Global analysis of phase locking in gene expression during cell cycle: the potential in network modeling
© Gao et al; licensee BioMed Central Ltd. 2010
Received: 18 July 2010
Accepted: 3 December 2010
Published: 3 December 2010
In nonlinear dynamic systems, synchrony through oscillation and frequency modulation is a general control strategy to coordinate multiple modules in response to external signals. Conversely, the synchrony information can be utilized to infer interaction. Increasing evidence suggests that frequency modulation is also common in transcription regulation.
In this study, we investigate the potential of phase locking analysis, a technique to study the synchrony patterns, in the transcription network modeling of time course gene expression data. Using the yeast cell cycle data, we show that significant phase locking exists between transcription factors and their targets, between gene pairs with prior evidence of physical or genetic interactions, and among cell cycle genes. When compared with simple correlation we found that the phase locking metric can identify gene pairs that interact with each other more efficiently. In addition, it can automatically address issues of arbitrary time lags or different dynamic time scales in different genes, without the need for alignment. Interestingly, many of the phase locked gene pairs exhibit higher order than 1:1 locking, and significant phase lags with respect to each other. Based on these findings we propose a new phase locking metric for network reconstruction using time course gene expression data. We show that it is efficient at identifying network modules of focused biological themes that are important to cell cycle regulation.
Our result demonstrates the potential of phase locking analysis in transcription network modeling. It also suggests the importance of understanding the dynamics underlying the gene expression patterns.
A major goal of systems biology is to integrate biological functions of individual genes in terms of their interactions. Time course gene expression profiling, which can capture the global transcriptional responses to signals during a biological process of interest, offers a major data source to achieve this goal .
In network modeling of gene expression data, assessing pair-wise relationships is often a starting point. In early days, correlation coefficient [2, 3], Euclidean distance, as well as their variations, such as partial correlations, empirical Bayes and bootstrap methods , were used. They are effective for computing direction free linear dependence when the data are independent. Networks constructed this way are essentially co-expression networks. While having the appeal of being simple and intuitive, correlation metrics have limitations when applied to time course data. They assume independence of the order of the data points, while in reality the data at each time step depend on the previous time points. Ignoring the inter-time point dependence not only loses sensitivity toward detecting interactions but could also lead to erroneous predictions .
Significant phase shift in the timing of expression changes have also been observed for highly associated genes . Some studies tried to identify the phase lag directly by shifting gene expression time series with respect to each other until the optimal alignment is reached. For instance, Qian et al proposed a local clustering approach based on optimal pair-wise alignment through dynamic programming ; Schmitt et al used the Pearson's correlation , Balasubramaniyan et al used the Spearman rank correlation , Pereda et al used cross correlation , to compute the maximum time-lagged similarity between two transcript profiles, and utilized the results to identify clusters. The degree of lag varies widely in different gene pairs, and these approaches need multiple runs to find the lag that best aligns each pair. The performance of the alignment depends on whether the lags are close to integer numbers of the sampling steps of the experiment.
More sophisticated methods were also developed. Aach and Church implemented both simple and interpolative time warping based on dynamic programming to identify an optimal alignment of two gene expression time series . Expanding this approach, Liu and Müller proposed a non-parametric time-synchronized iterative mean updating technique to construct modes of temporal structure in gene expression profiles . Bar-Joseph et al.[13, 14] developed an approach to align temporal data sets using piecewise spline fitting, extracting shift and stretch parameters for each data set. Butte et al. utilized digital signal-processing tools, including power spectral densities, coherence, transfer gain, and phase shift, to find pair-wise gene associations based on periodically expressed time-invariant gene profiles. More recently, a hidden Markov model based approach was utilized to infer the timing in gene expression changes under different experimental conditions .
