On the dynamics of random Boolean networks subject to noise: Attractors, ergodic sets and cell types

The asymptotic dynamics of random Boolean networks subject to random fluctuations is investigated. Under the influence of noise, the system can escape from the attractors of the deterministic model, and a thorough study of these transitions is presented. We show that the dynamics is more properly described by sets of attractors rather than single ones. We generalize here a previous notion of ergodic sets, and we show that the Threshold Ergodic Sets so defined are robust with respect to noise and, at the same time, that they do not suffer from a major drawback of ergodic sets. The system jumps from one attractor to another of the same Threshold Ergodic Set under the influence of noise, never leaving it. By interpreting random Boolean networks as models of genetic regulatory networks, we also propose to associate cell types to Threshold Ergodic Sets rather than to deterministic attractors or to ergodic sets, as it had been previously suggested. We also propose to associate cell differentiation to the process whereby a Threshold Ergodic Set composed by several attractors gives rise to another one composed by a smaller number of attractors. We show that this approach accounts for several interesting experimental facts about cell differentiation, including the possibility to obtain an induced pluripotent stem cell from a fully differentiated one by overexpressing some of its genes.     

A comparison of two sensitivity analysis techniques based on four bayesian models representing ecosystem services provision in the Argentine Pampas

Sensitivity analyses (SAs) identify how an output variable of a model is modified by changes in the input variables. These analyses are a good way for assessing the performance of probabilistic models, like Bayesian Networks (BN). However, there are several commonly used SAs in BN literature, and formal comparisons about their outcomes are scarce. We used four previously developed BNs which represent ecosystem services provision in Pampean agroecosystems (Argentina) in order to test two local sensitivity approaches widely used. These SAs were: 1) One-at-a-time, used in BNs but more commonly in linear modelling; and 2) Sensitivity to findings, specific to BN modelling. Results showed that both analyses provided an adequate overview of BN behaviour. Furthermore, analyses produced a similar influence ranking of input variables over each output variable. Even though their interchangeably application could be an alternative in our bayesian models, we believe that OAT is the suitable one to implement here because of its capacity to demonstrate the relation (positive or negative) between input and output variables. In summary, we provided insights about two sensitivity techniques in BNs based on a case study which may be useful for ecological modellers.     

Maximum Number of Fixed Points in Regulatory Boolean Networks

Boolean networks (BNs) have been extensively used as mathematical models of genetic regulatory networks. The number of fixed points of a BN is a key feature of its dynamical behavior. Here, we study the maximum number of fixed points in a particular class of BNs called regulatory Boolean networks, where each interaction between the elements of the network is either an activation or an inhibition. We find relationships between the positive and negative cycles of the interaction graph and the number of fixed points of the network. As our main result, we exhibit an upper bound for the number of fixed points in terms of minimum cardinality of a set of vertices meeting all positive cycles of the network, which can be applied in the design of genetic regulatory networks.     

Bayesian networks as a novel tool to enhance interpretability and predictive power of ecological models

In today's world, it is becoming increasingly important to have the tools to understand, and ultimately to predict, the response of ecosystems to disturbance. However, understanding such dynamics is not simple. Ecosystems are a complex network of species interactions, and therefore any change to a population of one species will have some degree of community level effect. In recent years, the use of Bayesian networks (BNs) has seen successful applications in molecular biology and ecology, where they were able to recover plausible links in the respective systems they were applied to. The recovered network also comes with a quantifiable metric of interaction strength between variables. While the latter is an invaluable piece of information in ecology, an unexplored application of BNs would be using them as a novel variable selection tool in the training of predictive models. To this end, we evaluate the potential usefulness of BNs in two aspects: (1) we apply BN inference on species abundance data from a rocky shore ecosystem, a system with well documented links, to test the ecological validity of the revealed network; and (2) we evaluate BNs as a novel variable selection method to guide the training of an artificial neural network (ANN). Here, we demonstrate that not only was this approach able to recover meaningful species interactions networks from ecological data, but it also served as a meaningful tool to inform the training of predictive models, where there was an improvement in predictive performance in models with BN variable selection. Combining these results, we demonstrate the potential of this novel application of BNs in enhancing the interpretability and predictive power of ecological models; this has general applicability beyond the studied system, to ecosystems where existing relationships between species and other functional components are unknown.     

