CEJP 3(2003)516{555

AnIntroduction to Physical Theoryof Molecular Evolution

BartoloLuque ¤ DepartamentoMatem¶ aticaAplicada y Estad¶³stica, EscuelaSuperior de Ingenieros Aeron¶ auticos, UniversidadPolit¶ ecnicade Madrid, PlazaCardenal Cisneros 3,Madrid 28040, Spain.

Received 5March2003; revised 10July

Abstract: This workis atutorial in Molecular Evolution fromthe point ofview of Physics. Wediscuss Eigen’s model, alink between evolutionary theory andphysics. We will begin by assumingthe existence of(macro)molecules orreplicators with the template property,that is, the capacity to self-replicate. According tothis assumption, information will be randomlygenerated and destroyed by mutations in the code(i.e., errors in the copyingprocess) andnew bits ofinformation will be­xed (madestable) by the existence ofan external pressure onthe system (i.e.,selection), andthe ability ofthe molecules to replicate themselves. Our aimis tobuild amodelin order todescribe molecularevolution fromas general a standpoint aspossible. As wewill see, even very simple modelsfrom the theoretical point ofview will havesurprisingly deep consequences. c Central EuropeanScience Journals. All rights reserved. ® Keywords:Mole cular Evolution, Quasi-species, Hypercycles, Biological Physics; Adaptationand Self-Organizing Systems;Soft Condense dMatter; Statistical Mechanics PACS(2000):87.10.+e, 87.23.-n, 89.75.-k

1Thebirth of information

The question of how lifearose onEarth isone of the most engaging intellectualchallenges in sciencetoday [1,2]. During recent yearswe havebeen ableto reconstruct atentative phylogenetictree starting from acommon ancestor by putting together the data provided both by Paleontology and MolecularBiology [3]. The particular historical order in which di®erent branches haveappeared and disappeared from this tree is,however, still subject to much debate. Furthermore, sincethe concretehistory oflifeon this planet isprobably

¤ E-mail:[email protected] B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 517 full ofaccidentalsituations, hardly reproducible by their own nature, itislikelythat this willcontinue to be the state of a®airs for along timeindeed. Unless we are willingto acceptEarth’s uniqueness inthe Universe,there must be aset ofgeneral principles that makelife a viablephenomenon [4].It isabout these generalprinciples that we should concern ourselves,and it isin this spirit that we willtry to tacklethe problems put forward in this introduction. The latenineteenth century Biology,developedin the midst of the industrial revo- lution, put forward avisionof livingsystems in which their internal mechanisms and energytransformations were the keyelements. However, the telecommunications domi- nated societyof today seeslife as adynamicalstate of matter organized by and around information. Energy transformations are not enough. Today we know that the genetic codestores information, and we are learning to decodeit. The letters of this alphabet are the chemicalbases that form the DNA, codons maybe thought ofaswords, operons as paragraphs. Thus, we can rephrase the old question about the origin oflifeas: What isthe origin of the information stored inlivingsystems, and why isit stable? [5] Afew preliminary conclusions can already be established by examiningthe physical and informational constraints that (pre)living systems need to achieve: First, one should observethat all° uctuations are absorbed, eliminated,in asystem in ° thermodynamic equilibrium.The spontaneous generation of order and information in this picture seemsa verydi± cult (if not impossible) feat to accomplish.Our ¯rst conclusion therefore isthat we willneed to modelprocesses that takeplace out of equilibrium [6]. As the second pieceof the puzzle,classical Thermodynamics tellsus about the un- ° stoppable increaseof entropy that takesplace in isolated systems,about information loss.Any way out of this second constraint needs thus to provide an energy° ux, an open system.In principle,this energy° ux could be used either to prevent the degradation of the information carrying units or to fabricate new replicasof them (in atimescaleshorter than their lifetime,obviously). In the prebiotic stage, how- ever,these units (prebiotic molecules)will be smalland easyto build. Therefore, in our context the latter route willtend to be the most e±cient way to prevent the information lossassociated with entropy increase. The third and central component of the picture isthe need for evolution. An im- ° mediate re°ection about what evolution means in the biologicalsense leads to the conclusion that we must havetwo di®erent mechanisms acting atoncein an evolving system.First, the information contained in itmust bemodi¯able, and must beable to change. Secondly,aparticular direction ofchange must beselectedby anexternal pressure or force exertedon the system,i.e., there must be a selection mechanism at work. Wewant to emphasizethat evolution does occur in everypopulation of entities with three characteristics: reproduction, heredity and mutation [2]. In this introduction we willassume the existenceof (macro)molecules or replicators with the template property,that is,the capacityto self-replicate.On the basis of this assumption, information willbe randomly generated and destroyed by mutations in the 518 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555 codei.e., errors in the copying process, and new bits of information willbe made stable due to the existenceof an external pressure on the system i.e.,selection, and the ability of the moleculesto replicatethemselves. The availableexperimental evidence points towards RNAchains as the components of this ancient prebiotic world on Earth: the so-called"RNA world" scenario [7].There are at leastthree main questions in the RNA world scenario:¯ rst the synthesis of ancient nucleotides,second the emergenceof the ¯rst self-replicating molecules,and third the error-catastrophe [8].In this introduction we shallfocus our attention onthe last point. What isofrealimportance to us, however, isthat until the work of Manfred Eigen [9]in the 70’sthere had been no convincing modelfor the evolution of complexityof this earlyreplicators. Eigen has created alink between evolutionary theory and physics.This area iswell-known today asthe Physical Theory of MolecularEvolution and isthe main focus of this work. Our aimwill be to tacklethe subject from atheoretical point ofview.Thus, we aimto build amodelwhich can describe molecularevolution from averygeneral framework. As we shall see,even simpletheoretical models havesurprisingly deep consequences. Afew words before we start: this work isnot an extensivereview and can beregarded as afriendly introduction to MolecularEvolution to those who liketo be initiated into the ¯eld.Many interesting issues are just mentioned and can be complemented with updated bibliography.Complex details havebeen removedfor the sakeof simplicity and clarity.Three excellentreviews which go in depth are:"Biological evolution and statistical physics" by Barbara Drossel [10],"Introduction to the statistical theory of Darwinian evolution" by Luca Peliti[11] and "Molecularreplicator dynamics" by B.M. R.Stadler and P.F.Stadler [12].

2Experimentalevolution?

