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Boolean network analysis through the joint use of linear algebra and algebraic geometry. (English) Zbl 1412.92096
Summary: Among the various phenomena that can be modeled by Boolean networks, i.e., discrete-time dynamical systems with binary state variables, gene regulatory interactions are especially well known. Therefore, the analysis of Boolean networks is critical, e.g., to identify genetic pathways and to predict the effects of mutations on the cell functionality. Two methodologies (i.e., the semi-tensor product and the Gröbner bases over finite fields) have recently been proposed to tackle the problem of determining cycles and attractors (with the corresponding basin of attraction) for such systems. Here, it is shown that, by suitably coupling methodologies taken from these two fields (i.e., linear algebra and algebraic geometry), it is not only possible to determine cycles and attractors, but also to find closed-form solutions of the Boolean network. Such a goal is pursued by finding an immersion that recasts the Boolean dynamics in a linear form and by computing the closed-form solution of the latter system. The effectiveness of this technique is demonstrated by fully computing the solutions of the Boolean network modeling the differentiation of the Th-lymphocyte, a type of white blood cells involved in the human adaptive immune system.
MSC:
92C40 Biochemistry, molecular biology
92C42 Systems biology, networks
14A99 Foundations of algebraic geometry
15A99 Basic linear algebra
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[1] Agnello, D.; Lankford, C. S.; Bream, J.; Morinobu, A.; Gadina, M.; O’shea, J. J.; Frucht, D. M., Cytokines and transcription factors that regulate t helper cell differentiation: new players and new insights, J. Clin. Immunol., 23, 3, 147-161, (2003)
[2] Akutsu, T.; Kosub, S.; Melkman, A. A.; Tamura, T., Finding a periodic attractor of a Boolean network, IEEE/ACM Trans. Comput. Biol. Bioinf., 9, 5, 1410-1421, (2012)
[3] Albert, R.; Barabási, A.-L., Dynamics of complex systems: scaling laws for the period of Boolean networks, Phys. Rev. Lett., 84, 24, 5660, (2000)
[4] Bansal, M.; Belcastro, V.; Ambesi-Impiombato, A.; Di Bernardo, D., How to infer gene networks from expression profiles, Mol. Syst. Biol., 3, 1, 78, (2007)
[5] Bansal, M.; Gatta, G. D.; Di Bernardo, D., Inference of gene regulatory networks and compound mode of action from time course gene expression profiles, BMC Bioinf., 22, 7, 815-822, (2006)
[6] Busetto, A. G.; Lygeros, J., Experimental design for system identification of Boolean control networks in biology, IEEE 53rd Annual Conference on Decision and Control (CDC), 5704-5709, (2014), IEEE
[7] Chaouiya, C.; Naldi, A.; Thieffry, D., Logical modelling of gene regulatory networks with GINsim, Bacterial Molecular Networks, 463-479, (2012), Springer
[8] Cheng, D.; Qi, H., A linear representation of dynamics of Boolean networks, IEEE Trans. Autom. Control, 55, 10, 2251-2258, (2010) · Zbl 1368.37025
[9] Cheng, D.; Qi, H.; Li, Z., Analysis and Control of Boolean Networks: A Semi-tensor Product Approach, (2010), Springer Science & Business Media
[10] Covert, M. W.; Schilling, C. H.; Famili, I.; Edwards, J. S.; Goryanin, I. I.; Selkov, E.; Palsson, B. O., Metabolic modeling of microbial strains in silico, Trends Biochem. Sci., 26, 3, 179-186, (2001)
[11] Cox, D. A.; Little, J.; O’Shea, D., Using Algebraic Geometry, (2006), Springer Science & Business Media
[12] Cox, D. A.; Little, J.; O’Shea, D., Ideals, Varieties, and Algorithms, (2015), Springer Science & Business Media
[13] Crick, F. H., On protein synthesis, (Sanders, F. K., Symposia of the Society for Experimental Biology: The Biological Replication of Macromolecules, 12, (1958)), 138-163
[14] Crick, F. H., Central dogma of molecular biology, Nature, 227, 5258, 561, (1970)
[15] Davidich, M.; Bornholdt, S., The transition from differential equations to Boolean networks: a case study in simplifying a regulatory network model, J. Theor. Biol., 255, 3, 269-277, (2008) · Zbl 1400.92207
[16] Dilão, R., The regulation of gene expression in eukaryotes: bistability and oscillations in repressilator models, J. Theor. Biol., 340, 199-208, (2014)
[17] Faugère, J.-C.; Ars, G., Comparison of XL and Gröbner basis algorithms over Finite Fields, (2004), INRIA, Ph.D. thesis
[18] Fornasini, E.; Valcher, M. E., Observability, reconstructibility and state observers of Boolean control networks, IEEE Trans. Autom. Control, 58, 6, 1390-1401, (2013) · Zbl 1369.93101
[19] Grayson, D. R., Stillman, M. E., 2018. Macaulay2, a software system for research in algebraic geometry. Available at https://faculty.math.illinois.edu/Macaulay2/.
