Rigorous Quantitative Analysis of Nonlinear Uncertain Biomolecular Systems using Validated Methods

The paper addresses the critical challenge of accurately characterising steady states in biomolecular systems, which are often complex, nonlinear, multistable and subject to significant uncertainties. Traditional numerical methods often fail to provide complete or guaranteed solutions under these conditions. To overcome these limitations, the research proposes and evaluates the application of interval analysis methodologies. We provided algorithms for interval Newton and interval Krawczyk methods for rigorously bounding all possible steady states (both stable and unstable) in multistable, multidimensional nonlinear systems. This study involves a comparative analysis of these two methods in conjunction with interval bisection and interval constraint propagation. We addressed numerical examples for an array of biologically plausible models, involving both feedback and feedforward gene networks. The work recommends the choice of the most suitable method for various types of biomolecular systems, ultimately offering a robust computational framework to understand cellular functions and design synthetic biological circuits.