Discontinuous Galerkin finite element method for simulation of a transcription process model

Abstract

The 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.