Journal of Dynamics and Differential Equations

, Volume 25, Issue 3, pp 563–604 | Cite as

Dynamics and Control at Feedback Vertex Sets. I: Informative and Determining Nodes in Regulatory Networks

  • Bernold Fiedler
  • Atsushi Mochizuki
  • Gen Kurosawa
  • Daisuke Saito
Article

Abstract

We consider systems of differential equations which model complex regulatory networks by a graph structure of dependencies. We show that the concepts of informative nodes (Mochizuki and Saito, J Theor Biol 266:323–335, 2010) and determining nodes (Foias and Temam, Math Comput 43:117–133, 1984) coincide with the notion of feedback vertex sets from graph theory. As a result we can determine the long-time dynamics of the entire network from observations on only a feedback vertex set. We also indicate how open loop control at a feedback vertex set, only, forces the remaining network to stably follow prescribed stable or unstable trajectories. We present three examples of biological networks which motivated this work: a specific gene regulatory network of ascidian cell differentiation (Imai et al., Science 312:1183–1187, 2006), a signal transduction network involving the epidermal growth factor in mammalian cells (Oda et al., Mol Syst Biol 1:1–17, 2005), and a mammalian gene regulatory network of circadian rhythms (Mirsky et al., Proc Natl Acad Sci USA 106:11107–11112, 2009). In each example the required observation set is much smaller than the entire network. For further details on biological aspects see the companion paper (Mochizuki et al., J Theor Biol, 2013, in press). The mathematical scope of our approach is not limited to biology. Therefore we also include many further examples to illustrate and discuss the broader mathematical aspects.

Keywords

Differential equations on graphs Reaction network Determining node Global attractor Takens embedding Biological network Gene regulation 

Notes

Acknowledgments

The first two authors gratefully acknowledge pleasant excesses of mutual hospitality during very enjoyable working visits. We are also indebted to Sze-Bi Hsu and NCTS Taiwan for generous hospitality and helpful comments. The relation with determining nodes was suggested by Abderrahim Azouani. Hiroshi Kokubu, Hiroe Oka, Genevieve Raugel and her insightful referee have greatly assisted with valuable further suggestions. Skillful and very patient typesetting was gracefully achieved by Margrit Barrett and Ulrike Geiger. This work was generously supported by the Deutsche Forschungsgemeinschaft, SFB 910 “Control of Self-Organizing Nonlinear Systems”.

