Dimension Reduction of Large Sparse AND-NOT Network Models

Finding the fixed points of a Boolean network from a positive feedback vertex set

MOTIVATION

In the modeling of biological systems by Boolean networks, a key problem is finding the set of fixed points of a given network. Some constructed algorithms consider certain structural properties of the regulatory graph like those proposed by Akutsu et al. and Zhang et al., which consider a feedback vertex set of the graph. However, these methods do not take into account the type of action (activation and inhibition) between its components.

RESULTS

In this article, we propose a new algorithm for finding the set of fixed points of a Boolean network, based on a positive feedback vertex set P of its regulatory graph and which works, by applying a sequential update schedule, in time O(2|P|·n2+k), where n is the number of components and the regulatory functions of the network can be evaluated in time O(nk), k≥0. The theoretical foundation of this algorithm is due a nice characterization, that we give, of the dynamical behavior of the Boolean networks without positive cycles and with a fixed point.

AVAILABILITY AND IMPLEMENTATION

An executable file of FixedPoint algorithm made in Java and some examples of input files are available at: www.inf.udec.cl/˜lilian/FPCollector/.

SUPPLEMENTARY INFORMATION

Supplementary material is available at Bioinformatics online.

Steady-State Design of Large-Dimensional Boolean Networks

Analysis and design of steady states representing cell types, such as cell death or unregulated growth, are of significant interest in modeling genetic regulatory networks. In this article, the steady-state design of large-dimensional Boolean networks (BNs) is studied via model reduction and pinning control. Compared with existing literature, the pinning control design in this article is based on the original node's connection, but not on the state-transition matrix of BNs. Hence, the computational complexity is dramatically reduced in this article from O(2ⁿ×2ⁿ) to O(2×2^r) , where n is the number of nodes in the large-dimensional BN and is the largest number of in-neighbors of the reduced BN. Finally, the proposed method is well demonstrated by a T-LGL survival signaling network with 18 nodes and a model of survival signaling in large granular lymphocyte leukemia with 29 nodes. Just as shown in the simulations, the model reduction method reduces 99.98% redundant states for the network with 18 nodes, and 99.99% redundant states for the network with 29 nodes.

Boolean network analysis through the joint use of linear algebra and algebraic geometry

Among the various phenomena that can be modeled by Boolean networks, i.e., discrete-time dynamical systems with binary state variables, gene regulatory interactions are especially well known. Therefore, the analysis of Boolean networks is critical, e.g., to identify genetic pathways and to predict the effects of mutations on the cell functionality. Two methodologies (i.e., the semi-tensor product and the Gröbner bases over finite fields) have recently been proposed to tackle the problem of determining cycles and attractors (with the corresponding basin of attraction) for such systems. Here, it is shown that, by suitably coupling methodologies taken from these two fields (i.e., linear algebra and algebraic geometry), it is not only possible to determine cycles and attractors, but also to find closed-form solutions of the Boolean network. Such a goal is pursued by finding an immersion that recasts the Boolean dynamics in a linear form and by computing the closed-form solution of the latter system. The effectiveness of this technique is demonstrated by fully computing the solutions of the Boolean network modeling the differentiation of the Th-lymphocyte, a type of white blood cells involved in the human adaptive immune system.

Modeling hormonal control of cambium proliferation

Rise of atmospheric CO₂ is one of the main causes of global warming. Catastrophic climate change can be avoided by reducing emissions and increasing sequestration of CO₂. Trees are known to sequester CO₂ during photosynthesis, and then store it as wood biomass. Thus, breeding of trees with higher wood yield would mitigate global warming as well as augment production of renewable construction materials, energy, and industrial feedstock. Wood is made of cellulose-rich xylem cells produced through proliferation of a specialized stem cell niche called cambium. Importance of cambium in xylem cells production makes it an ideal target for the tree breeding programs; however our knowledge about control of cambium proliferation remains limited. The morphology and regulation of cambium are different from those of stem cell niches that control axial growth. For this reason, translating the knowledge about axial growth to radial growth has limited use. Furthermore, genetic approaches cannot be easily applied because overlaying tissues conceal cambium from direct observation and complicate identification of mutants. To overcome the paucity of experimental tools in cambium biology, we constructed a Boolean network CARENET (CAmbium REgulation gene NETwork) for modelling cambium activity, which includes the key transcription factors WOX4 and HD-ZIP III as well as their potential regulators. Our simulations predict that: (1) auxin, cytokinin, gibberellin, and brassinosteroids act cooperatively in promoting transcription of WOX4 and HD-ZIP III; (2) auxin and cytokinin pathways negatively regulate each other; (3) hormonal pathways act redundantly in sustaining cambium activity; (4) individual cambium cells can have diverse molecular identities. CARENET can be extended to include components of other signalling pathways and be integrated with models of xylem and phloem differentiation. Such extended models would facilitate breeding trees with higher wood yield.

