Bio-polymerization processes like transcription and translation are central to proper function of a cell. The speed at which the bio-polymer grows is affected both by the number of pauses of elongation machinery, as well the number of bio-polymers due to crowding effects. In order to quantify these effects in fast transcribing ribosome genes, we rigorously show that a classical traffic flow model is the limit of a mean occupancy ODE model. We compare the simulation of this model to a stochastic model and evaluate the combined effect of the polymerase density and the existence of pauses on the instantaneous transcription rate of ribosomal genes.

A discontinuous Galerkin finite element method is used to approximate solutions to a classical traffic flow PDE. This PDE is used to model the biological process of transcription; the process of transferring genetic information from DNA either to mRNA or to rRNA. The transcription process is punctuated by short, frequent RNAP pauses which are incorporated into the model as traffic lights. These pauses cause a delay in the average transcription process. The DG solution of the nonlinear model is used to calculate the delay and to determine the effect of the pauses on the average transcription time. Numerical error measurements between the DG solution and the true solution (derived by the method of characteristics) are given for a simple model problem. It shows an excellent agreement in a neighborhood away from the shocks as well as O(∆x) convergence for the delay calculation. Preliminary parameter studies indicate that in a system with multiple pauses both the location and time duration of the pauses can significantly affect the average delay experienced by an RNAP.Mathematics subject classification: 65M60, 35R05, 35Q92, 35L65.

In this paper, we study the limitations imposed on the transcription process by the presence of short ubiquitous pauses and crowding. These effects are especially pronounced in highly transcribed genes such as ribosomal genes (rrn) in fast growing bacteria. Our model indicates that the quantity and duration of pauses reported for protein-coding genes is incompatible with the average elongation rate observed in rrn genes. When maximal elongation rate is high, pause-induced traffic jams occur, increasing promoter occlusion, thereby lowering the initiation rate. This lowers average transcription rate and increases average transcription time. Increasing maximal elongation rate in the model is insufficient to match the experimentally observed average elongation rate in rrn genes. This suggests that there may be rrn-specific modifications to RNAP, which then experience fewer pauses, or pauses of shorter duration than those in protein-coding genes. We identify model parameter triples (maximal elongation rate, mean pause duration time, number of pauses) which are compatible with experimentally observed elongation rates. Average transcription time and average transcription rate are the model outputs investigated as proxies for cell fitness. These fitness functions are optimized for different parameter choices, opening up a possibility of differential control of these aspects of the elongation process, with potential evolutionary consequences. As an example, a gene’s average transcription time may be crucial to fitness when the surrounding medium is prone to abrupt changes. This paper demonstrates that a functional relationship among the model parameters can be estimated using a standard statistical analysis, and this functional relationship describes the various trade-offs that must be made in order for the gene to control the elongation process and achieve a desired average transcription time. It also demonstrates the robustness of the system when a range of maximal elongation rates can be balanced with transcriptional pause data in order to maintain a desired fitness.