Volume 6 Supplement 3
A dynamic time order network for time-series gene expression data analysis
- Pengyue Zhang†1,
- Raphaël Mourad†1Email author,
- Yang Xiang2,
- Kun Huang2,
- Tim Huang3,
- Kenneth Nephew4,
- Yunlong Liu1 and
- Lang Li1
© Zhang et al.; licensee BioMed Central Ltd. 2012
Published: 17 December 2012
Typical analysis of time-series gene expression data such as clustering or graphical models cannot distinguish between early and later drug responsive gene targets in cancer cells. However, these genes would represent good candidate biomarkers.
We propose a new model - the dynamic time order network - to distinguish and connect early and later drug responsive gene targets. This network is constructed based on an integrated differential equation. Spline regression is applied for an accurate modeling of the time variation of gene expressions. Then a likelihood ratio test is implemented to infer the time order of any gene expression pair. One application of the model is the discovery of estrogen response biomarkers. For this purpose, we focused on genes whose responses are late when the breast cancer cells are treated with estradiol (E2).
Our approach has been validated by successfully finding time order relations between genes of the cell cycle system. More notably, we found late response genes potentially interesting as biomarkers of E2 treatment.
Breast cancer represents a major public health issue since it comprises 22.9% of all cancers in women and it is an important cause of death . Some breast cancers are sensitive to hormones such as estrogen (E2) . Thus it is possible to treat these cancers by blocking the effects of these hormones, using for instance tamoxifen . The discovery of biomarkers of the response to drugs is an important task in medical research because it helps know if a drug is effective for a specific patient and how it is metabolized by his organism. Biomarkers play thus an important role in personalized medicine, such as in the choice of the most relevant treatment.
Biomarkers often refer to proteins measured in the blood whose concentrations reflect the presence or the severity of the disease. In the case of estrogen treatment, biomarkers can be seen as parameters reflecting the effects of the drug on the patient. The biomarkers of hormone therapy of the breast cancer is not well developed. For instance, although tamoxifen's pharmacology mechanism is well known, its clinical biomarker is not well established yet. Understanding the cascade of estrogen signaling pathway is the key to study the potential biomarkers.
Gene expression-based biomarker discovery has demonstrated efficiency for breast cancer [4, 5]. Standard methods rely on computing correlations between gene expressions and drug treatment status. Simple statistical procedures are used such as t-tests to assess the significance of over- or under-expressions of genes before and after treatment in steady-state analysis . Clustering has also been successfully used for revealing particular patterns of expression .
Unfortunately standard methods might fail to reveal key biomarkers, since they do not take into account the temporal aspect of gene expression and the complex network of gene regulation. To tackle this issue, the analysis of time series data through dynamic networks represents efficient alternatives . In this context, three main approaches can be distinguished: dynamic Bayesian networks, information-theoretic networks and ordinary differential equations. Dynamic Bayesian networks (DBNs) have been successfully applied to infer causal gene networks [9, 10]. Conditional independences encoded in DBNs guarantee to infer direct relations between genes. The second approach consists in inferring the structure of dependences through an information-theoretic framework [11, 12]. Most notably, the data processing inequality principle helps discard the majority of indirect dependences without involving time consuming algorithms such as those for DBNs. The last method relies on ordinary differential equations (ODEs) [13, 14]. In this method, changes of gene expression are related to each other through a system of differential equations. Most notably, this method accurately and explicitly models the continuous time aspect of gene expression. Recently a combination of ODEs and DBNs has been proposed for taking into account both causal discovery (DBNs) and accurate modeling (ODEs) of gene expression .
Late response genes might represent relevant biomarkers because they are more stable over the time. Our approach relies on this biological aspect of biomarker discovery. To identify late response genes, we propose a new model based on a dynamic time order network (DTON). The model interpretation is simple and intuitive: it reflects which genes express in the early times and which ones in the late times after the hormone treatment. The DTON is constructed based on an integrated differential equation. Spline regression is applied for an accurate modeling of the time variation of gene expressions. A likelihood ratio test is implemented to infer the time order of any gene expression pair. The advantages of this modeling approach are numerous: (i) closed-form expressions of ODEs, (ii) accurate modeling of the time series data by using spline regression and by integrating differential equations, and (iii) model learning involving simple regressions quick to compute and only a few parameters have to be estimated. The method has been validated by successfully finding time order relations between genes of the cell cycle system. Most importantly, we found late response genes as candidate biomarkers of E2 treatment.
