Fig 1.
JTK_CYCLE compares all possible pair relations for a time series to those for a reference waveform.
(A) JTK_CYCLE tests for pairwise agreement between a reference (blue) and a signal (cyan) time series. Three discordant pairwise relationships are indicated by red lines. (B) The previous implementation compared a time series to a set of phase-shifted cosines. (C) We add a set of asymmetric waveforms to the reference. An example is shown here with the same phases as in A.
Fig 2.
Empirical p-values are uniformly distributed for the null model of JTK_CYCLE.
P-values versus their ranks from lowest (most significant) to highest (least significant) for JTK_CYCLE testing phases at 2 h intervals (green line) or phases and asymmetries at 2 h intervals (blue line) with time series consisting of Gaussian noise. Unbiased estimates should follow the black line (see text). (A) “Initial” p-values from JTK_CYCLE with multiple hypothesis testing underestimate the true p-values. (B) The Bonferroni correction results in p-values that are too high (less significant). (C) The Benjamini-Hochberg correction performs better than the Bonferroni correction but still results in p-values that are generally too high. (D) Empirical p-values that we calculate by permutation are close to uniformly distributed, as desired; their correspondence to the null model improves as the number of hypotheses tested increases.
Fig 3.
(A) Different waveforms simulated with a 24 h period. From left to right, cosine, ramp, step, and impulse (width at half-max is 2 h). Waveforms in figure may not be to scale. (B) Cosine in black, with Gaussian noise with standard deviation of 25% (blue) or 50% (green) of amplitude.
Fig 4.
AUROCs for simulated data with 50% noise (standard deviation of Gaussian noise as a percent of amplitude).
An AUROC value of 1 represents perfect discrimination between rhythmic and arrhythmic time series, and a value of 0.5 corresponds to random guessing. In each panel, the number of replicates increases from 1 to 4 replicates from left to right, and the number of sampled points per period is indicated by color. AUROC for single-replicate ANOVA (for which the method is undefined) is set at 0.5 exactly. Imp: impulse waveform, Cyclo: cyclohedron test, Address: address reduction, Stable: stable persistence, JTK: original JTK_CYCLE with Bonferroni correction, JTK_BH: JTK_CYCLE with Benjamini-Hochberg correction with symmetric triangle reference, eJTK: empirical JTK_CYCLE with symmetric triangle reference, JTK_BH_aby2: JTK_CYCLE with Benjamini-Hochberg correction and triangle references with asymmetries from 2 to 22 h by 2 h, eJTK_aby2: empirical JTK_CYCLE with triangle references with asymmetries from 2 to 22 h by 2 h.
Fig 5.
Higher numbers of replicates provide greater sensitivity compared to increased density of time points for the same number of samples.
Results shown are AUROC values for sine and ramp simulated data with 50% noise (see S4 Fig. for additional waveforms and sample numbers). “Points” refers to the number of time points per period (“Points 12” refers to 12 points per period) and “Replicates” refers to the number of replicates per time series (“Replicates 2” refers to 2 samples per time point). Together, “Points 12 Replicates 2” refers to a time series that consists of 12 time points per period with 2 replicates per time point. Abbreviations are the same as in Fig. 4.
Fig 6.
Empirical JTK_CYCLE outperforms the other methods in the presence and absence of asymmetric time series.
Simulated data with rhythmic time series without asymmetry (left, A and C) or with evenly distributed asymmetry (right, B and D) were tested with different methods. The cumulative histograms are plotted before (A and B) and after (C and D) Benjamini-Hochberg multiple hypothesis correction across time series. The vertical axis shows the number of time series with a p-value (P) (A and B) or false discovery rate (FDR, the Benjamini-Hochberg adjusted p-value) (C and D) below or equal to a significance threshold, shown on the horizontal axis. Results shown are for the second simulated dataset with 25% noise, but the effects of Benjamini-Hochberg correction are significantly greater at 50% noise (not shown). The method abbreviations are the same as those in Fig. 4. The legends of A and B correspond to C and D, respectively. The rightmost point on the horizontal axis is 0.2.
Fig 7.
Z-score normalization allows combining of time series from different datasets into smooth time series.
Pdp1 gene expression from metadata before (A) and after (B) Z-score normalization. Light gray crosses indicate individual replicates, and the black curve is the mean.
Fig 8.
Empirical JTK_CYCLE with asymmetry search of 4 h (eJTK_aby4) identifies more genes than ANOVA, F24, and the other JTK_CYCLE methods.
(A) The vertical axis shows the number of genes with a p-value below or equal to the horizontal axis for the methods indicated. The rightmost point on the horizontal axis is 0.2. (B) The Benjamini-Hochberg correction for testing multiple genes impacts the relative performance of the different methods. The rightmost point on the horizontal axis is 0.2. The colors are the same as in A. (C) The number of genes with Benjamini-Hochberg adjusted p-values below 0.05 (blue) and below 0.20 (red) with the different methods is shown. (D) A comparison of the intersection (below the diagonal) and union (above the diagonal) of genes identified as rhythmic with Benjamini-Hochberg adjusted p-values less than 0.05 for the different methods. JTK: the original JTK_CYCLE method with Bonferroni correction. JTK_BH: the JTK_CYCLE method with Benjamini-Hochberg correction. eJTK: the JTK_CYCLE method with empirical calculation of the p-values. “_aby4” refers to an asymmetry search every 4 h (at 4, 8, 12, 16, and 20 h).
Fig 9.
Manual grouping of annotation terms identified as enriched by DAVID.
The number of annotation terms enriched in the genes with Benjamini-Hochberg adjusted p-values less than 0.05 that are identified by each method are shown in grey shading and red numbers. Annotation terms were enriched with Benjamini-Hochberg adjusted p-values below 0.05 as identified by the DAVID web tool [51, 52]. Abbreviations are the same as in Fig. 8.
Fig 10.
Annotation terms identified by DAVID as enriched for rhythmic genes.
Rhythmic genes shown are those that are identified with eJTK_aby4 with a Benjamini-Hochberg adjusted p-value less than 0.05. The terms shown are those identified by the DAVID web tool [51, 52] as enriched with a Benjamini-Hochberg adjusted p-value less than 0.05. (A) The individual annotation terms are shown with their adjusted p-values and phase distributions. The red numbers refer to the number of genes in that annotation term with that phase. The horizontal axis of A is the same as that of B. (B) Total phase distribution of the cycling genes. (C) Total asymmetry distribution of the cycling genes.