Kousaku Okubo

Osamu Ogasawara

Shoko Kawamoto

Comptes Rendus Biologies (CRBIOL)

uency andells, theyroposeviding aionns le muscleDans desiers rangstochastiqueontraintesC. R. Biologies 326 (2003) 1097–1101Molecular biology and geneticsZipf’s law and human transcriptomes: an explanationwith an evolutionary modelOsamu Ogasawara, Shoko Kawamoto, Kousaku Okubo∗Division of Gene Expression analysis, The Center for Information Biology and DNA Data Bank of Japan, National Institute of Genetics,Mishima 411-8540, Shizuoka, JapanReceived 16 September 2003; accepted 23 September 2003Presented by François GrosAbstractDetailed analysis of human gene expression data reveals several patterns of relationship between transcript freqabundance rank. In muscle and liver, organs composed primarily of a homogeneous population of differentiated cobey Zipf’s law. In cell lines, epithelial tissue and compiled transcriptome data, only high-rankers deviate from it. We pan evolutionary process model during which expression level changes stochastically proportionally to its intensity, pronovel interpretation of transcriptome data and of evolutionary constraints on gene expression.To c te this article: O. Ogasawaraet al., C. R. Biologies 326 (2003). 2003 Published by Elsevier SAS on behalf of Académie des sciences.RésuméLoi de Zipf et transcriptomes humains : explication avec un modèle évolutif. L’analyse détaillée de données d’expressdes gènes humains révèle plusieurs types de relations entre la fréquence des transcrits et leur rang d’abondance. Daet le foie, organes composés principalement d’une population de cellules différenciées, elles obéissent à la loi de Zipf.lignées cellulaires, le tissu épithélial et des compilations de données de transcriptomes, seuls les transcrits des premen dévient. Nous proposons un modèle de processus évolutif lors duquel le niveau d’expression change de manière sproportionnellement à son intensité, permettant une nouvelle interprétation des données du transcriptome et des cévolutives sur l’expression génique.Pour citer cet article : O. Ogasawara et al., C. R. Biologies 326 (2003). 2003 Published by Elsevier SAS on behalf of Académie des sciences.Keywords: abundance; expression; frequency; transcriptome; Zipf’s lawMots-clés : abondance ; expression ; fréquence ; loi de Zipf ; transcriptomemee-3meat-uschnter-1. Relationship between transcript frequency andabundanceIn the genetics–linguistics analogy, a transcriptois a text in which a life plan is ‘expressed’ with a g* Corresponding author.E-mail address: kousaku@genomatrix.com (K. Okubo).1631-0691/$ – see front matter 2003 Published by Elsevier SAS ondoi:10.1016/j.crvi.2003.09.031nomic vocabulary. By analyzing SAGE tag [1] and′EST [2] data, we found that the human transcriptofollows the statistical constraint, characteristic for nural language, known as Zipf’s law [3]. In a corpof texts, Zipf’s law dictates that the frequency of eaword,f , and its abundance rank,r (r = 1 for the mostfrequent word,r = 2 for the second most frequeword, and so on) are related according to the powbehalf of Académie des sciences.1098 O. Ogasawara et al. / C. R. Biologies 326 (2003) 1097–1101data werean SAGEibraries.Fig. 1. Log frequency log rank plot (Zipf’s plot) of transcriptome data. The frequency of occurrence(f ) of each transcript in 3′ EST and SAGEtag (∗) collections representing various transcriptomes were plotted against the abundance rank(r). The broken line representsf = 0.1/r .(a) Organs with homogeneous populations of differentiated cells. For example, the most abundant transcript (r = 1) in liver, albumin, occurredabout 12% in EST data for liver. Gene names forr = 1–6 in liver are given. (b) Cell lines and complex tissues. (c) Compiled data from 51human EST sets, 31 mouse EST sets, and 64 SAGE tag sets. Gene names forr = 1–6 in compiled human transcriptome (3′ EST) are given.(d) Occurrence of 3′ EST in normalized libraries. The total tag occurrence for each data set is given in parentheses. The frequencyobtained fromhttp://bodymap.ims.u-tokyo.ac.jp/datasets/index.html(3′ EST) andftp://ncbi.nlm.nih.gov/pub/sage/(SAGE). The data for liverare combined data for two human liver libraries. The frequencies of total SAGE tags are obtained from re-analysis of all available humtags. Clustering 3′ ESTs for two representative normalized libraries in dbEST, 1N1B and 2NbHM, generated the data for normalized lre-softaslaw f ∞. r−k , with k ≈. 