  Dynamical systems modeling, particularly via systems of ordinary differential
equations, has been used to effectively capture the temporal behavior of
different biochemical components in signal transduction networks. Despite the
recent advances in experimental measurements, including sensor development and
'-omics' studies that have helped populate protein-protein interaction networks
in great detail, modeling in systems biology lacks systematic methods to
estimate kinetic parameters and quantify associated uncertainties. This is
because of multiple reasons, including sparse and noisy experimental
measurements, lack of detailed molecular mechanisms underlying the reactions,
and missing biochemical interactions. Additionally, the inherent nonlinearities
with respect to the states and parameters associated with the system of
differential equations further compound the challenges of parameter estimation.
In this study, we propose a comprehensive framework for Bayesian parameter
estimation and complete quantification of the effects of uncertainties in the
data and models. We apply these methods to a series of signaling models of
increasing mathematical complexity. Systematic analysis of these dynamical
systems showed that parameter estimation depends on data sparsity, noise level,
and model structure, including the existence of multiple steady states. These
results highlight how focused uncertainty quantification can enrich systems
biology modeling and enable additional quantitative analyses for parameter
estimation.
