Geometric interpretation of gene coexpression network analysis

THE MERGING OF NETWORK THEORY AND MICROARRAY DATA ANALYSIS TECHNIQUES HAS SPAWNED A NEW FIELD: gene coexpression network analysis. While network methods are increasingly used in biology, the network vocabulary of computational biologists tends to be far more limited than that of, say, social network theorists. Here we review and propose several potentially useful network concepts. We take advantage of the relationship between network theory and the field of microarray data analysis to clarify the meaning of and the relationship among network concepts in gene coexpression networks. Network theory offers a wealth of intuitive concepts for describing the pairwise relationships among genes, which are depicted in cluster trees and heat maps. Conversely, microarray data analysis techniques (singular value decomposition, tests of differential expression) can also be used to address difficult problems in network theory. We describe conditions when a close relationship exists between network analysis and microarray data analysis techniques, and provide a rough dictionary for translating between the two fields. Using the angular interpretation of correlations, we provide a geometric interpretation of network theoretic concepts and derive unexpected relationships among them. We use the singular value decomposition of module expression data to characterize approximately factorizable gene coexpression networks, i.e., adjacency matrices that factor into node specific contributions. High and low level views of coexpression networks allow us to study the relationships among modules and among module genes, respectively. We characterize coexpression networks where hub genes are significant with respect to a microarray sample trait and show that the network concept of intramodular connectivity can be interpreted as a fuzzy measure of module membership. We illustrate our results using human, mouse, and yeast microarray gene expression data. The unification of coexpression network methods with traditional data mining methods can inform the application and development of systems biologic methods.     

Estimating genomic coexpression networks using first-order conditional independence

We describe a computationally efficient statistical framework for estimating networks of coexpressed genes. This framework exploits first-order conditional independence relationships among gene-expression measurements to estimate patterns of association. We use this approach to estimate a coexpression network from microarray gene-expression measurements from Saccharomyces cerevisiae. We demonstrate the biological utility of this approach by showing that a large number of metabolic pathways are coherently represented in the estimated network. We describe a complementary unsupervised graph search algorithm for discovering locally distinct subgraphs of a large weighted graph. We apply this algorithm to our coexpression network model and show that subgraphs found using this approach correspond to particular biological processes or contain representatives of distinct gene families.     

B-cell ligand processing pathways detected by large-scale comparative analysis

The initiation of B-cell ligand recognition is a critical step for the generation of an immune response against foreign bodies.We sought to identify the biochemical pathways involved in the B-cell ligand recognition cascade and sets of ligands that trigger similar immunological responses.We utilized several comparative approaches to analyze the gene coexpression networks generated from a set of microarray experiments spanning 33 different ligands.First,we compared the degree distributions of the generated networks.Second,we utilized a pairwise network alignment algorithm,BiNA,to align the networks based on the hubs in the networks.Third,we aligned the networks based on a set of K_EGG pathways.We summarized our results by constructing a consensus hierarchy of pathways that are involved in B cell ligand recognition.The resulting pathways were further validated through literature for their common physiological responses.Collectively,the results based on our comparative analyses of degree distributions,alignment of hubs,and alignment based on KEGG pathways provide a basis for molecular characterization of the immune response states of B-cells and demonstrate the power of comparative approaches(e.g.,gene coexpression network alignment algorithms) in elucidating biochemical pathways involved in complex signaling events in cells.     

Path matching and graph matching in biological networks.

We develop algorithms for the following path matching and graph matching problems: (i) given a query path p and a graph G, find a path p' that is most similar to p in G; (ii) given a query graph G (0) and a graph G, find a graph G (0)' that is most similar to G (0) in G. In these problems, p and G (0) represent a given substructure of interest to a biologist, and G represents a large network in which the biologist desires to find a related substructure. These algorithms allow the study of common substructures in biological networks in order to understand how these networks evolve both within and between organisms. We reduce the path matching problem to finding a longest weighted path in a directed acyclic graph and show that the problem of finding top k suboptimal paths can be solved in polynomial time. This is in contrast with most previous approaches that used exponential time algorithms to find simple paths which are practical only when the paths are short. We reduce the graph matching problem to finding highest scoring subgraphs in a graph and give an exact algorithm to solve the problem when the query graph G (0) is of moderate size. This eliminates the need for less accurate heuristic or randomized algorithms.We show that our algorithms are able to extract biologically meaningful pathways from protein interaction networks in the DIP database and metabolic networks in the KEGG database. Software programs implementing these techniques (PathMatch and GraphMatch) are available at http://faculty.cs.tamu.edu/shsze/pathmatch and http://faculty.cs.tamu.edu/shsze/graphmatch.     

