Observations from recent biological experiments indicate that the motion of a transcription complex or a ribosome along the template is subject to multiple pauses of varying time durations. In the case of transcription of ribosomal RNA in bacteria, the density of elongating complexes is sufficiently high to experience traffic jams. Since the growth rate in a fast growing E. coli population is determined by the availability of ribosomes, which in turn is limited by ribosomal RNA transcription rates, the elongation process of ribosomal RNA is a key factor determining the E. coli growth rate. This research project uses a nonlinear hyperbolic partial differential equation (PDE) as a mathematical model for the transcription process of ribosomal RNA. The PDE takes the form of a nonlinear conservation law, and it was first proposed as a mathematical model for traffic patterns in the 1950's. Although this type of model was superseded in the study of traffic flow by models that take into account driver reactions, it is suitable for modeling elongation processes as it incorporates density dependent velocities and the presence of transcriptional pauses. The PDE model is used to estimate an instantaneous elongation rate in the presence of a non-uniform distribution of pauses, which are biologically important yet experimentally inaccessible quantities. The second major element of the research is the development of analytical as well as sophisticated numerical methods for verification, validation and sensitivity analysis of the model. Discontinuous Galerkin methods form the foundation of the computational schemes, allowing efficient numerical solution of systems of nonlinear conservation laws with discontinuous coefficients that encode the presence of pauses. These new tools will be used to determine the limits that transcriptional pauses impose on the mean and variance of the transcription rate of ribosomal RNA.
Maintenance of homeostasis, capacity to grow and ability to respond to stimuli are just a few of the characteristics of life that depend on each cell's ability to regulate production of molecules such as mRNA and proteins. Proteins are produced by ribosomes sliding along a long thin strand of messenger RNA (mRNA). In turn, mRNA is generated by a transcription complex sliding along the DNA template. These translation and transcription processes fall within the broad classification of "elongation processes." Although recent revolutionary advances in cell biology have improved scientists' qualitative understanding of life on a molecular level, efforts to construct predictive quantitative models are often limited by a lack of quantitative experimental data. However, one important system, transcription of ribosomal RNA, is sufficiently experimentally characterized to allow delineation of the limits it imposes on cellular physiology. The current work develops quantitative models of this transcription process in order to quantify limitations it imposes on bacterial growth rates. Connections between the molecular biophysics of elongation processes and the macroscopically observable growth rate are quantitatively studied. The current research combines careful modeling of a complex biological process with the development of new mathematical and numerical techniques to answer fundamental biological questions. The results may have a broad impact on our understanding of bacterial physiology and, through a better understanding of the driving mechanisms of bacterial growth, may have a long term impact on molecular biotechnology.