References

  1. 1.
    Arai, Z., Kalies, W., Kokubu, H., Mischaikow, K., Oka, H., Pilarczyk, P.: A database schema for the analysis of global dynamics of multiparameter systems. SIAM J. Appl. Dyn. Syst. 8, 757–789 (2009)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arnold, V.I.: Ordinary Differential Equations. MIT Press, Cambridge (1973)MATHGoogle Scholar
  3. 3.
    Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)MATHCrossRefGoogle Scholar
  4. 4.
    Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North Holland, Amsterdam (1992)MATHGoogle Scholar
  5. 5.
    Barboza, R., Chen, G.: On the global boundedness of the Chen system. Int. J. Bifurcation Chaos 21, 3373–3385 (2011)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Belykh, V.N., Belykh, I.V., Hasler, M.: Connection graph stability method for synchronized coupled chaotic systems. Phys. D 195, 159–187 (2004)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55(21), 1–19 (2008)MathSciNetGoogle Scholar
  8. 8.
    Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics, vol. 49. AMS Colloquium, Providence (2002)MATHGoogle Scholar
  9. 9.
    Constantin, P., Foias, C., Nicolaenko, B., Temam, R.: Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. Applied Mathematics and Sciences, vol. 70. Springer, New York (1989). AMS Colloquium, Providence (2002)Google Scholar
  10. 10.
    Eden, A., Foias, C., Nicolaenko, B., Temam, R.: Exponential Attractors for Dissipative Evolution Equations. Wiley, Chichester (1994)MATHGoogle Scholar
  11. 11.
    Foias, C., Temam, R.: Determination of the solutions of the Navier–Stokes equations by a set of nodal values. Math. Comput. 43, 117–133 (1984)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Foias, C., Titi, E.S.: Determining nodes, finite differences schemes and inertial manifolds. Nonlinearity 4, 135–153 (1991)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Golubitsky, M., Romano, D., Wang, Y.: Network periodic solutions: full oscillation and rigid synchrony. Nonlinearity 23, 3227–3243 (2010)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Mathematical Survey 25. AMS, Providence (1988)Google Scholar
  15. 15.
    Hale, J.K., Raugel, G.: Regularity, determining modes and Galerkin methods. J. Math. Pures Appl., IX 82, 1075–1136 (2003)MathSciNetMATHGoogle Scholar
  16. 16.
    Hale, J.K., Magalhães, L.T., Oliva, W.M.: Dynamics in Infinite Dimensions. Springer, New York (2002)MATHGoogle Scholar
  17. 17.
    Imai, K.S., Levine, M., Satoh, N., Satou, Y.: Regulatory blueprint for a chordate embryo. Science 312, 1183–1187 (2006)CrossRefGoogle Scholar
  18. 18.
    Joly, R.: Observation and inverse problems in coupled cell networks. Nonlinearity 25, 657–676 (2012)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Karp, R.M.: Reducibility among combinatorial problems. Kibernet Sb. 12, 16–83 (1975)MATHGoogle Scholar
  20. 20.
    Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. Mathematical Survey, vol. 176. AMS, Providence (2011)Google Scholar
  21. 21.
    Ladyzhenskaya, O.A.: Dynamical system generated by the Navier–Stokes equations. Sov. Phys. Dokl. 17, 647–649 (1972); translation from. Dokl. Akad. Nauk SSSR 205, 318–320 (1972)Google Scholar
  22. 22.
    Ladyzhenskaya, O.A.: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)MATHCrossRefGoogle Scholar
  23. 23.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)CrossRefGoogle Scholar
  24. 24.
    Mallet-Paret, J., Sell, G.R.: The principle of spatial averaging and inertial manifolds for reaction- diffusion equations. In: Nonlinear Semigroups, Partial Differential Equations and Attractors. Proceedings of the Symposium, Washington, DC (1985). Lecture Notes in Mathematics, vol. 1248, pp. 94–107. Springer, Berlin (1987)Google Scholar
  25. 25.
    Mirsky, H.P., Liu, A.C., Welsh, D.K., Kay, S.A., Doyle III, F.J.: A model of the cell-autonomous mammalian circadian clock. Proc. Natl. Acad. Sci. USA 106, 11107–11112 (2009)CrossRefGoogle Scholar
  26. 26.
    Mischaikow, K., Mrozek, M., Szymczak, A.: Chaos in the Lorenz equations: a computer assisted proof. III: classical parameter values. J. Differ. Equ. 169, 17–56 (2001)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Mochizuki, A.: Structure of regulatory networks and diversity of gene expression patterns. J. Theor. Biol. 250, 307–321 (2008)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Mochizuki, A., Saito, D.: Analyzing steady states of dynamics of bio-molecules from the structure of regulatory networks. J. Theor. Biol. 266, 323–335 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Mochizuki, A., Fiedler, B., Kurosawa, G., Saito, D.: Dynamics and control at feedback vertex sets. II: A faithful monitor to determine the diversity of molecular activities in regulatory networks. J. Theor. Biol. (2013, in press)Google Scholar
  30. 30.
    Oda, K., Matsuoka, Y., Funahashi, A., Kitano, H.: A comprehensive pathway map of epidermal growth factor receptor signaling. Mol. Syst. Biol. 1, 1–17 (2005)CrossRefGoogle Scholar
  31. 31.
    Pecora, L.M., Carroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109–2112 (1998)CrossRefGoogle Scholar
  32. 32.
    Raugel, G.: Global attractors. In: Fiedler, B. (ed.) Handbook of Dynamical Systems, vol. 2, pp. 885–982. Elsevier, Amsterdam (2002)Google Scholar
  33. 33.
    Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Springer, New York (2002)MATHCrossRefGoogle Scholar
  34. 34.
    Sparrow, C.: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Applied Mathematical Sciences, vol. 41. Springer, Berlin (1982)Google Scholar
  35. 35.
    Takens, F.: Reconstruction theory and nonlinear time series analysis. In: Broer, H., et al. (eds.) Handbook of Dynamical Systems, vol. 3, pp. 345–377. Elsevier, Amsterdam (2010)Google Scholar
  36. 36.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)MATHCrossRefGoogle Scholar
  37. 37.
    Tucker, W.: A rigorous ODE solver and Smale’s 14th problem. Found. Comput. Math. 2(1), 53–117 (2002)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Bernold Fiedler
    • 1
  • Atsushi Mochizuki
    • 2
  • Gen Kurosawa
    • 2
  • Daisuke Saito
    • 2
  1. 1.Fachbereich Mathematik und InformatikInstitut für MathematikBerlinGermany
  2. 2.Theoretical Biology LaboratoryRIKENWakoJapan

Personalised recommendations