An efficient approach of attractor calculation for large-scale Boolean gene regulatory networks

Boolean network models provide an efficient way for studying gene regulatory networks. The main dynamics of a Boolean network is determined by its attractors. Attractor calculation plays a key role for analyzing Boolean gene regulatory networks. An approach of attractor calculation was proposed in this study, which improved the predecessor-based approach. Furthermore, the proposed approach combined with the identification of constant nodes and simplified Boolean networks to accelerate attractor calculation. The proposed algorithm is effective to calculate all attractors for large-scale Boolean gene regulatory networks. If the average degree of the network is not too large, the algorithm can get all attractors of a Boolean network with dozens or even hundreds of nodes.

Logical Reduction of Biological Networks to Their Most Determinative Components

Boolean networks have been widely used as models for gene regulatory networks, signal transduction networks, or neural networks, among many others. One of the main difficulties in analyzing the dynamics of a Boolean network and its sensitivity to perturbations or mutations is the fact that it grows exponentially with the number of nodes. Therefore, various approaches for simplifying the computations and reducing the network to a subset of relevant nodes have been proposed in the past few years. We consider a recently introduced method for reducing a Boolean network to its most determinative nodes that yield the highest information gain. The determinative power of a node is obtained by a summation of all mutual information quantities over all nodes having the chosen node as a common input, thus representing a measure of information gain obtained by the knowledge of the node under consideration. The determinative power of nodes has been considered in the literature under the assumption that the inputs are independent in which case one can use the Bahadur orthonormal basis. In this article, we relax that assumption and use a standard orthonormal basis instead. We use techniques of Hilbert space operators and harmonic analysis to generate formulas for the sensitivity to perturbations of nodes, quantified by the notions of influence, average sensitivity, and strength. Since we work on finite-dimensional spaces, our formulas and estimates can be and are formulated in plain matrix algebra terminology. We analyze the determinative power of nodes for a Boolean model of a signal transduction network of a generic fibroblast cell. We also show the similarities and differences induced by the alternative complete orthonormal basis used. Among the similarities, we mention the fact that the knowledge of the states of the most determinative nodes reduces the entropy or uncertainty of the overall network significantly. In a special case, we obtain a stronger result than in previous works, showing that a large information gain from a set of input nodes generates increased sensitivity to perturbations of those inputs.

Boolean modeling techniques for protein co-expression networks in systems medicine

INTRODUCTION

Application of systems biology/systems medicine approaches is promising for proteomics/biomedical research, but requires selection of an adequate modeling type.

AREAS COVERED

This article reviews the existing Boolean network modeling approaches, which provide in comparison with alternative modeling techniques several advantages for the processing of proteomics data. Application of methods for inference, reduction and validation of protein co-expression networks that are derived from quantitative high-throughput proteomics measurements is presented. It's also shown how Boolean models can be used to derive system-theoretic characteristics that describe both the dynamical behavior of such networks as a whole and the properties of different cell states (e.g. healthy or diseased cell states). Furthermore, application of methods derived from control theory is proposed in order to simulate the effects of therapeutic interventions on such networks, which is a promising approach for the computer-assisted discovery of biomarkers and drug targets. Finally, the clinical application of Boolean modeling analyses is discussed. Expert commentary: Boolean modeling of proteomics data is still in its infancy. Progress in this field strongly depends on provision of a repository with public access to relevant reference models. Also required are community supported standards that facilitate input of both proteomics and patient related data (e.g. age, gender, laboratory results, etc.).