Contraction Theory for Systems Biology

  • Giovanni Russo
  • Mario di Bernardo
  • Jean Jacques Slotine


In this chapter, we present a theoretical framework for the analysis, synchronization and control of biochemical circuits and systems modeled by means of ODEs. The methodology is based on the use of contraction theory, a powerful concept from the theory of dynamical systems, ensuring convergence of all trajectories of a system of interest towards each other. After introducing contraction theory, we present some application to biochemical networks. Specifically, we introduce a graphical approach to verify if a system is contracting and apply it to synthesize networks of self-synchronizing Repressilators. We then present a more general analysis of quorum sensing networks.


Contraction theory Graphical algorithm Entrainment Synchronization 


  1. 1.
    Alon U (2007) Network motifs: theory and experimental approaches. Nature 450:450–461CrossRefGoogle Scholar
  2. 2.
    Anastassiou C, Montgomery SM, Barahona M, Buzsaki G, Koch C (2010) The effect of spatially inhomogeneous extracellular electric fields on neurons. J Neurosci 30:1925–1936CrossRefGoogle Scholar
  3. 3.
    Anetzberger C, Pirch T, Jung K (2009) Heterogeneity in quorum sensing-regulated bioluminescence of vibro harvey. Mol Microbiol 2:267–277CrossRefGoogle Scholar
  4. 4.
    Angeli D (2002) A Lyapunov approach to incremental stability properties. IEEE Trans Automat Contr 47:410–321MathSciNetCrossRefGoogle Scholar
  5. 5.
    Beebe DJ, Mensing G, Walker G (2002) Physics and applications of microfluidics in biology. Annu Rev Biomed Eng 4:261–286CrossRefGoogle Scholar
  6. 6.
    Bertsekas D, Tsitsiklis J (1989) Parallel and distributed computation: numerical methods. Prentice-Hall, Upper Saddle RiverMATHGoogle Scholar
  7. 7.
    Borenstein E, Ullman S (2008) Combined top–down/bottom–up segmentation. IEEE Trans Pattern Anal Mach Intell 30:2109–2125CrossRefGoogle Scholar
  8. 8.
    Boustani SE, Marre O, Behuret P, Yger P, Bal T, Destexhe A, Fregnac Y (2009) Network–state modulation of power-law frequency-scaling in visual cortical neurons. PLoS Comput Biol 5:e1000519CrossRefGoogle Scholar
  9. 9.
    Del Vecchio D, Ninfa AJ, Sontag ED (2008) Modular cell biology: retroactivity and insulation. Nat Mol Syst Biol 4:161CrossRefGoogle Scholar
  10. 10.
    Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403:335–338CrossRefGoogle Scholar
  11. 11.
    Garcia-Ojalvo J, Elowitz MB, Strogatz SH (2004) Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing. Proc Natl Acad Sci U S A 101:10955–10960MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gardner T, Cantor C, Collins J (2000) Construction of a genetic toggle in Escherichia coli. Nature 403:339–342CrossRefGoogle Scholar
  13. 13.
    George D, Hawkins J (2009) Towards a mathematical theory of cortical micro-circuits. PLoS Comput Biol 5:e1000532MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gigante G, Mattia M, Braun J, DelGiudice P (2009) Bistable perception modeled as competing stochastic integrations at two levels. PLoS Comput Biol 5:e1000430MathSciNetCrossRefGoogle Scholar
  15. 15.
    Godsil C, Royle G (2001) Algebraic graph theory. Springer, New YorkMATHGoogle Scholar
  16. 16.
    Gregor T, Fujimoto K, Masaki N, Sawai S (2010) The onset of collective behavior in social Amoebe. Science 328:1021–1025CrossRefGoogle Scholar
  17. 17.
    Hartman P (1961) On stability in the large for systems of ordinary differential equations. Canadian J Math 13:480–492MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Javaloyes J, Perrin M, Politi A (2008) Collective atomic recoil laser as a synchronization transient. Phys Rev E 78:011108CrossRefGoogle Scholar
  19. 19.
    Jouffroy J (2005) Some ancestors of contraction analysis. In: Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain pp 5450–5455Google Scholar
  20. 20.
    Kandel E, Schwartz J, Jessel T (2000) Principles of neural science, 4th edn. Oxford University Press, McGraw-Hill, New York, USAGoogle Scholar
  21. 21.
    Kobayashi H, Kaern M, Araki M, Chung K, Gardner T, Cantor C, Collins J (2004) Programmable cells: interfacing natural and engineered gene networks. Proc Natl Acad Sci USA 101:8414–8419CrossRefGoogle Scholar
  22. 22.
