The science of linguistics has fully as many facets


 Bernard James
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1 The lne of enes insiht review rtiles Dvid B. erls Bioinformtis Division, Genetis Reserh, GlxomithKline Phrmetils, 709 wedelnd Rod, PO Box 1539, Kin of Prssi, Pennsylvni 19406, UA (emil: Linisti metphors hve been woven into the fbri of molelr bioloy sine its ineption. The determintion of the hmn enome seqene hs broht these metphors to the forefront of the poplr imintion, with the ntrl extension of the notion of DNA s lne to tht of the enome s the book of life. Bt do these nloies o deeper nd, if so, n the methods developed for nlysin lnes be pplied to molelr bioloy? In ft, mny tehniqes sed in bioinformtis, even if developed independently, my be seen to be ronded in linistis. Frther interwevin of these fields will be instrmentl in extendin or nderstndin of the lne of life. The siene of linistis hs flly s mny fets nd fields s bioloy, nd like bioloy, wht my be lled its modern er n be tred to the 1950s 1. The dede tht nveiled the strtre of DNA lso witnessed revoltion in linistis led by Nom Chomsky, whose work rdilly diversified the field beyond its thenrrent fos on simply tloin the tl tternes of lne, to explorin the mehnisms by whih they re proded. eekin to identify the niversls t the ore of ll lnes, he posited new, enertive form of rmmr, or set of syntti rles, tht wold help to ont for the immense retivity in the prodtion of lne tht emeres so rpidly s individls develop 2. In prsit of his niversl rmmr, Chomsky reted wves tht wshed p on mny sientifi shores. Besides his profond inflene on theoretil linistis, his mthemtil pproh to the desription of lnes prompted brst of development in forml lne theory. This proded methods with widespred tility in ompter siene, from the speifition nd interprettion of ompter lnes to the fields of syntti pttern reonition, ntrl lne proessin nd speeh nderstndin 3. The Chomsky hierrhy of lne lsses hs proven espeilly drble s mens of strtifyin forml lnes ordin to their expressive power nd resltin ompttionl nd mthemtil omplexity (Box 1). Chomsky s inflene hs lso extended to onitive siene, nlyti philosophy nd even literry ritiism. The ommon experiene in nmber of fields is tht it is not only nlyti tehniqes derived from linistis, bt lso wht miht be lled linisti sensibility, tht n illminte nd inform other similrly omplex domins. Mthemtil linistis nd mromoleles In the 1980s, severl workers ben to follow vrios threds of Chomsky s ley in pplyin linisti methods to molelr bioloy. Erly reslts inlded the fndmentl observtion tht forml representtions old be pplied to bioloil seqenes 4 the extension of linisti formlisms in new, bioloilly inspired diretions 5 nd the demonstrtion of the tility of rmmrs in ptrin not only informtionl bt lso strtrl spets of mromoleles 6. Nlei id linistis From this work there followed series of mthemtil reslts onernin the linistis of nlei id strtre 7 9. These reslts derive from the ft tht folded RNA seondry strtre entils pirin between nleotide bses tht re t distne from eh other in the primry seqene, estblishin reltionships tht in linistis re lled dependenies. The most bsi seondrystrtre element is the stemloop, in whih the stem retes session of nested dependenies tht n be ptred in idelized form by the followin ontextfree bsepirin rmmr 7 (Box 1): (The in the lst rle indites tht n is simply ersed.) This rmmr ffords ny nd every derivtion of hirpin seqenes of form sh s the followin: Derivtions from this rmmr row otwrd from the entrl, retin the nested dependenies of the stem (Fi. 1), nloos to sh phenomen s nested reltive lses in ntrl lne (for exmple, The ene tht the sientist whom or rnt spported disovered enoded kinse ). In relisti stemloop, the derivtion wold terminte in n npired loop of t lest severl bses nd miht lso ontin, for exmple, nonwtson Crik bse pirs nd bles. Bt sh fetres re esily dded to the rmmr withot ffetin the fndmentl reslt tht ny lne onsistin of RNA seqenes tht fold into these bsi strtres reqires ontextfree expression 10. In ddition to stemloop strtres, rbitrrily brnhed folded strtres my be ptred by simply ddin to the rmmr bove rle, whose pplition retes bifrtions in the derivtion tree 7 (Fi. 1b). The bsepirin dependenies remin nonrossin, lthoh more omplited. The resltin rmmr is formlly mbios, menin tht there re rnteed to be seqenes in the lne for whih more thn one derivtion tree is possible 10. Ths, the strin n be derived s sinle hirpin or s brnhed strtre (Fi. 1, b). This linisti property of mbiity, refleted in ntrl lnes in sentenes tht n be synttilly prsed in more thn one wy (for exmple, he sw the mn with the telesope ), diretly models the bioloil phenomenon of lterntive seondry strtre 7. Althoh these models re only bstrtions of thermodynmilly determined proess, mbiity llows them to embody the ensemble of potentil seondry strtres, nd more speifi rmmrs n speify prtilr forms, sh s trnsfer RNA loverlefs 9. NATURE VOL NOVEMBER Ntre Pblishin Grop 211
