Asymptotic stability in probability for stochastic Boolean networks. (English) Zbl 1373.93369
Summary: In this paper, a new class of Boolean networks, called Stochastic Boolean Networks, is presented. These systems combine some features of the classical deterministic Boolean networks (the state variables admit two operation levels, either 0 or 1) and of Probabilistic Boolean Networks (at each time instant the transition map is selected through a random process), enriching the set of admissible dynamical behaviors, thanks to the set-valued nature of the transition map. Necessary and sufficient Lyapunov conditions are given to guarantee global asymptotic stability (resp., global asymptotic stability in probability) of a given set for a deterministic Boolean network with set-valued transition map (resp., for a Stochastic Boolean Network). A constructive procedure to compute a Lyapunov function (resp., stochastic Lyapunov function) relative to a given set for a deterministic Boolean network with set-valued transition map (resp., Stochastic Boolean Network) is reported.
MSC:
| 93E15 | Stochastic stability in control theory |
| 93D20 | Asymptotic stability in control theory |
| 90B15 | Stochastic network models in operations research |
References:
| [1] | Albert, R.; Barabási, A.-L., Dynamics of complex systems: Scaling laws for the period of Boolean networks, Physical Review Letters, 84, 24, 5660-5663 (2000) |
| [2] | Albert, R.; Othmer, H. G., The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in drosophila melanogaster, Journal of Theoretical Biology, 223, 1, 1-18 (2003) · Zbl 1464.92108 |
| [3] | Albert, I.; Thakar, J.; Li, S.; Zhang, R.; Albert, R., Boolean network simulations for life scientists, Source Code for Biology and Medicine, 3, 1-8 (2008) |
| [4] | Bansal, M.; Belcastro, V.; Ambesi-Impiombato, A.; Di Bernardo, D., How to infer gene networks from expression profiles, Molecular Systems Biology, 3, 78, 1-10 (2007) |
| [5] | Bansal, M.; Della Gatta, G.; Di Bernardo, D., Inference of gene regulatory networks and compound mode of action from time course gene expression profiles, BMC Bioinformatics, 22, 7, 815-822 (2006) |
| [6] | Bertsekas, D. P.; Tsitsiklis, J. N., Parallel and distributed computation: numerical methods. Vol. 23 (1989), Prentice hall: Prentice hall Englewood Cliffs, NJ · Zbl 0743.65107 |
| [7] | Bollobás, B., Modern graph theory. Vol. 184 (2013), Springer Science & Business Media |
| [8] | Bourbaki, N., General topology (1998), Springer Verlag · Zbl 0145.19302 |
| [10] | Caetano, M. A.L.; Yoneyama, T., Boolean network representation of contagion dynamics during a financial crisis, Physica A, 417, 1-6 (2015) · Zbl 1402.91939 |
| [11] | Chaves, M.; Albert, R.; Sontag, E. D., Robustness and fragility of Boolean models for genetic regulatory networks, Journal of Theoretical Biology, 235, 3, 431-449 (2005) · Zbl 1445.92173 |
| [12] | Cheng, D.; Qi, H., A linear representation of dynamics of Boolean networks, IEEE Transactions on Automatic Control, 55, 10, 2251-2258 (2010) · Zbl 1368.37025 |
| [13] | Conte, E.; Federici, A.; Zbilut, J. P., On a simple case of possible non-deterministic chaotic behavior in compartment theory of biological observables, Chaos, Solitons & Fractals, 22, 2, 277-284 (2004) · Zbl 1058.92001 |
| [14] | Cormen, T. H., Introduction to algorithms (2009), MIT · Zbl 1187.68679 |
| [15] | Datta, A.; Choudhary, A.; Bittner, M. L.; Dougherty, E. R., External control in Markovian genetic regulatory networks, Machine Learning, 52, 1-2, 169-191 (2003) · Zbl 1039.68161 |
| [16] | Easton, G.; Brooks, R. J.; Georgieva, K.; Wilkinson, I., Understanding the dynamics of industrial networks using Kauffman Boolean networks, Advances in Complex Systems, 11, 1, 139-164 (2008) · Zbl 1165.90354 |
| [17] | Eisen, M. B.; Spellman, P. T.; Brown, P. O.; Botstein, D., Cluster analysis and display of genome-wide expression patterns, Proceedings of the National Academy of Sciences, 95, 25, 14863-14868 (1998) |
| [18] | Friedman, N.; Linial, M.; Nachman, I.; Pe’er, D., Using Bayesian networks to analyze expression data, Journal of Computational Biology, 7, 3-4, 601-620 (2000) |
| [19] | Fristedt, B. E.; Gray, L. F., A modern approach to probability theory (1997), Birlchäuser · Zbl 0869.60001 |
| [20] | Ghysen, A.; Thomas, R., The formation of sense organs in drosophila: a logical approach, BioEssays, 25, 8, 802-807 (2003) |
| [21] | Goebel, R.; Sanfelice, R. G.; Teel, A. R., Hybrid dynamical systems: modeling, stability, and robustness (2012), Princeton Univ. Press · Zbl 1241.93002 |
| [23] | Grieb, M.; Burkovski, A.; Sträng, J. E.; Kraus, J. M.; Groß, A.; Palm, G.; Kühl, M.; Kestler, H. A., Predicting variabilities in cardiac gene expression with a Boolean network incorporating uncertainty, PloS One, 10, 7, e0131832 (2015) |
| [24] | Hamming, R. W., Error detecting and error correcting codes, Bell System Technical Journal, 29, 2, 147-160 (1950) · Zbl 1402.94084 |
| [25] | Harris, S. E.; Sawhill, B. K.; Wuensche, A.; Kauffman, S., A model of transcriptional regulatory networks based on biases in the observed regulation rules, Complexity, 7, 4, 23-40 (2002) |
| [26] | Heinrich, R.; Schuster, S., The regulation of cellular systems (2012), Springer Science & Business Media |
| [27] | 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 Bioinformatics, 12, 295, 1-11 (2011) |
| [28] | Jaynes, E. T., Probability theory: the logic of science (2003), Cambridge Univ. press · Zbl 0732.60002 |
| [29] | Kaitala, V.; Heino, M., Complex non-unique dynamics in simple ecological interactions, Proceedings of The Royal Society of London. Series B, 263, 1373, 1011-1015 (1996) |
| [30] | Kaitala, V.; Ylikarjula, J.; Heino, M., Non-unique population dynamics: basic patterns, Ecological Modelling, 135, 2, 127-134 (2000) |
| [31] | Karmarkar, N., A new polynomial-time algorithm for linear programming, (ACM symp. theory comput. (1984)), 302-311 · Zbl 0557.90065 |
| [32] | Kauffman, S. A., Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology, 22, 3, 437-467 (1969) |
| [33] | Kauffman, S.; Peterson, C.; Samuelsson, B.; Troein, C., Random Boolean network models and the yeast transcriptional network, Proceedings of the National Academy of Sciences, 100, 25, 14796-14799 (2003) |
| [34] | Kaushik, A. C.; Sahi, S., Boolean network model for GPR142 against Type 2 diabetes and relative dynamic change ratio analysis using systems and biological circuits approach, Systems and Synthetic Biology, 9, 1, 45-54 (2015) |
| [35] | Khalil, H. K., Nonlinear systems (1996), Prentice Hall · Zbl 0626.34052 |
| [36] | Lidl, R.; Niederreiter, H., Introduction to finite fields and their applications (1994), Cambridge Univ. press · Zbl 0820.11072 |
| [37] | Luenberger, D. G., Introduction to linear and nonlinear programming. Vol. 28 (1973), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0241.90052 |
| [38] | Margolin, A. A.; Nemenman, I.; Basso, K.; Wiggins, C.; Stolovitzky, G.; Favera, R. D.; Califano, A., ARACNE: An algorithm for the reconstruction of gene regulatory networks in a mammalian cellular context, BMC Bioinformatics, 7, 1, 1-15 (2006) |
| [39] | 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, International Journal of Control, 90, 3, 519-552 (2017) · Zbl 1388.49042 |
| [42] | Pal, R.; Datta, A.; Bittner, M. L.; Dougherty, E. R., Intervention in context-sensitive probabilistic Boolean networks, BMC Bioinformatics, 21, 7, 1211-1218 (2005) |
| [43] | Perdew, G. H.; Vanden Heuvel, J.; Peters, J. M., Regulation of gene expression (2014), Springer |
| [45] | Rockafellar, R. T.; Wets, R. J.-B., Variational analysis (2009), Springer Science & Business Media |
| [46] | Rosin, D. P., Dynamics of complex autonomous boolean networks (2015), Springer · Zbl 1305.94004 |
| [47] | Shmulevich, I.; Dougherty, E. R.; Zhang, W., Control of stationary behavior in probabilistic Boolean networks by means of structural intervention, Journal on Bioinformatics and Systems, 10, 4, 431-445 (2002) · Zbl 1099.92003 |
| [48] | Shmulevich, I.; Dougherty, E. R.; Zhang, W., Gene perturbation and intervention in probabilistic Boolean networks, BMC Bioinformatics, 18, 10, 1319-1331 (2002) |
| [49] | Subbaraman, A.; Teel, A. R., A converse Lyapunov theorem for strong global recurrence, Automatica, 49, 10, 2963-2974 (2013) · Zbl 1358.93163 |
| [50] | Teel, A. R., Lyapunov conditions certifying stability and recurrence for a class of stochastic hybrid systems, Annual Reviews in Control, 37, 1, 1-24 (2013) |
| [51] | Teel, A. R.; Hespanha, J. P.; Subbaraman, A., A converse Lyapunov theorem and robustness for asymptotic stability in probability, IEEE Transactions on Automatic Control, 59, 9, 2426-2441 (2014) · Zbl 1360.93640 |
| [52] | Thomas, R., Boolean formalization of genetic control circuits, Journal of Theoretical Biology, 42, 3, 563-585 (1973) |
| [53] | Upadhyay, R. K., Multiple attractors and crisis route to chaos in a model food-chain, Chaos, Solitons and Fractals, 16, 5, 737-747 (2003) · Zbl 1033.92038 |
| [54] | Yu, J.; Smith, V. A.; Wang, P. P.; Hartemink, A. J.; Jarvis, E. D., Advances to Bayesian network inference for generating causal networks from observational biological data, BMC Bioinformatics, 20, 18, 3594-3603 (2004) |
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