There are two well-known criticismsto the Darwinian evolutionary theory: the selection concept impliesa tautology,and that it cannot be checkedexperimentally .However, it isalready possible today to observe evolutionary processes in short timescales: the evolution ofthe in°uenza or RNAviruses [13]are wellknown examples. Oneof the ¯rst well-known examplewas the phago Q- ­ ,aRNAvirus that attacks bacteria. It stores its own information in RNAform and uses inversetranscriptasa to makeits complementary copy.In the 60’s,Haruna and Spiegelman cut o®the phago Q-­ ’scatalyst, the Q- ­ replicasa.This enzymeis the one responsible for the RNAcopy. If we combine together in atest tube Q- ­ replicasaalong with monomers likeA TP,GTP, UTP orCTP,the viralRNA no longer needs hosts for its self-reproduction. Inthis way it iseasy to de¯ne the ¯tness of amoleculeby simplycalculating its replication and decomposition rate. Thus, the tautology isavoided. Note that this isobviously not a natural environment for avirus. The capsule construction instructions stored inthe RNA havebecome unnecessary sincethe now reproduction capsid isnot needed any more,for example.As in the replicating process somemistakes may occur, the next generation of copiescould conceivablyobtain somereproduction bene¯ts in this new environment B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 519

due to mutations. Thus, for example,the now accessorycapsid instructions could be detached from the chain and dismissed, in order to shorten the length of the chain and increaseits reproduction rate. Aexperiment (see¯ gure 1) with the samedesign as the one just described was carried out by Spigelman and collaborators [14].A Q- ­ phage was inserted inthe ¯rst test tube. After afew hours, part of the population of the tube was extracted and subsequently inserted in anew test tube. The end result after seventyrepetitions of this process was that the chains had reached astable con¯guration inwhich they had lost their infective capabilities.The RNAvirus had only17% its original length, and was reproducing at arate 15 times faster than its initialreplication rate. Nowadays, molecularor viral evolution experiments similarto the one described here yieldspectacular results, and are providing us with awealth of information onevolution [13].

3TheEigen’s model

The modelthat we are going to introduce now follows the schemesuggested by Eigen and Schuster [15].W econsider asystem (a °ux reactor as depicted in¯gure 2(a) formed

by s molecularspecies: I1; I2; : : : ; Is with capacityfor self-replication.Rich energetic monomers · ¤ are introduced by valve1 and the products of moleculedegradation ( · ) are eliminatedby means ofvalve2. Let x (t) 0bethe concentration ofthe molecular i ¶ species Ii in the system.W edesignate the replicationrate of species Ii by Ai and its

degradationrate by Di.The process of moleculardegradation isa direct consequence of molecularinteractions, be they interaction of moleculeswith radiation or collisions amongst molecules,for example.W ewillhowever simplifythe book-keepingof allthese processes by assuming that moleculesdegrade ataconstant rate.

3.1 Simple replication

If we assume simplereplication and degradation in the system,we can write the kinetic reactions as: Ai Replication: · ¤ + Ii 2Ii. ° D ¡ ! Degradation: I i · . ° i ¡ ! If by valve1 we maintain · ¤ constant in this model the self-replicating units or replicators are totally independent. There are no interactions between di®erent species,and the population growth of agivenspecies does not a®ect the growth ofanother. Wewillthus havea continuous dynamicalequation for the concentrations of each species: dx i = (A D )x (t) = E x (t); i = 1; : : : ; s: (1) dt i ¡ i i i i Wherewe de¯ne E = A D as the productivity of species I .The solutions to this i i ¡ i i equation are verysimple:

xi(t) = xi(0) exp (Eit); i = 1; : : : ; s: (2) 520 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555

The productivity acts obviouslyas the exponential growth rate for the species,so that if

Ei < 0 species Ii willdisappear and if Ei > 0its population willgrow unboundedly.

In ¯gure 2(b) we seethis happening for four specieswith E1 = 0:8, E2 = 0:2, E3 = 0:1 and E = 0:5.All the speciesinitially have identical concentration x (0) = 0:25. ¡ 4 ¡ i Wecan seehow the two with positiveproductivity ( I1 and I2)grow without limit,while the two with negativeproductivity ( I3 and I4)disappear exponentially.Thus, with this modelwe obtain segregation of speciesinto two classes:only those specieswith Ei > 0 survive,while the rest disappear. This however does not correspond to selectionin the biologicalsense. If we want aselectionmechanism we need to introduce competition.

3.2 Replication and Selection

Astraightforward manner to introduce competition in our modelis to limitthe resources and/or spaceavailable. Equivalently ,we maykeep the total molecularpopulation con- stant (in the literature this constraint isreferred to as constantpopulation or CP). Al- though itisevident that inthe prebiotic world such aconstraint does not exist,we would liketo introduce aterm that mimicsnatural selectionand seewhat lessons can belearned from it.The CPcondition can expressed as:

s

xi(t) = 1; (3) Xi=1 where we haveset the constant equalto one merelyfor convenience.This constraint impliesthat there willbe no temporal variation inthe total molecularpopulation inside our reactor: s d x (t) = 0: (4) dt i hXi=1 i Physically,this can be arranged through an evacuation °ux ©( t),anon-chemicalrate of change modulated by the valve3 in picture 2(a). Introducing this constraint in our equations yields: dx i = (E ©(t))x (t); i = 1; : : : ; s (5) dt i ¡ i where we assume that the evacuation °ux of agivenspecies is proportional to its popu- lation size.If we sum allthe equations and use (3) and (4), we obtain:

s s dx i = (E ©(t))x (t) = 0; (6) dt i ¡ i Xi=1 Xi=1 and thus we havean expression for the temporal dependence ofthe °ux:

s ©(t) = E x (t) = E(t) : (7) i i h i Xi=1 The simplemeaning of this expression isthat if we want to keepthe population constant, at any instant we haveto evacuatemolecules at arate equalto the averageproductivity B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 521 at that instant E(t) .Thus, the outgoing °ux willbe equaland opposite to the rate at h i which the population would change inasystem without restrictions. The equations (5) can bewritten: dx i = (E E(t) )x (t); i = 1; : : : ; s (8) dt i ¡ h i i These equations are now non-linear because E(t) depends onallthe concentrations as h i we can seein (7). The solutions are not di±cult to compute analytically,but before we do that itwillbe useful to givesome qualitative considerations without explicitcalculus. Let us assume that initiallywe haveseveral species at equalconcentrations, with an initialmean productivity E(t = 0) .Obviously,someof the speciesinitially present will h i havea productivity below the mean.What equation 8tellsus isthat the concentrations of these moleculeswill subsequently start to decrease,while the concentrations of species with productivity abovethe mean willgrow. As timegoes on however,this process willin turn makethe mean productivity grow. But as the mean productivity grows, somespecies that initiallyhad productivities abovethe mean willfall below it,and their concentrations willstart to decrease,and so on. In this manner the averageproductivity willkeep growing, and di®erent specieswill keep disappearing, until E(t) reaches a h i valueequal to the highest productivity present in the system.At this point, onlythe speciescorresponding to this productivity willhave a non-zero concentration, and we can saythat this specieshas been selected,hence the selective equilibrium is reached. This fact isillustrated in ¯gures 3(a) and 3(b). In ¯gure 3(a) we start with four specieswith positiveproductivities and identicalinitial concentrations. Initially,the mean productivity is E(t = 0) = 0:45.W ecan seethat specieswith productivity below h i the mean productivity ( E1 = 0:1 and E2 = 0:3) immediatelystart to decreasetheir concentration. Meanwhile,the concentrations of those specieswith productivity above the initialmean ( E3 = 0:6 and E4 = 0:8) start to increasein time.As an outcome, the mean productivity increasestoo until it grows above E3 = 0:6,at which point the concentration of I3 alsostarts to decrease.Finally ,we reach astate in which allthe population ismade up from the specieswith the highest productivity, I4, with E4 = 0:8. At this point the averageproductivity isof course stabilizedat E(t > 30) = E = 0:8. h i 4 In ¯gure 3(b) we perform the sameexperiment, but with an initialcondition inwhich

