Identifying targets of multiple coregulating transcription factors from expression timeseries by Bayesian model comparison
 Michalis K Titsias†^{1}Email author,
 Antti Honkela†^{2, 3}Email author,
 Neil D Lawrence^{4}Email author and
 Magnus Rattray^{4}Email author
DOI: 10.1186/17520509653
© Titsias et al.; licensee BioMed Central Ltd. 2012
Received: 20 December 2011
Accepted: 30 May 2012
Published: 30 May 2012
Abstract
Background
Complete transcriptional regulatory network inference is a huge challenge because of the complexity of the network and sparsity of available data. One approach to make it more manageable is to focus on the inference of contextspecific networks involving a few interacting transcription factors (TFs) and all of their target genes.
Results
We present a computational framework for Bayesian statistical inference of target genes of multiple interacting TFs from highthroughput gene expression timeseries data. We use ordinary differential equation models that describe transcription of target genes taking into account combinatorial regulation. The method consists of a training and a prediction phase. During the training phase we infer the unobserved TF protein concentrations on a subnetwork of approximately known regulatory structure. During the prediction phase we apply Bayesian model selection on a genomewide scale and score all alternative regulatory structures for each target gene. We use our methodology to identify targets of five TFs regulating Drosophila melanogaster mesoderm development. We find that confident predicted links between TFs and targets are significantly enriched for supporting ChIPchip binding events and annotated TFgene interations. Our method statistically significantly outperforms existing alternatives.
Conclusions
Our results show that it is possible to infer regulatory links between multiple interacting TFs and their target genes even from a single relatively short time series and in presence of unmodelled confounders and unreliable prior knowledge on training network connectivity. Introducing data from several different experimental perturbations significantly increases the accuracy.
Keywords
Bayesian inference Gene regulation Transcription factor Gene regulatory network Systems biologyBackground
A major challenge for computational systems biology is the inference of gene regulatory networks (GRNs) from highthroughput data such as gene expression timeseries[1–5]. This is particularly challenging when the available timeseries are short (i.e. contain few time points) and multiple regulators interact through cooperative or competitive mechanisms. An important first step towards uncovering regulatory networks is the identification of the targets of regulatory factors, particularly transcription factor (TF) proteins which control the transcription rate of their target genes through DNAbinding associations. In this paper we develop a computational method to infer the targets of a set of coregulating TFs using expression timeseries data from a small number of conditions. Our method is based on first learning the nature of the TF activities by focussing on a wellcharacterised subnetwork of targets and then performing genomewide scans to locate other targets of the TFs. A flexible regulation model accounts for nonlinear response, TF interactions and protein/mRNA degradation. A Bayesian model scoring procedure provides a principled framework for comparing alternative regulation scenarios for each putative target gene and determining the statistical support for direct regulatortarget relationships.
An experimental approach to identifying TF targets might involve the design of mutant strains with the TF perturbed (knocked out, knocked down or overexpressed) and differences in the gene expression of all putative targets analyzed[6–8]. When considering multiple regulators such experiments are difficult to design since all combinations of regulators have to be probed. It can also be very difficult to differentiate between direct and indirect regulation from perturbation data. An alternative or complementary experimental approach is to discover the binding sites of regulating TFs of interest through chromatin immunoprecipitation (ChIP) experiments[9, 10] (ChIPchip or ChIPSeq). This provides an excellent means to identify direct TF regulation. However, not all binding events show a clear relationship with gene regulation[11] and bound enhancers that are not close to a promoter region may be difficult to assign to a particular target gene. To capture transient regulatory events it is necessary to carry out a ChIP experiment in timeseries[12] and this may be prohibitively costly and time consuming for multiple TFs. Gene expression timeseries data therefore remain an immensely useful resource for uncovering the functional significance of regulatory interactions and to help confirm enhancertarget relationships.
Many computational methods have been introduced to infer or “reverse engineer” GRNs from timeseries expression data[1, 3–5]. Many of the proposed methods focus on uncovering the regulatory network for a subset of regulatory genes that are assumed to form the core of a regulatory network. This subset is typically identified as a preprocessing step, e.g. all differentially expressed or periodic TFs. Popular methods include statespace models[13], dynamic Bayesian networks[14] and ordinary differential equation (ODE) models[15–17]; see[5] for a recent review and comparative assessment on real and synthetic timeseries datasets. A related but more constrained problem than GRN inference is the identification of the targets of one or a few TFs that are known to be of functional significance[18–20]. Such an approach can be applied to find targets genomewide without very substantial filtering to reduce the set of putative targets. This target identification problem is often not aimed at identifying the full GRN model since only a limited number of TFs may be considered. However, genomewide target identification is very useful for identifying regulated pathways or processes, or for prioritizing targets for further analysis (e.g. integrating with other evidence such as ChIP or in situ expression data) or further experiments (e.g. ChIP or perturbation experiments on highranking targets). An example is the work of Barenco et al.[18] who used Bayesian inference over a linear activation model to rank targets of a single TF. They considered the case of a TF activated by posttranslational modification in which case a small set of known targets are required to learn the TF activity prior to ranking putative targets. In subsequent work by Gao et al.[21], Gaussian process inference techniques were developed for the same model and for nonlinear generalisations (Hill kinetics activation and repression models)[21]. Honkela et al.[20, 22] extended the Gaussian process method for target ranking in the case of a TF under transcriptional control by including a model of TF translation. In this case a set of known targets is not required to fit the model.
