 Methodology article
 Open Access
Inferring gene regression networks with model trees
 Isabel A NepomucenoChamorro^{1}Email author,
 Jesus S AguilarRuiz^{2}Email author and
 Jose C Riquelme^{1}
https://doi.org/10.1186/1471210511517
© NepomucenoChamorro et al; licensee BioMed Central Ltd. 2010
 Received: 30 April 2010
 Accepted: 15 October 2010
 Published: 15 October 2010
Abstract
Background
Novel strategies are required in order to handle the huge amount of data produced by microarray technologies. To infer gene regulatory networks, the first step is to find direct regulatory relationships between genes building the socalled gene coexpression networks. They are typically generated using correlation statistics as pairwise similarity measures. Correlationbased methods are very useful in order to determine whether two genes have a strong global similarity but do not detect local similarities.
Results
We propose model trees as a method to identify gene interaction networks. While correlationbased methods analyze each pair of genes, in our approach we generate a single regression tree for each gene from the remaining genes. Finally, a graph from all the relationships among output and input genes is built taking into account whether the pair of genes is statistically significant. For this reason we apply a statistical procedure to control the false discovery rate. The performance of our approach, named REG NET, is experimentally tested on two wellknown data sets: Saccharomyces Cerevisiae and E.coli data set. First, the biological coherence of the results are tested. Second the E.coli transcriptional network (in the Regulon database) is used as control to compare the results to that of a correlationbased method. This experiment shows that REG NET performs more accurately at detecting true gene associations than the Pearson and Spearman zeroth and firstorder correlationbased methods.
Conclusions
REG NET generates gene association networks from gene expression data, and differs from correlationbased methods in that the relationship between one gene and others is calculated simultaneously. Model trees are very useful techniques to estimate the numerical values for the target genes by linear regression functions. They are very often more precise than linear regression models because they can add just different linear regressions to separate areas of the search space favoring to infer localized similarities over a more global similarity. Furthermore, experimental results show the good performance of REG NET.
Keywords
 Model Tree
 Benchmark Suite
 Global Similarity
 Linear Regression Function
 Gaussian Graphical Model
Background
In the area of microarray data analysis, inferring genegene interactions involved in biological function is a relevant task. Over the past few years several statistical and machine learning techniques have been proposed to carry out the inferring task of genegene interactions or gene regulatory networks. Clustering algorithm represents one of the first approaches to support the identification of regulatory modules [1, 2]. These approaches are motivated by a simple idea which is still widely used in functional genomic. It is called the guiltbyassociation heuristic: coexpression means coregulation, i.e. if two genes show similar expression profiles, they are supposed to follow the same regulatory regime.
In order to formalize the idea of similar expression, several statistical measures have been proposed as solution. In correlation methods, interactions are inferred using correlation statistics as pairwise similarity measures between gene expression profiles over multiple conditions, as for example in [3]. In this kind of methods, if the correlation between gene pairs is higher than a threshold value, then it is considered that these gene pairs interact directly in a signaling pathway and are relevant in a biological way [4–6]. These methods build gene coexpression networks, also known as gene association, gene interaction or gene relevance networks. These networks provide a framework for assigning biological function to group of genes as it was argued in [7]. Correlation coefficient is widely used as a way of obtaining an association measure between two random variables but does not provide a causal measure between them. However, correlation is still informative about the underlying structure [8]. The causal properties that can be inferred from correlations have been investigated in [9, 10].
Correlationbased methods are very useful to determine whether two genes have a strong global similarity over all conditions from the data set. This is an important constrain as there might exist a strong local similarity over a subset of conditions, which could not be detected with global similarity measures. In addition, many pairs of genes show similar behavior in gene expression profiles by chance even though they are not biologically related [11], i.e. the significance of the results should be assessed in interaction networks.
On the other hand, Gaussian graphical models (GGM) are a full conditional independence model. These models try to explain the correlation between two genes by the rest of the genes and they are a popular tool to represent gene association network [8, 12, 13]. Recently, [14] has proposed estimating partial correlations to attach lengths to the edges of the GGM, where the length of an edge is inversely related to the partial correlation between a gene pair. As a drawback, these models are hard to estimate if the number of samples is small compared to the number of variables. In contrast to GGMs, other models try to explain the correlation between two genes not by the rest of the genes, but only by single third genes. This idea can also be implemented using sparse Gaussian graphical model based on partial correlation [15] or conditional mutual information to test for firstorder independence [16–18].
