Constructing non-stationary Dynamic Bayesian Networks with a flexible lag choosing mechanism
- Yi Jia^{1} and
- Jun Huan^{1}Email author
https://doi.org/10.1186/1471-2105-11-S6-S27
© Huan and Jia; licensee BioMed Central Ltd. 2010
Published: 7 October 2010
Abstract
Background
Dynamic Bayesian Networks (DBNs) are widely used in regulatory network structure inference with gene expression data. Current methods assumed that the underlying stochastic processes that generate the gene expression data are stationary. The assumption is not realistic in certain applications where the intrinsic regulatory networks are subject to changes for adapting to internal or external stimuli.
Results
In this paper we investigate a novel non-stationary DBNs method with a potential regulator detection technique and a flexible lag choosing mechanism. We apply the approach for the gene regulatory network inference on three non-stationary time series data. For the Macrophages and Arabidopsis data sets with the reference networks, our method shows better network structure prediction accuracy. For the Drosophila data set, our approach converges faster and shows a better prediction accuracy on transition times. In addition, our reconstructed regulatory networks on the Drosophila data not only share a lot of similarities with the predictions of the work of other researchers but also provide many new structural information for further investigation.
Conclusions
Compared with recent proposed non-stationary DBNs methods, our approach has better structure prediction accuracy By detecting potential regulators, our method reduces the size of the search space, hence may speed up the convergence of MCMC sampling.
Keywords
Introduction
Recently non-stationary Bayesian network models have attracted significant research interests in modeling gene expression data. In non-stationary Bayesian networks, we assume that the underlying stochastic process that generates the gene expression data may change over time. Non-stationary Bayesian networks have advantage over conventional methods in applications where the intrinsic regulatory networks are subject to changes for adapting to internal or external stimuli. For example, gene expression profiles may go through dramatic changes in different development stages [1], or in the invasion process of viruses [2], or as response to changes of outside environment such as temperature and light intensity [3].
Recent work on non-stationary Bayesian networks could be found in [1, 2]. Robinson's method [1] used RJMCMC (Reversible Jump Markov Chain Monte Carlo) to sample underlying changing network structures, in which an extended BDe metric (Bayesian-Drichlet equivalent) is applied. And Grzegorczy et al. [2] proposed a non-homogeneous Bayesian network method to model non-stationary gene regulatory processes, in which they included a Gaussian mixture model based on allocation sampler technique [4], provided an extended non-linear BGe (Bayesian Gaussian likelihood equivalent) metric and employed MCMC (Markov Chain Monte Carlo) to collect samples.
There are several limitations on the existing non-stationary DBNs methods that are discussed above. First, the RJMCMC that is used in Robinson’s work [1] is a computationally expensive approach especially in dealing with gene networks. Second, mixture model used by Grzegorczy et al. avoided intensive computational issue by using MCMC, but it does not capture the underlying changing network structures over time. In addition, both methods used a fixed time delay τ = 1 that leads to a relatively low accuracy of prediction on network re-construction [5].
In this paper, we proposed a new non-stationary DBNs approach extending the work presented in [1] and [5]. Our method modified RJMCMC by employing a systematic approach to determine potential regulators. We designed a flexible lag determine mechanism by considering the delay in the gene expression changes between potential regulators and target genes. In this approach we efficiently reduce the model searching space, capture the dynamics of transcriptional time delay, and speed up computation with a fast convergence.
Related work
With a well-defined probabilistic semantics and the capability to handle hidden variables [6], Dynamic Bayesian Networks (DBNs) are widely used on regulatory network structure inference from noisy microarray gene expression data [7–16].
The early work of applying BNs to analyzing expression data could be found in [7, 8]. Many works have been done since then. Hartemink et al. extended the static BNs by including latent variables and annotated edges, and their work focused on scoring the models of regulatory network [10]. Considering the problem of information loss incurred by discretization of expression data, Imoto et al. proposed a continuous BNs and non-parametric regression model [12]. They used Laplace approximation to the marginal probability to infer a BNRC score as the scoring metric for network models. Further, Hartemink and Imoto extended their techniques to DBNs [11, 14]. Before the BNs, previous efforts at modeling genetic regulatory networks fell into two categories [9, 10]: fine-scale methods utilizing differential equations, and coarse-scale methods using clustering and boolean network models. BNs method is perceived as a good compromise of the two levels. With the challenging of small number of samples, researchers seek additional information such as transcriptional localization data [16], DNA sequences of promoter elements [13], and protein-protein interaction data [15] to improve the accuracy of gene networks reconstruction.