Bayesian Pathway Analysis of Cancer Microarray Data

High Throughput Biological Data (HTBD) requires detailed analysis methods and from a life science perspective, these analysis results make most sense when interpreted within the context of biological pathways. Bayesian Networks (BNs) capture both linear and nonlinear interactions and handle stochastic events in a probabilistic framework accounting for noise making them viable candidates for HTBD analysis. We have recently proposed an approach, called Bayesian Pathway Analysis (BPA), for analyzing HTBD using BNs in which known biological pathways are modeled as BNs and pathways that best explain the given HTBD are found. BPA uses the fold change information to obtain an input matrix to score each pathway modeled as a BN. Scoring is achieved using the Bayesian-Dirichlet Equivalent method and significance is assessed by randomization via bootstrapping of the columns of the input matrix. In this study, we improve on the BPA system by optimizing the steps involved in “Data Preprocessing and Discretization”, “Scoring”, “Significance Assessment”, and “Software and Web Application”. We tested the improved system on synthetic data sets and achieved over 98% accuracy in identifying the active pathways. The overall approach was applied on real cancer microarray data sets in order to investigate the pathways that are commonly active in different cancer types. We compared our findings on the real data sets with a relevant approach called the Signaling Pathway Impact Analysis (SPIA).     

The complex fluctuations of probabilistic Boolean networks

Probabilistic Boolean networks (PBNs) are extensions of Boolean networks (BNs), and both have been widely used to model biological systems. In this paper, we study the long-range correlations of PBNs based on their corresponding Markov chains. PBN states are quantified by the deviation of their steady-state distributions. The results demonstrate that, compared with BNs, PBNs can exhibit these dynamics over a wider and higher noise range. In addition, the constituent BNs significantly impact the generation of 1/f dynamics of PBNs, and PBNs with homogeneous steady-state distributions tend to sustain the 1/f dynamics over a wider noise range.     

Boolean dynamics of biological networks with multiple coupled feedback loops

Boolean networks have been frequently used to study the dynamics of biological networks. In particular, there have been various studies showing that the network connectivity and the update rule of logical functions affect the dynamics of Boolean networks. There has been, however, relatively little attention paid to the dynamical role of a feedback loop, which is a circular chain of interactions between Boolean variables. We note that such feedback loops are ubiquitously found in various biological systems as multiple coupled structures and they are often the primary cause of complex dynamics. In this article, we investigate the relationship between the multiple coupled feedback loops and the dynamics of Boolean networks. We show that networks have a larger proportion of basins corresponding to fixed-point attractors as they have more coupled positive feedback loops, and a larger proportion of basins for limit-cycle attractors as they have more coupled negative feedback loops.     

Growing Bayesian network models of gene networks from seed genes

MOTIVATION: For the last few years, Bayesian networks (BNs) have received increasing attention from the computational biology community as models of gene networks, though learning them from gene-expression data is problematic. Most gene-expression databases contain measurements for thousands of genes, but the existing algorithms for learning BNs from data do not scale to such high-dimensional databases. This means that the user has to decide in advance which genes are included in the learning process, typically no more than a few hundreds, and which genes are excluded from it. This is not a trivial decision. We propose an alternative approach to overcome this problem. RESULTS: We propose a new algorithm for learning BN models of gene networks from gene-expression data. Our algorithm receives a seed gene S and a positive integer R from the user, and returns a BN for the genes that depend on S such that less than R other genes mediate the dependency. Our algorithm grows the BN, which initially only contains S, by repeating the following step R + 1 times and, then, pruning some genes; find the parents and children of all the genes in the BN and add them to it. Intuitively, our algorithm provides the user with a window of radius R around S to look at the BN model of a gene network without having to exclude any gene in advance. We prove that our algorithm is correct under the faithfulness assumption. We evaluate our algorithm on simulated and biological data (Rosetta compendium) with satisfactory results.     

Probabilistic Boolean Networks: a rule-based uncertainty model for gene regulatory networks

MOTIVATION: Our goal is to construct a model for genetic regulatory networks such that the model class: (i) incorporates rule-based dependencies between genes; (ii) allows the systematic study of global network dynamics; (iii) is able to cope with uncertainty, both in the data and the model selection; and (iv) permits the quantification of the relative influence and sensitivity of genes in their interactions with other genes. RESULTS: We introduce Probabilistic Boolean Networks (PBN) that share the appealing rule-based properties of Boolean networks, but are robust in the face of uncertainty. We show how the dynamics of these networks can be studied in the probabilistic context of Markov chains, with standard Boolean networks being special cases. Then, we discuss the relationship between PBNs and Bayesian networks--a family of graphical models that explicitly represent probabilistic relationships between variables. We show how probabilistic dependencies between a gene and its parent genes, constituting the basic building blocks of Bayesian networks, can be obtained from PBNs. Finally, we present methods for quantifying the influence of genes on other genes, within the context of PBNs. Examples illustrating the above concepts are presented throughout the paper.     

