Improving the accuracy of expression data analysis in time course experiments using resampling
© Walter et al.; licensee BioMed Central Ltd. 2014
Received: 8 February 2014
Accepted: 16 October 2014
Published: 25 October 2014
As time series experiments in higher eukaryotes usually obtain data from different individuals collected at the different time points, a time series sample itself is not equivalent to a true biological replicate but is, rather, a combination of several biological replicates. The analysis of expression data derived from a time series sample is therefore often performed with a low number of replicates due to budget limitations or limitations in sample availability. In addition, most algorithms developed to identify specific patterns in time series dataset do not consider biological variation in samples collected at the same conditions.
Using artificial time course datasets, we show that resampling considerably improves the accuracy of transcripts identified as rhythmic. In particular, the number of false positives can be greatly reduced while at the same time the number of true positives can be maintained in the range of other methods currently used to determine rhythmically expressed genes.
The resampling approach described here therefore increases the accuracy of time series expression data analysis and furthermore emphasizes the importance of biological replicates in identifying oscillating genes. Resampling can be used for any time series expression dataset as long as the samples are acquired from independent individuals at each time point.
Even with decreasing costs for sequencing and microarray experiments, time series experiments are still expensive and require a large number of samples. Thus, most time series currently have a very limited number of biological replicates. This makes it difficult to identify genes that truly show time-dependent expression patterns (true positives) and genes that just seem to have similar patterns due to biological variance (false positives). The biological variance is likely to be relatively high, especially when samples are collected from higher eukaryotes, because animals and plants are usually sampled from different individuals to avoid perturbation artifacts during sampling. Thus, in most time course experiments, the samples at each time point are usually from different individuals, resulting in a high biological variance among samples. This is the main reason why sufficient numbers of replicates are necessary. Lee et al. proposed that three replicates are sufficient, but this number also depends on the type of experiment -. However, the importance of biological replicates is often neglected in time series experiments, especially when circadian rhythms in gene expression are examined using transcriptomics datasets.
Many organisms have an endogenous clock, known as a circadian clock, to coordinate daily activities. The output of the circadian clock has the period of approximately 24 h; for example, the body temperature and sleep-wake cycle in humans, leaf movement in Mimosa, and flower opening in night-blooming jasmine all show 24 h diurnal rhythms under both light/dark and approx. 24 h rhythms under constant conditions -. Although the molecular components of circadian clocks are not conserved between animals and plants, negative and positive feedback loops in transcriptional and post-translational levels are the core system of circadian clocks in both animals and plants . These multiple interlocked feedback loops confer stability and protection from stochastic perturbations on the complexity of the circadian system ,. To understand this complex network on a transcriptional level, time series microarrays have been frequently used to examine the oscillation of genes on a genomic scale ,.
Diurnal rhythms in transcript accumulation can be described in mathematical terms, including period, phase, and amplitude . There are several different algorithms that can be used to calculate these parameters from real data; they can furthermore be applied to identify oscillating genes in microarray or RNA-sequencing data. From the algorithms available we selected ARSER , HAYSTACK  as well as the algorithms implemented in BIODARE (http://www.biodare.ed.ac.uk/) ,. ARSER was selected as it has been shown to outperform earlier available algorithms such as COSOPT and Fisher’s G-Test . The BIODARE platform is not originally designed for the analysis of gene expression data as the maximal list length of datasets that can be submitted is limited to 2500. Thus, gene expression data has to be split into multiple datasets. Nevertheless BIODARE has the advantage of providing 6 additional different algorithms for the analysis.
Using these algorithms we show the influence of replicates and resampling on the accuracy of predictions of rhythmically expressed genes. Although we perform the analysis to identify oscillating genes in circadian expression datasets, the resampling method can be similarly used to improve the detection of other time dependent expression patterns as long as the samples are collected from different individuals at the specific time points.
Results and discussion
The determination of oscillating genes is a binary classification. There are only two possible outcomes: either a gene is rhythmically expressed or it is not. The accuracy of this classification can be estimated by a confusion matrix. There are four fundamental members of the matrix: true positives (expression profiles correctly classified as periodic), false negatives (expression profiles incorrectly classified as non-periodic), true negatives (expression profiles correctly classified as non-periodic), and false positives (expression profiles incorrectly classified as periodic). As the number of true negatives and false negatives can be directly calculated from the total number of oscillating and non-oscillating genes and the number of true- and false positive genes identified, we only analyzed true- and false-positives in our calculations. The total number of oscillating and non-oscillating genes was set to 8400 in our simulated datasets (see Methods section for details).
To calculate the performance of ARSER, HAYSTACK and the algorithms implemented in BIODARE, we simulated different conditions and wave forms for oscillating transcripts. To do this we used three different simulation procedures.
To simulate entrained, synchronized oscillations all simulations were done with a fixed period ranging from 22 to 28 h (LD-dataset). In contrast, the differences in free running period between different individuals under constant conditions were simulated by generating a dataset that contained 36 time courses that differed in period according to published standard deviations for individual cells  (LL-dataset). In addition we generated a time course based on a published ordinary equation model of the mammalian circadian model  (ODE-dataset) (see Methods section for details).
In the averaged time courses the number of true positives detected by ARSER is in most cases slightly higher than in the individual replicates but this again comes at the cost of a higher number of false positives. HAYSTACK and BIODARE FFT-NLLS show similar performance but HAYSTACK has more problems to detect oscillating genes in ODE-based simulations. As detected false positive genes can be experimentally quite costly in follow up studies, we wanted to improve the accuracy of the prediction without increasing the number of replicates or time points required as this too would be experimentally costly if not infeasible.
