Finding important hubs in scale-free gene networks

Transcription

1 Finding important hubs in scale-free gene networks Jing Hu 1, Yongling Song 2, and Su-Shing Chen 2 1 Department of Electrical and Computer Engineering, 2 Department of Computer Information Science and Engineering, University of Florida, Gainesville, FL 32611, USA *Corresponding author: suchen@cise.ufl.edu Abstract Several biological networks have been recently found to demonstrate scale-free and smallworld behavior instead of random graph characteristics. It is very important to find the hubs (genes, proteins, metabolites and so on) which dominate the topology of scale-free networks, since those are usually of great biological significance. In this paper, we will develop a general information theory method to construct scale-free networks from biological (gene, protein, metabolite) expression data, and find the important hubs by varying a threshold parameter family and deforming the network configurations. Thus such networks can be visualized at systematic levels. We will demonstrate the method by a microarray data set associated with a clinically relevant Spinal Cord Injury (SCI) case. The gene networks of two genetically and immunologically diverse rat strains Athymic Nude (AN) and Sprague Dawley (SD) are constructed. The topologies of the constructed gene networks are studied. It turns out that the gene networks for both rat strains demonstrate scale-free behavior in certain vertex degree ranges and have different topologies. By exploiting scale-free properties of the constructed gene networks, we find 36 important genes or hubs for AN strain and 51 for SD strain. Comparing those important genes identified, we find only 3 are in common for both strains. Bibliographies of authors: Jing Hu is a graduate student of Electrical and Computer Engineering Department. Yongling Song is a graduate student of Computer and Information Science and Engineering Department. Dr. Su-Shing Chen is a professor of Computer and Information Science and Engineering Department of the University of Florida. All three are members of the Systems Biology Laboratory of the university. 1

2 1 Introduction Recently, scale-free and small-world behaviors have been found in biological networks. Watts and Strogatz (1998) reported the architecture of the Caenorhabditis elegans nervous system to show significant small-world behavior. Fell and Wagner (2000) constructed a graph, defined by a vertex set consisting of all metabolites in Escherichia coli. Two metabolites were considered to be linked if they occurred in the same reaction. They found the graph to be sparse, with glutamate, coenzyme A, 2-oxoglutarate, pyruvate, and glutamine having the highest degree of connectivity. Jeong et al. (2000) comparatively analyzed metabolic networks of organisms representing all three domains of life and found that the topologies of these metabolic networks are best described by a scale-free model. More recently, Wuchty (2001) studied the topology of protein domain networks generated with data from the ProDom, Pfam, and Prosite domain databases. It was found that these networks exhibited small-world and scale-free topologies with a high degree of local clustering accompanied by a few long-distance connections. It is very important to find the hubs (genes, proteins, metabolites and so on) which dominate the topology of scale-free networks, since those are usually of great biological significance. For example, the metabolites found by Fell and Wagner (2000) with the highest degree of connectivity are viewed as a core of E. coli metabolism. Jeong et al. (2000) found that the ranking of the most connected metabolites is largely identical for all organisms. It has become increasingly important to study biology at systematic levels. One crucial step toward this goal is to reverse biological networks or pathways (for reviews see Rice and Stolovitzky 2004; van Someren et al. 2002) and to study the topology of those networks. Initial work in this area includes methods to identify metabolic network topologies based on correlation (Arkin and Ross 1995) and later on mutual information (Samoilov et al. 2001) of time-series measurements of species concentration. Other examples include relevance networks connecting genes with similar responses (Butte and Kohane 2000), networks based on protein-protein interactions (Uetz et al. 2000, Han et al. 2004) and reverse engineering of regulatory networks in human B cells (Basso 2005). All these methods have used a single threshold for the cutoff of correlation measures. In this paper, we will develop a general information theory method to construct scale-free networks from biological (gene, protein, metabolite) expression data, and find the important hubs by varying a threshold parameter family and deforming the network configurations. Thus such networks can be visualized at systematic levels. We will demonstrate the method by a microarray data set associated with a clinically relevant Spinal Cord Injury (SCI) case. The gene networks of two genetically and immunologically diverse rat strains Athymic Nude (AN) and Sprague Dawley (SD) are constructed. The topologies of the constructed gene networks are studied. It turns out that the gene networks 2

