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Mathematics
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Boolean Networks Models in Science and Engineering
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Boolean Networks Models in Science and Engineering

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Network Science".

Deadline for manuscript submissions: closed (16 April 2021) | Viewed by 24258

Special Issue Editors

Prof. Dr. Jose C. Valverde

E-Mail Website
Guest Editor
Department of Mathematics, University of Castilla‐La Mancha, Avda. de Espana, s/n, 02071 Albacete, Spain
Interests: dynamical systems; discrete mathematics; biomathematics; boolean networks
Prof. Dr. Juan A. Aledo

E-Mail Website
Guest Editor
Department of Mathematics, University of Castilla‐La Mancha, Avda. de Espana, s/n, 02071 Albacete, Spain
Interests: boolean networks; discrete mathematics; computational mathematics
Prof. Dr. Silvia Martínez

E-Mail Website
Guest Editor
Department of Mathematics, University of Castilla‐La Mancha, Plza. de la Universidad, 3, 02071 Albacete, Spain
Interests: discrete mathematics; boolean networks; mathematics education

Special Issue Information

Dear Colleagues,

As a generalization of other notions like cellular automata or Kauffman networks appeared in the last quarter of the twentieth century, the notion of Boolean networks has undergone a special development in the last decades. This is mainly due to its applications in Science and Engineering.

In this sense, several research groups of mathematicians are working and obtaining relevant results in this area that can be applied in other fields. We invite them to submit their latest research to our Special Issue, “Boolean Networks Models in Science and Engineering”, in the reputed journal Mathematics.

We hope to collect novel and interesting papers, both theoretical and practical, in this field. Submissions are welcome presenting new theoretical results, new algorithmic as well as new models or applications in Science and Engineering.

Potential topics include, but are not limited to:

  • Dynamics of Boolean network models
  • Algorithms, methods, and software for the study of Boolean network models
  • Boolean network models applied in science and engineering

Prof. Dr. Jose C. Valverde
Prof. Dr. Juan A. Aledo
Prof. Dr. Silvia Martínez
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Boolean network models
  • Combinatorial dynamics
  • Algorithms, methods and software
  • Application of Boolean algebra
  • Boolean functions

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Published Papers (8 papers)

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Editorial

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3 pages, 164 KiB  
Editorial
Boolean Networks Models in Science and Engineering
by Juan A. Aledo, Silvia Martinez and Jose C. Valverde
Mathematics 2021, 9(8), 867; https://doi.org/10.3390/math9080867 - 15 Apr 2021
Viewed by 1463
Abstract
As a generalization of other notions like cellular automata or Kauffman networks appeared in the last quarter of the twentieth century, the notion of Boolean networks has undergone a special development in recent decades [...] Full article
(This article belongs to the Special Issue Boolean Networks Models in Science and Engineering)