I4 isnot present. Wecan seethat the selectiveequilibrium is reached when species I3 saturates the population and the mean productivity stabilizesat 0 :6.T obreak up this state at t =25 we introduce an in¯nitesimal concentration of I4,and we seehow this speciesdisplaces I from the system until anew selectiveequilibrium at E(t > 30) = 0:8 3 h i has been reached. It isnot di±cult to compute explicitlythe solutions of equations 8.First, noticethat the steady state of the k-th equation, dxk=dt =0,corresponds to either xk = 0 or to E = E, where E = E(t ) isthe asymptotic valueof E(t) .If we consider anon- k h ! 1 i h i degenerate system,that is, E = E for all i; j,onlyone speciescan haveproductivity E = i 6 j i E.Thus, for the total system,the trivial¯ xedpoints de¯ned by ­ ¤ (0; 0; :::; 0; :::; 0) 0 ² and the set ­ ¤ (0; 0; :::; x¤ = 1; :::; 0); i = 1; : : : ; s provide the s +1global steady k ² f k g 522 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555 states for asystem of s species.It would therefore seemthat coexistenceis not possible. Wecan study the evolution of the system using classicalstability analysis.W eperturb the state of the system taking it slightlyaway from the ¯xedpoint ­ ¤ (0; 0; :::; x¤ = k ² k 1; :::; 0) and study the stability of the system against the perturbation. For this we need to compute the Jacobi matrix ofthe system.Its elementsare:

@ @ s L (t) = E E(t) )x (t) = E E x (t) x (t) ij @x i ¡ h i i @x i ¡ r r i j j r=1 h³ s i h³ X ´ i Ei r=1(r=i) Erxr(t) 2Eixi(t) if i = j = ¡ 6 ¡ : (9) E x (t) if i = j ½ ¡ j Pi 6

If we haveas solution ­ ¤ (0; 0; :::; x¤ = 1; :::; 0), then x = 0 for all r = k and we obtain k ² k r 6 E E if k = i = j i ¡ k 6 Ek if k = i = j Lij (­ k¤) = 8 ¡ (10) > Ej if k = i = j <> ¡ 6 0 otherwise > or, in matrix form, :> . . E1 Ek 0 . 0 . 0 ¡ . . 0 0 E2 Ek . 0 . 0 1 . ¡ ...... 0 L(­ ¤) = B C (11) k B . . C B E1 E2 . Ek . Es C B ¡ . ¡ . . ¡ . . ¡ C B ...... 0 C B C B C B 0 0 0 0 0 Es Ek C @ ¡ A Wecan compute the characteristic polynomialthrough the determinant

L(­ ¤ ) ¶ I = 0: (12) j k ¡ j Therefore, s P (¶ ) = (E ¶ ) (E E ¶ ) = 0; (13) k ¡ r ¡ k ¡ r=1=k Y6 whence the associated eigenvalueswill be ¶ = E E for j = k and ¶ = E . r r ¡ k 6 k ¡ k The ¯xedpoint generallyis locally stable if Re(¶ ) < 0.Naturally , ¶ here isreal.

Evidently this willbe the caseif and onlyif the point xk¤ represents the concentration of aspeciessuch that E > E for all r = k.Thus, onlythe best replicator survives. k r 6 Now we haveauthentic selectionat work, but alas,no evolution.Once a selective equilibriumhas been reached the system remains inthat state inde¯nitely (indeed, that iswhy it iscalled equilibrium!). The onlyway to introduce achange seemsto be by hand, aswe did in the experiment of¯gure 3(b). The next question isthus: How can we introduce noveltyin the system in an endogenous way? The answer is:easily ,ifonlywe remember that we can learn from our mistakes... B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 523

3.3 Replication, selection and mutation

If we want evolution we need mutation, we need errors in the copying process. Suppose that the replicationaccuracy of species I is Q [0; 1].Let us use this to represent the i ii 2 probability ofproducing aperfect copy.Thus, if Qii =1everycopy produced by species I willbe identicalto the original.If Q =1however there willbe anon-zero probability, i ii 6 1 Q ,for the copying process to havean error. Inthis case,the copying of amolecule ¡ ii of type Ii willproduce amoleculeof another species,say Ij.Let us therefore de¯ne Qij

as the probability that amolecule Ii produces, as the result of an erroneous replication

process, acopybelonging to species Ij .Thus, we willhave

s 1 Q = Q : (14) ¡ ii ij j=1=i X6 The kineticreactions willnow be: AiQii Replicationwithout error: · ¤ + I 2I . ° i ¡ ! i AiQij Replicationwith error: · ¤ + Ii Ii + Ij=i. ° ¡ ! 6 D Degradation: I i · . ° i ¡ ! It isnot di±cult to incorporate the e®ect of the mutation into our equations (5):

dx i = (A Q D )x (t) + A Q x (t) ©(t)x (t); i = 1; : : : ; s (15) dt i ii ¡ i i j ij j ¡ i j=i X6 The ¯rst term ofthe equation represents the increasein the population asaresult of the

production of correct replicas,the second represents the apparition of copiesof Ii as a consequenceof defectivereplication ofother species,while the third isthe °ux term used to implement the CPconstraint. Wecan again ¯nd avaluefor the °ux by adding allthe equations and taking into account that now:

s s s A Q x (t) = A (1 Q )x (t); i = 1; : : : ; s: (16) j ij j i ¡ ii i i=1 j=i i=1 X X6 X As was to beexpected,again the mean productivity equalsthe °ux.Using this, equation (15) can berewritten in amore convenient form as

dx s i = W x (t) E(t) x (t); i = 1; : : : ; s: (17) dt ij j ¡ h i i Xi=1 The coe±cients W = A Q D are the so-called selective values , while W = A Q ii i ii ¡ i ij j ij for i = j are known as the mutationvalues .Obviously,if Q = 1 then Q = 0 for all 6 ii ij j = i and the equations revert backto system 8.In fact if Q 0we can neglectthe 6 ij º mutational term. Then the solution to our equations willcontain asinglespecies that willdominate the selectiveequilibrium, as was seenbefore. Wecallthis selectivewinner 524 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555

