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Advances in Unconventional Computing pp 609–660Cite as

Computational Properties of Cell Regulatory Pathways Through Petri Nets

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Part of the book series: Emergence, Complexity and Computation ((ECC,volume 22))

Abstract

The paper develops a Petri net model of a negative feedback oscillator, Case 2a from Tyson et al. (Curr. Opin. Cell Biol. 15, 221–231, (2003), [48]), in order to be able to perform the holonomy decomposition of the automaton derived from its token markings and allowed transitions for a given initial state. The objective is to investigate the algebraic structure of the cascade product obtained from its holonomy components and to relate it to the behaviour of the physical system, in particular to the oscillations. The analysis is performed in two steps, first focusing on one of its component systems, the Goldbeter–Koshland ultrasensitive switch (Case 1c from Tyson et al. Curr. Opin. Cell Biol. 15, 221–231, (2003), [48]), in order to verify the validity of its differential model and, from this, to validate the corresponding Petri net through a stochastic simulation. The paper does not present new original results but, rather, discusses and critiques existing results from the different points of view of continuous and discrete mathematics and stochasticity. The style is one of a review paper or tutorial, specifically to make the material and the concepts accessible to a wide interdisciplinary audience. We find that the Case 2a model widely reported in the literature violates the assumptions of the Michaelis–Menten quasi-steady-state approximation. However, we are still able to show oscillations of the full rate equations and of the corresponding Petri net for a different set of parameters and initial conditions. We find that even the automata derived from very coarse Petri nets of Case 1c and Case 2a, with places of capacity 1, are able to capture meaningful biochemical information in the form of algebraic groups, in particular the reversibility of the phosphorylation reactions. Significantly, it appears that the algebraic structures uncovered by holonomy decomposition are a larger set than what may be relevant to a specific physical problem with specific initial conditions, although they are all physically possible. This highlights the role of physical context in helping select which algebraic structures to focus on when analysing particular problems. Finally, the interpretation of Petri nets as positional number systems provides an additional perspective on the computational properties of biological systems.

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Notes

  1. 1.

    http://www.biomicsproject.eu.

  2. 2.

    http://mpf.biol.vt.edu/lab_website/.

  3. 3.

    More precisely, if the capacity of the Petri net is 3, for example, meaning that each place can hold up to 3 tokens, then the base is \(3+1 = 4\) since the absence of tokens in a given place corresponds to a 0 digit for that place. So for a constant-capacity Petri net the base for this scheme is always given by the capacity \(+\)1.

  4. 4.

    Either theorem shows that more than one cascade of machines can emulate the same automaton, i.e. the decomposition is not unique. Uniqueness up to isomorphism is only guaranteed in the analogous and immensely simpler case of the decomposition of finite groups, known as the Jordan–Hölder theorem of elementary finite group theory.

  5. 5.

    A reset is a constant map from two or more states to a single state.

  6. 6.

    The search for the right variables of a problem was vigorously pursued in the empirical discipline of hydraulics and hydrodynamics in the 19th Century, which led to method of similarity analysis which was then later shown to be a branch of group theory. In other words, looking for the right variables of a problem and looking for its symmetries are often one and the same problem.

  7. 7.

    One of these inaccuracies is present in our own work: it appears in [9] and was explained and rectified in [11].

  8. 8.

    Strictly speaking, for the QSS\(_2\) assumption, since both QSS and tQSS rely on the QSS\(_1\) assumption.

  9. 9.

    Confusion may be caused by the use of ‘total’ in the name of this approximation to indicate only \(Y + [YX]\), whereas in (24.16) we used \(Y_T\) for all the species that involve Y in some way. The former is a subjective choice for the name of a variable that makes sense in reactions where the product is actually a different molecule rather than recognizably the same molecule in a phosphorylated state, as here.

  10. 10.

    As expected, the same result is obtained by substituting \(Y_p + [Y_p Z]\) for \(\hat{Y}_p\) in (24.50).

  11. 11.

    Or could mean inhibition if the dotted line ends with a flat segment instead of an arrow.

  12. 12.

    Rather than a statement of biochemistry, this should be seen as a dynamical systems interpretation of the observed behaviour.

  13. 13.

    A permutation-reset automaton is an automaton whose action is either a permutation of the state set or a constant map or reset (i.e. a map from the state set to a single state).

  14. 14.

    This metaphor is due to Attila Egri-Nagy.

  15. 15.

    Transformation semigroup elements composed of Petri net transitions are normally assumed to act on the left. This is done so that a string of transitions can be written and read from left to right.

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Acknowledgments

The author is grateful to Chrystopher L. Nehaniv and Attila Egri-Nagy for helpful discussions and for help with running SgpDec. The author is also grateful to Zoran Konkoli for discussions about the shortcomings of ODE models in the presence of low particle numbers and their generalizations based on stochastic methods, and to the TRUCE EU project that provided the context that enabled such discussions to take place. Finally, the author is grateful to Egon Börger for the many stimulating discussions about the similarities and differences between Petri nets and Abstract State Machines. The work reported in this article was funded by the BIOMICS EU project, contract number CNECT-ICT-318202. Its support is gratefully acknowledged.

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  1. Royal Society Wolfson Biocomputation Research Lab, School of Computer Science, University of Hertfordshire, Hatfield, UK

    Paolo Dini

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  1. Unconventional Computing Centre, University of the West of England Unconventional Computing Centre, Bristol, United Kingdom

    Andrew Adamatzky

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Dini, P. (2017). Computational Properties of Cell Regulatory Pathways Through Petri Nets. In: Adamatzky, A. (eds) Advances in Unconventional Computing. Emergence, Complexity and Computation, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-33924-5_24

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