2 insiht review rtiles Box 1 The Chomsky hierrhy nd forml lne theory Forml lne theory defines lnes to be nothin more thn sets of strins of symbols drwn from some lphbet. A rmmr is rlebsed pproh to speifyin lne, onsistin of set of rewritin rles tht tke forms sh s A xb. Here, pperse letters denote temporry or nonterminl symbols, whih do not or in the lphbet, wheres lowerse letters re terminl symbols tht do. The exmple rle speifies tht ny orrene of the nonterminl A my be repled by n x followed by B. Beinnin with strtin nonterminl, derivtion from rmmr onsists of series of rewritin steps tht ends when the lst nonterminl is eliminted. Consider the simple rmmr with n lphbet x nd y, nd ontinin the rles x nd y. This rmmr enertes ll strins beinnin with ny nmber of x s nd endin in sinle y. It prodes derivtions sh s x xx xxx xxxy, where eh doble rrow sinifies the pplition of sinlerrow rle. In this se there re three pplitions of the first rle followed by sinle pplition of the seond to prode terminl strin, one of the infinite nmber of sh strins in this lne. Any rmmr whose rles rewrite nonterminl s terminl followed by t most one nonterminl is lled relr, nd is sid to enerte relr lne. An eqivlent mens of enertin sh lnes is finitestte tomton (FA), notionl mhine sed to reson bot ompttion, bilt ot of sttes (irles; see fire opposite) whih re interonneted by trnsitions (rrows) tht emit symbols from the lphbet s they re trversed. Grmmrs tht llow ny rrnement of terminls nd nonterminls on the rihthnd sides of rles hve reter expressive power. They re lled ontextfree rmmrs, nd n enerte not only ll relr lnes, bt lso nonrelr lnes sh s strins of x s followed by the sme nmber of y s (for exmple, xxxxyyyy). h lnes nnot be speified by relr rmmr or FA bese these devies hve no mehnism for rememberin how mny x s were enerted when the time omes to derive the y s. This shortomin is remedied by mens of ontextfree rles sh s xy, whih lwys enerte n x nd y t the sme time. Alterntively, n tomton mented with pshdown store, memory devie tht pshes or pops symbols to or from stk drin trnsitions, lso provides sh ontin pbility. In either se, ontextfree lnes llow strins tht embody dependenies between terminls, sh s the reltionship mthin x s nd y s in the exmple, provided tht those dependenies n be drwn s nested, either stritly within or independent of eh other, bt never rossin. Even ontextfree rmmrs re indeqte for some lnes, for instne strins of onsetive x s, y s nd z s in eql nmber (for Lne Rersively enmerble lnes Contextsensitive lnes Contextfree lnes Relr lnes Atomton Trin mhine Liner bonded Pshdown (stk) Finitestte tomton Grmmr Unrestrited B A Context sensitive At A Context free Relr A A Reonition Undeidble Exponentil? Polynomil Liner exmple, xxxyyyzzz). This entils dependenies tht neessrily ross one nother, lled rossseril dependenies, nd to ptre these with rmmr reqires rles tht hve dditionl symbols on their lefthnd side (thoh never more thn on their rihthnd side). h ontextsensitive rles orrespond to tomt with more sophistited memory devie, tpe whose lenth is bonded in ertin wy, pon whih the mhine n red nd write symbols. Contextsensitive lnes inlde ll ontextfree lnes nd mny more, yet theoretilly there exist lnes otside even this set, lled rersively enmerble lnes, enerted by rmmrs of ompletely nrestrited form or by mhines with nbonded tpes best known s Trin mhines. In the fire bove, the lne lsses in the left olmn ontin extly those lnes tht n be enerted by the tomt nd rmmr types indited in the next two olmns. Eh level ontins ll of those below it. The rihthnd olmn illstrtes the ompttionl omplexity, in the enerl se, of reonizin whether strin belons in iven lne, showin how the time reqired rows s fntion of the lenth of the inpt strin. At the hihest level of the hierrhy, one is not even rnteed to be ble to rrive t n nswer by ompttionl mens. This is jst one indition of the trdeoff between the inrese in expressive power fforded by sendin the Chomsky hierrhy, nd the mthemtil nd lorithmi limittions tht invribly reslt 24.? Findin tht the lne of RNA is t lest ontextfree hs mthemtil nd ompttionl onseqenes, for exmple, for the ntre nd inherent performne bonds of ny lorithm delin with seondry strtre (Box 1). For instne, the fst, relrexpression serh tools sed ommonly in bioinformtis (sh s those in the poplr Perl sriptin lne) re rled ot, s in their stndrd form they speify only relr lnes. These onseqenes show the importne of hrterizin linisti domins in the ommon terminoloy nd methodoloy of forml lne theory, so s to onnet them immeditely to the welth of tools nd nderstndin lredy vilble. For this reson, reent bioinformtis textbooks hve devoted whole hpters to the reltionship of bioloil seqenes to the Chomsky hierrhy 11,12. In liht of these prtil onseqenes of linisti omplexity, sinifint findin is tht there exist phenomen in RNA tht in ft rise the lne even beyond ontextfree. The most obvios of these re solled nonorthodox seondry strtres sh s psedoknots, whih re pirs of stemloop elements in whih prt of one stem resides within the loop of the other (Fi. 1). This onfirtion indes rossseril dependenies in the resltin bse pirins, reqirin ontextsensitive expression (Box 1). Preditbly, iven this frther promotion in the Chomsky hierrhy, the need to enompss psedoknots within seondrystrtre reonition nd predition prorms hs sinifintly omplited lorithm desin 13. Another nonontextfree phenomenon tht ors in RNA is onseqene of lterntive seondry strtre, sh s tht seen in bteril ttentors, whih re reltory elements tht depend on swithin between onformtions in nsent mrna moleles. For ny rmmr reqired to simltneosly represent both onformtions, these mtlly exlsive options rete overlppin (nd ths rossseril) dependenies in the lternte bsepirin shemes 7 (Fi. 1d) Ntre Pblishin Grop NATURE VOL NOVEMBER 2002