the mastersequence .In general,however, Qii willnot be strictly equalto 1.Then the master sequencewill always be accompanied by anumber ofmutants. The set formed by this cloud ofmutants together with the master sequenceis known asthe quasispecies [16]. Figure4 graphicallyillustrates the concept. It isnot di±cult though to solvethe equations without neglectingthe mutational term [17].In matrix form, equations 17 takethe form: dX = (W E(t) )X: (18) dt ¡ h i Tosolvethis system it su±ces to perform an orthogonal change of variablesto take W to adiagonal form. If O isa matrix such that OWO¡ = ¶ I and U = OX,equation 18 transforms to: dU = (¶ I E(t) )U; (19) dt ¡ h i or, equivalently, du i = (¶ E(t) )u (t); i = 1; : : : ; s: (20) dt i ¡ h i i These equations are formally equivalentto 8.Now, the eigenvalues ¶ i act as the productivity Ei and the ui act as concentrations. Furthermore, it iseasyto show that: s E(t) = ¶ u : (21) h i i i Xi=1 That is:the mean productivity isinvariant under the orthogonal transformation. Solution of (20) isobviously identical, step-by-step, to the preceding case.Therefore, we know that onlyone ui willbe selected(i.e., the one with the highest ¶ i).Now, however, ui does not represent asinglespecies but alinear combination ofthem, aquasi-species.Thus, a quasi-speciesis a dynamicallystable distribution ofspecies.It willbe the quasi-species, rather than the singlespecies, which willdominate the selectiveequilibrium. Weillustrate, following reference [18],the quasi-speciesconcept in¯gure 5(a), where we show the results ofasimulation for a W given by 1 0:001 0:001 0:01 0:1 2 0:01 0:01 W = 0 1 0:001 0:1 3 0:01 B C B 0:001 0:001 0:001 4 C @ A Wehavechosen selectivevalues Wii = i purely for convenienceso that I4 has the highest selectivevalue. W estart our simulation with allthe population formed by species I1.

Subsequently,species I2, I3 and I4 appear by mutation. As we can see,when allthe mutational valuesare smallthe selectiveequilibrium state isdominated by the master sequenceand the presence ofmutants isvery small.

If,however, we choose higher valuesfor the Wij,such asfor example: 1 0:001 0:001 1 0:1 2 0:01 1 W = 0 ; 1 0:001 0:1 3 1 B C B 0:001 :001 0:001 4 C @ A B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 525 the net result isagain acoexistenceof allthe species,but now the relativeconcentrations of mutants isof the sameorder as that of the master sequence.F urthermore, for these high mutational valuesthe majority of the population in the selectiveequilibrium state isin the form of mutants.

4Tierra:silicon quasispecies

The entities taking part inthe type ofdynamics described in the previous sections do not necessarilyhave to belong to the realmof chemistry.Any object ableto self-replicate, mutate, and degrade, mayindeed follow the sameevolutive equations [19].A particularly interesting exampleis provided by computer programs. If one makesa computer code susceptible to mutations, (i.e.,one puts into it someerrors) in generalthe result willbe anon working code.T. Ray[20] solved this problem, known as brittleness ,designing a closedcode in assemblylanguage made up of ¯vebit instructions. Hewrote his code in away that amutation in any of these bits ledto other instruction with sensein the code.F or example,the \codon" 11111 in Tierra means \divide"in assemblylanguage. Asinglemutation maytake this codon to 01111, the \inc"instruction. In addition, the use of templates (patterns of instructions) for addressing purposes rather than absolute addresses could avoidthe brittleness problem. Acodon’s sequenceis a creature in Tierra. Each creature has aCPU time(it has someresources). CPU timeis limited (this isthe equivalentto the CPcondition). In the previous equations we havenot taken into account the e®ect of the ¯nite size of the replicant population (for advances inthe analyticaltreatment of ¯nite population in the quasi-speciesmodel see [21] and the problem of Muller’sRatchet [22]).In general, onlya tiny fraction ofallthe possible specieswill be represented inthe population atany time.Therefore, to begin with, the population structure willstabilize around aquasi- speciesdistribution formed by someof the speciesthat were initiallypresent. Obviously though this quasi-specieswill in general not be the one with the highest productivity, and thus mutation willin timebring on amaster sequencewith a¯tness higher than the dominant one.This higher ¯tness mutant willsubsequently oust the dominant quasi- speciesand generate anew metastable quasi-specieswith higher ¶ than the initialone, and so on, until we reach the quasi-specieswith the maximum ¶ available.Note that it isan optimization process, asearch for better solutions. In this sensethe quasi-species form an exampleof genetic algorithm [23]. 8 In ¯gure 6,a typicalTierra simulation with mutation rate q = 0:5 10¡ per bit is £ shown [24].The verticalaxis represents the most common ¯tness ofthe existingcreatures, the most popular genotype. Productivity isassociated with ¯tness. Timeis represented on the horizontal axisin units of 10 9 instructions executedby the CPU. As it isshown in the plot what we get isa punctuated picture of evolution:stasis periods, epochs when amaster sequenceand its cohort of mutants dominate, are interrupted by very abrupt, discontinuous changes. Anew mutant with higher ¯tness appears inthe system destabilizing the dominant quasi-speciesand taking the system to anew metastable 526 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555 selectiveequilibrium. Classically mutations correspond to statistical °uctuations, needed as acontinuous source of evolutionary novelty.The object ofselection,the target, isthe ’wildtype’ or master sequence.In the Eigen model the quasi-speciesis what isselected. Let us brie°y comment on someimportant characteristics ofthis evolutionary paths that generallyappear in many di®erent systems [19].In Tierra, itisclear that aseparation of timescales exists: the one for stasis being much greater than the jump or nucleation that acts as relaxation time.This isreminiscent of aSelf-Organized Criticality(SOC) [25].In ¯gure 7we represent the power spectrum of the ¯gure 8.The dashed lineis a ¯ ¯t [24] to P (f) f ¡ with ­ = 2:0 0:05.As we know, this kind of noiseis typical ¹ § of SOC, but its presence isnot asu±cient condition to warrant the presence of SOC behavior [26]. In the plot 8the integrated distribution of the timesof domination for eachquasi- species, N(t), is shown 1 1 M (½ ) = N(t)dt (22) ½ Z¿ The integrated distribution isdistributed with the sameexponent as N(t)but itismore signi¯cant statistically,especiallyfor asmallnumber of measurements. The plotted results havebeen obtained from 50simulations under identicalinitial conditions [24].In all,512 times were measured. Fitting the data yields ½ M (½ ) = C½ ¡ exp ¡ : (23) T ³ ´ with ¬ = 1:1 0:05 and acut-o®parameter T = 450 50.W e¯nd stasis periods and § § power laws exponents indicativeof SOC. The extraordinary di®erence is that now we are dealing with computer creatures or molecules,rather than with livinganimals that inhabited the World, the planet Tierra.

