Quantitative utilization of prior biological knowledge in the Bayesian network modeling of gene expression data
© Gao and Wang; licensee BioMed Central Ltd. 2011
Received: 1 January 2011
Accepted: 31 August 2011
Published: 31 August 2011
Bayesian Network (BN) is a powerful approach to reconstructing genetic regulatory networks from gene expression data. However, expression data by itself suffers from high noise and lack of power. Incorporating prior biological knowledge can improve the performance. As each type of prior knowledge on its own may be incomplete or limited by quality issues, integrating multiple sources of prior knowledge to utilize their consensus is desirable.
We introduce a new method to incorporate the quantitative information from multiple sources of prior knowledge. It first uses the Naïve Bayesian classifier to assess the likelihood of functional linkage between gene pairs based on prior knowledge. In this study we included cocitation in PubMed and schematic similarity in Gene Ontology annotation. A candidate network edge reservoir is then created in which the copy number of each edge is proportional to the estimated likelihood of linkage between the two corresponding genes. In network simulation the Markov Chain Monte Carlo sampling algorithm is adopted, and samples from this reservoir at each iteration to generate new candidate networks. We evaluated the new algorithm using both simulated and real gene expression data including that from a yeast cell cycle and a mouse pancreas development/growth study. Incorporating prior knowledge led to a ~2 fold increase in the number of known transcription regulations recovered, without significant change in false positive rate. In contrast, without the prior knowledge BN modeling is not always better than a random selection, demonstrating the necessity in network modeling to supplement the gene expression data with additional information.
our new development provides a statistical means to utilize the quantitative information in prior biological knowledge in the BN modeling of gene expression data, which significantly improves the performance.
Reverse engineering of genetic networks will greatly facilitate the dissection of cellular functions at the molecular level [1–3]. The time course gene expression study offers an ideal data source for transcription regulatory network modeling. However, in a typical microarray experiment usually up to tens of thousands of genes are measured in only several dozens or less samples, data from such experiments alone is significantly underpowered, leading to high rate of false positive predictions . Network reconstruction from microarray data is further limited by low data quality, noise and measurement errors .
Incorporating other types of data and existing knowledge of gene relationships into the network modeling process is a practical approach to overcome some of these problems. It has been proven that data integration and useful bias with relevant knowledge can improve the network prediction accuracy from gene expression data [6, 7]. Among the various approaches of network modeling, Bayesian Networks (BN) have shown great promise and are receiving increasing attention . BN is a graphic probabilistic model that describes multiple interacting quantities by a directed acyclic graph (DAG). The nodes in the network represent random variables (expression levels), and edges represent conditional dependencies between nodes . Learning a BN structure is to find a DAG that best matches the dataset, namely maximizing the posterior probability of DAG given data D: P (DAG|D). The sound probabilistic schematics allow BN to deal with the inherent stochasticity in gene expressions and the noise brought in by the microarray technology. Furthermore, BN is capable of integrating prior knowledge into the system in a natural way [9, 10].
A number of studies demonstrated that adding prior knowledge to BN improved the performance [4, 11–14]. Many sources of data and information are useful to supplement the gene expression data, and they can be incorporated at different steps of BN simulation, from prior structure definition to structure simulation and evaluation.
Known protein-DNA interaction or other clues of the relationships between transcription factors and their target genes are useful to transcription regulatory network inference. Hartemink et al. included data from the chromatin immunoprecipitation (ChIP) assay , and Tamada et al incorporated promoter sequence motif information , to define the prior probability of network structures. Information of other types of gene pair relationship has also been explored. Steele et al. developed a gene-pair association score from the correlation of their concept profiles derived from literature, and utilized that to define the prior structure probabilities . Larsen et al defined a Likelihood of Interaction (LOI) score, which measures the statistical significance of two genes interacting with each other according to their shared Gene Ontology (GO) information. They then restricted the candidate network edges (interactions) to those with significant p -values of LOI during the BN structure learning iterations [17, 18]. By doing so, the quantitative information of the likelihood is not fully utilized in the network modeling. Djebbari and Quackenbush utilized literature, high-throughput protein-protein interaction (PPI) data, or the combination of both to define the seed (initial) network structure. They observed an improved ability of the BN analysis to learn gene interaction networks from the expression data .
