Using Lomb-Scargle Periodograms to Identify Periodic Genes in ...

Using Lomb-Scargle Periodogramsto Identify Periodic Genes in SomitogenesisStowers Institute for Medical ResearchEarl F. GlynnScientific Programmer29 March 20061

Using Lomb-Scargle Periodogramsto Identify Periodic Genes in Somitogenesis• Periodic Patterns in Biology• Simple Periodic Gene Expression Model• Introduction to Lomb-Scargle Periodogram• Data Pipeline• Methodology Validation Study:Bozdech’s Plasmodium dataset• Application to Mary-Lee’s somitogenesis dataset• Conclusions2

Periodic Patterns in BiologyA vertebrate’s body plan: a segmented pattern.Segmentation is established during somitogenesis.Photograph taken at Reptile Gardens, Rapid City, SDwww.reptile-gardens.com3

Introduction to Lomb-Scargle Periodogram• What is a Periodogram?• Why Lomb-Scargle Instead of Fourier?• Example Using Cosine Expression Model• Mathematical Details• Lomb-Scargle Experiments- Single Dominant Frequency- Multiple Frequencies- Mixtures: Signal and Noise- Multiple Hypothesis Testing5

What is a Periodogram?• A graph showing frequency “power” for a spectrumof frequencies• “Peak” in periodogram indicates a frequency withsignificant periodicityPeriodic SignalPeriodogramLog 2(Expression)ComputationSpectral“Power”TimeFrequency6

Why Lomb-Scargle Instead of Fourier?Lomb-Scargle MethodWeights data pointsData can be unevenly sampledNo data imputationAny number of data pointsKnown statistical properties“p” valueNeed estimate of number of“independent frequencies” butexplore using continuumFourier MethodWeights frequency intervalsRequires uniform spacingMissing data imputed2 N points for FFT; 0 paddingPermutation tests needed toassess statistical propertiesAd hoc scoring rulesUsually only look at“independent” Fourierfrequencies7

Lomb-Scargle PeriodogramExample Using Cosine Expression ModelExpression-1.0 0.0 0.5 1.0Cosine Curve (N=48)N = 480 10 20 30 40A small valuefor the false-alarmprobability indicatesa highly significantperiodic signal.Time [hours]T = 1fNormalized Power Spectral Density0 5 10 20Lomb-Scargle PeriodogramPeriod at Peak = 48 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.050.00 0.05 0.10 0.15 0.20Probability0.0 0.4 0.8Peak Significancep = 3.3e-009 at Peak0.00 0.05 0.10 0.15 0.20Frequency [1/hour]Frequency [1/hour]Evenly-spaced time points8

Lomb-Scargle PeriodogramExample Using Noisy Cosine Expression ModelCosine Curve + Noise (N=48)Time Interval VariabilityExpression-1.0 0.0 1.0Frequency0 2 4 6 8N = 480 10 20 30 40-1.0 -0.5 0.0 0.5 1.0Time [hours]log10(delta T)Normalized Power Spectral Density0 5 10 20Lomb-Scargle PeriodogramPeriod at Peak = 45.7 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.050.00 0.05 0.10 0.15 0.20Probability0.0 0.4 0.8Peak Significancep = 2.54e-007 at Peak0.00 0.05 0.10 0.15 0.20Frequency [1/hour]Frequency [1/hour]Unevenly-spaced time points9

Lomb-Scargle PeriodogramExample Using NoiseNoise (N=48)Time Interval VariabilityExpression-1.0 0.0 0.5 1.0N = 480 10 20 30 40Frequency0 2 4 6 8-1.0 -0.5 0.0 0.5 1.0Time [hours]log10(delta T)Normalized Power Spectral Density0 5 10 20Lomb-Scargle PeriodogramPeriod at Peak = 7.4 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.050.00 0.05 0.10 0.15 0.20Probability0.0 0.4 0.8Peak Significancep = 0.973 at Peak0.00 0.05 0.10 0.15 0.20Frequency [1/hour]Frequency [1/hour]10

Lomb-Scargle Periodogram Mathematical DetailsP N (ω) has an exponential probability distribution with unit mean.Source: Numerical Recipes in C (2 nd Ed), p. 57711

Lomb-Scargle Periodogram Experiment:Single Dominant FrequencyCosine Curve (N=48)Expression-1.0 0.0 0.5 1.0N = 480 10 20 30 40Expression = Cosine(2pt/24)Time [hours]Normalized Power Spectral Density0 5 10 20Lomb-Scargle PeriodogramPeriod at Peak = 24 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.050.00 0.05 0.10 0.15 0.20Probability0.0 0.4 0.8Peak Significancep = 3.3e-009 at Peak0.00 0.05 0.10 0.15 0.20Frequency [1/hour]Frequency [1/hour]Single “peak” in periodogram. Single “valley” in significance curve.12

