 Research article
 Open Access
 Published:
MICOP: Maximal information coefficientbased oscillation prediction to detect biological rhythms in proteomics data
BMC Bioinformaticsvolume 19, Article number: 249 (2018)
Abstract
Background
Circadian rhythms comprise oscillating molecular interactions, the disruption of the homeostasis of which would cause various disorders. To understand this phenomenon systematically, an accurate technique to identify oscillating molecules among omics datasets must be developed; however, this is still impeded by many difficulties, such as experimental noise and attenuated amplitude.
Results
To address these issues, we developed a new algorithm named Maximal Information Coefficientbased Oscillation Prediction (MICOP), a sine curvematching method. The performance of MICOP in labeling oscillation or nonoscillation was compared with four reported methods using Mathews correlation coefficient (MCC) values. The numerical experiments were performed with timeseries data with (1) mimicking of molecular oscillation decay, (2) high noise and low sampling frequency and (3) onecycle data. The first experiment revealed that MICOP could accurately identify the rhythmicity of decaying molecular oscillation (MCC > 0.7). The second experiment revealed that MICOP was robust against highlevel noise (MCC > 0.8) even upon the use of lowsamplingfrequency data. The third experiment revealed that MICOP could accurately identify the rhythmicity of noisy onecycle data (MCC > 0.8). As an application, we utilized MICOP to analyze timeseries proteome data of mouse liver. MICOP identified that novel oscillating candidates numbered 14 and 30 for C57BL/6 and C57BL/6 J, respectively.
Conclusions
In this paper, we presented MICOP, which is an MICbased algorithm, for predicting periodic patterns in largescale timeresolved protein expression profiles. The performance test using artificially generated simulation data revealed that the performance of MICOP for decaying data was superior to that of the existing widely used methods. It can reveal novel findings from timeseries data and may contribute to biologically significant results. This study suggests that MICOP is an ideal approach for detecting and characterizing oscillations in timeresolved omics data sets.
Background
The circadian rhythm, which involves oscillations over a cycle lasting 24h, plays a critical role in biological systems [1]. Transcriptional negative feedback loops composed of clock genes are a key component of this mechanism [1,2,3]. These clock genes regulate downstream gene expression, leading to the 24h cyclic oscillation of various physiological phenomena such as cell division, energy metabolism, blood pressure, and sleep [4, 5]. Many molecules are involved in these systems, so comprehensive and multilayered approaches are required to clarify the complex systems. Thus, it is crucial to obtain a deep understanding of the circadian rhythm in order to understand biological systems.
The availability of biological timecourse data is key to elucidating circadian rhythms, but there are several difficulties in analyzing biological timeseries data. In particular, the accumulation of timeseries omics data via the technological innovation of mass spectrometry and DNA sequencers has led to the following problems: (1) low sampling frequency and (2) unstable oscillation. The first problem is derived from the generally low sampling frequency of omics datasets because comprehensive approaches such as proteomics and transcriptomics are often expensive and laborious. Several omics studies collected timecourse data every 2–4 h per day and estimated periodicity using 12 to 24 points [6,7,8,9]. This sampling frequency of omics data was relatively low compared with those for locomotor activity or tissue luminescence, which were provided every minute [10]. The second problem is the unstable oscillation (such as amplitude decay) of timecourse experimental values. There are various types of unstable oscillations in the expression pattern of genes and proteins. For example, previous reports assumed unstable oscillations such as cosine with outlier time points, cosine with a linear trend, cosine with an exponential trend, and decaying cosine as possible natural oscillation phenomena [11, 12]. These unstable oscillations hamper oscillation detection, in particular for amplitude decay, which is often observed in experimental systems and, is caused by degradation of the metabolic activity of cells and degradation of fluorescent protein [13]. Therefore, novel computational analysis that functions over the time course of omics studies with limited sampling points and amplitude decay should be developed.