Method
Structure Learning of Non-stationary Bayesian Networks
Bayesian networks (BNs) are a special case of probabilistic graphic models. A static BN is defined by an acyclic directed graph G and a complete joint probability distribution of its nodes P(X) = P(X_{1},…, X_{ n }). The graph G : G = {X, E} contains a set of variables X = {X_{1},…, X_{ n }}, and a set of directed edges E, defining the causal relations between variables. With a directed acyclic graph, the joint distribution of random variables X = {X_{1},…, X_{ n }} are decomposed as P(X_{1},…, X_{ n }) = ∏_{ i } P(X_{ i }|Π_{ i }), where Π_{ i } are the parents of the node (variable) X_{ i }.
The current application of DBNs to gene expression data assumes that the underlying stochastic process generating the data is stationary. Here we provide a new approach to capture the structural dynamics of non-stationary data.
In the following discussion, we specify the formula for calculating each component of Equation 4. The prior P(τ_{ max }|T) is 1 because we set the τ_{ max } value when we find the potential parents for each variable.
,where and s_{ i }, is the number of edges’ change between G_{ i }_{+1} and G_{ i }. We have no prior knowledge on P(G_{1}) and see the uniform distribution as the prior.
I_{ h } is a segment where a network structure G_{ h } and its corresponding lag value τ_{ h } work. are the parameters associated with the data of one segment I_{ h } corresponding to G_{ h }. is the probability density function of .
where is the conditional probability of the j th component’s value in the lag vector τ^{ T }.
Potential regulator detection
Structure sampling using RJMCMC
We choose sampling approaches rather than heuristic methods to search network structures due to the reason that microarray expression data are usually sparse, which makes the posterior probability of structures to be diffuse [9]. In this approach, a group of most likely structures could explain data better than a single one. We use a sampling method called RJMCMC (Reversible Jump Markov Chain Monte Carlo) to collect structure samples. The details of this method are available on [19].
Compared with the move types introduced in [1], we add one new move type called change lag and modify most of the existing operations by incorporating more restrictions. We also define a vector of time points L^{ T } = (L_{1},...,L_{ m }_{−1}), where L_{ i } : 1 ≤ i ≤ m − 1 is the start time point where G_{ i+ }_{1} is applied. We use Metropolis-Hastings algorithm for RJMCMC sampling [20]. The move set of our RJMCMC consists of 11 move types:
MT1: add edge to G_{ i }.
MT2: delete edge from G_{ i }.
MT3: add edge to ΔG_{ i }.
MT4: delete edge from ΔG_{ i }.
MT5: move edge between ΔG_{ i }s.
MT6: shift time, which changes a single L_{ i }’s value. This operation will trigger the checking of τ_{ i }’s value under the restriction of τ_{ i } ≤ L_{ i } − 2, where 1 ≤ i ≤ m − 1, and τ_{ m } ≤ T − 1.
MT7: change lag, which changes a single τ_{ i }’s value. This move type needs to follow the limitations showed on MT6.
MT8: merge ΔG_{ i } and ΔG_{ i+ }_{1}.
MT9: split ΔG_{ i }.
MT10: create new ΔG_{ i }.
MT11: delete ΔG_{ i }.
Both MT8 and MT9 operations will trigger the change of dimensions of L^{ T } and τ^{ T }. In MT8, the new component of τ^{ T } takes the least value of two merged components. Similarly with MT8 and MT9, M10 and M11 will change the dimensions of L^{ T } and τ^{ T }. MT1, MT3, MT10 and MT11 follow the restriction that the edges pointed to one target gene should have the origins from its potential regulators.
Experimental study and evaluation
We performed all the experiments on a cluster with 256 Intel Xeon 3.2 Ghz EM64T processors with 4 GB memory each. We implemented our method FLnsDBNs (Flexible Lag Non-Stationary Dynamic Bayesian Networks) in Matlab.
