17][18][19] . Thus computing sensitivities of multiscale systems using single-scale techniques is often a computationally intractable problem.…”

Global sensitivity analysis in stochastic simulators of uncertain reaction networks

“…The trajectories X(Y, c), X(Y, c), and X( Y, c) are then determined using the MNRM Algorithm 1 (see lines [6][7][8]; the statistic and correlations of the functional G(X) between these trajectories are subsequently accumulated, before proceeding to the next sample. Finally, the sensitivity indices of G are estimated when M samples have been computed.…”

“…The former essentially consists in the characterization of the variability in the statistical moments, based on their derivatives with respect to the kinetic parameters at some nominal values (see, for instance, Refs. [4][5][6][7].…”