1 for all languages [4]. In adouble logarithmic axis plot, such a relation is repsented by a linear dependence off as a function ofr with a slope of−1. In muscle and liver, the organcomposed primarily of a homogeneous populationdifferentiated cells, the frequency,f of each transcripand its abundance rank,r distributed very close tothe linef = 0.1/r (Fig. 1a). In other sources, suchO. Ogasawara et al. / C. R. Biologies 326 (2003) 1097–1101 1099iptondi-e-ntIn-iledofpf-ralsticopewbycis-ausetoannalhes:sticl tote,neshes byionbu--heofn.-rib-mehandis-ib-hear-meene,.n-).ofndu-he-alw,entsumof7.iewenesn-restip-m-cell lines and epithelial tissue, only high-rankers(r <100), which comprise less than 1% of the transcrvariety, deviated from this trend (Fig. 1b). Reductiof tissue-specific transcripts in cell lines and theirlution in complex cell populations explains such dviation, at least in part. Compiling data for differetranscriptomes affected the plot similarly (Fig. 1c).terestingly, the plot appeared universal to the comptranscriptomes at enough multiplicities, regardlessthe data sources. In normalized libraries [5], the Zilike structure was lost completely (Fig. 1d).2. An evolutionary model of expression levelvariationIn transcript abundance, the slope(|k|) is very closeto 1, similarly to the classical example in natulanguage, which suggests the underlying stochaprocess that robustly dictates the value of the slas well as the linearity in log–log plot. We knothat the abundance of each transcript is dictatedthe DNA sequence on the genome known aselements and the mutations in these elements calteration in its abundance. Accordingly it is naturalassume that the abundance distribution results fromevolutionary process. We propose here an evolutiomodel that explains linearity in log–log plot and tslope(= 1) with only simple assumptions as follow(1) the mutations in cis-elements cause a stochachange in the expression level that is proportionathe original level of expression; (2) at the initial stathe genome contains from a small number of gewith an arbitrary distribution of gene expression. Tgene number gradually increases over generationgaining new genes by gene duplication [6].The first assumption can be written as the relatfi(t + 1) = λi(t)fi(t), wherefi(t) is the expressionlevel of genei at generationt , andλi(t) is a randomvariable extracted from a time-independent distrition π(λ).After T generations, we obtain:(1)logfi(T ) = logfi(0) +T∑j=0logλi(j)If the random variableλi(t) is independent and identically distributed with finite mean and variance, tcentral limit theorem says that the distribution∑Tj=0 logFi(j) converges to a normal distributioFor sufficiently largeT , fi(T ) is well approximatedby the lognormal distribution which is extremely similar in shape to power-law distribution.As generations pass, the deviation of the distution becomes large, and expression level of sogenes becomes very small. If they become lower tthe minimal level, then the genes are regarded asappeared.Because we cannot determine the initial distrution fi(0) of formula (1), a priori, the model (1)should be extended slightly. At the initial state, tgenome contains a small number of genes with anbitrary distribution of gene expression. The genogrows over generations by gaining new genes by gduplication. While the number of gene(N) increasesthe total number of mRNA molecules in a cell(W)also increases, according to the following formula:∆N∆W= N(t + 1) − N(t)∑fi(t + 1) − ∑fi(t) = KwhereK is a constant positive value smaller than 1By simulation of this process, we show that trascriptome distribution converges to Zipf’s law (Fig. 2The evolutionary model is essentially the analogythe city size growth model, proposed by Blank aSolomon (2000) [7]. In their city size model, the poplation growth(∆W) is the cause of the increase in tnumber of cities(∆N). In our evolutionary model, increase in total mRNA(∆W) is driven by the growthof gene number(∆N).3. Interpretation of transcriptome dataZipf’s law predicts various important numericfeatures of the transcriptome. According to this la50% of transcripts in a differentiated cell represonly 83 mRNA species. Because the accumulatedof f for all transcripts is 1, the predicted numberdifferent transcripts in a cell is restricted to 12 36These values are in accordance with the classical vof an average transcriptome where less than 100 gare responsible for 50% of the cellular mRNA cotent and about 10 000 mRNA species comprise the[8]. Assuming that the plot for a compiled transcrtome (Fig. 1c) remains unchanged after further co1100 O. Ogasawara et al. / C. R. Biologies 326 (2003) 1097–1101eneration).rs were settoFig. 