The computational time of three methods
CMV | ArobidopsisThalianaT 20 | |
---|---|---|
RJnsDBNs | 9.06s | 333s |
ASnsDBNs | 457.53s | 13394s |
FLnsDBNs | 219.66s | 14034s |
Our experimental study is based on three data sets: (i) Bone Marrow-derived Macrophages gene expression time series data (Macrophages data set), (ii) Circadian regulation in Arabidopsis Thaliana gene expression time series data (Arabidopsis data set), and (iii) Drosophila muscle development gene expression time series data (Drosophila data set). To compare the results from different data sets, we follow the evaluation method introduced in [2, 9, 21]. For each data set, we first collect gold standard reference networks as the ground truth. For the Macrophages data set, such reference networks are available in [2, 22, 23]. For the Arabidopsis data set, we collect the network information from [3, 24–27]. For the Drosophila data set, there is no ground truth regarding the network structure. We compare our method with others by showing the commonality and differences. In case where we have ground truth network structure (the Bone Marrow data set and Arabidopsis data set), we use the area under receiver operating characteristic curve (AUROC) values to evaluate the performance. We obtained the ROC curves by postprocessing the posterior probabilities of directed edges and taking different cutoff thresholds in [0, 1]. If the posterior probability of an edge is greater than the threshold, we keep the edge. Otherwise, we do not keep the edge. With the ROC curves, we evaluate the performance of different methods by comparing the AUROC scores. In addition, for each data set, we show the posterior distribution of the number of segments and the locations of changepoints. In all of our experimental study, we find that the method FLnsDBNs produces compatible results with previous methods and demonstrates better network prediction performance in all the data sets. Before we discuss the details of experimental results, we present our data set first below.
Data sets
As mentioned briefly before, we evaluate our method on three data sets used in [1, 2]. We preprocess the original data sets by following Zhao’s work [28]. We set the values of a missed time point with the mean of its two neighbors; i.e., X_{ i },_{ t } = (X_{ i },_{ t }_{−1} + X_{ i }_{,}_{ t }_{+1})/2 if 1 < t < T. If the missed values are at the beginning or end, simply set the same value as its neighbor; i.e., X_{ i }_{,}_{ t } = X_{ i }_{,}_{ t }_{+1} if t = 1 or X_{ i }_{,}_{ t } = X_{ i }_{,}_{ t }_{−1} if t = T. In the following, we show the details of each data set.
Bone Marrow-derived Macrophages gene expression data. Interferon regulatory factors (IRFs) are proteins crucial for the mammalian innate immunity [29]. These transcription factors are central to the innate immune response to the infection by pathogenic organisms [23]. We use the Macrophage data sets sampled from three external conditions: (I) Infection with Cytomegalovirus (CMV), (II) Treatment with Interferon Gamma (IFN_{ γ }), and (III) Infection with Cytomegalovirus after pretreatment with IFN_{ γ }(CMV+IFN_{ γ }). Each data set has 3 genes: Irf1, Irf2 and Irf3, and contains 25 time points with the interval of 30 minutes. We use the network Irf 2 ↔ Irf 1 ← Irf 3 as the gold standard and assume the network never changes over the time.
Drosophila muscle development gene expression data. The original transcriptional profile on the life cycle of Drosophila melanogaster contains 4028 genes, nearly one third of all of the predicted Drosophila genes. The samples were collected over 66 time steps throughout the life cycle of Drosophila melanogaster consisting of four periods: embryonic, larval, pupal, and adulthood periods [31]. The intervals of sampling are not even, from overlapped 1 hour during the early embryonic period to multiple days in the adulthood. We choose 11 genes for analysis, which are eve, gfl/lmd, twi, mlc1, sls, mhc, prm, actn, up, myo61f, msp300. Those genes were reported to be related with the muscle development of Drosophila.
Experimental results
In this section, we compare the experimental results of three approaches: FLnsDBNs, RJnsDBNs, and ASnsDBNs on three data sets.
Comparison of AUROC values on Macrophage data
CMV | IFN _{ γ } | CMV + IFN _{ γ } | |
---|---|---|---|
RJnsDBNs | 1 | 0.7778 | 0.2222 |
ASnsDBNs | 1 | 0.6667 | 0.6667 |
FLnsDBNs | 1 | 0.8333 | 1 |
For the CMV + IFN_{ γ } data, all three methods identify 1 segment, which corresponds to a coexistence state between virus and its host cell [2, 32], and have the same range of the number of segments 1 ~ 3. In Table 2, we find that FLnsDBNs shows a much better network prediction with the AUROC score equal to 1 while in RJnsDBNs the AUROC score is equal to 0.2222 and in ASnsDBNs the AUROc score is equal to 0.6667. For the IFN_{ γ } data, there is a postulated transition with the immune activation under the treatment of IFNγ. FLnsDBNs infers 2 segments and finds two posterior peaks of transition time at 8 and 14.