This paper is organized as follows. Section Materials and methods first describes experiments and data preprocessing. Late response genes are defined and discussed. Then the dynamic time order network and its model learning are presented. It is described how dynamic time order relations between genes are inferred through a likelihood ratio test. The next section illustrates our method on real data analysis. Our model is validated with the well-known cell cycle system. Late response genes of E2 treatment are discovered. Finally, the last section concludes and points out promising perspectives.
Materials and methods
Experiment and data preprocessing
The gene expression data come from estrogen stimulated ZR_75_1 cells. G0 - G1 synchronization cells were treated with 10-8 M of 17 β - estradiol (E2). Then RNA was extracted from the cells before (0) or after 1, 2, 4, 6, 8, 12, 16, 20, 24, 28 and 32 hours of stimulation. For more details, the reader is referred to the original study . There are 48702 probes in the original study and some of them are duplicated. Duplicated probes are averaged. Then only highly differentially expressed genes are considered through the following method. Standard deviation and mean were computed for each mRNA. A gene is considered as not differentially expressed if its standard deviation over its mean is small. At this point, we chose 0.15 as threshold. Finally, we only kept 5003 genes with high variation of their expression. The logarithmic concentration ratio (LCR) at every time point is used. Let C t denotes the concentration at time point t for a gene, then the LCR at time point t would be log . The LCR indicates how much the concentration increases or decreases from the concentration at the first time point. In order to unify the variance for different genes, we standardized the LCRs at each time point.
Late response gene
In breast cancer cells, Cicatiello et al.  showed that all the major time dependent gene expression profile clusters follow two major patterns: (i) go up or down, then stay flat; and (ii) go up or down first, stay flat, then go down or up, respectively. These patterns can be captured by a natural cubic spline function divided in three parts using two knots. The early response genes are thus defined as either up- or down-regulated genes before 5.333 hours, following E2 stimulation. The late response genes are defined as either up- or down-regulated genes after 17.333 hours. The time points 5.333 hours and 17.333 hours represent the 33th and 67th percentiles of the sampling time points.
Biologically, we favor late response genes because of their clinical implications. To check whether a drug works in human, i.e. inhibiting or simulating the target, one or multiple reliable biomarkers are useful to indicate the drug effects. An early response gene may not be predictive for the long term effect of the drug. It is always desirable to use a biomarker that can predict a sustainable effect of the drug. Therefore, a late response gene represents a better biomarker than an early response one. In our dataset, responsive genes after 17.333 hours following E2 treatment are likely to be the best biomarkers.
The dynamic time order relationship
In Equation (2), F1(t) and F2(t) represent the cumulative expression of G1 and G2. The integration of the ODE can help to better distinguish which gene is firstly expressed in a non-trivial scenario, such as the one presented in Figure 1b. In this example, we can see that it is possible to infer the dynamic time order relation between G1 and G2 only during the early time (because only in the early time we observe a significant difference between the two rates). By integrating the ODE (see Equation (1)), the model can take into account all the variation of the gene LCR (in early and late times). Note that this dynamic time order relation does not imply any causal relation between two genes but only indicates which one is expressed after the other.
Natural cubic spline regression
with and respectively denote the residuals of the spline regression and their variance associated with the gene G i .
The time interval of our gene expression data is t ∈ [0; 32] hours. We divide the function f i into three parts using two knots at 5.333 hours and 17.333 hours. The decomposition of the cubic function using knots is presented in Additional file 1.
with . The value of j refers to the first, second or third interval.
The parameters β ij are learned by maximizing the likelihood in Equation 6 with constrains (see Additional file 1). There are 12 parameters in the cubic function. However, only 4 out of the 12 parameters are free, as constrains must be satisfied. If we set as the free parameters, we can solve parameters βi 1and βi 3(see Additional File 2).
Time order determination
where and are constant terms. They vary for different gene LCRs.
with . Vectors and are the LCRs for the pair of genes G1 and G2, and are the associate parameters in the model presented in Equation 10. Let F i (t) be the integration of f i (t) getting from Equation 11 and be the function values for F i (t) at each time point t. Then the predictor variable is X = (1, F1, F2).
