Fig 1 - available via license: Creative Commons Attribution 4.0 International
Content may be subject to copyright.
An intuitive idea of partitiong a Boolean network into two blocks and composing steady states found independently. A: A Boolean network G can be partitioned into two blocks P and Q which are connected by cp. Then, we construct two subnetworks P and Q′ where Q′ = Q∪{cp}, and cp is used to combine the local steady states found in each subnetwork. B: When there also exists a set of edges from cq of Q, we construct two subnetworks P′ and Q′ where P′ = P∪{cq} and Q′ = Q∪{cp}. Both cp and cq are used to combine the steady states of subnetworks.
Source publication
Boolean networks have been widely used to model biological processes lacking detailed kinetic information. Despite their simplicity, Boolean network dynamics can still capture some important features of biological systems such as stable cell phenotypes represented by steady states. For small models, steady states can be determined through exhaustiv...
Similar publications
![Fig. 6: Wiring of the MAPK logical model of [20]. The diagram contains...](profile/Andrzej-Mizera/publication/316428835/figure/fig1/AS:486506520420352@1493003601940/Wiring-of-the-MAPK-logical-model-of-20-The-diagram-contains-three-types-of-nodes_Q320.jpg)
Boolean networks is a well-established formalism for modelling biological systems. A vital challenge for analysing a Boolean network is to identify all the attractors. This becomes more challenging for large asynchronous Boolean networks, due to the asynchronous updating scheme. Existing methods are prohibited due to the well-known state-space expl...
The controllability analysis of Boolean networks (BNs), as models of biomolecular regulatory networks, has drawn the attention of researchers in recent years. In this paper, we aim at governing the steady-state behavior of BNs using an intervention method which can easily be applied to most real system, which can be modeled as BNs, particularly to...
Background
Advancements in gene expression technology allow acquiring cheap and abundant data for analyzing cell behavior. However, these technologies produce noisy, and often correlated, measurements on the transcriptional states of genes. The Boolean network model has been shown to be effective in capturing the complex dynamics of gene regulatory...
In systems biology, models of cellular regulatory processes such as gene regulatory networks or signalling pathways are crucial to understanding the behaviour of living cells. Available biological data are however often insufficient for full model specification. In this paper, we focus on partially specified models where the missing information is...
Boolean networks are a widely used qualitative modelling approach which allows the abstract description of a biological system. One issue with the application of Boolean networks is the state space
explosion problem which limits the applicability of the approach to large realistic systems. In this paper we investigate developing a compositional fra...
Citations
... In large gene regulatory networks (with more than 100 vertices, such as the one presented in this work), it is possible to adopt different approaches to define attractors to overcome the exponential growth of the state space size according to the increase in the number of network size. For example, one approach was to partition the network into small subnets, finding the attractors corresponding to these partitions and then combining them to build a stable state relative to the entire network (Hong et al., 2015). Another approach is to configure network input vertices with initial Boolean values representing their gene expression level (Cho et al., 2016). ...
... These initial conditions were the starting points of a trajectory, driven by the topology and the logic functions characterizing the network, whose evolution ended when reaching an attractor. This strategy is very time efficient, avoiding the nonpolynomial complexity of other strategies to find network attractors (Hong et al., 2015). The result obtained can be considered satisfactory given the percentage of attractors obtained. ...
Cancer is a genomic disease involving various intertwined pathways with complex cross-communication links. Conceptually, this complex interconnected system forms a network, which allows one to model the dynamic behavior of the elements that characterize it to describe the entire system's development in its various evolutionary stages of carcinogenesis. Knowing the activation or inhibition status of the genes that make up the network during its temporal evolution is necessary for the rational intervention on the critical factors for controlling the system's dynamic evolution. In this report, we proposed a methodology for building data-driven boolean networks that model breast cancer tumors. We defined the network components and topology based on gene expression data from RNA-seq of breast cancer cell lines. We used a Boolean logic formalism to describe the network dynamics. The combination of single-cell RNA-seq and interactome data enabled us to study the dynamics of malignant subnetworks of up-regulated genes. First, we used the same Boolean function construction scheme for each network node, based on canalyzing functions. Using single-cell breast cancer datasets from The Cancer Genome Atlas, we applied a binarization algorithm. The binarized version of scRNA-seq data allowed identifying attractors specific to patients and critical genes related to each breast cancer subtype. The model proposed in this report may serve as a basis for a methodology to detect critical genes involved in malignant attractor stability, whose inhibition could have potential applications in cancer theranostics.
... In large gene regulatory networks (with more than 100 vertices, such as the one presented in this work), it is possible to adopt different approaches to define attractors to overcome the exponential growth of the state space size according to the increase in the number of network size. For example, one approach was to partition the network into small subnets, finding the attractors corresponding to these partitions and then combining them to build a stable state relative to the entire network (Hong et al., 2015). Another approach is to configure network input vertices with initial Boolean values representing their gene expression level (Cho et al., 2016). ...