Lomb-Scargle Periodogram Experiment:Multiple FrequenciesSum of 3 Cosines (N=48)Expression-2 -1 0 1 2 3Expression =Cosine(2pt/48) +Cosine(2pt/24) +Cosine(2pt/ 8)N = 480 10 20 30 40Time [hours]Normalized Power Spectral Density0 5 10 20Lomb-Scargle PeriodogramPeriod at Peak = 21.8 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.050.00 0.05 0.10 0.15 0.20Probability0.0 0.4 0.8Peak Significancep = 0.00246 at Peak0.00 0.05 0.10 0.15 0.20Frequency [1/hour]Frequency [1/hour]Multiple peaks in periodogram. Corresponding valleys in significance curve. 13

Lomb-Scargle Periodogram Experiment:Multiple FrequenciesSum of 3 Cosines (N=48)Expression-2 0 2 4Expression =3*Cosine(2pt/48) +Cosine(2pt/24) +Cosine(2pt/ 8)N = 480 10 20 30 40Time [hours]Normalized Power Spectral Density0 5 10 20Lomb-Scargle PeriodogramPeriod at Peak = 48 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.050.00 0.05 0.10 0.15 0.20Probability0.0 0.4 0.8Peak Significancep = 2.37e-007 at Peak0.00 0.05 0.10 0.15 0.20Frequency [1/hour]Frequency [1/hour]“Weaker” periodicities cannot always be resolved statistically.14

Lomb-Scargle Periodogram Experiment:Multiple Frequencies: “Duty Cycle”50% 66.6% (e.g., human sleep cycle)duty cycle: 1/2duty cycle: 2/3Expression0.0 0.2 0.4 0.6 0.8 1.0N = 480 10 20 30 40Expression0.0 0.2 0.4 0.6 0.8 1.0N = 480 10 20 30 40Time [hours]Time [hours]Normalized Power Spectral Density0 5 10 15 20 25Lomb-Scargle PeriodogramPeriod at Peak = 24 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.05Probability0.0 0.2 0.4 0.6 0.8 1.0Peak Significancep = 2.54e-007 at PeakNormalized Power Spectral Density0 5 10 15 20 25Lomb-Scargle PeriodogramPeriod at Peak = 24 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.05Probability0.0 0.2 0.4 0.6 0.8 1.0Peak Significancep = 5.06e-006 at Peak0.0 0.1 0.2 0.3 0.4 0.50.0 0.1 0.2 0.3 0.4 0.50.0 0.1 0.2 0.3 0.4 0.50.0 0.1 0.2 0.3 0.4 0.5Frequency [1/hour]Frequency [1/hour]Frequency [1/hour]Frequency [1/hour]One peak with symmetric “duty cycle”.Multiple peaks with asymmetric cycle.15

Lomb-Scargle Experiment:Mixtures: Periodic Signal Vs. Noiselog(p) histogramFrequency0 50 100 150'p' Histogram for 5000 Simulated Expresson Profiles (N= 48 )p corresponding to max Periodogram Power Spectral Density100 % simulated periodic genesFrequency0 500 1000 1500'p' Histogram for 5000 Simulated Expresson Profiles (N= 48 )p corresponding to max Periodogram Power Spectral Density50 % simulated periodic genesFrequency0 500 1000 1500 2000'p' Histogram for 5000 Simulated Expresson Profiles (N= 48 )p corresponding to max Periodogram Power Spectral Density0 % simulated periodic genes-8 -6 -4 -2 0-8 -6 -4 -2 0-8 -6 -4 -2 0log10(p)log10(p)log10(p)100% periodic genes 50% periodic50% noise100% noise16

More False NegativesLomb-Scargle Experiment:Mixtures: Periodic Signal Vs. NoiseBonferroniHolmHochbergBenjamini &Hochberg FDRMultiple-Hypothesis TestingLog10(p)-8 -6 -4 -2 0Multiple Testing Correction Methods50 % simulated periodic genesbonferroniholmhochbergfdrnoneNoneMore False Positives0 1000 2000 3000 4000 5000Rank Order of Sorted p Values50% periodic, 50% noise17

Data Pipeline to Apply to Microarray Dataset1. Apply quality control checks to data2. Apply Lomb-Scargle algorithm to allexpression profiles3. Apply multiple hypothesis testing todefine “significant” periodic genes4. Analyze biological significance ofperiodic genes18

Methodology Validation Study:Bozdech’s Plasmodium datasetFrom Bozdech, et al, Fig. 1A, PLoS Biology, Vol 1, No 1, Oct 2003, p 3.Intraerythrocytic Developmental Cycle of Plasmodium falciparumCritical Assessment of Microarray Data Analysis Conference (CAMDA), Nov 2004http://research.stowers-institute.org/efg/2005/LombScargle/19

