Boolean Network Model Research Articles

Construction and analysis of gene-gene dynamics influence networks based on a Boolean model

BackgroundIdentification of novel gene-gene relations is a crucial issue to understand system-level biological phenomena. To this end, many methods based on a correlation analysis of gene expressions or structural analysis of molecular interaction networks have been proposed. They have a limitation in identifying more complicated gene-gene dynamical relations, though.ResultsTo overcome this limitation, we proposed a measure to quantify a gene-gene dynamical influence (GDI) using a Boolean network model and constructed a GDI network to indicate existence of a dynamical influence for every ordered pair of genes. It represents how much a state trajectory of a target gene is changed by a knockout mutation subject to a source gene in a gene-gene molecular interaction (GMI) network. Through a topological comparison between GDI and GMI networks, we observed that the former network is denser than the latter network, which implies that there exist many gene pairs of dynamically influencing but molecularly non-interacting relations. In addition, a larger number of hub genes were generated in the GDI network. On the other hand, there was a correlation between these networks such that the degree value of a node was positively correlated to each other. We further investigated the relationships of the GDI value with structural properties and found that there are negative and positive correlations with the length of a shortest path and the number of paths, respectively. In addition, a GDI network could predict a set of genes whose steady-state expression is affected in E. coli gene-knockout experiments. More interestingly, we found that the drug-targets with side-effects have a larger number of outgoing links than the other genes in the GDI network, which implies that they are more likely to influence the dynamics of other genes. Finally, we found biological evidences showing that the gene pairs which are not molecularly interacting but dynamically influential can be considered for novel gene-gene relationships.ConclusionTaken together, construction and analysis of the GDI network can be a useful approach to identify novel gene-gene relationships in terms of the dynamical influence.

BoolFilter: an R package for estimation and identification of partially-observed Boolean dynamical systems

BackgroundGene regulatory networks govern the function of key cellular processes, such as control of the cell cycle, response to stress, DNA repair mechanisms, and more. Boolean networks have been used successfully in modeling gene regulatory networks. In the Boolean network model, the transcriptional state of each gene is represented by 0 (inactive) or 1 (active), and the relationship among genes is represented by logical gates updated at discrete time points. However, the Boolean gene states are never observed directly, but only indirectly and incompletely through noisy measurements based on expression technologies such as cDNA microarrays, RNA-Seq, and cell imaging-based assays. The Partially-Observed Boolean Dynamical System (POBDS) signal model is distinct from other deterministic and stochastic Boolean network models in removing the requirement of a directly observable Boolean state vector and allowing uncertainty in the measurement process, addressing the scenario encountered in practice in transcriptomic analysis.ResultsBoolFilter is an R package that implements the POBDS model and associated algorithms for state and parameter estimation. It allows the user to estimate the Boolean states, network topology, and measurement parameters from time series of transcriptomic data using exact and approximated (particle) filters, as well as simulate the transcriptomic data for a given Boolean network model. Some of its infrastructure, such as the network interface, is the same as in the previously published R package for Boolean Networks BoolNet, which enhances compatibility and user accessibility to the new package.ConclusionsWe introduce the R package BoolFilter for Partially-Observed Boolean Dynamical Systems (POBDS). The BoolFilter package provides a useful toolbox for the bioinformatics community, with state-of-the-art algorithms for simulation of time series transcriptomic data as well as the inverse process of system identification from data obtained with various expression technologies such as cDNA microarrays, RNA-Seq, and cell imaging-based assays.

Robustness and relative stability of multiple attractors in\\ a stochastic Boolean network

