Symbolic Detection of Steady States of Autonomous Differential Biological Systems by Transformation into Block Triangular Form

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10849)


In this paper we propose a method for transforming a square polynomial set into block triangular form by using Tarjan’s algorithm. The proposed method is then applied to symbolic detection of steady states of autonomous differential biological systems which are usually sparse systems with a large number of loosely coupling variables. Two biological systems of 12 and 43 variables respectively are studied to illustrate the effectiveness of the proposed method.


Systems biology Differential biological system Steady state Block triangular form Sparsity 



The author would like to thank Yufei Gao and Yishan Cui for their help in the investigation on Tarjan’s algorithm and the biological database and the anonymous reviewers for their helpful comments which lead to improvement on this manuscript and potential enrichment in its extended version.


  1. 1.
    Allen, L.J.: Some discrete-time SI, SIR, and SIS epidemic models. Math. Biosci. 124(1), 83–105 (1994)CrossRefGoogle Scholar
  2. 2.
    Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symbolic Comput. 28(1–2), 105–124 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bock, H.G.: Numerical treatment of inverse problems in chemical reaction kinetics. In: Ebert, K.H., Deuflhard, P., Jäger, W. (eds.) Modelling of Chemical Reaction Systems. Springer Series in Chemical Physics, vol. 18, pp. 102–125. Springer, Heidelberg (1981). Scholar
  4. 4.
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. Ph.D. thesis, Universität Innsbruck, Austria (1965)Google Scholar
  5. 5.
    Cifuentes, D., Parrilo, P.A.: Exploiting chordal structure in polynomial ideals: A Gröbner bases approach. SIAM J. Discrete Math. 30(3), 1534–1570 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In: Brakhage, H. (ed.) Automata Theory and Formal Languages 2nd GI Conference. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975). Scholar
  7. 7.
    Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symbolic Comput. 12(3), 299–328 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Duff, I.S.: On algorithms for obtaining a maximum transversal. ACM Trans. Math. Softw. 7(3), 315–330 (1981)CrossRefGoogle Scholar
  9. 9.
    Duff, I.S., Erisman, A.M., Reid, J.K.: Direct Methods for Sparse Matrices. Oxford University Press, New York (1986)zbMATHGoogle Scholar
  10. 10.
    El Kahoui, M., Weber, A.: Deciding Hopf bifurcations by quantifier elimination in a software-component architecture. J. Symbolic Comput. 30(2), 161–179 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Faugère, J.C.: A new efficient algorithm for computing Gröbner bases (\({F_4}\)). J. Pure Appl. Algebra 139(1–3), 61–88 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Faugère, J.C., Mou, C.: Sparse FGLM algorithms. J. Symbolic Comput. 80(3), 538–569 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Faugère, J.C., Rahmany, S.: Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases. In: May, J.P. (ed.) Proceedings of ISSAC 2009, pp. 151–158. ACM (2009)Google Scholar
  14. 14.
    Faugère, J.C., Spaenlehauer, P.J., Svartz, J.: Sparse Gröbner bases: The unmixed case. In: Nabeshima, K., Nagasaka, K. (eds.) Proceedings of ISSAC 2014, pp. 178–185. ACM (2014)Google Scholar
  15. 15.
    Ferrell, J.E., Tsai, T.Y.C., Yang, Q.: Modeling the cell cycle: why do certain circuits oscillate? Cell 144(6), 874–885 (2011)CrossRefGoogle Scholar
  16. 16.
    Gatermann, K., Huber, B.: A family of sparse polynomial systems arising in chemical reaction systems. J. Symbolic Comput. 33(3), 275–305 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Laubenbacher, R., Sturmfels, B.: Computer algebra in systems biology. Am. Math. Monthly 116(10), 882–891 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li, C., Donizelli, M., Rodriguez, N., Dharuri, H., et al.: Biomodels database: an enhanced, curated and annotated resource for published quantitative kinetic models. BMC Syst. Biol. 4(1), 92 (2010)CrossRefGoogle Scholar
  19. 19.
    Li, X., Mou, C., Niu, W., Wang, D.: Stability analysis for discrete biological models using algebraic methods. Math. Comput. Sci. 5(3), 247–262 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mayr, E., Meyer, A.: The complexity of the word problems for commutative semigroups and polynomial ideals. Adv. Math. 46(3), 305–329 (1982)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mol, M., Patole, M.S., Singh, S.: Immune signal transduction in leishmaniasis from natural to artificial systems: role of feedback loop insertion. Biochim. Biophys. Acta Gen. Subj. 1840(1), 71–79 (2014)CrossRefGoogle Scholar
  22. 22.
    Mou, C., Bai, Y.: On the chordality of polynomial sets in triangular decomposition in top-down style (2018). arXiv:1802.01752
  23. 23.
    Niu, W., Wang, D.: Algebraic analysis of bifurcation and limit cycles for biological systems. In: Horimoto, K., Regensburger, G., Rosenkranz, M., Yoshida, H. (eds.) AB 2008. LNCS, vol. 5147, pp. 156–171. Springer, Heidelberg (2008). Scholar
  24. 24.
    Niu, W., Wang, D.: Algebraic approaches to stability analysis of biological systems. Math. Comput. Sci. 1(3), 507–539 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sturm, T., Weber, A., Abdel-Rahman, E.O., El Kahoui, M.: Investigating algebraic and logical algorithms to solve Hopf bifurcation problems in algebraic biology. Math. Comput. Sci. 2(3), 493–515 (2009)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, D.: Elimination Methods. Springer, Heidelberg (2001). Scholar
  28. 28.
    Wang, D., Xia, B.: Stability analysis of biological systems with real solution classification. In: Kauers, M. (ed.) Proceedings of ISSAC 2005, pp. 354–361. ACM Press (2005)Google Scholar
  29. 29.
    Yang, L., Xia, B.: Real solution classifications of parametric semi-algebraic systems. In: Proceedings of A3L 2005, pp. 281–289. Herstellung und Verlag, Norderstedt (2005)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Beijing Advanced Innovation Center for Big Data and Brain Computing/LMIB – School of Mathematics and Systems ScienceBeihang UniversityBeijingChina

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