Background
The study of gene regulatory mechanisms controlling cell proliferation and differentiation is central in developmental biology. Because all hematopoietic cells are easily obtained as individual cells, and due to high clinical interest, the development of lymphocytes is particularly well-studied 12. In mammals, all blood cells develop from pluri-potent, self-renewing hematopoietic stem cells (pHSC) of the bone marrow. In the classical model, these pHSC differentiate into common myelo-erythroid progenitors and common lymphoid progenitors 3. The latter give rise to all cells of the adaptive immune system, including T, B and natural killer cells, which are the focus of our work.
Lymphocytes are well characterized; they can be purified by fluorescence activated cell sorting (FACS) exploiting the large variety of cell surface antigens, which appear in specific order during differentiation as the result of a linear sequence of genomic rearrangements at the T and B cell receptor loci 45. Based on this, lineage-specific expression and roles of transcription factors have been studied extensively 126. It has been shown, for example, that Gata3 is required for CD4 T cell maturation and that Runx3 silences the CD4 gene in CD8 T cells. Very recently, a new class of regulatory RNAs, microRNAs, have been identified as being involved in lymphocyte cell development 789.
Several groups 45101112 have combined FACS mediated cell sorting and mRNA expression profiling to derive a more comprehensive picture of the lymphocytes in distinguishable developmental stages. Our interest focuses on these patterns of gene expression in the distinct stages of the developmental tree, the developmental profiles of genes; see Fig. 1 for a developmental tree. Observing such patterns, the first natural question to ask is whether further genes exhibit the same developmental profile; for example, are there other genes co-expressed with Gata3. It is reasonable to assume that genes with a prescribed pattern of expression, such as "up-regulated in proliferating cells", might be relevant for specific functions of cells in a particular stage of differentiation. Clearly, not all relevant developmental profiles are known beforehand, so clustering is the next logical step. Clustering allows us to divide genes into groups of similar developmental profiles, some of which will be irrelevant–genes expressed in all stages–others will differ in distinct branches of the developmental tree and thus indicate relevance for differentiation. Once the gamut of developmental profiles is determined, further questions can be addressed with statistical methods: which regulatory effects might cause differentiation, which subgroups of developmental stages share regulatory patterns or at which developmental stage is the difference in expression between two groups the largest. Prior work in this context relies on classical clustering methods, such as self-organizing maps 45, hierarchical clustering 12, or on performing tests of differential expression between cell types of interest 11. Further studies concentrated on small-scale data, where selected genes are used to infer regulatory networks. One such study applied a state-space model to infer networks of T cell activation 13. Troncale and colleagues adopted Petri Nets to model and infer regulatory networks of early pHSC development 14, while Basso and colleagues proposed a novel algorithm for a similar task 15.
<p>Figure 1</p>
Schematic view of lymphocyte cell development
Schematic view of lymphocyte cell development. Developmental stages are depicted as nodes and arrows indicate transition from one stage to another, i.e. specialization. Self-renewing hematopoietic stem cells give rise to T cells in the thymus (green), B cells in the bone marrow (blue) and natural killer cells (NK) via intermediate stages. DN stands for CD4-/CD8-double negative cells, DPL for CD4+/CD8+ double positive large cells, and DPS for CD4+/CD8+ double positive small cells. Cell surface antigens and rearrangement events are partially annotated. The expression data sets investigated in this paper are marked as follows: green ovals for TCell, blue ovals for BCell, and pink boxes for LymphoidTree. We do not investigate developmental stages and transitions depicted in grey.
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Classical clustering relies on distance functions between developmental profiles such as correlation or Euclidean distance, which neglect the dependence structure of the developmental tree (Fig. 1). As a matter of fact, the clustering result does not change if one permutes all the variables. Biology suggests however, that the very sequence of changes does matter as this exact sequence of events is what takes a cell from pluri-potent to, say, mature B-cell. Thus we propose dependence tree models–see 16 for the discrete variate version–to model expression during the course of development. Our model assumes that the dependence of gene expression between subsequent stages is the most relevant one for identification of co-expressed genes. We assume that gene expression has been measured for a sufficient number of stages, in particular those relevant for differentiation processes, and that the cell population in a particular stage is sufficiently pure. The disagreement between reality and our assumptions is subsumed as noise, which our method can successfully deal with on simulated data. If we consider all pairwise dependencies between developmental stages our model would be equivalent to a multivariate Gaussian distribution with full covariance matrix. Due to the complexity the estimation of such models is prone to over-fitting 1718. The dependence tree model represents a tradeoff between methods assuming independence between variables, such as k-means and hierarchical clustering, and complex models, such as multivariate Gaussians, which makes estimation more robust.
With one such tree we can find genes with a specified developmental profile, for example similar to the developmental profile of Gata3, by ranking genes in order of decreasing likelihood under the tree. To cluster developmental profiles we combine several trees with the same topology but with distinct parameters in a classical mixture model 17; tree topologies are taken from the biological literature. Thus we obtain a robust and flexible statistical model for clustering genome-wide mRNA expression data sets, which takes the inherent dependencies between developmental stages explicitly into account. The resulting clusters of genes sharing similar developmental expression profiles are well-suited for a subsequent search for common regulators such as transcription factors or microRNAs.
Our choice of model class is motivated by the successful application of mixtures of complex statistical models to the analysis of mRNA expression time-courses. There, models that take temporal dependencies into account, such as Splines 1920, Autoregressive models 21 or Hidden Markov models 22, outperform simpler models, which assume independence of the variables, for example k-means, self-organizing maps or hierarchical clustering.