... These initial conditions were the starting points of a trajectory, driven by the topology and the logic functions characterizing the network, whose evolution ended when reaching an attractor. This strategy is very time efficient, avoiding the nonpolynomial complexity of other strategies to find network attractors (Hong et al., 2015). The result obtained can be considered satisfactory given the percentage of attractors obtained. ...
Cancer is a genomic disease involving various intertwined pathways with complex cross-communication links. Conceptually, this complex interconnected system forms a network, which allows one to model the dynamic behavior of the elements that characterize it to describe the entire system's development in its various evolutionary stages of carcinogenesis. Knowing the activation or inhibition status of the genes that make up the network during its temporal evolution is necessary for the rational intervention on the critical factors for controlling the system's dynamic evolution. In this report, we proposed a methodology for building data-driven boolean networks that model breast cancer tumors. We defined the network components and topology based on gene expression data from RNA-seq of breast cancer cell lines. We used a Boolean logic formalism to describe the network dynamics. The combination of single-cell RNA-seq and interactome data enabled us to study the dynamics of malignant subnetworks of up-regulated genes. First, we used the same Boolean function construction scheme for each network node, based on canalyzing functions. Using single-cell breast cancer datasets from The Cancer Genome Atlas, we applied a binarization algorithm. The binarized version of scRNA-seq data allowed identifying attractors specific to patients and critical genes related to each breast cancer subtype. The model proposed in this report may serve as a basis for a methodology to detect critical genes involved in malignant attractor stability, whose inhibition could have potential applications in cancer theranostics.
... The study of large networks, in particular the computation of its steady states and attractors, remains a hard problem. Several algorithms have been proposed to determine the singletons in a Boolean network, using approaches such as constrained programming (Devloo et al. 2003), SAT formulas for special classes of functions (Akutsu et al. 2011), or partitioning the system into smaller networks (Hong et al. 2015). Using computational algebra, Veliz-Cuba et al. (2014) construct a very cost-efficient algorithm to compute all singletons in a Boolean network. ...
... The state transition graph of a large network is a computationaly expensive object, especially for asynchronous networks. Likewise, computing the attractors of large Boolean networks is a NP-hard problem (Akutsu et al. 2011;Hong et al. 2015). Here, we proposed a new method that computes the attractors of large Boolean networks which are constructed by assembling together sets of smaller, already known, modules, as is often the case in physics or biology, for instance. ...
Boolean models of physical or biological systems describe the global dynamics of the system and their attractors typically represent asymptotic behaviors. In the case of large networks composed of several modules, it may be difficult to identify all the attractors. To explore Boolean dynamics from a novel viewpoint, we will analyse the dynamics emerging from the composition of two known Boolean modules. The state transition graphs and attractors for each of the modules can be combined to construct a new asymptotic graph which will (1) provide a reliable method for attractor computation with partial information; (2) illustrate the differences in dynamical behavior induced by the updating strategy (asynchronous, synchronous, or mixed); and (3) show the inherited organization/structure of the original network’s state transition graph.
... In Tournier and Chaves (2013) a compositional approach for asynchronous Boolean networks is considered based on using a state graph approach. Various work has also considered extending the applicability of SAT based techniques by partitioning large models and then aggregating the results (see for example Guo et al., 2014;Hong et al., 2015). An algorithmic approach for efficient attractor identification in large-scale Boolean networks based on analysing subnetworks in a model is considered in Zhao et al. (2013), Cheng et al. (2012). ...
Boolean networks are a widely used qualitative approach for modelling and analysing biological systems. However, their application is restricted by the well-known state space explosion problem which means that modelling large-scale, realistic biological systems is challenging. In this paper we set out to facilitate the con- struction and analysis of large scale biological models by developing a formal framework for the composition of Boolean networks. The compositional approach we present is based on merging entities between Boolean net- works using a binary Boolean operator and we formalise the preservation of behaviour under composition using a notion of compatibility. We investigate characterising compatibility in terms of the composed models by developing a trace alignment property. In particular, we use a formalisation of the interference that can occur in a composed model to define an extended trace alignment property that we show completely characterises compatibility.
... Solving Boolean systems of equations is a core computational problem in fields such as verification of computing hardware, deduction, verification and synthesis of logic, constraint satisfaction problems, Multivariable Quadratic (MQ) equations over binary field, optimization under logic constraints, cryptanalysis, problems in graph theory etc. which arise in Computer engineering, Cryptology, Optimization and Integer programming [8,15,14,10,7]. Since recent times a most glaring new application of solving Boolean systems arises in Biological (Genetic Regulatory) Networks [21,22,24,23]. Two basic versions of such problems are the decision problem in which it is required to determine the consistency (or solvability) of the system and, presenting all solutions of the problem when the system is consistent. ...
