Sensitivity Measures for Oscillating Systems: Application to Mammalian Circadian Gene Network

Abstract

Vital physiological behaviors exhibited daily by bacteria, plants, and animals are governed by endogenous oscillators called circadian clocks. The most salient feature of the circadian clock is its ability to change its internal time (phase) to match that of the external environment. The circadian clock, like many oscillators in nature, is regulated at the cellular level by a complex network of interacting components. As a complementary approach to traditional biological investigation, we utilize mathematical models and systems theoretic tools to elucidate these mechanisms. The models are systems of ordinary differential equations exhibiting stable limit cycle behavior. To study the robustness of circadian phase behavior, we use sensitivity analysis. As the standard set of sensitivity tools are not suitable for the study of phase behavior, we introduce a novel tool, the parametric impulse phase response curve (pIPRC).

Oscillator Model Reduction Preserving the Phase Response: Application to the Circadian Clock

“…1 with respect to the parameter of interest p j . In Taylor et al (14), we showed that the phase evolution equation is a good predictor for the phase response to arbitrary signals. However, to use the phase evolution equation, the signal must be known or postulated a priori.…”

“…The second is the effect of p j through each state, a measure acquired by computing components of the cumulative phase sensitivity. The cumulative phase sensitivity (14,23,37) predicts the phase shift incurred by a longterm perturbation to p j , i.e., df/dp j (t) predicts the phase shift due to a perturbation to p j lasting from time 0 to time t (the implication is that the cumulative phase sensitivity is not a periodic measure-the effects of a perturbation accumulate over time, leading to greater phase shifts at time t 1 1 t than at time t 1 ). The long-term perturbation is defined according to df dp j ðtÞ ¼ +…”

“…The PRC maps signal arrival time to the resultant phase shift. To study the phase response capabilities of a model separate from the signal, we developed the parametric impulse phase response curve (pIPRC) (14), which is an infinitesimal analog to the PRC. For an arbitrary signal manifesting as the modulation of a model parameter, the pIPRC can be used to predict the response to that signal and can be said to characterize the phase behavior of the oscillator.…”

“…For circadian clock models, a stimulus is modeled as a time-varying perturbation to a parameter. In Taylor et al (14), using Eqs. 2 and 3 as our theoretical bases, we developed the parametric impulse phase response curve (pIPRC) and the accompanying phase evolution equation.…”

Pharmaceutical-based entrainment of circadian phase via nonlinear model predictive control

“…where ∂ ∂t ∂Φ γ ∂ p is the infinitesimal parametric PRC (ipPRC) for the oscillator on the limit cycle [37]. The formulation (9) is valid up to the first-order approximation in both duration and magnitude of the control, and is referred to as the phase-reduced model.…”