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International Conference on Computational Methods in Systems Biology

CMSB 2020: Computational Methods in Systems Biology pp 210–233Cite as

Classifier Construction in Boolean Networks Using Algebraic Methods

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Part of the book series: Lecture Notes in Computer Science ((LNBI,volume 12314))

Abstract

We investigate how classifiers for Boolean networks (BNs) can be constructed and modified under constraints. A typical constraint is to observe only states in attractors or even more specifically steady states of BNs. Steady states of BNs are one of the most interesting features for application. Large models can possess many steady states. In the typical scenario motivating this paper we start from a Boolean model with a given classification of the state space into phenotypes defined by high-level readout components. In order to link molecular biomarkers with experimental design, we search for alternative components suitable for the given classification task. This is useful for modelers of regulatory networks for suggesting experiments and measurements based on their models. It can also help to explain causal relations between components and phenotypes. To tackle this problem we need to use the structure of the BN and the constraints. This calls for an algebraic approach. Indeed we demonstrate that this problem can be reformulated into the language of algebraic geometry. While already interesting in itself, this allows us to use Gröbner bases to construct an algorithm for finding such classifiers. We demonstrate the usefulness of this algorithm as a proof of concept on a model with 25 components.

Supported by the DFG-funded Cluster of Excellence MATH+: Berlin Mathematics Research Center, Project AA1-4. Matías R. Bender was supported by the ERC under the European’s Horizon 2020 research and innovation programme (grant agreement No. 787840).

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Notes

  1. 1.

    Note that \(\lim _{n \rightarrow \infty } \frac{n!}{2^n} = \infty \), so it is more efficient to iterate through \(2^{n}\) candidate sets than through n! orderings.

  2. 2.

    An attractor of a Boolean network is a terminal strongly connected component of the corresponding state transition graph.

  3. 3.

    This follows from the identity \(f\cdot (f+g+f\cdot g)=f\) for \(f,g\in \mathbb {B}(n)\).

References

  1. Albert, R., Thakar, J.: Boolean modeling: a logic-based dynamic approach for understanding signaling and regulatory networks and for making useful predictions. Wiley Interdiscip. Rev.: Syst. Biol. Med. 6(5), 353–369 (2014)

    Google Scholar 

  2. Alexe, S., Blackstone, E., Hammer, P.L., Ishwaran, H., Lauer, M.S., Snader, C.E.P.: Coronary risk prediction by logical analysis of data. Ann. Oper. Res. 119(1–4), 15–42 (2003)

    Article  Google Scholar 

  3. Bonzanni, N., et al.: Hard-wired heterogeneity in blood stem cells revealed using a dynamic regulatory network model. Bioinformatics 29(13), i80–i88 (2013)

    Article  Google Scholar 

  4. Boros, E., Hammer, P.L., Ibaraki, T., Kogan, A., Mayoraz, E., Muchnik, I.: An implementation of logical analysis of data. IEEE Trans. Knowl. Data Eng. 12(2), 292–306 (2000)

    Article  Google Scholar 

  5. Brickenstein, M., Dreyer, A.: Polybori: a framework for gröbner-basis computations with Boolean polynomials. J. Symb. Comput. 44(9), 1326–1345 (2009)

    Article  Google Scholar 

  6. Brickenstein, M., Dreyer, A.: Gröbner-free normal forms for Boolean polynomials. J. Symb. Comput. 48, 37–53 (2013)

    Article  Google Scholar 

  7. Buchberger, B.: Applications of Gröbner bases in non-linear computational geometry. In: Rice, J.R. (ed.) Mathematical Aspects of Scientific Software. The IMA Volumes in Mathematics and its Applications, pp. 59–87. Springer, New York (1988). https://doi.org/10.1007/978-1-4684-7074-1_3

  8. Calzone, L., et al.: Mathematical modelling of cell-fate decision in response to death receptor engagement. PLOS Comput. Biol. 6(3), 1–15 (2010). https://doi.org/10.1371/journal.pcbi.1000702

    Article  MathSciNet  Google Scholar 

  9. Chaouiya, C., Remy, E., Mossé, B., Thieffry, D.: Qualitative analysis of regulatory graphs: a computational tool based on a discrete formal framework. In: Benvenuti, L., De Santis, A., Farina, L. (eds.) Positive Systems. Lecture Notes in Control and Information Science, vol. 294. Springer, Heidelberg. https://doi.org/10.1007/978-3-540-44928-7_17