Multiple phenotypic states and dynamic pathways always exist in biological networks, but their robustness against fluctuations and relative stability have not been fully recognized or carefully analyzed yet. Here we try to address these issues with a widely used stochastic Boolean network model. The stability of phenotypic states and dynamical pathways corresponds to the robustness and relative stability of attractors in such a stochastic Boolean network, which can be analyzed using the theory of exponentially perturbed Markov chains. It is already known that the logarithm of the expected exit time escaping from an attractor is proportional to the ``minimum activation energy barrier in the exponentially perturbed Markov chains. We first prove that within an exponentially perturbed Markov chain, all pathways between attractors with the minimum possible ``activation energy barrier have the same probability weights in the zero-noise limit, and therefore the relative stability just depends on the number of optimal paths. Another important implication of this theory is that, once the non-neglectable fluctuations are rather low, a phase transition phenomenon emerges: In one parameter region the probability weights of the coexisted phenotypic states and pathways are comparable with each other; whereas in some other parameter regions, the probability weight of certain phenotypic state or pathway can even dominate and become globally attractive. Finally, we apply the theory to an artificial model and the Siah-1/beta-catenin/p14/19 ARF loop of protein p53 dynamics. Our theory can also determine how the transition time and the number of optimal transition paths between the phenotypic states and parallel pathways depend on all the parameters, and help to identify those possibly more crucial nodes and the interactions between them in a biological network.

How causal analysis can reveal autonomy in models of biological systems.

Standard techniques for studying biological systems largely focus on their dynamical or, more recently, their informational properties, usually taking either a reductionist or holistic perspective. Yet, studying only individual system elements or the dynamics of the system as a whole disregards the organizational structure of the system-whether there are subsets of elements with joint causes or effects, and whether the system is strongly integrated or composed of several loosely interacting components. Integrated information theory offers a theoretical framework to (1) investigate the compositional cause-effect structure of a system and to (2) identify causal borders of highly integrated elements comprising local maxima of intrinsic cause-effect power. Here we apply this comprehensive causal analysis to a Boolean network model of the fission yeast (Schizosaccharomyces pombe) cell cycle. We demonstrate that this biological model features a non-trivial causal architecture, whose discovery may provide insights about the real cell cycle that could not be gained from holistic or reductionist approaches. We also show how some specific properties of this underlying causal architecture relate to the biological notion of autonomy. Ultimately, we suggest that analysing the causal organization of a system, including key features like intrinsic control and stable causal borders, should prove relevant for distinguishing life from non-life, and thus could also illuminate the origin of life problem.This article is part of the themed issue 'Reconceptualizing the origins of life'.

A protein phosphatase network controls the temporal and spatial dynamics of differentiation commitment in human epidermis.

Epidermal homeostasis depends on a balance between stem cell renewal and terminal differentiation. The transition between the two cell states, termed commitment, is poorly understood. Here, we characterise commitment by integrating transcriptomic and proteomic data from disaggregated primary human keratinocytes held in suspension to induce differentiation. Cell detachment induces several protein phosphatases, five of which - DUSP6, PPTC7, PTPN1, PTPN13 and PPP3CA - promote differentiation by negatively regulating ERK MAPK and positively regulating AP1 transcription factors. Conversely, DUSP10 expression antagonises commitment. The phosphatases form a dynamic network of transient positive and negative interactions that change over time, with DUSP6 predominating at commitment. Boolean network modelling identifies a mandatory switch between two stable states (stem and differentiated) via an unstable (committed) state. Phosphatase expression is also spatially regulated in vivo and in vitro. We conclude that an auto-regulatory phosphatase network maintains epidermal homeostasis by controlling the onset and duration of commitment.

The logic of the floral transition: Reverse-engineering the switch controlling the identity of lateral organs.

Much laboratory work has been carried out to determine the gene regulatory network (GRN) that results in plant cells becoming flowers instead of leaves. However, this also involves the spatial distribution of different cell types, and poses the question of whether alternative networks could produce the same set of observed results. This issue has been addressed here through a survey of the published intercellular distribution of expressed regulatory genes and techniques both developed and applied to Boolean network models. This has uncovered a large number of models which are compatible with the currently available data. An exhaustive exploration had some success but proved to be unfeasible due to the massive number of alternative models, so genetic programming algorithms have also been employed. This approach allows exploration on the basis of both data-fitting criteria and parsimony of the regulatory processes, ruling out biologically unrealistic mechanisms. One of the conclusions is that, despite the multiplicity of acceptable models, an overall structure dominates, with differences mostly in alternative fine-grained regulatory interactions. The overall structure confirms the known interactions, including some that were not present in the training set, showing that current data are sufficient to determine the overall structure of the GRN. The model stresses the importance of relative spatial location, through explicit references to this aspect. This approach also provides a quantitative indication of how likely some regulatory interactions might be, and can be applied to the study of other developmental transitions.

