Bulletin of Mathematical Biology

, Volume 73, Issue 4, pp 899–916 | Cite as

Algorithmic Global Criteria for Excluding Oscillations

  • Andreas Weber
  • Thomas Sturm
  • Essam O. Abdel-Rahman
Original Article

Abstract

We investigate algorithmic methods to tackle the following problem: Given a system of parametric ordinary differential equations built by a biological model, does there exist ranges of values for the model parameters and variables which are both meaningful from a biological point of view and where oscillating trajectories, can be found? We show that in the common case of polynomial vector fields known criteria excluding the existence of non-constant limit cycles lead to quantifier elimination problems over the reals.

We apply these criteria to various models that have been previously investigated in the context of algebraic biology.

Keywords

Oscillations Mass action kinetics 

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References

  1. Achatz, M., McCallum, S., & Weispfenning, V. (2008). Deciding polynomial-exponential problems. In ISSAC’08: Proceedings of the twenty-first international symposium on symbolic and algebraic computation, Linz/Hagenberg, Austria (pp. 215–222). New York: ACM. CrossRefGoogle Scholar
  2. Arnon, D. S., Collins, G. E., & McCallum, S. (1984). Cylindrical algebraic decomposition I: The basic algorithm. SIAM J. Comput., 13(4), 865–877. CrossRefMathSciNetGoogle Scholar
  3. Bonhoeffer, S., Coffin, J. M., & Nowak, M. A. (1997). Human immunodeficiency virus drug therapy and virus load. J. Virol., 71(4), 3275. Google Scholar
  4. Boulier, F., Lefranc, M., Lemaire, F., Morant, P., & Ürgüplü, A. (2007). On proving the absence of oscillations in models of genetic circuits. In H. Anai, H. Horimoto, & T. Kutsia (Eds.), Lecture notes in computer science: Vol. 4545. Algebraic biology (AB 2007), Castle of Hagenberg, Austria (pp. 66–80). Berlin: Springer. CrossRefGoogle Scholar
  5. 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. In K. Horimoto, G. Regensburger, M. Rosenkranz, & H. Yoshida (Eds.), Lecture notes in computer science: Vol. 5147. Algebraic biology (AB 2008)—third international conference, Castle of Hagenberg, Austria (pp. 56–64). Berlin: Springer. CrossRefGoogle Scholar
  6. Brown, C. W. (2004). QEPCAD B: A system for computing with semi-algebraic sets via cylindrical algebraic decomposition. ACM SIGSAM Bull., 38(1), 23–24. CrossRefGoogle Scholar
  7. Davenport, J. H., & Heintz, J. (1988). Real quantifier elimination is doubly exponential. J. Symb. Comput., 5(1–2), 29–35. CrossRefMATHMathSciNetGoogle Scholar
  8. De Jong, H., Geiselmann, J., Batt, G., Hernandez, C., & Page, M. (2004). Qualitative simulation of the initiation of sporulation in Bacillus subtilis. Bull. Math. Biol., 66(2), 261–299. CrossRefMathSciNetGoogle Scholar
  9. Dolzmann, A., & Gilch, L. A. (2004). Generic Hermitian quantifier elimination. In J. A. C. Bruno Buchberger (Ed.), Lecture notes in computer science: Vol. 3249. Artificial intelligence and symbolic computation: 7th international conference, AISC 2004, Linz, Austria (pp. 80–93). Berlin: Springer. Google Scholar
  10. Dolzmann, A., & Sturm, T. (1997a). REDLOG: Computer algebra meets computer logic. ACM SIGSAM Bull., 31(2), 2–9. CrossRefMathSciNetGoogle Scholar
  11. Dolzmann, A., & Sturm, T. (1997b). Simplification of quantifier-free formulae over ordered fields. J. Symb. Comput., 24(2), 209–231. CrossRefMATHMathSciNetGoogle Scholar
  12. Dolzmann, A., Sturm, T., & Weispfenning, V. (1998a). A new approach for automatic theorem proving in real geometry. J. Autom. Reason., 21(3), 357–380. CrossRefMATHMathSciNetGoogle Scholar
  13. Dolzmann, A., Sturm, T., & Weispfenning, V. (1998b). Real quantifier elimination in practice. In B. H. Matzat, G.-M. Greuel, & G. Hiss (Eds.), Algorithmic algebra and number theory (pp. 221–247). Berlin: Springer. Google Scholar
  14. Dolzmann, A., Seidl, A., & Sturm, T. (2004). Efficient projection orders for CAD. In J. Gutierrez (Ed.), Proceedings of the 2004 international symposium on symbolic and algebraic computation (ISSAC 2004) (pp. 111–118). New York: ACM. CrossRefGoogle Scholar
  15. El Kahoui, M., & Weber, A. (2000). Deciding Hopf bifurcations by quantifier elimination in a software-component architecture. J. Symb. Comput., 30(2), 161–179. CrossRefMATHGoogle Scholar
