Gene Regulatory Network Reconstruction Using Conditional Mutual Information
© K.-C. Liang and X. Wang. 2008
Received: 30 December 2007
Accepted: 22 May 2008
Published: 4 June 2008
The inference of gene regulatory network from expression data is an important area of research that provides insight to the inner workings of a biological system. The relevance-network-based approaches provide a simple and easily-scalable solution to the understanding of interaction between genes. Up until now, most works based on relevance network focus on the discovery of direct regulation using correlation coefficient or mutual information. However, some of the more complicated interactions such as interactive regulation and coregulation are not easily detected. In this work, we propose a relevance network model for gene regulatory network inference which employs both mutual information and conditional mutual information to determine the interactions between genes. For this purpose, we propose a conditional mutual information estimator based on adaptive partitioning which allows us to condition on both discrete and continuous random variables. We provide experimental results that demonstrate that the proposed regulatory network inference algorithm can provide better performance when the target network contains coregulated and interactively regulated genes.
The prediction of the functions of genes and the elucidation of the gene regulatory mechanisms have been an important topic of genomic research. The advances in microarray technology over the past decade have provided a wealth of information by allowing us to observe the expression levels of thousands of genes at once. With the increasing availability of gene expression data, the development of tools that can more accurately predict gene-to-gene interactions and uncover more complex interactions between genes has become an intense area of research.
Gene Clustering Algorithms
Some of the first attempts at determining gene regulations are based on the gene expression clustering algorithms. These algorithms determine genes that are likely to be coregulated by grouping genes that exhibit similar gene expressions under the same conditions. Different clustering algorithms differ in the metric used to measure similarity between gene expressions, and how the metric is used to cluster into groups similarly expressed genes . In , a hierarchical clustering algorithm using a correlation coefficient metric is proposed. The K-means algorithm has also been applied to partition genes into different clusters . Other clustering algorithms such as self-organizing map (SOM) , mutual-information-based algorithms , and graph-theory-based algorithms  have also been proposed.
While gene clustering algorithms allow us to discover genes that are coregulated, they do not reveal much of the underlying biological mechanism such as the regulatory pathways. In recent years, many models have been proposed attempting to understand how individual genes interact with each other to govern the diverse biological processes in the cell. In [8–10], gene regulatory network inference based on graphical models is proposed. A graphical model depicts the relationships among nodes in a graph which are considered as random variables. Links between nodes represent dependence of the two variables. For network inference based on the graphical Gaussian model , the nodes with corresponding random variables are assumed to be jointly distributed according to the multivariate Gaussian distribution , with mean vector and covariance matrix . In , the gene-to-gene interaction is predicted from expression data using Bayesian networks, another type of graphical model. The dependence relationship between the variables is denoted by a directed acyclic graph where the nodes are associated with the variables , , and the nodes are linked if a dependent relationship exists between the two corresponding variables. Given a set of expression values , the algorithm selects the graph that best describes by choosing the graph that maximizes a scoring function based on the Bayes' rule . In , gene regulatory network reconstruction based on the dynamic Bayesian network is proposed to support cycles in the network, and time-series data in .
Relevance Network Algorithms
Another method that is related to graphical model is called relevance network. Relevance networks are based on the idea of "covariance graph" where a link exists between genes and , if and only if the corresponding gene expressions of and are marginally dependent . Different measures of dependence have been used in relevance-network-based algorithms. In , the correlation coefficient is used to represent the dependence between two genes, and in both , mutual information is used to measure the nonlinear relationship between the expressions of two genes. Since these metrics are computed from a finite number of samples, a threshold is often imposed so that two nodes are connected if the computed metric between the two nodes is above the threshold. In , entropy and joint entropy are first computed based on the histogram, then the mutual information of and is computed by . In , the proposed ARACNE algorithm uses the Gaussian kernel estimator to estimate the mutual information between the expressions and of genes and . Before estimating from the observed expressions and using the Gaussian kernel estimator, and are copula-transformed to take values between 0 and 1. This step is performed so that the expression data are transformed to uniform distribution, and arbitrary artifacts from microarray processing are removed. In gene regulatory networks, if gene regulates , which in turn regulates , then and will also be highly correlated. Using methods based on relevance network, a link will often be incorrectly inferred between and due to the high correlation measures. In , ARACNE tries to resolve this problem by using the data processing inequality (DPI). From DPI, if , and form a Markov chain (denoted as ), then . For a triplet of genes where the estimated mutual information of all three pairs of genes exceed the threshold, the link with the lowest mutual information is removed by ARACNE in the DPI step.
