An Efficient Algorithm for Computing Attractors of Synchronous And Asynchronous Boolean Networks

Biological networks, such as genetic regulatory networks, often contain positive and negative feedback loops that settle down to dynamically stable patterns. Identifying these patterns, the so-called attractors, can provide important insights for biologists to understand the molecular mechanisms underlying many coordinated cellular processes such as cellular division, differentiation, and homeostasis. Both synchronous and asynchronous Boolean networks have been used to simulate genetic regulatory networks and identify their attractors. The common methods of computing attractors are that start with a randomly selected initial state and finish with exhaustive search of the state space of a network. However, the time complexity of these methods grows exponentially with respect to the number and length of attractors. Here, we build two algorithms to achieve the computation of attractors in synchronous and asynchronous Boolean networks. For the synchronous scenario, combing with iterative methods and reduced order binary decision diagrams (ROBDD), we propose an improved algorithm to compute attractors. For another algorithm, the attractors of synchronous Boolean networks are utilized in asynchronous Boolean translation functions to derive attractors of asynchronous scenario. The proposed algorithms are implemented in a procedure called geneFAtt. Compared to existing tools such as genYsis, geneFAtt is significantly faster in computing attractors for empirical experimental systems.

Availability

The software package is available at https://sites.google.com/site/desheng619/download.     

A Tutorial on Analysis and Simulation of Boolean Gene Regulatory Network Models

Driven by the desire to understand genomic functions through the interactions among genes and gene products, the research in gene regulatory networks has become a heated area in genomic signal processing. Among the most studied mathematical models are Boolean networks and probabilistic Boolean networks, which are rule-based dynamic systems. This tutorial provides an introduction to the essential concepts of these two Boolean models, and presents the up-to-date analysis and simulation methods developed for them. In the Analysis section, we will show that Boolean models are Markov chains, based on which we present a Markovian steady-state analysis on attractors, and also reveal the relationship between probabilistic Boolean networks and dynamic Bayesian networks (another popular genetic network model), again via Markov analysis; we dedicate the last subsection to structural analysis, which opens a door to other topics such as network control. The Simulation section will start from the basic tasks of creating state transition diagrams and finding attractors, proceed to the simulation of network dynamics and obtaining the steady-state distributions, and finally come to an algorithm of generating artificial Boolean networks with prescribed attractors. The contents are arranged in a roughly logical order, such that the Markov chain analysis lays the basis for the most part of Analysis section, and also prepares the readers to the topics in Simulation section.     

A SAT-based algorithm for finding attractors in synchronous Boolean networks

This paper addresses the problem of finding attractors in synchronous Boolean networks. The existing Boolean decision diagram-based algorithms have limited capacity due to the excessive memory requirements of decision diagrams. The simulation-based algorithms can be applied to larger networks, however, they are incomplete. We present an algorithm, which uses a SAT-based bounded model checking to find all attractors in a Boolean network. The efficiency of the presented algorithm is evaluated by analyzing seven networks models of real biological processes, as well as 150,000 randomly generated Boolean networks of sizes between 100 and 7,000. The results show that our approach has a potential to handle an order of magnitude larger models than currently possible.     

An in silico target identification using Boolean network attractors: Avoiding pathological phenotypes

Target identification aims at identifying biomolecules whose function should be therapeutically altered to cure the considered pathology. An algorithm for in silico target identification using Boolean network attractors is proposed. It assumes that attractors correspond to phenotypes produced by the modeled biological network. It identifies target combinations which allow disturbed networks to avoid attractors associated with pathological phenotypes. The algorithm is tested on a Boolean model of the mammalian cell cycle and its applications are illustrated on a Boolean model of Fanconi anemia. Results show that the algorithm returns target combinations able to remove attractors associated with pathological phenotypes and then succeeds in performing the proposed in silico target identification. However, as with any in silico evidence, there is a bridge to cross between theory and practice. Nevertheless, it is expected that the algorithm is of interest for target identification.     