Many analytical approaches to predict molecules with oscillating levels from timeseries data have been developed. These algorithms were classified into timedomain and frequencydomain methods [14]. Typical timedomain methods are based on cosine curvebased pattern matching and their simple algorithm helps biologists to evaluate their analytical results [14]. For example, COSOPT and chisquared periodogram are algorithms employing curve fitting and autocorrelation, respectively [15, 16]. Hughes et al. developed a nonparametric approach using rank by the nonparametric Jonckheere–Terpstra (JT) test and obtained the strength of correlation by Kendal’s tau test (JTK) [17]. However, they have disadvantages, such as sensitivity to noise and outliers, and being able to detect only cosine wavelike curves; as such, there is a need for a novel algorithm that can overcome these obstacles. Meanwhile, frequencydomain methods based on spectral analysis are strongly noiseresistant and modelindependent [14]. Fisher’s Gtest estimates periodicity by calculating the periodogram of experimental data and calculating the Pvalue using Fisher’s Gstatistic [18]. Autoregressive spectral (ARS) analysis is an approach combining timedomain and frequencydomain methods, used to identify molecules with rhythmically oscillating levels in largescale timeresolved profiles by autoregressive spectral analyses [19, 20]. Similarly, an approach combining autocorrelation and spectral analysis after removing noise from raw data with a digital filter was also proposed [21]; however, frequencydomain methods are limited by the low sampling frequency and short time period in omics experiments, which means that they are often insufficient to predict the periodicity of largescale omics datasets [22]. Therefore, developed approaches to characterize oscillating molecules in biological data have been used with success and have contributed to our understanding of biological systems; meanwhile, it has been shown that each method sometimes produces inconsistent results because of noise, sampling rate, and waveform [23]. A novel oscillation prediction method compatible with omics experiments, having a low sampling frequency, was required, for which quantitative evaluation of the performance could also be achieved.
This study developed Maximal Information Coefficient (MIC)based Oscillating Prediction (MICOP) for analyzing timeseries omics datasets with highlevel noise and possible decay. MICOP offers unsurpassed performance to identify and characterize oscillating molecules in omics datasets.
Methods
Datasets
Timeresolved data from biological samples are generally obtained every 2–6 h per day [6,7,8,9]. Therefore, we simulated timeseries data containing 6–24 points for two cycles for a performance test. Half of these artificially simulated data did not feature oscillation, while the other half did. For oscillating data, to mimic experimental data, noise according to the normal distribution (average = 0, standard deviation = 0–0.6) was added to the sin curve. The decaying timeseries datasets were designed so that the value of the peak in the second cycle is onethird of the value of the peak in the first cycle. The nonoscillating data were random numerical data. Proteomics datasets of C57BL/6 J and C57BL/6, which was already normalized, were downloaded from journal websites [8, 9]. The simulated data released by Wu et al. are included in MetaCycle, as described below [23, 24].
Design
A conceptual diagram of MICOP is shown in Fig. 1. The MIC belongs to the nonparametric exploration class, and the score indicates the strength of the linear or nonlinear association between variables. First, the mutual information for a scatterplot of X and Y is calculated as:
Where p(X) and p(Y) are marginal probability distribution functions of X and Y, and p(X,Y) is joint probability distribution function. Then, to compare the values from different grids and to obtain normalized values between 0 and 1, MIC is divided by the lesser number of X and Y bins. MIC is calculated as;
The algorithm calculates the MIC value between the reference sin curve and experimental data. The same sin curve was used for all input traces. The script for MICOP and its performance test is provided as an R script. The Pvalues were calculated from the frequency of each MIC value of experimental data and the MIC values that were calculated from the random numbers. The MIC represents the strength of association between the two variables. The MIC between the reference sin curve and targeted data, such as experimental data or simulated data, was calculated using the following steps. Step 1: Grids with different resolutions are introduced to separate the different areas of the scatter plot of the two variables. Step 2: Maximized mutual information at each resolution is selected. Step 3: The mutual information is normalized for each resolution. Step 4: The maximum value among all division methods is MIC. Step 5: to calculate the Pvalue, MIC between the reference curve and 1000 nonoscillating timeseries datasets, which comprised random values, was calculated. We compared MIC values and enumerated the occurrences (k) when the MIC score exceeded the score calculated. k/1000 was taken as the Pvalue of the MICOP. Then, we compute the Pvalue as;
where I is the indicator function, and X_{pi} and Y_{pi} is the ith permutated version of X and Y, respectively. If the datasets have missing points, MIC is calculated without the point.
Performance test
To test the performance of MICOP, the periodicity of simulated data was determined by MICOP, JTK, ARS, and LS. To compare the precision and sensitivity of MICOP, the MCC was compared [25]. MCC values were calculated as below:
where TP is the number of true positives, TN is the number of true negatives, FP is the number of false positives, FN is the number of false negatives. The false discovery rate is widely used and is calculated from true positive and false positive values. In contrast, MCC is more informative as a value evaluating the performance of the classification method because it is calculated from true positive, false positive, true negative, and false negative values.