3 for both rat strains demonstrate scale-free behavior in certain vertex degree ranges and have different topologies. By exploiting scale-free properties of the constructed gene networks, we find 36 important genes or hubs for AN strain and 51 for SD strain. Comparing those important genes identified, we find only 3 are in common for both strains. 2 Methods we develop a general information theory method to construct networks from biological (gene, protein, metabolite) expression data, and find the important hubs by varying a threshold parameter family and deforming the network configurations. 2.1 Entropy, relative entropy and mutual information The entropy of an expression pattern is a measure of the information content in that pattern. Given a random vector X and its probability distribution P(X = x i ), i = 1,,N x, where N x is the number of possible values X can take, the entropy is defined as H(X) = N x i=1 P(X = x i )lnp(x = x i ). (1) Higher entropy for X variables means that their expression levels are more uniformly distributed. The joint entropy of X and Y is a measure of the total uncertainty contained in X and Y, and is defined as H(X, Y ) = N x N y i=1 j=1 P(X = x i, Y = y j )lnp(x = x i, Y = y j ), (2) where N y is the number of possible values Y can take. The mutual information between X and Y is a measure of information about X (or Y ) contained in Y (or X), and is given by I(X; Y ) = H(Y ) H(X Y ) = H(X) H(Y X) = H(X) + H(Y ) H(X, Y ) = N x i=1 N y j=1 P(X = x i, Y = y j )ln P(X = x i, Y = y j ) P(X = x i )P(Y = y j ) (3) Note that mutual information is always non-negative, i.e., I(X; Y ) 0 (Cover 1991). A mutual information at zero means that the joint distribution of expression values holds no more information than the objects considered separately. A higher mutual information between two objects means that one object is non-randomly associated with the other. In this way, mutual information can be used as a metric between two objects related to their degree of independence. It can be assumed that the higher mutual information is between two objects, the more likely it is they have a biological relationship (Butte and Kohane 2000). 3

4 In practice, a histogram technique to calculate entropy and mutual information is usually employed. The probabilities in Eq. (3) are estimated by the corresponding histograms, i.e., P(X = x i, Y = y j ) = (x i, y j ), N (4) P(X = x i ) = (x i) N, (5) P(Y = y j ) = (y j) N, (6) where N is the total number of samples, and (x i ) denotes the number of occurrences of the x i. Due to the relatively small sample size of the expression data, the probabilities estimated by the corresponding histograms may be sensitive to the number of bins chosen. When the number of bins changes, the value for the mutual information may change dramatically. To mitigate the effect of bin size choosing, we normalize the mutual information I normalized (X; Y ) = I(X; Y ) H(X)H(Y ) (7) Note that I(X; Y ) H(X), and I(X; Y ) H(Y ), thus the normalized mutual information is between 0 and 1. We shall always work with the normalized mutual information hereafter. 2.2 Construction of biological networks Given an expression data set, thousands of objects (genes, proteins and metabolites) have been simultaneously measured under a variety of conditions. Suppose the total number of objects measured is N g. Measurements of all objects are compared against each other, resulting in a total of N g (N g 1)/2 pair-wise calculations of mutual information. Each object is completely connected to every other object with a calculated mutual information. We then choose a threshold mutual information (TMI) and display only those objects that are linked to others with a mutual information higher than the threshold. Out of the completely connected network, we are left with objects that are more strongly connected to each other than the TMI. How to choose a proper TMI is a non-trivial problem. One usual way is to perform permutations of expression measurements many times, e.g., 20 or 30 times, and recalculate a distribution of the permuted pair-wise mutual information for each permutation. Then the many permuted distributions are averaged, and one good choice for the TMI is the largest mutual information value in the averaged permuted distribution. This has been employed by many researchers, such as Butte and Kohane (2000) and Basso (2005). However, sometimes this way of determining TMI is too conservative, with only a few pairs of objects having mutual information higher than the TMI. To overcome this, we do not use a fixed TMI, rather we choose different TMIs and check the consistency of the networks constructed under these TMIs. This will be discussed in more details below. 4