ASnsDBNs and RJnsDBNs infer only one segment, even though the two methods identify a differnt posterior peak at the location around 5. On the assessment of the predicted network structures, the AUROC scores are 0.8333 in FLnsDBNs, 0.7778 in RJnsDBNs, and 0.6667 in ASnsDBNs. In all of three Macrophages data sets, our approach shows the best network prediction accuracy.
For each Macrophages data set using FLnsDBNs and RJnsDBNs methods, we find that the posterior probability distributions of any edge do not change much across different segments. This finding is consistent with the assumption that the underlying network does not change through the time.
The experimental results on Arabidopsis data. On the Arabidopsis data, we use a larger number of iterations in the MCMC sampling because the data set is much larger than the Macrophages data. We run 10,000 iterations for burn-in and then take additional 990,000 iterations to collect samples. The sample collection of FLnsDBNs on the Arabidopsis data takes about 4 hours.
Comparison of AUROC values on Arabidopsis data
ArabidopsisT 20 | ArabidopsisT 28 | |
---|---|---|
RJnsDBNs | 0.5070 | 0.5773 |
ASnsDBNs | 0.5929 | 0.5641 |
FLnsDBNs | G1:0.6138; G2:0.6150 | G1:0.6558; G2:0.6628 |
Comparison of TP|FP = 5 values on Arabidopsis data
ArabidopsisT 20 | ArabidopsisT 28 | |
---|---|---|
RJnsDBNs | 2 | 6 |
ASnsDBNs | 4 | 3 |
FLnsDBNs | G1:8; G2:8 | G1:11; G2:11 |
The experimental results on Drosophila data. For the Drosophila data, We run 10,000 iterations for burn-in and then take additional 990,000 iterations to collect samples. The sample collection of FLnsDBNs on the Drosophila data takes about 10 hours.
These four predictions share many similarities and also show some difference. We find that the gene msp-300 may play a key role in the cluster of these 11 genes. myo-61f is only predicted to be a regulated gene by msp-300 in [33], but other three methods show that myo-61f is another key gene in this cluster. In [33], myo-61f is correlated with twi, sls, mlc1, mhc and msp-300. In RJnsDBNs (KNKT), myo-61f serves as the regulators of prm, up and sls. Our approach predicts that myo-61f regulates four genes: sls, prm, actn, and msp-300. FLnsDBNs, [33] and [34] all agree that there are regulation relationships between myo-61f and msp300, while RJnsDBNs (KNKT) did not identify this interaction. Different from the prediction of RJnsDBNs (KNKT), Our approach finds that twi is not separated from other genes and actn serves as the parents of other genes, which is consistent with the networks in [33]. In Figure 10E, twi is the regulator of sls, and actn regulates sls, prm and gfl. We also notice that the regulating effects of myo-61f and msp-300 on other genes intensify over the time. Nearly different from all of three methods, our approach finds that twi and gfl/lmd are regulators of other genes while only [33] sees twi as a regulator. gfl/lmd and twi are direct upstream regulators of mef2[35, 36] that directly regulates some target myosin family genes at all stages of muscle development [37], such as mhc and mlc1. Evidence show the cooperative binding of twi and Mef2 or gfl/lmd and Mef2 to these target genes are attractive models [35, 37]. It indicates that a co-regulation role of twi and gfl/lmd with Mef2 to other muscle development genes may exist. The prediction of our method shows this biological behavior. Currently the reference regulatory network on the muscle development of Drosophila melanogaster is not available and the relevant biological literatures are limited. Further biological researches and experiments are needed to verify the regulatory networks.
Conclusion
In this paper we introduced a new non-stationary DBNs method and applied our approach on three time series microarray gene expression data. Our new DBNs method uses a systematic way to determine potential regulators and takes a flexible lag choosing mechanism. Our experimental study demonstrated that compared with recent proposed non-stationary DBNs methods, our approach has better structure prediction accuracy. By detecting potential regulators, our method reduces the size of the search space, hence may speed up the convergence of MCMC sampling.
Declarations
Acknowledgements
This work is partially supported by NSF IIS award 0845951. Data sets and softwares are provided by Dr. Grzegorczy at the University of Edinburgh, UK and Mr. Robinson at Duke University.
This article has been published as part of BMC Bioinformatics Volume 11 Supplement 6, 2010: Proceedings of the Seventh Annual MCBIOS Conference. Bioinformatics: Systems, Biology, Informatics and Computation. The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2105/11?issue=S6.
Authors’ Affiliations
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