BN is a graphical model to capture complex relationships among a set of random variables {X ₁, X ₂,...,X_n } encoding the Markov assumption, each node representing a variable. In the context of gene network modeling, each node represents a gene, while gene interactions are represented by directed edges between nodes. Each variable X_i in the DAG is conditionally independent of its non-descendants given its set of parents. Mathematically the joint distribution of the DAG can be decomposed into a product form as:

P ( DAG ) = P ( X 1 , X 2 , … , X n ) = ∏ i = 1 n P ( X i ∕ Π i )

where Π _i denotes the parent set of the variable X_i . This is referred as the chain rule for BNs 9 10 . Learning a BN structure is to find a DAG that best matches the dataset, namely maximizing the posterior probability of DAG given data P (DAG|D). Here we adopt the sampling-based approach to Bayesian inference, and sample network structures from a candidate edge reservoir with the MCMC network learning method. In the reservoir the edge representation is proportional to the likelihood of the two genes being functionally linked according to prior knowledge. This way, the edges between the strongly-related gene pairs have higher chance to be proposed as part of the candidate network. The overall design is given in Figure 1. The major steps included:

1. Determine the probability of functional link p _link between each gene pairs

1.1 Calculate GO schematic similarity

1.2 Calculate p value of PubMed co-citation.

1.3 Integrate GO and PubMed information using the Naïve BN to determine p _link.

2. Construct candidate network edge reservoir in which copy number of each edge is proportional to the p _link of the corresponding gene pair.

3. Learn network structure using the MCMC algorithm through sampling the candidate network edge reservoir.

At each step of the iteration, the proposed network is retained with an acceptance probability that is determined by the relative posterior of the proposed versus current network, penalized by the network complexity 29 30 . In calculating the posterior we use the BDe (Bayesian Dirichlet equivalence) scoring metric 10 31 . The prior distribution is assumed to be uniform.

To evaluate the performance of our BN algorithm, and the benefit of adding prior knowledge, we compare it to two alternative approaches: (1) Plain BN. In each iteration, a new network is proposed by randomly changing one edge in the current network. (2) The method developed by Husmeier and Werhli 22 23 .

GO annotation and gene citation database (PubMed) were downloaded from ftp://ftp.ncbi.nlm.nih.gov/gene/DATA. Schematic similarity in GO taxonomy was first calculated for each gene pair using the approach proposed by Cao et al 32 , which calculates the shared information content of the GO terms. The value of this measure ranges between [0 1], with 0 being no similarity, and 1 being maximum similarity. The GO similarity between each gene pair is defined to be the maximum schematic similarity of all the GO terms they share.

For a given pair of genes, the total number of PubMed abstracts in which each gene appears (n and m, respectively), and in which both appear (k) were determined. The probability of co-citation frequency observed by random chance is calculated by

p PubMed ( # of co - citation ≥ k | n , m , N ) = 1 - ∑ i = 0 k - 1 p ( i | n , m , N )

where p ( i | n , m , N ) = n ! ( N - n ) ! m ! ( N - m ) ! ( n - i ) ! i ! ( m - i ) ! ( N - n - m + i ) ! N ! , and N is the total number of abstracts in PubMed 1 2 .

We used the Naïve Bayesian network to integrate the GO and co-citation information, and a simple Bayesian naïve classifier to predict the functional linkage probability p_link for all gene pairs. Note that the prior knowledge of functional linkage is undirected, i.e. p_link (i, j) = p_link (j, i). An edge sampling reservoir was constructed, in which the number of replicates for the edge between gene i and j N (Edge _{i , j} ) is in proportion to their p _link :

N ( Edg e i , j ) = Ceil ( 1 0 × p link ( i , j ) )

where Ceil(x) is the smallest integer no less than x. In this definition, any gene pair will be represented at least once and at most 10 times. The edges of gene pairs with higher p _link will appear more frequently in the edge reservoir, and hence enjoy a higher chance to be selected during the network structure learning.

Our BN simulation algorithm is implemented in Matlab utilizing Kevin Murphy's BNT package 33 bnt.googlecode.com, and is summarized in Table 1. Note that steps 1 and 3.1 contain unique features that separate our approach from others. The source code is available upon request. The networks were visualized with Cytoscape 34 .

The distribution of p_link for yeast gene pairs is given in Figure 2. Note that only a small proportion of gene pairs share high values of p_link , for about 92% of the gene pairs this value is less than 0.2. This indicates that most gene pairs share no functional linkage, consistent with the fact that gene networks are usually sparse. The candidate edge reservoir is constructed according to equation 3, and the MCMC samples this distribution to propose new candidate network structure at each iteration. In Figure 2 we have also included the distribution for gene pairs predicted to be interacting to each other, with and without the prior knowledge. Among all possible gene pairs, only ~8% with p_link 0.6. In contrast, this proportion increases to 28% among the predicted interactions. It indicates that the prior knowledge did affect the outcome of the BN learning. The results from the other data sets are similar.

The assumption of incorporating prior knowledge of functional linkage is that they can help network modeling. Existing data from yeast revealed that genes sharing the same GO attribute interact genetically more often than expected by chance (p < 0.05) 43 44 . In a very conservative estimate, over ~12% of the genetic interactions are comprised of genes with identical GO annotation (a 12 fold enhancement over what expected by chance, p < 10^-12); and over 27% are between genes with similar or identical GO annotations (an 8 fold enhancement, p < 10^-10).