Application of Bayesian networks on large-scale biological data

The investigation of the interplay between genes, proteins, metabolites and diseases plays a central role in molecular and cellular biology. Whole genome sequencing has made it possible to examine the behavior of all the genes in a genome by high-throughput experimental techniques and to pinpoint molecular interactions on a genome-wide scale, which form the backbone of systems biology. In particular, Bayesian network (BN) is a powerful tool for the ab-initial identification of causal and non-causal relationships between biological factors directly from experimental data. However, scalability is a crucial issue when we try to apply BNs to infer such interactions. In this paper, we not only introduce the Bayesian network formalism and its applications in systems biology, but also review recent technical developments for scaling up or speeding up the structural learning of BNs, which is important for the discovery of causal knowledge from large-scale biological datasets. Specifically, we highlight the basic idea, relative pros and cons of each technique and discuss possible ways to combine different algorithms towards making BN learning more accurate and much faster.     

Probabilistic logic methods and some applications to biology and medicine

For the computational analysis of biological problems-analyzing data, inferring networks and complex models, and estimating model parameters-it is common to use a range of methods based on probabilistic logic constructions, sometimes collectively called machine learning methods. Probabilistic modeling methods such as Bayesian Networks (BN) fall into this class, as do Hierarchical Bayesian Networks (HBN), Probabilistic Boolean Networks (PBN), Hidden Markov Models (HMM), and Markov Logic Networks (MLN). In this review, we describe the most general of these (MLN), and show how the above-mentioned methods are related to MLN and one another by the imposition of constraints and restrictions. This approach allows us to illustrate a broad landscape of constructions and methods, and describe some of the attendant strengths, weaknesses, and constraints of many of these methods. We then provide some examples of their applications to problems in biology and medicine, with an emphasis on genetics. The key concepts needed to picture this landscape of methods are the ideas of probabilistic graphical models, the structures of the graphs, and the scope of the logical language repertoire used (from First-Order Logic [FOL] to Boolean logic.) These concepts are interlinked and together define the nature of each of the probabilistic logic methods. Finally, we discuss the initial applications of MLN to genetics, show the relationship to less general methods like BN, and then mention several examples where such methods could be effective in new applications to specific biological and medical problems.     

Inferring Boolean network states from partial information

Networks of molecular interactions regulate key processes in living cells. Therefore, understanding their functionality is a high priority in advancing biological knowledge. Boolean networks are often used to describe cellular networks mathematically and are fitted to experimental datasets. The fitting often results in ambiguities since the interpretation of the measurements is not straightforward and since the data contain noise. In order to facilitate a more reliable mapping between datasets and Boolean networks, we develop an algorithm that infers network trajectories from a dataset distorted by noise. We analyze our algorithm theoretically and demonstrate its accuracy using simulation and microarray expression data.     

Multi-class subcellular location prediction for bacterial proteins

Two algorithms, based onBayesian Networks (BNs), for bacterial subcellular location prediction, are explored in this paper: one predicts all locations for Gram+ bacteria and the other all locations for Gram- bacteria. Methods were evaluated using different numbers of residues (from the N-terminal 10 residues to the whole sequence) and residue representation (amino acid-composition, percentage amino acid-composition or normalised amino acid-composition). The accuracy of the best resulting BN was compared to PSORTB. The accuracy of this multi-location BN was roughly comparable to PSORTB; the difference in predictions is low, often less than 2%. The BN method thus represents both an important new avenue of methodological development for subcellular location prediction and a potentially value new tool of true utilitarian value for candidate subunit vaccine selection.     

