# On three genetic repressilator topologies

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## 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 that can also be called regular behaviour: they have a single asymptotically stable steady state with small amplitude damping 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.

## Keywords

Repressilator models Genetic oscillator Steady states Computer algebra**Mathematica**

**Singular**

**QeHopf**Designable repressor

## Notes

### Acknowledgements

Maša Dukarić and Valery Romanovski are supported by the Slovenian Research Agency (Program P1-0306 and Project NI-0063) and by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme, FP7-PEOPLE-2012-IRSES-316338. The work has also been partially supported by the Hungarian-Slovenian cooperation projects TÉT_16-1-2016-0070 and BI-HU-17-18-011. Roman Jerala and Tina Lebar are supported by Slovenian Research Agency project J1-6740 and program P4-0176. Tina Lebar is partially supported by the UNESCO-L’OREAL national fellowship “For Women in Science”. János Tóth also acknowledges the support by the National Research, Development and Innovation Office (SNN 125739).

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