MONTE CARLO SIMULATION MODELS OF EARLY KEY GENETIC SYSTEMS

Transcription

1 MONTE CALO SIMULATION MODELS OF EALY KEY GENETIC SYSTEMS Eas Zntzaras, Mauro Santos 2 and Eörs Szathmáry Bomathematcs Unt, Medca Schoo, Unversty of Thessay, 22 Papayraz Str., 422 Larsa, Greece and Natona Agrcutura esearch Foundaton, Agaeas Str., 2 Athens, Greece (e-ma: zntza@nagref.gr 2 Departament de Genètca de Mcrobooga, Grup de Booga Evoutva, Unverstat Autònoma de Barceona, 08 Beaterra, Barceona, Span (e-ma: mauro.santos@uab.es Coegum Budapest, Insttute for Advanced Study, Szentháromság u. 2, H-04 Budapest, Hungary and Department of Pant Taxonomy and Ecoogy, Eötvös Unversty, 2 Ludova tér, H-08 Budapest, Hungary (e-ma: szathmary@cobud.hu Summary Egen proposed a mathematca mode to descrbe the mutpcaton of NA-e sequences. Ths mode shown the foowng paradox: the amount of nformaton that can be transmtted and mantaned by natura seecton depends on the accuracy of repcaton. To escape from the paradox Egen hmsef proposed the hypercyce: a cyccay couped system of autocataytc and cross-cataytc moecuar mutuasts, where each member heps the foowng member and receves hep from the precedng one. But such a system s evoutonary unstabe when mutatons (moecuar parastes are taen nto account because t s not an evoutonary unt. Then the cooperatng networs of genes shoud have been encapsuated n a protoce (ce-e structure. However, when the ce was nvented so there was no need for hypercyces snce a smpe pacage of competng genes, descrbed by the stochastc corrector mode (SCM, coud have provded an aternatve souton. In ths mode the tempates are repcated by a nonspecfc repcase and free to compete wthn each compartment because they can reap the benefts of a common metabosm dfferenty. The rate of growth and dvson of the compartments depend on the nds of moecues they contan, and dvson of abor and cooperaton between moecues woud evove. We present Monte Caro smuaton modes whch descrbe protoce popuatons that encose a hypercycc (HPC or a non-hypercycc (SCM system when they are facng reastc mutaton rates. Numerca resuts ndcate both systems are vabe, however, a popuaton of SCM protoces can toerate hgh deeterous mutaton rates and reaches an equbrum mutatona oad ower than that n a popuaton of HPC protoces. Based on the SCM we propose a Monte Caro mode to nvestgate the effect of sex n the vabty of the popuaton. We chaenge the dea that the emergence of sex was due to recombnatona reparng of mutant genes and we show that sex was benefca as a sde effect of the baance between fuson, mutaton and seecton.

2 Introducton Egen ( shown how the accuracy of repcaton n the eary NA (tempetes word paced mts on the sze of the genome that coud be mantaned by seecton. As a way out to the error catastrophe probem, Egen & Schuster ( offered the hypercyce (HPC as a mode of nformaton ntegraton n whch an arbtrary number of repcators are ned together n a cataytc oop. In the HPC each member serves as a tempate that can catayze ts own repcaton and, second, each member receves hep (repcase from the precedng member of the cyce. epcaton of each member thus depends on the product of ts own concentraton and that of the precedng one. When mutatons (sefsh mutants-parastes that are better targets for repcaton are taen nto account, HPCs are eventuay destroyed (Maynard Smth,. In order the evouton to contnue s requred the HPC;s to be encapsuate n to compartments (protoces. However, we do not now f the pacage of a hypercycc system nto compartments was a necessary ntermedate stage of evouton. A more economc aternatve system such as the stochastc corrector mode (SCM (Szathmáry & Demeter, 8, whch descrbes the dynamcs of genes encapsuated n a reproductve protoce, coud be another. In the SCM there s no hypercycc coupng among the tempates because they are repcated by a non-specfc repcase. The SCM assumes that there s an optma tempate composton that ensures the fastest growth and dvson of protoces. The behavor of the system depends on two types of stochastcty: ( repcaton of tempates wthn protoces, and ( random assortment of tempates nto offsprng protoces. Even though tempates compete wthn compartments, seecton on stochastcay produced offsprng varants can rescue the popuaton from extncton. We present Monte Caro smuaton modes whch descrbe the evouton protoce popuatons that encose a hypercycc (HPC or a non-hypercycc (SCM system when a constant nput of deeterous mutatons s aowed (Zntzaras et a Based on the SCM we propose a Monte Caro mode to nvestgate the effect of sex as a reparng mechansm n the vabty of a protoce popuaton (Santos et a