The target identification methods of Barenco et al.[18] and Honkela et al.[20] are restricted to the case of a single regulating TF. This is a useful simplification when data are limited but often TFs interact to regulate their targets through cooperative or competitive processes. Methods that ignore such interactions may have reduced accuracy in identifying targets and cannot be used to identify coregulation of targets by multiple TFs. Other methods have been developed which allow for regulation by multiple regulators. A popular method is the Inferelator[15] which is based on fitting an ODE model with a sigmoidal nonlinear regulation function to all putative regulatortarget interactions. Sparse regression techniques are used to identify the regulatory network by setting the influence of unsupported links to zero. The Inferelator was one of the top performing methods for GRN inference in recent Dialogue for Reverse Engineering Assessments and Methods (DREAM) competitions for network inference[23] (DREAM 3[24] and DREAM 4[25]). Unlike other GRN inference methods for timeseries data, such as statespace models[13], dynamic Bayesian networks[14] and other ODEbased methods[17], the Inferelator can be used for the more limited target identification task since it models the single layer targetregulator network in a decoupled manner. The highly efficient methods for inference developed for the Inferelator allows the model to be applied to large sets of regulating TFs, making this an attractive and highly practical tool for target inference from timeseries data. The method is also quite general and can incorporate steadystate expression data from perturbation experiments.
A confounding aspect when applying target prediction models for a limited number of regulators is the presence of TFs that are unknown or other unmeasurable influences on the system. We show on simulated data that, despite the presence of such confounding influences, our model can reconstruct the influence of multiple regulators of interest. We also show how data from additional conditions can easily be incorporated to improve inference when available.
Results and discussion
Overview of the method
Our approach is based on three main components: i) the use of ODEs to model transcription, translation and mRNA/protein decay, ii) a known set of TFs that regulate transcription and iii) datadriven inference of the model parameters and network structures by using a fully Bayesian statistical method[26]. To infer TF activities over time, which can be considered functional parameters in our model, we extend previously developed Gaussian process inference techniques[20, 21] to the case of multiple TFs interacting through a nonlinear regulation function. Here we provide a brief description of the methodology and introduce notation that is useful for the presentation of the results. A detailed description is given in Methods and the supplementary information.
In the transcription equation, the TFs can jointly modulate the mRNA production rate of a target gene through the response function G(·) (see Methods). The equation also models mRNA degradation with rate d_{ j } while b_{ j } represents a basal production rate and s_{ j } is a sensitivity parameter. The response function takes a sigmoidal form that nonlinearly transforms the TF activities so that saturation effects are taken into account and the TFs can competitively or cooperatively activate or repress transcription[27]. The response function also depends on parameters θ_{ j }which determine the network structure and regulation model coefficients. These parameters include weights that can effectively model n th order reactions, thus approximating the effect of, for example, TF dimerisation. Similarly, the translation equation explains the production rate of the active TF protein as a function of its mRNA while accounting for the protein degradation with rate δ_{ i }. We assume that the main ratelimiting step in production of active TF protein is transcription. Thus the TF activity can be considered equivalent to the TF protein concentration. This is thought to be a reasonable assumption for TFs in the Drosophila embryonic developmental system considered later[28] but in other systems TFs may be primarily regulated by posttranslational modifications. In the Drosophila system there is significant evidence for dimerisation of the TFs, see e.g.[29–32], but no evidence of regulation by other posttranslational modifications. In systems where TF activity is actively regulated by posttranslational modification, e.g. through phosphorylation by a signalling pathway, then the above translation model would not correctly model changes in the concentration of active TF protein in the nucleus. However, the modelling framework that we propose can still be applied by removing the translation equations and modelling the TF protein activity as a driving latent function; see[21] for examples of this approach to TF activity inference.
 1.
Training phase: Here, we use the dynamical models to estimate the TF activities, p _{ i }(t), by using a small set of training genes. The approximate structure of this subnetwork is assumed to be given so that for these genes the regulating TFs are known to some degree. All other model parameters are unknown and are inferred from the data. In this phase both the transcription model and the translation model are used to estimate the TFs. Observations associated with both the mRNA of the training genes and the TF mRNAs are required. The training phase could be carried out without the translation model in cases where TF protein activity is regulated by posttranslational modification. Extensive experimentation with artificial data reveals that, when appropriate, combining a translation model with TF mRNA observations greatly aids in estimation of the TF activities.
 2.
Prediction phase: Once the TF activities have been estimated, each test gene (for which the regulating TFs are unknown) is processed independently and the parameters (θ _{∗},d _{∗},b _{∗},s _{∗}) are inferred. Here, only the transcription model is needed while the translation model is irrelevant. This phase is applied on a genomewide scale and aims to identify the regulating TFs for each test gene.
The above phases can be applied to a situation where prior biological knowledge provides information only about a small set of wellstudied genes for which the regulating TFs are known to some degree. These genes are treated as the training data that are used to infer the activity profiles of the TFs. Typically, a full genomewide list of targets of the TFs is unknown. This is the motivation behind the second phase which applies the trained models for genomewide prediction of network links between genes and TFs. An important property of the second phase is that it is trivially parallelizable which allows for fast computations. The algorithms for fitting the models are based on Bayesian probabilistic inference and details are given in the supplementary information.
We will illustrate our methodology using mesoderm development in embryonic Drosophila melanogaster. First, though, we create an artificial example that highlights the difficulties inherent in inference of transcription networks directly from data.
Synthetic data