Bayesian networks try to explain the dependence between genes if there are no subset of other genes that explain the dependency [19]. An example of Bayesian networks can be found in [20] where a stochastic expectation and maximization algorithm is used to learn a probabilistic model, and regression trees are used to learn graph topologies that maximize Bayesian scores. Recently, [21] has revised the approach before using an ensemble method, and [22] has incorporated prior knowledge from literature on Bayesian networks. Also, several approaches have been developed to build Boolean networks [23], or to infer regulatory rules [24, 25] using machine learning principles.
In this paper, we present a novel method inspired by model trees as a way to detect linear dependencies between genes and to set a group of genegene dependencies. From that set, our method provides as genegene interactions all those significant dependencies in a statistical sense. Then, it builds undirected dependency graphs (UDGs) from these genegene interactions. Furthermore, our method analyzes which dependencies between genes are considered as a discovery by means of the Benjamini and Yekutieli procedure [26]. This statistical procedure enables the control of the expected proportion of false discoveries among all the discoveries made. One of the main contributions of our approach is that it addresses the issue of searching for local similarities arising from conditional regulatory relationships instead of global similarities.
The remainder of this paper is organized as follows. In Section Method, a detailed explanation of the methodology and the algorithm are presented. In Section Results and Discussion, experimental results tested on an in silico benchmark suite of datasets, yeast and E.coli data are provided. Finally, Section Conclusions summarizes the most relevant conclusions and future research directions.
Method
Correlation methods are focused on the global match of two gene expression profiles, analyzing each possible pair of genes. Instead, our approach analyzes each gene in an iterative way. At each iteration a gene is taken as target gene and the remaining genes as input for splitting the search space. In each subspace generated by that division, a linear model is built to identify a linear dependency between the target gene and a subgroup of genes, i.e. the target gene expression values are estimated by this subgroup of genes involved in that linear model. As a consequence, the dependency between the genes is not calculated for the complete gene expression profile, but for a localized subspace of the profiles using M5' model tree algorithm.
The second step implies the extraction of the set of genegene dependencies from the forest of trees obtained by the previous step. Specifically, our approach considers which hypothetical evidences of genegene dependency exist between the target gene and every gene participating in the linear regression functions of the target gene.
Finally, the third step involves learning a graph model of gene coexpression network by assessing the significance of the set of hypothetical evidences. Many sets of genes show similar behavior in expression profile by chance even though they do not share the same biological function. Therefore, the aim of this step is to minimize the number of false discoveries among all those discoveries made in the previous step. For that reason, we apply a statistical procedure to control the false discovery rate instead of the increase of type I error when a family of hypotheses is being tested simultaneously. The reliability of our method is strengthened by applying the BenjaminiYekutieli statistical procedure to assess the significance of the results.
Building model trees
The first work on regression trees dates from [28], although the most popular reference is the seminal work of [29]. Later on, [30] introduced the system M5. It builds multivariate trees using linear regression functions at the leaves. M5' is introduced in [31], a rational reconstruction of Quinlan's M5 algorithm. Throughout the description of model tree, we will refer to gene as attribute, and sample as instance space.
Extracting genegene dependencies
This step extracts a set of dependencies between the target gene and the genes involved in the linear regression functions from each tree. Correlationbased methods extract genegene dependencies by computing a similarity score for each pair of genes. These methods are based on the assumption that two genes show similar expression profiles if they follow the same regulatory regime, i.e. coexpression hints at coregulation [11]. Our approach analyzes each gene as a target by taking into account the remaining genes as inputs to obtain linear models that estimate the expression value of that target gene. We assume that the genes involved in these linear models control or influence the target expression value and they follow the same regulatory regime. This influence can be explained when several genes fit a specific area of the space, which leads to an evidence for dependency.
Let LM be a multivariate linear model of a M5' tree defined by $LM:{g}_{x}={\displaystyle {\sum}_{i}{\lambda}_{i}{g}_{{y}_{i}}}$, where g_{ x }belongs to the set of target genes, ${g}_{{y}_{i}}$, is a gene involved in the linear regression that belongs to the set of genes, and λ_{ i }is a coefficient of the linear model. Our approach considers that an hypothetical evidence of dependency or expression pattern exists between g_{ x }and every ${g}_{{y}_{i}}$, which will be statistically tested in the next step.
