# 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).

## References

- 1.Allwright DJ (1977) A global stability criterion for simple control loops. J Math Biol 4(4):363–373CrossRefGoogle Scholar
- 2.Arányi P, Tóth J (1977) A full stochastic description of the Michaelis-Menten reaction for small systems. Acta Biochim Biophys Acad Sci Hung 12(4):375–388PubMedGoogle Scholar
- 3.Boros B (2017) Existence of positive steady states for weakly reversible mass-action systems. arXiv:1710.04732
- 4.Bratsun D, Volfson D, Tsimring LS, Hasty J (2005) Delay-induced stochastic oscillations in gene regulation. Proc Natl Acad Sci USA 102(41):14593–14598CrossRefGoogle Scholar
- 5.Brown CW (2004) QEPCAD B: a system for computing with semi-algebraic sets via cylindrical algebraic decomposition. ACM SIGSAM Bull 38(1):23–24CrossRefGoogle Scholar
- 6.Buchberger B (2006) Bruno Buchberger’s PhD thesis 1965: an algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. J Symb Comput 41(3–4):475–511CrossRefGoogle Scholar
- 7.Collins GE (1975) Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In: Second GI conference, automata theory and formal languages. Lecture Notes in Computer Science, vol 33, pp 134–183Google Scholar
- 8.Cong L, Zhou R, Kuo Y, Cunniff M, Zhang F (2012) Comprehensive interrogation of natural TALE DNA-binding modules and transcriptional repressor domains. Nat Commun 3:968CrossRefGoogle Scholar
- 9.Cox D, Little J, O'shea D (2007) Ideals, varieties, and algorithms, vol 3. Springer, New YorkCrossRefGoogle Scholar
- 10.Decker W, Laplagne S, Pfister G, Schonemann HA (2010) SINGULAR 3-1 library for computing the prime decomposition and radical of ideals, primdec.libGoogle Scholar
- 11.Decker W, Laplagne S, Pfister G, Schönemann HA (2012) SINGULAR 3-1-6—a computer algebra system for polynomial computations. http://www.singular.uni-kl.de
- 12.Dilão R (2014) The regulation of gene expression in eukaryotes: bistability and oscillations in repressilator models. J Theor Biol 340:199–208CrossRefGoogle Scholar
- 13.Dolzmann A, Sturm T (1997) Redlog: computer algebra meets computer logic. ACM Sigsam Bull 31(2):2–9CrossRefGoogle Scholar
- 14.El Kahoui M, Weber A (2000) Deciding Hopf bifurcations by quantifier elimination in a software-component architecture. J Symb Comput 30(2):161–179CrossRefGoogle Scholar
- 15.Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403(6767):335–338CrossRefGoogle Scholar
- 16.Érdi P, Lente G (2016) Theory and (Mostly) systems biological applications. Springer Series in Synergetics. Springer, New YorkGoogle Scholar
- 17.Érdi P, Tóth J (1989) Mathematical models of chemical reactions. Theory and applications of deterministic and stochastic models. Princeton University Press, PrincetonGoogle Scholar
- 18.Fraser A, Tiwari J (1974) Genetical feedback-repression: II. Cyclic genetic systems. J Theor Biol 47(2):397–412CrossRefGoogle Scholar
- 19.Gaber R, Lebar T, Majerle A, Šter B, Dobnikar A, Benčina M, Jerala R (2014) Designable DNA-binding domains enable construction of logic circuits in mammalian cells. Nat Chem Biol 10(3):203–208CrossRefGoogle Scholar
- 20.Garg A, Lohmueller JJ, Silver PA, Armel TZ (2012) Engineering synthetic TAL effectors with orthogonal target sites. Nucleic Acids Res 40(15):7584–7595CrossRefGoogle Scholar
- 21.Gianni P, Trager B, Zacharias G (1988) Gröbner bases and primary decomposition of polynomial ideals. J Symb Comput 6(2–3):149–167CrossRefGoogle Scholar
- 22.Goodwin BC (1965) Oscillatory behavior in enzymatic control processes. Adv Enzyme Regul 3:425–437CrossRefGoogle Scholar
- 23.Griffith JS (1968) Mathematics of cellular control processes I. Negative feedback to one gene. J Theor Biol 20(2):202–208CrossRefGoogle Scholar
- 24.Guantes R, Poyatos JF (2006) Dynamical principles of two-component genetic oscillators. PLoS Comput Biol 2(3):e30CrossRefGoogle Scholar
- 25.Jacob F, Monod J (1961) Genetic regulatory mechanisms in the synthesis of proteins. J Mol Biol 3(3):318–356CrossRefGoogle Scholar
- 26.Joshi B, Shiu A (2013) Atoms of multistationarity in chemical reaction networks. J Math Chem 51(1):153–178Google Scholar
- 27.Kiani S, Beal J, Ebrahimkhani MR, Huh J, Hall RN, Xie Z, Li Y, Weiss R (2014) CRISPR transcriptional repression devices and layered circuits in mammalian cells. Nat Methods 11(7):723–726CrossRefGoogle Scholar