Reanalysis of proteomics data
To verify the practicality of MICOP, we reanalyzed the published timeseries data [8, 9, 26]. Briefly, these are proteome datasets of mouse liver sampled every 3 h for 2 days, and simulated data which are two cycles containing 20 molecules [26]. The MIC and Pvalue were calculated as described in the Design section.
Programming language, packages, and statistical analysis
R language (ver. 3.3.2) was used for all analyses [27]. Three different random seeds were used; rnorm function was used to generate random numbers according to a normal distribution and runif function was used to generate uniform random numbers. The performance of each method was compared to MICOP by TukeyKramer test. The Pvalues were corrected by the Benjamini–Hochberg procedure for multiple testing. A graphical package named ggplot 2 (ver. 2.2.0) was used to draw figures. The Minerva package (ver. 1.4.3) was used to calculate the MIC score, and binning range to calculate MIC score was 0.6, which is a default value of the R library. The MetaCycle package (1.1.0) was used for periodicity judgment by ARS, JTK, and LS [21, 23, 24].
Results
Comparison of MICOP and existing methods for decaying data
To test the performance of MICOP, JTK, ARSER, and LombScargle (LS) for mimicking the decaying timeresolved data, the Matthews correlation coefficient (MCC) values were calculated to differentiate significantly oscillating data from nonoscillating data using timeseries simulation data, including 100 sets of oscillating data and 100 sets of nonoscillating ones (Fig. 2, Additional file 1) [17, 20]. Twoway ANOVA with Method and sampling frequency as factors revealed significant effects of Method (F = 631.8, P < 0.005), sampling frequency (F = 810.1, P < 0.005) and Method x sampling frequency interaction (F = 122.9, P < 0.005). MCC values were 0.72 (P < 0.005), 0.40 (P < 0.005), 0.082 (P < 0.005), and 0.00 (P < 0.005) for MICOP, ARS, JTK, and LS, respectively, when the sampling interval was 4 h (Fig. 2). The MCC values increased as the sampling frequency increased, and these values became almost equal to 1 in all methods at 1h interval sampling. The MCC values of MICOP were 0.7 or more at all sampling frequencies and were the highest at a sampling interval of 1–3 h, followed by ARS and JTK. LS did not function as a classifier at a sampling interval of 1–3 h.
Comparison of MICOP and existing methods for noisy or lowsamplingfrequency or onecycle data
We compared the accuracy of MICOP and existing methods for timeseries data containing noise and having a low sampling frequency without attenuation (Fig. 3a and b, Additional file 2). Initially, we quantitatively evaluated the degradation of classification performance due to the noise of MICOP (Fig. 3a). Twoway ANOVA with Method and noise level as factors revealed significant effects of Method (F = 1099.4, P < 0.005), noise level (F = 643.2, P < 0.005) and method x noise level interaction (F = 475.5, P < 0.005). The MCC values were 0.8 or more, except for LS, in all conditions, even if the noise was 0.500; however, LS did not function as a classifier when the noise was 0.375 or more.
The performance of MICOP as a classifier for lowsamplingfrequency unattenuated data was also quantitatively evaluated (Fig. 3b). Twoway ANOVA with Method and sampling frequency as factors revealed significant effects of Method (F = 424.3, P < 0.005), sampling frequency (F = 447.7, P < 0.005) and Method x sampling frequency interaction (F = 142.2, P < 0.005). The MCC values increased in all methods as the sampling interval decreased, and were equal to 1 in all four methods at a sampling interval of 1 h. LS did not function as a classifier at sampling intervals of 3–4 h. The MCC values of MICOP were 0.7 or more under all conditions.
We compared the accuracy of MICOP and existing methods for onecycle data (Fig. 4). Among all conditions (method, noise, and sampling frequency), determination accuracies using onecycle were lower than those using two cycles. All methods did not work under all conditions at the 4h sampling frequency. Meanwhile, MICOP and JTK showed high performances under sampling conditions ≤3 h.
Reanalysis of previously reported timeresolved proteomics datasets
We reanalyzed the timeseries proteome data for mouse liver reported by Mauvoisin et al. using C57BL/6 and those reported by Robles et al. using C57BL/6 J, as well as simulated data released by Wu et al. (Fig. 5, Table 1, Table 2) [8, 9, 23]. The numbers of significantly oscillating proteins assessed by standard harmonic regression were 9 (the F test for multilinear regression, P < 0.01), 9 (Fisher’s exact test, P < 0.01), and 3 (P < 0.01) for biological data in the original work. Meanwhile, 32, 22, and 5 proteins were judged as being significantly oscillating for C57BL/6 J, C57BL/6, and Wu’s simulated data by MICOP, respectively (P < 0.05). The numbers of proteins judged to be significantly oscillating in both the original work and MICOP were 2, 8, and 2 for biological data, respectively. The numbers of proteins judged as being significantly oscillating for the three abovementioned tests only by MICOP were 30, 14, and 3 for biological data, respectively.