For discrete variates, dependence trees were first proposed by Chow and Liu 16, who showed that efficient computation is possible. Mixtures of trees were first proposed and applied in image recognition problems 23, where more efficient versions of the structure learning algorithm for sparse data sets became necessary. In bioinformatics, mixtures of trees were applied to infer mutation events in HIV strains 24. We present an extension of the dependence trees to continuous variates, requiring modifications to the densities and provide a framework for robust clustering based on mixtures. To the best of our knowledge, there is no prior work on genome-scale mRNA expression analysis in which the developmental tree structure is taken into account. Both the biological application and our approach of combining tree models with mixture estimation for this purpose is novel. However, the main methodological ingredients are well-established. Our advanced statistical framework allows us to identify clusters of genes with similar developmental profiles. We detect interesting groups of genes not found using standard techniques, such as self-organizing maps 25, in developing lymphoid cells. Results on simulated data show the conditions under which our method has a technical advantage. From our clustering results we can identify plausible regulatory roles of microRNAs known to be involved in hematopoiesis. We provide a graphical user interface and a web database of clustering results; see 26 for implementations, a tutorial on how to use the tools, and a web database with the results presented below. Our findings suggest that our framework is well-suited for analysis of genome-wide expression data from detailed cell development studies.
Results/Discussion
In the next two sections, we describe the dependence trees and how they are combined in a mixture to find groups of developmental profiles. Subsequently, we present the results of the application of our method to three lymphoid cell datasets. In the last subsection, we analyze the groups of genes, given by our mixture of dependence trees (MixDTrees) results, for common microRNA binding sites patterns, in order to gain insights into regulatory function of microRNAs.
Dependence trees
The main assumption behind the dependence trees (DTree) is that expression levels of a particular developmental stage depend primarily on expression levels of the immediately preceding stage. For example, cf. Fig. 2, we can approximate the joint probability density function (pdf) of four random variables (XA, XB, XC, XD) by
<p>Figure 2</p>
Example of a simple developmental tree and a cluster of developmental profiles
Example of a simple developmental tree and a cluster of developmental profiles. On the left, we depict a simple development tree, where arrows represent dependencies between variables. Above each tree variable, we depict a distribution related to it. On the right, we display the gene expression values (y-axis) in the distinct development stages (x-axis). Each line corresponds to the developmental profile of a given gene of a particular path of the tree in the left, as in a time-course plot. Distinct paths have distinct colors, in correspondence with the tree on the left. In this particular example, we have the path A, B and C in green and B and D in red. By superimposing the lines corresponding to paths B to C and B to D, we can contrast the differences in expression values of genes in these two alternative differentiation pathways.
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p [XA, XB, XC, XD] ≈ p [XA]p [XB|XA]p [XC|XB]p [XD|XB].
In other words, we condition the probability of a given variable on its immediate predecessor, in accordance with the tree structure shown in Fig. 2. There, also a cluster of hypothetical genes with similar developmental profiles is depicted (Fig. 2, right). The genes display average expression in stage A, up-regulation in stage B, down-regulation in stage C and up-regulation in stage D. Furthermore, the genes have clearly distinct expression intensities, but similar relative expression changes. Genes strongly over-expressed in B are also strongly under-expressed in C and strongly expressed in D. These dependencies are reflected in the correlation between these stages. For example, A and B (or B and D) are positively correlated, and stages B and C are negatively correlated. A statistical model for such developmental profiles has to include these dependencies between subsequent stages, as it is provided by dependence trees. Let X = (X1, ..., Xu, ..., XL) be a L-dimensional continuous random vector where the variable Xu denotes the expression values of the developmental stage u and x = (x1, ..., xL) denotes a realization of X representing a developmental profile of a gene. We represent a tree by its predecessor or parent map, pa {1, ..., L} ↦ {1, ..., L} for which we assume without loss of generality that 1 < pa(u) < u and pa(1) = 1. Then we can write for the probability density function (pdf) of a conditional
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We denote the model parameters by θ = (τ1, ..., τu, ... τL) and the DTree by the tuple (X, pa, θ). Note, that a DTree can also be viewed as an approximation of the joint distribution of a L-dimensional continuous random vector by a product of L - 1 second order distributions 16.
We use conditional Gaussian density functions 27 as conditional densities, denoted by p [xu|xpa(u), τu] in Eq. 2. Hence, for a given developmental profile x and a non-root developmental stage u with pa(u) = v, the pdf takes the form
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where τu = (μu, wu|v, σu|v2
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For a given expression data set consisting of measurements for N genes at L developmental stages, let xi = (xi1, ..., xiu, ..., xiL) be the developmental profile of gene i, and xiu be the expression value of the gene i in development stage u for 1 ≤ i ≤ N and 1 ≤ u ≤ L. As derived in the Protocol in the Additional data file 1, the maximum likelihood estimates (MLE) for the parameters of the conditional Gaussian are
<p>Additional data file 1</p>
Protocol. This file contains information on software implementations, derivations of estimation formulas and additional experiments with simulated data.
Click here for file
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These terms can be computed from the sufficient statistics
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The conditional normal distribution can be seen as estimating a linear fit between Xu and Xv, where wu|v> 0 indicates a positive linear correlation and wu|v<0 a negative linear correlation between variables; wu|v = 0 if the variables are independent. Furthermore, wu|v and σu|v2
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A very simple, but useful application, is to query the developmental profiles from a data set with a tree model. By defining the model parameters in an interactive manner, we can compute the likelihood (Eq. 2) of all expression profiles xi. rank them accordingly, and list the m most likely profiles (see 26 for the tool description and tutorial). This interactive tool allows biological experts to find genes following a developmental profile of interest.