We examined whether p_link can potentially discriminate interacting gene pairs from non-interacting ones, using the receiver operating characteristic (ROC) curve. ROC is a graphical plot of the sensitivity versus (1-specificity), namely the fraction of true positives versus the fraction of false positives, as the discrimination threshold of a classifier is varied. The area under curve (AUC) reflects the performance. The ROC of a random classifier would be a 45° line with AUC = 0.5. Figure 3 presents the ROC plot for the nine yeast cell cycle regulating transcription factors (TF): Fkh1, Fkh2, Ndd1, Mcm1, Ace2, Swi5, Mbp1, Swi4, and Swi6, and their targets identified using the ChIP-chip technology 45 . The AUC of 0.6064 indicating that is positively correlated with interaction and therefore useful in interaction inference.

In Figure 4A we plot the acceptance ratio versus number of MCMC steps in the yeast cell cycle dataset. Obviously in the later steps the probability to accept the new proposed DAG is small and flattens. The results from the other datasets are similar. In addition, the MCMC simulation was repeated 20 times with independent initializations, and consistency in the marginal posterior probabilities was examined. We found that they correlated well between different runs: 0.83 ± 0.11 for the simulated dataset, 0.68 ± 0.10 for the yeast data set, and 0.51 ± 0.26 for the mouse pancreas dataset. Figure 4B presents the scatter plot of the edge posterior probability from two typical runs that simulate the yeast dataset.

In our network inference, the MCMC learning simulation is repeated 20 times with independent initializations and an interaction will be considered in the final network if it is observed more than 15 times. Our new BN algorithm was first tested in a simulated time course (50 time points) gene expression dataset of an artificial network generated using SynTReN 46 . This network contains 76 genes, of which 24 act as regulators with a total of 124 regulatory relationships (i.e. 124 edges). The results are summarized in Table 3, 2n^d column. It demonstrates that incorporating the functional linkage as prior knowledge allows the identification of a significantly higher number, 21 versus 14, of the true gene-gene relationships compared with the plain BN modeling of gene expression data only. A random network of the same number of edges was also created for the 76 genes 47 . The improvement of BN with prior knowledge over random is significant (p < 0.01, Table 3), while without prior knowledge it is not (p~0.2, Table 3).

Next the new algorithm was applied to one of the Stanford yeast cell cycle data http://genome-www.stanford.edu/cellcycle/, where the cells from a cdc15 temperature sensitive mutant were studied 48 . To evaluate the performance, we compared the predicted interactions from our algorithm to the annotated interactions in BIND http://bind.ca 49 , and the transcription regulation predicted by the ChIP-chip data 45 . Tables 4, 5 list the benchmark interactions for the 107 yeast cell cycle genes that were recovered by the BN modeling. The statistical results are summarized in Table 3, columns 3-4.

Evidently, our method is capable of identifying a higher number of the positive benchmarks compared with the plain BN without prior knowledge. When evaluated with the BIND annotation, the number of correctly identified interactions doubled from 13 to 26 (p~0.13, χ²~2.28). The plain BN actually did not perform better than random selection (p~0.11). In contrast, BN with prior knowledge performed significantly better than random selection with χ² = 24.5, p < 0.001. When evaluated with the ChIP-chip data, the story is similar. The number of correctly identified gene regulatory relationships increased from 11 to 23 with the addition of prior knowledge (p < 0.01, χ² = 6.71). Without the prior knowledge, the plain BN is not different from random selection (p~0.1).

Figure 5A-5C shows the ROC curves that give a more quantitative view of the performance of BN with/without prior knowledge, and of the Werhli and Husmeier's algorithm 22 23 , in detecting TF-target gene interactions. Incorporation of prior knowledge significantly improved the performance with higher AUC. Our algorithm performed slightly better than Werhli and Husmeier's.

We also validated our algorithm using a mammal dataset. The experiment profiled gene expression changes in pancreas during embryonic development or during compensatory growth after partial pancreatectomy. Elucidating the networks is key to understand the complex nature of pancreas development and function 50 51 . A number of efforts have been made to manually annotate the key transcription factors and the gene networks they regulate based on low-throughput data, nicely reviewed by Servitja and Ferrer 52 . In Table 6, we list the 24 experimentally confirmed gene-gene regulatory relationships 52 , and their network is depicted in Figure 6A. With prior knowledge BN modeling of the expression data is able to recover half of them (12), as shown in Figure 6C and Table 6. In contrast, the plain BN is only able to identify 6 of them (Figure 6B). This is again a ~two-fold enhancement. In Figures 7A-7C the ROC curves are presented. Incorporation of prior knowledge significantly improved the ability to detect known interactions. Our algorithm performed comparably to Werhli and Husmeier's.

In Additional file 1, we listed the GO similarity and PubMed co-citation of the gene pairs with known regulatory relationships that were missed by plain BN. Clearly, almost all of them have high GO similarity and share a significant number of co-citations. Adding the functional linkage as prior knowledge helped to recover them.