Discussion
Although many algorithms have been developed to extract molecules with rhythmic oscillation in their levels from largescale timeseries data derived from mass spectrometry systems or DNA sequencers, it is known that the accuracy and sensitivity of such methods depend on noise, sampling frequency, and waveform. In particular, the discussion of the prediction power in conditions of decaying oscillation was insufficient. In this research, we provide MICOP, which is classified as a timedomain method, and demonstrate that the algorithm is particularly effective for detecting decaying oscillation.
We compared the detection power of MICOP and previously reported algorithms for decaying oscillation. We revealed that, in terms of the power for detection decaying oscillation, MICOP outperformed other algorithms (Fig. 2). In particular, MICOP showed a clear advantage when the sampling frequency was low. This is because MIC can effectively detect nonlinear associations like associations between decaying oscillation and the reference sin curve (Fig. 1). Although we compared the performance for only cosine wave, additional experiment with peak wave or complex wave is also important. ARS showed high performance following MICOP because detrending at preprocessing seemed to cancel out the decay of timeseries data. JTK was the tool with the third best detection power, although high performance was expected because it was based on Kendall’s tau, which is a measure of rank correlation, and it did not depend on amplitude. This indicates that MICOP has excellent performance for decaying oscillation, and suggests that an MICbased approach that can detect nonlinear associations is useful to detect decaying oscillation.
Moreover, we compared the MCC values for all methods on data containing gradual Gaussian noise to test the noise resistance (Fig. 3a). As a result, MICOP showed equal performance to JTK and ARS in the range of standard deviation of 0.125–0.500. This indicated that the performance of MICOP for noisy data is equal to that of the existing methods. This result suggests that the robustness to noise of MICOP is the same as that of wellknown ARS and JTK, while the high performance of LS was limited to conditions with a low noise level. This numerical experiment revealed that the noise resistance of MICOP is the same as that of other widely used methods.
Clarifying the relationship between accuracy and sampling frequency in analyzing omics data, for which increasing the number of sampling points seems difficult, is important for determining the experimental design. As expected, with increase in the sampling frequency, the MCC values tended to increase (Figs. 2 and 3b). The fact that the ARS, JTK, and LS could characterize oscillation and nonoscillation in almost all cases when the sampling interval was 2 h or less is similar to the findings in original research studies of various methods and research comparing them [11, 28]. This suggested that a high sampling frequency improved accuracy; therefore, sampling frequency should be as high as experimental constraints allow.
We applied MICOP and existing methods for onecycle of data (Fig. 4). As expected, accuracy decreased for all methods when onecycle was used. However, MICOP and JTK showed high MCC values among methods under this condition. Also, MICOP seems to outperformed JTK under limited conditions which is low sampling frequency and high noise for onecycle data (Fig. 4). Human omics data often have lower sampling frequencies, high noise levels, and only onecycle. Our results suggest that MICOP and JTK have considerable potential for analyzing human omics datasets.
We reanalyzed the timeseries proteomics data of C57BL/6 J and C57BL/6 to test the performance of MICOP and explore additional candidates of proteins with rhythmic change in their expression profile [8, 9]. These datasets include the mouse liver proteome data obtained by sampling every 3 h for 2 days, for which the analysis of the peptides was performed with a mass spectrometer. Approximately, 3000 protein types were detected in each study. Proteins that were detected in both MICOP and the original studies numbered 2 and 8 for C57BL/6 J and C57BL/6, respectively (Fig. 5). This actual application for proteomics data suggests that MICOP can obtain results in a manner approximately similar to the existing methods. Specifically, the MICOP results were consistent with those in the original articles regarding these commonly identified proteins. Furthermore, the proteins that were uniquely identified with MICOP were numbered 30 and 14 for C57BL/6 J and C57BL/6, respectively (Table 1, Table 2). These results strongly suggest that MICOP is a powerful tool to detect proteins with rhythmic changes in their expression levels from timeresolved proteomics data.
