Stochastic Boolean networks: An efficient approach to modeling gene regulatory networks

ABSTRACT: BACKGROUND: Various computational models have been of interest due to their use in the modelling of gene regulatory networks (GRNs). As a logical model, probabilistic Boolean networks (PBNs) consider molecular and genetic noise, so the study of PBNs provides significant insights into the understanding of the dynamics of GRNs. This will ultimately lead to advances in developing therapeutic methods that intervene in the process of disease development and progression. The applications of PBNs, however, are hindered by the complexities involved in the computation of the state transition matrix and the steady-state distribution of a PBN. For a PBN with n genes and N Boolean networks, the complexity to compute the state transition matrix is O(nN22n) or O(nN2n) for a sparse matrix. RESULTS: This paper presents a novel implementation of PBNs based on the notions of stochastic logic and stochastic computation. This stochastic implementation of a PBN is referred to as a stochastic Boolean network (SBN). An SBN provides an accurate and efficient simulation of a PBN without and with random gene perturbation. The state transition matrix is computed in an SBN with a complexity of O(nL2n), where L is a factor related to the stochastic sequence length. Since the minimum sequence length required for obtaining an evaluation accuracy approximately increases in a polynomial order with the number of genes, n, and the number of Boolean networks, N, usually increases exponentially with n, L is typically smaller than N, especially in a network with a large number of genes. Hence, the computational complexity of an SBN is primarily limited by the number of genes, but not directly by the total possible number of Boolean networks. Furthermore, a time-frame expanded SBN enables an efficient analysis of the steady-state distribution of a PBN. These findings are supported by the simulation results of a simplified p53 network, several randomly generated networks and a network inferred from a T cell immune response dataset. An SBN can also implement the function of an asynchronous PBN and is potentially useful in a hybrid approach in combination with a continuous or single-molecule level stochastic model. CONCLUSIONS: Stochastic Boolean networks (SBNs) are proposed as an efficient approach to modelling gene regulatory networks (GRNs). The SBN approach is able to recover biologically-proven regulatory behaviours, such as the oscillatory dynamics of the p53-Mdm2 network and the dynamic attractors in a T cell immune response network. The proposed approach can further predict the network dynamics when the genes are under perturbation, thus providing biologically meaningful insights for a better understanding of the dynamics of GRNs. The algorithms and methods described in this paper have been implemented in Matlab packages, which are attached as Additional files.     

An effective data mining technique for reconstructing gene regulatory networks from time series expression data

Recent development in DNA microarray technologies has made the reconstruction of gene regulatory networks (GRNs) feasible. To infer the overall structure of a GRN, there is a need to find out how the expression of each gene can be affected by the others. Many existing approaches to reconstructing GRNs are developed to generate hypotheses about the presence or absence of interactions between genes so that laboratory experiments can be performed afterwards for verification. Since, they are not intended to be used to predict if a gene in an unseen sample has any interactions with other genes, statistical verification of the reliability of the discovered interactions can be difficult. Furthermore, since the temporal ordering of the data is not taken into consideration, the directionality of regulation cannot be established using these existing techniques. To tackle these problems, we propose a data mining technique here. This technique makes use of a probabilistic inference approach to uncover interesting dependency relationships in noisy, high-dimensional time series expression data. It is not only able to determine if a gene is dependent on another but also whether or not it is activated or inhibited. In addition, it can predict how a gene would be affected by other genes even in unseen samples. For performance evaluation, the proposed technique has been tested with real expression data. Experimental results show that it can be very effective. The discovered dependency relationships can reveal gene regulatory relationships that could be used to infer the structures of GRNs.     

A biological approach to computational models of proteomic networks

Computational modeling is useful as a means to assemble and test what we know about proteins and networks. Models can help address key questions about the measurement, definition and function of proteomic networks. Here, we place these biological questions at the forefront in reviewing the computational strategies that are available to analyze proteomic networks. Recent examples illustrate how models can extract more information from proteomic data, test possible interactions between network proteins and link networks to cellular behavior. No single model can achieve all these goals, however, which is why it is critical to prioritize biological questions before specifying a particular modeling approach.     

From gene regulation to gene function: regulatory networks in bacillus subtilis

Bacillus subtilis is a sporulating Gram-positive bacterium that lives primarily in the soil and associated water sources. The publication of the B. subtilis genome sequence and subsequent systematic functional analysis and gene regulation programmes, together with an extensive understanding of its biochemistry and physiology, makes this micro-organism a prime candidate in which to model regulatory networks in silico. In this paper we discuss combined molecular biological and bioinformatical approaches that are being developed to model this organism's responses to changes in its environment.     