Robustness and state-space structure of Boolean gene regulatory models

Robustness to perturbation is an important characteristic of genetic regulatory systems, but the relationship between robustness and model dynamics has not been clearly quantified. We propose a method for quantifying both robustness and dynamics in terms of state-space structures, for Boolean models of genetic regulatory systems. By investigating existing models of the Drosophila melanogaster segment polarity network and the Saccharomyces cerevisiae cell-cycle network, we show that the structure of attractor basins can yield insight into the underlying decision making required of the system, and also the way in which the system maximises its robustness. In particular, gene networks implementing decisions based on a few genes have simple state-space structures, and their attractors are robust by virtue of their simplicity. Gene networks with decisions that involve many interacting genes have correspondingly more complicated state-space structures, and robustness cannot be achieved through the structure of the attractor basins, but is achieved by larger attractor basins that dominate the state space. These different types of robustness are demonstrated by the two models: the D. melanogaster segment polarity network is robust due to simple attractor basins that implement decisions based on spatial signals; the S. cerevisiae cell-cycle network has a complicated state-space structure, and is robust only due to a giant attractor basin that dominates the state space.     

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.     

Intervention in a family of Boolean networks

MOTIVATION: Intervention in a gene regulatory network is used to avoid undesirable states, such as those associated with a disease. Several types of intervention have been studied in the framework of a probabilistic Boolean network (PBN), which is a collection of Boolean networks in which the gene state vector transitions according to the rules of one of the constituent networks and where network choice is governed by a selection distribution. The theory of automatic control has been applied to find optimal strategies for manipulating external control variables that affect the transition probabilities to desirably affect dynamic evolution over a finite time horizon. In this paper we treat a case in which we lack the governing probability structure for Boolean network selection, so we simply have a family of Boolean networks, but where these networks possess a common attractor structure. This corresponds to the situation in which network construction is treated as an ill-posed inverse problem in which there are many Boolean networks created from the data under the constraint that they all possess attractor structures matching the data states, which are assumed to arise from sampling the steady state of the real biological network. RESULTS: Given a family of Boolean networks possessing a common attractor structure composed of singleton attractors, a control algorithm is derived by minimizing a composite finite-horizon cost function that is a weighted average over all the individual networks, the idea being that we desire a control policy that on average suits the networks because these are viewed as equivalent relative to the data. The weighting for each network at any time point is taken to be proportional to the instantaneous estimated probability of that network being the underlying network governing the state transition. The results are applied to a family of Boolean networks derived from gene-expression data collected in a study of metastatic melanoma, the intent being to devise a control strategy that reduces the WNT5A gene's action in affecting biological regulation. AVAILABILITY: The software is available on request. SUPPLEMENTARY INFORMATION: The supplementary Information is available at http://ee.tamu.edu/~edward/tree     

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.     

A Full Bayesian Approach for Boolean Genetic Network Inference

Boolean networks are a simple but efficient model for describing gene regulatory systems. A number of algorithms have been proposed to infer Boolean networks. However, these methods do not take full consideration of the effects of noise and model uncertainty. In this paper, we propose a full Bayesian approach to infer Boolean genetic networks. Markov chain Monte Carlo algorithms are used to obtain the posterior samples of both the network structure and the related parameters. In addition to regular link addition and removal moves, which can guarantee the irreducibility of the Markov chain for traversing the whole network space, carefully constructed mixture proposals are used to improve the Markov chain Monte Carlo convergence. Both simulations and a real application on cell-cycle data show that our method is more powerful than existing methods for the inference of both the topology and logic relations of the Boolean network from observed data.     

Algorithms and Complexity Analyses for Control of Singleton Attractors in Boolean Networks

A Boolean network (BN) is a mathematical model of genetic networks. We propose several algorithms for control of singleton attractors in BN. We theoretically estimate the average-case time complexities of the proposed algorithms, and confirm them by computer experiments. The results suggest the importance of gene ordering. Especially, setting internal nodes ahead yields shorter computational time than setting external nodes ahead in various types of algorithms. We also present a heuristic algorithm which does not look for the optimal solution but for the solution whose computational time is shorter than that of the exact algorithms.     

