Computing parametric sensitivities for oscillators has a now well-understood subtlety associated with the indeterminacy of phase. A less universal, but still vexing, subtlety arises when an oscillator is described by a system of differential equations with "hidden" conservation constraints (HCC's); defined as weighted sums of state variables that are time-invariant. If there are HCC's, as is commonly the case for models of biochemical oscillators but rarely the case for practical circuit oscillators, the now-standard approach to computing parametric sensitivities can yield incorrect results. In addition, the monodromy matrix (the matrix of state sensitivities over one oscillation period), is often defective in a way that interferes with the usual approach to computing oscillator phase noise. In this paper we analyze the HCC case, and show that by augmenting the standard sensitivity approach with explicit HCC's, one can recover the correct parametric sensitivities. In addition, we prove that there is a typically satisfied condition that guarantees that a system with HCCs will have a defective monodromy matrix. A deliberately "flawed" ring oscillator circuit and a cyanobacterial circadian clock biochemical oscillator are used to demonstrate the parametric sensitivity problem and its resolution, and to show the issue of the defective monodromy matrix.

Mechanism-based chemical kinetic models are increasingly being used to describe biological signaling. Such models serve to encapsulate current understanding of pathways and to enable insight into complex biological processes. One challenge in model development is that, with limited experimental data, multiple models can be consistent with known mechanisms and existing data. Here, we address the problem of model ambiguity by providing a method for designing dynamic stimuli that, in stimulus-response experiments, distinguish among parameterized models with different topologies, i.e., reaction mechanisms, in which only some of the species can be measured. We develop the approach by presenting two formulations of a model-based controller that is used to design the dynamic stimulus. In both formulations, an input signal is designed for each candidate model and parameterization so as to drive the model outputs through a target trajectory. The quality of a model is then assessed by the ability of the corresponding controller, informed by that model, to drive the experimental system. We evaluated our method on models of antibody-ligand binding, mitogen-activated protein kinase (MAPK) phosphorylation and de-phosphorylation, and larger models of the epidermal growth factor receptor (EGFR) pathway. For each of these systems, the controller informed by the correct model is the most successful at designing a stimulus to produce the desired behavior. Using these stimuli we were able to distinguish between models with subtle mechanistic differences or where input and outputs were multiple reactions removed from the model differences. An advantage of this method of model discrimination is that it does not require novel reagents, or altered measurement techniques; the only change to the experiment is the time course of stimulation. Taken together, these results provide a strong basis for using designed input stimuli as a tool for the development of cell signaling models.