    Kuznetsov A, Kaern M, Kopell N (2004) Synchrony in a population of hysteresis-based genetic oscillators. SIAM J Appl Math 65:392–425MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Lewis DC (1949) Metric properties of differential equations. Am J Math 71:294–312MATHCrossRefGoogle Scholar
  24. 24.
    Lohmiller W, Slotine JJE (1998) On contraction analysis for non-linear systems. Automatica 34:683–696MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Lohmiller W, Slotine JJE (2000) Nonlinear process control using contraction theory. AIChE J 46:588–596CrossRefGoogle Scholar
  26. 26.
    Lohmiller W, Slotine JJ (2005) Contraction analysis of non-linear distributed systems. Int J Control 78:678–688MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Miller M, Bassler B (2001) Quorum sensing in bacteria. Annu Rev Microbiol 55:165–199CrossRefGoogle Scholar
  28. 28.
    Nadell CD, Xavier J, Levin SA, Foster KR (2008) The evolution of quorum sensing in bacteria biofilms. PLoS Comput Biol 6:e14Google Scholar
  29. 29.
    Nardelli C, Bassler B, Levin S (2008) Observing bacteria through the lens of social evolution. J Biol 7:27CrossRefGoogle Scholar
  30. 30.
    Ng W, Bassler B (2009) Bacterial quorum-sensing network architectures. Ann Rev Genet 43:197–222CrossRefGoogle Scholar
  31. 31.
    Pavlov A, Pogromvsky A, van de Wouv N, Nijmeijer H (2004) Convergent dynamics, a tribute to Boris Pavlovich Demidovich. Syst Control Lett 52:257–261MATHCrossRefGoogle Scholar
  32. 32.
    Pesaran B, Pezaris J, Sahani M, Mitra P, Andersen R (2002) Temporal structure in neuronal activity during working memory in macaque parietal cortex. Nature 5:805–811Google Scholar
  33. 33.
    Pham QC, Tabareau N, Slotine JJE (2009) A contraction theory approach to stochastic incremental stability. IEEE Trans Automat Contr 54:816–820MathSciNetCrossRefGoogle Scholar
  34. 34.
    Prindle A, Hasty J (2010) Stochastic emergence of groupthink. Science 328:987–988CrossRefGoogle Scholar
  35. 35.
    Russo G, di Bernardo M (2009) An algorithm for the construction of synthetic self synchronizing biological circuits. In: International symposium on circuits and systems. Taipei, Taiwan pp 305–308Google Scholar
  36. 36.
    Russo G, di Bernardo M (2009) Contraction theory and the master stability function: linking two approaches to study synchronization in complex networks. IEEE Trans Circuit Syst II 56:177–181CrossRefGoogle Scholar
  37. 37.
    Russo G, di Bernardo M (2009) How to synchronize biological clocks. J Comput Biol 16:379–393MathSciNetCrossRefGoogle Scholar
  38. 38.
    Russo G, Slotine J (2010) Global convergence of quorum-sensing networks, to be published in Phys Rev E 82:041919Google Scholar
  39. 39.
    Russo G, di Bernardo M, Slotine J (2011) A graphical algorithm to prove contraction of nonlinear circuits and systems. IEEE Trans Circuit Syst I 58:336–348CrossRefGoogle Scholar
  40. 40.
    Russo G, di Bernardo M, Sontag E (2010) Global entrainment of transcriptional systems to periodic inputs. PLoS Comput Biol 6:e1000739CrossRefGoogle Scholar
  41. 41.
    Slotine J (2003) Modular stability tools for distributed computation and control. Int J Adapt Control Signal Process 17:397–416MATHCrossRefGoogle Scholar
  42. 42.
    Slotine JJE, Li W (1990) Applied nonlinear control. Prentice-Hall, Englewood CliffsGoogle Scholar
  43. 43.
    Stokkan KA, Yamazaki S, Tei H, Sakaki Y, Menaker M (2001) Entrainment of the circadian clock in the liver by feeding. Science 19:490–493CrossRefGoogle Scholar
  44. 44.
    Tabareau N, Slotine J, Pham Q (2010) How synchronization protects from noise. PLoS Comput Biol 6:e1000637MathSciNetCrossRefGoogle Scholar
  45. 45.
    Toth R, Taylor AF, Tinsley MR (2006) Collective behavior of a population of chemically coupled oscillators. J Phys Chem 110:10170–10176Google Scholar
  46. 46.
    Wang W, Slotine JJE (2005) On partial contraction analysis for coupled nonlinear oscillators. Biol Cyber 92:38–53MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    You L 3rd, Cox RS, Weiss R, Arnold FH (2004) Programmed population control by cell-cell communication and regulated killing. Nature 428:868–871CrossRefGoogle Scholar
  48. 48.
    Yu G, Slotine JJE (2009) Visual grouping by oscillator networks. IEEE Trans Neural Netw 20:1871–1884CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Giovanni Russo
    • 1
  • Mario di Bernardo
  • Jean Jacques Slotine
  1. 1.Department of Systems and Computer EngineeringUniversity of Naples Federico IINapoliItaly

Personalised recommendations