Linear and non-linear multivariate analysis and signal processing techniques were also introduced to analyze time series microarray data . Several studies used pair wise mutual information to infer interactions and regulatory relationships between genes [17, 18]. This method assumes a fixed time delay, which might not be true across different experimental conditions. In frequency domain time series analysis, causality and interrelationship among the components can be studied using coherence and partial coherence. Graphical models based on such analysis have been studied by Butte et al and Salvador et al. However, Albo et al showed that partial coherence-based causality measures are sensitive to measurement noise.
Apparently, more studies are needed to fully utilize the dynamics underlying the temporal gene expression pattern, and to better understand the complex spatial-temporal architecture of transcriptome. Recently, increasing evidence, including those from the advancement of single-cell time course gene expression profiling technologies , suggest that like other complex dynamic systems in nature, synchrony through oscillation and frequency modulation is a general strategy for an organism to coordinate the transcription of multiple target genes in responses to external signals [22–26]. Examples include the p53-Mdm2 feedback loop [24, 25], the NF-κB signaling pathway , and calcium responsive pathways . These further emphasize the need of new methods to study and utilize the dynamics. The oscillations in gene expression, like other oscillations in biological systems , are most often pulsatile or relaxed oscillations rather than harmonic, thus calling for mathematical methods rooted from phase space analysis [29, 30].
In this study, we investigate the potential of network inference using the phase locking analysis technique . This approach is based on the following concepts originating from nonlinear dynamics [29, 30]: if two time series interact with each other, there will be a process of rhythmic adjustment resulting from the interaction, leading to phase locking. Phase-locked oscillators progress through their trajectories in phase space at the same pace (1:1 locking), or rational ratios with respect to each other (m:n locking, m and n being integers). Conversely, such phase locking phenomenon can be utilized to infer interaction between two dynamic systems, even for weak interactions . Recently Kim et al clustered genes of synchronized oscillatory pattern (1:1 phase locking) during yeast cell cycle, and observed that genes in the same cluster were closely associated, as evidenced by the sharing of GO terms and BioGRID interactions . In this study we will apply the phase locking analysis to the Stanford yeast cell cycle data [33, 34], and examine the phase locking (including higher order locking) between transcription factors and targets, between gene pairs with prior evidence of other types of interaction, and between cell cycle genes. Based on the results, we will propose a new network inference approach utilizing the phase locking index, and examine the modular structure of the networks constructed and the biological themes shared by genes in the network modules.
Phase locking of interacting genes
Distribution of λ
Phase locking between the cell cycle regulating TFs and their targets
Following the original gene expression study of cell cycles [33, 34], several groups have investigated yeast transcription binding using the ChIP-chip technology [35, 36]. These data provide useful information of which genes are potentially transcription regulation targets of each TF. We have obtained the data from Simon et al, where the promoter binding by the 9 known cell cycle regulating TFs were studied . Both ChIP-chip and microarray data are noisy, and we found no direct quantitative dependence of λ or r on the binding p-value (r < 0.1).
Phase locking of BioGRID gene pairs
To further investigate the potential of phase locking analysis in network inference, we examined phase locking between gene pairs that have evidence of other types of interaction according to BioGRID http://www.thebiogrid.org. BioGRID is a freely accessible database of physical (protein-protein) and genetic interactions, curated from high-throughput data and literature [37, 38]. Of all possible gene pairs in our data sets, ~53,000 are annotated in BioGRID. We constructed five Bootstrapping  sets that consisted of the same number of BioGRID gene pairs, randomly sampled from all possible pairs. The distribution of phase locking index was examined in each group. We found that the distribution for the BioGRID pairs is skewed toward higher λ than the Bootstrapping sets (p < 1e-5, KS test).
Phase locking among cell cycle genes
Phase lags and m:n phase locking
Agreement between the 4 datasets
Network inference using phase locking
In our study so far, we have demonstrated that phase locking in expression changes is a good indicator of interaction. It is therefore natural to utilize it to construct gene interaction networks.