Although mass spectrometrybased approaches have been used for proteomelevel studies of circadian rhythms, completely measuring mouse proteomes remains difficult. A comprehensive transcriptome analysis with parallel sequencers has revealed that ~ 15–20% of mouse liver mRNA significantly oscillates [29]. However, in these proteome studies of C57BL/6 and C57BL/6 J, significantly oscillating protein are rare (< 1% of detected total proteins; FDR < 0.05), a result inconsistent with those of mouse proteome studies. Multiple factors can explain this pattern. Typical clock protein known as principle oscillators such as CRY1, CRY2, PER2, REVERBα and CLOCK have comparatively low expression levels and are not detected in these studies [8, 9]. In addition, nonGaussian experimental noise which is specific to MS measurement hampers the application of statistical test on proteins [30]. These problems may be improved by analyzing higher quality proteome datasets with modern technologies [31, 32]. Some core circadian proteins such as CRY1, CRY2, PER2, REVERVα and CLOCK could be detected in recently published proteome datasets [31, 32]. Thus, the development of proteome analysis technology may resolve discrepancies between results of transcriptome analysis and proteome analysis, and clarify connections within the circadian rhythm transcription and translation network.
We present a new list of proteins that oscillate by MICOP (Tables 1 and 2). The accuracy of these estimates is difficult to ascertain. Interestingly, when examining expression patterns of genes encoding these proteins, we estimated that the proteins were new oscillating molecules in MICOP. In addition, a large fraction of candidates was presumed to oscillate in a previous transcriptome analysis [29]. Two independent studies which measured both transcriptome and proteome of human samples revealed that only 30% of mRNAprotein correlation had statistically significant [33, 34]. This fact suggested that even if mRNA abundance is oscillating, protein abundance may not be always oscillating. However, about 90% of mRNAprotein correlation showed positive, hence rhythmic mRNA expression suggests the possibility of protein oscillation [34]. An overlap between reanalyzed proteomics data by MICOP and transcriptome analysis showed a consistent result.
MICOP accuracy tends to be low for data that do not perfectly fit a sine curve. The periodicity that MICOP can detect is subject to the shape of the reference curve, so changing the reference curve is necessary to detect asymmetric waveforms including saw toothlike shapes like RAIN [30]. Furthermore, adjusting the false discovery rate is essential for accurate prediction, since MICOP repeats the hypothesis tests. In addition, verification with additional data such as periodic peak wave or overlapping sine wave is necessary in order to evaluate the accuracy of MICOP more precisely. Judgments of phase and cycle are possible in principle, but we did not perform them; therefore, this should be considered in future studies. Mutual information increased when sample size was small and correlation between two variables was null, even when the variables were random [35]. We solved this issue in MICOP by determining the Pvalue with the Monte Carlo method. When the time points (sample size) are small, the criterion for calculating the Pvalue increases, and when the time points are large, the criterion for calculating the Pvalue decreases (Additional file 3). In this paper, we presented MICOP, which is an MICbased algorithm, for predicting periodic patterns in largescale timeresolved protein expression profiles. The performance test using artificially generated simulation data revealed that the performance of MICOP for decaying data was superior to that of the existing widely used methods. Additionally, we indicated that MICOP is compatible with noisy data obtained with a low sampling frequency. Furthermore, the performance test using actual mouse proteomics data suggested that MICOP may be able to provide novel findings from proteomics data. Specifically, it can reveal novel findings from timeseries data and may contribute to biologically significant results. This study suggests that MICOP is an ideal approach for detecting and characterizing oscillations in timeresolved omics data sets.
Conclusion
In this paper, we presented MICOP, which is an MICbased algorithm, for predicting periodic patterns in largescale timeresolved protein expression profiles. The performance test using artificially generated simulation data revealed that the performance of MICOP for decaying data was superior to that of the existing widely used methods. Additionally, we indicated that MICOP is compatible with noisy data obtained with a low sampling frequency. Furthermore, the performance test using actual mouse proteomics data suggested that MICOP may be able to provide novel findings from proteomics data. Specifically, it can reveal novel findings from timeseries data and may contribute to biologically significant results. This study suggests that MICOP is an ideal approach for detecting and characterizing oscillations in timeresolved omics data sets.
Abbreviations
 ARS:

Autoregressive spectral estimation
 FN:

False Negative
 FP:

False positive
 JTK:

Jonckheere–Terpstra (JT) test and obtained the strength of correlation by Kendal’s tau test
 LS:

LombScargle
 MCC:

Mathews correlation coefficient
 MIC:

Maximal information coefficient
 MICOP:

Maximal information coefficientbased oscillation prediction
 MINE:

Maximal informationbased nonparametric estimation
 TN:

True negative
 TP:

True positive
References
 1.
