Sensitivity and Identifiability Analyses

I expect that SA of stochastic and multiscale models will be important in traditional tasks such as the identification of rate-determining steps and parameter estimation. I propose that SA will also be a key tool in controlling errors in information passing between scales. For example, within a multiscale framework, one could identify what features of a coarse-level model are affected from a finer scale model and need higher-level theory to improve accuracy of the overall multiscale simulation. Next a brief overview of SA for deterministic systems is given followed by recent work on SA of stochastic and multiscale systems. 

SA determines the change in a response R as a result of a perturbation in one of the parameters P of the model. Parameters of a model can be any conceivable ones. For example, in a MD simulation, parameters could be all factors appearing in the intermolecular potential. Since the magnitude of various parameters can be very different, it is common to compute a normalized sensitivity coefficient (NSC) defined as 

SA of SODEs describing chemically reacting systems was introduced early on, in the case of white noise added to an ODE (Dacol and Rabitz, 1984). In addition to expected values (time or ensemble average quantities), SA of variances or other correlation functions, or even the entire pdf, may also be of interest. In other words, in stochastic or multiscale systems one may also be interested in identifying model parameters that mostly affect the variance of different responses. In many experimental systems, the noise is due to multiple sources as a result, comparison with model-based SA for parameter estimation needs identification of the sources of experimental noise for meaningful conclusions. 

One of the difficulties in performing SA of stochastic or more generally multiscale models is that a closed form equation does not often exist. As a result, brute force SA has so far been the method of choice, which, while possible, is computationally intensive. As suggested in Raimondeau et al. (2003), since the response obtained is noisy, one has to introduce relatively large perturbations to ensure that the responses are reliable, so that meaningful SA results are obtained. For most complex systems, local SA may not be feasible. However, I do not see this being an impediment since SA is typically used to rank-order the 

In our group we have used SA in lattice 2D and 3D KMC in order to identify key parameters for parameter estimation from experimental data (see corresponding section below). Finite difference approximations of NSC were employed (Raimondeau et al., 2003 Snyder and Ylachos, 2004). Drews et al. (2003a) motivated by extraction of parameters for Cu electrodeposition, obtained an expression for the sensitivity coefficient, analogous to Eq. (9), that minimizes the effect of noise on the NSC assuming that the variance of the stochastic correction is unaffected by the perturbation. 

---

Doyle and co-workers have used sensitivity and identifiability analyses in a complex genetic regulatory network to determine practically identifiable parameters (Zak et al., 2003), i.e., parameters that can be extracted from experiments with a certain confidence interval, e.g., 95%. The data used for analyses were based on simulation of their genetic network. Different perturbations (e.g., step, pulse) were exploited, and an identifiability analysis was performed. An important outcome of their analysis is that the best type of perturbations for maximizing the information content from hybrid multiscale simulations differs from that of the deterministic, continuum counterpart model. The implication of this interesting finding is that noise may play a role in systems-level tasks. 

---