scholarly journals Matrix Analysis of Synchronous Boolean Networks

2021 ◽  
Vol 6 (2) ◽  
pp. 598-610
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Adnan Ahmad Alsogati
Keyword(s):  
Biological Networks ◽  
Transition Matrix ◽  
Matrix Multiplication ◽  
Transient Behavior ◽  
Galois Field ◽  
Boolean Networks ◽  
Matrix Analysis ◽  
Cyclic Behavior ◽  
State Matrix ◽  
Underlying Basis

The synchronous Boolean network (SBN) is a simple and powerful model for describing, analyzing, and simulating cellular biological networks. This paper seeks a complete understanding of the dynamics of such a model by employing a matrix method that relies on relating the network transition matrix to its function matrix via a self-inverse state matrix. A recursive ordering of the underlying basis vector leads to a simple recursive expression of this state matrix. Hence, the transition matrix is computed via multiplication of binary matrices over the simplest finite (Galois) field, namely the binary field GF(2), i.e., conventional matrix multiplication involving modulo-2 addition, or XOR addition. We demonstrate the conceptual simplicity and practical utility of our approach via an illustrative example, in which the transition matrix is readily obtained, and subsequently utilized (via its powers, characteristic equation, minimal equation, 1-eigenvectors, and 0-eigenvectors) to correctly predict both the transient behavior and the cyclic behavior of the network. Our matrix approach for computing the transition matrix is superior to the approach of scalar equations, which demands cumbersome manipulations and might fail to predict the exact network behavior. Our approach produces result that exactly replicate those obtained by methods employing the semi-tensor product (STP) of matrices, but achieves that without sophisticated ambiguity or unwarranted redundancy.

scholarly journals The Eigenvectors of the Transition Matrix as Predictors of the Dynamics of a Synchronous Boolean Network

2020 ◽  
pp. 80-99
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Adnan Ahmad Alsogati
Keyword(s):  
Characteristic Equation ◽  
Transition Matrix ◽  
Boolean Network ◽  
Partial Information ◽  
Cyclic Behavior ◽  
Main Concern ◽  
State Matrix ◽  
Matrix Methods ◽  
Network Transition ◽  
Function Matrix

The synchronous Boolean network model is a simple and powerful tool in describing, analyzing and simulating cellular biological networks. This paper seeks a complete understanding of the dynamics of such a model by utilizing conventional matrix methods, rather than scalar methods, or matrix methods employing the non-conventional semi-tensor products (STP) of matrices. The paper starts by relating the network transition matrix to its function matrix via a self-inverse (involutary) state matrix, which has a simple recursive expression, provided a recursive ordering is employed for the underlying basis vector. Once the network transition matrix is obtained, it can be used to generate a wealth of information including its powers, characteristic equation, minimal equation, 1-eigenvectors, and 0-eigenvectors. These might be used to correctly predict both the transient behavior and (more importantly) the cyclic behavior of the network. In a short-cut partial variant of the proposed approach, the step of computing the transition matrix might be by-passed. The reason for this is that the transition matrix and the function matrix are similar matrices that share the same characteristic equation and hence the function matrix might suffice when only the partial information supplied by the characteristic equation is all that is needed. We demonstrate the conceptual simplicity and practical utility of our approach via two illustrative examples. The first example illustrates the computation of 1-eigenvectors (that can be used to identify loops or attractors), while the second example deals with the evaluation of 0-eigenvectors (that can be used to explore transient chains). Since attractors are the main concern in the underlying model, then analysis of the Boolean network might be confined to the determination of 1-eigenvectors only.

