Abstract
One of the typical properties of biological systems is the law of conservation of mass, that is, the property that the mass must remain constant over time in a closed chemical reaction system. However, it is known that Boolean networks, which are a promising model of biological networks, do not always represent the conservation law. This paper thus addresses a kind of conservation law as a generic property of Boolean networks. In particular, we consider the problem of finding network structures on which, for any Boolean operation on nodes, the number of active nodes, i.e., nodes whose state is one, is constant over time. As a solution to the problem, we focus on the strongly-connected network structures and present a necessary and sufficient condition.
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References
S. Kauffman. Homeostasis and differentiation in random genetic control networks. Nature, 1969, 224(5215): 177–178.
L. A. Amaral, A. Diaz-Guilera, A. A. Moreira, et al. Emergence of complex dynamics in a simple model of signaling networks. Proceedings of the National Academy of Sciences, 2004, 101(44): 15551–15555.
S. E. Harris, B. K. Sawhill, A. Wuensche, et al. A model of transcriptional regulatory networks based on biases in the observed regulation rules. Complexity, 2002, 7(4): 23–40.
T. Akutsu, M. Hayashida, T. Tamura. Algorithms for inference, analysis and control of Boolean networks. Algebraic Biology. Lecture Notes in Computer Science. Berlin: Springer, 2008: 1–25.
D. Cheng, H. Qi, Z. Li. Analysis and Control of Boolean Networks: A Semi-tensor Product Approach, London: Springer, 2011.
T. Akutsu, S. Miyano, S. Kuhara. Identification of genetic networks from a small number of gene expression patterns under the Boolean network model. Proceedings of the Pacific Symposium on Biocomputing, Hawaii: World Scientific, 1999: 17–28.
T. Akutsu, S. Kuhara, O. Maruyama, et al. Identification of genetic networks by strategic gene disruptions and gene overexpressions under a Boolean model. Theoretical Computer Science, 2003, 298(1): 235–251.
B. Drossel, T. Mihaljev, F. Greil. Number and length of attractors in a critical Kauffman model with connectivity one. Physical Review Letters, 2005, 94 (8): https://doi.org/10.1103/PhysRevLett.94.088701.
A. Mochizuki. An analytical study of the number of steady states in gene regulatory networks. Journal of Theoretical Biology, 2005, 236(3): 291–310.
S. Zhang, M. Hayashida, T. Akutsu, et al. Algorithms for finding small attractors in Boolean networks. Journal on Bioinformatics and Systems Biology, 2007: https://doi.org/10.1155/2007/20180.
D. Cheng, H. Qi, Z. Li, et al. Stability and stabilization of Boolean networks. International Journal of Robust and Nonlinear Control, 2011, 21(2): 134–156.
H. Li, Y. Wang. Consistent stabilizability of switched Boolean networks. Neural Networks, 2013, 46: 183–189.
K. Kobayashi, K. Hiraishi. Design of Boolean networks based on prescribed singleton attractors. Proceedings of the 13th European Control Conference, Strasbourg, France: IEEE, 2014: 1504–1509.
H. Li, Y. Wang, Z. Liu. Stability analysis for switched Boolean networks under arbitrary switching signals. IEEE Transactions on Automatic Control, 2014, 59(7): 1978–1982.
S. Azuma, T. Yoshida, T. Sugie. Stability analysis of Boolean networks with partial information. Proceedings of the 21th International Symposium on Mathematical Theory of Networks and Systems, Groningen, The Netherlands, 2014: 1395–1397.
S. Azuma, T. Yoshida, T. Sugie. Structural monostability of activation-inhibition Boolean networks. IEEE Transactions on Control of Network Systems, 2017, 4(2): 179–190.
S. Chen, Y. Wu, L. Wang. A note on monostability and bistability of Boolean networks based on structure graphs. SICE Annual Conference, Kanazawa, Japan: IEEE, 2017: 1153–1158.
S. Azuma, T. Yoshida, T. Sugie. Structural oscillatority analysis of Boolean networks. IEEE Transactions on Control of Network Systems, 2019, 6(2): 464–473.
S. Azuma, T. Yoshida, T. Sugie. Structural bistability analysis of flower-shaped and chain-shaped Boolean networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 2020: DOI https://doi.org/10.1109/TCBB.2019.2917196.
D. Cheng, H. Qi. Controllability and observability of Boolean control networks. Automatica, 2009, 45(7): 1659–1667.
Y. Zhao, D. Cheng, H. Qi. Input-state incidence matrix of Boolean control networks and its applications. Systems & Control Letters, 2010, 59(12): 767–774.
D. Laschov, M. Margaliot. Controllability of Boolean control networks via the Perron-Frobenius theory. Automatica, 2012, 48(6): 1218–1223.
T. Akutsu, M. Hayashida, W. Ching, et al. Control of Boolean networks: hardness results and algorithms for tree structured networks. Journal of Theoretical Biology, 2007, 244(4): 670–679.
D. Laschov, M. Margaliot. A maximum principle for single-input Boolean control networks. IEEE Transactions on Automatic Control, 2011, 56(4): 913–917.
K. Kobayashi, K. Hiraishi. Optimal control of gene regulatory networks with effectiveness of multiple drugs: A Boolean network approach. BioMed Research International, 2013: DOI https://doi.org/10.1155/2013/246761.
K. Kobayashi, K. Hiraishi. ILP/SMT-based method for design of Boolean networks based on singleton attractors. IEEE/ACM Computational Biology and Bioinformatics, 2014, 11(6): 1253–1259.
K. Kobayashi, K. Hiraishi. Structural control of probabilistic Boolean networks and its application to design of real-time pricing systems. IFAC Proceedings Volumes, 2014, 47(3): 2442–2447.
N. Bof, E. Fornasini, M. E. Valcher. Output feedback stabilization of Boolean control networks. Automatica, 2015, 57: 21–28.
J. Aracena. Maximum number of fixed points in regulatory Boolean networks. Bulletin of Mathematical Biology, 2008, 70(5): 1398–1409.
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This work was supported by Grant-in-Aid for Scientific Research (B) #17H03280 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
Shun-ichi AZUMA received the B.Eng. degree in Electrical Engineering from Hiroshima University, Higashi Hiroshima, Japan, in 1999, and the M.Eng. and Ph.D. degrees in Control Engineering from Tokyo Institute of Technology, Tokyo, Japan, in 2001 and 2004, respectively. He is currently a Professor in the Department of Mechanical Systems Engineering, Graduate School of Engineering, Nagoya University, Nagoya, Japan. Prior to joining Nagoya University, he was an Assistant Professor and an Associate Professor in the Department of Systems Science, Graduate School of Informatics, Kyoto University, Uji, Japan, from 2005 to 2011 and from 2011 to 2017, respectively. He serves as Associate Editors of IFAC Journal Automatica from 2014, IEEE Transactions on Automatic Control from 2019, and so on. His research interests include analysis and control of hybrid systems and applications to systems biology.
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Azuma, Si. Robust network structures for conserving total activity in Boolean networks. Control Theory Technol. 18, 143–147 (2020). https://doi.org/10.1007/s11768-020-9202-6
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DOI: https://doi.org/10.1007/s11768-020-9202-6