The purpose of our experiment with simulated data is to predict the set of regulating TFs for each gene using artificially generated mRNA measurements. Since the groundtruth network links are known, we can make a rigorous assessment of the ability of the model to identify the target genes of each of the TFs, as well as an assessment of the ability to predict nonregulation. The modelling is split into two distinct phases as described in the previous section. In the training phase, 30 genes with approximately known connectivity were used for learning the TF profiles. Specifically, to make the training phase more realistic we added 15% noise to the groundtruth network links in these 30 training genes. This resulted in 16 links between TFs and genes (in the initial groundtruth network structure) to change so that some of these links falsely became active and others were removed (i.e. from active they became inactive). Notice that this noise in the network links adds an extra modelmismatch in addition to the presence of the UNK TF which is not part of the model. In the prediction phase these profiles were used to rank other potential targets of the TFs from the remaining 1000 genes. Full details on how the data have been generated are given in Methods, while the dataset is provided together with software that is available online.
To assess the predictive ability of the model with respect to the amount of information present in the data, we consider three experiments. In the first experiment only data from the first experimental condition are used, in the second experiment only data from the second experimental condition are used, while in the third experiment all data from both conditions are considered.
Using data from one experimental condition
Here, we assume the synthetic mRNA data are produced by a single experimental condition, i.e. either the first or the second condition mentioned earlier. When considering the first condition the true TF profiles for ANT, BEE, CAR and UNK are shown in the left plot of Figure3(a) and the corresponding TF mRNA functions are shown in Additional file1: Figure S1(a). The remaining three plots in Figure3(a) show the TF activities estimated in the training phase by using 30 genes with approximately known network connectivity and unknown model parameters. The coloured solid lines show the estimated means and the shaded areas represent 95% posterior credible regions around the estimated means. Plots showing how the model fits the mRNA data in the training phase are presented in Additional file1: Figures S2 and S6 and all corresponding ODE parameters are shown in Additional file1: Figures S9 and S10.
From the above experiment we can conclude that it is rather difficult to accurately predict network links between TFs and genes from experimental data obtained under conditions that do not disambiguate sufficiently the functionality of the TFs during the transcription process. Roughly speaking, the “similarity” of some TFs causes the observed mRNA data to be well explained by alternative hypotheses associated with the presence/absence of these similar TFs and makes it hard to statistically identify which of those TFs were actually driving the regulation process.
We now consider a second series of observed mRNA measurements associated with an alternative simulated experimental condition comprising a perturbation of the biological system that better disambiguates the two (previously overlapping) TFs in terms of their influence in gene transcription. We first use only these new data instead of the data associated with the first experimental condition. This alternative perturbation changes significantly the protein activity for BEE as shown on the left plot in Figure3(b), while ANT, CAR and UNK are assumed to behave similarly to the first experimental condition. The estimated TFs are shown in the plots of the remaining three columns of Figure3(b) and model fits in the training mRNA data for this second condition are plotted in Additional file1: Figures S3 and S7 and all associated ODE parameters are shown in Additional file1: Figures S11 and S12. The blue ROC curves in Figure4 show predictive performance when using this second experimental condition. As the blue curves indicate, the performance now improves compared to the results obtained by using the first condition (red curves). This is expected since the second condition disambiguates more efficiently the TF activities than the first condition. In the next section we will see that the performance can be further improved when the models are fitted simultaneously to data from both experimental conditions.
Combining the data from both experimental conditions
In our third experiment we fit the models using all data from both experimental conditions. Figure3(c) shows the TFs that generated the mRNA data for both experimental conditions (plots in the first column from the left) and the estimated TFs (plots in the remaining three columns). Each row of Figure3(c) corresponds to each of the two conditions. Model fits in the training mRNA data are plotted in Additional file1: Figures S4 and S8 and all associated ODE parameters are shown in Additional file1: Figures S13 and S14.
Including data from both experimental conditions allows for a more confident estimation of the TF profiles. To see this, we can contrast the second up to fourth plots in the first row of Figure3(c) with the corresponding plots of Figure3(a)(b). The credible regions when simultaneously using both experimental conditions are significantly smaller, which implies higher confidence.
Furthermore, we obtain a significant increase in the predictive performance when identifying network links. As the green coloured ROC curves in Figure4 reveal, the performance when predicting single network links is significantly improved. Finally, we can exploit the ability of the model to predict a simultaneous regulation of the target gene by two or more TFs. Additional file1: Figure S5 displays the predictive ROC curves for all three TF pairs in this example.
Drosophila data
Prediction of network connections
Once the TF activities have been estimated, we use the model to predict the regulator TFs for a set of 6003 test genes which exclude the 92 genes used in the training phase. A webbased browser that displays how the model fits the mRNA data of test genes is available online at[34]. Full posterior probabilities of all alternative models for all test genes are included in Additional file2. This set includes all genes in the data that are not classified as weakly expressed according to the criterion explained previously[20]. We followed an approach to evaluation of predictive performance similar to one described in[20]. A number of predictions is evaluated by considering for each gene a predicted set of regulators correct if all TFs in the set had evidence of binding within 2000 base pairs of the corresponding gene in the ChIPchip data in[12]. Different methods can be compared based on the corresponding percentage enrichments. It should be noted that this validation is still far from perfect since bound enhancers can regulate transcription from a distance greater than the conservative limit considered here. We also perform similar evaluation using TFgene links in the Drosophila Interaction Database (DroID)[35]. This database in not specific to development and may thus include links that are not active in our data. We only include the 5521 test genes with some predicted TF regulators in the database. We compare two variants of our proposed method to a maximumlikelihoodbased baseline method, the Inferelator 1.1[15] and a simpler sparse regression approach (see Methods).
Linkspecific ChIP evaluation bootstrap results
Predicting regulation by single TFs  