Thus in Equations 12 and 13, values of y i (left hand side) come from data and values of X (right hand side) result from the integration of the NCSR functions. For the pair of genes G1 and G2, the model in equation 12 represents the dynamic time order relation G2 → G1 and the model in equation 13 represents the dynamic time order relation G1 → G2.
Pairwise regressions are then computed for all pairs of genes and the log-likelihoods are calculated (see Additional file 4). In order to find whether a pair of genes has a dynamic time order relation, we look at their log-likelihood difference. If two genes present a dynamic time order relation, the regression relying on the true relation will have a better log-likelihood value than the regression based on the wrong relation, as Equations 12 and 13 represent two different dynamic time orders.
After determining the time order relationships, an n-by-n adjacency matrix (n is the number of genes) is constructed whose weights are the previously computed log-likelihood differences. In the matrix, for a couple of genes, only the positive log-likelihood difference value is kept and the negative (symmetric) log-likelihood difference value is set to 0. This adjacency matrix represents the complete directed graph of time order relationships.
When the network is small (less than one hundred nodes), it is interesting to keep as much as possible information about time order relations. The best strategy in this case is fine tune a threshold used to remove non-significant edges. For this purpose, a simple and efficient approach is the use of the median or other quantiles of the distribution of log-likelihood difference values. Then a simplification step is used to remove redundant edges. For instance, when one observes A → B and B → C, then A → C is considered as redundant and is removed. For graph drawing, the Sugiyama's algorithm  provides a hierarchical display which is particularly relevant for reflecting time order relations.
When the network is huge, such as the genome-wide network from the microarray data, the previous approach cannot be used. The reason is that a low threshold value will create a network highly connected which is too complex to manipulate and to visualize, whereas a high threshold value will lead to a graph with many connected components from which it will only be possible to infer time orders between connected genes. To tackle this issue, we compute the so-called maximum weight spanning tree (MWST). This graph presents several advantages: (i) its tree shape is a very simple structure easy to manipulate and visualize, and (ii) every node is connected by a path such that we can access to the time order relation between each gene. Besides, the MWST can be quickly computed in O(n2logn) through the Prim's algorithm . Regarding graph drawing, the Sugiyama's algorithm cannot be used when the graph is too huge. Instead we prefer to display it using an algorithm specific to tree drawing, the Bubble tree algorithm .
Biological interpretation of the model
Our learning method is implemented in R. The R source code is available on request. For graph drawing and display, the software Tulip (http://tulip.labri.fr/TulipDrupal/) was used. It is a user-friendly tool able to deal with about one million nodes.
Results and discussion
Reproducing the cell cycle temporal system
List of the 10 most important incoming-edge hubs.
Number of incoming edges
We also search in the literature if the late response genes identified with our method can be good candidate biomarkers. Since biomarkers are molecules that are observed in cancer patients but not in healthly people, there are likely to be genes overexpressed after E2 treatment. Among the overexpressed hubs of the network, CALB2, PDZK1, MT2A and FANCD2 are well-known in the literature as diagnostic marker of breast cancer and E2 response [22–25]. Besides, C20orf160 is reported in the Genes-to-Systems Breast Cancer Database (http://www.itb.cnr.it/breastcancer//index.html). WDR51A (also called POC1A) is found associated with breast cancer in The Human Protein Atlas (http://www.proteinatlas.org/).
Based on experimentations carried out on time-series gene expression data, our dynamic time order network has been shown to efficiently distinguish and connect early and late response genes. First, our model has faithfully reproduced the cell cycle temporal system. Over the 27 time order relations inferred, 89% correspond to the state-of-art network, 11% cannot be checked, but no one are false. Second, our approach has been successfully applied to a genome-wide level. The learning method has been able to process five thousands genes and the network simplification through the maximum weighted spanning tree provided a graphical display of the huge network. Most notably, several incoming-edge hubs showing very high connectivity have been discovered. All these hubs showed late gene response profiles. Regarding those which are overexpressed over the time, they have been reported as biomarkers of breast cancer and E2 response in the literature and databases.
The comparison of results with other approaches is not straightforward, since our method is the only one dedicated to identify late response genes. When compared with standard methods in gene expression analysis, our approach yielded specific results, contrary to agglomerative hierarchical clustering. Moreover it does not need any complex thresholding such as with a t-test strategy. It is worth noting that all genes identified with DTON showed late responses, while this is not the case with the t-test strategy. Besides, our approach is based on the comparison of gene expression integrals combined with cubic spline regression, thus offering an accurate assessment of time order relations.
