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Abstract This report investigates a stabilization method for first order hyperbolic differential equations applied to DNA transcription modeling. It is known that the usual unstabilized finite element method contains spurious oscillations for nonsmooth solutions. To stabilize the finite element method the authors consider adding to the first order hyperbolic differential system a stabilization term in space and time filtering. Numerical analysis of the stabilized finite element algorithms and computations describing a few biological settings are studied herein. 

We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one describes the dependence of delay on state, and the other is the feedback nonlinearity. Both increasing and decreasing nonlinearities are considered. Our analysis is exhaustive both analytically and numerically as we examine the bifurcations of the system for various combinations of increasing and decreasing nonlinearities. We identify rich bifurcation patterns including Bautin, Bogdanov–Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depend on the derivative signs of nonlinearities. Our analysis confirms many of these patterns in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold. Perhaps one of the most surprising findings is the existence of a Hopf bifurcation to a periodic solution when the nonlinearity is monotone increasing and the time delay is a decreasing function of the state variable. 

The focus of this paper is the development, numerical simulation and parameter analysis of a model of the transcription of ribosomal RNA in highly transcribed genes. Inspired by the well-known classic Lighthill-Whitham-Richards (LWR) traffic flow model, a linear advection continuum model is used to describe the DNA transcription process. In this model, elongation velocity is assumed to be essentially constant as RNA polymerases move along the strand through different phases of gene transcription. One advantage of using the linear model is that it allows one to quantify how small perturbations in elongation velocity and inflow parameters affect important biology measures such as Average Transcription Time (ATT) for the gene. The ATT per polymerase is the amount of time an individual RNAP spends traveling through the DNA strand. The numerical treatment for model simulations includes introducing a low complexity and time accurate method by adding a simple linear time filter to the classic upwind scheme. This improved method is modular and requires a minimal modification of adding only one line of code resulting in increased accuracy without increased computational expense. In addition, it removes the overdamping of upwind. A stability condition for the new algorithm is derived, and numerical computations illustrate stability and convergence of the filtered scheme as well as improved ATT estimation.