5Modelingquasi-species in a computer:the errorcatastrophe

Up until now we havenot made any reference to the chemicalnature of our self-replicative molecules.If we takeribozimes for instance (RNAwith self-catalyticcapacity which in fact havebeen proposed as candidates for the prebiotic stages on Earth [27]),we would be speaking of chains ofmonomers. Agivenmolecule of length N can be described as a sequenceof variables( s1; s2; :::; sN ), where the si can takefour valuesin order to describe the four types of monomers that can occupythe locus i of the sequence.(Thus for instance s = 2would implythat the ¯rst monomer iscitosine C, s = 1that it isurine U, 1 ¡ 1 ¡ s1 =1guanine G,and s1 =2adenine, to choosebut one particular valueassignation). A moleculeformed by N bases can take4 N di®erent con¯gurations, and this con¯gurations willform the sequencespace. Each possible moleculewill have a productivity associated to it:its ¯tness. The geometricalrepresentation ofthe multidimensional spaceof allthe possible sequences,with their associated ¯tness represented as ’heights’ isthe so-called ¯tness landscape .An idea¯ rst introduced by SewallW right and later extended by several authors [28]. B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 527

Following [29],in order to simulate the quasi-speciesmodel we willmake a simplifying approximation of our moleculesas follows: eachmolecule will be asequenceof N = 50 monomers, and eachmonomer willbelong to onlytwo possible types, purines (either G or Aindistinctly) anpyrimidines (CorU).Thus the si willtake only two values(-1, +1), and the whole chain ismade up of asequenceof plus and minus ones (1 ; 1; 1; :::). There ¡ are then 250 possible con¯gurations inthe sequencespace. In this spacewe willde¯ ne a distance between sequences:the Hamming distance. Westart with apopulation of sequenceswhere allthe strings are 1111 :::1111, i.e. onlyplus ones,and they are the best-adapted type (that is,the one with higher ¯tness). This master sequencereplicates itself ten timesfaster than the rest, with allthe other sequencesreplicating at the same¯ xedrate, of say1 for example.The ¯tness landscape isthen verysimple: a peakwith height 10 for the master sequenceand aplatform with identicalvalue 1 for the rest of the sequences.In the literature this ¯tness landscape is known asthe singlepeak landscape. In this picture, point mutation naturally corresponds to the substitution ofa+1for a-1 (or viceversa) at locus i as the chain replicates.As the system evolvesin timethe chains replicateand point-mutate, moving away from the master copyin the spaceof sequences,but lower replication rates for most mutants willtend to keepthe copiesclose to the master sequence.W ede¯ne the error rate for the copying of individual symbols as 1 q.That is,the probability that a 1changes to 1(or a1to 1) in the copying ¡ ¡ ¡ process of everymolecule is 1 q.Thus, the probability for astring I to produce a ¡ i mutant o®spring Ij at aHamming distance d away will be

N N d d w = q ¡ (1 q) (24) ij d ¡ µ ¶ Wewilldesignate the master sequenceconcentration attime t by x0(t),the combined concentrations of allthe single-error mutants by x1(t),and in general,the combined concentration of mutants at adistance d by xd(t).In this notation the CPcondition 50 takesthe form d=0 xd(t) = 1. Westart the simulations with x (t)=1,i.e.,all the initialstrings belong to the master P 0 sequence,and letthe system evolveto the quasi-speciesequilibrium. Figure 9 represents the distribution ofequilibriumconcentrations as afunction of the error rate asit ranges from 1 q =0(error free) to 0 :06.Only in the error free casewe are ledto an all-or-none ¡ selection.As the error rate increases,the concentration ofthe master sequencedecreases rapidly.Neverthelessthe quasi-speciesremains stable, there isa consensus sequence , i.e., it ispossible to recuperate the information statisticallyabout the master sequence.This isso until we reach avalueof (1 q) = 0:046.This value,known asthe errorthreshold , ¡ marks asharp transition. Beyond it,the distribution ofchains suddenly changes and all the information about the original master sequenceis lost. The process clearlyseems to resemblea phase transition. This can beeseen more clearlyin the logarithmic plot (¯gure 9down). Belowthe error threshold we havea non-trivial distribution, the quasi-species,which as we can see, can beregarded asthe master sequencewith acometlike tail of mutants accompanying 528 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555 it,while beyond the error threshold we havea uniform distribution, arandom population. The biologicalinterpretation of this sharp dependence of the modelon the mutational rate isamazing. RNAviruses o®er anopportunity for exploring long term evolution under controlled conditions [30].Indeed, letus suppose that our chains represent RNAviruses genomes. Wecatalog new viralentities viatheir consensus sequence,or \wild type",(master se- quencein our jargon). For apopulation of viruses in general,the consensus sequence isextracted viastatistical methods (population sampling). Weknow, however,that the probability of ¯nding avirus representative of the consensus sequenceis negligible. Ac- cording to our model,this fact isindicates that viruses havea high mutational rate. But it cannot be too high avaluebecause the consensus sequencewould altogether disap- pear, and the virus for instance would looseits infectivecapabilities. The valuesof the mutational rates thus indicate that many RNAviruses ¯nd themselvesjust at the edge of the error catastrophe. What selectiveadvantage do they ¯nd craning their necksout into the abyss? Viruses are identi¯ed and eliminatedby immune systems.If the virus shows itself in many di®erent variations the immune system willbe in deep trouble to recognizeand eliminatethem all,and the virus willget to escapeits prosecutors. Viruses therefore placethemselves exactly at apoint (valueof its mutational rate) where the mutant diversityis biggest whilethe biologicalidentity isstill preserved. Foradditional information about the selectivebene¯ t ofhigher mutation rate see[31]. Now that we know this natural strategy though we can playdirty with the virus. If we push avirus alittlefurther, if we increaseits mutational rate externally,then itwill fallbeyond the error catastrophe, losing its infectivecapacity (and indeed its identity). This ideahas been successfullyapplied by E.Domingo et al.[32] to the aftosa virus in vitro.An amazing successof molecularevolutionary theory.

6Fromliving systems to magnets: the Isingconnection

It isaremarkable fact that there isa well-known system instatistical physics that seems to closelyresemble the type of dynamics that we havejust described. Indeed, a1- dimensional Isingchain of length N can be described in terms of N si’s, (s1; s2; :::; sN ), with s = 1.As the chain evolvesin time,nearby spins tend to alignin the samedirec- i § tion to achievethe leastenergetic con¯ guration, whereas thermal °uctuations willtend to makespins °ip(a closeanalogy to point mutations). In1985-6, LeuthÄausser [33]took this conceptual analogy between quasi-speciesand Isingmodel a step forward, demonstrating that the similaritybetween the two systems could be put in arigorous form. Hethus opened adoor to the possibility of applying the wellestablished framework ofequilibrium phase transitions to understand the error threshold and the existenceof quasi-species[34] (Mappings with other statistical systems havebeen proposed. For examplewith direct polymers in random media[35]). The basicidea is rather simple,as it isoften the case.Indeed, the constant population constraint does not fundamentally alter the dynamics of the system,amounting to amere B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 529