3 The Monte Caro mode The basc boogca unt n our mode s a protoce wth two dfferent tempates that stand for metaboc genes (M (Gant essenta for survvorshp and are repcated by genes ( wth a repcase functon (Fg.. These tempates may dffer n ther rate of repcaton wthn a ce and n ther contrbuton to ce growth and dvson. Fgure. Schematc representaton of compartments (protoces that ntegrate genetc nformaton va the hypercyce (HPC or the stochastc corrector mode (SCM. M stands for metaboc tempates (genes that mae a drect contrbuton to protoce ftness, stands for repcase tempates, and µ s for repcaton rates. In the mode, at generaton t 0, a popuaton s started wth K=0 dentca protoces wth n tempates (genes. Each tempate s assumed to consst of three mutabe stes (nuceotdes. A protoce s randomy chosen accordng to ts reatve ftness for tempate repcaton wth a deeterous mutaton rate per nuceotde per gene equa to u. The ce tempates are randomy repcated accordng to ts probabty of repcaton and the protoce s then turned bac to the popuaton whenever ts tempate number s ess than 2n, otherwse t dvdes as before. The procedure contnues unt the popuaton sze ncreases to 2K, then haf of the ces are dscarded at random and the start of a new generaton s assumed. In a runs the process was contnued for up to 00 generatons (Zntzaras et a The fow chart of the smuaton mode s as foows (for the nvovement of sex, see beow:

4 Popuaton of K = 00 ces When sex s nvoved Generaton g 0 Yes No Choose one ce at random accordng to ce ftness Seectvey repcate a tempate andomy fuse two ve protoces wth P fus Number of genes 2n? Yes No andomy assort the tempates n two daughter ces No Number of ces 2K? Yes andomy dscard haf of the ces 4

5 Protoce ftness At the protoce eve, the ftness functon we used was: d = g w = d, [] A = where d s the number of dfferent metaboc gene types, g s the number of copes of the th metaboc gene type, and g = A A j s the tota contrbuton of a ts j varants to protoce s j= ftness ( - ( υ cj υ A e e j =, [2] - 2 ( υ e where υ s the number of nuceotdes of gene, and c j s the number of correct (wd type nuceotdes. The ftness of the protoce exponentay decreases from w max = to 0 dependng on the number of mutant nuceotdes per gene (Fg. 2. Fgure 2. Ftness functon of a protoce accordng to the number of deeterous mutant nuceotdes n metaboc genes essenta for survvorshp. The HPC mode Two types of tempates are present, M and M 2 2 (two dfferent joned parts (genes. At generaton t 0 a protoces start wth copes of each tempate,.e. a tota of 60 genes. After a partcuar tempate repcase (. s randomy chosen (the dot stands for, 2, t w bnd a tempate (M.. aso chosen at random and eventuay repcates t n accordance wth ts target

6 affnty toward the repcase. epcaton occurs seray aong the tempate and can tae pace n ether two ways: sef-repcaton of tempates by ther specfc repcase, or repcaton by a dfferent repcase. The dynamcs are: ( = µ (, M =, 2; 0,.., ; 0,..,. j( = µ j(, M j( ; j; j =, 2. where the µ s are the repcaton rates determned by the entres n (, matrces (see beow. The probabtes of repcatng a tempate depend on the prevous growth rates and are cacuated as foows: P(M ( = ( ( j ; = j( =, P(M j( = j( Therefore, repcaton of each member depends on ts own concentraton and that of the precedng one. j j(. The SCM To smuate a conceptuay anaogous verson for the SCM, we assumed that each protoce has three nds of tempates: a non-specfc repcase (, and two target metaboc genes (M, M 2 ; see Fg.. At generaton t 0 a protoces start wth 20 copes of each tempate,.e. a tota of 60 genes as before. The repcaton process can aso tae pace n ether two ways: repcaton of target metaboc tempates or sef-repcaton of (sampng of tempates s aso done wthout repacement. The dynamcs are: ( = µ j(, =, 2; j =, 0,.., ; 2; 0,..,. d( ( ( m = µ s(, m ( ( m ; s =, 2; m = 0,..,. The probabtes of repcatng a tempate are cacuated as: M ( ; = = j; 6