Imoto et al formulated an novel approach to incorporate prior biological knowledge within the BN framework by adopting the energy concepts from statistical physics [20, 21], which was later further extended by Husmeier and Werhli [22, 23]. In this approach an energy function was first defined to measure the agreement between a candidate network and the prior biological knowledge, and prior distribution of network structure is hence calculated using the Gibbs distribution in a canonical ensemble. Using this approach, the two groups examined several types of prior knowledge, including PPI, protein-DNA interaction, binding site information, literature, and Kyoto Encyclopedia of Genes and Genomes (KEGG) pathways [22–24]. The algorithms were validated using yeast gene expression data [20, 21], and synthetic data .
Existing studies often utilize prior knowledge to construct the prior distribution of network, or initial network structure. It has been demonstrated that the sampling method during simulation also affects the performance of BN structure learning . Though prior knowledge has been utilized to bias the sampling step, it is normally done through restricting the search space to sub regions, for instance, only simulate candidate structures whose significance is above a certain threshold according to prior knowledge [17, 18].
In searching for the network structure (DAG) that maximize P (DAG|D), the Markov Chain Monte Carlo (MCMC) approach is regarded better than greedy searching algorithms, especially for the microarray data with small sample size where there is often no single structure that is prominently better than others . In this study we propose a new approach to incorporate prior knowledge in a quantitative way to bias the MCMC simulation of candidate structure. It utilizes information of functional linkage between gene pairs, assuming that functionally linked genes are likely to interact with each other. It is known that interacting proteins or genes often share similar function, and participate in the same biological pathways and processes . Interaction has been utilized to infer functional linkage and annotate gene functions . Increasing evidence suggests that the reverse is also frequently true . In our algorithm a probability score is first calculated that measures how likely two genes are functionally linked based on prior knowledge; A candidate edge reservoir is then constructed where the number of copies of each edge is proportional to this probability score; The reservoir is in turn used for sampling candidate network structure during the MCMC simulation. This way the quantitative information of the potential gene pair link predicted by prior knowledge is retained.
We will consider two type of prior knowledge: co-citation in PubMed literature and similarity in ontological annotation according GO http://www.geneontology.org/. We will demonstrate they both contain information of functional linkage. The performance of the new algorithm is evaluated using a synthetic data set as well as data from two real microarray experiments: the yeast cell cycle study, and the mouse pancreas development/growth study. We will demonstrate that including the prior knowledge significantly improves the performance of BN modeling of gene expression data.
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.
Construct candidate network edge reservoir in which copy number of each edge is proportional to the p link of the corresponding gene pair.
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 schematic similarity and significance of PubMed co-citation
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, 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.
Construction of the candidate network edge reservoir
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 plink will appear more frequently in the edge reservoir, and hence enjoy a higher chance to be selected during the network structure learning.
Implemention of the new BN structure learning algorithm
n: number of nodes in the network.
D: discretized expression data matrix.
BurnIn: number of steps to take before drawing sample networks for evaluation. Default value: 50 times the size of the sampling reservoir.
n_iteration: number of iterations. Default value: 80 times the size of the sampling reservoir.
Δ_samples: interval of sample networks being collected from the chain after burn-in. Default
maxFanIn: maximum number of parents of a node.
A set of DAGs after reaching the max iteration step.
An average DAG in the form of a matrix.
1. Create a sampling edge reservoir based on p link .
2. Set all elements of the adjacency matrix for the initial DAG to 0.
3. for loop_index = 1: n_iteration do
(1) randomly select a element edge(i,j) from the edge sampling reservoir, corresponding to gene pair (i,j).
(2) if edge(i,j) exists in the current DAG, delete the edge; else if edge(j,i) exists in the current DAG, reverse edge(j,i) to edge(j,i); else add edge(i,j). We name these operations as "delete", "reverse" and "add", respectively.