5 2.3 Finding important hubs in constructed biological networks by TMI algorithm A constructed network can be viewed as a graph Ω, defined by a pair (W, E), where W = {s i }, i = 1,,N is the set of N vertices (genes, proteins, neurons, etc.) and E = {{s i, s j }} is the set of edges/connections between vertices. The adjacency matrix ξ i j indicates the interaction between two vertices s i, s j Ω. For the purpose of studying the topology of networks, it suffices to set the elements of adjacency matrix as binary, with ξ i j = 1 indicating that an interaction exists between two vertices or ξ i j = 0 indicating that the interaction is absent. The pair-wise mutual information between two vertices is used as the correlation measure. If the mutual information is greater than or equal to the TMI, then ξ i j = 1, otherwise ξ i j = 0. Based on the adjacency matrix, we can calculate the degree k of each vertex, which is defined as the number of other vertices to which it is linked. Then we can compute the degree distribution f (k) and check whether the distribution follows a distribution comparable to a power-law distribution, that is, f (k) k γ. (8) If Eq. (8) holds, then the network under study is scale-free. This type of networks displays a high degree of robustness against errors. However, these networks are highly vulnerable to perturbations of the highly connected nodes. In order to accurately estimate the parameter γ in Eq. (8) from finite experimental data, it is often customary to work with the complementary cumulative distribution function, given by k P(K k) = 1 f (t)dt k (γ 1). (9) It is clear that if one takes log-log of Eq. (9), then the parameter γ is the slope of a linear relation plus 1. If the networks under study are scale-free, then one is usually most interested in those objects with relatively high degree of connectivity. Those important objects correspond to the rightmost end of the power-law relation in the degree distribution. We develop a 1-parameter family of different TMIs that has a stable and consistent structure of networks constructed under these TMIs. The family will deform to important hubs of the given biological network. The TMI Algorithm is detailed as follows: Input: Expression data set Output: The 1-parameter biological network family under different TMIs with colored hubs in the initial network Step 1. Compute the entropy of expression patterns and the mutual information from expression patterns for each pair 5

6 of objects. Step 2. Set a tentative threshold parameter e. Step 3. Choose the objects above the threshold e. Step 4. Construct biological network N(e) depending on the threshold e. Step 5. Increase threshold e to a higher threshold e and repeat steps 3 and 4, until the network structure become not consistent. Step 6. Color objects in the converged network N as potential hubs. Step 7. Output the 1-paramter family of networks with hubs colored in the initial network. We will provide an example of the algorithm in the following sections. 3 Results 3.1 SCI microarray data We apply the general information theory method to study the transcriptional expression profiles obtained after a clinically relevant SCI in two genetically and immunologically diverse rat strains (Velardo 2004). Two groups of female adult AN and SD rats received midline contusions using the New York University impactor. These rats were used to interrogate Affymetrix U34A rat genome GeneChips and were killed at 1, 3, 10, 30, and 90 days after injury. Microarray data were collected with 8799 probe sets under 44 different experiment conditions including normal and 1 through 90 days postinjury. Among the 44 measurements, 23 are for SD strain and 21 for AN strain. Raw data files and more detailed description about the data can be found at Construction of gene networks associated with SCI We perform some pre-processing to alleviate the effect of noise in the SCI microarray data. We first take logarithm with base two for each data entry. If original data is zero, the mean of the 44 conditions is assigned to it. Then we compute standard deviation for each probes over the 44 conditions and filter out those probes with standard deviation less than After these steps, we retain 3060 probes. Then we separate the 44 conditions for the two strains of rats, 21 for AN and 23 for SD, to form two subsets of data. Within each subset, we calculate the pair-wise mutual information for all the 3060 probes retained, with the number of bins chosen as 3. We calculate the mean and standard deviation of the total (obtained by 3060 (3060 1)/2) pair-wise mutual information for each subset. The mean value for AN and SD strains is and , respectively, while the standard deviation is and , respectively. For each subset, we choose three different TMIs, TMI i = 6