7 P(M ( = ( + d( ( m, P( ( m = d( ( m + d( ( m. epcaton rates The repcaton rates (µ s are smpy the products of the target affntes mutped by the repcase actvtes, whch are specfc vaues wth a sgmod pattern dependng on the number of mutated stes (Fg.. Fgure. Assocaton between repcase actvty and the number of mutated stes n. The repcase actvty s a coumn vector wth the vaues,, and 0 for a repcase wth = 0,, 2, and deeterous mutant nuceotdes, respectvey. The putatve target affntes of a set of wd type and/or randomy mutated tempates are defned by the entres n 4 4 matrces (Zntzaras et a. 2002: M(0. (0 M(0.( M(0. (2 M(0. ( M M M M (. (0 (.( (. (2 (. ( M(2. (0 M(2.( M(2. (2 M(2. ( M(. (0 M(.( M(. (2 M(. ( where the fgures n parentheses ndcate number of mutant nuceotdes and the dot n.( stands for, j. In the case of no mutaton we have up to four target affntes n the HPC and up to three n the SCM. Dependng on ther reatve vaues, we can mode for sefsh and/or cooperatve tempates. Target affntes for sef-repcaton of M and M 2 2 n HPC were consdered to be equa and the same as the target affnty for sef-repcaton of n the SCM, and target

8 affntes for repcaton by a dfferent repcase (.e., M ( j( ; j were assumed to be dfferent. Therefore, we have to defne three 4 4 matrces of target affntes. The target affntes were obtaned from unformy dstrbuted random numbers over the nterva (0, wth the constrant µ 2 < µ = µ 2 > µ 2, whch corresponds to a sefsh repcase n the SCM. A other entres n the matrces of target affntes when deeterous mutatons are ncorporated n the mode were aso obtaned from unformy dstrbuted random numbers but wth no constrants. Sex between protoces The effect of sex n the eary stages of evouton was nvestgated usng the SCM snce ths s an effcent nformaton ntergrator system and can toerate hgher nput of deeterous mutatons than a hypercyce system (Santos et a In the Monte Caro mode, sex was ncuded after the popuaton reaches ts equbrum under protoce seecton whch happens after the frst 0 generatons (see fow chart above. Sex was defned by the random fuson of two protoces wth non-zero ftness foowed by the random assortment nto two offsprng ces. The fuson taes pace wth probabty P fus =0 per generaton. In ths sexua phase the protocces recombne ther genomes. Ths phase s separate form, and precedes, the tempate repcaton. Numerca esuts Twenty ndependent runs were obtaned for each set of condtons, and numerca resuts are potted n Fgs. 4- for mutaton rates up to 2 per nuceotde per repcaton. Both HPC and SCM popuatons can survve when u = 2 assumng a compartment of ~60 genes. Beow u, HPC performs better than SCM. The reason s that wth ow mutaton rates the assortment oad s hgher than the mutaton oad and stronger n the SCM than n the HPC. As u ncreases from, the average ftness of HPC protoces progressvey decreases, whe the fa n average ftness of SCM protoces s far ess cear-cut. When u = HPC protoces qucy coapse wthn the frst 20 generatons. The mutatona oad (Hadane L s aways ower n the SCM than n the HPC. Wth u = the mutatona oad can become stabe at n the SCM, whereas n the HPC s amost. The reason for a hgher mutatona oad n the HPC s that any of ts members s a sefrepcatng NA-e moecue that aso catayses specfcay the repcaton of the next member. Therefore, for every tempate M a poor repcase can easy htchhe a fttest wd type metaboc gene. Both repcases ( and 2 need to be smutaneousy functona n the HPC due to ts doube autocataytc nature. The man cause for the seectve nferorty of the HPC at hgh mutaton rates s due to the non-functona repcases whch are n strong nage dsequbrum wth one of the metaboc genes. There are two mportant sources of genetc 8