Bozdech’s Plasmodium dataset:1. Apply Quality Control ChecksGlobal views of experiment.Remove certain outliers.20

Bozdech’s Plasmodium dataset:1. Apply Quality Control ChecksMany missing data points require imputation for Fourier analysis.21

Bozdech’s Plasmodium dataset:2. Apply Lomb-Scargle AlgorithmPhasePeriodic Expression Patterns Phaseopfi17638Time Interval Variabilityi3518_1Time Interval Variability-4 -2 0 20 10 20 30 40-2 -1 0 10 10 20 30 40N = 460 10 20 30 40-1.0 -0.5 0.0 0.5 1.0N = 460 10 20 30 40-1.0 -0.5 0.0 0.5 1.00 5 10 15 20 25Lomb-Scargle PeriodogramPeriod at Peak = 45.7 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.050.0 0.2 0.4 0.6 0.8 1.0Peak Significancep = 1.19e-008 at Peak0 5 10 15 20 25Lomb-Scargle PeriodogramPeriod at Peak = 45.7 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.050.0 0.2 0.4 0.6 0.8 1.0Peak Significancep = 1.48e-008 at Peak0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.200.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20Examples of highly-significant periodic expression profiles.22

Bozdech’s Plasmodium dataset:2. Apply Lomb-Scargle AlgorithmAperiodic/Noise Expression Patternsj167_5Time Interval Variabilityf35105_2Time Interval Variability-0.5 0.0 0.5 1.0N = 350 10 20 30 400 5 10 15 20 25-1.0 -0.5 0.0 0.5 1.0-1.0 -0.5 0.0 0.5 1.0 1.5N = 450 10 20 30 400 10 20 30 40-1.0 -0.5 0.0 0.5 1.00 5 10 15 20 25Lomb-Scargle PeriodogramPeriod at Peak = 17.8 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.050.0 0.2 0.4 0.6 0.8 1.0Peak Significancep = 0.998 at Peak0 5 10 15 20 25Lomb-Scargle PeriodogramPeriod at Peak = 32 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.050.0 0.2 0.4 0.6 0.8 1.0Peak Significancep = 0.516 at Peak0.00 0.05 0.10 0.15 0.200.00 0.05 0.10 0.15 0.200.00 0.05 0.10 0.15 0.200.00 0.05 0.10 0.15 0.2023

Bozdech’s Plasmodium dataset:2. Apply Lomb-Scargle AlgorithmSmall “N”f58149_1Time Interval Variabilityn170_1Time Interval Variability-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5N = 390 10 20 30 400 5 10 15 20 25 30-1.0 -0.5 0.0 0.5 1.0-3 -2 -1 0 1 2N = 320 10 20 30 400 5 10 15 20 25 30-1.0 -0.5 0.0 0.5 1.00 5 10 15 20 25Lomb-Scargle PeriodogramPeriod at Peak = 48 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.050.0 0.2 0.4 0.6 0.8 1.0Peak Significancep = 8.54e-006 at Peak0 5 10 15 20 25Lomb-Scargle PeriodogramPeriod at Peak = 64 hoursp = 1e-06p = 1e-05p = 1e-04p = 0.001p = 0.01p = 0.050.0 0.2 0.4 0.6 0.8 1.0Peak Significancep = 2.74e-005 at Peak0.00 0.05 0.10 0.15 0.20N=390.00 0.05 0.10 0.15 0.200.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20N=3224

Bozdech’s Plasmodium dataset:2. Apply Lomb-Scargle AlgorithmSignal and Noise Mixture'p' histogramComplete Bozdech set of 6875 probesNumber of Probes0 50 100 150 200Periodic ProbesAperiodic Probes or NoiseMissing NoiseSpike?Frequency0 500 1000 1500 2000'p' Histogram for 5000 Simulated Expresson Profiles (N= 48 )p corresponding to max Periodogram Power Spectral Density0 % simulated periodic genes-8 -6 -4 -2 0log10(p)-8 -6 -4 -2 0histogram-log10p.pdf 2004-11-06 10:26log10(p)25

Bozdech’s Plasmodium dataset:3. Apply Multiple-Hypothesis TestingMore False NegativesMultiple Testing Correction Methods(Using R's p.adjust methods)BonferroniHolmHochbergBenjamini &Hochberg FDRLog10(p)-8 -6 -4 -2 0bonferroniholmhochbergfdrnoneSignificanceα = 1E-4None0 1000 2000 3000 4000 5000 6000 7000Rank Order of Sorted p ValuesMore False Positivesp-adjust.pdf 2004-11-06 10:1226

Bozdech’s Plasmodium dataset:4. Analyze Biological SignificanceBozdech Complete Set6875 probesSmall "N"1795Qualilty Control Set5080Overview Set243 501 36111084355Lomb-ScarglePeriodic3719BozdechPeriodic27