(3) check whether the newly proposed DAG remains acyclic and satisfy the maxFanIn rules to nodes (i,j). If not, keep the current DAG and give up proposed DAG, go to (1).
(4) calculate log value of the marginal likelihood (LL)* of the expression data D of node j and its parents given the current DAG (LL_old) or the proposed DAG (LL_new) and define bf1 = exp(LL_new - LL_old).
(5) if the operation is "delete" or "add", bf2 = 1; if the operation is "reverse", calculate bf2 for node i in same way as for node j in (4).
(6) calculate the prior probability* of current DAG (prior_old) and propose DAG (prior_new); calculate the Metropolis-Hastings ratio (RHM) of the two DAGs; generate a random number u between 0 to 1, if bf1*bf2*prior_new/prior_old<u*RHM, keep the current DAG and give up proposed DAG, go to (1).
(7) when loop_index>BurnIn and (loop_index-BurnIn) is exactly divisible by Δ _samples, record the proposed DAG and its posterior probability.
4. End of loop, calculate the average DAG in the form of a matrix, where the elements are given by the averaged edges of all recorded DAGs weighted by their posterior probabilities.
Utility of GO similarity and PubMed co-citation in discovering functional linkage between gene pairs
to describe the utility of data D in functional linkage inference. An LLS close to 0 suggest that the data is not more informative than random pairing, whilst higher positive values of LLS indicates that data D contains more information of functional linkage.
We adopted equation (4) to evaluate whether GO schematic similarity and PubMed co-citation were useful in identifying functional linkage. The KEGG http://www.genome.ad.jp/KEGG and Munich Information Center for Protein Sequences (MIPS, mips.gsf.de/) database [1, 2] were used to construct the benchmarks of functional linkage. These databases were chosen for their high quality . In this study we utilized yeast and mouse gene expression data to validate our algorithm. For each species, the positive control set consists of randomly sampled 5% (43,761 for yeast, and 35,424 for mouse) of all gene pairs that are in the same KEGG pathways . The choice of 5% rather than all is to lower the computational complexity. The negative control set was constructed with gene pairs that encode proteins localized in different cellular compartments, with the underlying assumption that they are functionally unrelated and do not interact with each other. Four categories in the MIPS annotation  were utilized: 70.03 cytoplasm, 70.10 nucleus, 70.16 mitochondrion, and 70.27 extracellular/secretion proteins.
Again we only kept 5% of all possible gene pairs, totaling 112,693 for yeast and 531,089 for mouse, respectively. The same benchmark sets were also utilized to train the Naïve Bayesian classifier when calculating plink.
GO and PubMed citation contain information of functional linkage
GO similarity LLS, yeast
We found that there is a marginal dependence between the GO similarity and PubMed co-citation (Fisher's Z test, p~0.1). Theoretically naïve Bayesian classifier is optimal when the attributes are independent given class. However, empirical studies have shown that the classifier still performs well in many domains when there is moderate attribute dependences . The weak dependence between them indicates that the naïve Bayesian Network is an appropriate choice to integrate their information . Interestingly, the GO and MIPS categories, which are both functional annotations, also only depend weakly on each other. This may be because the present annotations are far from being perfect and complete .
Utility of functional linkage information to interaction network modeling
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).
Convergence of simulation
Validation using simulated data
The improvement in network modeling with the addition of prior knowledge
Yeast cell cycle study, benchmark from BIND
Yeast cell cycle study, benchmark from ChIP-chip
Mouse pancreas study
Number of genes
Number of established regulations
Number of possible regulations
76*75 = 5700
107*106/2 = 5671*
9*106 = 954
36*35 = 1260
Number of known regulations recovered with (without) prior knowledge
Total number of regulations predicted, with (without) prior knowledge
Improvement over plain BN
χ 2 = 0.36,
χ 2 = 2.28, p < 0.13
χ 2 = 0.04, p~0.84
χ 2 = 0.98,
Improvement: over random selection
χ 2 = 7.32,
p < 0.01
χ 2 = 24.5, p < 0.001
χ 2= 6.71, p < 0.01
χ 2 = 2.87,
p < 0.09
Plain BN over random
χ 2 = 1.58,
χ 2 = 2.42, p~0.11
χ 2 = 1.6, p~0.2
χ 2 = 0.01, p~0.8
Validation using the yeast cell cycle data
Predicted yeast gene regulatory relationships that are annotated in BIND
BN with prior knowledge
BN Without prior knowledge
Predicted yeast gene regulatory relationships that are confirmed by ChIP-chip
BN with prior knowledge
BN without prior knowledge
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).