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

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.

     

Floral morphogenesis: stochastic explorations of a gene network epigenetic landscape

In contrast to the classical view of development as a preprogrammed and deterministic process, recent studies have demonstrated that stochastic perturbations of highly non-linear systems may underlie the emergence and stability of biological patterns. Herein, we address the question of whether noise contributes to the generation of the stereotypical temporal pattern in gene expression during flower development. We modeled the regulatory network of organ identity genes in the Arabidopsis thaliana flower as a stochastic system. This network has previously been shown to converge to ten fixed-point attractors, each with gene expression arrays that characterize inflorescence cells and primordial cells of sepals, petals, stamens, and carpels. The network used is binary, and the logical rules that govern its dynamics are grounded in experimental evidence. We introduced different levels of uncertainty in the updating rules of the network. Interestingly, for a level of noise of around 0.5-10%, the system exhibited a sequence of transitions among attractors that mimics the sequence of gene activation configurations observed in real flowers. We also implemented the gene regulatory network as a continuous system using the Glass model of differential equations, that can be considered as a first approximation of kinetic-reaction equations, but which are not necessarily equivalent to the Boolean model. Interestingly, the Glass dynamics recover a temporal sequence of attractors, that is qualitatively similar, although not identical, to that obtained using the Boolean model. Thus, time ordering in the emergence of cell-fate patterns is not an artifact of synchronous updating in the Boolean model. Therefore, our model provides a novel explanation for the emergence and robustness of the ubiquitous temporal pattern of floral organ specification. It also constitutes a new approach to understanding morphogenesis, providing predictions on the population dynamics of cells with different genetic configurations during development.     

Inferring gene regulatory networks with time delays using a genetic algorithm

Recently a state-space model with time delays for inferring gene regulatory networks was proposed. It was assumed that each regulation between two internal state variables had multiple time delays. This assumption caused underestimation of the model with many current gene expression datasets. In biological reality, one regulatory relationship may have just a single time delay, and not multiple time delays. This study employs Boolean variables to capture the existence of the time-delayed regulatory relationships in gene regulatory networks in terms of the state-space model. As the solution space of time delayed relationships is too large for an exhaustive search, a genetic algorithm (GA) is proposed to determine the optimal Boolean variables (the optimal time-delayed regulatory relationships). Coupled with the proposed GA, Bayesian information criterion (BIC) and probabilistic principle component analysis (PPCA) are employed to infer gene regulatory networks with time delays. Computational experiments are performed on two real gene expression datasets. The results show that the GA is effective at finding time-delayed regulatory relationships. Moreover, the inferred gene regulatory networks with time delays from the datasets improve the prediction accuracy and possess more of the expected properties of a real network, compared to a gene regulatory network without time delays.     

A framework to find the logic backbone of a biological network

Background

Cellular behaviors are governed by interaction networks among biomolecules, for example gene regulatory and signal transduction networks. An often used dynamic modeling framework for these networks, Boolean modeling, can obtain their attractors (which correspond to cell types and behaviors) and their trajectories from an initial state (e.g. a resting state) to the attractors, for example in response to an external signal. The existing methods however do not elucidate the causal relationships between distant nodes in the network.

Results

In this work, we propose a simple logic framework, based on categorizing causal relationships as sufficient or necessary, as a complement to Boolean networks. We identify and explore the properties of complex subnetworks that are distillable into a single logic relationship. We also identify cyclic subnetworks that ensure the stabilization of the state of participating nodes regardless of the rest of the network. We identify the logic backbone of biomolecular networks, consisting of external signals, self-sustaining cyclic subnetworks (stable motifs), and output nodes. Furthermore, we use the logic framework to identify crucial nodes whose override can drive the system from one steady state to another. We apply these techniques to two biological networks: the epithelial-to-mesenchymal transition network corresponding to a developmental process exploited in tumor invasion, and the network of abscisic acid induced stomatal closure in plants. We find interesting subnetworks with logical implications in these networks. Using these subgraphs and motifs, we efficiently reduce both networks to succinct backbone structures.