3 insiht review rtiles b d ε ε ε Fire 1 Grmmrstyle derivtions of idelized versions of RNA strtres., A stem; b, brnhed strtre;, psedoknot; nd d, lterntive seondry strtres of n ttentor. The trees for nd b re rphil depitions of derivtions from rmmrs iven in the text. By onvention, strtin nonterminl is t the root of the tree nd ives rise to brnhes for eh symbol to whih it rewrites in the orse of the derivtion. The strin derived n be red by trin the frontier or lef nodes of the tree, left to riht (dshed ble lines). For nd d, derivtion trees re not expliitly indited bese of the omplexity of the ontextsensitive rmmrs reqired 7. The sme strins re lso shown in liner fshion, with dependenies indited between terminls derived t the sme steps. Usin formlisms lled treedjoinin rmmrs nd their vrints 14, whih re onsidered to be mildly ontextsensitive nd reltively trtble, it is possible to enompss wide rne of RNA seondry strtres 15. Additionlly, new types of rmmrs hve been invented to del with sh bioloil exmples 16,17. Ntrl lnes seem to be beyond ontextfree s well, bsed on linisti phenomen entilin rossseril dependenies 18, lthoh in both domins sh phenomen seem to be less ommon thn nested dependenies. Ths, by one mesre t lest, nlei ids my be sid to be t bot the sme level of linisti omplexity s ntrl hmn lnes. Protein linistis There hs been less tivity in modellin proteins with linisti methods, perhps bese they re viewed s hvin riher bsi repertoire of intertions nd onformtions thn nlei ids, nd perhps lso more of sense of emerent properties. Yet rmmrs n be extrordinrily detiled nd nned (while reminin mneble bese of their inherently modlr nd hierrhil desin), nd moreover need not ptre every spet of strtre to be sefl. In ft, the omprehensiveness nd proper role of rmmrs remins s mh n isse for ntrl lne s it miht prove to be for proteins, s does the qestion of whether exemplrs of either lne re sseptible of ompositionl semntis (tht is, one for whih the menin or fntion of the whole n be bilt p in rlebsed fshion from tht ssoited with its prts) 3. In ny se there is deidedly linisti flvor to ertin bstrted depitions of protein strtre, sh s domin shemtis (for exmple, the MART system, whih portrys the hihly vrible rrnements of mobile domins 19 ) or topoloy rtoons (for exmple, the TOP system, whih nnottes dependenies between seondry strtrl elements, inldin positionl nd hirl reltionships 20 ). peifi spets of protein strtre hve been modelled expliitly with rmmrs. eondry strtrl elements, nd in prtilr the hydroen bondin between strnds in sheet, my be rryed in ntiprllel fshion, retin nested dependenies by nloy with stemloop strtres in RNA, or in prllel fshion, whih retes rossseril dependenies. h rrnements hve been represented sin stohsti tree rmmrs 21, whih re relted to treedjoinin rmmrs nd whih hve lso been shown to enerte rne of onfirtions of sheets tht orresponds well to tht seen in ntre (A. Joshi, personl ommnition). Another rmmrbsed pproh, sin tools from rph theory, ws shown reently to be pble of enertin preponderne of the lss of ll folds from jst for bsi rles 22. Mthemtiins re onerned with losre properties of lnes, tht is, whether they remin t the sme level of the Chomsky hierrhy when vrios opertions re performed on their ontents 9. imple ontention of strins is solled relr opertion, wheres insertion of one strin in nother is ontextfree opertion, insofr s it never ses dependenies to ross, bt only frther nests them. Neither opertion rises ontextfree lne beyond ontextfree, nor (it n be shown) do series of bioloil opertions sh s replition nd reombintion 10. However, trnslotion of sements of strin my rete rossseril dependenies where none existed before, nd ths the blok movements typil of enomi rerrnements my onstitte n pwrd fore in the Chomsky hierrhy tht is inherent in evoltion 10. Nevertheless, within proteins we see evidene tht t the level of domins (if not sperseondry strtre) there is in reltive srity of nonontextfree forms (Fi. 2). This is perhps ttribtble not only to the reter omplexity of the enomi hnes reqired, bt lso to the enereti brriers tht miht be ntiipted in foldin knotlike rossseril dependenies, by nloy with diffilties they pose in linisti nlysis. In liht of this, it is interestin tht the speil se of irlr permttions (tht is, hedtotil rerrnements), to whih protein domins seem more prone 23, do in ft preserve ontextfree stts from mthemtil perspetive 24. Compttionl linistis nd enes The reslts smmrized bove ll relte to strtrl spets of mromoleles, tht is, ftors inherent in their biophysil behvior nd independent of ny informtion they ontin. Yet enes do onvey informtion, nd frthermore this informtion is ornized in hierrhil strtre whose fetres re ordered, onstrined nd relted in mnner nloos to the syntti strtre of sentenes in ntrl lne. It is ths not srprisin tht nmber of themes, both expliit nd impliit, hve fond their wy from ompttionl linistis to ompttionl bioloy. One impliit theme is onverene between orniztionl shemes in the two fields. Lne proessin is often oneived s proeedin from (1) the lexil level, t whih individl words from NATURE VOL NOVEMBER Ntre Pblishin Grop 213