2. Simulated emergrence of Zipf-like distribution in transcriptome with evolutionary model. (a) Distribution after 50 generations, (b) after100 generations, (c) after 200 generations. The continuous line is the Zipf’s law function (f = a/r). The distribution converged to Zipf’s lawwithin a few hundred generations. The number of genes at initial state was 50, and the number was increased exponentially (2% par gBut the initial state and the exponent parameter values have no substantial influence on the result of the distribution. Other parameteasK = 0.001,π(λ) was set as the normal distributionN(µ,σ2) with µ = 1.0, σ2 = 0.05. Again, the resultant distribution was not sensitivethe exact value of the parameters.n-heintign-ngi-an-ls.tial,cityber-delsran--aly,ri-ro-o-thereeryandincanu,o,e,z,k,a,gel-urep:Acidsrt,y 1-h,pilation of different transcriptomes (∑10 000r=1 f = 0.70for SAGE tags), the predicted total number of trascripts in a whole body is approximately 210 000. Tfurther benefit of the discovery of such a constrais that it may eventually assist researchers in desing experiments with tissue mRNA and interpretiresults from them. In transcript profiling by DNA mcroarray, for example, the relation off andr can beconsidered as the relation between sensitivity of trscript detection and the number of hybridizing signaApplicability of Zipf’s law has been reported noonly for natural languages but also in several soceconomical and behavioral phenomena, such assize and income [3], and clicks in World Wide Wesurfing [9]. Although there have been many controvsies, in essence, three kinds of mathematical mohave survived to explain such phenomena: Thedom network model [10] for WWW, Mandelbrot’s optimization model [11] for linguistical Zipf’s law andmultiplicative model applied to city size [7]. Recentpower-law distributions were also found in wide vaety of genomic properties, such as populations of ptein families [6,12–14], protein folds [6,14], pseudgenes [14] and protein domains [15,16], althoughslope is not always close to 1 in these cases.After this work was submitted, two papers wepublished reporting that Zipf’s law holds in a widvariety of eukaryotes, supporting our evolutionamodel [17,18]. In one of these papers, FurusawaKaneko [18] applied a random network modelmRNA production to explain Zipf’s distribution, inwhich they assumed that transcriptional regulationbe modeled as an analogy of a catalytic reaction.References[1] V.E. Velculescu, S.L. Madden, L. Zhang, A.E. Lash, J. YC. Rago, A. Lal, C.J. Wang, G.A. Beaudry, K.M. CiriellB.P. Cook, M.R. Dufault, A.T. Ferguson, Y. Gao, T.C. HH. Hermeking, S.K. Hiraldo, P.M. Hwang, M.A. LopeH.F. Luderer, B. Mathews, J.M. Petroziello, K. PolyaL. Zawel, W. Zhang, X. Zhang, W. Zhou, F.G. HaluskJ. Jen, S. Sukumar, G.M. Landes, G.J. Riggins, B. Vostein, K.W. Kinzler, Analysis of human transcriptomes, NatGenet. 23 (1999) 387–388.[2] T. Hishiki, S. Kawamoto, S. Morishita, K. Okubo, BodyMaa human and mouse gene expression database, NucleicRes. 28 (2000) 136–138.[3] G.K. Zipf, Human Behavior and the Principle of Least EffoAddison-Wesley, Cambridge, UK, 1949.[4] J.L. Casti, Bell curves and monkey languages, Complexit(1995) 12–15.[5] L.D. Hillier, G. Lennon, M. Becker, M.F. Bonaldo, B. Chiapelli, S. Chissoe, N. Dietrich, T. DuBuque, A. Favello, W. GisO. Ogasawara et al. / C. R. Biologies 326 (2003) 1097–1101 1101,ge,a-tagsldaryn,h a79–w,80alei-ringteinon15ein,er-ies,a,leol.in3.icsells,ys.M. Hawkins, M. Hultman, T. Kucaba, M. Lacy, M. Le, N. LeE. Mardis, B. Moore, M. Morris, J. Parsons, C. PranL. Rifkin, T. Rohlfing, K. Schellenberg, M. Marra, Genertion and analysis of 280 000 human expressed sequenceGenome Res. 6 (1996) 806–828.[6] J. Qian, N.M. Luscombe, M. 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Zipf's law and human transcriptomes: an explanation with an evolutionary model

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Zipf's law and human transcriptomes: an explanation with an evolutionary model

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