  10. Cheng, D., Qi, H.: Controllability and observability of Boolean control networks. Automatica 45(7), 1659–1667 (2009)

    Article  MathSciNet  Google Scholar 

  11. Chikalov, I., et al.: Logical analysis of data: theory, methodology and applications. In: Three Approaches to Data Analysis. Intelligent Systems Reference Library, vol. 41. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-28667-4_3

  12. Cox, D.A., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. UTM. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-16721-3

    Book  MATH  Google Scholar 

  13. Cox, D.A., Little, J., O’Shea, D.: Using Algebraic Geometry (2004)

    Google Scholar 

  14. Dickenstein, A., Millán, M.P., Shiu, A., Tang, X.: Multistationarity in structured reaction networks. Bull. Math. Biol. 81(5), 1527–1581 (2019). https://doi.org/10.1007/s11538-019-00572-6

    Article  MathSciNet  MATH  Google Scholar 

  15. Drton, M., Sturmfels, B., Sullivant, S.: Lectures on Algebraic Statistics. Oberwolfach Seminars, Birkhäuser Basel (2009). https://www.springer.com/gp/book/9783764389048

  16. Faugère, J.C., Gianni, P., Lazard, D., Mora, T.: Efficient computation of zero-dimensional Gröbner bases by change of ordering. J. Symb. Comput. 16(4), 329–344 (1993)

    Article  Google Scholar 

  17. Faugère, J.-C., Joux, A.: Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using Gröbner bases. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 44–60. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45146-4_3

    Chapter  Google Scholar 

  18. Fauré, A., Vreede, B.M., Sucena, É., Chaouiya, C.: A discrete model of drosophila eggshell patterning reveals cell-autonomous and juxtacrine effects. PLoS Comput. Biol. 10(3), e1003527 (2014)

    Article  Google Scholar 

  19. Gao, S., Platzer, A., Clarke, E.M.: Quantifier elimination over finite fields using Gröbner bases. In: Winkler, F. (ed.) CAI 2011. LNCS, vol. 6742, pp. 140–157. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21493-6_9

    Chapter  Google Scholar 

  20. Germundsson, R.: Basic results on ideals and varieties in finite fields. Tech. rep. S-581 83 (1991)

    Google Scholar 

  21. Gonzalez, A.G., Naldi, A., Sánchez, L., Thieffry, D., Chaouiya, C.: GINsim: a software suite for the qualitative modelling, simulation and analysis of regulatory networks. Biosystems 84(2), 91–100 (2006). https://doi.org/10.1016/j.biosystems.2005.10.003. http://www.sciencedirect.com/science/article/pii/S0303264705001693

    Article  Google Scholar 

  22. González, A., Chaouiya, C., Thieffry, D.: Logical modelling of the role of the Hh pathway in the patterning of the drosophila wing disc. Bioinformatics 24(16), i234–i240 (2008)

    Article  Google Scholar 

  23. Hammer, P.L., Bonates, T.O.: Logical analysis of data - an overview: from combinatorial optimization to medical applications. Ann. Oper. Res. 148(1), 203–225 (2006)

    Article  Google Scholar 

  24. Jarrah, A.S., Laubenbacher, R.: Discrete models of biochemical networks: the toric variety of nested canalyzing functions. In: Anai, H., Horimoto, K., Kutsia, T. (eds.) AB 2007. LNCS, vol. 4545, pp. 15–22. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73433-8_2

    Chapter  Google Scholar 

  25. Laubenbacher, R., Stigler, B.: A computational algebra approach to the reverse engineering of gene regulatory networks. J. Theor. Biol. 229(4), 523–537 (2004). https://doi.org/10.1016/j.jtbi.2004.04.037. http://www.sciencedirect.com/science/article/pii/S0022519304001754

    Article  MathSciNet  MATH  Google Scholar 

  26. Le Novere, N.: Quantitative and logic modelling of molecular and gene networks. Nat. Rev. Genet. 16(3), 146–158 (2015)

    Article  Google Scholar 

  27. Millán, M.P., Dickenstein, A., Shiu, A., Conradi, C.: Chemical reaction systems with toric steady states. Bull. Math. Biol. 74(5), 1027–1065 (2012)

    Article  MathSciNet  Google Scholar 

  28. Minato, S.I.: Zero-suppressed BDDs for set manipulation in combinatorial problems. In: Proceedings of the 30th International Design Automation Conference, pp. 272–277. ACM (1993)

    Google Scholar 

  29. Mishchenko, A.: An introduction to zero-suppressed binary decision diagrams. In: Proceedings of the 12th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning, vol. 8, pp. 1–15. Citeseer (2001)

