Skip to main content
Springer Nature Link
Log in

A Control Theory for Boolean Monomial Dynamical Systems

  • Published:
Discrete Event Dynamic Systems Aims and scope Submit manuscript

Abstract

Recently criteria for determining when a certain type of nonlinear discrete dynamical system is a fixed point system have been developed. This theory can be used to determine if certain events modeled by those systems reach a steady state. In this work we formalize the idea of a “stabilizable” discrete dynamical system. We present necessary and sufficient conditions for a Boolean monomial dynamical control system to be stabilizable in terms of properties of the dependency graph associated with the system. We use the equivalence of periodicity of the dependency graph and loop numbers to develop a new O(n 2logn) algorithm for determining the loop numbers of the strongly connected components of the dependency graph, and hence a new O(n 2logn) algorithm for determining when a Boolean monomial dynamical system is a fixed point system. Finally, we show how this result can be used to determine if a Boolean monomial dynamical control system is stabilizable in time O(n 2logn).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Aho AV, Hopcroft JE, Ullman JD (1975) The design and analysis of computer algorithms. Addison-Wesley, Reading

    Google Scholar 

  • Bollman D, Colón-Reyes O, Orozco E (2007) Fixed points in discrete models for regulatory genetic networks. EURASIP J Bioinformatics Syst Biol 2007:8. Article ID 97356. doi:10.1155/2007/97356

  • Cassandras C, Lafortune S (1998) Introduction to discrete event systems. Springer, New York

    Google Scholar 

  • Colón-Reyes O (2005) Monomial dynamical systems over finite fields. ProQuest, Cambridge

    Google Scholar 

  • Colón-Reyes O, Laubenbacher R, Pareigis B (2004) Boolean monomial dynamical systems. Ann Comb 8:425–439

    Article  MATH  MathSciNet  Google Scholar 

  • Colón-Reyes O, Jarrah A, Laubenbacher R, Sturmfels B (2006) Monomial dynamical systems over finite fields. J Complex 16:333–342

    Google Scholar 

  • Denardo EV (1977) Periods of connected networks and powers of nonnegative matrices. Math Oper Res 2:20–24

    Article  MATH  MathSciNet  Google Scholar 

  • Elspas B (1959) The theory of autonomous linear sequential networks. IRE Trans Circuit Theory CT-6:45–60

    Google Scholar 

  • Fuller GN, Rhee CH, Hess KR (1999) Reactivation of insulin-like growth factor binding protein 2 expressions in glioblastoma multiforme. Cancer Res 59:4228–4232

    Google Scholar 

  • Jarvis JP, Shier DR (1999) Graph-theoretic analysis of finite Markov chains. In: Shier DR, Wallenius KT (eds) Applied mathematical modeling: a multidisciplinary approach, chapter 13. CRC, Boca Raton

    Google Scholar 

  • Jarrah A, Vastani H, Duca K, Laubenbacher R (2004) An optimal control problem for in vitro virus competition. In: Proc. of the 43rd IEEE conference on decision and control, Bahamas

  • Just W (2009) The steady state system problem is NP-hard even for Monotone quadratic Boolean dynamical systems. Ann Comb (preprint)

  • Laubenbacher R, Stigler B (2004) A computational algebra approach to the reverse engineering of gene regulatory networks. J Theor Biol 229:523–537

    Article  MathSciNet  Google Scholar 

  • Laubenbacher R, Stigler B (2005) Mathematical tools for systems biology. http://faculty.smu.edu/bstigler/sb-workshop.pdf

  • Lidl R, Niederreiter H (1977) Finite fields, encyclopedia of math and its appl, vol 20. Cambridge University Press, London

    Google Scholar 

  • Shmulevich I, Dougherty ER, Zhang W (2002) Gene perturbation and intervention in probabilistic Boolean networks. Bioinformatics 18(10):1319–1331

    Article  Google Scholar 

  • Tarjan R (1972) Depth-first search and linear graph algorithms. SIAM J Comput 1(2):146–160

    Article  MATH  MathSciNet  Google Scholar 

  • Yildrim N, Mackey M (2003) Feedback regulation in the lactose operon: a mathematical modeling study of comparison with experimental data. Biophys J 84:2841–2851

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Edgar Delgado for introducing us to the subject of BMCSs and for his help in proving Theorem 4.7. The authors also want to thank the reviewers for their helpful suggestions. The figures in this paper were prepared using the dynamic visualizer of the Virginia Bioinformatics Institute (http://dvd.vbi.vt.edu). The fourth author was partially supported by a UPR-Río Piedras FIPI grant and by Grant Number P20RR016470 from the National Center for Research Resources (NCRR), a component of the National Institutes of Health (NIH).

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, P.O Box 9018, Mayagüez, 00681-9018, Puerto Rico

    Dorothy Bollman, Omar Colón-Reyes & Victor A. Ocasio

  2. Department of Computer Science, University of Puerto Rico at Rio Piedras, Rio Piedras, 00931-3328, Puerto Rico

    Edusmildo Orozco

Authors
  1. Dorothy Bollman
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Omar Colón-Reyes
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Victor A. Ocasio
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Edusmildo Orozco
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Dorothy Bollman.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bollman, D., Colón-Reyes, O., Ocasio, V.A. et al. A Control Theory for Boolean Monomial Dynamical Systems. Discrete Event Dyn Syst 20, 19–35 (2010). https://doi.org/10.1007/s10626-009-0086-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10626-009-0086-3

Keywords