Theses and Dissertations at Montana State University (MSU)
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Item Mathematical modeling for transcription of DNA with pausing : stochastic model with torque, and diffusive transport model(Montana State University - Bozeman, College of Letters & Science, 2016) Heberling, Tamra Lindsey; Chairperson, Graduate Committee: Lisa DavisIn fast-transcribing prokaryotic genes, like an rrn gene in Escherichia coli, many RNA polymerases (RNAPs) transcribe the DNA simultaneously. Active elongation of RNAPs is often interrupted by pauses, which has been observed to cause RNAP traffic jams; yet some studies indicate that elongation seems to be faster in the presence of multiple RNAPs than elongation by a single RNAP. We propose that an interaction between RNAPs via the torque produced by RNAP on helically twisted DNA can explain this apparent paradox. We have incorporated the torque mechanism into a stochastic model and simulated transcription both with and without torque. Simulation results illustrate that the torque causes shorter pause durations and fewer collisions between polymerases. Our results suggest that the torsional interaction of RNAPs is an important mechanism in maintaining fast transcription times, and that transcription should be viewed as a cooperative group effort by multiple polymerases. In an effort to further understand transcription, we investigate the Brownian ratchet model for nucleotide translocation. We model elongation as diffusive particle transport in a tilted periodic potential. To incorporate the RNAP pauses, a second periodic potential is added to the first. We present a formula for the mean escape time from a tilted, periodic potential composed of multiple periodic functions as the product of the mean escape time from each individual periodic function. This formula is extended to an arbitrary finite number of periodic functions. Two examples using truncated Fourier series are presented and analyzed.Item Discontinuous Galerkin finite element method for simulation of a transcription process model(Montana State University - Bozeman, College of Letters & Science, 2013) Thorenson, Jennifer Rae; Chairperson, Graduate Committee: Lisa DavisThe classical traffic flow PDE from the 1950s is used to model the biological process of transcription; the process of transferring genetic information from DNA to mRNA, in an E. coli gene. Polymerase elongating along the DNA strand encounter frequent but short pauses which are incorporated into the transcription model as several traffic lights. These pauses result in a delay in the transcription time and a delay function is defined to quantify this effect. Numerical simulations of the PDE model are conducted using a discontinuous Galerkin finite element method (DG) formulation. The entropy satisfying weak solution of the PDE model with a single pause is derived using the method of characteristics. This weak solution is used to show convergence of the DG formulation even though the flux function is not smooth. Once convergence of the DG solution is established for one pause, the numerical simulation for multiple pauses is used to calculate the delay due to the pauses and determine their effect on the overall transcription time. Preliminary parameter studies show a complex relationship between pause location and delay values. To determine the effect of pause clustering on protein production, an ongoing research goal is optimization of the delay function with respect to pause location. For preliminary work on this optimization problem, a DG formulation used to solve a sensitivity equation for a linear hyperbolic PDE with a spatial interface parameter is derived to gain insight for the more complicated nonlinear traffic flow PDE.