(time dependent) normalization of the concentrations. Thus we mayas welldo away with it,which iseasily accomplished by the change ofvariables

t s dt0 j=1 Ajxj(t0) yi(t) = xi(t)e 0 (25) Wewillidentify everypossible chain con¯guration with adi®erent species.Thus, for chains with ¸ sites,the matrix W willnow be of order 2 º (2º 2º).In these variables £ the equation of motion for the system simplyreads, inmatrix form, dy = Wy: (26) dt This type ofequation isof course verywell known inPhysics. W isanalogous to the negativeof the Hamiltonian in Euclidean Quantum Mechanics(possibly up to afactor). The elementsof W are allreal and greater than or equalto zero,since they represent mutation rates. In practice,although W need not be symmetric,the Wij are chosen so that allthe eigenvaluesof W are real(inclusion of an imaginary part in the eigenvalues willmerely produce damped or ampli¯ed oscillations).If the highest eigenvalue ¶ 0 is unique and isolated then the evolution operator e t willpro ject any initialpopulation into a¯nal state givenby the eigenvectorof ¶ 0 (i.e.,the ground state of the equivalent quantum theory). That is,

y(t ) = e ty(0) v ; (27) ! 1 ! where v isthe eigenvectorcorresponding to ¶ 0.Therefore, by observing the equivalence with Quantum Mechanicswe havere-derived our main,previous result. Although this conclusion could havebeen reached by mereinspection ofequation 26,the realadvantage of the analogy with Euclidean Quantum Mechanicslies in exploitingthe deep connection of the latter with Statistical Mechanics,where awealth of techniques havebeen developed in order to explorethe structure of the ground state and the equilibriumproperties of systems with many (possibly in¯nite) degrees offreedom. The fundamental tool used to describe asystem in Statistical Physics isthe partition function ¯ Z = tr e¡ (28) in which ­ isthe inversetemperature (in natural units), H the Hamiltonian ofthe system, and the trace istaken overall the possible states of the system. Z provides ameans to determine the structure of the lowest energy(analogous to the highest ¯tness) state ofthe system.In particular, aswe saw in the chapter devoted to the Isingmodel, in the caseof latticemodels with ¯nite range interactions (for examplenearest neighbors) it ispossible to write the partition function with periodic boundary conditions inaparticularly simple way in terms ofthe transfermatrix ,

Z = tr TN ; (29) in which N isa measure of the sizeof the system and T isa much simpler matrix than ¯ the original e¡ ,typicallyinvolving only the interaction ofanelementwith its nearest neighbors. 530 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555

Returning to our problem, letus think in terms of areplication process driven by discrete timesteps. Accordingly,letus assume that the population changes discretely from one generation to the next,the timeinterval needed to go from one generation to the next being ¢t. Thus y(0) isthe ¯rst generation population, y(¢t)the second, and so on. Figure10 depicts the time-ordered layout of successivegenerations starting from asinglemolecule, the up and down arrows ateachpoint identifying the type of monomer present ateachlocus of the chain. The evolution operator that advances the initialchain forward in time n generations willthen evidentlybe

e n¢t = (e ¢t)n; (30) or, (e )n; (31) by setting ¢t =1.Thus, our timeevolution operator can alsobe recast inthe form ofa simpler transfer matrix up to apower related to the system size.Note however that the involvedsize scale is now the time scale.This impliesthat we willbe concerned only about the surface properties of the system described by this transfer matrix (that is, the properties ofthe system attime t)and not about the behavior of the bulk (i.e.,the previous generations). Onecould argue that although going from eqn.(28) to (29) involvesa considerable simpli¯cation, the step from (30) to (31) does not, sinceafter allin both caseswe have to deal with e .The tricklies however in realizing(as LeuthÄausser did) that W is in fact already formally identicalto the transfer matrix for the 2-dimensional Isingmodel. Indeed, it iseasy to convinceoneself that for chains ofsize ¸ we willhave:

º d d W = A q ¡ ij (1 q) ij ; (32) ij i ¡ where Ai isthe replication rate of chain i, q isthe probability that aparticular locus of the chain iscopied correctly into the next generation and dij isthe Hamming distance between chains i and j (that is,the number ofelementsof the chain that haveto change to go from i to j.The Hamming distance however can bewritten as

º 1 d = ¸ s(i)s(j) (33) ij 2 ¡ l l ³ Xl=1 ´ (the reader can checkthis by using asimpleexample with ¸ =4for instance). Substi- tuting this into the expression for Wij and performing alittlerearrangement then yields

1 ¸ s(i)s(j) 1 q ¡ 2 l=1 l l W = A [q(1 q)]º=2 ¡ ; (34) ij i ¡ q µ ¶ or, º W = A [q(1 q)]º=2 exp K s(i)s(j) ; (35) ij i ¡ ¡ l l µ Xl=1 ¶ B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 531 where 1 1 q K = ln ¡ : (36) 2 q µ ¶ It su±ces now to compare this expression with the transfer matrix ofthe 2-d Isingmodel for rows of length ¸

º T = exp ( ­ E(¼ (i); ¼ (i))) exp ­ J ¼ (i)¼ (j) : (37) ij ¡ ¡ l l ³ Xl=1 ´ In this expression ¼ (i) and ¼ (j) labelthe con¯gurations i and j (out of the 2º possible con¯gurations) of two neighboring spin rows, E(¼ (i); ¼ (i)))isthe interaction energybe- tween spins of one row, and J isthe strength ofthe ferromagnetic coupling. Comparison of these two expressions shows that K includes both the ferromagnetic case( K < 0 with 1=2 q 1being equivalentto J < 0) and the antiferromagnetic one ( K > 0 with µ µ 0 q 1=2equivalentto J > 0. Since q isalways close to unity however,we will µ µ be dealing with the ¯rst caseonly .Also,the transition from abinary to aquaternary system alsoleads to anexactsolution, although this does not alter the basicconclusions presented here. It isof utmost physicalimportance to observe the equivalencebetween temperature and mutation rate. Indeed, 1 1 q ln ¡ : (38) T ¹ q ¯ ¯ ¯ ¯ In the thermodynamic limit,the 2-d Ising model¯ isknown¯ to gothrough aphase transition ¯ ¯ at acriticaltemperature Tc that separates anergodic phase at T > Tc,inwhich any ¯nite region of availablephase spacehas anon-vanishing probability to be explored,from a phase of broken ergodicityat T < Tc,in which the phase spaceis broken into disjoint sets.In the latter casephase spacedecomposes into aset ofdisjoint sets: ifthe system is initiallyin one of those sets,it remains there at alater time.The precisecorrespondence between expressions 35and 37,and the equivalencebetween T and q immediatelyallows us to recognizethe error catastrophe: the ergodic phase corresponds to the disappearance of the master copyand the spreading of the system overall the spaceof sequences, whereas the non-ergodic phase corresponds to the existenceof the quasi-species.The whole formulation developedto study equilibriumphase transitions, spin glasses,etc, becomesthus availableto us in order to study how evolution proceeds in more complex and realistic¯ tness landscapes [36].In [37]have given examples of error catastrophes in mutation-replication models which are not phase transitions. In [37]athe author argues that in the models for which the error catastrophe isa true phase transition, this isdue to the \singularity" of the potential being used, namely,that there isone ¯tmaster sequenceand allothers are equallyun¯ t. In his more realisticmodels there isa\smooth transition" from the master sequenceto the most un¯t sequences.The author realized that one does not need to enforce CPat all,since it isonlythe relativeproportions ofthe di®erent sequencesthat matter, and not the total number ofindividuals in the population. Thus one can use linear algebra to study quasi-species.In this way the already known 532 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555 results can be re-derived as wellas somenew ones,having to do with the appearance and timeevolution ofaquasi-speciesafter infection and the relation between the ¯tness of the sequencesand their relativeabundances. Infact, the study ofcomplexlandscapes actuallyinvolves new aspects of the quasi-speciesmodel under activeinvestigation. A concrete exampleis the dynamic ¯tness landscape: situations in which the replication or/and decayrates ofthe replicators change overtime [38]. Another isneutral evolution. It isbelieved that many mutations of amoleculeare evolutionarilyneutral in the sense that they do not alter the ¯tness to perform the function for which it has been selected [39].Despite the long history of the concept, many aspects of neutral evolution are still not wellunderstood, and there are many interesting investigations in this line[40].