4 insiht review rtiles Fire 2 Protein domin rrnements nd the Chomsky hierrhy. hown re bkbone strtres for, t msle pyrvte kinse (1pkm in Protein Dt Bnk; mins short minoterminl domin) nd b, Esherihi oli Dmltodextrin bindin protein (1omp in Protein Dt Bnk). At the bottom re shems of the domin reltionships, with doble rrows onnetin sements prtiiptin in the sme domin. The pper, rboxyterminl (ble) domin of 1pkm tthes by wy of simple ontention, whih is relr opertion ommonly seen in proteins. The entrl redndreen / brrel, however, is interrpted in the middle by n insertion of the lower (orne) domin, ontextfree opertion insofr s it ths retes stritly nested dependeny between the divided domin sements (s wold ny nmber of domin insertions t ny point). Insertions re less ommon thn ontentions, bt still firly freqent. The two min domins of 1omp, on the other hnd, seem to be interleved, ths retin rossseril dependenies tht re neessrily ontextsensitive. Whether the Cterminl (ble) sement is involved flly in the lower domin s ore, however, is open to qestion; in ny se, tre interleved strtrl domins seem to be very rre. The dshed ellipses in the bkbone dirms illstrte tht the nmber of rossovers between domins (1, 2 nd 3, respetively) is inditive of the level in the Chomsky hierrhy of the resltin domin rrnement. b Insertion Contention Interlevin liner inpt strem (of, for exmple, phonemes or hrters) re reonized nd hrterized; to (2) the syntti level, t whih words re roped nd relted hierrhilly ordin to rmmr rles to form strtrl desription; to (3) the semnti level, t whih some representtion of menin is ssined to the resltin strtre, derived from tht of its individl lexil elements; nd finlly to (4) the prmti level, t whih lne is viewed in lrer ontext enompssin the roles nd interreltionships of sentenes (nd ertin referenes within them sh s pronons) in n overll disorse or diloe 3. This proression mps netly nd meninflly onto one sed widely in bioloy, of seqene to strtre to fntion to role 25. In prtilr, the distintion between syntx nd semntis (fmosly exemplified by Chomsky with his rmmtil yet meninless Colorless reen ides sleep friosly 2 ) is pertinent to bioloy. Consider two types of seqene: strin of words, nd sement of enome. A prsin step my be seen s determinin whether the words form rmmtil sentene, or, notionlly, whether the enomi seqene will spport the prodtion of polypeptide ordin to rles impliit in the trnsriptionl nd trnsltionl mhinery of the ell; in both ses the proesses re mehnil, in ft lrely proessive. Then, n interprettive step determines whether the resltin sentene is meninfl, ordin to lws of loi nd experiene, or whether the polypeptide will fold into ompt ore nd orient its side hins so s to do sefl work, proess overned by lws of thermodynmis nd biohemistry. Mtted enes tht re expressed bt do not llow for fntionl fold my be sid to pss the first test bt not the seond. The ntrl history of enefindin lorithms offers nother illstrtion. In the 1980s, detetin enes (in wht enomi seqene ws then extnt) ws stritly lexil ffir. Alorithms simply snned n inpt seqene nd within movin window ssessed its odin potentil on the bsis of sttistil mesres sh s olionleotide freqenies nd periodiities. It ws lso possible to detet sinls sh s pttive splie sites, in s individl lexil elements. Then, in the erly 1990s, prorms ben to pper tht ssembled lexil elements hierrhilly nd imposed onstrints of distintly syntti st. (Ths, jst s sentene onstitents mst ree s to nmber, ender, tense, nd so on, so hd pttive exons to mintin redin frme ross whole enes.) Indeed, one prorm tht performed reditbly t the time ws bsed expliitly on ene rmmr nd enerlprpose prser ( prorm tht determines if n inpt is vlid instne of ny iven rmmr nd, if so, prodes treestrtred desription of the prse) 26. One dvnte of linisti ene reonition ws the ntrl ommodtion of mbiity in the form of mltiple trnsripts ttribtble, for exmple, to lterntive spliin. Another dvnte ws verstility: the sme prser, bt with different rmmrs sbstitted, ws effetive in reonizin sh fetres s trna enes nd rop I introns, inldin seondry strtre extendin to psedoknots 27. Yet nother re in whih rmmrs hve proven pt is in the speifition of ene reltory elements, with their hihly vrible distribtion of disprte fetres. This se, in ft, ws one of the first sested bioloil pplitions of Chomskystyle rmmrs 28 nd remins n tive re of reserh 29,30. Althoh hvin the dvnte of flexibility, enerlprpose prsers nnot ompete in effiieny with prormmin tht is stomized to prtilr domin, espeilly one tht does not retly benefit from the pity of rmmrs to speify vritions on theme with ese. (Enlish rmmr wold be sperflos if every sentene were ptterned on the sme bsi delrtive templte.) Conseqently, ltterdy enefindin lorithms, whih hve the stndrd model ene strtre hrdwired, do not mke se of rmmrs per se. However, wht hs insted beome dominnt tehniqe in the nlysis of bioloil seqenes, the hidden Mrkov model (HMM), lso tres its pediree to linisti roots nd inherits different set of dvntes. An HMM is vriety of tomton nnotted with probbility vles tht overn its behvior 3. They were first widely deployed in the field of speeh reonition nd more reently hve fond their wy into nmber of pplitions for the nlysis of bioloil seqenes, beinnin with protein fmily profiles 11. HMM Ntre Pblishin Grop NATURE VOL NOVEMBER 2002
