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<p>This dissertation includes three projects in mathematical biology: (1) Mathematical
modeling of the circadian clock and dopamine, (2) Mathematical analysis of a circadian
clock model, and (3) Mathematical modeling of sex differences in one-carbon metabolism.
Circadian rhythms, dopamine, and metabolism are all important aspects of human physiology
and we use methods from dynamical systems and scientific computing to investigate
these systems at the molecular level. </p><p>First, we create a mathematical model
to understand the influences of the mammalian circadian clock on the neurotransmitter
dopamine (DA). The circadian clock circuitry in the suprachiasmatic nucleus (SCN)
drives 24-hour rhythms in many important physiological processes, including the dopaminergic
system. DA imbalances have been linked to various neurological and psychiatric conditions
such as Parkinson's disease, attention-deficit/hyperactivity disorder (ADHD), and
mood disorders. Previous studies have shown that these conditions are often accompanied
by disrupted circadian rhythms, but it has not been well understood why. We use our
mathematical model to explain the mechanisms by which the circadian clock influences
DA in the brain. We show that the model corresponds well with a host of experimental
data and accurately predicts daily variation in extracellular DA. </p><p>The model
is comprised of a system of nonlinear ordinary differential equations with solutions
displaying a remarkably robust 24-hour period consistent with the biology. We further
investigate different dynamical behaviors in the model, including periodicity, quasiperiodicity,
and decoupling. We show that these behaviors are biologically meaningful and may help
to explain clinically observed circadian or dopaminergic disruptions. In addition,
experiments have suggested several links between the mammalian clock and one-carbon
metabolism (OCM). OCM is essential for the synthesis of DNA and proteins, and we
use mathematical modeling to understand how the biochemical reactions in OCM are influenced
by sex hormones and micronutrients like folate, vitamin B12, and vitamin B6.</p>
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