9 oad n the SCM protoces: assortment oad and mutatona oad. The assortment oad coud be easy reduced n a popuaton of SCM protoces ntay heterogeneous for target affntes. When µ S = µ 2 = µ 2 the average ftness for SCM protoces wth no mutaton stabzes at 6. Smary, the average ftness stabzes at 0. wth u as hgh as 2 (.e., L 0.. Now the reatvey sma decrease n average ftness s ony due to the mutatona oad (c.f., L 8 n Fg.. Therefore, a popuaton of SCM protoces w be vabe n spte of a hgh nput of deeterous aees and ths provdes a souton to the error threshod probem. When sex s ncorporated n the SCM, the average ftness (Fg. 6 for a hgh mutaton rate u=2 stabzes at ~0., ndcatng that sex has no substanta effect n the vabty of the evouton of the frst NAs. u = 0 HPC contnuous B u = 0 HPC contnuous M, M2, 2 M M M, M2, 2.0 u = HPC contnuous M M 2 2 M, M2, 2.0 u = 2 HPC contnuous M M 2 2 M, M2, 2 Fgure 4. esuts of the Monte Caro smuatons for deeterous mutaton rates per nuceotde per repcaton up to 2 n HPC protoces. For each condton, tweve ndependent runs were acheved and a trajectores (dotted nes for the average ftness of the popuaton are ndcated together wth the tota average (sod ne. Each pot embodes the average number of wd type copes of each gene n the protoces thorough generatons (ogarthmc scae.

10 .0 u = 0 SCM contnuous M M2 B.0 u = 0 SCM contnuous M M2.0 u = SCM contnuous M M2.0 u = 2 SCM contnuous M M2 Fgure. esuts of the Monte Caro smuatons for deeterous mutaton rates per nuceotde per repcaton up to 2 n SCM protoces. es n a tweve ndependent runs (dotted nes are ndcated together wth the tota average from the survvng runs (sod ne. Pots for the average number of wd type copes of each gene n the protoces are aso shown (ogarthmc scae. Fgure 6. esuts of the Monte Caro smuatons for deeterous mutaton rates per nuceotde per repcaton u=2 n SCM protoces consderng the sex. 0

11 Concusons It has been sad that once ceuarzaton appeared, t soved the very probem that ntated the deveopment of hypercyce theory and, therefore, we no onger need hypercyces at a. But as noted above, there have been nether rgorous comparsons of hypercycc versus nonhypercycc protoce popuatons, nor detaed anayses of ther toerabe deeterous mutaton rates. Two man resuts have emerged from our Monte Caro smuatons: ( both HPC- and SCM protoces coud toerate a very hgh nput of deeterous mutatons, and ( the mutatona oad s ower n a popuaton of SCM protoces. For HPC protoces we have assumed the mnmum hypercyce of ony two members and, therefore, our concuson that they can cope wth hgh mutatona oad shoud be taen wth cauton. The stochastc corrector mode not ony aows to ntegrate genetc nformaton but aso to overcome the danger of nformaton decay before repcaton was reasonaby accurate. We have now strong reasons to use t as a reatvey robust system to modeng the frst major transtons n evouton (Maynard Smth and Szathmary. Thus, the Monte Caro mode n ths paper w aso be used to anayze mportant probems such as the roe sex payed n the dynamcs of fragmented genomes, and the spread of nage. When sex was ntroduced n the stochastc corrector Monte Caro mode, ts effect was not mportant for the survva of the popuaton ndcatng that n the eary stages of evouton sex was actng as a sde effect. eferences Egen M. ( Naturwssenschaften 8, Egen M. & Schuster P. ( The Hypercyce: A Prncpe of Natura Sef-Organzaton (Sprnger-Verag, Bern. Gánt, T. ( Bosystems, -2. Hadane, J.B.S. ( Am. Nat., -4. Maynard Smth, J. ( Nature (London 280, Maynard Smth, J. & Szathmáry, E. ( The Major Transtons n Evouton (W.H. Freeman, Oxford, U.K.. Santos M., Zntzaras E. and Szathmáry, E. (200 Org. Lfe Evo. Bosph. (accepted. Szathmáry, E. & Demeter, L. (8 J. Theor. Bo. 28, Zntzaras, E., Santos, M. and Szathmáry, E. (2002 J. Theor. Bo. 2, 6-8.