Highly connected genes tend to be essential genes
Genes in network modules have focused biological themes
Top GO Biological Processes shared by the genes in the 7 modules shown in Figure 12, at p < 0.01.
GO Accession #
cellular biosynthetic process
macromolecule biosynthetic process
ribonucleoprotein complex biogenesis
macromolecule biosynthetic process
cellular biosynthetic process
cellular protein metabolic process
protein metabolic process
cellular macromolecule metabolic process
rRNA metabolic process
ribonucleoprotein complex biogenesis
RNA metabolic process
nucleobase, nucleoside, nucleotide and nucleic acid metabolic process
ribonucleoprotein complex biogenesis
mitotic cell cycle
cell cycle process
cellular biosynthetic process
cell cycle phase
regulation of cell cycle
DNA-dependent DNA replication
rRNA metabolic process
The importance of phase space information
Synchrony through oscillation is a common, maybe the most efficient, way to coordinate regulation in complex nonlinear dynamic systems. It is also a ubiquitous phenomenon in biological systems, where oscillations are observed in all organisms across a wide range of temporal and spatial scales, and are believed to play an important role in maintaining homeostasis and delivering encoded information [22, 46, 47]. Examples include the synchronized oscillations in interneuron networks, pulsatile endocrine hormone secretion, circadian oscillators, Somite segmentation, and innumerable others. Higher order phase locking occurs frequently, reflecting the multi-stability of complex systems, and is believed important to function. For instance, in the study of cardiorespiratory synchronization, when plotting the instantaneous respiratory phase at the occurrence of a heartbeat versus time, Schäfer and co-workers found n:1 synchronization between heart and respiration . In a study with anesthetized rats, Stefanovska et al. further observed lengthy synchronization epochs, and transitions from one ratio to another. They suggested that such transitions might be useful in monitoring depth of anesthesia [49, 50].
Increasing evidence suggests that, as in many other complex systems in nature, oscillation and frequency modulation is also a general strategy for an organism to coordinate multi-gene responses to external signals . It is found that the transcription factor activities, rather than levels of transcription factor expression, mediate transcriptional regulations . In the negative feedback loop between the tumor suppressor p53 and the oncogene Mdm2, p53 is expressed in a series of discrete pulses after DNA damage, leading to oscillations in Mdm2 [24, 25]. The amplitude of the oscillations was much more variable than the period, suggesting strong temporal regulation. In the NF-κB signaling pathway, NF-κB (RelA) localization showed asynchronous oscillations following cell stimulation that decreased in frequency with increased IκBα transcription. Transcription of target genes depended on its oscillation persistence, and thus the functional consequences of NF-κB signaling likely depend on temporal characteristics of the oscillations . In yeast cells it has been shown for several calcium stress responsive TFs (Crz1 and Msn2) that calcium concentration controls the frequency, but not the duration, of their oscillatory localization bursts, and the oscillation propagates to the expression of downstream genes. It has been argued that such frequency modulation of localization bursts ensures proportional expression of multiple target genes across a wide dynamic range of expression levels .
These facts all imply the importance to study synchrony in expression oscillation, to understand the information encoded and the underlying interaction/regulation mechanisms. Data from these studies [22–27] also indicate that oscillations in gene expression, like most other oscillations in biological systems, are often pulsatile rather than harmonic. Therefore, mathematical methods rooted from phase space analysis are desirable. The latter can potentially lead to new efficient network modeling algorithms, and help to understand the complex spatial-temporal architecture of transcriptome.