The output of this step is a set of genegene dependencies (Q in Figure 2) that are potential interactions for the problem under study.
Building the gene regression network
After obtaining the set of genegene interactions, the significance of these results must be assessed. The authors in [32] have shown that for microarrays studies, the expected proportion of false discoveries among all the discoveries made (socalled false discovery rate, FDR) is more important than the low number of false discoveries or the small probability of making at least one false positive (calculated by means of adjustments of pvalues). For this reason we apply a statistical procedure in order to control the number of type I errors (connections inferred which do not correspond to a connection in the real network, also called false positives) among the number of discoveries when a family of hypotheses is tested simultaneously.
Once the set of genegene dependencies (D) has been provided, our approach builds a graph Q of interactions defined as a tuple (N, E) of N nodes and E edges. We will denote by g_{ x }~g_{ y }an hypothetical genegene dependency. Our approach takes several g_{ x }~g_{ y }from D and the genes g_{ x }and g_{ y }are mapped as two nodes in the set of nodes N, and the dependency is mapped as an edge of the set E. This step, to decide which g_{ x }~g_{ y }is mapped onto an edge, i.e. which dependency is considered as a discovery, is carried out by means of the BenjaminiYekutieli (BY) procedure.
If no such i exists, none of the hypotheses will be rejected. This procedure controls the proportion of false discoveries (FDR) among all the discoveries.
In this context, we will say that g_{ x }~g_{ y }is not an interaction in Q* if and only if there is not any significant monotonic relationship between the two variables, i.e. H_{0} : ρ_{ xy }≈ 0 (where ρ is a correlation measure), taking into account the subspace of the input data identified by the leaf of the linear model in the M5' tree. If this null hypothesis is rejected at the significance level represented by α, this dependency is mapped into the graph. To test whether a significant monotonic relationship exists, we use the Kendall's τ (under the subspace or subset of gene expression samples) as nonparametric measure of association [33].
Algorithm
In order to formalize the algorithm, named REG NET, several definitions are required.
Definition 1 (Microarray)
Let M be the microarray data, defined as$M=(\mathcal{C},\mathcal{G},\mathcal{L})$, where$\mathcal{C}=\{{c}_{1},{c}_{2},\mathrm{...},{c}_{n}\}$is a finite set of experimental conditions, $\mathcal{G}=\{{g}_{1},{g}_{2},\mathrm{...},{g}_{m}\}$is a finite set of genes, and$\mathcal{L}=({v}_{ij})$is a n × m gene expression matrix, where v_{ ij }= ℓ (c_{ i }, g_{ j }) given by the level function$\ell :\mathcal{C}\times \mathcal{G}\to \mathbb{R}$.
Definition 2 (Partition)
A partition Π of a set S is a nonempty collection of nonempty subsets of S, Π = {π_{ i }}_{i = 1}_{,..., p}such that ⋃π_{ i }= S and π_{ i }⋂ π_{ j }= ∅ when i ≠ j for i, j = 1,..., p. The set of partitions of S is denoted by PART(S).
Definition 3 (Model Tree)
A model tree MT_{ j }is aimed at estimating the values of the level function ℓ for the column j, i.e. for the target gene g_{ j }, MT_{ j }= {(ψ_{ i }, ϕ_{ i })}_{ i }_{=1}_{,..., q}, where$\cup {\psi}_{i}\in \text{PART(}\mathcal{C}\text{)}$, and ϕ_{ i }is a linear function defined on a subset of genes${\Omega}_{i}\subset \mathcal{G}\left\{{g}_{j}\right\}$, i.e., ϕ_{ i }:Ω_{ i }→ ℝ. Therefore, each function ϕ_{ i }will be applied in a subspace of conditions ψ_{ i }to locally estimate the level function of the gene g_{ j }.
Definition 4 (Forest)
The forest of model trees FT is the collection of every model tree MT generated from each gene g_{ j }, 1 ≤ j ≤ m, FT = {MT_{1}, MT_{2},..., MT_{ m }}, where each MT_{ j }is built by minimizing the error ε at estimating the level function for gene g_{ j }and the conditions within ψ_{ i }by means of the functions ϕ_{ i }.