  16. El Kahoui, M., & Weber, A. (2002). Symbolic equilibrium point analysis in parameterized polynomial vector fields. In Ganzha, V. G., Mayr, E. W. & Vorozhtsov, E.V. (Eds.), Computer algebra in scientific computing (CASC) (pp. 71–83), Yalta, Ukraine, Sept. 2002. Google Scholar
  17. Érdi, P., & Tóth, J. (1989). Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models. Manchester: Manchester University Press. MATHGoogle Scholar
  18. Feckan, M. (2001). A generalization of Bendixson’s criterion. Proc. Am. Math. Soc., 129(11), 3395–3400. CrossRefMATHMathSciNetGoogle Scholar
  19. Fiedler, B., & Hsu, S.-B. (2009). Non-periodicity in chemostat equations: a multi-dimensional negative Bendixson–Dulac criterion. J. Math. Biol., 59(2), 233–253. CrossRefMATHMathSciNetGoogle Scholar
  20. Fussmann, G. F., Ellner, S. P., Shertzer, K. W., Hairston, J., & Nelson, G. (2000). Crossing the Hopf bifurcation in a live predator-prey system. Science, 290(5495), 1358–1360. CrossRefGoogle Scholar
  21. Gilch, L. A. (2003). Effiziente Hermitesche Quantorenelimination. Diploma thesis, Universität Passau, 94030 Passau, Germany, Sept. 2003. Google Scholar
  22. Glass, L., & Pasternack, J. S. (1978). Prediction of limit cycles in mathematical models of biological oscillations. Bull. Math. Biol., 40(1), 27–44. MathSciNetGoogle Scholar
  23. Griffith, J. S. (1968). Mathematics of cellular control processes. I. negative feedback to one gene. J. Theor. Biol., 20, 202–208. CrossRefGoogle Scholar
  24. Hasty, J., McMillen, D., Isaacs, F., Collins, J. J. et al. (2001). Computational studies of gene regulatory networks: in numero molecular biology. Nat. Rev. Genet., 2(4), 268–279. CrossRefGoogle Scholar
  25. Hong, H., Liska, R., & Steinberg, S. (1997). Testing stability by quantifier elimination. J. Symb. Comput., 24(2), 161–187. CrossRefMATHMathSciNetGoogle Scholar
  26. Jäger, W., So, J. W. H., Tang, B., & Waltman, P. (1987). Competition in the gradostat. J. Math. Biol., 25, 23–42. CrossRefMATHMathSciNetGoogle Scholar
  27. Lasaruk, A., & Sturm, T. (2007a). Weak integer quantifier elimination beyond the linear case. In V. G. Ganzha, E. W. Mayr, & E. V. Vorozhtsov (Eds.), Lecture notes in computer science: Vol. 4770. Computer algebra in scientific computing. Proceedings of the CASC 2007 (pp. 275–294). Berlin: Springer. CrossRefGoogle Scholar
  28. Lasaruk, A., & Sturm, T. (2007b). Weak quantifier elimination for the full linear theory of the integers. A uniform generalization of Presburger arithmetic. Appl. Algebra Eng. Commun. Comput., 18(6), 545–574. CrossRefMATHMathSciNetGoogle Scholar
  29. Li, M. Y., & Muldowney, J. S. (1995). Lower bounds for the Hausdorff dimension of attractors. J. Dyn. Differ. Equ., 7(3), 457–469. CrossRefMATHMathSciNetGoogle Scholar
  30. Mincheva, M., & Roussel, M. R. (2007). Graph-theoretic methods for the analysis of chemical and biochemical networks. I. multistability and oscillations in ordinary differential equation models. J. Math. Biol., 55(1), 61–86. CrossRefMATHMathSciNetGoogle Scholar
  31. Muldowney, J. S. (1990). Compound matrices and ordinary differential equations. Rocky Mt. J. Math., 20(4), 857–872. CrossRefMATHMathSciNetGoogle Scholar
  32. Niu, W., & Wang, D. (2008). Algebraic approaches to stability analysis of biological systems. Math. Comput. Sci., 1(3), 507–539. CrossRefMATHMathSciNetGoogle Scholar
  33. Novak, B., Pataki, Z., Ciliberto, A., & Tyson, J. J. (2001). Mathematical model of the cell division cycle of fission yeast. J. Nonlinear Sci., 11(1), 277–286. MATHGoogle Scholar
  34. Seidl, A. Cylindrical decomposition under application-oriented paradigms. Doctoral dissertation, Universität Passau, 94030 Passau, Germany, Dec. 2006. Google Scholar
  35. Sensse, A., Hauser, M. J. B., & Eiswirth, M. (2006). Feedback loops for Shil’nikov chaos: The peroxidase-oxidase reaction. J. Chem. Phys., 125, 014901. CrossRefGoogle Scholar
  36. Smith, H. L., Tang, B., & Waltman, P. (1991). Competition in an n-vessel gradostat. SIAM J. Appl. Math., 51(5), 1451–1471. CrossRefMATHMathSciNetGoogle Scholar
  37. Strzebonski, A. (2000). Solving systems of strict polynomial inequalities. J. Symb. Comput., 29(3), 471–480. CrossRefMATHMathSciNetGoogle Scholar
  38. Strzebonski, A. W. (2006). Cylindrical algebraic decomposition using validated numerics. J. Symb. Comput., 41(9), 1021–1038. CrossRefMATHMathSciNetGoogle Scholar