Returning to the example in Fig. 2, the model estimates given the tree and developmental profiles are
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.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqadeabfaaaaaqaaGGaciab=r8a0naaBaaaleaacqWGbbqqaeqaaaGcbaGaeyypa0dabaGaeiikaGIaf8hVd0MbaKaadaWgaaWcbaGaemyqaeeabeaakiabcYcaSiqbdEha3zaajaWaaSbaaSqaaiabdgeabbqabaGccqGGSaalcuWFdpWCgaqcamaaDaaaleaacqWGbbqqaeaacqaIYaGmaaGccqGGPaqkaeaacqGH9aqpaeaacqGGOaakcqGHsislcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqGGSaalcqaIWaamcqGGSaalcqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqGGPaqkcqGGSaalaeaacqWFepaDdaWgaaWcbaGaemOqaieabeaaaOqaaiabg2da9aqaaiabcIcaOiqb=X7aTzaajaWaaSbaaSqaaiabdkeacbqabaGccqGGSaalcuWG3bWDgaqcamaaBaaaleaacqWGcbGqcqGG8baFcqWGbbqqaeqaaOGaeiilaWIaf83WdmNbaKaadaqhaaWcbaGaemOqaiKaeiiFaWNaemyqaeeabaGaeGOmaidaaOGaeiykaKcabaGaeyypa0dabaGaeiikaGIaeGimaaJaeiOla4IaeGyoaKJaeG4naCJaeiilaWIaeGOmaiJaeiOla4IaeGOmaiJaeiilaWIaeGimaaJaeiOla4IaeGimaaJaeGOmaiJaeiykaKIaeiilaWcabaGae8hXdq3aaSbaaSqaaiabdoeadbqabaaakeaacqGH9aqpaeaacqGGOaakcuWF8oqBgaqcamaaBaaaleaacqWGdbWqaeqaaOGaeiilaWIafm4DaCNbaKaadaWgaaWcbaGaem4qamKaeiiFaWNaemOqaieabeaakiabcYcaSiqb=n8aZzaajaWaa0baaSqaaiabdoeadjabcYha8jabdkeacbqaaiabikdaYaaakiabcMcaPaqaaiabg2da9aqaaiabcIcaOiabgkHiTiabicdaWiabc6caUiabiMda5iabiMda5iabcYcaSiabgkHiTiabicdaWiabc6caUiabiodaZiabcYcaSiabicdaWiabc6caUiabicdaWiabigdaXiabcMcaPiabcYcaSiabbccaGiabbggaHjabb6gaUjabbsgaKbqaaiab=r8a0naaBaaaleaacqWGebaraeqaaaGcbaGaeyypa0dabaGaeiikaGIaf8hVd0MbaKaadaWgaaWcbaGaemiraqeabeaakiabcYcaSiqbdEha3zaajaWaaSbaaSqaaiabdseaejabcYha8jabdkeacbqabaGccqGGSaalcuWFdpWCgaqcamaaDaaaleaacqWGebarcqGG8baFcqWGcbGqaeaacqaIYaGmaaGccqGGPaqkaeaacqGH9aqpaeaacqGGOaakcqaIWaamcqGGUaGlcqaI0aancqaI1aqncqGGSaalcqaIWaamcqGGUaGlcqaI1aqncqaIZaWmcqGGSaalcqaIWaamcqGGUaGlcqaIWaamcqaIXaqmcqGGPaqkcqGGUaGlaaaaaa@C668@
As expected, wB|A and wD|B are positive, indicating a linear dependence between these variables. On the other hand wC|B is negative.
Mixtures of dependence trees
In order to find clusters of co-expressed genes, we combine several dependence trees (DTree) in a mixture. Each DTree is a representation of a cluster or group of genes with similar developmental profiles; that is, each DTree models distinct patterns of gene expression in the course of development (see Fig. 3 for an example). The differentiation of cells is conveniently represented as a developmental tree and the structure or topology of this tree is well-known for most data sets under investigation. Consequently, all trees in a mixture share the same topology. A mixture of dependence trees accommodates overlapping clusters while reflecting the inherent dependencies between stages. Throughout this paper we refer to the presented method as well as to the resulting model as MixDTrees.
<p>Figure 3</p>
Example of a mixture of four dependence trees with the topology defined in Fig. 2
Example of a mixture of four dependence trees with the topology defined in Fig. 2. Each of the trees models distinct developmental profiles found in an example data set. Furthermore, clusters may have distinct sizes proportional to their αi's. Note also that it is not necessary that clusters have distinct expression values in branching stages. For example, stages C and D have similar expression values for cluster 3 and 4. This can be interpreted as the genes being equally expressed in the two alternative lineages.
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More formally, we combine a set of K DTrees in a mixture model f(x|Θ)=∑k=1Kαkp[x|θk]
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGMbGzcqGGOaakcqWG4baEcqGG8baFcqqHyoqucqGGPaqkcqGH9aqpdaaeWaqaaGGaciab=f7aHnaaBaaaleaacqWGRbWAaeqaaOGaemiCaaNaei4waSLaemiEaGNaeiiFaWNae8hUde3aaSbaaSqaaiabdUgaRbqabaGccqGGDbqxaSqaaiabdUgaRjabg2da9iabigdaXaqaaiabdUealbqdcqGHris5aaaa@4902@, where Θ = (θ1, ..., θK, α1, ..., αK), θk denotes the parameters of the k-th DTree and αk is proportional to the number of developmental profiles assigned to the k-th Dtree; as usual αk ≥ 0 and ∑k=1Kαk=1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaaGGaciab=f7aHnaaBaaaleaacqWGRbWAaeqaaOGaeyypa0JaeGymaedaleaacqWGRbWAcqGH9aqpcqaIXaqmaeaacqWGlbWsa0GaeyyeIuoaaaa@3853@. To avoid over-fitting of the tree models, in particular for components with low component priors αk–that is, a small number of assigned genes–we propose a maximum-a-posteriori (MAP) approach, which regularizes the estimates from Eq. 5 and Eq. 6. Given this preferable characteristic, MAP estimates are used in all MixDTrees experiments, unless otherwise stated. Note also, the parameters of the mixture are estimated with the Expectation-Maximization (EM) algorithm 28 (see Methods section for EM and MAP details).
