Biological clocks generate rhythms with periods from seconds to months in many organisms and control many processes that are critical to the survival of the organism. Many rhythms in biology are the result of rhythms in the mRNA and protein abundances at the cellular level that are then synchronized within the organism. To better understand biological clocks, mathematical and computational modeling is crucial as these tools provide directions for experimental procedures, test hypotheses within these models, and corroborate new biological revelations. Therefore, we apply novel mathematical and computational techniques to obtain insights into biological clocks at the molecular level. First, we generalize and analyze an ODE model of the repressilator with an arbitrary number of genes. Previous models of the repressilator were derived from assumptions that are biologically restrictive. These previous models assume first-order transcription, translation, and degradation, with rates equivalent among genes, mRNAs, and proteins, respectively. This assumption, however, is not consistent with current biological knowledge. Accordingly, we propose a new repressilator model allowing for differing transcription, translation, and degradation terms. We show that, under conditions on these new functions, there is still a unique steady state when an odd number of genes are in the network. We also show that, with an odd number of genes, either the model converges to the steady state or to a periodic orbit. Finally, we give a counterexample to a recent conjecture proposed by Tyler et al. Taken together, our results advance the theoretical study of cyclic gene repression by generalizing the current repressilator models. Next, we derive a new transcription-rate function that arises from more reasonable biological assumptions than the traditionally used transcription-rate function. Furthermore, we analyze the qualitative differences in the period, amplitude, and phase of a model with our new transcriptionrate function and a model with the old transcription-rate function. Our numerical simulations reveal drastic differences in the period, amplitude, and phase of the protein waveforms between the two models. Finally, we present novel algorithms to address the parameter estimation of the repressilator. Parameter estimation of models of biological oscillators is inherently challenging due to the nonlinearity present in the system. We illustrate two specific challenges by attempting to fit the parameters of the repressilator using traditional techniques. We then revisit two standard parameter estimation procedures and show how they fail to generate accurate parameter estimates. Next, we present three novel algorithms, two that take as input time-course data of the repressilator and output parameter estimates. The third algorithm takes as input time-course data of the repressilator along with a specific parameter estimate and outputs the period of the model solution with that parameter estimate as well as model solution values at the time points of the data. Ultimately, we show that our new procedures are more accurate than the two standards in the field of computational biology and a commonly used global optimization procedure, the particle swarm algorithm.