Mohawk JA, Green CB, Takahashi JS. Central and peripheral circadian clocks in mammals. Annu Rev Neurosci. 2012;35:445–62.
 2.
Koike N, Yoo SH, Huang HC, Kumar V, Lee C, Kim TK, Takahashi JS. Transcriptional architecture and chromatin landscape of the core circadian clock in mammals. Science. 2012;338(6105):349–54.
 3.
Partch CL, Green CB, Takahashi JS. Molecular architecture of the mammalian circadian clock. Trends Cell Biol. 2014;24(2):90–9.
 4.
Weitzman ED, Fukushima D, Nogeire C, Roffwarg H, Gallagher TF, Hellman L. Twentyfour hour pattern of the episodic secretion of cortisol in normal subjects. J Clin Endocrinol Metab. 1971;33(1):14–22.
 5.
Kennaway DJ, Voultsios A, Varcoe TJ, Moyer RW. Melatonin in mice: rhythms, response to light, adrenergic stimulation, and metabolism. Am J Physiol Regul Integr Comp Physiol. 2002;282(2):R358–65.
 6.
Kasukawa T, Sugimoto M, Hida A, Minami Y, Mori M, Honma S, Honma K, Mishima K, Soga T, Ueda HR. Human blood metabolite timetable indicates internal body time. Proc Natl Acad Sci U S A. 2012;109(37):15036–41.
 7.
Minami Y, Kasukawa T, Kakazu Y, Iigo M, Sugimoto M, Ikeda S, Yasui A, van der Horst GT, Soga T, Ueda HR. Measurement of internal body time by blood metabolomics. Proc Natl Acad Sci U S A. 2009;106(24):9890–5.
 8.
Robles MS, Cox J, Mann M. Invivo quantitative proteomics reveals a key contribution of posttranscriptional mechanisms to the circadian regulation of liver metabolism. PLoS Genet. 2014;10(1):e1004047.
 9.
Mauvoisin D, Wang J, Jouffe C, Martin E, Atger F, Waridel P, Quadroni M, Gachon F, Naef F. Circadian clockdependent and independent rhythmic proteomes implement distinct diurnal functions in mouse liver. Proc Natl Acad Sci U S A. 2014;111(1):167–72.
 10.
Ono D, Honma K, Honma S. Circadian and ultradian rhythms of clock gene expression in the suprachiasmatic nucleus of freely moving mice. Sci Rep. 2015;5:12310.
 11.
Deckard A, Anafi RC, Hogenesch JB, Haase SB, Harer J. Design and analysis of largescale biological rhythm studies: a comparison of algorithms for detecting periodic signals in biological data. Bioinformatics. 2013;29(24):3174–80.
 12.
Agostinelli F, Ceglia N, Shahbaba B, SassoneCorsi P, Baldi P. What time is it? Deep learning approaches for circadian rhythms. Bioinformatics. 2016;32(12):i8–i17.
 13.
UkaiTadenuma M, Yamada RG, Xu H, Ripperger JA, Liu AC, Ueda HR. Delay in feedback repression by cryptochrome 1 is required for circadian clock function. Cell. 2011;144(2):268–81.
 14.
Chudova D, Ihler A, Lin KK, Andersen B, Smyth P. Bayesian detection of nonsinusoidal periodic patterns in circadian expression data. Bioinformatics. 2009;25(23):3114–20.
 15.
Harmer SL, Hogenesch JB, Straume M, Chang HS, Han B, Zhu T, Wang X, Kreps JA, Kay SA. Orchestrated transcription of key pathways in Arabidopsis by the circadian clock. Science. 2000;290(5499):2110–3.
 16.
Straume M. DNA microarray time series analysis: automated statistical assessment of circadian rhythms in gene expression patterning. Methods Enzymol. 2004;383:149–66.
 17.
Hughes ME, Hogenesch JB, Kornacker K. JTK_CYCLE: an efficient nonparametric algorithm for detecting rhythmic components in genomescale data sets. J Biol Rhythm. 2010;25(5):372–80.
 18.
Wichert S, Fokianos K, Strimmer K. Identifying periodically expressed transcripts in microarray time series data. Bioinformatics. 2004;20(1):5–20.
 19.
