Controllability of reaction systems
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Controllability of reaction systems Sergiu Ivanov1 · Ion Petre2,3 Received: 17 July 2020 / Accepted: 6 October 2020 / Published online: 19 November 2020 © Springer Nature Singapore Pte Ltd. 2020
Abstract Controlling a dynamical system is the ability of changing its configuration arbitrarily through a suitable choice of inputs. It is a very well-studied concept in control theory, with wide-ranging applications in medicine, biology, social sciences and engineering. We introduce in this article the concept of controllability of reaction systems as the ability of transitioning between any two states through a suitable choice of context sequences. We show that the problem is PSPACE-hard. We also introduce a model of oncogenic signalling based on reaction systems and use it to illustrate the intricacies of the controllability of reaction systems. This study opens up a new line of research on the dynamic properties of reaction systems and it introduces a new, intricate biomedical model based on reaction systems. Keywords Reaction systems · Controllability · Oncogenic signalling · Computational complexity
1 Introduction Reaction systems are a biologically inspired model of computing originally introduced in [15]. They capture two fundamental interactions typically present between biochemical entities—activation and inhibition. Reaction systems are dynamical systems: reactions transform a set of reactants into a set of products provided that none of its inhibitors are present, which are then transformed further into other products, etc. The reactions in reaction systems are governed by two fundamental principles: the threshold principle and the nonpermanency principle. The threshold principle stipulates that when a resource is available it is available in unlimited amounts. This defines reaction systems as a qualitative modelling framework: their states are plain sets of species, bearing no quantitative information about amounts or * Sergiu Ivanov sergiu.ivanov@univ‑evry.fr Ion Petre [email protected] 1
IBISC, Université Évry, Université Paris-Saclay, Évry‑Courcouronnes, France
2
Department of Mathematics and Statistics, University of Turku, Turku, Finland
3
National Institute for Research and Development in Biological Sciences, Bucharest, Romania
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concentrations. This also means that concurrent competition for resources is not implicitly included in the underlying semantics, but rather must be elucidated via some dedicated reactions and species. The non-permanency principle states that if a resource is not explicitly sustained/produced by reactions, it will disappear. The next state of a reaction system only consists of the species explicitly produced by the reactions enabled in the previous state. Reaction systems are open systems: there is a notion of context that adds to the current state in each step of its dynamic process. The next state is produced by the reactions applied to the previous state plus the species added by the context. Since their introduction in 200
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