While relevance-network-based methods such as ARACNE perform well when the interactions in the gene regulatory network are between pairs of genes, they are unable to completely discover interactions that are results of the joint regulation of the target gene by two or more genes. The XOR interactive regulation is one such interaction that can be recognized only by exploiting the conditional dependence between variables of interest. Using conditional mutual information (CMI), it is possible to detect the XOR and other nonlinear interactive regulation by two genes.
Several recent works have attempted to incorporate information theoretic measures for more than two variables in regulatory network discovery. In , a CMI measure where the conditioning variable takes discrete values in two states (high and low) is proposed to discovery the transcriptional interactions in the human B lymphocytes. In , methods based on both MI and CMI have also been proposed to decrease the false positive rate for the detection of the interactions. In , the conditional coexpression model is introduced, and the CMI is used as a measure of conditional coexpression. In , an extension of the context likelihood of relatedness (CLR) algorithm , called "synergy augmented CLR" is proposed. The technique uses the recently developed information theoretic concept of synergy  to define a numerical score for a transcriptional interaction by identifying the most synergistic partner gene for the interaction. In this work, we propose a relevance-network-based gene regulatory network inference algorithm similar to , using information theoretic measure to determine the relationship between triplets of genes.
Here, we make use of both mutual information and conditional mutual information as measures of dependence between gene expressions. The main focus of this work is to discover the potential interactions between genes by adapting the relevance network model, which is also used in . The inference of the connectivity, or the "wiring" of the network, is also an important aspect of biological network inference. The proposed network inference algorithm uses an adaptive partitioning scheme to estimate the mutual information between and conditioned on , where can be either discrete or continuous. We show that using both mutual information and conditional mutual information allows us to more accurately detect correlations due to interactive regulation and other complex gene-to-gene relationships. In this work, our primary focus is on the detection of Boolean interactive regulation and other interactions which cause incorrect inferences, such as coregulation and indirect regulation. The experimental results show that the proposed network inference algorithm can successfully detect these types of regulation, and outperform two commonly used algorithms, BANJO and ARACNE.
The remainder of the paper is organized as follows. In Section 2, we present the system model for regulatory network inference. In Section 3, we present the adaptive partitioning algorithms for estimating mutual information and conditional mutual information as well as our proposed network inference algorithm based on MI-CMI. In Section 4, we present experimental results. Section 5 concludes the paper.
2. System Model
Suppose that the given set of genes form a regulatory network, where each node of the network is represented by a gene. Associated with each node, is a random variable with unknown steady-state distribution from which the expressions of are generated. We assume that for gene , we have the vector of steady-state gene expressions , where is the gene expression of gene under condition .
In the relevance network approach, two nodes are connected when they exhibit high degrees of interaction according to the chosen metric. Using metrics such as correlation coefficient and mutual information, high degrees of interaction between two genes typically indicate that one of the genes is directly or indirectly regulating the other gene, or the two genes are being coregulated by another gene. In relevance networks, indirect regulation and coregulated genes often are the cause of false positive links. ARACNE, as discussed in the previous section, removes indirect regulation by the application of DPI. However, ARACNE and other network inference algorithms based only on correlation coefficient or mutual information are unable to identify genes that are being coregulated, particularly if they are coregulated by the same mechanism. For example, in Figure 1(b), both and are regulated by an AND interaction of and . Using correlation coefficient or mutual information as metric will always result in a high interaction between and , and in most cases, greater than the interaction between the regulated gene and either one of the regulating genes, whereas using DPI will result in a false positive link.
