LARGE SYSTEMS OF MANY INTERCONNECTED DYNAMICAL UNITS: GENE NETWORK INFERENCE, EPIGENETIC HERITABILITY, AND EMERGENT BEHAVIOR IN OSCILLATOR SYSTEMS
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In this thesis, which consists of three parts, we investigate problems related to systems biology and collective behavior in complex systems.
The first part studies genetic networks that are inferred using gene expression data. Here we use established transcriptional regulatory interactions (TRIs) in combination with microarray expression data from both Escherichia coli (a prokaryote) and Saccharomyces cerevisiae (a eukaryote) to assess the accuracy of predictions of coregulated gene pairs and TRIs from observations of coexpressed gene pairs. We find that highly coexpressed gene pairs are more likely to be coregulated than to share a TRI for Saccharomyces cerevisiae, while the incidence of TRIs in highly coexpressed gene pairs is higher for Escherichia coli. The data processing inequality (DPI) of information theory has previously been applied for the inference of TRIs. We consider the case where a transcription factor gene is known to regulate two genes (one of which is a transcription factor gene) that are known not to regulate one another. According to the DPI if certain conditions hold, the non-interacting gene pairs should have the smallest mutual information among all pairs in the triplets. While we observe that this is sometimes the case for Escherichia coli, we find that it is almost always not the case for Saccharomyces cerevisiae, thus indicating that the assumed conditions under which the DPI was derived do not hold.
The second part of this dissertation is related to the dynamical process of epigentic heritability. Epigenetic modifications to histones may promote either activation or repression of the transcription of nearby genes. Recent experimental studies show that the promoters of many lineage-control genes in stem cells have "bivalent domains" in which the nucleosomes contain both active (H3K4me3) and repressive (H3K27me3) marks. Here we formulate a mathematical model to investigate the dynamic properties of bivalent histone modification patterns, and we predict some interesting and potentially experimental observable features.
The third part of this dissertation studies dynamical systems in which a large number $N$ of identical Landau-Stuart oscillators are globally coupled via a mean-field. Previously, it has been observed that this type of system can exhibit a variety of different dynamical behaviors including clumped states (in which each oscillator is in one of a small number of groups for which all oscillators in each group have the same state which is different from group to group), as well as extensive chaos (a situation in which all oscillators have different states and the macroscopic dynamics of the mean field is chaotic). One of our foci is the transition between clumped states and extensive chaos as the system is subjected to slow adiabatic parameter change. We observe and analyze explosive discontinuous transitions between the clumped states and the extensively chaotic states. Also, we study the fractal structures of the extensively chaotic attractors.