As stated in the introduction, the problem approached here is closely related to gene expression time-course analysis. There is a vast amount of literature on models and clustering methods suitable for time-courses 18192021222930. Lately, attention has been given to the fact that these time-courses have usually few time points 3132, a characteristic previously ignored. This aspect is also essential to our application, since the number of distinguishable developmental stages is usually small, for example at most seven in our data sets. Note that a single chain of subsequent development stages, such as the stages of B-cell differentiation in Fig. 1, is by definition a tree. While dependence trees are indeed also suitable for time-courses, the complex dependency structures necessary due to branching of the developmental tree into distinct lineages prevents the use of time-course models, as there is no effective way of incorporating the necessary extensions into these models 1922. In the context of mixtures, our method represents an alternative to the parameterization of the covariance matrix of a mixture of multivariate Gaussians 17. With MLE, the dependence tree model essentially imputes zeros in the covariance matrix reducing the number of parameters to the order of L. If we would consider all the covariances between observations for L developmental stages; it would be straightforward to represent the data distribution by a L-variate Gaussian model with full covariance matrix. However, the estimates for the L2 parameters are often unreliable even for small values of L and the parameter estimation is prone to over-fit to outliers often found in noisy and scarce data. In fact, mixtures of Gaussians with full covariance matrix were outperformed by simpler parameterizations of the covariance matrices in the context of gene expression time courses 18.
Application in lymphocyte cell development
We apply our method to obtain MixDTrees for the data sets TCell, BCell, LymphoidTree, and SIM (see Methods section for details) and compare our clustering results to previous work. Our data is complemented with information from OMIM 33, the Gene Ontology database 34 and from literature. For TCell and BCell, we use the same number of clusters as Hoffmann and colleagues (20) 4535 and for LymphoidTree we apply the BIC criterion 36 (see Fig. S4 in Additional data file 2), which also resulted in an optimal choice of 20 clusters. As discussed in Dependence trees section, a simple way to check for similarities in the expression between developmental stages is to compute the correlation matrix of the data set at hand (see Mixtures of dependence trees estimation section).
<p>Additional data file 2</p>
Supplementary Figures. Figures 1, 2 and 3 contains all clusters results from MixDTrees on BCell, TCell and LymphoidTree, and Figure 4 contains BIC results from LymphoidTree. Figures 5 and 6 contain comparisons between microRNA enrichment with MixDTrees-MAP and SOM in TCell and BCell, Figures 7 and 8 depict the empirical cumulative distribution function (cdf) of microRNA enrichment p-values from TCell and BCell, and Figures 9 and 10 contain comparisons between microRNA enrichment with MixDTrees-MAP and MixDTrees-MLE in TCell and BCell. Figure 11 describes the cluster size distribution of clustering results in TCell and BCell.
Click here for file
T cell development (TCell)
TCell is a gene expression data set from seven differentiation stages of the T cell development (see Methods section and Fig. 1 for details). The only branch in this tree is the final differentiation of DPS precursors into CD4 single positive SP4 cells and CD8 single positive SP8 cells. Most clusters show a distinctive pattern of differential expression along the developmental path but do not differ between SP4 and SP8 cells (clusters 4, 7, 11, 13, 14, 15, 16, 19, and 20). The most drastic changes occur at the DPL stage in which the cells are proliferating and subsequently start to rearrange the TCRα-locus. This is also reflected in the overall correlation matrix (Table S1 in Additional data file 3). Although the expression values of all neighboring stages are positively correlated, the correlation between the DPL stage and the DPS stage is much smaller in comparison to the double negative stages, all of which are relatively highly correlated. The correlation matrix suggests that SP4 and SP8 cells are more similar to each other than to their precursor DPS cells, which is expected since the two types of mature T cells share many cellular functions 4. The largest differences with respect to SP4 and SP8 are found in clusters 5 and 18 (Fig. 4). In cluster 5, cell-cycle genes are clearly enriched. In contrast, cluster 18 mainly contains regulatory proteins involved in transcription and signaling (see Fig. 4).
<p>Additional data file 3</p>
Supplementary Tables. Tables 1, 2 and 3 contains correlation matrices from BCell, TCell and LymphoidTree datasets; Tables 4, 5 and 6 contains enriched microRNA and gene targets from SOM results on TCell and BCell and from MixDTrees-MAP results on LymphoidTree; Tables 7, 8, 9 contains microRNA enrichment p-values for BCell, TCell and LymphoidTree on MixDTrees-MAP results; Tables 10 and 11 contains microRNA enrichment p-values for BCell and TCell on SOM results; Tables 12 and 13 contain the contingency tables comparing clusters from MixDTrees-MAP and SOM with BCell and TCell datasets; and Tables 14 and 15 contain the contingency tables comparing clusters from MixDTrees-MAP and MixDTrees-MLE with BCell and TCell datasets.
Click here for file
<p>Figure 4</p>
Selected clusters from MixDTrees on Tcell
Selected clusters from MixDTrees on Tcell. We depict the clusters 5, 8 and 18 found in TCell, expression values on the y-axis, and cell types on the x-axis. Lines corresponding to developmental profile values between stages DN2, DN3, DN4, DPL, DPS and SP4 are in green and between DPS and SP8 in red.
