  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.
