Design of Parameter Intervals to Meet Steady-State Specifications in Biomolecular Circuits using Interval Analysis

It is often desired, in the analysis or the design of biomolecular circuits, to search for parameters that satisfy certain specifications, such as on the number of the steady states or on their magnitudes. These problems are challenging because of the presence of multiple parameters, the nonlinear mapping between the parameters and the steady states, as well as the specifications themselves, such as when multiple steady states are desired. Typically, exact analytical solutions are limited and numerical approaches, if they converge, may not capture all solutions. We used Interval Analysis versions of Bisection and Constraint Propagation to obtain rigorous and guaranteed estimates of circuit parameters for the design problem. We established criteria that rule out the existence of design solutions in a given parameter space. We presented algorithms to rigorously bound all solutions and developed variants that enclosed the solutions as accurately as required. These theoretical results were illustrated on benchmark feedback and feedforward circuits. These results should aid in the analysis and design of biomolecular circuits as well as in other contexts with similar modelling frameworks. The rigorous nature of the results may be particularly useful for resource optimization and safety criticality in synthetic biology applications, such as in therapeutic applications and drug delivery.