For the dataset from Na et al.  147 transcripts were found in all 3 initial replicates (Figure 5C). In our resampled dataset 796 transcripts were identified as oscillating when we require a consensus of 10, 183 genes remain if we require a consensus of 20.
In this analysis, we conclude that in comparison to single replicates and averaged datasets, our resampling method improves the detection of oscillating transcripts without increasing the number of false positives. The resampling method particularly outperformed the average method to reduce the number of false positive transcripts. Furthermore, the resampling method shows that biological replicates are important to accurately identify true oscillating transcripts using time series gene expression datasets, and that the average method may result in a large number of false positives. To reliably identify oscillating transcripts, resampled datasets should be generated from at least 3 experimental samples per time point.
Simulated time series
where SNR?=?2 is the signal-to-noise ratio; ? is the period in the range of 22 and 28 hours; ? is phase (0-28 h with 0.1 h intervals); and ?t is the normally distributed noise term (mean =0 standard deviation =1).
Desynchronizing individuals under constant condition (for example constant light (LL)) were simulated using the above formula but with a fixed period that was randomly selected for each of the 36 initially simulated time courses. The periods were normally distributed with a mean of 25 hours and a standard deviation of 3 h according to published experimental data .
For more realistic circadian simulations we used the ODE-model of the mammalian circadian oscillators by Leloup et al. . We first generated time courses for all variable model species and then generated phase shifted copies thereof. Phase shifts had 0.1 h intervals. From these time courses, datasets with 4 h sampling intervals were generated. Normally distributed white noise was added as for the cosine wave simulations. To simulate non-periodic time series, we used normally distributed white noise with the same mean and standard deviation as above.
The simulations described above were repeated 36 times for each type of data. If not described otherwise we generated 3 initial datasets from these time courses by randomly selecting once from each original simulation to generate a new 4 hour interval time course. This mimics the sampling procedure from different individuals in real experiments.
Python scripts used to generate time series and initial datasets are provided as Additional file 1.
ARSER, HAYSTACK and BIODARE
Recently, Yang and Su developed the algorithm ARSER, which combines frequency domain and time domain analyses . The algorithm first removes any linear trend from time series data (data preprocessing), and then the period is determined by AR spectral analysis (period detection). Because the period can differ from 24 h depending on the experimental conditions, the algorithm takes a range from 20 h to 28 h into account. With each period, ARSER employs harmonic regression to determine the four cyclic parameters: period, amplitude, mean level, and phase (rhythm modeling). Finally, false discovery rate (FDR) q-values are calculated for multiple comparisons and the output was filtered and only those transcripts with a q-value greater than 0.05 were consider in the analysis.
To exclude the possibility that our results depend on the chosen algorithm, the analysis was repeated with the HAYSTACK algorithm  and the FFT-NLLS algorithm implemented in BIODARE ,. HAYSTACK was designed to find periodic patterns in any large-scale dataset representing at least three data points. The web version and 120 cycling patterns are available at http://haystack.mocklerlab.org/. The HAYSTACK algorithm compares gene expression profiles with predefined cycling patterns. Different cutoffs are used to detect oscillating patterns in gene expression. The most important parameter is the correlation coefficient. The higher value means a higher correlation between the experimental data and the predefined models. A coefficient of +1 indicates perfect positive correlation. Other cutoff values are the fold change and p-value, and these values are used to achieve statistical significance. The HAYSTACK algorithm searches for at least six different patterns, including “asymmetric,” “rigid,” “spike,” “cosine,” “sine,” and “box-like” patterns. The models that most successfully identify rhythmically expressed genes are “cosine” and “spike.”
BIODARE and the implemented algorithms are described elsewhere . Shortly, FFT-NLLS (Fast Fourier Transform - Non-Linear Least Square) is a curve fitting method which models a sum of cosine functions and calculates confidence levels for period, phase and amplitude. The BIODARE FFT-NLLS algorithm detection was limited to period range from 20 to 28 h to match the period range of ARSER. Linear detrending was applied.
The artificially generated time series dataset consists of 12 time points at 4 h sampling intervals, representing 48 h of observation. To generate resampled datasets, expression values of each gene were randomly selected from (if not stated otherwise) three initial replicate time series, and the values were combined to generate the new resampled dataset. Each expression value has an equal probability of selection, and the time points are treated independently of one another. If not stated otherwise, the procedure was repeated 36 times, and we created 36 different resampled datasets (python script provided as Additional file 1). Each resampled dataset was analyzed by the ARSER algorithm with the stringency threshold (q-value) set to 0.05. HAYSTACK algorithm was used with the following parameter: p-value?=?0.05; fold change?=?2.0, correlation cutoff?=?0.8; and background cutoff?=?0.01. Using the oscillating transcripts detected the consensus between the 36 resampled datasets were calculated.
We thank Emily Wheeler for language editing and Tomasz Zielinski for his support that allowed us to use BIODARE for gene expression data analysis. This work is supported by European Research Council advanced grant ClockworkGreen (No. 293926) to ITB, the Global Research Lab program (2012055546) from the National Research Foundation of Korea, Human Frontier Science Program (RGP0002/2012), and the Max Planck Society.
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