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Minimization and equivalence in multi-valued logical models of regulatory networks


Multi-valued logical models can be used to describe biological networks on a high level of abstraction based on the network structure and logical parameters capturing regulatory effects. Interestingly, the dynamics of two distinct models need not necessarily be different, which might hint at either only non-functional characteristics distinguishing the models or at different possible implementations for the same behaviour. Here, we study the conditions allowing for such effects by analysing classes of dynamically equivalent models and both structurally maximal and minimal representatives of such classes. Finally, we present an efficient algorithm that constructs a minimal representative of the respective class of a given multi-valued model.

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  • Bloniarz PA, Hunt HB III, Rosenkrantz DJ (1984) Algebraic structures with hard equivalence and minimization problems. J ACM 31(4):879–904

    Article  MATH  MathSciNet  Google Scholar 

  • Dennunzio A, Formenti E, Provillard J (2012) Non-uniform cellular automata: classes, dynamics, and decidability. Inf Comput 215:32–46

    Article  MATH  MathSciNet  Google Scholar 

  • Gallet E, Manceny M, Le Gall P, Ballarini P (2014) An LTL model checking approach for biological parameter inference. In: Merz S, Pang J (eds) Formal methods and software engineering, lecture notes in computer science, vol 8829, Springer International Publishing, pp 155–170

  • Gershenson C (2004) Introduction to random Boolean networks. arXiv:nlin/0408006

  • Kauffman S (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 22(3):437–467

    Article  Google Scholar 

  • Klarner H, Streck A, Šafránek D, Kolčák J, Siebert H (2012) Parameter identification and model ranking of Thomas networks. In: Gilbert D, Heiner M (eds) Computational methods for systems biology, lecture notes in computer science, vol 7605, Springer, Heidelberg, pp 207–226

    Chapter  Google Scholar 

  • Naldi A, Remy E, Thieffry D, Chaouiya C (2011) Dynamically consistent reduction of logical regulatory graphs. Theor Comput Sci 412(21):2207–2218

    Article  MATH  MathSciNet  Google Scholar 

  • Siebert H (2008) Local structure and behavior of Boolean bioregulatory networks. In: Horimoto K, Regensburger G, Rosenkranz M, Yoshida H (eds) Algebraic biology, lecture notes in computer science, vol 5147, Springer, Heidelberg, pp 185–199

    Google Scholar 

  • Streck A, Siebert H (2014) Equivalences in multi-valued asynchronous models of regulatory networks. In: Wąs J, Sirakoulis GC, Bandini S (eds) Cellular automata, lecture notes in computer science, vol 8751, Springer International Publishing, pp 571–575

  • Thomas R (1991) Regulatory networks seen as asynchronous automata: a logical description. J Theor Biol 153(1):1–23

    Article  Google Scholar 

  • Thomas R (2013) Remarks on the respective roles of logical parameters and time delays in asynchronous logic: an homage to El Houssine Snoussi. Bull Math Biol 75(6):896–904

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Adam Streck.

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Streck, A., Lorenz, T. & Siebert, H. Minimization and equivalence in multi-valued logical models of regulatory networks. Nat Comput 14, 555–566 (2015).

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