Advantages and limitations of the phase locking analysis
Additional to the theoretical appeals, we believe that phase locking analysis has several advantages in clustering genes of similar patterns and in network modeling. Firstly, compared to approaches that primarily rely on the similarities in the amplitude domain patterns, phase locking utilizes the dynamics underlying the temporal pattern, which is more robust against noise. This is particularly appealing in network modeling of gene expression data, as they usually contain high noise. Also the transcript abundance measurements often reflect a compressed, even altered representation of the true expression changes due to technical issues [51–54]. These can significantly mask the true patterns in amplitude changes. In contrast, phase locking analysis, which focuses on the timing of the changes, will be less affected by the noise and the technical issues in the microarray gene expression study. In fact, it is known that noisy coupled nonlinear dynamic systems may synchronize in phase whilst their amplitudes remain uncorrelated [31, 55]. In our analysis, we have seen ample examples where interacting gene pairs (according to BioGRID or ChIP-chip) exhibit obvious phase locking, but have very low correlation (See Figure 9 and Additional files 2, 4 and 5).
Secondly, in phase locking analysis, the phase lags in gene expression changes between different genes are automatically accounted for, and the performance is not affected by the amount of lag. On the other hand, the performance of alignment approaches depends on whether the lag is close to an integer number of the time steps of the experiment, and they need to be adjusted for each pair as the lags of different gene pairs vary greatly.
Thirdly, phase locking does not require the two time series to have the same dynamic time scale, or the same frequency. It is known that some pathways or gene groups in a cell respond to external signals at a much faster time scale than others . High order m:n phase locking analysis can take care of such interacting gene pairs, whilst they would be missed by the alignment method.
A limitation of the phase locking analysis is its reliance on the temporal spectrum to accurately derive the instantaneous phase, which could significantly affect results when either the number of sampling points or the sampling frequency is too low. Note that higher sampling frequencies are needed to detect high order phase locking. We have carried out a set of numerical simulations and observed significant deterioration in performance when the number of sampling points is reduced to ~5 or lower (data not shown). This limitation is not unique to the phase locking analysis. All pair-wise alignment approaches suffer from the same limitations as they all rely on adequate sample size to make a good assessment of whether the expression patterns of the pair are similar or not. There is also a caveat with the application of phase locking analysis to network modeling. Gene interactions or phase locking occurs inside each cell. High throughput time series gene expression studies commonly measure a population of cells all at one time, effectively averaging the expressions of each gene across the whole cell population. In processes where there is high synchrony in the whole cell population, such as the cell cycle study presented here, phase locking between gene expression changes that occur inside each cell is preserved at population level, and can be detected from population measurements. In other biological processes, where cellular heterogeneity plays a key role, information of the signaling dynamics and phase locking inside each cell could be lost in population-level measurement. Again, in such situations, the performance of the co-expression alignment approaches to detect interaction will also be affected.
Lastly, the recent advancements in single-cell techniques has enabled the generation of time-series gene expression measurements in a large number of individual living cells . We believe that phase locking analysis will be particularly suitable for such data. The dynamic information at the level of individual living cells will be critical to unravel how a genetic network operates at the systems level.
A major challenge in systems biology is to reconstruct gene networks that are involved in basic cellular processes, and to understand how alterations to the network affect functions and consequently phenotypes. Interactions between genes can result in expression amplitude variations as well as temporal patterns. Therefore, network inference utilizing temporal domain information deserves more attention. In this study, we investigated the potential of the phase locking analysis in network modeling of time course gene expression data. We demonstrated that interacting gene pairs, including transcription regulation interaction, protein-protein, or genetic interaction, are more likely to exhibit phase locking in their gene expression changes, and vice versa. Among the phase locked pairs, up to ~10% are contributed from higher order locking, and the relative phase difference spans across the whole range of [-π, π]. Based on these findings, we constructed interaction networks and revealed that genes with higher network degrees are more likely to be essential genes. Utilizing the phase-locking index based topological overlapping matrix, we further investigated the modular structures in the network. We showed that genes forming network modules are more likely to be essential genes than scattered genes in the network, and members of the same module tend to be involved in the same biological functions and processes. In view of the importance of the frequency domain signal in transcription regulation, we believe that the phase locking analysis can potentially lead to new network modeling approaches and help to understand the dynamic designs of the intracellular signaling networks.