5 insiht review rtiles Orrene of Pfm protein domins in. erevisie enome ( ) Protein kinse domin nd the I to r trnsporter domin WD domin Helise onserved domin of my is tht RNAreonition motif Rnk order of freqeny Fire 3 Distribtions of the nmber of orrenes of Pfm protein domins (ble sqres) in the enome of the yest hromyes erevisie, nd of words (red dimonds) in hkespere s Romeo nd Jliet, in both ses sorted in rnk order from left to riht. The most freqently orrin domins nd words re lbelled. In both ses (nd in mny other enomes nd texts) the rves re ood fits to powerlw distribtion known s Zipf s lw, whih reltes the freqeny to the inverse of the rnk. rhitetres embody wht monts to syntx nd se n ssoited set of lorithms to refine nd employ the model. HMMs with sophistited domin models form the bsis for severl ledin ene finders, inldin Genn 31 nd Genie 32, nd the enefindin pplition hs driven frther refinement of the method s well. The reent mrked trend in ompttionl bioloy towrds probbilisti methods sh s HMMs hs mirrored similr trn in ntrl lne proessin, whih hs been inviorted by shift towrds finitestte nd stohsti pprohes 3. The se of HMMs in the two fields hs been ompred diretly in reent review 33. The tomt ssoited with HMMs re t the lowest rn of the Chomsky hierrhy nd re ths indeqte for sh nonrelr fetres s the seondry strtre in trna. This shortomin hs been ddressed by ddin probbilities to ontextfree rmmrs to rete stohsti ontextfree rmmrs nd then dptin the HMM lorithms to work with the resltin dt strtres 34. h systems hve proven sefl not only in trna detetion 35, bt lso in vriety of relted bioloil pplitions 36 39, nd hve even been extended to nonontextfree strtres 40. Historil linistis nd evoltion Lon before Chomsky s revoltion, historil linistis ws the dominnt disipline in the field 41, driven lrely by n inresinly systemti ttempt to ont for the desent of modern lnes from hypothesized protoindoeropen lne first proposed in Of this work Drwin himself noted tht the formtion of different lnes nd of distint speies, nd the proofs tht both hve been developed throh rdl proess, re riosly prllel 42. These prllels hve sine inspired mny thors. Dwkins onept of memes s replitin ltrl frments nderoin drwinin seletion enompsses lne hne 43, s does reent synthesis of forml lne theory, lernin theory nd evoltionry dynmis 44. tron nloies between the evoltion of lnes nd of speies hve even formed the bsis for serios sientifi rments inst retionism 45. Cvlliforz hs omprehensively explored how popltion enetis n id nderstndin of lne evoltion from demorphi perspetive 46, nd bioloil phyloenetireonstrtion tehniqes hve lso been pplied to lnes Orrene of words in Romeo nd Jliet ( ) Amon the methods linists themselves hve sed to drw fmily trees of lnes hs been the sttistil omprison of voblries, or lexiosttistis 41. This pproh posits tht, ross mny lnes, there is bsi, ore set of ontes (essentilly, word ortholoes ) reltin to niversl hmn experiene nd reltively resistnt to hne. In the 1950s, wdesh estblished 200 sh onepts (for exmple, I, this, not, person, fish, blood, e, knee, lod, montin nd ood) nd, bsed on similrity of orrespondin words in different lnes, derived qntittive mesres of overll lne reltedness 48. He frther proposed tht lne diverenes old be dted in this mnner by ssmin onstnt rte of lexil hne, tehniqe lled lottohronoloy. Althoh ontroversil, this is lerly ehoed in the notion of the evoltionry molelr lok. Indeed, the need to ont for vryin rtes of hne in different words nd proteins hs been reonized independently in eh field 49. The ompiltion of ore voblries from mltiple lnes resembles efforts to ssemble miniml ene sets presmed sffiient to spport life (by one estimte, nmberin bot 300) by tkin intersetions of mltiple enomes, nd similr tions hve been noted in their se nd interprettion 50. For instne, from the ft tht Frenh hs no word for shllow one old not onlde tht the lne is impoverished, ny more thn the pprent bsene of iven enzyme neessrily rles ot ertin metboli pity. Comprisons of ene ontents ross phyloeny hve been sed in wys tht miht hve been drwn diretly from the lexiosttistil litertre. Exmples inlde the olletion of lsters of ortholoos rops 51 nd the se of deree of overlp of ene omplements (s opposed to individl seqene similrities) s bsis for phyloeny onstrtion s well s preditor of protein fntion 55. Both fields ontend with omplitions introded by synonyms nd flse ontes ( fx mis ) on the one hnd, nd on the other, nonortholoos ene displement nd fntionl shifts, while reent theory onernin retilte evoltion hrkens bk to wellstdied phenomen of lne mixtre sh s reoliztion 56. Words themselves rise nd evolve by mehnisms tht hve been ompred to bioloil drivers of diversity, sh s mttion nd reombintion (lled blendin by linists) 57. One mehnism they lerly hve in ommon is ompondin. The tomi nits of linisti menin re morphemes, typilly stems nd ffixes tht ombine to form words, wheres lexil nits re lexemes, whih my be sinle words or omponds nd ertin nitry phrses 3. In like mnner, proteins re onsidered to omprise one or more fntionl domins, nd reent stdy