Sensitivity analysis of oscillating dynamical systems with applications to the mammalian circadian clock

Abstract

The work presented in this thesis consists of two major parts. In Chapter 2, the theory for sensitivity analysis of oscillatory systems is developed and discussed. Several contributions are made, in particular in the precise definition of phase sensitivities and in the generalization of the theory to all types of autonomous oscillators. All methods rely on the solution of a boundary value problem, which identifies the periodic orbit. The choice of initial condition on the limit cycle has important consequences for phase sensitivity analysis, and its influence is quantified and discussed in detail. The results are exact and efficient to compute compared to existing partial methods. The theory is then applied to different models of the mammalian circadian clock system in the following chapters. First, different types of sensitivities in a pair of smaller models are analyzed. The models have slightly different architectures, with one having an additional negative feedback loop compared to the other. The differences in their behavior with respect to phases, the period and amplitude are discussed in the context of their network architecture. It is found that, contrary to previous assumptions in the literature, the additional negative feedback loop makes the model less "flexible" in at least one sense that was studied here. The theory was also applied to larger, more detailed models of the mammalian circadian clock, based on the original model of Forger and Peskin. Between the original model's publication in 2003 and the present time, several key advances were made in understanding the mechanistic detail of the mammalian circadian clock, and at least one additional clock gene was identified. These advances are incorporated in an extended model, which is then studied using sensitivity analysis. Period sensitivity analysis is performed first and it was found that only one negative feedback loop dominates the setting of the period.

(cont.) This was an interesting one-to-one correlation between one topological feature of the network and a single metric of network performance. This led to the question of whether the network architecture is modular, in the sense that each of the several feedback loops might be responsible for a separate network function. A function of particular interest is the ability to separately track "dawn" and "dusk", which is reported to be present in the circadian clock. The ability of the mammalian circadian clock to modify different relative phases --defined by different molecular events -- independently of the period was analyzed. If the model can maintain a perceived day -- defined by the time difference between two phases -- of different lengths, it can be argued that the model can track dawn and dusk separately. This capability is found in all mammalian clock models that were studied in this work, and furthermore, that a network-wide effort is needed to do so. Unlike in the case of the period sensitivities, relative phase sensitivities are distributed throughout several feedback loops. Interestingly, a small number of "key parameters" could be identified in the detailed models that consistently play important roles in the setting of period, amplitude and phases. It appears that most circadian clock features are under shared control by local parameters and by the more global "key parameters". Lastly, it is shown that sensitivity analysis, in particular period sensitivity analysis, can be very useful in parameter estimation for oscillatory systems biology models. In an approach termed "feature-based parameter fitting", the model's parameter values are selected based on their impact on the "features" of an oscillation (period, phases, amplitudes) rather than concentration data points. It is discussed how this approach changes the cost function during the parameter estimation optimization, and when it can be beneficial.

(cont.) A minimal model system from circadian biology, the Goodwin oscillator, is taken as an example. Overall, in this thesis it is shown that the contributions made to the theoretical understanding of sensitivities in oscillatory systems are relevant and useful in trying to answer questions that are currently open in circadian biology. In some cases, the theory could indicate exactly which experiments or detailed mechanistic studies are needed in order to perform meaningful mathematical analysis of the system as a whole. It is shown that, provided the biologically relevant quantities are analyzed, a network-wide understanding of the interplay between network function and topology can be gained and differences in performance between models of different size or topology can be quantified.