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Control Theory and Technology http://link.springer.com/journal/11768

Robust network structures for conserving total activity in Boolean networks Shun-ichi AZUMA Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan Received 24 December 2019; revised 18 March 2020; accepted 19 March 2020

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. Keywords: Boolean network, robust network structure, structural analysis, conservation law DOI https://doi.org/10.1007/s11768-020-9202-6

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Introduction

Boolean networks are a mathematical model of network systems with binary-state nodes [1]. The models are frequently used for analyzing the qualitative properties of biological networks such as gene regulatory networks [2, 3]. When we employ a Boolean network to express a biological network, it is important to preserve the essential properties of the biological system in the corresponding Boolean network. For example, one of the typical prop-

erties 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, there is a Boolean network such that the conservation law does not hold. In this paper, we address a kind of conservation law as a generic property of Boolean networks. More precisely, we consider the problem of finding network structures in 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

E-mail: [email protected]. Tel.: +81-52-789-2745. 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. © 2020 South China University of Technology, Academy of Mathematics and Systems Science, CAS and Springer-Verlag GmbH Germany, part of Springer Nature

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S. Azuma / Control Theory Tech, Vol. 18, No. 2, pp. 143–147, May 2020

focus on the strongly connected network structures and present a necessary and sufficient condition. Finally, we summarize the relation between our results and the existing results. So far, various problems have been addressed for Boolean networks [4, 5], such as identification [6, 7], attractor analysis and stability [8–19], c