Parametric sensitivity of biochemical networks is an indispensable tool for studying system robustness properties, estimating network parameters, and identifying targets for drug therapy. For discrete stochastic representations of biochemical networks where Monte Carlo methods are commonly used, sensitivity analysis can be particularly challenging, as accurate finite difference computations of sensitivity require a large number of simulations for both nominal and perturbed values of the parameters. In this paper we introduce the common random number ͑CRN͒ method in conjunction with Gillespie's stochastic simulation algorithm, which exploits positive correlations obtained by using CRNs for nominal and perturbed parameters. We also propose a new method called the common reaction path ͑CRP͒ method, which uses CRNs together with the random time change representation of discrete state Markov processes due to Kurtz to estimate the sensitivity via a finite difference approximation applied to coupled reaction paths that emerge naturally in this representation. While both methods reduce the variance of the estimator significantly compared to independent random number finite difference implementations, numerical evidence suggests that the CRP method achieves a greater variance reduction. We also provide some theoretical basis for the superior performance of CRP. The improved accuracy of these methods allows for much more efficient sensitivity estimation. In two example systems reported in this work, speedup factors greater than 300 and 10 000 are demonstrated.

Characterizing the sensitivity to infinitesimally small perturbations in parameters is a powerful tool for the analysis, modeling, and design of chemical reaction networks. Sensitivity analysis of networks modeled using stochastic chemical kinetics, in which a probabilistic description is used to characterize the inherent randomness of the system, is commonly performed using Monte Carlo methods. Monte Carlo methods require large numbers of stochastic simulations in order to generate accurate statistics, which is usually computationally demanding or in some cases altogether impractical due to the overwhelming computational cost. In this work, we address this problem by presenting the regularized pathwise derivative method for efficient sensitivity analysis. By considering a regularized sensitivity problem and using the random time change description for Markov processes, we are able to construct a sensitivity estimator based on pathwise differentiation (also known as infinitesimal perturbation analysis) that is valid for many problems in stochastic chemical kinetics. The theoretical justification for the method is discussed, and a numerical algorithm is provided to permit straightforward implementation of the method. We show using numerical examples that the new regularized pathwise derivative method (1) is able to accurately estimate the sensitivities for many realistic problems and path functionals, and (2) in many cases outperforms alternative sensitivity methods, including the Girsanov likelihood ratio estimator and common reaction path finite difference method. In fact, we observe that the variance reduction using the regularized pathwise derivative method can be as large as ten orders of magnitude in certain cases, permitting much more efficient sensitivity analysis than is possible using other methods.

Overexpression of the inducible heat shock protein 70, Hsp72, has broadly cytoprotective effects and improves outcome following stroke. A full understanding of how Hsp72 protects cells against injury is elusive, though several distinct mechanisms are implicated. One mechanism is its anti-inflammatory effects. We study the effects of Hsp72 overexpression on activation of the transcription factor NF-κB in microglia combining experimentation and mathematical modeling, using TNFα to stimulate a microglial cell line stably overexpressing Hsp72. We find that Hsp72 overexpression reduces the amount of NF-κB DNA binding activity, activity of the upstream kinase IKK, and amount of IκBα inhibitor phosphorylated following TNFα application. Simulations evaluating several proposed mechanisms suggest that inhibition of IKK activation is an essential component of its regulatory activities. Unexpectedly we find that Hsp72 overexpression reduces the initial amount of the RelA/p65 NF-κB subunit in cells, contributing to the attenuated response. Neither mechanism in isolation, however, is sufficient to attenuate the response, providing evidence that Hsp72 relies upon multiple mechanisms to attenuate NF-κB activation. An additional observation from our study is that the induced expression of IκBα is altered significantly in Hsp72 expressing cells. While the mechanism responsible for this observation is not known, it points to yet another means by which Hsp72 may alter the NF-κB response. This study illustrates the multi-faceted nature of Hsp72 regulation of NF-κB activation in microglia and offers further clues to a novel mechanism by which Hsp72 may protect cells against injury.

BackgroundActivation of the NF-κB transcription factor and its associated gene expression in microglia is a key component in the response to brain injury. Its activation is dynamic and is part of a network of biochemical species with multiple feedback regulatory mechanisms. Mathematical modeling, which has been instrumental for understanding the NF-κB response in other cell types, offers a valuable tool to investigate the regulation of NF-κB activation in microglia at a systems level.ResultsWe quantify the dynamic response of NF-κB activation and activation of the upstream kinase IKK using ELISA measurements of a microglial cell line following treatment with the pro-inflammatory cytokine TNFα. A new mathematical model is developed based on these data sets using a modular procedure that exploits the feedback structure of the network. We show that the new model requires previously unmodeled dynamics involved in the stimulus-induced degradation of the inhibitor IκBα in order to properly describe microglial NF-κB activation in a statistically consistent manner. This suggests a more prominent role for the ubiquitin-proteasome system in regulating the activation of NF-κB to inflammatory stimuli. We also find that the introduction of nonlinearities in the kinetics of IKK activation and inactivation is essential for proper characterization of transient IKK activity and corresponds to known biological mechanisms. Numerical analyses of the model highlight key regulators of the microglial NF-κB response, as well as those governing IKK activation. Results illustrate the dynamic regulatory mechanisms and the robust yet fragile nature of the negative feedback regulated network.ConclusionsWe have developed a new mathematical model that incorporates previously unmodeled dynamics to characterize the dynamic response of the NF-κB signaling network in microglia. This model is the first of its kind for microglia and provides a tool for the quantitative, systems level study the dynamic cellular response to inflammatory stimuli.