7Theinformational crisis

Returning to equations (17), we know that if Qii =1there willbe no evolution.It thus seemsintuitively evident that adecreaseof Qii willincrease the noveltyin the system and the `velocity’of its evolutionary process. But aswe saw, Qii can not decreasebelow the error threshold without compromising the stability of the quasi-species.W ecan make this fact explicitby taking apolymer chain of ¸ nucleotides (a master sequence)and considering its reproduction as aneasy branchingprocesses [41]. Let q bethe copyfaith- fulness per nucleotide.Then our chain willbe copiedwith exact¯ delitywith probability qº (qº = Q ),and acopywill have at leastone \bad" nucleotide with probability (1 qº ). ii ¡ If we de¯ne the survivalprobability ofachain to hydrolysis inone generation as w, then (1 w)willbe the probability of disappearing viahydrolysis. Note that in one genera- ¡ tion amoleculewill have 0 o®springs with probability (1 w)(the moleculedegrades by ¡ hydrolysis), 1with probability w(1 qº)(the moleculesurvives but its onlyo® spring is ¡ amutation, therefore onlyone copyof itselfremains atthe end of the process), and have 2descendants with probability wqº (caseof survivaland faithful copy).The population of agiven\correct" polynucleotide evolvesfollowing abranching processes of mean:

m = 0(1 w) + 1w(1 qº ) + 2wqº = w(1 + qº ) (39) ¡ ¡ The extinction issecure if: w(1 + qº ) 1, i.e., if µ 1 w qº ¡ : (40) µ w Thus, for aninde¯nite survivalof the moleculewe must have 1 1 w ¸ < log ¡ ; (41) log q w if q 1then we can approximate log q q 1 to get º º ¡ 1 w ¸ < log : 1 q 1 w ¡ ¡ This condition imposes alimitto the size ¸ of the moleculein order to avoidextinction, that is,a limiton the amount ofinformation that can bestored inthe master sequence. B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 533

Let us study this inthe framework of our di®erential equations. Wewilltry to deter- minethe concentration xm(t)of the master sequence.If we suppose that the contribution to the master sequenceconcentration coming from mutations from other speciesis neg- ligible,then from the equation (17) we get: dx m (W E(t) )x (t): (42) dt º mm ¡ h i m

dxm Thus inthe selectiveequilibrium, dt =0,we willhave E(t ) = W : (43) h ! 1 i mm Let us de¯ne, following [42],the mean productivity of allthe mutants exceptthe master sequenceas:

s s j=m Ej(t)xj (t) j=1 Ej (t)xj(t) Em(t)xm(t) Ej=m(t) = 6 s = ¡ : (44) 6 h i j=m xj (t) 1 xm(t) P 6 P ¡ Then the master sequenceconcentration P can be expressed as

E(t) Ej=m(t) xm(t) = h i ¡ h 6 i; (45) Em(t) Ej=m(t) ¡ h 6 i and in the selectiveequilibrium ( t ) ! 1

Wmm Ej=m( ) xm¤ = ¡ h 6 1 i : (46) Em( ) Ej=m( ) 1 ¡ h 6 1 i

Weseethat, as Qii decreasesthe mutant cloud becomesmore and more important: its mean productivity Ej=i(t) grows until it reachesthe selectivevalue of the master h 6 i sequence Wii.At this moment the master sequenceis doomed to extinction.The in- formation stored in it islost. The error catastrophe has taken place.The value Qii for which the error catastrophe takesplace thus corresponds of course to the error threshold,

Qc.Equating expression 46 to zero,we get

Dm + Ej=m 1 Qc = h 6 i = (47) Am £ where we havede¯ ned £,the superiority of the master sequenceas the inverseof Qc. º Sinceas previouslystated, Qii = q ,at the threshold log £ log £ ¸ = (48) c ¡ log q º 1 q ¡ Thus, in order to increasethe length of the master sequencewithout getting into the error catastrophe either we should increasethe superiority (increasing the replication rate Am,decreasing the mean productivity of the mutants Ej=m and/or decreasing the h 6 i degradation rate Dm)orwe increasethe ¯delity q. In the next table [43]we show the maximum possible sizesof di®erent types of chains as afunction of avarietyof valuesof q and £.Together with them, the approximate 534 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555

Qualityfactor SuperiorityMaximum size Example

q £ ¸ 2 14 tRNA precursors 0:95 20 60 (¸ 80) º 200 106 2 1386 RNA ofphage Q ­ 0.9995 20 5991 (¸ 4500) º 200 10597 2 0:7 106 Escherichia coli £ 0.999995 20 3:0 106 (¸ 4 106) £ º £ 200 5:3 106 £ 2 0:7 109 Homosapiens £ 0.99999995 20 3:0 109 (¸ 3 109) £ º £ 200 5:3 109 £ Table 1 Maximumpossibles sizes of di¬erent examples of chains as a function ofa varietyof valuesof quality factor q andsuperiority £ . sizesof tRNA,RNA of the phage Q ­ ,and DNAchains ofEschericiacoli and of human beings are shown. The earlyreplicators necessarilyhave a smallsize. The examples chosen show an increasein the complexityof the replicating entities as the sizesof their corresponding chains grow. Grosso modo,one could saythat longer chains store more information. Note how if one does not want to fallprey to the error catastrophe, one should enormously increasethe qualityfactor. Ifwe assume the averagepoint mutation rate to be approximately constant for allmolecules, the onlyway to increasethe quality factor isthrough codecorrectors, enzymes.However, a DNAofextraordinary complexity (and length!) isneeded in order to be ableto reproduce and havethe enzymesat our disposal, giventheir prodigious specialties.Whence we ¯nd ourselvesat aGordian knot, achicken-eggtype of dilemma:What came¯ rst, the corrector enzymesor the DNA capable of making them? Itseemsto beimpossible to increasethe qualityfactor without long sequences,but itseemsimpossible to replicatelong sequenceswithout increasing the qualityfactor. A cul de sac for the increaseof complexityin our model? Aninsuperable crisisof information?

