Bayesian parameter estimation for dynamical models in systems biology

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.Author summary: Mathematical models of biological signal transduction networks have been widely used to capture the temporal behavior of such systems. Calibrating these models to increasingly available experimental data is essential to ensure that models accurately portray biological phenomena. However, measurement noise, the inability to measure all biochemical species in a system, and the lack of detailed knowledge about all reactions make model calibration difficult and can introduce errors. In this study, we propose a principled and complete computational framework to enable model calibration in the face of these challenges. Therein, we quantify any uncertainties (potential errors) in the calibrated model. We apply the framework to a series of example models demonstrating various dynamic regimes common in biology (limit cycles and steady states) to highlight how our method provided additional context and insights to modeling-based studies.