Steady-state parameter sensitivity in stochastic modeling via trajectory reweighting

Abstract

Parameter sensitivity analysis is a powerful tool in the building and analysis of biochemical network models. For stochastic simulations, parameter sensitivity analysis can be computationally expensive, requiring multiple simulations for perturbed values of the parameters. Here, we use trajectory reweighting to derive a method for computing sensitivity coefficients in stochastic simulations without explicitly perturbing the parameter values, avoiding the need for repeated simulations. The method allows the simultaneous computation of multiple sensitivity coefficients. Our approach recovers results originally obtained by application of the Girsanov measure transform in the general theory of stochastic processes [A. Plyasunov and A. P. Arkin, J. Comp. Phys. 221, 724 (2007)]. We build on these results to show how the method can be used to compute steady-state sensitivity coefficients from a single simulation run, and we present various efficiency improvements. For models of biochemical signaling networks the method has a particularly simple implementation. We demonstrate its application to a signaling network showing stochastic focussing and to a bistable genetic switch, and present exact results for models with linear propensity functions.

Malliavin Weight Sampling: A Practical Guide

“…Similar expressions are given in [5,7]. Thus, the Malliavin weight q λ is not fixed by the state point S but by the entire trajectory of states that have led to state point S. Since many different trajectories can lead to S, many values of q λ are possible for the same state point S. The average q λ (t) S is actually the expectation value of the Malliavin weight, averaged over all trajectories that reach state point S at time t. This can be used to obtain an alternative proof that q λ S = ∂ ln P/∂λ.…”

“…The rules for the propagation of Malliavin weights have been derived for the kinetic Monte-Carlo algorithm [4,7], for the Metropolis Monte-Carlo scheme [5] and for both underdamped and overdamped Brownian dynamics [8]. Here we present a generic theoretical framework, which encompasses these algorithms and also allows extension to other stochastic simulation schemes.…”

“…For this estimate to be accurate, we require that ∆t is long enough that C(∆t) has reached its plateau value; this typically means that ∆t should be longer than the typical relaxation time of the system's dynamics. The correlation function approach is discussed in more detail in [7,8].…”

Efficiency of the Girsanov Transformation Approach for Parametric Sensitivity Analysis of Stochastic Chemical Kinetics

“…We also investigate a modified GT method inspired by the work in [28], which we call the centered Girsanov transformation (CGT) method in which we replace the estimator f (X(t, c))Z(t, c) with (f (X(t, c)) − E(f (X(t, c))))Z(t, c). Since Z(t, c) has zero mean this new estimator has the same mean as the original one and hence is also unbiased.…”

“…So it is adequate to study the variance of (f (X(t, c)) − E(f (X(t, c))))Z(t, c). In the formula used in [28] for the ultimate estimator, Z (i) above were replaced by Z (i) −Z whereZ was the sample mean of Z (i) . When the sample size N s is large, both ultimate estimators are similar.…”