Annotating gene function by combining expression data with a modular gene network

MOTIVATION: A promising and reliable approach to annotate gene function is clustering genes not only by using gene expression data but also literature information, especially gene networks. RESULTS: We present a systematic method for gene clustering by combining these totally different two types of data, particularly focusing on network modularity, a global feature of gene networks. Our method is based on learning a probabilistic model, which we call a hidden modular random field in which the relation between hidden variables directly represents a given gene network. Our learning algorithm which minimizes an energy function considering the network modularity is practically time-efficient, regardless of using the global network property. We evaluated our method by using a metabolic network and microarray expression data, changing with microarray datasets, parameters of our model and gold standard clusters. Experimental results showed that our method outperformed other four competing methods, including k-means and existing graph partitioning methods, being statistically significant in all cases. Further detailed analysis showed that our method could group a set of genes into a cluster which corresponds to the folate metabolic pathway while other methods could not. From these results, we can say that our method is highly effective for gene clustering and annotating gene function.     

Analysis of Gene Coexpression by B-Spline Based CoD Estimation

The gene coexpression study has emerged as a novel holistic approach for microarray data analysis. Different indices have been used in exploring coexpression relationship, but each is associated with certain pitfalls. The Pearson's correlation coefficient, for example, is not capable of uncovering nonlinear pattern and directionality of coexpression. Mutual information can detect nonlinearity but fails to show directionality. The coefficient of determination (CoD) is unique in exploring different patterns of gene coexpression, but so far only applied to discrete data and the conversion of continuous microarray data to the discrete format could lead to information loss. Here, we proposed an effective algorithm, CoexPro, for gene coexpression analysis. The new algorithm is based on B-spline approximation of coexpression between a pair of genes, followed by CoD estimation. The algorithm was justified by simulation studies and by functional semantic similarity analysis. The proposed algorithm is capable of uncovering both linear and a specific class of nonlinear relationships from continuous microarray data. It can also provide suggestions for possible directionality of coexpression to the researchers. The new algorithm presents a novel model for gene coexpression and will be a valuable tool for a variety of gene expression and network studies. The application of the algorithm was demonstrated by an analysis on ligand-receptor coexpression in cancerous and noncancerous cells. The software implementing the algorithm is available upon request to the authors.     

Local coexpression domains of two to four genes in the genome of Arabidopsis

Expression of genes in eukaryotic genomes is known to cluster, but cluster size is generally loosely defined and highly variable. We have here taken a very strict definition of cluster as sets of physically adjacent genes that are highly coexpressed and form so-called local coexpression domains. The Arabidopsis (Arabidopsis thaliana) genome was analyzed for the presence of such local coexpression domains to elucidate its functional characteristics. We used expression data sets that cover different experimental conditions, organs, tissues, and cells from the Massively Parallel Signature Sequencing repository and microarray data (Affymetrix) from a detailed root analysis. With these expression data, we identified 689 and 1,481 local coexpression domains, respectively, consisting of two to four genes with a pairwise Pearson's correlation coefficient larger than 0.7. This number is approximately 1- to 5-fold higher than the numbers expected by chance. A small (5%-10%) yet significant fraction of genes in the Arabidopsis genome is therefore organized into local coexpression domains. These local coexpression domains were distributed over the genome. Genes in such local domains were for the major part not categorized in the same functional category (GOslim). Neither tandemly duplicated genes nor shared promoter sequence nor gene distance explained the occurrence of coexpression of genes in such chromosomal domains. This indicates that other parameters in genes or gene positions are important to establish coexpression in local domains of Arabidopsis chromosomes.     

A polynomial time solvable formulation of multiple sequence alignment.

Since traditional multiple alignment formulations are NP-hard, heuristics are commonly employed to find acceptable alignments with no guaranteed performance bound. This causes a substantial difficulty in understanding what the resulting alignment means and in assessing the quality of these alignments. We propose an alternative formulation of multiple alignment based on the idea of finding a multiple alignment of k sequences which preserves k - 1 pairwise alignments as specified by edges of a given tree. Although it is well known that such a preserving alignment always exists, it did not become a mainstream method for multiple alignment since it seems that a lot of information is lost from ignoring pairwise similarities outside the tree. In contrast, by using pairwise alignments that incorporate consistency information from other sequences, we show that it is possible to obtain very good accuracy with the preserving alignment formulation. We show that a reasonable objective function to use is to find the shortest preserving alignment, and, by a reduction to a graph-theoretic problem, that the problem of finding the shortest preserving multiple alignment can be solved in polynomial time. We demonstrate the success of this approach on three sets of benchmark multiple alignments by using consistency-based pairwise alignments from the first stage of two of the best performing progressive alignment algorithms TCoffee and ProbCons and replace the second heuristic progressive step of these algorithms by the exact preserving alignment step. We apply this strategy to TCoffee and show that our approach outperforms TCoffee on two of the three test sets. We apply the strategy to a variant of ProbCons with no iterative refinements and show that our approach achieves similar or better accuracy except on one test set. We also compare our performance to ProbCons with iterative refinements and show that our approach achieves similar or better accuracy on many subcategories even without further refinements. The most important advantage of the preserving alignment formulation is that we are certain that we can solve the problem in polynomial time without using a heuristic. A software program implementing this approach (PSAlign) is available at http://faculty.cs.tamu.edu/shsze/psalign.     