Bozdech’s Plasmodium dataset:4. Analyze Biological Significance“Phaseograms”Probes Ordered by PhaseProbes Ordered by PhaseTimeLomb-Scargle Results4355 ProbesTimeBozdech: “Overview” Dataset2714 genes, 3395 probes28

Bozdech’s Plasmodium dataset:Bozech’s Ad Hoc ScoringpowerMAX vs Power at Peak FrequencyPeriodicGenes29

Bozdech’s Plasmodium dataset:Bozech’s Ad Hoc Scoring Vs Lomb-Scargle p values30

Bozdech’s Plasmodium dataset:Bozdech’s “Phase” Vs. Peak of Smoothed Time Series31

Mary-Lee’s Somitogenesis DatasetWe wanted to validate the Lomb-Scargle method withBozdech’s dataset before applying to our somitogenesisproblem, since the Fourier technique could not be used:Scargle (1982):“surprising result is that the … spectrum of a process canbe estimated … [with] only the order of the samples …”32

Somitogenesis DatasetCosine (N = 10 )Time Interval VariabilityCosine Curve-1.0 0.0 0.5 1.0Frequency0 2 4 6 8N100.1p0 20 40 60 80 100Time [minutes]-6 -4 -2 0 2 4 6log10(delta T)170.006Normalized Power Spectral Density0 2 4 6 8 10Lomb-Scargle Periodogramperiod = 119 minutes0.005 0.015 0.025Probability0.0 0.4 0.8p = 0.1058Peak Significance0.005 0.015 0.02520480.0023E-9Frequency [1/minute]Frequency [1/minute]33

Somitogenesis DatasetMary-Lee, Science Club, 200434

Somitogenesis DatasetMary-Lee, Science Club, 200435

Somitogenesis DatasetDo we care about time order, or only periodic genes?22,690Affy Probesets36

Somitogenesis DatasetDo we care about time order, or only periodic genes?ReorderPlan: Use only known periodic genesTimeOrderingExperiment1.doc, Nov 200437

Somitogenesis DatasetPerturbation (“Jitter”) and Permutation TestsOrder Nveau.pptPerturbation (times):27.75443 33.08618 42.6607 11.04462 -0.6288767 18.55415 12.93994 44.90501 59.98529 67.2538 46.87758 55.91625 74.7259178.9921 83.98062 104.9910 78.12926 77.38472 111.5297 109.2331Permutation (order for fixed times):7 5 6 8 4 3 2 1 | 11 12 9 10 | 13 | 17 15 16 18 14 | 20 1938

Somitogenesis DatasetPerturbation (“Jitter”) and Permutation TestsBestRank ProductJitterSpreadHistograms.pdf, Feb 2005WorstBestRank ProductPermuteSpreadHistograms.pdfWorst 39

Somitogenesis DatasetPerturbation (“Jitter”) and Permutation TestsPermutePerturbComparison.xls, Jan 200540

Somitogenesis DatasetPerturbation (“Jitter”) and Permutation TestsTop 50: Somites Vs Random100009000800070006000SomitesTop 50 Score50004000300020001000Random01 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97RankStatComparison1000.xls: Top50, Jan 200541

Somitogenesis DatasetPerturbation (“Jitter”) and Permutation TestsSomites Vs Random350003000025000RandomLogRankSum2000015000100005000SomitesStatComparison1000.xls: TopLogRankSum01 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97Rank42

Somitogenesis DatasetHow many periodic genes are in the dataset?Approx Noise Cutoff43

Somitogenesis DatasetWhat periodicities are present?Periodogram ClusteringHierarchical Clustering4 7 241 9320.1 0PeriodogramClustering.docAug 2004F1 F5 F9 F13 F17 F21 F25 F29 F33 F37119 min 59 min44

Somitogenesis DatasetUnconstrained Permutations to Estimate False Hit Rate(10,000 Permutations, 17-point time series, 7544 Affy probes)Use p-value cutoffs from “baseline” with permutationsAbout 35-40 cyclic genes detected?FinalStatsPeriodRange.xls, March 200645

Conclusions• Lomb-Scargle periodogram is effective tool toidentify periodic gene expression profiles• Results comparable with Fourier analysis• Lomb-Scargle can help when data are missingor not evenly spaced46

Conclusions• Conclusions should not be drawn using theindividual p-value calculated for each profile. Amultiple comparison procedure False DiscoveryRate (FDR) must be used to control the error rate.• Expression profiles may be more complex thansimple cosine curves• Power spectra of non-sinusoid rhythms may bedifficult to interpret47

AcknowledgementsPourquie LabOlivier PourquiéMary-Lee DequéantBioinformaticsArcady MushegianGalina GlaskoJie Chen48