On three genetic repressilator topologies

TL;DR

This paper introduces and analyzes three novel mathematical models of genetic repressilator topologies with added positive feedback loops, revealing their simple, stable dynamics and absence of oscillations or bifurcations.

Contribution

It presents new models of repressilator topologies with positive feedback, analyzed using advanced symbolic algorithms, showing their stable behavior and lack of complex oscillations.

Findings

All models have a single stable steady state.

Models exhibit small amplitude or no oscillations.

Hopf bifurcation is excluded in the 3D system.

Abstract

Novel mathematical models of three different repressilator topologies are introduced. As designable transcription factors have been shown to bind to DNA non-cooperatively, we have chosen models containing non-cooperative elements. The extended topologies involve three additional transcription regulatory elements---which can be easily implemented by synthetic biology---forming positive feedback loops. This increases the number of variables to six, and extends the complexity of the equations in the model. To perform our analysis we had to use combinations of modern symbolic algorithms of computer algebra systems Mathematica and Singular. The study shows that all the three models have simple dynamics what can also be called regular behaviour: they have a single asymptotically stable steady state with small amplitude oscillations in the 3D case and no oscillation in one of the 6D cases and damping oscillation in the second 6D case. Using the program QeHopf we were able to exclude the presence of Hopf bifurcation in the 3D system.