Takalo R, Hytti H, Ihalainen H. Tutorial on univariate autoregressive spectral analysis. J Clin Monit Comput. 2005;19(6):401–10.
 20.
Yang R, Su Z. Analyzing circadian expression data by harmonic regression based on autoregressive spectral estimation. Bioinformatics. 2010;26(12):i168–74.
 21.
Levine JD, Funes P, Dowse HB, Hall JC. Signal analysis of behavioral and molecular cycles. BMC Neurosci. 2002;3:1.
 22.
Langmead CJ, Yan AK, McClung CR, Donald BR. Phaseindependent rhythmic analysis of genomewide expression patterns. J Comput Biol. 2003;10(3–4):521–36.
 23.
Wu G, Zhu J, Yu J, Zhou L, Huang JZ, Zhang Z. Evaluation of five methods for genomewide circadian gene identification. J Biol Rhythm. 2014;29(4):231–42.
 24.
Wu G, Anafi RC, Hughes ME, Kornacker K, Hogenesch JB. MetaCycle: an integrated R package to evaluate periodicity in large scale data. Bioinformatics. 2016;32(21):3351–3.
 25.
Matthews BW. Comparison of the predicted and observed secondary structure of T4 phage lysozyme. Biochim Biophys Acta. 1975;405(2):442–51.
 26.
Wu G, Anafi RC, Hughes ME, Kornacker K, Hogenesch JB. MetaCycle: an integrated R package to evaluate periodicity in large scale data. Bioinformatics. 2016;32(21):3351–53.
 27.
Team RC. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing; 2017.
 28.
Cohen GB, Rangan VS, Chen BK, Smith S, Baltimore D. The human thioesterase II protein binds to a site on HIV1 Nef critical for CD4 downregulation. J Biol Chem. 2000;275(30):23097–105.
 29.
Zhang R, Lahens NF, Ballance HI, Hughes ME, Hogenesch JB. A circadian gene expression atlas in mammals: implications for biology and medicine. Proc Natl Acad Sci U S A. 2014;111(45):16219–24.
 30.
Thaben PF, Westermark PO. Detecting rhythms in time series with RAIN. J Biol Rhythm. 2014;29(6):391–400.
 31.
Robles MS, Humphrey SJ, Mann M. Phosphorylation is a central mechanism for circadian control of metabolism and physiology. Cell Metab. 2017;25(1):118–27.
 32.
Wang J, Mauvoisin D, Martin E, Atger F, Galindo AN, Dayon L, Sizzano F, Palini A, Kussmann M, Waridel P, et al. Nuclear proteomics uncovers diurnal regulatory landscapes in mouse liver. Cell Metab. 2017;25(1):102–17.
 33.
Gry M, Rimini R, Stromberg S, Asplund A, Ponten F, Uhlen M, Nilsson P. Correlations between RNA and protein expression profiles in 23 human cell lines. BMC Genomics. 2009;10:365.
 34.
Zhang B, Wang J, Wang X, Zhu J, Liu Q, Shi Z, Chambers MC, Zimmerman LJ, Shaddox KF, Kim S, et al. Proteogenomic characterization of human colon and rectal cancer. Nature. 2014;513(7518):382–7.
 35.
Steuer R, Kurths J, Daub CO, Weise J, Selbig J. The mutual information: detecting and evaluating dependencies between variables. Bioinformatics. 2002;18(Suppl 2):S231–40.
 36.
MollerLevet CS, Archer SN, Bucca G, Laing EE, Slak A, Kabiljo R, Lo JC, Santhi N, von Schantz M, Smith CP, et al. Effects of insufficient sleep on circadian rhythmicity and expression amplitude of the human blood transcriptome. Proc Natl Acad Sci U S A. 2013;110(12):E1132–41.
 37.
Pembroke WG, Babbs A, Davies KE, Ponting CP, Oliver PL. Temporal transcriptomics suggest that twinpeaking genes reset the clock. Elife. 2015;4. https://www.ncbi.nlm.nih.gov/pubmed/26523393.
 38.
Goh BC, Wu X, Evans AE, Johnson ML, Hill MR, Gimble JM. Food entrainment of circadian gene expression altered in PPARalpha−/− brown fat and heart. Biochem Biophys Res Commun. 2007;360(4):828–33.
 39.
Hughes ME, DiTacchio L, Hayes KR, Vollmers C, Pulivarthy S, Baggs JE, Panda S, Hogenesch JB. Harmonics of circadian gene transcription in mammals. PLoS Genet. 2009;5(4):e1000442.
 40.