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Hoffmann and colleagues used self-organizing maps (SOM) to cluster the expression profiles 4535. From now on, we refer to Hoffman and colleagues' results simply as SOM. In our analysis we observe clusters with similar developmental profiles, which we define as the average over the gene expression profiles of a cluster. As expected, there is not a one-to-one relationship between the two clusterings. While the single gene profiles are similar since we used analogous normalization and filtering procedures (see Methods section), the actual gene clusterings differ (see Table S12 in Additional data file 3). An objective assessment of clustering quality on developmental data is impossible due to lack of benchmarking data. Furthermore, there is no agreement in the literature on a methodology to validate clustering results 37. In order to demonstrate that our method is able to extract additional biological information, we concentrate our discussion on clusters of distinct developmental profiles that could not be detected by SOM 4. For such a cluster we assign functions to genes using the GO term annotation and complementary literature. Ideally, the functions of all genes of the cluster would match the cellular processes of the particular developmental stage at which these genes are over-expressed. Additionally, if some of these genes are of unknown function then the developmental profile can help to generate hypotheses about their functional role. In our analysis we find that genes of cluster 8 are over-expressed in DN3 and DN4 cells, a developmental profile that has not been previously discovered (Fig. 4). With SOM, the genes of this cluster are dispersed over the two clusters (see Table S12 in Additional data file 3). Out of the 30 genes of cluster 8 seven are related to vesicle transport, or to the Golgi/ER system. Additionally, we find five cell-cycle related genes, three involved in mitochondrial function, and seven genes of other functions, which are mainly involved in signaling. These findings agree with the functions of DN3 and DN4 cells, which is the transport of precursor receptor molecules to the cell surface membrane and the initiation of proliferation. This demonstrates that our method is able to identify functionally relevant gene sets even if the expression changes are not as large as for the DPL stage, for example. The complete results, including gene expression plots, analysis of GO-term and microRNA enrichment, can be found in our web database 26.
B cell development (BCell)
In a similar approach to the TCell study, we investigated gene expression for five consecutive stages during B cell development (see Methods section and 45 for details). The correlation matrix of BCell suggests dependencies between gene expression values of successive stages, with the largest correlation between pre-BI and large pre-BII cells and between immature and mature B cells (see Table S2 in Additional data file 2). When we compare, as in the TCell set, our clustering results to those of Hoffmann and colleagues 4, we observe similar average developmental profiles although the contingency table indicates differences in the cluster compositions (Table S13 in Additional data file 3). Clusters 3, 5 and 6, for example, contain genes that are up-regulated in pre-BI and large pre-BII cells and down-regulated in later developmental stages (Fig. 5). Consistent with the phenotype of these cells, the function assigned to the genes of this cluster are mainly related to proliferation. GO categories that are associated with mitosis, cell-cycle and chromatin remodeling are clearly overrepresented in these clusters (see our web database 26).
<p>Figure 5</p>
Selected clusters from from MixDTrees on Bcell
Selected clusters from from MixDTrees on Bcell. We depict clusters 3, 5, 6 and 20 found in BCell, expression values on the y-axis, and cell types on the x-axis. Lines corresponding to developmental profile values between between all stages are in red.
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Cluster 20 shows an average developmental profile that was not detected with SOM 45. The genes of this cluster are down-regulated in pre-BI cells, in which the first rearrangement of the DH and JHsegments on the H chain loci has taken place, and up-regulated in all the following developmental stages (Fig. 5). With SOM 4, these 23 genes are found distributed over the four clusters 11, 13, 14 and 17 (Table S12 in Additional data file 3). The most palpable common function of many cluster 20 genes is the regulation of survival and apoptosis during B cell development. The gene products Nfkbia, Traf5 and the Src-family protein tyrosine kinases Lyn and Syk are known regulators of NF-kappa B activity, which in turn has been found to be involved in B cell fate decision and survival 383940. Similarly, Krupel-like factor 2 (Klf2) protects cells against TNF-alpha induced apoptosis 41. Furthermore, Icam-2 and Rhoh, whose encoding genes are two other members of cluster 20, regulate the adhesiveness of primary B cells depending on their activation state and protect them from apoptosis 3342.
Lymphoid tree (LymphoidTree)
LymphoidTree combines data sets of several studies 101112, and the resulting tree contains expression measurements from lymphoid cells of six developmental stages, namely hematopoietic stem cells, pro-B, pre-B, and immature B cells, mature SP4 T cells, and natural killer (NK) cells. This integration of data is possible because the studies were carried out on the same array platform. Although the developmental tree is far less detailed compared to TCell and BCell, we still gain insights on differences between the cell lineages. As expected, the correlation matrix shows that the expression patterns of the three B cell stages are more highly correlated among each other then expression patterns of different lineages. Moreover, the overall expression of SP4 cells and NK cells is positively correlated. The resulting clusters provide a basis to hypothesize about early developmental decisions and suggest target genes for further investigations. For example cluster 11 contains genes that are strongly up-regulated in NK cells, weakly induced in the SP4 cells and not expressed in the precursor B cells (Fig. 6). Many of the cluster 11 genes are well known to be expressed in NK cells, as for example the cell surface receptor genes Cd244, Klra1, and Crtam 3343. Among the lesser known genes is the one that codes for the Pu.1 related transcription factor SpiC, which has already been found to be temporarily expressed during B cell development 44. In contrast, cluster 19 contains genes that are up-regulated in SP4 cells and in all B cell precursors but not in NK cells (Fig. 6). Important functions during B and T cell maturation are reflected by genes in this cluster, like the bruton tyrosine kinase Btk, the transcription factor Pou2af1, which is involved in immunoglobulin gene regulation, and the DNA repair genes Trp53bp1 and Pnkp 33.
<p>Figure 6</p>
Selected clusters from from MixDTrees on LymphoidTree
Selected clusters from from MixDTrees on LymphoidTree. We depict clusters 11 and 19 found in LymphoidTree, expression values on the y-axis, and cell types on the x-axis. Lines corresponding to developmental profile values between stages HSC, pro-B, pre-B and immature B cell are in read, between HSC and NK cells in blue, and between HSC and SP4 cells in green.