Discrimination of singleton and periodic attractors in Boolean networks

Determining the minimum number of sensor nodes to observe the internal state of the whole system is important in analysis of complex networks. However, existing studies suggest that a large number of sensor nodes are needed to know the whole internal state. In this paper, we focus on identification of a small set of sensor nodes to discriminate statically and periodically steady states using the Boolean network model where steady states are often considered to correspond to cell types. In other words, we seek a minimum set of nodes to discriminate singleton and periodic attractors. We prove that one node is not necessarily enough but two nodes are always enough to discriminate two periodic attractors by using the Chinese remainder theorem. Based on this, we present an algorithm to determine the minimum number of nodes to discriminate all given attractors. We also present a much more efficient algorithm to discriminate singleton attractors. The results of computational experiments suggest that attractors in realistic Boolean networks can be discriminated by observing the states of only a small number of nodes.

Expected Number of Fixed Points in Boolean Networks with Arbitrary Topology.

Boolean network models describe genetic, neural, and social dynamics in complex networks, where the dynamics depend generally on network topology. Fixed points in a genetic regulatory network are typically considered to correspond to cell types in an organism. We prove that the expected number of fixed points in a Boolean network, with Boolean functions drawn from probability distributions that are not required to be uniform or identical, is one, and is independent of network topology if only a feedback arc set satisfies a stochastic neutrality condition. We also demonstrate that the expected number is increased by the predominance of positive feedback in a cycle.

Boolean analysis reveals systematic interactions among low-abundance species in the human gut microbiome.

The analysis of microbiome compositions in the human gut has gained increasing interest due to the broader availability of data and functional databases and substantial progress in data analysis methods, but also due to the high relevance of the microbiome in human health and disease. While most analyses infer interactions among highly abundant species, the large number of low-abundance species has received less attention. Here we present a novel analysis method based on Boolean operations applied to microbial co-occurrence patterns. We calibrate our approach with simulated data based on a dynamical Boolean network model from which we interpret the statistics of attractor states as a theoretical proxy for microbiome composition. We show that for given fractions of synergistic and competitive interactions in the model our Boolean abundance analysis can reliably detect these interactions. Analyzing a novel data set of 822 microbiome compositions of the human gut, we find a large number of highly significant synergistic interactions among these low-abundance species, forming a connected network, and a few isolated competitive interactions.

Quorum-Sensing Systems in the Plant Growth-Promoting Bacterium Paraburkholderia phytofirmans PsJN Exhibit Cross-Regulation and Are Involved in Biofilm Formation.

Quorum-sensing systems play important roles in host colonization and host establishment of Burkholderiales species. Beneficial Paraburkholderia species share a conserved quorum-sensing (QS) system, designated BraI/R, that controls different phenotypes. In this context, the plant growth-promoting bacterium Paraburkholderia phytofirmans PsJN possesses two different homoserine lactone QS systems BpI.1/R.1 and BpI.2/R.2 (BraI/R-like QS system). The BpI.1/R.1 QS system was previously reported to be important to colonize and produce beneficial effects in Arabidopsis thaliana plants. Here, we analyzed the temporal variations of the QS gene transcript levels in the wild-type strain colonizing plant roots. The gene expression patterns showed relevant differences in both QS systems compared with the wild-type strain in the unplanted control treatment. The gene expression data were used to reconstruct a regulatory network model of QS systems in P. phytofirmans PsJN, using a Boolean network model. Also, we examined the phenotypic traits and transcript levels of genes involved in QS systems, using P. phytofirmans mutants in homoserine lactone synthases genes. We observed that the BpI.1/R.1 QS system regulates biofilm formation production in strain PsJN and this phenotype was associated with the lower expression of a specific extracytoplasmic function sigma factor ecf26.1 gene (implicated in biofilm formation) in the bpI.1 mutant strain.

