Deterministic and stochastic mathematical modeling and computer simulation of the mammalian intracellular circadian cloc

Physiological systems within the cell are often so complex that simple diagrams and intuition are insufficient to relate detailed biochemical data with overall system behavior. Fortunately mathematical modeling and in particular simulation, can step in where intuition fails. This thesis is about the mathematical modeling and simulation of circadian clocks. We begin with a review and revision of many of the techniques currently used to mathematically represent and analyze biological processes within a cell. Particular focus is placed on the implicit assumptions typically made in mathematical terms representing processes like transcription or enzyme action, and in some cases we offer improved models.;A mathematical model of the mammalian circadian (near 24-hour period) clock is developed which incorporates a wide range of experimental data, and is by far the most detailed mathematical model of a circadian clock yet derived. Since the specific biochemical rates of reactions in the model have not yet been experimentally determined, we estimate the parameters of the model as an inverse problem on the experimentally determined time courses of mRNAs and Proteins within the clock. The model is accurate in its predictions with respect to mutations and can be used to understand key questions about clock structure and phase resetting.;Based on an experimental estimate of the number of molecules of key proteins within the mammalian circadian clock, we can directly, without ambiguity, simulate our model of the mammalian circadian clock with stochastic molecular interactions. Amazingly, interactions with promoters on the time scale of seconds are required for accurate 24-hour timekeeping. The stochasticity of our model follows the central limit theorem. Finally we find that non-redundant gene duplication can increase immunity to molecular noise by allowing for more frequent interactions between transcription factors and promoters.