Research

Jump to: Editorial

18 pages, 1084 KiB  
Article
Random Networks with Quantum Boolean Functions
by Mario Franco, Octavio Zapata, David A. Rosenblueth and Carlos Gershenson
Mathematics 2021, 9(8), 792; https://doi.org/10.3390/math9080792 - 7 Apr 2021
Cited by 6 | Viewed by 3169
Abstract
We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean network where the functions associated with the nodes [...] Read more.
We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean network where the functions associated with the nodes are quantum Boolean functions. We study some properties of this novel model and, using a quantum simulator, we study how the dynamics change in function of connectivity of the network and the set of operators we allow. For some configurations, this model resembles the behavior of reversible Boolean networks, while for other configurations a more complex dynamic can emerge. For example, cycles larger than 2N were observed. Additionally, using a scheme akin to one used previously with random Boolean networks, we computed the average entropy and complexity of the networks. As opposed to classic random Boolean networks, where “complex” dynamics are restricted mainly to a connectivity close to a phase transition, quantum Boolean networks can exhibit stable, complex, and unstable dynamics independently of their connectivity. Full article
(This article belongs to the Special Issue Boolean Networks Models in Science and Engineering)
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19 pages, 1712 KiB  
Article
Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements
by Marco Montalva-Medel, Thomas Ledger, Gonzalo A. Ruz and Eric Goles
Mathematics 2021, 9(6), 600; https://doi.org/10.3390/math9060600 - 11 Mar 2021
Cited by 5 | Viewed by 5445
Abstract
In Veliz-Cuba and Stigler 2011, Boolean models were proposed for the lac operon in Escherichia coli capable of reproducing the operon being OFF, ON and bistable for three (low, medium and high) and two (low and high) parameters, representing the concentration ranges of [...] Read more.
In Veliz-Cuba and Stigler 2011, Boolean models were proposed for the lac operon in Escherichia coli capable of reproducing the operon being OFF, ON and bistable for three (low, medium and high) and two (low and high) parameters, representing the concentration ranges of lactose and glucose, respectively. Of these 6 possible combinations of parameters, 5 produce results that match with the biological experiments of Ozbudak et al., 2004. In the remaining one, the models predict the operon being OFF while biological experiments show a bistable behavior. In this paper, we first explore the robustness of two such models in the sense of how much its attractors change against any deterministic update schedule. We prove mathematically that, in cases where there is no bistability, all the dynamics in both models lack limit cycles while, when bistability appears, one model presents 30% of its dynamics with limit cycles while the other only 23%. Secondly, we propose two alternative improvements consisting of biologically supported modifications; one in which both models match with Ozbudak et al., 2004 in all 6 combinations of parameters and, the other one, where we increase the number of parameters to 9, matching in all these cases with the biological experiments of Ozbudak et al., 2004. Full article
(This article belongs to the Special Issue Boolean Networks Models in Science and Engineering)
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16 pages, 645 KiB  
Article
Parallel One-Step Control of Parametrised Boolean Networks
by Luboš Brim, Samuel Pastva, David Šafránek and Eva Šmijáková
Mathematics 2021, 9(5), 560; https://doi.org/10.3390/math9050560 - 6 Mar 2021
Cited by 6 | Viewed by 2932
Abstract
Boolean network (BN) is a simple model widely used to study complex dynamic behaviour of biological systems. Nonetheless, it might be difficult to gather enough data to precisely capture the behavior of a biological system into a set of Boolean functions. These issues [...] Read more.
Boolean network (BN) is a simple model widely used to study complex dynamic behaviour of biological systems. Nonetheless, it might be difficult to gather enough data to precisely capture the behavior of a biological system into a set of Boolean functions. These issues can be dealt with to some extent using parametrised Boolean networks (ParBNs), as this model allows leaving some update functions unspecified. In our work, we attack the control problem for ParBNs with asynchronous semantics. While there is an extensive work on controlling BNs without parameters, the problem of control for ParBNs has not been in fact addressed yet. The goal of control is to ensure the stabilisation of a system in a given state using as few interventions as possible. There are many ways to control BN dynamics. Here, we consider the one-step approach in which the system is instantaneously perturbed out of its actual state. A naïve approach to handle control of ParBNs is using parameter scan and solve the control problem for each parameter valuation separately using known techniques for non-parametrised BNs. This approach is however highly inefficient as the parameter space of ParBNs grows doubly exponentially in the worst case. We propose a novel semi-symbolic algorithm for the one-step control problem of ParBNs, that builds on symbolic data structures to avoid scanning individual parameters. We evaluate the performance of our approach on real biological models. Full article