hypothesizes nient nteedent domin sements, reltin these expliitly to linisti vrition 58. There is more thn srfe similrity to sh onventions, insofr s these re ll elements tht re srmised to ombine nd ressort in the orse of evoltion, ffordin ombintoril diversity, nd some of the sme tehniqes hve been pplied in their nlysis. For instne, qntittive pproh to the ssoition of words is ollotion nlysis. Here, the freqeny of oorrene of words in text is not only sefl heristi in stohsti prsin 3, bt lso provides les in lexil semnti stdies, for whih omponds hve been lssified into sh teories s non+non onstitents, idioms, nd so forth 59. This tehniqe hs been reinvented in the ontin of ene fsions ross mny enomes s preditor, for exmple, for protein protein intertions or prtiiption of proteins in ommon pthwys 60. In both ses, prtil implementtions ll for sh steps s filterin of promisos elements tht re less preditive of ommon fntion or menin 61. Literry linistis nd the enome Wht miht be lled literry linistis inldes prsits rnin from stylistis to textl nlysis to literry ritiism. Althoh seeminly t opposite poles from the hrd siene of molelr bioloy, these tivities re t some level not so different from the inresinly hermeneti role of the bioinformtiin, insofr s NATURE VOL NOVEMBER Ntre Pblishin Grop 215
6 insiht review rtiles both re onerned with omprin texts, detetin sbtle ptterns nd reltionships, nd elidtin theme nd vrition 25. Nor is textl ritiism devoid of qntittive methods; onern with isses sh s thorship ttribtion nd thentiity hs enendered n tive disipline of sttistil literry stdies ided by omptin 62,63. The most pervsive theme in ll sh work is the stdy of word freqenies in texts, the mthemtil nlysis of whih oriintes with the linist G. K. Zipf, who first observed powerlw distribtion reltin word s freqeny of orrene to the inverse of its position in the rnk orderin of those freqenies 64 (Fi. 3). Mndelbrot elborted on this insiht, proposin reltionship between wht hs ome to be known s Zipf s lw nd presmed frtl ntre of lnes 65. Apprent instnes of powerlw behvior hve now been observed in mny fets of molelr bioloy, inldin olionleotide freqenies 66, sizes of ene fmilies 67 (inldin psedoenes 68 ),distribtions of protein 69 nd RNA 70 folds, nd even levels of ene expression 71. As in the linisti se, severl explntions for these powerlw behviors hve been proposed, inldin their mthemtil reltionship to slefree networks 72 sh s miht be expeted in metboli pthwys 73 nd protein intertion mps 74, nd models for how they miht rise in the evoltion of protein fmilies 69, ll of whih evine omprison to properties of words. Textl ritiism shres both ols nd methods with bioinformtis. peiesspeifi distribtions of olionleotides re mon the sinls (lled style mrkers by linists 62 ) tht hve been sed in thorship ttribtion of enome sements thoht to rise by horizontl trnsmission between speies (for exmple, pthoeniity islnds in bteri 75 ), nd in hekin the thentiity of loned seqenes possibly ontminted by forein mteril 76. Word freqenies nd mny other style mrkers hve been nlysed in litertre sin sh tools s lsterin 77, prinipl omponents nlysis 78, nerl networks 79, spport vetor mhines 80 nd eneti lorithms 81, ll of whih re now bein pplied s well to trnsript freqenies inferred from mirorry experiments. A reent review of these methods pplied to ene expression omes fll irle by sin lsterin lorithm to rop nd lssify rtiles on the topi bsed on word freqenies 82, fory into wht hs been termed bibliomis 33,83. The omplexity of hmn nd bioloilseqene lnes t lexil level hs been ompred expliitly by Trifonov nd oworkers 84. Usin metris desined to detet the extent of overlppin odes, they sest tht seqene lnes re more lyered, with mltiple sinls refletin, for exmple, different elllr proesses, nd ths more omplex insofr s the odes my onstrin or interfere with one nother 85. (Extreme exmples re virl enomes with overlppin, frmeshifted odin reions.) It shold be noted, however, tht hmn lne is not sinle ode s sested by Trifonov, bt involves lyerin t mltiple levels. An obvios illstrtion is poetry, where lexil nd syntti ommodtions re often mde for sh overlid onstrints s rhyme sheme, metre nd verse form, nd even hiher orders of metphor, mood nd theme witness the virslike eonomy of hik. h sperposition in lnes is even treted formlly, insofr s ontextfree lnes re not losed nder intersetion nd ths my be driven hiher in the Chomsky hierrhy by lyerin 10 ; speifi instne is the view of psedoknot s the intersetion of two stemloop strtres 40. A brnh of textl ritiism lled stemmtis is onerned with the ry of texts, possibly nient, tht exist in mltiple forms for resons rnin from printers errors to thoril revisions to frmentry sores. For mnsripts opied mny times by sribes, there hs even been mthemtil modellin of opyin errors for prposes of estimtin pirwise distnes lon pth from ommon nestor 86 ; bioloilly motivted lorithms hve been enlisted in this se to elidte the provenne of Cher s Cnterbry Tles 87. However, the very fondtion of these lorithms in bioloil ldistis repitltes older, similr methods from stemmtis nd linistis, s ws lredy reonized qrterentry o 88. One postmodern (nd ths ntithoritrin) shool of textl ritiism promotes the ide of eneti text, dynmi onept tht enompsses ll versions nd even sores of text throh time 89, lrely bndonin the onept of min version nd thereby reqirin new orniztionl prdims nd ompttionl ids 90. The eneti text tht is the enome srely presents similr hllenes, nd the mny ommonlities (s well s the instrtive differenes) between ntrl nd bioloil lnes my ths form the bsis for shrin tools, tehniqes nd wys of thinkin bot omplex systems, on mny different levels. doi: /ntre Aithison, J. Linistis (NTC/Contemporry Pblishin, Chio, 1999). 