The insufficiencies of using only mutual information or correlation coefficient as discussed above naturally lead us to the use of conditional mutual information as the metric of choice in our proposed regulatory network inference algorithm. For Figure 1(a), it is clear that the interaction between and and that between and can be detected by and . To resolve false positives due to coregulated genes recall that the conditional mutual information measures the reduction of information provided about by observing conditioned on having observed . An example of Figure 1(a) can be seen in . In Figure 1(b), coregulation of and can be recognized by the fact that if and are regulated by the same biological mechanism, and , since having observed or , no more information is provided about by observing , or information provided about by observing , respectively. On the other hand, having observed , which regulates both and , the information provided about by observing is reduced, and we have . Thus, by considering both the mutual information and conditional mutual information, we are able to reduce the amount of false positive links due to coregulation. Example of Figure 1(b) can be seen in .
From the above discussion, in the next section, we develop a relevance-network-based regulatory network inference algorithm that utilizes both mutual information and conditional mutual information to predict interactions between genes from the observed gene expression data. It is clear that we need efficient estimators that can accurately compute mutual information and conditional mutual information from data. Moreover, the conditional mutual information estimator should be able to support both discrete and continuous conditioning variables to allow for wider ranging uses.
3. MI-CMI Regulatory Network Inference Algorithm
There are several mutual information estimators such as the Gaussian kernel estimator and the equipartition estimator  but each has its weakness. The Gaussian kernel estimator requires a smoothing window that needs to be optimized for different underlying distributions, thus increasing the estimator complexity. While the equipartition estimator is simple in nature, the different grids in a partition often have variable efficiency in terms of contribution to the mutual information estimate due to the underlying sample distribution. In this section, we make use of an adaptive partitioning mutual information estimator proposed in  and extend it to estimating conditional mutual information. These estimators are then employed in building our MI-CMI-based relevance network for regulatory network inference.
3.1. Adaptive Partitioning Mutual Information Estimator
Let us consider a pair of random variables and taking values in and , both of which are assumed to be the real line for simplicity. For each random variable, we have samples and . From the samples we wish to obtain an estimate of the mutual information .
For mutual information estimators that partition the samples according to equal length or equiprobable partition, many of the grids may turn out to be inefficient due to the distribution of the samples. For example, let and , where is uniformly distributed on . Hence, the samples fall on a unit circle; and grids inside the circle do not contribute to the estimation of the mutual information between and . Therefore, a partitioning scheme that can adaptively change the number, size, and placement of the grids is more efficient in estimating mutual information. In the following, we describe a mutual information estimator proposed in  that adaptively partitions the observation space based on the unknown underlying distributions of the samples.
In the adaptive partitioning scheme, the sample space is divided into rectangular grids of varying sizes depending on the underlying distributions. A grid denoted as has the -axis range and -axis range . Furthermore, the set containing all the grids of the partitioning is denoted as .
Thus, mutual information can be estimated by computing the relative sample frequency on appropriately placed rectangular grids.
We now give the adaptive partitioning algorithm that constructs an asymptotic sufficient sequence of partitions for mutual information estimation.
Quadrant sample counts in each grid after third-level partition, and result of chi-squared test.
3.2. Conditional Mutual Information Estimator
This is done by simply dividing the samples into bins according to the value takes, and taking the weighted summation of the mutual information in each bin. In the case of conditioning on a continuous random variable, however, the partitioning of is often not so clear. Next, we propose a modification to the adaptive partitioning estimator that also adaptively partitions the z-axis to allow the estimation of conditional mutual information when the conditioned random variable is continuous.
in the definition of vanishes if and only if , that is, and are independent in the cuboid . In the following, we propose an adaptive partitioning scheme that partitions the given samples into cuboids, where in each cuboid the conditional distributions of and given are independent. Similar to Algorithm 1, we use the Pearson's chi-square test to determine the independence of the samples.
We now present the algorithm for estimating the conditional mutual information with continuous conditioning variable.
Compared to the estimation of conditional mutual information for discrete conditioning variable, we can see that instead of grouping samples into subsets where samples belonging in the same subset have the same values for the discrete-valued conditioning variable, here we group samples based on the adaptively determined partitioning of on the z-axis. The problem of estimating the conditional mutual information is thus broken down into estimating the mutual information for each group of samples, where the samples are grouped by which they belong to.