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Simulated data (SIM)
We demonstrate with simulated data that our novel method outperforms established methods, such as SOM, k-means and mixture of Gaussians, when inferring tree components in complex mixtures for varying levels of dependence between the individual variates. The dependence is reflected in the magnitude of wu|v, k (Eq. 5) of a tree. By sampling these parameters from different intervals, [-ε, ε ], [-0.5, 0.5], [-1, 1], [-1.0, -0.5] ∪ [0.5, 1] and [-1, -1 + ε] ∪ [1 – ε, 1], we can create mixtures with components ranging from independent models to highly dependent ones. We generate a data set for each sampled mixture. We used MixDTrees, mixture of Gaussians, k-means and SOM to compute clusters, which we can compare to the classes used in data generation to compute specificity and sensitivity of the clustering solutions. Method performance is evaluated with a paired t-test. Details are given in Methods section.
We observe that the MixDTrees with MAP estimates (MixDTrees-MAP) have a higher specificity and sensitivity than k-means and SOM in all experimental settings (Fig. 7 top) (p-value below 0.005). In the (almost) independent case (wu|v, k ∈ [-ε, ε]), this is not expected, since the data agrees well with the assumptions of k-means and SOM. This also explains the large standard deviations of MixDTrees-MAP in that case. As expected, the MixDTrees-MAP clearly improves the cluster recovery in settings with pronounced dependence structure, while the performance of k-means and SOM deteriorates slightly. In comparison to others mixture model methods (Fig. 7 bottom), MixDTrees-MAP also obtains a significantly higher specificity and sensitivity in almost all experimental settings. The mixture of Gaussians with diagonal covariance matrices performs well in the independent case (1), which meets the model assumptions, but it has poor results in experiments with higher dependence (p-values below 0.05 for settings 3, 4 and 5). The mixture of Gaussians with full covariance matrix (MG-Full) has a reasonable sensitivity in all settings, but very poor specificity (p-value below 0.05 in settings 3, 4 and 5 for specificity and in all settings for specificity). The reason for these results is that MG-Full tends to populate some clusters with few data points, a problem known as spurious local maxima 17. Note that we use a MAP estimate of MG-Full to mitigate this problem. Even though there are other methods for detection of spurious local maxima in MG-Full, which could lead to better specificity, this would require extensions of the EM method, and consequently slower convergence 17. On the other hand, MixDTrees, which has a lower computational running time than MG-Full, achieves good results without the need of any extension. MixDTrees with MLE estimates (MixDTrees-MLE) has good overall performance, but is outperformed by MixDTrees-MAP in all cases, except experimental settings 1 and 5 (p-value below 0.05 for settings 2, 3 and 4). In experimental setting 5, where data is highly dependent, by definition, both methods work similarly well. Nevertheless, such high dependency would never be found in real data sets, since noise in the data obfuscates dependencies between variables. Additionally, we performed further experiments with simulated data to evaluate the robustness of the method with respect to noise (see Additional data file 1). There, MT-MAP maintains good sensitivity and specificity of cluster recovery even for high noise levels.
<p>Figure 7</p>
Results of SIM
Results of SIM. We display the mean sensitivity (left plots) and mean specificity (right plots) against five experimental settings: (1) wu|v, k ∈ [-ε, ε ] (independent data), (2) wu|v, k ∈ [-0.5, 0.5], (3) wu|v, k ∈ [-1, 1], (4) wu|v, k ∈ [-1.0, -0.5] ∪ [0.5, 1] and (5) wu|v, k ∈ [-1, -1 + ε] ∪ [1 – ε, 1]. The dependence increases with experiment number. On the top plots, k-means results are displayed in blue, SOM in green and mixture of dependence trees with MAP estimation (MixDTrees) in red. On the bottom plots, mixture of Gaussians with full covariance matrices are displayed in yellow, mixture of Gaussians with diagonal covariance matrices in purple, Mixture of dependence trees with MLE estimation in light blue (MixDTrees-MLE) and mixture of dependence trees with MAP estimation (MixDTrees-MAP) in red.
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This demonstrates that the MixDTrees is a superior alternative to SOM and k-means in all cases. In relation to other mixture models, MixDTrees represents a good tradeoff between a complex model class such as multivariate Gaussian with full covariance matrices and the simple Gaussian with diagonal covariance matrices. Furthermore, MAP estimates of the MixDTrees represent a more robust alternative to the MLE counterpart.
MicroRNA target discovery
LympMIR contains a set of 17 microRNAs that are potentially involved in lymphocyte cell development (for details see Methods section). It has been proposed that microRNAs bind target mRNAs specifically via base pairing, which subsequently leads to interference with the translational machinery or mRNA degradation, and thus can control whole groups of genes simultaneously 45. Recent microarray studies have demonstrated that the microRNA expression negatively correlates with mRNA target expression in a tissue specific manner 464748.
Having identified a cluster of co-expressed genes during lymphoid development we ask whether a certain microRNA could be a potential regulator of this cluster (see Fig. 8). For this task we first obtain lists of potential target genes for each microRNA from the miRBase Targets database 49, which contains predictions made by sequence based methods. Given our clustering results, we use the statistic of the Chi-Square Test 50 to obtain a list of microRNAs, whose potential targets are overrepresented in a cluster. This is an analogous approach to finding Gene Ontology 51 terms over-represented in a cluster of genes. Given a set of n genes, we count the number c of genes in a given cluster, the number t of genes identified as targets for a given microRNA and the number h of genes that are both in the cluster and are targets of the microRNA. The resulting p-value reflects the statistical significance of observing a count h, given n, c and t. A lower p-value indicates a higher "microRNA enrichment", and, consequently, a better result. By choosing a p-value cutoff, we can construct a list of enriched microRNAs for each cluster as well as a list of target genes related to the enriched microRNAs. Note, that the statistics for microRNA-binding are not well developed; intricate dependencies introduced by sequence similarities among the microRNAs and the target genes exist and complicate the analysis. As we also consider a manually selected set of microRNAs, we choose a somewhat relaxed p-value cutoff, foregoing multiple testing corrections 52, followed by a careful biological evaluation. For the following discussions we restrict our result set to clusters that contain at least four target genes in total.
