BACKGROUND: Boolean networks (BNs) provide an effective modelling formalism for various complex biochemical phenomena. Their long term behaviour is represented by attractors-subsets of the state space towards which the BN eventually converges. These are then typically linked to different biological phenotypes. Depending on various logical parameters, the structure and quality of attractors can undergo a significant change, known as a bifurcation. We present a methodology for analysing bifurcations in asynchronous parametrised Boolean networks. RESULTS: In this paper, we propose a computational framework employing advanced symbolic graph algorithms that enable the analysis of large networks with hundreds of Boolean variables. To visualise the results of this analysis, we developed a novel interactive presentation technique based on decision trees, allowing us to quickly uncover parameters crucial to the changes in the attractor landscape. As a whole, the methodology is implemented in our tool AEON. We evaluate the method's applicability on a complex human cell signalling network describing the activity of type-1 interferons and related molecules interacting with SARS-COV-2 virion. In particular, the analysis focuses on explaining the potential suppressive role of the recently proposed drug molecule GRL0617 on replication of the virus. CONCLUSIONS: The proposed method creates a working analogy to the concept of bifurcation analysis widely used in kinetic modelling to reveal the impact of parameters on the system's stability. The important feature of our tool is its unique capability to work fast with large-scale networks with a relatively large extent of unknown information. The results obtained in the case study are in agreement with the recent biological findings.

Faculty of Informatics Masaryk University Brno Czechia

Zobrazit více v PubMed

Kauffman SA. Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol. 1969;22(3):437–67. doi: 10.1016/0022-5193(69)90015-0. PubMed DOI

Chatain T et al. Boolean networks: beyond generalized asynchronicity. In: International workshop on cellular automata and discrete complex systems, 29–42. Springer, Cham, 2018. 10.1007/978-3-319-92675-9_3.

Fisher J et al. Synthesising executable gene regulatory networks from single-cell gene expression data. In: Computer Aided Verification, 544–560. Springer, Cham, 2015. 10.1007/978-3-319-21690-4_38.

Su C et al. Controlling large Boolean networks with temporary and permanent perturbations. In: Formal Methods, 707–724. Springer, Cham, 2019. 10.1007/978-3-030-30942-8_41.

Le Novère N. Quantitative and logic modelling of molecular and gene networks. Nature reviews. Genetics 2015. 10.1038/nrg3885 PubMed PMC

Baudin A, et al. Controlling large Boolean networks with single-step perturbations. Bioinformatics. 2019;35(14):558–567. doi: 10.1093/bioinformatics/btz371. PubMed DOI PMC

Feillet C, et al. Phase locking and multiple oscillating attractors for the coupled mammalian clock and cell cycle. Proc Natl Acad Sci. 2014;111(27):9828–9833. doi: 10.1073/pnas.1320474111. PubMed DOI PMC

Garg A, et al. Synchronous versus asynchronous modeling of gene regulatory networks. Bioinformatics. 2008;24(17):1917–1925. doi: 10.1093/bioinformatics/btn336. PubMed DOI PMC

Klemm K, Bornholdt S. Stable and unstable attractors in Boolean networks. Phys Rev E. 2005;72(5):055101. doi: 10.1103/PhysRevE.72.055101. PubMed DOI

Saadatpour A, et al. Attractor analysis of asynchronous Boolean models of signal transduction networks. J Theor Biol. 2010;266(4):641–656. doi: 10.1016/j.jtbi.2010.07.022. PubMed DOI

Zou YM. Boolean networks with multiexpressions and parameters. IEEE/ACM Trans Comput Biol Bioinf. 2013;10:584–592. doi: 10.1109/TCBB.2013.79. PubMed DOI

Beneš N et al. Formal analysis of qualitative long-term behaviour in parametrised Boolean networks. In: International conference on formal engineering methods, 353–369. Springer, Cham, 2019. 10.1007/978-3-030-32409-4_22