The discovery of biomarkers is one of the application of our model. The distinction between early and late response genes is also an important application in developmental biology where the understanding of the temporal aspect of gene expression is a key issue such as for cell differentiation. For the moment, we mainly focused on the identification of late response genes. The use of another graph modeling would be more efficient for pointing out early response genes than the MWST which tends to display incoming-edge hubs.
The authors are grateful to the three anonymous referees for constructive comments and help in improving their manuscript. This work is supported by National Cancer Institute awards CA113001 (to T.H-M.H. and K.P.N.). Yang Xiang was supported by the National Science Foundation under Grant #1019343 to the Computing Research Association for the CIFellows Project.
This article has been published as part of BMC Systems Biology Volume 6 Supplement 3, 2012: Proceedings of The International Conference on Intelligent Biology and Medicine (ICIBM) - Systems Biology. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/6/S3.
- International Agency for Research on Cancer: World cancer report. 2008Google Scholar
- Osborne CK, Yochmowitz MG, Knight WA, McGuire WL: The value of estrogen and progesterone receptors in the treatment of breast cancer. Cancer. 1980, 46 (Suppl 12): 2884-2888.View ArticlePubMedGoogle Scholar
- Jordan VC: Tamoxifen (ICI46,474) as a targeted therapy to treat and prevent breast cancer. British Journal of Pharmacology. 2006, 147 (Suppl 1): S269-S276.PubMed CentralPubMedGoogle Scholar
- van't Veer LJ, Dai H, van de Vijver MJ, He YD, Hart AAM, Mao M, Peterse HL, van der Kooy K, Marton MJ, Witteveen AT, Schreiber GJ, Kerkhoven RM, Roberts C, Linsley PS, Bernards R, Friend SH: Gene expression profiling predicts clinical outcome of breast cancer. Nature. 2002, 415 (6871): 530-536. 10.1038/415530a.View ArticleGoogle Scholar
- Sotiriou C, Pusztai L: Gene-expression signatures in breast cancer. The New England Journal of Medicine. 2009, 360 (8): 790-800. 10.1056/NEJMra0801289. [http://dx.doi.org/10.1056/NEJMra0801289] 10.1056/NEJMra0801289View ArticlePubMedGoogle Scholar
- Hibbs K, Skubitz KM, Pambuccian SE, Casey RC, Burleson KM, Oegema TR, Thiele JJ, Grindle SM, Bliss RL, Skubitz AP: Differential gene expression in ovarian carcinoma: identification of potential biomarkers. The American Journal of Pathology. 2004, 165 (2): 397-414. 10.1016/S0002-9440(10)63306-8.PubMed CentralView ArticlePubMedGoogle Scholar
- Golub TR, Slonim DK, Tamayo P, Huard C, Gaasenbeek M, Mesirov JP, Coller H, Loh ML, Downing JR, Caligiuri MA, Bloomfield CD, Lander ES: Molecular classification of cancer: Class discovery and class prediction by gene expression monitoring. Science. 1999, 286 (5439): 531-537. 10.1126/science.286.5439.531.View ArticlePubMedGoogle Scholar
- Bansal M, Belcastro V, Ambesi-Impiombato A, di Bernardo D: How to infer gene networks from expression profiles. Molecular Systems Biology. 2007, 3: 78-PubMed CentralView ArticlePubMedGoogle Scholar
- Kim S, Imoto S, Miyano S: Dynamic Bayesian network and nonparametric regression for nonlinear modeling of gene networks from time series gene expression data. BioSystems. 2004, 75 (1-3): 57-65. 10.1016/j.biosystems.2004.03.004.View ArticlePubMedGoogle Scholar
- Zou M, Conzen SD: A new dynamic Bayesian network (DBN) approach for identifying gene regulatory networks from time course microarray data. Bioinformatics. 2005, 21: 71-79. 10.1093/bioinformatics/bth463.View ArticlePubMedGoogle Scholar
- Basso K, Margolin AA, Stolovitzky G, Klein U, Dalla-Favera R, Califano A: Reverse engineering of regulatory networks in human B cells. Nature Genetics. 2005, 37 (4): 382-390. 10.1038/ng1532.View ArticlePubMedGoogle Scholar