Top 50  Top 100  Top 200  
P32  P2  ML  Inf  P32  P2  ML  Inf  P32  P2  ML  Inf  
P32  ***  *  P32  **  **  P32  **  **  
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Predicting regulation by TF pairs  
Top 50  Top 100  Top 200  
P32  P4  ML  Inf  P32  P4  ML  Inf  P32  P4  ML  Inf  
P32  **  +  P32  *  .  P32  +  *  
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ML  ML  ML  
Inf  .  Inf  Inf  
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P32  P4  ML  Inf  P32  P4  ML  Inf  P32  P4  ML  Inf  
P32  *  **  P32  +  ***  P32  **  
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ML  ML  .  ML  +  
Inf  Inf  Inf  
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Inf 
For the TFpair regulator rankings, we compute the marginal posterior probabilities for all possible pairs of TFs for each gene. The counterpart of Posterior2 now includes four models: the pair, both partners individually and the null, and is denoted by “Posterior4”. Otherwise the ranking lists are computed exactly as in the case of singleTF regulators but now for the 10×6003 possible TFpair models. Figure6(b) displays the results. The figure again shows statistically highly significant enrichment of binding of predicted regulators near the corresponding target genes. The enrichment is lower than it was for singleTF predictions, which is expected since the task of identifying regulating pairs of TFs is harder but may also be partly due to an increased number of false negatives in the validation data. The Bayesian methods based on posterior probabilities are consistently more accurate than the maximum likelihood baseline. In most cases the more restricted set of models seems to yield better results. Nevertheless, there are some TFs for which the opposite is true, as illustrated by the corresponding results, broken down for each TF, that are shown in Additional file1: Figures S16 and S17. This may be because the more restricted posterior probabilities are less sensitive to misspecification of prior probabilities of network links. Currently all TFs are assumed to regulate every gene with prior probability 0.5, which is unrealistic. Unfortunately it is nontrivial to construct better alternatives without significant extra information because the TFs are heavily correlated. We did not wish to use the ChIP data for constructing such a prior since this was required as independent data for validating the results.
Full model ChIP evaluation bootstrap results
Validating both positive and negative predictions  