Barclay JL, Husse J, Bode B, Naujokat N, MeyerKovac J, Schmid SM, Lehnert H, Oster H. Circadian desynchrony promotes metabolic disruption in a mouse model of shiftwork. PLoS One. 2012;7(5):e37150.
 41.
Gossan N, Zeef L, Hensman J, Hughes A, Bateman JF, Rowley L, Little CB, Piggins HD, Rattray M, BootHandford RP, et al. The circadian clock in murine chondrocytes regulates genes controlling key aspects of cartilage homeostasis. Arthritis Rheum. 2013;65(9):2334–45.
 42.
Miller BH, McDearmon EL, Panda S, Hayes KR, Zhang J, Andrews JL, Antoch MP, Walker JR, Esser KA, Hogenesch JB, et al. Circadian and CLOCKcontrolled regulation of the mouse transcriptome and cell proliferation. Proc Natl Acad Sci U S A. 2007;104(9):3342–7.
 43.
Tsuchiya H, da Costa KA, Lee S, Renga B, Jaeschke H, Yang Z, Orena SJ, Goedken MJ, Zhang Y, Kong B, et al. Interactions between nuclear receptor SHP and FOXA1 maintain oscillatory homocysteine homeostasis in mice. Gastroenterology. 2015;148(5):1012–23. e1014
 44.
Janich P, Arpat AB, CasteloSzekely V, Lopes M, Gatfield D. Ribosome profiling reveals the rhythmic liver translatome and circadian clock regulation by upstream open reading frames. Genome Res. 2015;25(12):1848–59.
 45.
Vollmers C, Schmitz RJ, Nathanson J, Yeo G, Ecker JR, Panda S. Circadian oscillations of proteincoding and regulatory RNAs in a highly dynamic mammalian liver epigenome. Cell Metab. 2012;16(6):833–45.
 46.
Geyfman M, Kumar V, Liu Q, Ruiz R, Gordon W, Espitia F, Cam E, Millar SE, Smyth P, Ihler A, et al. Brain and muscle Arntlike protein1 (BMAL1) controls circadian cell proliferation and susceptibility to UVBinduced DNA damage in the epidermis. Proc Natl Acad Sci U S A. 2012;109(29):11758–63.
 47.
EckelMahan KL, Patel VR, de Mateo S, OrozcoSolis R, Ceglia NJ, Sahar S, DilagPenilla SA, Dyar KA, Baldi P, SassoneCorsi P. Reprogramming of the circadian clock by nutritional challenge. Cell. 2013;155(7):1464–78.
 48.
Renaud J, Dumont F, Khelfaoui M, Foisset SR, Letourneur F, Bienvenu T, Khwaja O, Dorseuil O, Billuart P. Identification of intellectual disability genes showing circadian clockdependent expression in the mouse hippocampus. Neuroscience. 2015;308:11–50.
Funding
This work was supported by research funds from the Yamagata Prefectural Government and by research funds from Tsuruoka City, Japan.
Availability of data and materials
The scripts for analysis were uploaded on the following URL. https://docs.google.com/document/d/1bN44qAJFP9O6BTTA_0ameil9py0LS3rcXKT2cxbKTwY/edit?usp=sharing.
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HI conducted the bioinformatics analyses. MS supervised the project. HI and MS wrote the manuscript. MT supported the writing of the manuscript. All authors have read and approved the final manuscript.
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Correspondence to Masahiro Sugimoto.
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Additional files
Additional file 1:
Wide range comparison of MCC values of MICOP, ARS, JTK, and LS for decaying data. Sampling interval and noise level were gradually adjusted. The bar indicates MCC values (1 indicates a perfect prediction, 0 indicates a random prediction, and − 1 indicates a prediction in complete disagreement). (PDF 75 kb)
Additional file 2:
Widerange comparison of MCC values of MICOP, ARS, JTK, and LS for nondecaying data. Sampling interval and noise level were gradually adjusted. The bar indicates MCC values (1 indicates a perfect prediction, 0 indicates a random prediction, and − 1 indicates a prediction in complete disagreement). (PDF 75 kb)
Additional file 3:
MonteCarlo simulation to calculate Pvalues. MIC values were calculated between random numbers. The xaxis indicates sample number (N time points) and the yaxis indicates MIC. The error bar indicates the standard deviation (N = 1000). The red color represents random values and the blue color represents the significance threshold (5%). (PDF 68 kb)
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Keywords
 Circadian rhythm
 Mutual information
 Proteomics