2. Chomsky, N. yntti trtres (Moton, The He, 1957). 3. Jrfsky, D. & Mrtin, J. H. peeh nd Lne Proessin (Prentie Hll, Upper ddle River, NJ, 2000). 4. Brendel, V. & Bsse, H. G. Genome strtre desribed by forml lnes. Nlei Aids Res. 12, (1984). 5. Hed, T. Forml lne theory nd DNA: n nlysis of the enertive pity of speifi reombinnt behviors. Bll. Mth. Biol. 49, (1987). 6. erls, D. B. in Pro. 7th Ntl Conf. Artif. Intell (AAAI Press, Menlo Prk, CA, 1988). 7. erls, D. B. The linistis of DNA. Am. i. 80, (1992). 8. erls, D. B. in Loi Prormmin: Pro. North Am. Conf. (eds Lsk, E. & Overbeek, R.) (MIT Press, Cmbride, MA, 1989). 9. erls, D. B. in Artifiil Intelliene nd Molelr BioloyCh. 2 (ed. Hnter, L.) (AAAI Press, Menlo Prk, CA, 1993). 10.erls, D. B. in Mthemtil pport for Molelr Bioloy (eds FrhColton, M., Roberts, F.., Vinron, M. & Wtermn, M.) (Amerin Mthemtil oiety, Providene, RI, 1999). 11.Drbin, R., Kroh, A., Mithison, G. & Eddy,. Bioloil eqene Anlysis: Probbilisti Models of Proteins nd Nlei Aids (Cmbride Univ. Press, Cmbride, 1998). 12.Bldi, P. & Brnk,. Bioinformtis: The Mhine Lernin Approh (MIT Press, Cmbride, MA, 2001). 13.Lynso, R. B. & Pedersen, C. N. RNA psedoknot predition in enerybsed models. J. Compt. Biol. 7, (2000). 14.Joshi, A. in Ntrl Lne Proessin: Psyholinisti, Compttionl nd Theoretil Perspetives (eds Dowty, D., Krttnen, L. & Zwiky, A.) (Chio Univ. Press, New York, 1985). 15.Uemr, Y., Hsew, A., Kobyshi,. & Yokomori, T. Treedjoinin rmmrs for RNA strtre predition. Theor. Compt. i. 10, (1999). 16.erls, D. B. trin Vrible Grmmr: loi rmmr formlism for DNA seqenes. J. Loi Prorm. 24, (1995). 17.Rivs, E. & Eddy,. R. The lne of RNA: forml rmmr tht inldes psedoknots. Bioinformtis 16, (2000). 18.hieber,. Evidene inst the ontextfreeness of ntrl lne. Linist. Phil. 8, (1985). 19.hltz, J., Milpetz, F., Bork, P. & Pontin, C. P. MART, simple modlr rhitetre reserh tool: identifition of sinllin domins. Pro. Ntl Ad. i. UA 95, (1998). 20.Westhed, D. R., lidel, T. W., Flores, T. P. & Thornton, J. M. Protein strtrl topoloy: tomted nlysis nd dirmmti representtion. Protein i. 8, (1999). 21.Abe, N. & Mmitsk, H. Preditin protein seondry strtre sin stohsti tree rmmrs. Mhine Lern. 29, (1997). 22.Przytyk, T., rinivsn, R., & Rose, G. D. Rersive domins in proteins. Protein i. 11, (2002). 23.Jn, J. & Lee, B. Cirlrly permted proteins in the protein strtre dtbse. Protein i. 10, (2001). 24.Hoproft, J. E. & Ullmn, J. D. Introdtion to Atomt Theory, Lnes, nd Compttion (AddisonWesley, Redin, MA, 1979). 25.erls, D. B. Redin the book of life. Bioinformtis 17, (2001). 26.Don,. & erls, D. B. Gene strtre predition by linisti methods. Genomis 23, (1994). 27.erls, D. B. Linisti pprohes to bioloil seqenes. Compt. Appl. Biosi. 13, (1997). 28.ColldoVides, J. A trnsformtionlrmmr pproh to the stdy of the reltion of ene expression. J. Theor. Biol. 136, (1989). 29.Rosenbleth, D. A. et l. yntti reonition of reltory reions in Esherihi oli. Compt. Appl. Biosi. 12, (1996). 30.Len,., Mellish, C. & Robertson, D. Bsi Gene Grmmrs nd DNAChrtPrser for lne proessin of Esherihi oli promoter DNA seqenes. Bioinformtis 17, (2001). 31.Bre, C. & Krlin,. Predition of omplete ene strtres in hmn enomi DNA. J. Mol. Biol. 268, (1997). 32.Reese, M. G., Klp, D., Tmmn, H. & Hssler, D. Genie ene findin in Drosophil melnoster. Genome Res. 10, (2000). 33.Yndell, M. D. & Mjoros, W. H. Genomis nd ntrl lne proessin. Ntre Rev. Genet. 3, (2002). 34.kkibr, Y. et l. tohsti ontextfree rmmrs for trna modelin. Nlei Aids Res. 22, (1994). 35.Lowe, T. M. & Eddy,. R. trnasne: prorm for improved detetion of trnsfer RNA enes in enomi seqene. Nlei Aids Res. 25, (1997). 36.Rivs, E. & Eddy,. R. Nonodin RNA ene detetion sin omprtive seqene nlysis. BMC Bioinformtis 2, 8 (2001). 37.Kndsen, B. & Hein, J. RNA seondry strtre predition sin stohsti ontextfree rmmrs nd evoltionry history. Bioinformtis 15, (1999). 38.Brown, M. P. mll sbnit ribosoml RNA modelin sin stohsti ontextfree rmmrs. Pro. Int. Conf. Intell. yst. Mol. Biol. 8, (2000). 39.Holmes, I. & Rbin, G. M. Pirwise RNA strtre omprison with stohsti ontextfree rmmrs. P. ymp. Bioompt (2002). 40.Brown M. & Wilson C. RNA psedoknot modelin sin intersetions of stohsti ontext free rmmrs with pplitions to dtbse serh. P. ymp. Bioompt (1996) Ntre Pblishin Grop NATURE VOL NOVEMBER 2002
