Mathematical Modeling And Dynamical Analysis Of Two Biological Oscillators

The circadian clock and the cell cycle are two important biological oscillators in organisms. The circadian clock is an endogenous system which orchestrates the daily rhythms in physiology and behaviors. The cell cycle consists of a series of processes that lead to the cell division and proliferation. Disruption of the circadian rhythmicity and of the cell cycle can profoundly affect fitness. Experimental observations showed that both these oscillators are complex systems which include multiple redundant genes and integrated feedback loops. Mathematical modeling represents a powerful approach to interpret the mechanisms underlying these complex systems and their oscillatory behavior. Furthermore, the predictions made by the mathematical models provide a useful guide for further experiments.To explore the dynamical behaviors of the circadian clock and the cell cycle, as well as their underlying mechanisms, we use interdisciplinary methods in this thesis, which include(1) mathematical modeling,(2) bifurcation and stability analysis,(3) numerical simulations and qualitative predictions,(4) experimental validations of the theoretical predictions. The main novel results obtained in this thesis pertain to the following pints:(1) A new theoretical framework for the role of transcriptional regulation in the mammalian circadian clock:Experimental biologists found that some genes in the mammalian circadian clock and the downstream related systems are controlled by regulatory elements called E/E’ box, D box and RRE. The transcriptions regulated by these cis-elements are rhythmic. Most genes in the circadian clock are co-regulated by multiple rhythmic cis-elements. Although the expression pattern of a specific gene can be measured, the general rule of the transcriptional regulation in the mammalian circadian clock remains elusive.In the second chapter, we build mathematical models describing the transcriptional regulations and obtain general rules for gene expression in the mammalian circadian clock. This work provides a theoretical foundation for the study of the circadian clock and the downstream related systems.(2) Role of an auxiliary positive feedback loop in the oscillatory mechanism of the mammalian circadian clock:In the mammalian circadian clock, a negative feedback loop with time delay plays a primary role in the central mechanism generating the circadian oscillation. Recent experimental evidence showed that the Rev-erbα/ Cry1 auxiliary positive loop can also affect the circadian clock. However, the effect of this auxiliary loop on the oscillation of the circadian clock remains obscure.In the third chapter, we employ a differential equation with two time delays to describe the co-regulation of the primary negative feedback loop and the Rev-erbα/ Cry1 auxiliary positive loop. Using the Hopf bifurcation theory, we consider the two time delays as bifurcation parameters and find the conditions for sustained oscillations in the mammalian circadian clock.(3) Mechanism of period determination in the mammalian circadian clock based on transcriptional regulation:The period represents a major feature of the circadian clock. In some previous studies, the modifications and the degradations of proteins in the circadian clock were considered as the main factors affecting the period length. Recent experimental evidence showed that the clock period can also be significantly changed when the cis-elements are affected. However, the mechanism of the period determination based on the transcriptional regulation is still unclear. To address this issue, in the fourth chapter we build a new mathematical model of the mammalian circadian clock. Through a combined mathematical-experimental approach, we find that the Rev-erbα/Cry1 auxiliary positive loop can contribute to the period of the circadian clock. Also, if the post-translational time delay is assumed to be fixed, we show that the intensity ratio of the primary loop to the auxiliary loop can determine the period length. Furthermore, this ratio-regulating pattern can favor the period robustness of the circadian clock. These predictions are all confirmed by our experiments.(4) Mechanism of the size checkpoint incorporating spatial regulation in fission yeast:During proliferation, yeast cells need a specific control mechanism to maintain their proper size. In previous studies, the cell size was coupled to mitosis through the relationship between the cell size and the synthesis rate of Cyclin B, a protein involved in cell cycle progression. Recent experimental results revealed that a spatial regulation induced by the protein Pom1 can influence mitosis initiation. However, the effect of this spatial regulation on the size checkpoint is still unclear.In the fifth chapter, we build a mathematical model of the cell cycle in the fission yeast, which includes the Pom1 spatial regulation. The bifurcation analysis shows that a bistable response associated with the coexistence of two stable steady states can perform the function of the size checkpoint. Numerical simulations suggest that the spatial regulation induced by Pom1 protein can resist interference from different sources and thereby enhances the robustness of the size checkpoint. Finally, through bifurcation and stability analysis we obtain the theoretical conditions for the emergence of the size checkpoint in the presence of spatial regulation.The theoretical studies performed in this thesis contribute to clarify the mechanism of both the circadian clock and the cell cycle, in addition to providing novel design strategies for building synthetic biological oscillators.