<p>Figure 8</p>
Strategy to identify enriched microRNAs
Strategy to identify enriched microRNAs. Strategy to identify microRNAs and their target genes overrepresented in groups of co-expressed genes (indicated left) as part of a post-transcriptional regulatory mechanism. In the middle mRNAs clustered according to our mixture results are depicted and potential microRNA binding sites in their 3'UTRs are symbolized.
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In summary, in TCell our target prediction scheme detects significant enrichment for eleven out of the 17 initial microRNAs in four out of the 20 clusters (Table 1). In these four clusters we detect in total 35 candidate target genes, which is a considerable reduction of the set of 229 targets that have been predicted by sequence based methods alone 49. For BCell these numbers are respectively, eleven out of the 17 microRNAs, four out of the 20 clusters, and 29 out of the 273 predicted targets (Table 1). In particular, we find the five microRNA families miR-15, miR-181, miR-221, miR-26, and miR-142-3p to be enriched in both TCell and BCell by our criterion. See Table S6 in Additional data file 3 for microRNA enrichment in LymphoidTree and Table S7, Table S8, Table S9, for p-values of microRNA enrichment of all data sets. As mentioned earlier, the BCell clusters 3, 5, and 6 show a similar expression profile. We find that cluster 5 of the results of the TCell set overlaps substantially with clusters 3 and 5 of BCell (Table 1). In TCell cluster 5 we find miR-15a, miR-181a, miR-26a, miR-24, and miR-221 as potential regulators and 20 potential target genes, seven of which are also present among the 18 BCell candidate genes of clusters 3 and 5. The developmental profiles of the clusters of both lineages show strikingly analogous phenotypical features, namely up-regulation in the proliferating large cell populations (DN4, DPL, large pre-BII) and from then on strict down-regulation. In TCell cluster 5 there are eight genes and in the BCell clusters 3 and 5 there are nine target genes that are known to be involved in DNA metabolism, cell-cycle and mitosis (Table 1). This suggests a regulatory role for the identified microRNAs in reducing the transcript levels of genes that are important for cell proliferation. This is supported by the fact that a similar role for microRNA was found in Drosophila germline stem cells 53.
<p>Table 1</p>
List of LympMIR enriched in the clusters from MixDTrees on data sets TCell and BCell
Cluster ID
MicroRNA
Target Genes
TCell 3
miR-222
Elovl6, Nme1, Rcn1, Rps3
TCell 5
miR-15a1, miR-181a2, miR-2213,
2410015N17Rik4, Alad1,4, Atpif11,5, Aurkb2, Cdc25a1, Chek11
miR-244, miR-26a5
Cks1b2,4, Cks25, Eed2, H2afx4, Kpnb13, Mcm53, Nasp3,5, Pex72, Psmd122, Ranbp52, Rars1, Tk13, Trip131, Uchl55
TCell 10
miR-142-3p6, miR-1507
Gfi16, Marcks6, Msh66, Pp11r7, Psmc16,7
TCell 11
miR-1468, miR-169, miR-181b10
Atp1b310, Ipo49, Klhdc210, Mrpl308, Orc5l8, Tuba49
BCell 3
miR-181b1, miR-181c2, miR-26a3
Atpif13, Aurkb1,2, Cbx13, Cdc45l2, Cks1b1,2, Cks23, Cox5a3, Hmgb21,2, Melk1,2, Ttk1,2, Uchl53
BCell 5
miR-15a4, miR-15b5, miR-2216,
Cdca44,5, Chek14,5, Mcm47, Nasp6, Nfyb6, Smc4l17
miR-2237
Tuba2
4,5,7
BCell 6
miR-1558, miR-1919
Ctps9, Ddx18, Hint19, Mcm28, Phf178, Prdx49, SNrpd19
BCell 19
miR-142-3p14, miR-34215
2410002F23Rik14, H2-Eb114, Ltb15, Tap214,15
We display the cluster and data set id, the list of microRNA and list of target genes, with p-values <0.05 and at least four target genes per cluster. Genes involved in cell proliferation or DNA repair are depicted in bold. The indices indicates to which microRNA a gene is related, when there is more than one enriched microRNA in a cluster.
At the individual gene level we identify some candidate microRNA targets for further detailed analysis: the three known genes (H2-Eb1, Ltb, Tap2) of BCell cluster 19 are all involved in the antigen presentation by MHC class II molecules 3354. In the context of the cell cycle, Chek1 (clusters TCell 5 and BCell 5) and Cdc25a (cluster TCell 5) are important for the transition between G1/S and G2/M phases 55.
Furthermore, both genes are candidate targets of the same microRNA, miR-15a, which is related to apoptosis in chronic lymphoid leukemia cells 56. Another interesting gene codes for the nuclear factor Y (Nfyb; cluster BCell 5), which regulates Hoxb4 57, Cdc34 58 and the major histocompatibility complex in mice 59. These are all important genes for lymphoid development. The mRNA of the growth factor independence-1 transcription factor (Gfi1; cluster TCell 10) is a potential target of miR-142-3p with a function in the restriction of cell proliferation and maintenance of the functional integrity of lymphocyte cells 60. Moreover, Gfi1 is implicated in the transition from CD4/CD8 double negative to double positive T cells 61.