(This article belongs to the Special Issue Boolean Networks Models in Science and Engineering)
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14 pages, 309 KiB  
Article
Coexistence of Periods in Parallel and Sequential Boolean Graph Dynamical Systems over Directed Graphs
by Juan A. Aledo, Luis G. Diaz, Silvia Martinez and Jose C. Valverde
Mathematics 2020, 8(10), 1812; https://doi.org/10.3390/math8101812 - 16 Oct 2020
Cited by 8 | Viewed by 3334
Abstract
In this work, we solve the problem of the coexistence of periodic orbits in homogeneous Boolean graph dynamical systems that are induced by a maxterm or a minterm (Boolean) function, with a direct underlying dependency graph. Specifically, we show that periodic orbits of [...] Read more.
In this work, we solve the problem of the coexistence of periodic orbits in homogeneous Boolean graph dynamical systems that are induced by a maxterm or a minterm (Boolean) function, with a direct underlying dependency graph. Specifically, we show that periodic orbits of any period can coexist in both kinds of update schedules, parallel and sequential. This result contrasts with the properties of their counterparts over undirected graphs with the same evolution operators, where fixed points cannot coexist with periodic orbits of other different periods. These results complete the study of the periodic structure of homogeneous Boolean graph dynamical systems on maxterm and minterm functions. Full article
(This article belongs to the Special Issue Boolean Networks Models in Science and Engineering)
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9 pages, 347 KiB  
Article
An Algorithm for Counting the Fixed Point Orbits of an AND-OR Dynamical System with Symmetric Positive Dependency Graph
by Mauro Mezzini and Fernando L. Pelayo
Mathematics 2020, 8(9), 1611; https://doi.org/10.3390/math8091611 - 18 Sep 2020
Cited by 3 | Viewed by 1841
Abstract
In this paper we present an algorithm which counts the number of fixed point orbits of an AND-OR dynamical system. We further extend the algorithm in order to list all its fixed point orbits (FPOs) in polynomial time on the number of FPOs [...] Read more.
In this paper we present an algorithm which counts the number of fixed point orbits of an AND-OR dynamical system. We further extend the algorithm in order to list all its fixed point orbits (FPOs) in polynomial time on the number of FPOs of the system. Full article
(This article belongs to the Special Issue Boolean Networks Models in Science and Engineering)
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14 pages, 274 KiB  
Article
On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions
by Juan A. Aledo, Ali Barzanouni, Ghazaleh Malekbala, Leila Sharifan and Jose C. Valverde
Mathematics 2020, 8(7), 1088; https://doi.org/10.3390/math8071088 - 3 Jul 2020
Cited by 7 | Viewed by 2319
Abstract
In this paper, based on previous results on AND-OR parallel dynamical systems over directed graphs, we give a more general pattern of local functions that also provides fixed point systems. Moreover, by considering independent sets, this pattern is also generalized to get systems [...] Read more.
In this paper, based on previous results on AND-OR parallel dynamical systems over directed graphs, we give a more general pattern of local functions that also provides fixed point systems. Moreover, by considering independent sets, this pattern is also generalized to get systems in which periodic orbits are only fixed points or 2-periodic orbits. The results obtained are also applicable to homogeneous systems. On the other hand, we study the periodic structure of parallel dynamical systems given by the composition of two parallel systems, which are conjugate under an invertible map in which the inverse is equal to the original map. This allows us to prove that the composition of any parallel system on a maxterm (or minterm) Boolean function and its conjugate one by means of the complement map is a fixed point system, when the associated graph is undirected. However, when the associated graph is directed, we demonstrate that the corresponding composition may have points of any period, even if we restrict ourselves to the simplest maxterm OR and the simplest minterm AND. In spite of this general situation, we prove that, when the associated digraph is acyclic, the composition of OR and AND is a fixed point system. Full article
(This article belongs to the Special Issue Boolean Networks Models in Science and Engineering)
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13 pages, 292 KiB  
Article
On the Lyapunov Exponent of Monotone Boolean Networks †
by Ilya Shmulevich
Mathematics 2020, 8(6), 1035; https://doi.org/10.3390/math8061035 - 24 Jun 2020
Cited by 2 | Viewed by 2282
Abstract
Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consider networks comprised of monotone Boolean functions and derive [...] Read more.
Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consider networks comprised of monotone Boolean functions and derive asymptotic formulas for the Lyapunov exponent of almost all monotone Boolean networks. The formulas are different depending on whether the number of variables of the constituent Boolean functions, or equivalently, the connectivity of the Boolean network, is even or odd. Full article
(This article belongs to the Special Issue Boolean Networks Models in Science and Engineering)
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