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Reaction Kinetics, Mechanisms and Catalysis

, Volume 126, Issue 1, pp 3–30 | Cite as

On three genetic repressilator topologies

  • Maša Dukarić
  • Hassan Errami
  • Roman Jerala
  • Tina Lebar
  • Valery G. Romanovski
  • János TóthEmail author
  • Andreas Weber
Article
  • 27 Downloads

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. 1.
    Allwright DJ (1977) A global stability criterion for simple control loops. J Math Biol 4(4):363–373Google Scholar
  2. 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):375–388Google Scholar
  3. 3.
    Boros B (2017) Existence of positive steady states for weakly reversible mass-action systems. arXiv:1710.04732
  4. 4.
    Bratsun D, Volfson D, Tsimring LS, Hasty J (2005) Delay-induced stochastic oscillations in gene regulation. Proc Natl Acad Sci USA 102(41):14593–14598Google Scholar
  5. 5.
    Brown CW (2004) QEPCAD B: a system for computing with semi-algebraic sets via cylindrical algebraic decomposition. ACM SIGSAM Bull 38(1):23–24Google Scholar
  6. 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):475–511Google Scholar
  7. 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. 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 3:968Google Scholar
  9. 9.
    Cox D, Little J, O'shea D (2007) Ideals, varieties, and algorithms, vol 3. Springer, New YorkGoogle Scholar
  10. 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. 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. 12.
    Dilão R (2014) The regulation of gene expression in eukaryotes: bistability and oscillations in repressilator models. J Theor Biol 340:199–208Google Scholar
  13. 13.
    Dolzmann A, Sturm T (1997) Redlog: computer algebra meets computer logic. ACM Sigsam Bull 31(2):2–9Google Scholar
  14. 14.
    El Kahoui M, Weber A (2000) Deciding Hopf bifurcations by quantifier elimination in a software-component architecture. J Symb Comput 30(2):161–179Google Scholar
  15. 15.
    Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403(6767):335–338Google Scholar
  16. 16.
    Érdi P, Lente G (2016) Theory and (Mostly) systems biological applications. Springer Series in Synergetics. Springer, New YorkGoogle Scholar
  17. 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. 18.
    Fraser A, Tiwari J (1974) Genetical feedback-repression: II. Cyclic genetic systems. J Theor Biol 47(2):397–412Google Scholar
  19. 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):203–208Google Scholar
  20. 20.
    Garg A, Lohmueller JJ, Silver PA, Armel TZ (2012) Engineering synthetic TAL effectors with orthogonal target sites. Nucleic Acids Res 40(15):7584–7595Google Scholar
  21. 21.
    Gianni P, Trager B, Zacharias G (1988) Gröbner bases and primary decomposition of polynomial ideals. J Symb Comput 6(2–3):149–167Google Scholar
  22. 22.
    Goodwin BC (1965) Oscillatory behavior in enzymatic control processes. Adv Enzyme Regul 3:425–437Google Scholar
  23. 23.
    Griffith JS (1968) Mathematics of cellular control processes I. Negative feedback to one gene. J Theor Biol 20(2):202–208Google Scholar
  24. 24.
    Guantes R, Poyatos JF (2006) Dynamical principles of two-component genetic oscillators. PLoS Comput Biol 2(3):e30Google Scholar
  25. 25.
    Jacob F, Monod J (1961) Genetic regulatory mechanisms in the synthesis of proteins. J Mol Biol 3(3):318–356Google Scholar
  26. 26.
    Joshi B, Shiu A (2013) Atoms of multistationarity in chemical reaction networks. J Math Chem 51(1):153–178Google Scholar
  27. 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):723–726Google Scholar
  28. 28.
    Kiss K, Tóth J (2009) $n$-Dimensional ratio-dependent predator-prey systems with memory. Differ Equ Dyn Syst 17(1–2):17–35Google Scholar
  29. 29.
    Kuznetsov A, Afraimovich V (2012) Heteroclinic cycles in the repressilator model. Chaos Solitons Fract 45(5):660–665Google Scholar
  30. 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):1050–1058Google Scholar
  31. 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 5:5007Google Scholar
  32. 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):5180–5187Google Scholar
  33. 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):905–937Google Scholar
  34. 34.
    Nagy AL, Papp D, Tóth J (2012) ReactionKinetics—a mathematica package with applications. Chem Eng Sci 83:12–23Google Scholar
  35. 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):698–710Google Scholar
  36. 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):315–347Google Scholar
  37. 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):1173–1183Google Scholar
  38. 38.
    Romanovski V, Shafer D (2009) The center and cyclicity problems: a computational algebra approach. Birkhäuser, BostonGoogle Scholar
  39. 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):113–117Google Scholar
  40. 40.
    Sturm T (2007) ${ Redlog}$ online resources for applied quantifier elimination. Acta Acad Abo B 67(2):177–191Google Scholar
  41. 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):493–515Google Scholar
  42. 42.
    Thieffry D, Thomas R (1997) Qualitative analysis of gene networks. In: Biocomputing’98—proceedings of the pacific symposium, pp 77–88Google Scholar
  43. 43.
    Tigges M, Marquez-Lago TT, Stelling J, Fussenegger M (2009) A tunable synthetic mammalian oscillator. Nature 457(7227):309–312Google Scholar
  44. 44.
    Tóth J, Li G, Rabitz H, Tomlin AS (1997) The effect of lumping and expanding on kinetic differential equations. SIAM J Appl Math 57:1531–1556Google Scholar
  45. 45.
    Tóth J, Nagy AL, Papp D (2018) Reaction kinetics: exercises, programs and theorems. Springer, BerlinGoogle Scholar
  46. 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):126–129Google Scholar
  47. 47.
    Tyler J, Shiu A, Walton J (2018) Revisiting a synthetic intracellular regulatory network that exhibits oscillations, pp 1–25. arXiv:1808.00595
  48. 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. 49.
    Wang R, Jing Z, Chen L (2005) Modelling periodic oscillation in gene regulatory networks by cyclic feedback systems. Bull Math Biol 67(2):339–367Google Scholar
  50. 50.
    Widder S, Macía J, Solé R (2009) Monomeric bistability and the role of autoloops in gene regulation. PloS ONE 4(4):e5399Google Scholar
  51. 51.
    WRI (2018) Mathematica 11.3. http://www.wolfram.com
  52. 52.
    Yang X (2002) Generalized form of Hurwitz-Routh criterion and Hopf bifurcation of higher order. Appl Math Lett 15(5):615–621Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Center for Applied Mathematics and Theoretical PhysicsUniversity of MariborMariborSlovenia
  2. 2.Institut für Informatik IIUniversität BonnBonnGermany
  3. 3.Department for Synthetic Biology and ImmunologyNational Institute of ChemistryLjubljanaSlovenia
  4. 4.Faculty of Electrical Engineering and Computer ScienceUniversity of MariborMariborSlovenia
  5. 5.Faculty of Natural Science and MathematicsUniversity of MariborMariborSlovenia
  6. 6.Department of Mathematical AnalysisBudapest University of Technology and EconomicsBudapestHungary
  7. 7.Chemical Kinetics LaboratoryEötvös Loránd UniversityBudapestHungary

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