Mathematics

Editorial

Jump to: Research

3 pages, 164 KiB

Open AccessEditorial

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 1225

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

Open AccessArticle

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 - 07 Apr 2021

Cited by 6 | Viewed by 2740

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

2^{N}

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)

► Show Figures

Figure 1

19 pages, 1712 KiB

Open AccessArticle

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 4 | Viewed by 4668

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)

► Show Figures

Figure 1

16 pages, 645 KiB

Open AccessArticle

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 - 06 Mar 2021

Cited by 6 | Viewed by 2402

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)

► Show Figures

Figure 1

14 pages, 309 KiB

Open AccessArticle

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 5 | Viewed by 2834

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)

► Show Figures

Figure 1

9 pages, 347 KiB

Open AccessArticle

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 1571

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)

► Show Figures

Figure 1

14 pages, 274 KiB

Open AccessArticle

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 - 03 Jul 2020

Cited by 5 | Viewed by 2009

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)

► Show Figures

Figure 1

13 pages, 292 KiB

Open AccessArticle

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 1945

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)

Journal Menu

Journal Browser

Boolean Networks Models in Science and Engineering

Share This Special Issue

Special Issue Editors

Special Issue Information

Keywords

Published Papers (8 papers)

Editorial

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI