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Sib. Zh. Ind. Mat., 2003, Volume 6, Number 2, Pages 64–80 (Mi sjim448)  

This article is cited in 31 scientific papers (total in 31 papers)

Problems in the theory of the functioning of genetic networks

V. A. Likhoshvaia, Yu. G. Matushkina, S. I. Fadeevb

a Institute of cytology and genetics Siberian Branch of the Russian Academy of Sciences
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We consider hypothetical genetic networks (HGN), which are theoretical objects constructed from elementary units of two types: matrix elements and regulatory mechanisms. Their operation is described by autonomous systems consisting of$n$ equations and related equations of a special form with retarded arguments. We give examples of HGN that have stable states, limit cycles, and attractors of a complicated nature. We introduce four classes of HGN that correspond to these regulatory relations and determine their canonical representations. For symmetric canonical HGN we formulate an $(n,k)$-criterion that enables us to predict in a certain parameter range their limit regimes of operation, i.e., to predict for the corresponding models the number and character of the stable limit states in them. In the case of HGN of the first and second classes, we formulate a conjecture on the one-to-one correspondence between the stable states and the 1-bases of directed structural graphs of these networks. In the case of HGN of the third and fourth classes, we formulate a conjecture on the one-to-one correspondence between the stable states and the $1m$-bases of the directed structural graphs.

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UDC: 575:577.21:519.61
Received: 12.03.2003

Citation: V. A. Likhoshvai, Yu. G. Matushkin, S. I. Fadeev, “Problems in the theory of the functioning of genetic networks”, Sib. Zh. Ind. Mat., 6:2 (2003), 64–80

Citation in format AMSBIB
\Bibitem{LikMatFad03}
\by V.~A.~Likhoshvai, Yu.~G.~Matushkin, S.~I.~Fadeev
\paper Problems in the theory of the functioning of genetic networks
\jour Sib. Zh. Ind. Mat.
\yr 2003
\vol 6
\issue 2
\pages 64--80
\mathnet{http://mi.mathnet.ru/sjim448}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2041576}
\zmath{https://zbmath.org/?q=an:1039.94543}


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    6. Vasilenko N.L., Balueva K.E., Likhoshvai V.A., Nevinsky G.A., Matushkin Yu.G., “Modeling and construction of molecular trigger in E-coli”, Proceedings of the Fourth International Conference on Bioinformatics of Genome Regulation and Structure, 2004, 155–158  isi
    7. Kogai V.V., Fadeev S.L., Likhoshvai V.A., “About Numerical Investigation of Auto-Oscillations in Hypothetical Gene Networks”, Proceedings of the Fourth International Conference on Bioinformatics of Genome Regulation and Structure, Vol 2, Russian Acad Sci Siberian Branch, 2004, 73–76  isi
    8. Peshkov I.M., Likhoshvai V.A., Matushkin Yu.G., Fadeev I S., “On Research Into Hypothetical Networks of Ecological Nature”, Proceedings of the Fourth International Conference on Bioinformatics of Genome Regulation and Structure, Vol 2, Russian Acad Sci Siberian Branch, 2004, 128–130  isi
    9. E. P. Volokitin, S. A. Treskov, “Bifurkatsiya Andronova–Khopfa v modeli gipoteticheskoi gennoi seti”, Sib. zhurn. industr. matem., 8:1 (2005), 30–40  mathnet  mathscinet; E. P. Volokitin, S. A. Treskov, “The Andronov–Hopf bifurcation in a model of a hypothetical gene regulatory network”, J. Appl. Industr. Math., 1:1 (2007), 127–136  crossref
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    11. Kolchanov N., Hofestaedt R., “Bioinformatics of Genome Regulation and Structure II - Preface”, Bioinformatics of Genome Regulation and Structure II, eds. Kolchanov N., Hoffestaedt R., Milanesi L., Springer, 2006, XV+  isi
    12. Fadeev I S., Korolev V.K., “Program Package Hgnet for Computational Studies of Hypothetical Gene Networks”, Proceedings of the Fifth International Conference on Bioinformatics of Genome Regulation and Structure, Vol 3, eds. Kolchanov N., Hofestadt R., Russian Acad Sci Siberian Branch, 2006, 47–51  isi
    13. Yu. A. Gaidov, V. P. Golubyatnikov, “O nekotorykh nelineinykh dinamicheskikh sistemakh, modeliruyuschikh nesimmetrichnye gennye seti”, Vestn. NGU. Ser. matem., mekh., inform., 7:2 (2007), 19–27  mathnet
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    17. V. P. Golubyatnikov, I. V. Golubyatnikov, “O mnogomernykh modelyakh funktsionirovaniya gennykh setei”, Sib. zhurn. industr. matem., 13:3 (2010), 13–18  mathnet  mathscinet; V. P. Golubyatnikov, I. V. Golubyatnikov, “On multidimensional models of gene networks”, J. Appl. Industr. Math., 5:3 (2011), 343–347  crossref
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    22. D. N. Shtokalo, “O predelnom perekhode v modeli mnogostadiinogo mnogoetapnogo sinteza veschestva”, Sib. zhurn. industr. matem., 15:4 (2012), 135–146  mathnet  mathscinet
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    26. A. S. Kornienko, “Struktura funktsionalnykh grafov dlya tsirkulyantov s lineinymi bulevymi funktsiyami v vershinakh”, PDM, 2014, no. 1(23), 84–95  mathnet
    27. Glyzin S.D. Kolesov A.Yu. Rozov N.Kh., “Periodic Two-Cluster Synchronization Modes in Completely Connected Genetic Networks”, Differ. Equ., 52:2 (2016), 157–176  crossref  mathscinet  isi  elib
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