Definition 5 (Association)
A gene g is potentially associated with the gene g_{ j }(g ~ g_{ j }) if g appears in any of the Ω_{ i }of the corresponding functions ϕ_{1}, ϕ_{2},..., ϕ_{ q }defined at the leaves of the model tree MT_{ j }, whose target gene is g_{ j }. Each function ϕ_{ i }involves a set of genes Ω_{ i }related to g_{ j }, and therefore, all the genes associated with g_{ j }, represented as$\u25b3({g}_{j})={\cup}_{i=1}^{q}{\Omega}_{i}$, constitute potential associations.
Definition 6 (Gene Regression Network)
The input is the gene expression matrix M, a threshold value θ to prune the model trees generated, and the significance level α for the BenjaminiYekutieli procedure. The output is a graph of interactions Q* among the genes in $\mathcal{G}$.
Regarding the computational complexity of REG NET, the cost of building the forest of trees is m times the cost of building a M5' tree, i.e. O(m^{2}nlog(n)), where m is the number of genes and n the experimental conditions; extracting the hypothetical dependencies is an iterative process which has a linear complexity O(m); and finally, the BY procedure involves sorting the pvalues calculated before, i.e., O(mlog(m)). Consequently, the overall cost of the algorithm is O(m^{2}nlog(n)).
Results and Discussion
The robustness of the methodology is shown by means of the analysis on an in silico benchmark suite of datasets, the Saccharomyces Cerevisiae cell cycle and the E. coli data set.
In silico benchmark suite of datasets
We tested our approach on a published in silico benchmark suite of datasets [34]. The goal is the prediction of network structure from the given in silico gene expression dataset. We use this suite as a blind performance test to compare our approach REG NET against several benchmark methods.
We used the simulated steadystate gene expression datasets reported in DREAM4 (In silico Network Challenge) [35]. The challenge is to infer 5 networks of size 100 hidden in 15 different experiments of microarray. For each network, the GNW tool [36] is used to simulate three different experiments of microarray: the steadystate levels of singlegene knockouts (deletions); knockdowns experiments by reducing the transcription rate of the corresponding gene by half; multifactorial experiment where each expression profile could be extracted from a patient.
For network inference, we applied several benchmark methods:

A heuristic algorithm for learning highdimensional dependency networks from genomic data. We used the GeneNet R package to infer causal networks based on partial correlations. GeneNet implements the methods of [37] and [38] for learning largescale gene dependency networks.

WeightedLASSO for structured network inference implemented in the Simone R package [39] and [40]. This algorithm uses the GLasso procedure to estimate a sparse inverse covariance matrix using a lasso (L1) penalty.

For learning Bayesian networks (BN) we used the R package named Deal[41] and the R package named G1DBNhttp://cran.rproject.org/web/packages/G1DBN.
Results reported here were obtained from GeneNet, Simone and G1DBN. The task of learning Bayesian Networks (BN) from data is NPhard with respect to the number of network vertices, i.e. Bayesian methods are computationally intractable for a huge number of genes. The Deal algorithm for learning BN was unsuitable to obtained networks because of the number of genes in the input microarray (100 genes). The G1DBN was suitable to obtain networks because this algorithm performs Dynamic BN inference using first order conditional dependencies as heuristic.
Our approach outperformed the results reported by G1DBN and SIMONE in all the data set (knockout, knockdown and multifactorial experiments of microarray). In general, our approach showed higher accuracy. Only in five out of fifteen data sets, out approach did not outperform the results obtained by GeneNet.
Saccharomyces Cerevisiae dataset
We use Saccharomyces Cerevisiae cell cycle expression data set [42], which contains 2884 genes and 17 experimental conditions. In the first experiment, the effect of pruning and nonpruning the forest of model trees is compared. Simplifying the forest involves rejecting all the M5' trees that have a relative error greater than a threshold. For both experiments a level α = 0.05 is fixed for the statistical BY procedure. To analyze the biological coherence of the results we use Gene Ontology attributes to characterize the resulted genes derived from our algorithm. We use FuncAssociate [43] to provide a measure (pvalue) that determines whether the set of genes obtained is due to chance, or instead, to common biological behavior. Furthermore, this tool calculates appropriate corrections for multiple hypothesis testing, such as WestfallYoung [44].