- Margolin AA, Nemenman I, Basso K, Wiggins C, Stolovitzky G, Dalla Favera R, Califano A: ARACNE: an algorithm for the reconstruction of gene regulatory networks in a mammalian cellular context. BMC bioinformatics. 2006, 7 (Suppl 1): S7-10.1186/1471-2105-7-S1-S7.PubMed CentralView ArticlePubMedGoogle Scholar
- Sontag E, Kiyatkin A, Kholodenko BN: Inferring dynamic architecture of cellular networks using time series of gene expression, protein and metabolite data. Bioinformatics. 2004, 20 (12): 1877-1886. 10.1093/bioinformatics/bth173.View ArticlePubMedGoogle Scholar
- Bansal M, Gatta GD, di Bernardo D: Inference of gene regulatory networks and compound mode of action from time course gene expression profiles. Bioinformatics. 2006, 22 (7): 815-822. 10.1093/bioinformatics/btl003.View ArticlePubMedGoogle Scholar
- Li Z, Li P, Krishnan A, Liu J: Large-scale dynamic gene regulatory network inference combining differential equation models with local dynamic Bayesian network analysis. Bioinformatics. 2011, 27 (19): 2686-2691. 10.1093/bioinformatics/btr454.View ArticlePubMedGoogle Scholar
- Cicatiello L, Mutarelli M, Grober OM, Paris O, Ferraro L, Ravo M, Tarallo R, Luo S, Schroth GP, Seifert M, Zinser C, Chiusano ML, Traini A, De Bortoli M, Weisz A: Estrogen receptor α controls a gene network in luminal-like breast cancer cells comprising multiple transcription factors and microRNAs. The American Journal of Pathology. 2010, 176 (5): 2113-2130. 10.2353/ajpath.2010.090837.PubMed CentralView ArticlePubMedGoogle Scholar
- Green P, Silverman BW: Nonparametric regression and generalized linear models: A roughness penalty approach. 1993, Chapman and Hall/CRC, 1Google Scholar
- Sugiyama K, Tagawa S, Toda M: Methods for visual understanding of hierarchical system structures. IEEE Transactions on Systems, Man, and Cybernetics. 1981, 11 (2): 109-125. [http://dx.doi.org/10.1109/TSMC.1981.4308636]View ArticleGoogle Scholar
- Prim RC: Shortest connection networks and some generalizations. Bell System Technical Journa. 1957, 36: 1389-1401.View ArticleGoogle Scholar
- Grivet S, Auber D, Domenger JP, Melancon G: Bubble tree drawing algorithm. International Conference on Computer Vision and Graphics. 2004, 633-641.Google Scholar
- Rousseeuw P: Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. Journal of Computational and Applied Mathematics. 1987, 20: 53-65. [http://dx.doi.org/10.1016/0377-0427(87)90125-7]View ArticleGoogle Scholar
- Schummer M, Green A, Beatty JD, Karlan BY, Karlan S, Gross J, Thornton S, McIntosh M, Urban N: Comparison of breast cancer to healthy control tissue discovers novel markers with potential for prognosis and early detection. PLoS ONE. 2010, 5 (2): e9122-10.1371/journal.pone.0009122.PubMed CentralView ArticlePubMedGoogle Scholar
- Ghosh MG, Thompson DA, Weigel RJ: PDZK1 and GREB1 are estrogen-regulated genes expressed in hormone-responsive breast cancer. Cancer Research. 2000, 60 (22): 6367-6375.PubMedGoogle Scholar
- Kim HG, Kim JY, Han EH, Hwang YP, Choi JH, Park BH, Jeong HG: Metallothionein-2A overexpression increases the expression of matrix metalloproteinase-9 and invasion of breast cancer cells. FEBS Letters. 2011, 585 (2): 421-428. 10.1016/j.febslet.2010.12.030.View ArticlePubMedGoogle Scholar
- Zhang B, Chen R, Lu J, Shi Q, Zhang X, Chen J: Expression of FANCD2 in sporadic breast cancer and clinicopathological analysis. Journal of Huazhong University of Science and Technology - Medical Sciences-. 2010, 30 (3): 322-325. 10.1007/s11596-010-0350-7.View ArticleGoogle Scholar
- Collins K, Jacks T, Pavletich NP: The cell cycle and cancer. Proceedings of the National Academy of Sciences of the United States of America. 1997, 94 (7): 2776-2778. 10.1073/pnas.94.7.2776.PubMed CentralView ArticlePubMedGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.