Qualitative investigation of a gene model using computer algebra algorithms

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• 作者：F. Boulier (4)
  M. Han (1)
  F. Lemaire (4)
  V. G. Romanovski (1) (2) (3)
  
  4. LIFL ; Universite Lille 1 ; 59655 ; Villeneuve d鈥橝scq ; France
  1. Department of Mathematics ; Shanghai Normal University ; Shanghai ; 200234 ; China
  2. Center for Applied Mathematics and Theoretical Physics ; University of Maribor ; Krekova 2 ; Maribor ; SI-2000 ; Slovenia
  3. Faculty of Natural Sciences and Mathematics ; University of Maribor ; Koroska c. 160 ; Maribor ; SI-2000 ; Slovenia
  
• 刊名：Programming and Computer Software
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：41
• 期：2
• 页码：105-111
• 全文大小：254 KB
• 参考文献：1. Boulier, F, Lefranc, M, Lemaire, F, Morant, P-E, Urguplu, A (2007) On proving the absence of oscillations in models of genetic circuits. Algebraic Biology 4545: pp. 66-80 CrossRef
  2. Andronov, A.A., Leontovich, E.A., Gordon, I.M., and Maier, A.G., / Theory of Bifurcations of Dynamic Systems on a Plane. andrleont.pdf" class="a-plus-plus">wwwf.imperial.ac.uk/~dturaev/andrleont.pdf.
  3. Errami, H, Eiswirth, M, Grigoriev, D, Seiler, WM, Sturm, T, Weber, A (2013) Efficient methods to compute Hopf bifurcations in chemical reaction networks using reaction coordinates. Comput. Algebra Sci. Computing 8136: pp. 88-99 CrossRef
  4. Boulier, F, Lefranc, M, Lemaire, F, Morant, P-E (2008) Applying a rigorous quasi-steady state approximation method for proving the absence of oscillations in models of genetic circuits. Algebraic Biology 5147: pp. 56-64 CrossRef
  5. Boulier, F, Lefranc, M, Lemaire, F, Morant, P-E (2011) Model reduction of chemical reaction systems using elimination. Math. Comput. Sci.. pp. 289-301
  6. Decker, W., Greuel, G.-M., Pfister, G., and Shonemann, H., Singular 3-1-6: A computer algebra system for polynomial computations. http://www.singular.unikl.de.
  7. Decker, W, Laplagne, S, Pfister, G, Schonemann, HA (2010) primdec.lib Singular 3-1-6 library for computing the primary decomposition and radical of ideals.
  8. Darboux, G (1878) Memoire sur les equations differentielles algebriques du premier ordre et du premier degre (Melanges). Bull. Sci. Math.. pp. 60-96
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  13. Collins, GE (1975) Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. Proc. 2nd GI Conf. on Automata Theory and Formal Languages. pp. 134-183
  14. Chen, C, Davenport, JH, Lemaire, F, Moreno Maza, M, Phisanbut, N, Xia, B, Xiao, R, Xie, Y Solving semi-algebraic systems with the RegularChains library in Maple. In: Raschau, S eds. (2011) Proc. 4th Int. Conf. on Mathematical Aspects of Computer Science and Information Sciences (MACIS 2011). pp. 38-51
  15. Kuznetsov, YA (1995) Elements of Applied Bifurcation Theory. Springer, New York
  16. Liapunov, AM (1966) Stability of Motion. Academic, New York
  17. Romanovski, VG, Mencinger, M, Fercec, B (2013) Investigation of center manifolds of three-dimensional systems using computer algebra. Program. Comput. Software 39: pp. 67-73 CrossRef
  
• 刊物类别：Computer Science
• 刊物主题：Computer Science, general
  Software Engineering, Programming and Operating Systems
  Operating Systems
  Software Engineering
  Artificial Intelligence and Robotics
  Russian Library of Science
  
• 出版者：MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC.
• ISSN：1608-3261

文摘

Using algorithms and packages for computer algebra, we investigate a three-dimensional autonomous system of ordinary differential equations (ODEs) used in [1] to simulate the dynamics of a gene. A computational approach based on elimination theory algorithms is proposed to find invariant surfaces of multidimensional polynomial differential equation systems; this approach allows one to reduce the investigation of the system dynamics to studying the dynamics of a lower-order system. In addition, an effective approach based on the Lyapunov function is proposed to investigate the Andronov-Hopf bifurcation; this approach is used to find such bifurcations in the model under study.