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Validating only positive predictions  
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Running times
MAP32  Baseline  Regression  Inferelator 

5911.50  2236.36  0.96  0.25 
Linkspecific DroID evaluation bootstrap results
Predicting regulation by single TFs  

Top 50  Top 100  Top 200  
P32  P2  ML  Inf  P32  P2  ML  Inf  P32  P2  ML  Inf  
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Predicting regulation by TF pairs  
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P32  P4  ML  Inf  P32  P4  ML  Inf  P32  P4  ML  Inf  
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P32  P4  ML  Inf  P32  P4  ML  Inf  P32  P4  ML  Inf  
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Full model DroID evaluation bootstrap results
Validating both positive and negative predictions  

Top 200  Top 400  Top 800  
MAP  ML  Reg  Inf  MAP  ML  Reg  Inf  MAP  ML  Reg  Inf  
MAP  ***  ***  ***  MAP  ***  ***  ***  MAP  ***  ***  ***  
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Reg  Reg  Reg  
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MAP  ML  Reg  Inf  MAP  ML  Reg  Inf  MAP  ML  Reg  Inf  
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Validating only positive predictions  
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Parameter estimates
Inferred protein degradation rates
Degr. (1/h)  halflife (h)  

TIN  (0.81, 1.19, 1.66)  (0.42, 0.58, 0.86) 
BIN  (0.62, 0.80, 1.08)  (0.64, 0.87, 1.12) 
TWI  (3.79, 4.62, 5.79)  (0.12, 0.15, 0.18) 
BAP  (3.08, 5.41, 8.27)  (0.08, 0.13, 0.23) 
MEF2  (0.57, 0.80, 1.67)  (0.41, 0.86, 1.20) 
Discussion
It may be thought that a typical short timeseries expression dataset contains only very limited information about the structure of a GRN. In a metaanalysis of methods proposed in the DREAM 2 competition[37], the authors in[4] found timeseries data to be much less informative for network inference than data from a similar number of perturbation experiments. However, in the datasets considered there many of the timeseries experiments are rather uninformative about expression changes given the level of noise in the data and uninformative selection of sampled time points. We would argue that the success of a method for analysis of timeseries data will depend greatly on how informative the profiles of the regulatory species are. In our synthetic example we clearly demonstrated how inference is sensitive to confounding by highly similar temporal profiles of regulating TFs, so it is certainly desirable to have access to data from diverse experimental conditions where available. Yet with an animal system the available perturbations may be severely limited and the wildtype under normal conditions is of great interest for understanding healthy function. Methods for learning the structure of a regulatory network from one or a few short time course experiments are of great practical importance for uncovering a conditionspecific GRN.
Many methods for the inference of GRNs from gene expression data require much more data, and data from a much greater diversity of experiments, than we consider here[1–3]. However, several approaches have been proposed for identifying the targets of a specific TF given data from timeseries experiments collected under one or two conditions[18–20]. The methods in[18] and[20] rank targets by fitting simple linear activation differential equation models for a single regulating TF. These methods do not account for the more general and realistic scenario of nonlinear regulation by multiple TFs. The method in[19] does allow for regulation by other unknown factors, modelled by fitting a sparse linear regression model, but assumes measurements of the TF protein are available. Here we introduced a much more general method, where a model of nonlinear regulation by multiple TFs is used to predict which set of TFs regulate each putative target on a genomewide scale. Bayesian inference methods provide a principled approach for (i) dealing with an underdetermined inference problem by Bayesian parameter averaging, (ii) scoring alternative networks by Bayesian model selection and (iii) predicting TFtarget associations by Bayesian model averaging. Our results demonstrate that even with very limited timeseries information the method is able to correctly identify which of the closely related TFs regulate the given target. This is clearly a more challenging task that is not addressed in[18] and[20]. Additional information, even just independent estimates of decay rates of different transcripts, would certainly make the task easier, as demonstrated in[38] and also our results on synthetic data.
mRNA degradation rates
mRNA degrad. rates  Protein degrad. rates 