Validation using mouse pancreas development data
Established pancreas gene regulatory relationships that are identified by BN modeling
Known regulatory relationship
Identified by BN modeling with prior knowledge
Identified by the plain BN without prior knowledge
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.
In this study we proposed a new algorithm to quantitatively utilize prior biological knowledge in the network modeling of gene expression data. First the functional linkage of gene pairs was assessed based on multiple data sources using the naïve Bayesian classifier. The result was then utilized to construct a candidate network edge reservoir, where the number of replicate edges between each gene pair was proportional to their function linkage probability. During simulation new candidate network structure was formed by sampling from this reservoir at each iteration. Since the edges of gene pairs with stronger functional linkage had more representations in the reservoir, these biologically meaningful edges enjoyed a preferential treatment in network simulation. With both the simulated and real gene expression data, we demonstrated that incorporating the prior knowledge significantly improved the network modeling performance. More information of the gene interaction network could be extracted from the microarray data with higher accuracy. In contrast, in all datasets, without the prior knowledge, though the number of benchmark regulations recovered is more than a random selection, the improvement is not statistically significant, demonstrating the necessity to supplement the gene expression data with additional information. This finding that plain BN did not perform better than random selection was not unexpected, similar observations was recently reported for a number of publically available reverse-engineering algorithms when gene expression data is the sole source of information .
Our algorithm provides a practical way to integrate the probabilistic biological knowledge that is different from previous efforts by others . The quantitative nature makes it capable to handle soft constraints. Using the approach by Werhli and Husmeier for instance [22, 23], we differ in several key steps. First, they encode multiple sources of prior knowledge in a weighted sum via an energy function; we integrate information from multiple sources through a Bayesian classifier. Furthermore, in our approach the MCMC samples from a candidate edge distribution defined by the prior knowledge, rather than from the network posterior distribution where the network prior is defined by the prior knowledge. Our algorithm utilizes the prior knowledge at interaction level, while theirs at the network level. Finally the Werhli and Husmeier approach is more computational intensive. To reduce the computational complexity, they sum over all parent configurations of each node and limit the number of parents of each node to 3 or less; the complexity of this operation is (where N is size of the network, and m the maximum FanIn) . We find that it is still memory consuming for networks of moderate or large sizes. For instance, a Dell Optiplex 755 with 2GHZ DUO CPU, 3.25 GB RAM ran out of memory when simulating the 107-gene yeast network. Our algorithm does not have this problem.
We used two sources of prior evidence of functional linkage to assist network modeling: the PubMed co-citation and GO schematic similarity. However, our framework by design allows the integration of other types of data or knowledge, for instance, high throughput genomic data including PPI and ChIP-chip; gene-gene relationships derived from advanced methods including text mining , database curation, and computational modeling of sequence information; and many other sources. It has been demonstrated that the degree of improvement brought in by prior knowledge highly depends on the quality of the information being added . Low quality prior knowledge could even lower the performance of BN . Presently, most of the available prior knowledge each on its own suffers from high false positive rate and being incomplete, which can limit their efficacy in network modeling. Integration of data from different sources and utilizing their consensus provides an effective means to deal with this issue [1, 2]. A caveat here is, when considering more sources of data, the inter-dependency among them need to be scrutinized more carefully, and maybe a more sophisticated integration method than the naïve Bayesian classifier is needed.