Abou-Jaoudé W, et al. Logical modeling and dynamical analysis of cellular networks. Front Genet. 2016;7:94. doi: 10.3389/fgene.2016.00094. PubMed DOI PMC

Dubrova E, Teslenko M. A SAT-based algorithm for finding attractors in synchronous Boolean networks. IEEE/ACM Trans Comput Biol Bioinf. 2011;8(5):1393–1399. doi: 10.1109/TCBB.2010.20. PubMed DOI

Tamura T, Akutsu T. Detecting a singleton attractor in a Boolean network utilizing SAT algorithms. IEICE Trans Fund Electron Commun Comput Sci. 2009;E92.A(2):493–501. 10.1587/transfun.E92.A.493

Devloo V, et al. Identification of all steady states in large networks by logical analysis. Bull Math Biol. 2003;65(6):1025–1051. doi: 10.1016/S0092-8240(03)00061-2. PubMed DOI

Akutsu T et al. Integer programming-based methods for attractor detection and control of Boolean networks. In: IEEE Conference on Decision and Control, 2009:5610–5617. 10.1109/CDC.2009.5400017

Qu H et al. Improving BDD-based attractor detection for synchronous Boolean networks. In: Proceedings of the 7th Asia-Pacific Symposium on Internetware. Internetware ’15, 212–220. Association for Computing Machinery, New York, NY, USA, 2015. 10.1145/2875913.2875925.

Naldi A et al. Decision diagrams for the representation and analysis of logical models of genetic networks. In: Computational Methods in Systems Biology. Springer, Cham, 2007:233–247. 10.1007/978-3-540-75140-3_16.

Klarner H, et al. Computing maximal and minimal trap spaces of Boolean networks. Nat Comput. 2015;14(4):535–544. doi: 10.1007/s11047-015-9520-7. DOI

Harvey I, Bossomaier T. Time out of joint: attractors in asynchronous random Boolean networks. In: Proceedings of the fourth european conference on artificial life. MIT Press, Cambridge, 1997:67–75

Guo W, et al. A parallel attractor finding algorithm based on Boolean satisfiability for genetic regulatory networks. PLoS ONE. 2014;9(4):1–10. doi: 10.1371/journal.pone.0094258. PubMed DOI PMC

Mushthofa M et al. Computing attractors of multi-valued gene regulatory networks using fuzzy answer set programming. In: IEEE international conference on fuzzy systems, 2016:1955–1962 . 10.1109/FUZZ-IEEE.2016.7737931

Chatain T et al. Characterization of reachable attractors using Petri net unfoldings. In: Computational Methods in Systems Biology. Springer, Cham, 2014:129–142. 10.1007/978-3-319-12982-2_10

Zhang S-Q, et al. Algorithms for finding small attractors in Boolean networks. EURASIP J Bioinform Syst Biol. 2007;2007:4–4. doi: 10.1155/2007/20180. PubMed DOI PMC

Choo S-M, Cho K-H. An efficient algorithm for identifying primary phenotype attractors of a large-scale Boolean network. BMC Syst Biol. 2016;10(1):95. doi: 10.1186/s12918-016-0338-4. PubMed DOI PMC

Cheng D et al. Analysis and Control of Boolean Networks: a Semi-tensor Product Approach. Springer, London, 2010. 10.1007/978-0-85729-097-7

Liu X, et al. Gapore: Boolean network inference using a genetic algorithm with novel polynomial representation and encoding scheme. Knowl-Based Syst. 2021;228:107277. doi: 10.1016/j.knosys.2021.107277. DOI

Zhong J, et al. Pinning control for stabilization of boolean networks under knock-out perturbation. IEEE Trans Autom Control. 2021 doi: 10.1109/TAC.2021.3070307. DOI

Acernese A, et al. Reinforcement learning approach to feedback stabilization problem of probabilistic boolean control networks. IEEE Control Syst Lett. 2020;5(1):337–342. doi: 10.1109/LCSYS.2020.3001993. DOI