(5%,median,95%)  (ANT, BEE, CAR, UNK) 
(0.123, 0.610, 4.807)  (0.994, 0.945, 0.640, 1.200) 
Inference of continuoustime TF activity profiles from short timeseries is an illposed problem. We resolve this through introduction of a Gaussian process prior that effectively assumes smoothness of the underlying functions[21]. While this assumption appears reasonable for the TFs studied here, there are situations where the TF is activated very rapidly through signalling, e.g. in a sensory GRN[39]. In these situations an alternative model better suited for fast transitions such as that presented in[40] may be preferable. Alternatively, the Gaussian process could be transformed to provide a sharper switching behaviour by passing it through a sigmoidal nonlinearity (cf. Gaussian process classification[41]) and the current inference methodology would remain applicable.
Carrying out Bayesian inference over nonlinear systems with functional parameters is very challenging. For parameter inference we have made use of stateoftheart methods for Markov chain Monte Carlo (MCMC) over functional degrees of freedom[42]. We have developed a novel fast method for calculating the Bayesian evidence score that allows us to carry out genomewide model scoring (see supplementary information). Our method is very easily parallelizable within the prediction phase and can therefore be considered a practical contribution to the functional genomics toolkit.
The data used here are very limited and therefore one must accept that the method will make many false predictions. To improve accuracy, predictions based on the analysis of expression data can be combined with evidence from complementary sources (ChIP data, in situ expression data, sequence motifs) to identify a confident regulatory network structure. For example,[20] show how the accuracy of modelbased prediction improves greatly when additional evidence from spatial expression data is considered. The Bayesian framework presented here provides a very natural means for integrating other sources of data or prior knowledge for network inference. For example, it would be straightforward to associate alternative regulatory structures (e.g. those in Figure2(b)) with different prior probabilities derived from ChIPchip binding patterns. These priors could be used to reweight the Bayesian model averaging scheme used to calculate the probability of network structures. We do not pursue this approach here because we want independent ChIPchip validation of our method’s performance. Alternatively, given timeseries ChIP data, one could include binding observations directly in the model. This would have the advantage that one could model measurement errors for both the expression and ChIP experiments.
Conclusion
We have introduced a computational approach for genomewide inference of the targets of multiple regulating TFs given timeseries gene expression data. Using a time course measuring changes in wildtype expression during the embryonic development of Drosophila we were able to show that the method makes predictions which are significantly enriched for TF and TFpair binding identified using ChIPchip experiments on the same system. Our method works by fitting and scoring differential equation models of transcriptional regulation. Initially we use the model to infer the temporal pattern of TF protein activity given a small subnetwork of mostly known structure. Subsequently we score alternative target gene regulation models to make genomewide target predictions. By using a fully Bayesian procedure we are able to automatically balance model complexity with data fit when scoring alternative models. Our method is readily parallelizable in the prediction phase, making it a practical tool for genomewide network inference. On artificial data we showed that our method is able to cope with the existence of unknown regulating TFs that are not modelled and we showed that data from more diverse experimental conditions can help disambiguate between TFs that have similar profiles in a single condition. However, as our Drosophila example shows, even a single wildtype time course can be highly informative about the underlying regulatory network if the TFs of interest are changing over time. By combining the model predictions with other independent sources of evidence, e.g. from ChIP and spatial expression patterns, it will be possible to identify a confident conditionspecific regulatory network.
Availability
Software and a webbased browser displaying results in the Drosophila experiment are both available online at[34].
Methods
Dynamical models
Here, the Idimensional realvalued vector${\mathbf{w}}_{j}={\left[{w}_{j1}\dots {w}_{\mathit{\text{ji}}}\right]}^{\top}$ stores the interaction weights between the j^{th} target gene and the I TFs. These interaction weights quantify the network links so that when w_{ ji }= 0 the link between the j^{th} gene and the i^{th} TF is absent. When w_{ ji }is negative or positive the TF acts as a repressor or activator respectively. w_{j 0} is a realvalued bias parameter. The set of scalar parameters θ_{ j } in the response function G(·) is defined to be θ_{ j }= {w_{ j }w,_{j 0}}. Since the transcription ODE model is linear with respect to m_{ j }(t), it can be solved explicitly as shown in the supplementary information. The above transcription ODE model generalizes previous singleTF models that were used to estimate the concentration function of a single latent TF[18, 21, 43]. While a sigmoidal form for the response function G(·) was considered in all our experiments, our algorithms could easily be adapted to handle different forms for G(·).
Furthermore, the simple linear translation equation can be solved explicitly as shown in the supplementary information. Finally, the parameters {θ_{ j },d_{ j },b_{ j },s_{ j },δ_{ i }} are model parameters in the ODEs which need to be estimated under the constraint that {d_{ j },b_{ j },s_{ j },δ_{ i }} attain nonnegative real values, while θ_{ j }= {w_{ j },w_{j 0}} can attain both positive and negative real values. When we search for TFs that act only as activators, w_{ j } is constrained to be nonnegative.
A more detailed description of the ODE models is given in section 2 of the supplementary information.
Training modelling phase