[20] Hamming, R. W., Error detecting and error correcting codes, Bell Labs Tech. J., 29, 2, 147-160, (1950) · Zbl 1402.94084
[21] Hinkelmann, F.; Brandon, M.; Guang, B.; McNeill, R.; Blekherman, G.; Veliz-Cuba, A.; Laubenbacher, R., ADAM: analysis of discrete models of biological systems using computer algebra, BMC Bioinf., 12, 1, 295, (2011)
[22] Hinkelmann, F.; Murrugarra, D.; Jarrah, A. S.; Laubenbacher, R., A mathematical framework for agent based models of complex biological networks, Bull. Math. Biol., 73, 7, 1583-1602, (2011) · Zbl 1225.92001
[23] Isidori, A., Nonlinear Control Systems, (2013), Springer Science & Business Media · Zbl 0569.93034
[24] Kauffman, S. A., Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theor. Biol., 22, 3, 437-467, (1969)
[25] Kobayashi, K.; Hiraishi, K., Optimization-based approaches to control of probabilistic Boolean networks, Algorithms, 10, 1, 31, (2017) · Zbl 1461.93291
[26] Kobayashi, K.; Imura, J.-I.; Hiraishi, K., Polynomial-time algorithm for controllability test of a class of Boolean biological networks, EURASIP J. Bioinf. Syst.Biol., 2010, 1, 210685, (2010)
[27] Melkman, A. A.; Akutsu, T., An improved satisfiability algorithm for nested canalyzing functions and its application to determining a singleton attractor of a Boolean network, J. Comput. Biol., 20, 12, 958-969, (2013)
[28] Mendoza, L., A network model for the control of the differentiation process in Th cells, Biosystems, 84, 2, 101-114, (2006)
[29] Menini, L.; Possieri, C.; Tornambe, A., Boolean network representation of a continuous-time system and finite-horizon optimal control: application to the single-gene regulatory system for the lac operon, Int. J. Control, 90, 3, 519-552, (2017) · Zbl 1388.49042
[30] Menini, L.; Tornambe, A., Deformations for linear periodic discrete-time systems: the adjoint normal form, Int. J. Control, 86, 7, 1248-1257, (2013) · Zbl 1278.93158
[31] Menini, L.; Tornambe, A., Immersion and Darboux polynomials of Boolean networks with application to the pseudomonas syringae hrp regulon, IEEE 52nd Annual Conference on Decision and Control (CDC), 4092-4097, (2013), IEEE
[32] Menini, L.; Tornambe, A., Observability and dead-beat observers for boolean networks modeled as polynomial discrete-time systems, IEEE 52nd Annual Conference on Decision and Control (CDC), 4428-4433, (2013), IEEE
[33] Meyer, C. D., Matrix Analysis and Applied Linear Algebra, (2000), Siam
[34] Murphy, K. M.; Reiner, S. L., Decision making in the immune system: the lineage decisions of helper T cells, Nat. Rev. Immunol., 2, 12, 933, (2002)
[35] Müssel, C.; Hopfensitz, M.; Kestler, H. A., BoolNet—an R package for generation, reconstruction and analysis of Boolean networks, Bioinformatics, 26, 10, 1378-1380, (2010)
[36] Papin, J. A.; Price, N. D.; Wiback, S. J.; Fell, D. A.; Palsson, B. O., Metabolic pathways in the post-genome era, Trends Biochem. Sci., 28, 5, 250-258, (2003)
[37] Possieri, C.; Teel, A. R., Asymptotic stability in probability for stochastic Boolean networks, Automatica, 83, 1-9, (2017) · Zbl 1373.93369
[38] Remy, E.; Ruet, P.; Mendoza, L.; Thieffry, D.; Chaouiya, C., From logical regulatory graphs to standard Petri nets: dynamical roles and functionality of feedback circuits, Transactions on Computational Systems Biology VII, 56-72, (2006), Springer
[39] Rohr, C.; Marwan, W.; Heiner, M., Snoopy—a unifying Petri net framework to investigate biomolecular networks, Bioinformatics, 26, 7, 974-975, (2010)
[40] Shah, O. S.; Chaudhary, M. F.A.; Awan, H. A.; Fatima, F.; Arshad, Z.; Amina, B.; Ahmed, M.; Hameed, H.; Furqan, M.; Khalid, S., ATLANTIS-attractor landscape analysis toolbox for cell fate discovery and reprogramming, Sci. Rep., 8, 1, 3554, (2018)
[41] Veliz-Cuba, A., Reduction of Boolean network models, J. Theor. Biol., 289, 167-172, (2011) · Zbl 1397.92265
[42] Veliz-Cuba, A.; Aguilar, B.; Hinkelmann, F.; Laubenbacher, R., Steady state analysis of Boolean molecular network models via model reduction and computational algebra, BMC Bioinf., 15, 1, 221, (2014)
[43] Veliz-Cuba, A.; Aguilar, B.; Laubenbacher, R., Dimension reduction of large sparse AND-NOT network models, Electron. Notes Theor. Comput. Sci., 316, 83-95, (2015) · Zbl 1352.92069
[44] Veliz-Cuba, A.; Jarrah, A. S.; Laubenbacher, R., Polynomial algebra of discrete models in systems biology, BMC Bioinf., 26, 13, 1637-1643, (2010)
[45] Wuensche, A., Exploring Discrete Dynamics, (2011), Luniver Press · Zbl 1280.68009
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