    Google Scholar 

  30. Murrugarra, D., Veliz-Cuba, A., Aguilar, B., Laubenbacher, R.: Identification of control targets in Boolean molecular network models via computational algebra. BMC Syst. Biol. 10(1), 94 (2016). https://doi.org/10.1186/s12918-016-0332-x

    Article  Google Scholar 

  31. Naldi, A., Remy, É., Thieffry, D., Chaouiya, C.: Dynamically consistent reduction of logical regulatory graphs. Theor. Comput. Sci. 412(21), 2207–2218 (2011). https://doi.org/10.1016/j.tcs.2010.10.021. http://www.sciencedirect.com/science/article/pii/S0304397510005839

    Article  MathSciNet  MATH  Google Scholar 

  32. Samaga, R., Klamt, S.: Modeling approaches for qualitative and semi-quantitative analysis of cellular signaling networks. Cell Commun. Signal. 11(1), 43 (2013)

    Article  Google Scholar 

  33. Samaga, R., Saez-Rodriguez, J., Alexopoulos, L.G., Sorger, P.K., Klamt, S.: The logic of EGFR/ERBB signaling: theoretical properties and analysis of high-throughput data. PLoS Comput. Biol. 5(8), e1000438 (2009)

    Article  Google Scholar 

  34. Sánchez, L., Chaouiya, C., Thieffry, D.: Segmenting the fly embryo: logical analysis of the role of the segment polarity cross-regulatory module. Int. J. Dev. Biol. 52(8), 1059–1075 (2002)

    Article  Google Scholar 

  35. Sato, Y., Inoue, S., Suzuki, A., Nabeshima, K., Sakai, K.: Boolean Gröbner bases. J. Symb. Comput. 46(5), 622–632 (2011)

    Article  Google Scholar 

  36. Sturmfels, B.: Gröbner Bases and Convex Polytopes, vol. 8. American Mathematical Society, Providence (1996)

    MATH  Google Scholar 

  37. Thobe, K., Sers, C., Siebert, H.: Unraveling the regulation of mTORC2 using logical modeling. Cell Commun. Signal. 15(1), 6 (2017)

    Article  Google Scholar 

  38. Veliz-Cuba, A.: An algebraic approach to reverse engineering finite dynamical systems arising from biology. SIAM J. Appl. Dyn. Syst. 11(1), 31–48 (2012)

    Article  MathSciNet  Google Scholar 

  39. Veliz-Cuba, A., Aguilar, B., Hinkelmann, F., Laubenbacher, R.: Steady state analysis of Boolean molecular network models via model reduction and computational algebra. BMC Bioinform. 15(1), 221 (2014). https://doi.org/10.1186/1471-2105-15-221

    Article  Google Scholar 

  40. Veliz-Cuba, A., Jarrah, A.S., Laubenbacher, R.: Polynomial algebra of discrete models in systems biology. Bioinformatics 26(13), 1637–1643 (2010). https://doi.org/10.1093/bioinformatics/btq240

    Article  MATH  Google Scholar 

  41. Vera-Licona, P., Jarrah, A., Garcia-Puente, L.D., McGee, J., Laubenbacher, R.: An algebra-based method for inferring gene regulatory networks. BMC Syst. Biol. 8(1), 37 (2014)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Freie Universität Berlin, Berlin, Germany

    Robert Schwieger, Heike Siebert & Christian Haase

  2. Technische Universität Berlin, Berlin, Germany

    Matías R. Bender

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  1. Robert Schwieger
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Corresponding author

Correspondence to Robert Schwieger .

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Oxford, Oxford, UK

    Alessandro Abate

  2. Department of Computer and Information Science, University of Konstanz, Konstanz, Germany

    Tatjana Petrov

  3. Department of Computer Science, Saarland University, Saarbrücken, Germany

    Verena Wolf

Appendix

Appendix

Table 3. 9 different representations of classifier for NonACD (see Sect. 6).
Full size table
Table 4. 17 different representations of the classifier for apoptosis (see Sect. 6)
Full size table
Table 5. 84 different representations of the classifier of survival of the cell. (see Sect.  6)
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Schwieger, R., Bender, M.R., Siebert, H., Haase, C. (2020). Classifier Construction in Boolean Networks Using Algebraic Methods. In: Abate, A., Petrov, T., Wolf, V. (eds) Computational Methods in Systems Biology. CMSB 2020. Lecture Notes in Computer Science(), vol 12314. Springer, Cham. https://doi.org/10.1007/978-3-030-60327-4_12

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  • DOI: https://doi.org/10.1007/978-3-030-60327-4_12

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