The dynamical models contain a set of unknown quantities: the transcription model parameters${\{{\mathit{\theta}}_{j},{d}_{j},{b}_{j},{s}_{j}\}}_{j=1}^{J}$, where J is the number of target genes, the unobserved TF protein activities${\left\{{p}_{i}\right(t\left)\right\}}_{i=1}^{I}$ and the TF protein degradation rates${\left\{{\delta}_{i}\right\}}_{i=1}^{I}$. To estimate these quantities in the training modelling phase we consider a Bayesian probabilistic approach. More precisely, the observed mRNA data are used to construct likelihood functions that explain how the data are generated from the dynamical models. Together with the mRNA data for each training gene j we also have a binary vector${\mathbf{x}}_{j}\in {\{0,1\}}^{I}$ that specifies the regulatory network structure for that gene so that x_{ ji }= 1 indicates the presence of the link between the gene and the i TF, while x_{ ji }= 0 indicates the absence of the link. Prior distributions are assigned to all unknown quantities. The prior over each protein activity p_{ i }(t) was defined through the translation ODE and the placement of a suitable prior on the TF mRNA function, f_{ i }(t), through the use of Gaussian processes; see e.g.[41]. Bayesian inference in the training modelling phase was performed by Markov chain Monte Carlo (MCMC) techniques[44] where all the above unknown quantities were inferred using suitable MCMC updates.
A more detailed description of the training modelling phase is given in section 3 of the supplementary information.
Prediction modelling phase
where Y indicates the data used in the training modelling phase. To obtain the above, we need to compute the predictive density p(y_{∗}x_{∗}Y) for any possible combination of regulating TFs, i.e. any value of x_{∗}, together with the associated probabilities p(x_{∗}Y). While p(x_{∗}Y) could be computed by the frequencies of the known connectivity vectors in the training genes, this is unreliable since the small set of training genes may not be representative about the prior distribution of links between TFs and genes. Therefore, we set these probabilities to uniform values so that the posterior probability in Equation (1) becomes proportional to its predictive density value p(y_{∗}x_{∗}Y). This latter quantity is intractable since it requires an integration over the parameters (θ_{∗},d_{∗},b_{∗},s_{∗}). We approximate it using a novel fast approximation to a marginal likelihood, described in detail in section 4.1 in the supplementary information, that follows ideas similar to Chib’s approximation[45].
Given the estimated probabilities p(x_{∗}y_{∗},Y), with${\mathbf{x}}_{\ast}\in {\{0,1\}}^{I}$, any query related to the regulating TFs of target gene ∗ can be answered. For instance, in the results we made use of the following quantities:
Similarly we can compute the marginal probability p(x_{∗i} = 1,x_{∗j} = 1y_{∗},Y) for a pair of links.
A more detailed description of the prediction modelling phase is given in section 4 of the Supplementary Information.
The “Maximum Likelihood Baseline” method
This method, that was used in the experiments in Drosophila, follows exactly the same structure as the Bayesian approach with the following two differences. Firstly, the model parameters (such as kinetic parameters in the ODEs) were not treated using a Bayesian manner and instead they were obtained based on maximum likelihood which provides point estimates. Secondly, each protein function, p_{ i }(t), was deterministically estimated by the translation ODE model and by setting the driving TF mRNA function, f_{ i }(t), to a piecewise linear interpolation function computed from the TF mRNA observations. Apart from the above differences, prediction using the baseline method is done exactly analogously to the Bayesian case.
The “Regression” method
where m_{ jn } is the observed mRNA of the target gene at time t_{ n }${\left\{{f}_{\mathit{\text{in}}}\right\}}_{i=1}^{I}$ the corresponding observed TF mRNA values,$({\left\{{w}_{\mathit{\text{ji}}}\right\}}_{j=1}^{I},{w}_{j0})$ are parameters to be inferred and ε_{ n } is Gaussian noise. Notice that,${\left\{{w}_{\mathit{\text{ji}}}\right\}}_{j=1}^{I}$ are interaction weights and w_{j 0} is a bias parameter. Network inference in this linear model reduces to finding the nonzero interaction weights. This problem would typically require sparse optimization methods based on ℓ_{1} regularization as considered in[46]. However, in our case such algorithms are not needed since the number of TFs is small (I = 5) and hence we can enumerate all possible 32 regression models and select the best model using crossvalidation. In the results reported in Figure7, we firstly computed for each gene the MSE scores on heldout data (using 12fold cross validation) for all 32 models. Subsequently, we selected the model with the smallest MSE score for each gene and finally we ranked all genes based on the latter MSE scores (in ascending order) to produce the rankings shown in Figure7.
Application of the Inferelator 1.1
We compared our method against Inferelator 1.1[15] which is available for download athttp://err.bio.nyu.edu/inferelator/. This is the most recent version for which source code is available and which can be easily used for new data. We set each gene in its own cluster but otherwise used the default settings. We interpreted the maximum of the absolute values β_{ i } of all weights corresponding to a specific regulator alone or in combination with another as the counterpart of the posterior probability for ranking the predictions. For pairs, the corresponding value was$\text{max}\left(\right{\beta}_{3},min(\left{\beta}_{1}\right,\left{\beta}_{2}\right\left)\right)$, where β_{1} and β_{2} are the weights of the components (x_{1}x_{2}) of the pair and β_{3} is the weight of$min({x}_{1},{x}_{2})$ (see Eq. (6) in[15]). Combining information from independent and interaction terms like this significantly increased the performance of the method. Ranking by β_{ i } was also used in DREAM3 challenge submission of the Inferelator team[47].
Preprocessing of the Drosophila data
As previously described[20].
Training set for the Drosophila data
The training set was constructed from the training set of 310 ChIP cisregulatory modules (CRMs) collected in[12] (Additional file1: Table S8). The modules were mapped to genes using the CRM activity database in[12] (Additional file1: Table S4). Multiple CRMs for a gene were combined by taking the union of detected binding. Weakly expressed genes as defined in[20] were excluded, leaving a training set of 92 genes with wellcharacterised TF binding profiles.