In order to relate our approach with 45, we also perform a microRNA enrichment analysis with the results of SOM (see Table S4 and S5 in the Additional data file 3). In TCell there is little overlap between the microRNA targets, with the exception of SOM cluster 6, which is a subset of targets genes from cluster 5 from MixDTrees. We also compare the p-values obtained by both methods in a procedure similar to the one performed in 31. For TCell, MixDTrees results in lower p-values in nine out of 14 microRNAs (see Fig. S5 in Additional data file 2). In BCell, gene targets found with SOM are partially a subset of the ones encountered with MixDTrees; 14 out of 24 targets genes in BCell SOM are also detected by MixDTrees (Table S5 in the Additional data file 3). For BCell, (Fig. S6 in Additional data file 2), MixDTrees obtains lower p-values in 8 out of 14 microRNAs. Even though SOM obtains lower p-values for microRNAs found to be enriched with both methods, MixDTrees detects seven enriched microRNA not significantly enriched in SOM. An inspection of the cumulative distribution function of these p-values also reinforces the view that MixDTrees is more sensitive in detecting enriched microRNAs than SOM in BCell (Fig. S8 in Additional data file 2). Overall, the results suggests a higher sensitivity of MixDTrees-MAP in finding groups of microRNA targets sharing similar expression patterns compared to SOM. Additionally, we performed microRNA enrichment p-value comparison between MixDTrees-MAP and MixDTrees-MLE for both data sets (see Additional data file 2 Fig. S9 and S10). For TCell, MixDTrees-MAP achieves a higher enrichment for nine out of 14 microRNAs; while for BCell, six out of 13 microRNAs. In summary, clusters computed according to MAP have an increased enrichment for TCell and a slightly lowered enrichment for BCell. A manual inspection of the contingency table comparing the clusters from MAP and MLE (Additional data file 3 Table S15) and in the cluster size distributions (Additional data file 2 Fig. S11) shows that MixDTrees-MLE has a tendency to produce spurious, small clusters as a result of over-fitting, a known disadvantage of MLE estimates 17. Note that the resulting p-values decrease drastically as a function of the cluster size, making a clustering which joins clusters appear preferable. Enrichment analysis should be used cautiously to compare clusterings, if the cluster size distributions are not similar, as it is the case for the MLE results. This and the results on simulated data supports our preference of MixDTrees-MAP over MixDTrees-MLE.
Conclusion
The regulatory processes behind cell proliferation and differentiation are of central interest to developmental biologists and clinicians alike and are frequently the focus of large-scale studies to investigate gene expression along paths of differentiation. To make full use of this data in a principled manner we present a novel statistical framework which models gene expression in the course of development. By combining the dependence trees in a classical mixture model, we facilitate interactive querying and visualization of data and, more importantly, the detection of possibly overlapping clusters of co-expressed genes, which provide a basis for the identification of key players in the regulatory mechanism and their mode of action.
In particular, we detect interesting groups of genes not found by classical clustering methods such as SOM. By incorporating microRNA binding data, we show how to identify complex regulatory relationships. Compared to an analysis based only on sequence, we predict a manageable number of plausible microRNA targets. Moreover, our method offers some insights into the biological role of predicted microRNAs, by the inspection of the developmental profiles of gene targets associated with one microRNA. A comparison with SOM indicates that our approach is more sensitive for finding co-expressed genes on which the same microRNA can have a regulatory effect.
Extensions to accommodate further types of data are straightforward. Binding sites of transcription factors can be analyzed completely analogous to the microRNA analysis. If expression levels of microRNAs in developmental stages investigated in TCell or BCell were available, we could incorporate a target prediction framework 62. Furthermore, we can simply apply established techniques 63646566 to extend our mixture model to integrate heterogeneous data–sequence information, protein interaction, genotype, phenotype data–and semi-supervised extensions to mixture estimation can be applied to make use of biological knowledge about functional similarities and regulatory relationships 226768. This is of highest relevance, because the identification of regulatory modules is actually feasible compared to the automated inference of regulatory networks 69. Once a statistical model is obtained, further detailed questions about the significance of differences, or the most likely stage, at which differentiation occurs can be easily answered.
Fascinating extensions are possible, even when one only considers gene expression data and the basic method. None of the currently publicly available data sets offers both a tree with a large number of branches and a detailed view of all, in particular early, development stages (70 concentrates on mature and immature cells in final development stages); combining data from several microarray platforms suffers from the usual problems. Hence, we concentrate on two smaller but detailed studies covering several stages of T cell and B cell development 45, and a tree containing three lineages of lymphoid cells. Note that in the latter several cell types of intermediary development stages are not measured. Nevertheless, our analysis indicates that our method takes advantage of the tree structure information in detecting relevant differences of gene expression in these lineages. This also reinforces the importance of the creation of expression compendia, such as the one in 70, where many intermediary stages of differentiation of the developmental tree are also present. Such data will be of great value as computational methods can exploit characteristics intrinsic to cell development.
Lastly, developmental biologists are still redrawing developmental trees with the discovery of new intermediary stages and "alternative" paths of development 123; a particular developmental stage might also be formed by a mixture of distinct cell types not well characterized yet. As an example of an alternative path, there has been evidence that DN1 T cells can be originated not only from the lymphoid progenitor as depicted in Fig. 1, but also from the earlier multipotent progenitor cells 3. It is an exciting prospect to infer branches and stages of a developmental tree from gene expression data, ideally per functional module. This structure learning (see 16 for discrete data) can be incorporated in the EM-based parameter estimation. In conclusion, our results suggest that the mixture of dependence trees provides a natural and powerful representation of developmental gene expression data. Furthermore, our results reinforce the importance of the creation of detailed and heterogeneous data sets for helping elucidate the regulatory mechanisms of development.