Investigating meta-approaches for reconstructing gene networks in a mammalian cellular context

The output of state-of-the-art reverse-engineering methods for biological networks is often based on the fitting of a mathematical model to the data. Typically, different datasets do not give single consistent network predictions but rather an ensemble of inconsistent networks inferred under the same reverse-engineering method that are only consistent with the specific experimentally measured data. Here, we focus on an alternative approach for combining the information contained within such an ensemble of inconsistent gene networks called meta-analysis, to make more accurate predictions and to estimate the reliability of these predictions. We review two existing meta-analysis approaches; the Fisher transformation combined coefficient test (FTCCT) and Fisher's inverse combined probability test (FICPT); and compare their performance with five well-known methods, ARACNe, Context Likelihood or Relatedness network (CLR), Maximum Relevance Minimum Redundancy (MRNET), Relevance Network (RN) and Bayesian Network (BN). We conducted in-depth numerical ensemble simulations and demonstrated for biological expression data that the meta-analysis approaches consistently outperformed the best gene regulatory network inference (GRNI) methods in the literature. Furthermore, the meta-analysis approaches have a low computational complexity. We conclude that the meta-analysis approaches are a powerful tool for integrating different datasets to give more accurate and reliable predictions for biological networks.     

A Bayesian regression approach to the inference of regulatory networks from gene expression data

MOTIVATION: There is currently much interest in reverse-engineering regulatory relationships between genes from microarray expression data. We propose a new algorithmic method for inferring such interactions between genes using data from gene knockout experiments. The algorithm we use is the Sparse Bayesian regression algorithm of Tipping and Faul. This method is highly suited to this problem as it does not require the data to be discretized, overcomes the need for an explicit topology search and, most importantly, requires no heuristic thresholding of the discovered connections. RESULTS: Using simulated expression data, we are able to show that this algorithm outperforms a recently published correlation-based approach. Crucially, it does this without the need to set any ad hoc threshold on possible connections.     

Reconstructing gene regulatory networks: from random to scale-free connectivity

The manipulation of organisms using combinations of gene knockout, RNAi and drug interaction experiments can be used to reveal regulatory interactions between genes. Several algorithms have been proposed that try to reconstruct the underlying regulatory networks from gene expression data sets arising from such experiments. Often these approaches assume that each gene has approximately the same number of interactions within the network, and the methods rely on prior knowledge, or the investigator's best guess, of the average network connectivity. Recent evidence points to scale-free properties in biological networks, however, where network connectivity follows a power-law distribution. For scale-free networks, the average number of regulatory interactions per gene does not satisfactorily characterise the network. With this in mind, a new reverse engineering approach is introduced that does not require prior knowledge of network connectivity and its performance is compared with other published algorithms using simulated gene expression data with biologically relevant network structures. Because this new approach does not make any assumptions about the distribution of network connections, it is suitable for application to scale-free networks.     

Deconvoluting lung evolution: from phenotypes to gene regulatory networks

Speakers in this symposium presented examples of respiratoryregulation that broadly illustrate principles of evolution fromwhole organ to genes. The swim bladder and lungs of aquaticand terrestrial organisms arose independently from a commonprimordial "respiratory pharynx" but not from each other. Pathwaysof lung evolution are similar between crocodiles and birds buta low compliance of mammalian lung may have driven the developmentof the diaphragm to permit lung inflation during inspiration.To meet the high oxygen demands of flight, bird lungs have evolvedseparate gas exchange and pump components to achieve unidirectionalventilation and minimize dead space. The process of "screening"(removal of oxygen from inspired air prior to entering the terminalunits) reduces effective alveolar oxygen tension and potentiallyexplains why nonathletic large mammals possess greater pulmonarydiffusing capacities than required by their oxygen consumption.The "primitive" central admixture of oxygenated and deoxygenatedblood in the incompletely divided reptilian heart is actuallyco-regulated with other autonomic cardiopulmonary responsesto provide flexible control of arterial oxygen tension independentof ventilation as well as a unique mechanism for adjusting metabolicrate. Some of the most ancient oxygen-sensing molecules, i.e.,hypoxia-inducible factor-1alpha and erythropoietin, are up-regulatedduring mammalian lung development and growth under apparentlynormoxic conditions, suggesting functional evolution. Normalalveolarization requires pleiotropic growth factors acting viahighly conserved cell–cell signal transduction, e.g.,parathyroid hormone-related protein transducing at least partlythrough the Wingless/int pathway. The latter regulates morphogenesisfrom nematode to mammal. If there is commonality among thesediverse respiratory processes, it is that all levels of organization,from molecular signaling to structure to function, co-evolveprogressively, and optimize an existing gas-exchange framework.     

A multi-objective differential evolutionary approach toward more stable gene regulatory networks

Models are of central importance in many scientific contexts. Mathematical and computational modeling of genetic regulatory networks promises to uncover the fundamental principles of living systems. Biological models, such as gene regulatory models, can help us better understand interactions among genes and how cells regulate their production of proteins and enzymes. One feature shared among living systems is their ability to cope with perturbations and remain stable, a property that is the result of evolutionary fine-tuning over many generations. In this study we use random Boolean networks (RBNs) as an abstract model of gene regulatory systems. By applying Differential Evolution (DE), an evolution-based optimization technique, we produce networks with increased stability. DE requires relatively few user-specified parameters, has fast convergence and does not rely on initial conditions to find the global minima within multi-dimensional search spaces.