Cheng X, et al. Discrimination of attractors with noisy nodes in boolean networks. Automatica. 2021;130:109630. doi: 10.1016/j.automatica.2021.109630. DOI

Zhong J, et al. Steady-state design of large-dimensional boolean networks. IEEE Trans Neural Netw Learn Syst. 2020;32(3):1149–1161. doi: 10.1109/TNNLS.2020.2980632. PubMed DOI

Shah OS, et al. ATLANTIS - attractor landscape analysis toolbox for cell fate discovery and reprogramming. Sci Rep. 2018;8(1):3554. doi: 10.1038/s41598-018-22031-3. PubMed DOI PMC

Benque D et al. BMA: visual tool for modeling and analysis of biological networks. In: Computer Aided Verification. Springer, Cham, 2012:686–692. 10.1007/978-3-642-31424-7_50

Müssel C, et al. BoolNet-an R package for generation, reconstruction and analysis of Boolean networks. Bioinformatics. 2010;26(10):1378–1380. doi: 10.1093/bioinformatics/btq124. PubMed DOI

Schwab JD, Kestler HA. Automatic screening for perturbations in Boolean networks. Front Physiol. 2018;9:431. doi: 10.3389/fphys.2018.00. PubMed DOI PMC

Klarner H, et al. PyBoolNet: a python package for the generation, analysis and visualization of Boolean networks. Bioinformatics. 2016;33(5):770–772. doi: 10.1093/bioinformatics/btw682. PubMed DOI

Berntenis N, Ebeling M. Detection of attractors of large Boolean networks via exhaustive enumeration of appropriate subspaces of the state space. BMC Bioinform. 2013;14:361. doi: 10.1186/1471-2105-14-361. PubMed DOI PMC

Helikar T, et al. The cell collective: toward an open and collaborative approach to systems biology. BMC Syst Biol. 2012;1:96. doi: 10.1186/1752-0509-6-96. PubMed DOI PMC

Klamt S, et al. Structural and functional analysis of cellular networks with Cell NetAnalyzer. BMC Syst Biol. 2007;1(1):2. doi: 10.1186/1752-0509-1-2. PubMed DOI PMC

Mizera A, et al. ASSA-PBN: a toolbox for probabilistic Boolean networks. IEEE/ACM Trans Comput Biol Bioinf. 2018;15(4):1203–1216. doi: 10.1109/TCBB.2017.2773477. PubMed DOI

Chaouiya C et al. Logical modelling of gene regulatory networks with GINsim. In: Bacterial Molecular Networks. Springer, Cham, 2012:463–479. 10.1007/978-1-61779-361-5_23 PubMed

Streck A et al. Comparative statistical analysis of qualitative parametrization sets. In: Hybrid Systems Biology. Springer, Cham, 2015:20–34. 10.1007/978-3-319-26916-0_2

Abou-Jaoudé W, Monteiro PT. On logical bifurcation diagrams. J Theor Biol. 2019;466:39–63. doi: 10.1016/j.jtbi.2019.01.008. PubMed DOI

Abou-Jaoudé W, et al. From structure to dynamics: frequency tuning in the p53-mdm2 network. J Theor Biol. 2009;258(4):561–577. doi: 10.1016/j.jtbi.2009.02.005. PubMed DOI

Beneš N et al. AEON: attractor bifurcation analysis of parametrised Boolean networks. In: Computer Aided Verification. Springer, Cham, 2020:569–581. 10.1007/978-3-030-53288-8_28

Benes N et al. AEON 2021: bifurcation decision trees in Boolean networks. In: Computational Methods in Systems Biology. Springer, Cham, 2021:230–237. 10.1007/978-3-030-85633-5_14

Barnat J et al. Detecting attractors in biological models with uncertain parameters. In: Computational Methods in Systems Biology. Springer, Cham, 2017: 40–56. 10.1007/978-3-319-67471-1_3

Beneš N et al. A model checking approach to discrete bifurcation analysis. In: Formal Methods. Springer, Cham, 2016:85–101. 10.1007/978-3-319-48989-6_6