7 insiht review rtiles 41.Cmpbell, L. Historil Linistis: An Introdtion (MIT Press, Cmbride, MA, 1999). 42.Drwin, C. The Desent of Mn (John Mrry, London, 1871). 43.Dwkins, R. The elfish Gene (Oxford Univ. Press, Oxford, 1976). 44.Nowk, M. A., Komrov, N. L. & Niyoi, P. Compttionl nd evoltionry spets of lne. Ntre 417, (2002). 45.Pennok, R. T. Tower of Bbel: The Evidene inst the New Cretionism (Brdford/MIT Press, Cmbride, MA, 1999). 46.Cvlliforz, L. L. Genes, Peoples, nd Lnes (North Point Press, New York, 2000). 47.Wrnow T. Mthemtil pprohes to omprtive linistis. Pro. Ntl Ad. i. UA 94, (1997). 48.wdesh, M. Lexiosttistil dtin of prehistori ethni ontts: with speil referene to North Amerin Indins nd Eskimos. Pro. Am. Phil. o. 96, (1952). 49.Krskl, J. B., Dyen, I. & Blk, P. in Lexiosttistis in Geneti Linistis (ed. Dyen, I.) (Moton, The He, 1973). 50.Mshein, A. The miniml enome onept. Crr. Opin. Genet. Dev. 9, (1999). 51.Ttsov, R. L., Glperin, M. Y., Ntle, D. A. & Koonin, E. V. The COG dtbse: tool for enomesle nlysis of protein fntions nd evoltion. Nlei Aids Res. 28, (2000). 52.nel, B., Bork P, & Hynen, M. A. Genome phyloeny bsed on ene ontent. Ntre Genet. 21, (1999). 53.Teki, F., Lzno, A., & Djon, B. The enomi tree s reveled from whole proteome omprisons. Genome Res. 9, (1999). 54.Lin, J. & Gerstein, M. Wholeenome trees bsed on the orrene of folds nd ortholos: implitions for omprin enomes on different levels. Genome Res. 10, (2000). 55.Pellerini, M. et l. Assinin protein fntions by omprtive enome nlysis: protein phyloeneti profiles. Pro. Ntl. Ad. i. UA 96, (1999). 56.MWhorter, J. H. The Power of Bbel: A Ntrl History of Lne (Freemn, New York, 2001). 57.erls, D. B. From Jbberwoky to enome: Lewis Crroll nd ompttionl bioloy. J. Comp. Biol. 8, (2001). 58.Lps, A. N., Pontin, C. P. & Rssell, R. B. On the evoltion of protein folds: re similr motifs in different protein folds the reslt of onverene, insertion, or relis of n nient peptide world? J. trt. Biol. 134, (2001). 59.MKeown, K. R. & Rdev, D. R. in A Hndbook of Ntrl Lne Proessin (eds Dle, R., Moisl, H. & omers, H.) (Dekker, New York, 2000). 60.Mrotte, E. M. et l. Detetin protein fntion nd proteinprotein intertions from enome seqenes. iene 285, (1999). 61.mdj, F. Retrievin ollotions from text: XTRACT. Compt. Linist. 19, (1993). 62.Rdmn, J. The stte of thorship ttribtion stdies: some problems nd soltions. Compt. Hmnities 31, (1998). 63.Brnbrook, G. Lne nd Compters (Edinbrh Univ. Press, Edinbrh, 1996). 64.Zipf, G. K. Hmn Behvior nd the Priniple of Lest Effort (AddisonWesley, Boston, MA, 1949). 65.Mndelbrot, B. The Frtl Geometry of Ntre (Freemn, n Frniso, 1983). 66.Mnten, R. N. et l. Linisti fetres of nonodin DNA seqenes. Phys. Rev. Lett. 73, (1994). 67.Hynen, M. A. & vn Nimween, E. The freqeny distribtion of ene fmily sizes in omplete enomes. Mol. Biol. Evol. 15, (1998). 68.Hrrison, P. M. & Gerstein, M. tdyin enomes throh the eons: protein fmilies, psedoenes nd proteome evoltion. J. Mol. Biol. 318, (2002). 69.Qin, J., Lsombe, N. M. & Gerstein, M. Protein fmily nd fold orrene in enomes: powerlw behvior nd evoltionry model. J. Mol. Biol. 313, (2001). 70.hster, P., Fontn, W., tdler, P. F. & Hofker, I. L. From seqenes to shpes nd bk: se stdy in RNA seondry strtres. Pro. R. o. Lond. B 255, (1994). 71.Hoyle, D. C., Rttry, M., Jpp, R. & Brss, A. Mkin sense of mirorry dt distribtions. Bioinformtis 18, (2002). 72.Rzhetsky, A. & Gomez,. M. Birth of slefree molelr networks nd the nmber of distint DNA nd protein domins per enome. Bioinformtis 17, (2001). 73.Jeon, H. et l. The lresle orniztion of metboli networks. Ntre 407, (2000). 74.Prk, J., Lppe, M. & Teihmnn,. A. Mppin protein fmily intertions: intrmolelr nd intermolelr protein fmily intertion repertoires in the PDB nd yest. J. Mol. Biol. 307, (2001). 75.GriVllve,., Rome, A. & Pl, J. Horizontl ene trnsfer in bteril nd rhel omplete enomes. Genome Res. 10, (2000). 76.White, O. et l. A qlity ontrol lorithm for DNA seqenin projets. Nlei Aids Res. 21, (1993). 77.Hoover, D. I. ttistil stylistis nd thorship ttribtion: n empiril investition. Lit. Linist. Compt. 16, (2001). 78.Binono, J. N. G. & mith, M. W. A. The pplition of prinipl omponent nlysis to stylometry. Lit. Linist. Compt. 14, (1999). 79.Hoorn, J. F., Frnk,. L., Kowlzyk, W. & vn der Hm, F. Nerl network identifition of poets sin letter seqenes. Lit. Linist. Compt. 14, (1999). 80.Leopold, E. & Kindermnn, J. Text teoriztion with spport vetor mhines. How to represent texts in inpt spe? Mhine Lern. 46, (2002). 81.Holmes, D. I. & Forsyth, R.. The Federlist revisited: new diretions in thorship ttribtion. Lit. Linist. Compt. 10, (1995). 82.Altmn, R. B. & Ryhdhri,. Wholeenome expression nlysis: hllenes beyond lsterin. Crr. Opin. trt. Biol. 11, (2001). 83.erls D. B. Minin the bibliome. Phrmoenomis J. 1, (2001). 84.Popov, O., el, D. M. & Trifonov, E. N. Linisti omplexity of protein seqenes s ompred to texts of hmn lnes. Biosystems 38, (1996). 85.Trifonov, E. N. Interferin ontexts of reltory seqene elements. Compt. Appl. Biosi. 12, (1996). 86.penser, M. & Howe, C. Estimtin distnes between mnsripts bsed on opyin errors. Lit. Linist. Compt. 16, (2001). 87.Brbrook, A. C., Howe, C. J., Blke, N. & Robinson, P. The phyloeny of the Cnterbry Tles. Ntre 394, 839 (1998). 88.Pltnik, N. I. & Cmeron, H. D. Cldisti methods in textl, linisti, nd phyloeneti nlysis. yst. Zool. 26, (1977). 89.Tnselle, G. T. Litertre nd Artifts (Bibliorphil oiety of the University of Virini, Chrlottesville, VA, 1998). 90.Ferrer, D. Hypertextl representtion of literry workin ppers. Lit. Linist. Compt. 10, (1995). Aknowledements I thnk P. Arwl, A. Lps, N. Odendhl nd K. Rie for helpfl omments on the mnsript. NATURE VOL NOVEMBER Ntre Pblishin Grop 217
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