Kernel hierarchical gene clustering from microarray expression data

MOTIVATION: Unsupervised analysis of microarray gene expression data attempts to find biologically significant patterns within a given collection of expression measurements. For example, hierarchical clustering can be applied to expression profiles of genes across multiple experiments, identifying groups of genes that share similar expression profiles. Previous work using the support vector machine supervised learning algorithm with microarray data suggests that higher-order features, such as pairwise and tertiary correlations across multiple experiments, may provide significant benefit in learning to recognize classes of co-expressed genes. RESULTS: We describe a generalization of the hierarchical clustering algorithm that efficiently incorporates these higher-order features by using a kernel function to map the data into a high-dimensional feature space. We then evaluate the utility of the kernel hierarchical clustering algorithm using both internal and external validation. The experiments demonstrate that the kernel representation itself is insufficient to provide improved clustering performance. We conclude that mapping gene expression data into a high-dimensional feature space is only a good idea when combined with a learning algorithm, such as the support vector machine that does not suffer from the curse of dimensionality. AVAILABILITY: Supplementary data at www.cs.columbia.edu/compbio/hiclust. Software source code available by request.     

Metabolic networks are NP-hard to reconstruct

High-throughput data from various omics and sequencing techniques have rendered the automated metabolic network reconstruction a highly relevant problem. Our approach reflects the inherent probabilistic nature of the steps involved in metabolic network reconstruction. Here, the goal is to arrive at networks which combine probabilistic information with the possibility to obtain a small number of disconnected network constituents by reduction of a given preliminary probabilistic metabolic network. We define automated metabolic network reconstruction as an optimization problem on four-partite graph (nodes representing genes, enzymes, reactions, and metabolites) which integrates: (1) probabilistic information obtained from the existing process for metabolic reconstruction from a given genome, (2) connectedness of the raw metabolic network, and (3) clustering of components in the reconstructed metabolic network. The practical implications of our theoretical analysis refer to the quality of reconstructed metabolic networks and shed light on the problem of finding more efficient and effective methods for automated reconstruction. Our main contributions include: a completeness result for the defined problem, polynomial-time approximation algorithm, and an optimal polynomial-time algorithm for trees. Moreover, we exemplify our approach by the reconstruction of the sucrose biosynthesis pathway in Chlamydomonas reinhardtii.     

A network perspective on the topological importance of enzymes and their phylogenetic conservation

Background  

A metabolic network is the sum of all chemical transformations or reactions in the cell, with the metabolites being interconnected by enzyme-catalyzed reactions. Many enzymes exist in numerous species while others occur only in a few. We ask if there are relationships between the phylogenetic profile of an enzyme, or the number of different bacterial species that contain it, and its topological importance in the metabolic network. Our null hypothesis is that phylogenetic profile is independent of topological importance. To test our null hypothesis we constructed an enzyme network from the KEGG (Kyoto Encyclopedia of Genes and Genomes) database. We calculated three network indices of topological importance: the degree or the number of connections of a network node; closeness centrality, which measures how close a node is to others; and betweenness centrality measuring how frequently a node appears on all shortest paths between two other nodes.     

Extracting three-way gene interactions from microarray data

MOTIVATION: It is an important and difficult task to extract gene network information from high-throughput genomic data. A common approach is to cluster genes using pairwise correlation as a distance metric. However, pairwise correlation is clearly too simplistic to describe the complex relationships among real genes since co-expression relationships are often restricted to a specific set of biological conditions/processes. In this study, we described a three-way gene interaction model that captures the dynamic nature of co-expression relationship between a gene pair through the introduction of a controller gene. RESULTS: We surveyed 0.4 billion possible three-way interactions among 1000 genes in a microarray dataset containing 678 human cancer samples. To test the reproducibility and statistical significance of our results, we randomly split the samples into a training set and a testing set. We found that the gene triplets with the strongest interactions (i.e. with the smallest P-values from appropriate statistical tests) in the training set also had the strongest interactions in the testing set. A distinctive pattern of three-way interaction emerged from these gene triplets: depending on the third gene being expressed or not, the remaining two genes can be either co-expressed or mutually exclusive (i.e. expression of either one of them would repress the other). Such three-way interactions can exist without apparent pairwise correlations. The identified three-way interactions may constitute candidates for further experimentation using techniques such as RNA interference, so that novel gene network or pathways could be identified.