Conclusions

The logic representation identifies the causal relationships between distant nodes and subnetworks. This knowledge can form the basis of network control or used in the reverse engineering of networks.

     

An approximation method for solving the steady-state probability distribution of probabilistic Boolean networks

MOTIVATION: Probabilistic Boolean networks (PBNs) have been proposed to model genetic regulatory interactions. The steady-state probability distribution of a PBN gives important information about the captured genetic network. The computation of the steady-state probability distribution usually includes construction of the transition probability matrix and computation of the steady-state probability distribution. The size of the transition probability matrix is 2(n)-by-2(n) where n is the number of genes in the genetic network. Therefore, the computational costs of these two steps are very expensive and it is essential to develop a fast approximation method. RESULTS: In this article, we propose an approximation method for computing the steady-state probability distribution of a PBN based on neglecting some Boolean networks (BNs) with very small probabilities during the construction of the transition probability matrix. An error analysis of this approximation method is given and theoretical result on the distribution of BNs in a PBN with at most two Boolean functions for one gene is also presented. These give a foundation and support for the approximation method. Numerical experiments based on a genetic network are given to demonstrate the efficiency of the proposed method.     

Inferring qualitative relations in genetic networks and metabolic pathways

MOTIVATION: Inferring genetic network architecture from time series data of gene expression patterns is an important topic in bioinformatics. Although inference algorithms based on the Boolean network were proposed, the Boolean network was not sufficient as a model of a genetic network. RESULTS: First, a Boolean network model with noise is proposed, together with an inference algorithm for it. Next, a qualitative network model is proposed, in which regulation rules are represented as qualitative rules and embedded in the network structure. Algorithms are also presented for inferring qualitative relations from time series data. Then, an algorithm for inferring S-systems (synergistic and saturable systems) from time series data is presented, where S-systems are based on a particular kind of nonlinear differential equation and have been applied to the analysis of various biological systems. Theoretical results are shown for Boolean networks with noises and simple qualitative networks. Computational results are shown for Boolean networks with noises and S-systems, where real data are not used because the proposed models are still conceptual and the quantity and quality of currently available data are not enough for the application of the proposed methods.     

Generating Boolean networks with a prescribed attractor structure

MOTIVATION: Dynamical modeling of gene regulation via network models constitutes a key problem for genomics. The long-run characteristics of a dynamical system are critical and their determination is a primary aspect of system analysis. In the other direction, system synthesis involves constructing a network possessing a given set of properties. This constitutes the inverse problem. Generally, the inverse problem is ill-posed, meaning there will be many networks, or perhaps none, possessing the desired properties. Relative to long-run behavior, we may wish to construct networks possessing a desirable steady-state distribution. This paper addresses the long-run inverse problem pertaining to Boolean networks (BNs). RESULTS: The long-run behavior of a BN is characterized by its attractors. The rest of the state transition diagram is partitioned into level sets, the j-th level set being composed of all states that transition to one of the attractor states in exactly j transitions. We present two algorithms for the attractor inverse problem. The attractors are specified, and the sizes of the predictor sets and the number of levels are constrained. Algorithm complexity and performance are analyzed. The algorithmic solutions have immediate application. Under the assumption that sampling is from the steady state, a basic criterion for checking the validity of a designed network is that there should be concordance between the attractor states of the model and the data states. This criterion can be used to test a design algorithm: randomly select a set of states to be used as data states; generate a BN possessing the selected states as attractors, perhaps with some added requirements such as constraints on the number of predictors and the level structure; apply the design algorithm; and check the concordance between the attractor states of the designed network and the data states. AVAILABILITY: The software and supplementary material is available at http://gsp.tamu.edu/Publications/BNs/bn.htm