7 Table 1: The thresholds chosen for constructing gene networks corresponding to AN and SD strains threshold AN SD TMI TMI TMI mean + (i + 1) standard deviation, i = 1,2,3. We apply the TMI Algorithm to these genes. Specifically, the values of TMIs chosen for constructing gene networks corresponding to AN and SD strains are listed in Table 1. Note that we do not choose too small or too large thresholds for mutual information, since too small thresholds may lead to too many false functional connections between genes, while too large thresholds do not give higher reliability of functional interactions. Due to the large size of the constructed networks, here we show the result with relatively high threshold. One example for the constructed AN gene network deformed from threshold 0.7 to 0.8 is shown in Fig. 1, with nodes representing objects under study and lines between nodes representing hypothetical associations of objects. The important objects are red colored in the Figure. 3.3 Finding important genes in constructed gene networks For constructed gene networks, we calculate the cumulative degree distribution P(K k) and estimate the parameter γ. Fig. 2 shows the cumulative degree distributions for AN and SD strains. The three dashed curves from right to left correspond to TMI 1 to TMI 3 chosen for constructing gene networks. The red solid lines indicate the networks exhibit scale-free behavior in certain vertex degree ranges. The estimated values for the parameter γ are listed in Table 2. From Table 2, we observe that the γ value for the AN gene networks is always a bit larger than the corresponding that for the SD gene networks. We are most interested in those genes with relatively high degree of connectivity, that is, the genes corresponding to the rightmost end of the right lines in Fig. 2. For the two strains of gene networks, the vertex degree range identified from Figs. 2(a, b), and the number of genes found within the corresponding range for different TMIs used are listed in Table 2. We believe that the common genes found by setting different TMIs for each strain have high reliability to be the true hubs of underlying biological networks. For the AN and SD gene networks, we identify 36 and 51 such hubs, respectively. Due to the page limitation, we do not give the detailed information about those important genes here. It is interesting to compare those 36 and 51 important genes identified in the AN and SD gene networks. We find 7

8 Figure 1: One example for the constructed AN gene network deformed from threshold 0.7 to 0.8. Table 2: The genes with top number of links connected to them in the reconstructed biological networks associated with SD and AN strains Threshold AN SD γ degree range num of genes γ degree range num of genes TMI TMI TMI num of common genes: 36 num of common genes: 51 only 3 of them are in common for both strains, as listed in Table 3. Biologically, the two strains have indicated strong differences. 4 Summary and discussions We have developed a practical method for constructing biological networks of objects (genes, proteins and metabolites) and finding important hubs. and applied it to two genetically and immunologically diverse rat strains (AN and SD) associated with a clinically relevant SCI. We will apply this method to other gene, protein and metabolic networks. We will also collaborate with experimentalists to validate our findings and explore further results of important hubs. 8

9 10 0 (a) 10 0 (b) P(K k) 10 2 P(K k) k k Figure 2: The cumulative degree distributions for gene networks corresponding to (a) AN and (b) SD strain. The three dashed curves from right to left correspond to TMI 1 to TMI 3 chosen for constructing networks. The red solid lines indicate the networks exhibit scale-free behavior. Table 3: Common genes in both AN and SD gene networks Probe Set ID UniGene ID Gene Title Gene Symbol AF030087UTR 1 g at Rn activity and neurotransmitter-induced early gene 2 (ania-2) mrna, 3 UTR J05499 at Rn glutaminase 2 (liver, mitochondrial) Gls2 L13619 at Rn.772 insulin induced gene 1 Insig1 References [1] Arkin, A., and Ross, J. (1995) Statistical construction of chemical reaction mechanism from measured timeseries. J. Phys. Chem., 99, [2] Bareyre, F.M., and Schwab, M.E. (2003) Inflammation, degeneration and regeneration in the injured spinal cord: insights from DNA microarrays. Trends Neurosci., 26, [3] Basso, K., Margolin, A.A., Stolovitzky G., Klein, U., Dalla-Favera, R., and Califano, A. (2005) Reverse engineering of regulatory networks in human B cells. Nature genetics, 37, [4] Butte, A.J., and Kohane, I.S. (2000) Mutual information relevance networks: functional genomic clustering using pairwise entropy measurements. Pac. Symp. Biocomput., 2000, [5] Cover, T.M., and Thomas, J.A. (1991) Elements of information theory, Wiley, New York. 9

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