Beneš N et al. Parallel parameter synthesis algorithm for hybrid CTL. Sci Comput Program. 2020;185. 10.1016/j.scico.2019.102321

Beneš N et al. Computing bottom SCCs symbolically using transition guided reduction. In: International conference on computer aided verification. Springer, Cham, 2021:505–528. 10.1007/978-3-030-81685-8_24

Shmulevich I, et al. Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks. Bioinformatics. 2002;18(2):261–274. doi: 10.1093/bioinformatics/18.2.261. PubMed DOI

Kuznetsov YA. Elements of Applied Bifurcation Theory. Springer, New York, 1998. 10.1007/978-1-4757-3978-7

Wang R-S, et al. Boolean modeling in systems biology: an overview of methodology and applications. Phys Biol. 2012;9(5):055001. doi: 10.1088/1478-3975/9/5/055001. PubMed DOI

Bryant RE. Graph-based algorithms for Boolean function manipulation. IEEE Trans Comput. 1986;35(8):677–691. doi: 10.1109/TC.1986.1676819. DOI

Safavian SR, Landgrebe D. A survey of decision tree classifier methodology. IEEE Trans Syst Man Cybern. 1991;21(3):660–674. doi: 10.1109/21.97458. DOI

Kent JT. Information gain and a general measure of correlation. Biometrika. 1983;70(1):163–173. doi: 10.2307/2335954. DOI

Hojyo S, et al. How COVID-19 induces cytokine storm with high mortality. Inflamm Regen. 2020;40(1):1–7. doi: 10.1186/s41232-020-00146-3. PubMed DOI PMC

Ostaszewski M, et al. COVID-19 disease map, building a computational repository of SARS-CoV-2 virus-host interaction mechanisms. Sci Data. 2020;7:136. doi: 10.1038/s41597-020-0477-8. PubMed DOI PMC

Aghamiri SS, et al. Automated inference of Boolean models from molecular interaction maps using CaSQ. Bioinformatics. 2020;36(16):4473–4482. doi: 10.1093/bioinformatics/btaa484. PubMed DOI PMC

Papin JA, et al. Reconstruction of cellular signalling networks and analysis of their properties. Nat Rev Mol Cell Biol. 2005;6(2):99–111. doi: 10.1038/nrm1570. PubMed DOI

Thakar J, et al. Network model of immune responses reveals key effectors to single and co-infection dynamics by a respiratory bacterium and a gastrointestinal helminth. PLoS Comput Biol. 2012;8(1):1002345. doi: 10.1371/journal.pcbi.1002345. PubMed DOI PMC

Polak ME, et al. Petri net computational modelling of langerhans cell interferon regulatory factor network predicts their role in T cell activation. Sci Rep. 2017;7(1):1–13. doi: 10.1038/s41598-017-00651-5. PubMed DOI PMC

Wang K et al. Stability and bifurcation of genetic regulatory networks with delays. Neurocomputing. 2010;73(16):2882–92. 10.1109/ChiCC.2014.6896981. 10th Brazilian Symposium on Neural Networks (SBRN2008).

Grieco L, et al. Integrative modelling of the influence of MAPK network on cancer cell fate decision. PLoS Comput Biol. 2013;9(10):1003286. doi: 10.1371/journal.pcbi.1003286. PubMed DOI PMC

Fu Z, et al. The complex structure of GRL0617 and SARS-CoV-2 PLpro reveals a hot spot for antiviral drug discovery. Nat Commun. 2021;12(1):1–12. doi: 10.1038/s41467-020-20718-8. PubMed DOI PMC

Gonzalez-Cotto M, et al. TREML4 promotes inflammatory programs in human and murine macrophages and alters atherosclerosis lesion composition in the apolipoprotein e deficient mouse. Front Immunol. 2020;11:397. doi: 10.3389/fimmu.2020.00397. PubMed DOI PMC