Skip to main content

Advertisement

SpringerLink
Log in

Robustness, canalyzing functions and systems design

  • Original Paper
  • Published:
Theory in Biosciences Aims and scope Submit manuscript

Abstract

We study a notion of knockout robustness of a stochastic map (Markov kernel) that describes a system of several input random variables and one output random variable. Robustness requires that the behaviour of the system does not change if one or several of the input variables are knocked out. Gibbs potentials are used to give a mechanistic description of the behaviour of the system after knockouts. Robustness imposes structural constraints on these potentials. We show that robust systems can be described in terms of suitable interaction families of Gibbs potentials, which allows us to address the problem of systems design. Robustness is also characterized by conditional independence constraints on the joint distribution of input and output. The set of all probability distributions corresponding to robust systems can be decomposed into a finite union of components, and we find parametrizations of the components.

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

Access this article

Log in via an institution

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

  • Albert I, Thakar J, Li S, Zhang R, Albert R (2008) Boolean network simulations for life scientists. Source Code Biol Med 3:16

    Article  PubMed Central  PubMed  Google Scholar 

  • Ay N, Krakauer DC (2007) Geometric robustness theory and biological networks. Theory Biosci 125(2):93–121

    PubMed  Google Scholar 

  • Birolini A (2010) Reliability engineering: theory and practice. 6th edn. Springer, Berlin

  • Boldhaus G, Bertschinger N, Rauh J, Olbrich E, Klemm K (2010) Robustness of Boolean dynamics under knockouts. Phys Rev E 82(2):021916

    Article  Google Scholar 

  • Drton M, Sturmfels B, Sullivant S (2009) Lectures on algebraic statistics. 1st ed., ser. Oberwolfach Seminars. Birkhäuser, Basel, vol. 39.

  • Fink A (2011) The binomial ideal of the intersection axiom for conditional probabilities. J Algeb Comb 33(3):455–463

    Article  Google Scholar 

  • Herzog J, Hibi T, Hreinsdóttir F, Kahle T, Rauh J (2010) Binomial edge ideals and conditional independence statements. Adv Appl Math 45(3):317–333

    Article  Google Scholar 

  • Jarrah A.S, Raposa B, Laubenbacher R (2007) Nested canalyzing, unate cascade, and polynomial functions. Physica D 233(2):167–174

    Article  PubMed Central  PubMed  Google Scholar 

  • Kauffman S (1993) The origins of order: self-organization and selection in evolution. Oxford University Press, Oxford

  • Kauffman S, Peterson C, Samuelsson B, Troein C (2003) Random Boolean network models and the yeast transcriptional network. PNAS 100(25):14796–14799

    Google Scholar 

  • Kauffman S, Peterson C, Samuelsson B, Troein C (2004) Genetic networks with canalyzing Boolean rules are always stable. PNAS 101(49):17102–17107

    Google Scholar 

  • Lauritzen SL (1996) Graphical Models. In: Oxford Statistical Science Series, 1st edn. Oxford University Press, Oxford

  • Moreira AA, Amaral LN (2005) Canalizing Kauffman networks: nonergodicity and its effect on their critical behavior. Phys Rev Lett 94:218702

    Article  PubMed Central  PubMed  Google Scholar 

  • Ohtani M (2011) Graphs and ideals generated by some 2-minors. Commun Algebra 39(3):905–917

    Article  Google Scholar 

  • Rauh J (2013) Generalized binomial edge ideals. Adv Appl Math 50(3):409–414

    Article  Google Scholar 

  • Schuster P, Fontana W, Stadler PF, Hofacker IL (1994) From sequences to shapes and back: a case study in RNA secondary structures. Proc Roy Soc Lond B 255:279–284

    Article  CAS  Google Scholar 

  • Swanson I, Taylor A (2011) Minimal primes of ideals arising from conditional independence statements. J Algebra. Preprint: arXiv:1107.5604v3

  • de Visser JAGM, Hermisson J, Wagner GP, Meyers LA, Bagheri-Chaichian H, Blanchard JL, Chao L, Cheverud JM, Elena SF, Fontana W, Gibson G, Hansen TF, Krakauer D, Lewontin RC, Ofria C, Rice SH, von Dassow G, Wagner A, Whitlock MC (2003) Perspective: evolution and detection of genetic robustness. Evolution 57(9):1959–1972

  • Wagner G, Stadler PF (2003) Quasi-independence, homology and the unity of type: a topological theory of characters. J Theor Biol 220:505–527

    Article  PubMed  Google Scholar 

Download references

Acknowledgments

This work has been supported by the Volkswagen Foundation and the Santa Fe Institute. Nihat Ay thanks David Krakauer and Jessica Flack for many stimulating discussions on robustness. We thank the reviewers for their detailed remarks which helped us to improve our manuscript.

Author information

Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany

    Johannes Rauh & Nihat Ay

  2. Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM , 87501, USA

    Nihat Ay

Authors
  1. Johannes Rauh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nihat Ay
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Johannes Rauh.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rauh, J., Ay, N. Robustness, canalyzing functions and systems design. Theory Biosci. 133, 63–78 (2014). https://doi.org/10.1007/s12064-013-0186-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12064-013-0186-3

Keywords

Search

Navigation