The influence of canalization on the robustness of Boolean networks

Time- and state-discrete dynamical systems are frequently used to model molecular networks. This paper provides a collection of mathematical and computational tools for the study of robustness in Boolean network models. The focus is on networks governed by k-canalizing functions, a recently introduced class of Boolean functions that contains the well-studied class of nested canalizing functions. The variable activities and sensitivity of a function quantify the impact of input changes on the function output. This paper generalizes the latter concept to c-sensitivity and provides formulas for the activities and c-sensitivity of general k-canalizing functions as well as canalizing functions with more precisely defined structure. A popular measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the c-sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.

Synchronization in mutual-coupled temporal Boolean networks

In this paper, we first propose a mutual-coupled temporal Boolean networks model and then investigate complete synchronization in mutual-coupled temporal Boolean networks. The mutual-coupled temporal Boolean networks model with logical relationship is converted into an algebraic form based on a semi-tensor product. Necessary and sufficient conditions are derived to realize synchronization based on the algebraic forms. An example illustrates the effectiveness of the theoretical analysis.

The stability of Boolean network with transmission sensitivity

Boolean network has been widely used in modeling biological systems and one of the key problems is its stability in response to small perturbations. Based on the hypothesis that the states of all nodes are homogenously updated, great progress has been made in previous works. In real biological networks, however, the updates of genes typically show much heterogeneity. To address such conditions, we introduce transmission sensitivity into Boolean network model. By the method of semi-annealed approximation, we illustrate that in a homogenous network, the critical condition of stability has no connection with its transmission sensitivity. As for heterogeneous networks, it reveals that correlations between network topology and transmission sensitivity can have profound effects on the its stability. This result shows a new mechanism that affects the stability of Boolean network, which could be used to control the dynamics in real biological systems.

Erratum to: Probabilistic Boolean network modeling and model checking as an approach for DFMEA for manufacturing systems

Erratum to: Probabilistic Boolean network modeling and model checking as an approach for DFMEA for manufacturing systems

A novel mutual information-based Boolean network inference method from time-series gene expression data.

BackgroundInferring a gene regulatory network from time-series gene expression data in systems biology is a challenging problem. Many methods have been suggested, most of which have a scalability limitation due to the combinatorial cost of searching a regulatory set of genes. In addition, they have focused on the accurate inference of a network structure only. Therefore, there is a pressing need to develop a network inference method to search regulatory genes efficiently and to predict the network dynamics accurately.ResultsIn this study, we employed a Boolean network model with a restricted update rule scheme to capture coarse-grained dynamics, and propose a novel mutual information-based Boolean network inference (MIBNI) method. Given time-series gene expression data as an input, the method first identifies a set of initial regulatory genes using mutual information-based feature selection, and then improves the dynamics prediction accuracy by iteratively swapping a pair of genes between sets of the selected regulatory genes and the other genes. Through extensive simulations with artificial datasets, MIBNI showed consistently better performance than six well-known existing methods, REVEAL, Best-Fit, RelNet, CST, CLR, and BIBN in terms of both structural and dynamics prediction accuracy. We further tested the proposed method with two real gene expression datasets for an Escherichia coli gene regulatory network and a fission yeast cell cycle network, and also observed better results using MIBNI compared to the six other methods.ConclusionsTaken together, MIBNI is a promising tool for predicting both the structure and the dynamics of a gene regulatory network.

Adaptive Local Information Transfer in Random Boolean Networks.

Living systems such as gene regulatory networks and neuronal networks have been supposed to work close to dynamical criticality, where their information-processing ability is optimal at the whole-system level. We investigate how this global information-processing optimality is related to the local information transfer at each individual-unit level. In particular, we introduce an internal adjustment process of the local information transfer and examine whether the former can emerge from the latter. We propose an adaptive random Boolean network model in which each unit rewires its incoming arcs from other units to balance stability of its information processing based on the measurement of the local information transfer pattern. First, we show numerically that random Boolean networks can self-organize toward near dynamical criticality in our model. Second, the proposed model is analyzed by a mean-field theory. We recognize that the rewiring rule has a bootstrapping feature. The stationary indegree distribution is calculated semi-analytically and is shown to be close to dynamical criticality in a broad range of model parameter values.