This paper develops a parallel computational solver for computing all satifying assignments of a Boolean system of equations defined by Boolean functions of several variables. While there are well known solvers for satisfiability of Boolean formulas in CNF form, these are designed primarily for deciding satisfiability of the formula and do not address the problem of finding all satisfying solutions. Moreover development of parallel solvers for satisfiability problems is still an unfinished problem of Computer Science. The solver proposed in this paper is aimed at representing all solutions of Boolean formulas even without the CNF form with a parallel algorithm. Algorithm proposed is applied to Boolean functions in algebraic normal form (ANF). The algorithm is based on the idea to represent the satisfying assignments in terms of a complete set of impli-cants of the Boolean functions appearing as factors of a Boolean formula. The algorithm is effective mainly in the case when the factors of the formula are sparse (i.e. have a small fraction of the total number of variables). This allows small computation of a complete set of implicants of individual factors one at a time and reduce the formula at each step. An algorithm is also proposed for finding a complete set of orthogonal implicants of functions in ANF. An advantages of this algorithm is that all solutions can be represented compactly in terms of implicants. Finally due to small and distributed computation at every step as well as computation in terms of independent threads, the solver proposed in this paper is expected to be useful for developing heuristics for a well scalable parallel solver for large size problems of Boolean satisfiability over large number of processors.
... Los estados transitorios consecutivos ( ), ( + 1), … ( + ) son denominados como atractores con periodo si ( ) = ( + ). Un atactor, entonces, queda definido como un estado de punto fijo (o estado estable) si = 1, mientras que cualquier atractor con periodo > 1 queda definido como un atractor cíclico (Changki Hong, 2015). ...
The thesis tries to determine that there is no direct relation between child eork and school dropout. For these assumption, diferent categories of child work are presented and then confirmed through a boolean network using data from the Child Labor Module in Mexico.
... To tackle such a problem, there have been several attempts to reduce the original Boolean network model by eliminating some nodes or logically simplifying Boolean functions [17][18][19][20][21][22][23][24][25][26][27], or focusing only on point attractors [28,29]. Another attempt was partitioning the original large Boolean network into smaller blocks and reconstructing the original attractors by integrating the local attractors of partitioned blocks [30,31]. ...
Background
Boolean network modeling has been widely used to model large-scale biomolecular regulatory networks as it can describe the essential dynamical characteristics of complicated networks in a relatively simple way. When we analyze such Boolean network models, we often need to find out attractor states to investigate the converging state features that represent particular cell phenotypes. This is, however, very difficult (often impossible) for a large network due to computational complexity.
Results
There have been some attempts to resolve this problem by partitioning the original network into smaller subnetworks and reconstructing the attractor states by integrating the local attractors obtained from each subnetwork. But, in many cases, the partitioned subnetworks are still too large and such an approach is no longer useful. So, we have investigated the fundamental reason underlying this problem and proposed a novel efficient way of hierarchically partitioning a given large network into smaller subnetworks by focusing on some attractors corresponding to a particular phenotype of interest instead of considering all attractors at the same time. Using the definition of attractors, we can have a simplified update rule with fixed state values for some nodes. The resulting subnetworks were small enough to find out the corresponding local attractors which can be integrated for reconstruction of the global attractor states of the original large network.
Conclusions
The proposed approach can substantially extend the current limit of Boolean network modeling for converging state analysis of biological networks.
Electronic supplementary material
The online version of this article (doi:10.1186/s12918-016-0338-4) contains supplementary material, which is available to authorized users.
Recently a new compositional framework for constructing and analysing Boolean networks was presented based on merging entities using Boolean connectives. While this framework provides a good basis for engineering Boolean networks, its practical application is limited by the restricted composition structures allowed and the lack of support for attractor analysis. In this paper we significantly extend this compositional framework by developing a new general structure for compositions and by providing new techniques for compositionally identifying the attractors of a Boolean network. The results presented are important as they support ongoing work to use the framework for engineering biological systems and also provide a new basis for analysing Boolean networks which helps to address the practical limitations imposed by the state space explosion problem.
IntroductionSarcopenia is a clinical syndrome characterized by the reduction in muscle mass, strength and physical ability. Although proximal femur fractures are one of the major burdens affecting the ageing population, distal radius fractures are equally important for frequency, clinical and social consequences. The aim of this study is to evaluate the incidence of sarcopenia in distal radius fractures and clinical implications in functional recovery.Materials and methodsScopus and PubMed search was performed to find relationship between sarcopenia and distal radius fractures. Literature search was performed between 2009 and 2019 including clinical trials and clinical studies related to “sarcopenia and distal radius fracture” and “sarcopenia and wrist fracture”. After identification, studies were screened and analysed through the Oxford Level of Evidence.ResultsAccording to the inclusion and exclusion criteria, five articles were included. Four articles analysed the incidence of sarcopenia and its role as a risk factor in patients with distal radial fractures, while one article focused on sarcopenia and clinical results of surgical treatment of distal radius fractures.Incidence of sarcopenia in patients older than 50 years with distal radius fracture varied between 29.7% and 31.7%. Patients with distal radial fractures did not show a significant inferior muscle mass than control group in examined population. Functional results of surgery were significantly inferior in sarcopenic patients than control group (no sarcopenia).Conclusions
About 30% of patients older than 50 years with distal radius fracture suffered by sarcopenia; sarcopenic patients surgically treated had worse clinical results than no sarcopenic patients. Further studies with larger samples are needed to confirm these preliminary results.