On construction of stochastic genetic networks based on gene expression sequences

Reconstruction of genetic regulatory networks from time series data of gene expression patterns is an important research topic in bioinformatics. Probabilistic Boolean Networks (PBNs) have been proposed as an effective model for gene regulatory networks. PBNs are able to cope with uncertainty, corporate rule-based dependencies between genes and discover the sensitivity of genes in their interactions with other genes. However, PBNs are unlikely to use directly in practice because of huge amount of computational cost for obtaining predictors and their corresponding probabilities. In this paper, we propose a multivariate Markov model for approximating PBNs and describing the dynamics of a genetic network for gene expression sequences. The main contribution of the new model is to preserve the strength of PBNs and reduce the complexity of the networks. The number of parameters of our proposed model is O(n2) where n is the number of genes involved. We also develop efficient estimation methods for solving the model parameters. Numerical examples on synthetic data sets and practical yeast data sequences are given to demonstrate the effectiveness of the proposed model.     

The pace of evolution across fitness valleys

How fast does a population evolve from one fitness peak to another? We study the dynamics of evolving, asexually reproducing populations in which a certain number of mutations jointly confer a fitness advantage. We consider the time until a population has evolved from one fitness peak to another one with a higher fitness. The order of mutations can either be fixed or random. If the order of mutations is fixed, then the population follows a metaphorical ridge, a single path. If the order of mutations is arbitrary, then there are many ways to evolve to the higher fitness state. We address the time required for fixation in such scenarios and study how it is affected by the order of mutations, the population size, the fitness values and the mutation rate.     

Control of Boolean networks: hardness results and algorithms for tree structured networks

Finding control strategies of cells is a challenging and important problem in the post-genomic era. This paper considers theoretical aspects of the control problem using the Boolean network (BN), which is a simplified model of genetic networks. It is shown that finding a control strategy leading to the desired global state is computationally intractable (NP-hard) in general. Furthermore, this hardness result is extended for BNs with considerably restricted network structures. These results justify existing exponential time algorithms for finding control strategies for probabilistic Boolean networks (PBNs). On the other hand, this paper shows that the control problem can be solved in polynomial time if the network has a tree structure. Then, this algorithm is extended for the case where the network has a few loops and the number of time steps is small. Though this paper focuses on theoretical aspects, biological implications of the theoretical results are also discussed.     

Evolution of Escherichia coli in different carbon environments for 2,000 generations

Cellular energetics is thought to have played a key role in dictating all major evolutionary transitions in the history of life on Earth. However, how exactly cellular energetics and metabolism come together to shape evolutionary paths is not well understood. In particular, when an organism is evolved in different energy environments, what are the phenomenological differences in the chosen evolutionary trajectories, is a question that is not well understood. In this context, starting from an Escherichia coli K‐12 strain, we evolve the bacterium in five different carbon environments—glucose, arabinose, xylose, rhamnose and a mixture of these four sugars (in a predefined ratio) for approximately 2,000 generations. At the end of the adaptation period, we quantify and compare the growth dynamics of the strains in a variety of environments. The evolved strains show no specialized adaptation towards growth in the carbon medium in which they were evolved. Rather, in all environments, the evolved strains exhibited a reduced lag phase and an increased growth rate. Sequencing results reveal that these dynamical properties are not introduced via mutations in the precise loci associated with utilization of the sugar in which the bacterium evolved. These phenotypic changes are rather likely introduced via mutations elsewhere on the genome. Data from our experiments indicate that evolution in a defined environment does not alter hierarchy in mixed‐sugar utilization in bacteria.     

Boolean Network Model for Cancer Pathways: Predicting Carcinogenesis and Targeted Therapy Outcomes

A Boolean dynamical system integrating the main signaling pathways involved in cancer is constructed based on the currently known protein-protein interaction network. This system exhibits stationary protein activation patterns – attractors – dependent on the cell''s microenvironment. These dynamical attractors were determined through simulations and their stabilities against mutations were tested. In a higher hierarchical level, it was possible to group the network attractors into distinct cell phenotypes and determine driver mutations that promote phenotypic transitions. We find that driver nodes are not necessarily central in the network topology, but at least they are direct regulators of central components towards which converge or through which crosstalk distinct cancer signaling pathways. The predicted drivers are in agreement with those pointed out by diverse census of cancer genes recently performed for several human cancers. Furthermore, our results demonstrate that cell phenotypes can evolve towards full malignancy through distinct sequences of accumulated mutations. In particular, the network model supports routes of carcinogenesis known for some tumor types. Finally, the Boolean network model is employed to evaluate the outcome of molecularly targeted cancer therapies. The major find is that monotherapies were additive in their effects and that the association of targeted drugs is necessary for cancer eradication.