Efficient Characterization of Parametric Uncertainty of Complex (Bio)chemical Networks

Parametric uncertainty is a particularly challenging and relevant aspect of systems analysis in domains such as systems biology where, both for inference and for assessing prediction uncertainties, it is essential to characterize the system behavior globally in the parameter space. However, current methods based on local approximations or on Monte-Carlo sampling cope only insufficiently with high-dimensional parameter spaces associated with complex network models. Here, we propose an alternative deterministic methodology that relies on sparse polynomial approximations. We propose a deterministic computational interpolation scheme which identifies most significant expansion coefficients adaptively. We present its performance in kinetic model equations from computational systems biology with several hundred parameters and state variables, leading to numerical approximations of the parametric solution on the entire parameter space. The scheme is based on adaptive Smolyak interpolation of the parametric solution at judiciously and adaptively chosen points in parameter space. As Monte-Carlo sampling, it is “non-intrusive” and well-suited for massively parallel implementation, but affords higher convergence rates. This opens up new avenues for large-scale dynamic network analysis by enabling scaling for many applications, including parameter estimation, uncertainty quantification, and systems design.Author Summary: In various scientific domains, in particular in systems biology, dynamic mathematical models of increasing complexity are being developed and analyzed to study biochemical reaction networks. A major challenge in dealing with such models is the uncertainty in parameters such as kinetic constants; how to efficiently and precisely quantify the effects of parametric uncertainties on systems behavior remains a question. Addressing this computational challenge for large systems, with good scaling up to hundreds of species and kinetic parameters, is important for many forward (e.g., uncertainty quantification) and inverse (e.g., system identification) problems. Here, we propose a sparse, deterministic adaptive interpolation method tailored to high-dimensional parametric problems that allows for fast, deterministic computational analysis of large biochemical reaction networks. The method is based on adaptive Smolyak interpolation of the parametric solution at judiciously chosen points in high-dimensional parameter space, combined with adaptive time-stepping for the actual numerical simulation of the network dynamics. It is “non-intrusive” and well-suited both for massively parallel implementation and for use in standard (systems biology) toolboxes.