Zheng, Likun2011-11-042011-11-042011-09https://hdl.handle.net/11299/117555University of Minnesota Ph.D. dissertation. September 2011. Major: Mathematics. Advisor: Hans G. Othmer. 1 computer file (PDF); ix, 145 pages, appendices p. 130-145.Stochasticity is inherent in biochemical systems. Noise can come from internal sources such as the random motion and reactions of molecules, and external sources such as environmental fluctuations. The main purpose of this thesis is to study how fluctuations propagate in biological systems. First, we focus on how a signaling molecule called a ligand searches for and binds to its target (receptor). There exist membrane proteins that can bind to the ligand molecule and localize it near receptors, affecting the association between it and the receptor. Our analysis shows that although the membrane protein can concentrate the ligand molecule near the receptor surface, the membrane protein has to pass the localized ligand molecule to the receptor fast enough, in order to enhance signaling. Otherwise, the membrane protein inhibits signaling. Moreover, we also study the effect of localization on signal specificity. In particular, we discuss how the membrane proteins bind to ligand molecules and distribute them to different downstream signaling pathways. Upon ligand binding to receptors, bound receptors can initiate the downstream network, which may finally lead to gene expression. We then study how the noise from the initiation step of transcription propagates in the elongation step. Elongation can be interrupted by the pauses of the transcription complex on the DNA sequence. We give a condition under which the pause of the transcription complex can cause bursts of mRNA production. Finally, we use stochastic simulations to study dorsal-ventral patterning in Drosophila numerically. Our results indicate that a feedback loop can stabilize the determination of the amnioserosa boundary. We then propose a detailed single cell system for the downstream network in nuclei. Our analysis of time scales of reactions and molecular transport shows the phosphorylation of Mad and transport of mRNA across the nuclear membrane are the major limiting steps in the signal transduction pathway. Simulations results show noises are amplified at these limiting steps.en-USGene controlPattern formationSignalingStochastic fluctuationsMathematicsThesis or Dissertation