Bootstrap significance testing of ranking method performance differences
100,000fold bootstrap resampling was used to assess statistical significance of performance differences between different ranking methods. For each fold, the set of testing genes was resampled with replacement from the full set of 6003 genes. Topranked predictions within the resampled set were evaluated as usual and the fraction of folds where each method outperformed each other was tabulated.
Reduced training set for the Drosophila data using a robustified model
Since in the Drosophila data the target genes can be influenced by unknown factors that are not part of the model, we considered a robustified training procedure that filtered out genes not explained by the model. This procedure allowed us to reduce the initial set of 92 genes to 25 genes and was carried out as follows. Firstly, we performed a training phase using all 92 genes so that the likelihood functions had both preprocessing noise variances and additive genespecific adaptive variances. Then, genes having large inferred adaptive variances, which indicates that these genes cannot be explained well by the fiveTF model, are excluded so that finally a subset of 25 genes was retained. Then, the whole training phase was repeated using only the selected genes and without the additive variances this time. The selection involved setting a threshold, which was set to 0.01, so that genes having estimated adaptive variance larger than this threshold were excluded. The threshold value was chosen to be smaller than the average value of the preprocessing variances, which represent estimates of the actual observation noise in the gene expression measurements.
Robust fitting was also used in the prediction phase so that each test gene was fitted using a likelihood function in which the variance parameter was the sum of a fixed preprocessing noise variance and an adaptive variance. Again this allowed us to compensate for the model mismatch and the presence of other confounding factors which, while they could regulate the gene expression, are not part of the model. More details on the robustified fitting are given in Section 3 and 5.2 of the Supplementary Material.
DroID validation
We downloaded the TFgene interaction database from DroID (http://www.droidb.org, release 2011_11). Genes with no interactions in the database were excluded from the validation to avoid possible problems due to annotation incompatibilities.
Generation of the synthetic data
We generated synthetic mRNA timeseries data that correspond to 1030 target genes and four transcription factors: ANT, BEE, CAR and UNK. The TF activities are depicted in the first column of Figure3(c). For both experimental conditions, the TF activities have been generated by simulating the translation ODE equation by assuming certain profiles for the TF mRNA functions,${\left\{{f}_{i}\right(t\left)\right\}}_{i=1}^{4}$, which were chosen to have the profiles shown in Additional file1: Figure S1, and with protein degradation rates 0.994, 0.945, 0.640, 1.2 for the four TFs respectively. ANT, BEE and CAR are assumed to be known factors for which observations of their TF mRNA activities are available. UNK is assumed to be a confounding factor whose presence and origin is not known. Given these TF mRNA functions,${\left\{{f}_{i}\right(t\left)\right\}}_{i=1}^{3}$, noisy “observations” were obtained at ten nonuniformly spaced time points, t_{ k }∈ {0,1,2,3,5,7,9,11,14,18}, by adding zeromean Gaussian noise with variance 0.025 f_{ i }(t_{ k }) to the value f_{ i }(t_{ k }). Negative values were truncated to zero.
To generate mRNA observations for the target genes, we simulated the transcription ODE, given the known TF activities and by using model parameters (θ_{ j },d_{ j },b_{ j },s_{ j }) selected as follows. Each interaction weight w_{ ji } for the TFs ANT, BEE and CAR was selected from the distribution$0.5\mathcal{N}(0.5,1)+0.5{\delta}_{0}$ which with 0.5 probability sets the interaction weight to zero and with equal probability selects a value drawn from a Gaussian distribution with mean 0.5 and unit variance. The interaction weight for UNK was selected from$0.25\mathcal{N}(0.5,1)+0.75{\delta}_{0}$. Notice that when w_{ ji }= 0, the i^{th} TF does not regulate the j^{th} gene. The above procedure generates random sets of regulating TFs so that on average each target gene has approximately two regulating TFs. Each bias parameter w_{j 0} was drawn from the Gaussian$\mathcal{N}(0,1)$. The kinetic parameters (d_{ j },b_{ j },s_{ j }) plus an initial condition parameter a_{ j } (see supplementary information) were selected randomly from an empirical distribution obtained by applying the dynamical models to the 6095 genes (the 92 training genes plus the 6003 test genes) in the Drosophila data. This was done to obtain kinetic parameters that produce realistic mRNA profiles that closely resemble real gene expression data. Summaries of the values of these parameters are given in Table7. Given the above simulated mRNA functions the observations are obtained at the ten nonuniformly spaced time points, mentioned earlier, by adding zeromean Gaussian noise with variance 0.025 m_{ j }(t_{ k }) to the value m_{ j }(t_{ k }). Negative values were truncated to zero.
Notes
Abbreviations
 ChIP:

Chromatin immunoprecipitation
 CRM:

cisregulatory module
 DREAM:

Dialogue for Reverse Engineering Assessments and Methods
 GRN:

Gene Regulatory Network
 MAP:

Maximum a posteriori
 MCMC:

Markov chain Monte Carlo
 MSE:

Mean squared error
 ODE:

Ordinary differential equation
 ROC:

Receiver operating characteristic
 TF:

Transcription factor.
Declarations
Acknowledgements
MT, MR and NL acknowledge support from EPSRC Grant No EP/F005687/1 “Gaussian Processes for Systems Identification with Applications in Systems Biology”. AH was supported by Postdoctoral Researcher’s Project No 121179 of the Academy of Finland. The work was also supported under the European ERASysBio+ initiative project “SYNERGY” through BBSRC Grant No BB/I004769/2 (to MR and NL) and the Academy of Finland project No 135311 (to AH). This work was supported in part by the IST Programme of the European Community, under the PASCAL2 Network of Excellence, IST2007216886. This publication only reflects the authors’ views.
Authors’ Affiliations
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