MORO: a Cytoscape app for relationship analysis between modularity and robustness in large-scale biological networks.

BackgroundAlthough there have been many studies revealing that dynamic robustness of a biological network is related to its modularity characteristics, no proper tool exists to investigate the relation between network dynamics and modularity.ResultsAccordingly, we developed a novel Cytoscape app called MORO, which can conveniently analyze the relationship between network modularity and robustness. We employed an existing algorithm to analyze the modularity of directed graphs and a Boolean network model for robustness calculation. In particular, to ensure the robustness algorithm’s applicability to large-scale networks, we implemented it as a parallel algorithm by using the OpenCL library. A batch-mode simulation function was also developed to verify whether an observed relationship between modularity and robustness is conserved in a large set of randomly structured networks. The app provides various visualization modes to better elucidate topological relations between modules, and tabular results of centrality and gene ontology enrichment analyses of modules. We tested the proposed app to analyze large signaling networks and showed an interesting relationship between network modularity and robustness.ConclusionsOur app can be a promising tool which efficiently analyzes the relationship between modularity and robustness in large signaling networks.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-016-0363-3) contains supplementary material, which is available to authorized users.

Attractor landscape analysis of colorectal tumorigenesis and its reversion.

BackgroundColorectal cancer arises from the accumulation of genetic mutations that induce dysfunction of intracellular signaling. However, the underlying mechanism of colorectal tumorigenesis driven by genetic mutations remains yet to be elucidated.ResultsTo investigate colorectal tumorigenesis at a system-level, we have reconstructed a large-scale Boolean network model of the human signaling network by integrating previous experimental results on canonical signaling pathways related to proliferation, metastasis, and apoptosis. Throughout an extensive simulation analysis of the attractor landscape of the signaling network model, we found that the attractor landscape changes its shape by expanding the basin of attractors for abnormal proliferation and metastasis along with the accumulation of driver mutations. A further hypothetical study shows that restoration of a normal phenotype might be possible by reversely controlling the attractor landscape. Interestingly, the targets of approved anti-cancer drugs were highly enriched in the identified molecular targets for the reverse control.ConclusionsOur results show that the dynamical analysis of a signaling network based on attractor landscape is useful in acquiring a system-level understanding of tumorigenesis and developing a new therapeutic strategy.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-016-0341-9) contains supplementary material, which is available to authorized users.

An efficient algorithm for identifying primary phenotype attractors of a large-scale Boolean network.

BackgroundBoolean 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.ResultsThere 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.ConclusionsThe proposed approach can substantially extend the current limit of Boolean network modeling for converging state analysis of biological networks.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-016-0338-4) contains supplementary material, which is available to authorized users.

Identification of control targets in Boolean molecular network models via computational algebra.

BackgroundMany problems in biomedicine and other areas of the life sciences can be characterized as control problems, with the goal of finding strategies to change a disease or otherwise undesirable state of a biological system into another, more desirable, state through an intervention, such as a drug or other therapeutic treatment. The identification of such strategies is typically based on a mathematical model of the process to be altered through targeted control inputs. This paper focuses on processes at the molecular level that determine the state of an individual cell, involving signaling or gene regulation. The mathematical model type considered is that of Boolean networks. The potential control targets can be represented by a set of nodes and edges that can be manipulated to produce a desired effect on the system.ResultsThis paper presents a method for the identification of potential intervention targets in Boolean molecular network models using algebraic techniques. The approach exploits an algebraic representation of Boolean networks to encode the control candidates in the network wiring diagram as the solutions of a system of polynomials equations, and then uses computational algebra techniques to find such controllers. The control methods in this paper are validated through the identification of combinatorial interventions in the signaling pathways of previously reported control targets in two well studied systems, a p53-mdm2 network and a blood T cell lymphocyte granular leukemia survival signaling network. Supplementary data is available online and our code in Macaulay2 and Matlab are available via http://www.ms.uky.edu/~dmu228/ControlAlg.ConclusionsThis paper presents a novel method for the identification of intervention targets in Boolean network models. The results in this paper show that the proposed methods are useful and efficient for moderately large networks.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-016-0332-x) contains supplementary material, which is available to authorized users.