Note that the complexity of the Gaussian kernel estimator is known to be . However, the complexity of the adaptive partitioning estimator is dependent upon the joint distribution of the variables. For example, suppose and are independent and identically distributed uniform distributions. To compute from pairs of will take on average only the four initializing grids, since the sample pairs are typically uniformly distributed in each of the grids, and no further subpartitions are necessary according to the chi-squared test. On the other hand, suppose that and , where is uniformly distributed between , it will take many more subpartitions to obtain uniform distribution of the samples on each of the resulting grids. From our experience, for samples of and jointly Gaussian pairs, the Gaussian kernel estimator takes about 2 minutes to compute the MI, whereas for the adaptive partitioning algorithm, the time is between 2.5 to 3 minutes, on MATLAB code running on a Pentium 4 2.54 GHz machine. However, this is without taking into consideration the overhead required by the Gaussian estimator to compute the smoothing window.
3.3. Gene Regulatory Network Inference Algorithm
To infer a gene regulatory network that has various interactive regulations and coregulations, we propose a strategy of using both mutual information and conditional mutual information to reconstruct the regulatory network. In our proposed algorithm, we first use mutual information as metric to build regulatory network similarly to  to capture most of the direct regulations. To decrease the complexity of the algorithm by avoiding computing conditional mutual information for all triplets, while still allowing us to detect most of the causes for false positives and false negatives, we only compute the CMI for triplets of genes where either all three genes are connected, or all three genes are not connected. The decrease in complexity would depend on several factors. Once the pairwise MI threshold is chosen, the triplets that have one or two connections between the three genes indicate that the pairwise MI is sufficient for the determination of the interaction between the three genes, and the use of CMI is not necessary. Thus, instead of computing the CMI for all triplets of genes, CMI needs to be computed only for those triplets that are completely connected or completely unconnected. The amount of decrease in complexity would then depend on the ratio of triplets that have only one or two connections, which would depend on the actual connectivities between the genes, and the threshold selected for the pairwise mutual information phase of the algorithm.
The MI-CMI gene regulatory network inference algorithm is as follows.
- (iv)Detecting interactive regulation: for any triplet of genes where , compute the conditional mutual information estimate , and using Algorithm 2.
If all three of the CMI estimates are greater than , this may indicate that the two regulating genes may have had some prior interactions, or there is an XOR interaction between the 3 genes. Thus, we apply the DPI to remove the link with the weakest estimated CMI, and the links corresponding to the two largest estimated CMI are set to .
4. Experimental Results
In this section, we present simulation results to demonstrate the performance of the algorithms discussed in Section 3. We first illustrate the performance of Algorithm 2 for estimating the conditional mutual information of jointly Gaussian random variables. Next, we consider the performance of Algorithm 1 for estimating mutual information, by implementing the regulatory network inference algorithm in , but replacing the Gaussian kernel mutual information estimator employed there with Algorithm 1. Finally, we compare the network inference performance of Algorithm 3 with that of ARACNE  and BANJO  on synthetic networks.
4.1. Conditional Mutual Information of Jointly Gaussian Random Variables
To assess the accuracy of Algorithms 1 and 2 for the estimation of gene regulatory networks, we consider estimating the pairwise and conditional mutual information of multivariate Gaussian distributions. In our simulation, we compare the MI and CMI estimates of Algorithms 1 and 2 with those of the b-spline estimators. A b-spline MI estimator is proposed in  which divides the sample range into a number of bins. Contrary to the approach in the classical histogram estimators, where each sample contributes only to the bin it is in, for the b-spline estimator, the weight of a sample is spread to the bins. In the case of a third-order b-spline estimator, for a sample located in bin , the sample is assigned to the bins , , and , and the weight of the sample in each bin is computed using the b-spline coefficients. Here, we modify the b-spline estimator as proposed in  to estimate the 3-way MI so that the CMI can be obtained with the relationship .
Comparison of the estimated MI of bivariate Gaussian distribution with different correlation coefficient using Algorithm 1 and b-spline algorithm.
