Quasi-Stable Structures in Circular Gene Networks

  • S. D. Glyzin
  • A. Yu. Kolesov
  • N. Kh. Rozov


A new mathematical model is proposed for a circular gene network representing a system of unidirectionally coupled ordinary differential equations. The existence and stability of special periodic motions (traveling waves) for this system is studied. It is shown that, with a suitable choice of parameters and an increasing number m of equations in the system, the number of coexisting traveling waves increases indefinitely, but all of them (except for a single stable periodic solution for odd m) are quasistable. The quasi-stability of a cycle means that some of its multipliers are asymptotically close to the unit circle, while the other multipliers (except for a simple unit one) are less than unity in absolute value.


mathematical model circular gene network repressilator traveling wave asymptotics quasi-stability quasi-buffer phenomenon system of ordinary differential equations periodic solutions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. B. Elowitz and S. Leibler, “A synthetic oscillatory network of transcriptional regulators,” Nature 403, 335–338 (2000).CrossRefGoogle Scholar
  2. 2.
    A. N. Tikhonov, “Systems of differential equations containing small parameters in the derivatives,” Mat. Sb. 31 (3), 575–586 (1952).MathSciNetGoogle Scholar
  3. 3.
    E. P. Volokitin, “On limit cycles in the simplest model of a hypothetical genetic network,” Sib. Zh. Ind. Mat. 7 (3), 57–65 (2004).MathSciNetzbMATHGoogle Scholar
  4. 4.
    O. Buse, A. Kuznetsov, and R. A. Peréz, “Existence of limit cycles in the repressilator equations,” Int. J. Bifurcation Chaos 19 (12), 4097–4106 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    O. Buse, A. Pérez, and A. Kuznetsov, “Dynamical properties of the repressilator model,” Phys. Rev. E 81 (066206) 066206–1–066206–7 (2010).MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. A. Likhoshvai, Yu. G. Matushkin, and S. I. Fadeev, “Problems in the theory of the functioning genetic networks,” Sib. Zh. Ind. Mat. 6 (2), 64–80 (2003).MathSciNetGoogle Scholar
  7. 7.
    V. A. Likhoshvai, G. V. Demidenko, S. I. Fadeev, Yu. G. Matushkin, and N. A. Kolchanov, “Mathematical simulation of regulatory circuits of gene networks,” Comput. Math. Math. Phys. 44 (12), 2166–2183 (2004).MathSciNetzbMATHGoogle Scholar
  8. 8.
    S. I. Fadeev and V. A. Likhoshvai, “On hypothetic genetic networks,” Sib. Zh. Ind. Mat. 6 (3), 134–153 (2003).MathSciNetzbMATHGoogle Scholar
  9. 9.
    A. Yu. Kolesov, N. Kh. Rozov, and V. A. Sadovnichii, “Periodic solutions of travelling-wave type in circular gene networks,” Izv. Math. 80 (3), 523–548 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “Existence and stability of the relaxation cycle in a mathematical repressilator model,” Math. Notes 101 (1), 71–86 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. Yu. Kolesov and Yu. S. Kolesov, Relaxation Oscillations in Mathematical Models of Ecology (Am. Math. Soc., Providence, 1997).zbMATHGoogle Scholar
  12. 12.
    S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “Relaxation self-oscillations in Hopfield networks with delay,” Izv. Math. 77 (2), 271–312 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “Periodic traveling-wave-type solutions in circular chains of unidirectionally coupled equations,” Theor. Math. Phys. 175 (1), 499–517 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “The buffer phenomenon in ring-like chains of unidirectionally connected generators,” Izv. Math. 78 (4), 708–743 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yu. M. Romanovskii, N. V. Stepanova, and D. S. Chernavskii, Mathematical Modeling in Biophysics (Inst. Komp’yut. Issled., Moscow, 2003) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • S. D. Glyzin
    • 1
    • 2
  • A. Yu. Kolesov
    • 1
  • N. Kh. Rozov
    • 3
  1. 1.Faculty of MathematicsYaroslavl State UniversityYaroslavlRussia
  2. 2.Scientific Center in ChernogolovkaRussian Academy of SciencesChernogolovkaRussia
  3. 3.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia

Personalised recommendations