  39. Strzebonski, A. (2008). Real root isolation for exp-log functions. In ISSAC’08: Proceedings of the twenty-first international symposium on symbolic and algebraic computation, Linz/Hagenberg, Austria (pp. 303–314). New York: ACM. CrossRefGoogle Scholar
  40. Sturm, T. (2006). New domains for applied quantifier elimination. In V. G. Ganzha, E. W. Mayr, & E. V. Vorozhtsov (Eds.), Lecture notes in computer science: Vol. 4194. Computer algebra in scientific computing: 9th international workshop, CASC 2006, Chisinau, Moldova, 11–15 September 2006. Berlin: Springer. Google Scholar
  41. Sturm, T. (2007). Redlog online resources for applied quantifier elimination. Acta Acad. Abo., Ser. B, 67(2), 177–191. Google Scholar
  42. Sturm, T., & Weber, A. (2008). Investigating generic methods to solve Hopf bifurcation problems in algebraic biology. In K. Horimoto, G. Regensburger, M. Rosenkranz, & H. Yoshida (Eds.), Lecture notes in computer science: Vol. 5147. Algebraic biology—third international conference (AB 2008), Castle of Hagenberg, Austria. Berlin: Springer. Google Scholar
  43. Sturm, T., Weber, A., Abdel-Rahman, E. O., & El Kahoui, M. (2009). Investigating algebraic and logical algorithms to solve Hopf bifurcation problems in algebraic biology. Math. Comput. Sci., 2(3), 493–515. Special issue on ‘Symbolic Computation in Biology’. CrossRefMATHMathSciNetGoogle Scholar
  44. Tarski, A. (1951). A decision method for elementary algebra and geometry (2nd revised edn.). Berkeley: University of California Press. Reprinted in B. F. Caviness and J. R. Johnson (Eds.), Quantifier elimination and cylindrical algebraic decomposition. Texts and monographs in symbolic computation (pp. 24–84). Berlin: Springer (1998). MATHGoogle Scholar
  45. Thomas, R., Thieffry, D., & Kaufman, M. (1995). Dynamical behaviour of biological regulatory networks—I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state. Bull. Math. Biol., 57(2), 247–276. MATHGoogle Scholar
  46. Toth, J. (1987). Bendixson-type theorems with applications. Z. Angew. Math. Mech., 67, 31–35. CrossRefMATHMathSciNetGoogle Scholar
  47. Tuckwell, H. C., & Wan, F. Y. M. (2004). On the behavior of solutions in viral dynamical models. Biosystems, 73(3), 157–161. CrossRefGoogle Scholar
  48. Tyson, J. J., Chen, K., & Novak, B. (2001). Network dynamics and cell physiology. Nat. Rev. Mol. Cell Biol., 2(12), 908–916. CrossRefGoogle Scholar
  49. van den Driessche, P., & Zeeman, M. L. (1998). Three-dimensional competitive Lotka–Volterra systems with no periodic orbits. SIAM J. Appl. Math., 58(1), 227–234. CrossRefMATHMathSciNetGoogle Scholar
  50. Weispfenning, V. (1988). The complexity of linear problems in fields. J. Symb. Comput., 5(1&2), 3–27. CrossRefMATHMathSciNetGoogle Scholar
  51. Weispfenning, V. (1994). Quantifier elimination for real algebra—the cubic case. In Proceedings of the international symposium on symbolic and algebraic computation (pp. 258–263). New York: ACM. CrossRefGoogle Scholar
  52. Weispfenning, V. (1997a). Quantifier elimination for real algebra—the quadratic case and beyond. Appl. Algebra Eng. Commun. Comput., 8(2), 85–101. CrossRefMATHMathSciNetGoogle Scholar
  53. Weispfenning, V. (1997b). Simulation and optimization by quantifier elimination. J. Symb. Comput., 24(2), 189–208. Special issue on applications of quantifier elimination. CrossRefMATHMathSciNetGoogle Scholar
  54. Weispfenning, V. (1998). A new approach to quantifier elimination for real algebra. In B. Caviness & J. Johnson (Eds.), Texts and monographs in symbolic computation, quantifier elimination and cylindrical algebraic decomposition (pp. 376–392). New York: Springer. Google Scholar
  55. Weispfenning, V. (2000). Deciding linear-transcendental problems. In V. G. Ganzha, E. W. Mayr, & E. V. Vorozhtsov (Eds.), Proceedings of the 3rd workshop on computer algebra in scientific computing, CASC’2000 (pp. 423–437), Samarkand, Uzbekistan, 5–9 October 2000. Berlin: Springer. Google Scholar

Copyright information

© Society for Mathematical Biology 2011

Authors and Affiliations

  • Andreas Weber
    • 1
  • Thomas Sturm
    • 2
  • Essam O. Abdel-Rahman
    • 1
  1. 1.Institut für Informatik IIUniversität BonnBonnGermany
  2. 2.Departamento de Matemáticas, Estadística y Computación, Facultad de CienciasUniversidad de CantabriaSantanderSpain

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