- 28.Kiss K, Tóth J (2009) $n$-Dimensional ratio-dependent predator-prey systems with memory. Differ Equ Dyn Syst 17(1–2):17–35CrossRefGoogle Scholar
- 29.Kuznetsov A, Afraimovich V (2012) Heteroclinic cycles in the repressilator model. Chaos Solitons Fract 45(5):660–665CrossRefGoogle Scholar
- 30.Lebar T, Jerala R (2016) Benchmarking of TALE-and CRISPR/dCas9-based transcriptional regulators in mammalian cells for the construction of synthetic genetic circuits. ACS Synth Biol 5(10):1050–1058CrossRefGoogle Scholar
- 31.Lebar T, Bezeljak U, Golob A, Jerala M, Kadunc L, Pirš B, Stražar M, Vučko D, Zupančič U, Benčina M, Forstnerič V, Gaber R, Lonzarić J, Majerle A, Oblak A, Smole A, Jerala R (2014) A bistable genetic switch based on designable DNA-binding domains. Nat Commun 5:5007CrossRefGoogle Scholar
- 32.Lohmueller JJ, Armel TZ, Silver PA (2012) A tunable zinc finger-based framework for Boolean logic computation in mammalian cells. Nucleic Acids Res 40(11):5180–5187CrossRefGoogle Scholar
- 33.Müller S, Hofbauer J, Endler L, Flamm C, Widder S, Schuster P (2006) A generalized model of the repressilator. J Math Biol 53(6):905–937CrossRefGoogle Scholar
- 34.Nagy AL, Papp D, Tóth J (2012) ReactionKinetics—a mathematica package with applications. Chem Eng Sci 83:12–23CrossRefGoogle Scholar
- 35.Nissim L, Perli SD, Fridkin A, Perez-Pinera P, Lu TK (2014) Multiplexed and programmable regulation of gene networks with an integrated RNA and CRISPR/Cas toolkit in human cells. Mol Cell 54(4):698–710CrossRefGoogle Scholar
- 36.Orlov VN, Rozonoer LI (1984) The macrodynamics of open systems and the variational principle of the local potential II. Applications. J Frankl Inst 318(5):315–347CrossRefGoogle Scholar
- 37.Qi LS, Larson MH, Gilbert LA, Doudna JA, Weissman JS, Arkin AP, Lim WA (2013) Repurposing CRISPR as an RNA-guided platform for sequence-specific control of gene expression. Cell 152(5):1173–1183CrossRefGoogle Scholar
- 38.Romanovski V, Shafer D (2009) The center and cyclicity problems: a computational algebra approach. Birkhäuser, BostonGoogle Scholar
- 39.Sipos T, Tóth J, Érdi P (1974) Stochastic simulation of complex chemical reactions by digital computer, I. The model. React Kinet Catal Lett 1(1):113–117CrossRefGoogle Scholar
- 40.Sturm T (2007) ${ Redlog}$ online resources for applied quantifier elimination. Acta Acad Abo B 67(2):177–191Google Scholar
- 41.Sturm T, Weber A, Abdel-Rahman EO (2009) Investigating algebraic and logical algorithms to solve Hopf bifurcation problems in algebraic biology. Math Comput Sci 2(3):493–515CrossRefGoogle Scholar
- 42.Thieffry D, Thomas R (1997) Qualitative analysis of gene networks. In: Biocomputing’98—proceedings of the pacific symposium, pp 77–88Google Scholar
- 43.Tigges M, Marquez-Lago TT, Stelling J, Fussenegger M (2009) A tunable synthetic mammalian oscillator. Nature 457(7227):309–312CrossRefGoogle Scholar
- 44.Tóth J, Li G, Rabitz H, Tomlin AS (1997) The effect of lumping and expanding on kinetic differential equations. SIAM J Appl Math 57:1531–1556CrossRefGoogle Scholar
- 45.Tóth J, Nagy AL, Papp D (2018) Reaction kinetics: exercises, programs and theorems. Springer, BerlinCrossRefGoogle Scholar
- 46.Tsai TY, Choi YS, Ma W, Pomerening JR, Tang C, Ferrell JEJ (2008) Robust, tunable biological oscillations from interlinked positive and negative feedback loops. Science 321(5885):126–129CrossRefGoogle Scholar
- 47.Tyler J, Shiu A, Walton J (2018) Revisiting a synthetic intracellular regulatory network that exhibits oscillations, pp 1–25. arXiv:1808.00595
- 48.Vol’pert AI, Hudjaev SI (1985) Analysis in classes of discontinuous functions and the equations of mathematical physics. Martinus Nijhoff Publishers, Dordrecht. In Russian: Nauka, Moscow, (1975)Google Scholar
- 49.Wang R, Jing Z, Chen L (2005) Modelling periodic oscillation in gene regulatory networks by cyclic feedback systems. Bull Math Biol 67(2):339–367CrossRefGoogle Scholar
- 50.Widder S, Macía J, Solé R (2009) Monomeric bistability and the role of autoloops in gene regulation. PloS ONE 4(4):e5399CrossRefGoogle Scholar
- 51.WRI (2018) Mathematica 11.3. http://www.wolfram.com
- 52.Yang X (2002) Generalized form of Hurwitz-Routh criterion and Hopf bifurcation of higher order. Appl Math Lett 15(5):615–621CrossRefGoogle Scholar