A number of different approaches have been developed to integrate multiple sources of prior information in the BN modeling of gene expression data, at the different steps of the simulation process [4, 11–14]. It would be of interest to compare the efficiency of the different approaches, investigate whether the optimal approach depends on the types of prior knowledge, and if the different approaches can be combined for a most efficient utilization of prior knowledge in network modeling.
In this paper we proposed a new algorithm to integrate and utilize the prior biological knowledge in the BN modeling of gene expression data. Our study demonstrated that incorporating prior knowledge at the step of network structure simulation is an efficient way to preserve the quantitative information in it, and to improve the performance of network modeling.
Preparation of gene expression data for algorithm validation
The simulated time course gene expression dataset was generated using SynTReN  for a artificial network with 76 genes, of which 24 act as regulators with a total of 124 regulatory relationships (i.e. 124 edges). The total number of time points is 50. All parameters of SynTReN were set to default values , except number of correlated inputs, which was set to 50%. The topological structure and inner interacting relationships are sampled from the characteristics of the yeast transcriptional network, therefore the results will be indicative of the algorithm performance on real data.
Yeast cell cycle study
The 107 Yeast cell cycle genes that were simulated for their network structure
Mouse pancreas development and regeneration after damage
The pancreas development and growth expression data was downloaded from the RNA Abundance Database http://www.cbil.upenn.edu/RAD, with study IDs 2 and 1790. Study 2 profiled mouse pancreas gene expression at six different developmental time points: embryonic day 14.5, 16.5, 18.5, at birth, at postnatal day 7, and at adulthood. 4 samples at E14.5, and 6 at all the following time points, totaling 34 samples. Study 1790 profiled gene expression in mice pancreas following partial pancreatectomy and Exendin-4 treatment. Exendin-4 is a glucagon-like peptide-1 receptor agonist that augments the pancreatic islet beta-cell mass by increasing beta-cell neogenesis and proliferation and by reducing apoptosis. Mice underwent 50% pancreatectomy or sham operation, and received Exendin-4 or vehicle every 24 hours. 3-4 animals from each group were sacrificed at each time point of 12, 24 and 48 hr after operation, together with 4 animals that received no operation, totaling 46 samples. Because the two studies each only contain a few time points, we combined their data for network modeling . Replicate samples under the same condition at the same time point were averaged.
The 36 mouse genes chosen to reconstruct interaction networks during pancreas development and growth
Digitization of gene expression data
Expression data were further discretized into three levels. In each data set, we calculated the mean (μ) and standard deviation (SD) of expression across all time points for each gene. Each expression value is then assigned to 0, 1 or 2 according to whether the value is less than μ-SD, between μ-SD and μ+SD, or above μ+SD.
Prior data of interaction and transcription binding
Annotations of known yeast gene interaction were downloaded from the Biomolecular Interaction Network Database (BIND, http://bind.ca), a database designed to store full descriptions of interactions, molecular complexes and pathways . BIND includes both directed (such as protein-DNA interaction) and un-directed (such as protein-protein interaction) interactions. Therefore when comparing to BIND annotations, we ignored direction.
Simon et al studied the transcription regulation of yeast genes by 9 cell cycle regulating transcription factors (TF): Fkh1, Fkh2, Ndd1, Mcm1, Ace2, Swi5, Mbp1, Swi4, and Swi6, using the ChIP-chip technology . These nine TFs are among the 107 cell cycle genes that we performed network modeling. The data were downloaded from http://staffa.wi.mit.edu/cgi-bin/young_public/navframe.cgi?s=17%26;f=downloaddata. For each TF, the study derived a binding p-value for each gene which reflects the likelihood that the TF binds to the promoter of this gene. We constructed a positive control target set for each TF that consists of those with p < 0.001, a negative control target set for each TF that consists of those with p > 0.1. Note that the transcription binding data provide directed information.
List of abbreviations used
area under curve
directed acyclic graph
Markov Chain Monte Carlo
receiver operating characteristic
This work was supported in part by National Institute of Diabetes and Digestive and Kidney Diseases Grant R01DK080100 (XW).
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