Saccharomyces Cerevisiae data. Experiment I.
N  Padj  GO Attribute 

38  < 0.001  0005830: cytosolic ribosome 
42  < 0.001  0005840: ribosome 
37  < 0.001  0003735: structural constituent of ribosomal protein 
46  < 0.001  0030529: ribonucleoprotein complex 
20  < 0.001  0005843: cytosolic small ribosomal subunit 
20  < 0.001  0015935: small ribosomal subunit 
17  < 0.001  0015934: large ribosomal subunit 
Saccharomyces Cerevisiae data. Experiment II.
N  padj  GO Attribute 

21  < 0.001  0005830: cytosolic ribosome 
23  < 0.001  0005840: ribosome 
21  < 0.001  0003735: structural constituent of ribosomal protein 
22  < 0.001  0005198: structural molecule activity 
23  < 0.001  0030529: ribonucleoprotein complex 
11  < 0.001  0005843: cytosolic small ribosomal subunit 
11  < 0.001  0016283: eukaryotic 48S initiation complex 
11  < 0.001  0016282: eukaryotic 43S preinitiation complex/eukaryotic 43S preinitiation complex 
25  < 0.001  0005829: cytosol 
11  < 0.001  0015935: small ribosomal subunit 
10  < 0.001  0005842: cytosolic large ribosomal subunit 
24  < 0.001  0009059: macromolecule biosynthesis 
23  < 0.001  0006412: protein biosynthesis 
10  < 0.001  0015934: large ribosomal subunit 
4  < 0.001  0000028: ribosomal small subunit assembly and maintenance 
In summary, the use of constrains to provide more accurate model trees does not have negative influence on the quality of results. Selecting the best M5' trees from the forest reduces the size of the gene network without decreasing the quality of the results from a biological perspective.
Escherichia coli dataset
The predictive performance of our approach was tested using Escherichia coli (E.coli) gene expression database from [45]. The E.coli gene expression database M^{3}^{ D }(Many Microbe Microarrays Database) is used and E_coli_v 3_Build_ 3 from T. Gardner Lab is built. This dataset consists of 524 arrays from 13 different collections corresponding to various conditions. The experiments were carried out on Affymetrix GeneChip E.coli Antisense Genome arrays, containing 4292 gene probes. A RMA normalization procedure was performed on the data prior to the application of our approach and the benchmark method.
Our approach REG NET and a gene relevance network method based on Partial Correlation were applied. Firstly, REG NET was applied several times with different values as a threshold of pruning phase: 25%, 50% and 100%. Second, the method proposed in [8] is used to provide partial Pearson and Spearman correlations (zeroth and first order correlations, with level α = 0.001, are calculated). Partial correlation coefficients quantify the correlation between two variables when conditioning on one or several other variables, which seems closer to causal relationships.
We chose the E.coli K12 transcriptional network in the Regulon database, version 6.3 [46] as true gene interaction network. From this transcriptional network we derived a gene association graph of 3288 interactions.
Conclusions
Inferring any type of relationship from data is a difficult task, particularly when nonlinearity is present. Gene networks provide a framework to analyze regulation and causality.
Our approach, named REG NET, generates new hypothesis of interactions among genes from gene expression data, and differs from correlationbased methods in that the relationship between one gene and others is calculated simultaneously, and statistically validated when all these genes show linear dependency only in a region of the space. Our method is based on the idea that, given some control genes which define subspaces of the input data, multivariate linear models can be estimated for the target gene. REG NET strongly favours localized similarities over more global similarity, which it is one of the major drawbacks of correlationbased methods.
Experimental results show the good performance of REG NET. The first experiment, with yeast cell cycle data, is consistent with Gene Ontology. The aim of the second experiment is to check the ability of finding true gene associations from gene expression data in comparison with E.coli transcriptional network from Regulon database.
In general, REG NET is a powerful method to hypothesize on unknown relationships, and therefore, on genes potentially related to biological functions.
Declarations
Acknowledgements
This research work is partially supported by the Ministry of Science and Innovation, projects TIN200768084C0200, PCI2006A70575, and by the Junta de Andalucia, projects P07TIC02611 and TIC200. We are grateful to the anonymous reviewers who provided valuable feedback on our manuscript.
Authors’ Affiliations
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