Comparison of the estimated CMI of trivariate Gaussian distribution with different covariance matrices using Algorithm 2 and the modified b-spline algorithm.
As a comparison, we performed CMI estimation of covariance matrices 1 and 4 using b-spline estimator with 20 bins. In , it is shown that the b-spline method has similar performance to that of the kernel density estimator (KDE), and the MI computed has the same level of significance. However, the KDE is shown to be more computationally intensive than the b-spline method. Thus in our comparisons, we only included the results from the b-spline method. For matrix 4, the b-spline estimator now converges to the correct analytical value. However, for matrix 1, the b-spline estimator does not converge to zero as the estimator with 10 bins does. This illustrates the drawback of using the b-spline estimators for MI and CMI estimation. The accuracy of the b-spline estimators depend on the choice for its parameters. On the other hand, Algorithms 1 and 2 are nonparametric, and do not need any prior knowledge of the underlying distributions to produce good estimates.
Looking more closely at CMI estimation, for small sample size and large CMI value, Algorithm 2 has a negative bias. As the sample size increases, the bias quickly reduces. On the other hand, when the true CMI value is small, Algorithm 2 tends to overestimate. It should be noted that estimating the CMI from a finite number of samples for a distribution with zero conditional correlation coefficient will typically result in a nonzero value. Nevertheless, the estimation results are still reasonably accurate, even for only 100 samples, so that conditional independence can be easily detected.
4.2. Regulatory Networks with Only Direct Regulation
Next, we implemented the algorithm described in  by replacing the Gaussian kernel MI estimator there with Algorithm 1. The modified algorithm is then compared with the original ARACNE algorithm in . The purpose of this comparison is to show that the adaptive partitioning MI estimator is a valid alternative for the Gaussian kernel estimator. Specifically, we constructed 25 synthetic regulatory networks, each with 20 one-to-one gene regulations, using NetBuilder . To compare the network inference performance, we adopt the same metrics as used in —recall and precision. Recall, defined as , where is the number of true positive links and is the number of false negative links, measures the ratio of correctly identified links out of total number of links. Precision, defined as , where is the number of false positive links, measures the ratio of correctly predicted links out of total predicted links. The values and relationship between the two metrics change with the selected threshold value, . At low , more links will be admitted as gene interactions, potentially capturing more true links, resulting in high recall values. However, as more links are included, the number of false positives also increases, which decreases the precision. On the other hand, when is high, only links with high interactions are admitted, and they in most cases represent true interactions between genes, thus improving the precision. However, true interactions that exhibit lower interaction are not admitted, resulting in a decrease in recall.
4.3. Regulatory Networks with Coregulation and Interactive Regulation
In Figure 9, we give an example of a network discovered by each algorithm. For the MI-CMI algorithm, we randomly permute for each gene the expressions across the different conditions, similar to what is done in . We performed 30 such permutations, and for each permutation we computed the pairwise mutual information using Algorithm 1 for all possible pairs. The highest observed mutual information out of the 30 permutations is used as the threshold for both MI-CMI algorithm and ARACNE. Results for BANJO were obtained using the default parameters.
We have proposed a new gene regulatory network inference algorithm that employs both mutual information and conditional information to discover possible direct and interactive regulations between genes, and to eliminate false links due to indirect regulations and coregulation. The mutual information and conditional mutual information are estimated from the expression data using an adaptive partitioning estimator. We have shown that the proposed network inference method outperforms BANJO and ARACNE when the underlying regulatory network contains coregulated or interactively regulated genes. In this work, we have focused on the discovery of the joint regulation of a gene by two other genes. It is possible to extend this work to joint regulation by multiple genes by modifying the proposed conditional mutual information estimator to a higher order. However, doing so would pose several computational problems. As the dimension of the CMI increases, increasing number of samples is needed to maintain the same level of accuracy. Also, as the dimension of the CMI increases, the number of sets of genes to be tested also increases, thus rendering this method impractical for brute force computation of all possible sets of genes. One possibility to reduce the amount of computations needed is to take into consideration the constraints placed on the possible connectivities from known biochemical reactions between the genes involved. This can be a future direction for research in this area.
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