Algorithmic and Stochastic Representations of Gene Regulatory Networks and Protein-Protein Interactions

2019 ◽  
Vol 19 (6) ◽  
pp. 413-425 ◽  
Author(s):  
Athanasios Alexiou ◽  
Stylianos Chatzichronis ◽  
Asma Perveen ◽  
Abdul Hafeez ◽  
Ghulam Md. Ashraf
Keyword(s):  
Protein Interactions ◽  
Biological Networks ◽  
Regulatory Networks ◽  
Review Paper ◽  
Boolean Networks ◽  
Diagnostic Tools ◽  
Complex Nature ◽  
Cellular Interactions ◽  
Protein Protein Interactions ◽  
Deterministic Models

Background:Latest studies reveal the importance of Protein-Protein interactions on physiologic functions and biological structures. Several stochastic and algorithmic methods have been published until now, for the modeling of the complex nature of the biological systems.Objective:Biological Networks computational modeling is still a challenging task. The formulation of the complex cellular interactions is a research field of great interest. In this review paper, several computational methods for the modeling of GRN and PPI are presented analytically.Methods:Several well-known GRN and PPI models are presented and discussed in this review study such as: Graphs representation, Boolean Networks, Generalized Logical Networks, Bayesian Networks, Relevance Networks, Graphical Gaussian models, Weight Matrices, Reverse Engineering Approach, Evolutionary Algorithms, Forward Modeling Approach, Deterministic models, Static models, Hybrid models, Stochastic models, Petri Nets, BioAmbients calculus and Differential Equations.Results:GRN and PPI methods have been already applied in various clinical processes with potential positive results, establishing promising diagnostic tools.Conclusion:In literature many stochastic algorithms are focused in the simulation, analysis and visualization of the various biological networks and their dynamics interactions, which are referred and described in depth in this review paper.

Transition Matrix Analysis of System Outages

2007 ◽  
Vol 19 (19) ◽  
pp. 1529-1531 ◽  
Author(s):  
D. Yevick ◽  
M. Reimer
Keyword(s):  
Transition Matrix ◽  
Matrix Analysis

Transition matrix analysis of earthquake magnitude sequences

2005 ◽  
Vol 24 (1) ◽  
pp. 33-43 ◽  
Author(s):  
M LOVALLO ◽  
V LAPENNA ◽  
L TELESCA
Keyword(s):  
Transition Matrix ◽  
Earthquake Magnitude ◽  
Matrix Analysis

scholarly journals Controlling large Boolean networks with single-step perturbations

2019 ◽  
Vol 35 (14) ◽  
pp. i558-i567 ◽  
Author(s):  
Alexis Baudin ◽  
Soumya Paul ◽  
Cui Su ◽  
Jun Pang
Keyword(s):  
Biological Networks ◽  
Boolean Network ◽  
Real Life ◽  
Extended Period ◽  
Boolean Networks ◽  
Single Step ◽  
Divide And Conquer ◽  
Supplementary Information ◽  
Initial State ◽  
Minimal Subset

Abstract Motivation The control of Boolean networks has traditionally focussed on strategies where the perturbations are applied to the nodes of the network for an extended period of time. In this work, we study if and how a Boolean network can be controlled by perturbing a minimal set of nodes for a single-step and letting the system evolve afterwards according to its original dynamics. More precisely, given a Boolean network (BN), we compute a minimal subset Cmin of the nodes such that BN can be driven from any initial state in an attractor to another ‘desired’ attractor by perturbing some or all of the nodes of Cmin for a single-step. Such kind of control is attractive for biological systems because they are less time consuming than the traditional strategies for control while also being financially more viable. However, due to the phenomenon of state-space explosion, computing such a minimal subset is computationally inefficient and an approach that deals with the entire network in one-go, does not scale well for large networks. Results We develop a ‘divide-and-conquer’ approach by decomposing the network into smaller partitions, computing the minimal control on the projection of the attractors to these partitions and then composing the results to obtain Cmin for the whole network. We implement our method and test it on various real-life biological networks to demonstrate its applicability and efficiency. Supplementary information Supplementary data are available at Bioinformatics online.

Simplified transition matrix analysis of the hinge model

2009 ◽  
Vol 26 (3) ◽  
pp. 710 ◽  
Author(s):  
David Yevick ◽  
Michael Reimer ◽  
Maurice O'Sullivan
Keyword(s):  
Transition Matrix ◽  
Matrix Analysis ◽  
Hinge Model
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