Gene expression data
Yeast cell cycle gene expression data were downloaded from the Yeast Cell Cycle project at the Stanford University http://genome-www.stanford.edu/cellcycle/data/rawdata/. These studies profiled expression changes in 6178 genes at ~20 time points under each condition following alpha factor arrest (18 time points from 0-119 minutes), elutriation ELU (14 time points from 0-390 minutes), and arrest of a cdc15 (24 time points from 10-290 minutes) and a cdc28 (28 time points from 0-160 minutes) temperature sensitive mutant [33, 34]. Many genes have missing data points. The cdc28 data is the most severely affected, ~80% of genes contains at least 1 missing values. For the other three datasets, it ranged 6-27%. In this study, we kept genes that had at most 1 missing data point in each data set for further analysis. Among all 6178 genes profiled, 144 are annotated by the Gene Ontology (GO, http://www.geneontology.org/) to be involved in the biological process of cell cycle (Since the start of our study, more genes have been annotated to be involved in this biological process. The number is now higher than 144). They are termed cell cycle genes in this study.
ChIP-chip data of transcription binding
For each gene i, the Z-score of the sig across the 9 TFs is also determined to examine binding specificity.
We constructed a positive control target set for each TF that consists of those with sig > 3 (significant binding), and the Z-score > 1.5 (the binding is specific). The number of targets for each TF ranges from 18-54 for the alpha factor arrest data set, 12-50 for the cdc15 dataset, 1-21 for the cdc28 dataset, and 19-65 for the ELU data set. A negative control non-target set is constructed for each TF that includes all genes with sig < 1 (p > 0.1). This set consists of over 3,000 genes for each TF in the alpha factor and cdc15 datasets, over 875 for each TF in the cdc28 dataset, and over 4,000 for each TF in the ELU dataset.
Phase locking analysis
constrained around a constant value Ψ0, where m and n are integers.
In a perfect locking λm, n= |exp(i Ψ0)| = 1, and λm, n→ 0 when Ψm,n(t) is randomly distributed.
where arg is a mathematical function operating on complex numbers and gives the angle. Note that the value of λm, n is not affected by the value of the relative phase difference Ψ0 ; the two time series can have any amount of phase lag.
Here max is used because the two genes are considered phase locked if the value of anyone of the λ m, n 's, is high. We do not think there is a need to consider higher order than 4 due to the limited number of time points in these datasets, and the noise in microarray data. In addition, higher order locking is less common and probably unstable in the presence of noise . We have also investigated several other measures of phase locking, including the Shannon's entropy and the intensity of the first Fourier mode of the distribution of Ψm, n, . No significant difference in predictions was found. Therefore, in this study we will only report the results of λ.
Where λ0 is a threshold. The network degree of each gene i thus can be calculated by k i = ∑ j A ij . In this study, λ0 is chosen to be μ + 2σ (i.e. Z-score = 2), where μ and σ are the mean standard deviation of λ between random gene pairs. Namely, gene pairs with the value of their phase locking index at least two standard deviations above mean of all pairs are considered significantly phase locked. In the 4 yeast cell cycle data sets, λ0 ~0.75 - 0.80. When we compare phase locking networks with the networks predicted using the correlation coefficient r, a same Z = 2 cutoff was used.
where n i = ∑ j λ ij is the node connectivity. T measures the sharing of first degree phase locked neighbors, and is designed to identify the modular structure in interaction networks. Hierarchical clustering is then performed using T ij as the similarity measure to identify the network modules.
List of abbreviations used
area under curve
Cumulative Distribution Fraction
- KS test:
receiver operating characteristic
This work was supported in part by National Institute of Diabetes and Digestive and Kidney Diseases Grant R01DK080100 (X.W.); National Institute of Allergy and Infectious Diseases Grant R01AI078713 (M.J.H.); and a pilot grant from the Component on Analytic & Epidemiologic Genetics of the UAB Heflin Center for Genomic Science.
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