8Thehypercycle

In ¯gure 11(a) we seea group ofdi®erent quasi-species.As we know, in our modelquasi- speciescompete amongst them for selectiveequilibrium. Only one quasi-specieswin, and B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 535 onlyits information iskept. The information carried by the rest ofquasi-speciesis lost. However,if we want to solvethe informational crisiswe willneed the coexistenceof many quasi-species.As was proposed by M.Eigen and P.Schuster [44],cooperation can bethe way out to leavethe information crisisbehind us: mutual catalysisbetween quasi-species isa possible solution. As we shallnow see,higher order interactions between quasi- speciesmake coexistence possible. The model,know as thehypercycle [45],is formed by aset ofcatalyticreactions where somequasi-species catalyze the self-replication of other quasi-species. The hypercyclemodel applies to any catalyticchain or cyclein abroad sense(i.e., chemicalreactions, ecologicalinteractions, etc...).In ageneral form, we can write the hypercycleequation as[46] dq i = ¡ (q(t); k) ©(t)q (t); i = 1; : : : ; n (49) dt i ¡ i where q(t) = (qi(t); : : : ; qn(t))isa vectorof temporal concentrations of n quasi-species, k = (ki; : : : ; kn)isa vectorof kineticconstants and ©( t)isa °ux term to account for n the CPconstraint. Using the fact that the total concentration satis¯es i=1 qi(t) = 1;, adding up allthe equations and setting the sum ofthe derivativesequal to zerowe obtain P avaluefor the °ux: n

©(t) = ¡i(q(t); k); Xi=1 and the equations can berewritten as dq n i = ¡ (q(t); k) ¡ (q(t); k) q (t) i = 1; : : : ; n: (50) dt i ¡ j i ³Xj=1 ´ The typicalansatz for ¡i is: n n n

¡i = ki + kijqj (t) + kijlqj(t)ql(t) + : : : (51) Xj=1 Xj=1 Xl=1 where the contributions of order zero,¯ rst, second, etc.appear explicitly. If we couplethe quasi-speciesas in ¯gure 11(b) in acyclicalcatalytic chains, we obtain:

¡1(t) = k1q1(t)

¡i(t) = kiqi(t)qi 1(t); i = 2; : : : ; n (52) ¡ The net e®ect of this type of coupling isthat the growth ofeachquasi-species is promoted by its predecessor in the chain.It can be shown [46]that this coupling usually does not result in the coexistenceof allthe quasi-speciesin the chain (and indeed it isnot di±cult to convinceoneself that this must bethe case).This isalso the casefor linealchains with branches. The next simplestructure that we can study isa cyclicchain (this the hypercycle proper), that is,a structure likethe one depicted in ¯gure 11(c). Mathematically,

¡1(t) = k1q1(t)qn(t)

¡i(t) = kiqi(t)qi 1(t); i = 2; : : : ; n (53) ¡ 536 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555

If any ki isequal to zerothe hypercyclebecomes an acyclicalchain, and we go back to the previous case(52). The cyclicalchain that we havejust introduced isthe so- calledelemental hypercycle. Contrary to the acyclicalchains above,in this casethe coexistenceof speciesis possible sincethere isno net benefactor in the cycle:all the elementscontribute and they allbene¯ t. Let us consider somedynamical properties ofelementalhypercycles. The basicequa- tions are: n dq1 = k1q1(t)qn(t) q1(t) k1q1(t)qn(t) + krqr(t)qr 1(t)) dt ¡ ¡ r=2 ³ Xn ´ dqi = kiqi(t)qi 1(t) qi(t) k1q1(t)qn(t) + krqr(t)qr 1(t)) ; i = 2; : : : ; n (54) dt ¡ ¡ ¡ ³ Xr=2 ´ In ¯gure 12 we show the numerical solutions of these equations. The left hand side ¯gures represent relativeconcentrations ofthe quasi-speciesin time,while the ¯gures on the right represent the phase portrait for q1(t) and q2(t).For allthese caseswe have taken allthe coupling constants to be ki = 1:0.In (a) and (b) we havethe equations 54 with n =2and initialconditions q(t = 0) = (0:99; 0:01). As we seethe concentrations of the two quasi-speciestend to an equilibrium value.This isdepicted in the q (t) q (t) 1 ¡ 2 plane as aattractive focus. In (c) and (d) we take n =3with initialconditions q(t = 0) = (0:98; 0:01; 0:01). We observe damped oscillationsthat quicklytend to afocus equilibriumin aspiral way. In (e) and (f),the hypercycleis formed by n =4quasi-species.W ecan seethat after atransient state with damping, inthe ¯nal state the concentrations oscillateperiodically (as we run through the cycle).In the q (t) q (t)plane we can seethis as aspiral 1 ¡ 2 approximation to alimitcycle. In ¯gure 13(a) and 12(b) we take n =5and initialconditions q(t =0) =(0.96,0.01, 0.01,0.01, 0.01). Weseehow the dynamics isdominated again by periodic oscillations, while in the q (t) q (t)¯gure we seethat the limitingcycle is reached rapidly. 1 ¡ 2 For values n 5the existenceof alimitingcycle seems to hold, although this cannot ¶ be shown analytically.In (c) and (d) we take n =10 with initialconditions q(t = 0) = (0:91; 0:01; :::; 0:01) and again we obtain cyclicbehavior. As the ¯gures show, for n = 2 and n =3the concentrations tend to aconstant value, whereas for n 4in generalwe willhave oscillations and coexistenceis always possible ¶ for allthe quasi-species(something which isnot alwaysthe casefor n < 4). In (e) and (f) we take n =3with initialconditions q(t = 0) = (0:98; 0:01; 0:01) and asymmetric coupling constants k = (1; 1; 0:1). Weseethat the results are similarto the ones depicted above. As the reader mayimagine the elementalhypercycles are the simplest system that can avoidthe information crisis.The catalyticterms can bemore sophisticated and the order of the ansatz for ¡can however be higher. These combinatorial possibilitiesopen up a true bestiarium of systems in which dynamicalbehaviors of alltypes appear, including chaos. B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 537

The hypercyclesolution to the information crisis,however, has two main handi- caps [18].The ¯rst one isthat selectionof hypercyclesoperates ina`onceand for all’basis. If amutant hypercyclewith selectiveadvantage appears it willnot displacethe presently dominating cycle,in contrast to the quasi-speciesmodel (remember Tierra for example). Whythis di®erence between the metastability ofhypercyclesand quasi-species?With no CPconditions the quasi-speciesgrows exponentially,thus the population becomesin¯ nite in in¯nite time.But it iseasy to see[46] that for elementalhypercycles without CPthe increaseof the population ishyperbolic.Thus an in¯nite population isreached in a¯nite time.This fact isa hard blow for the model.The pricefor eliminatingthe limitedsize constraint imposed by the error threshold isthe incapacityfor evolution!The second handicap isthe instability ofhypercyclesagainst parasites. Aparasite in the model isa branch in the elementalhypercycle. It isaquasi-speciesthat bene¯ts from catalysisbut which does not contribute to the cycle.It seemsthat the hypercycledoes not support parasites, itisrendered unstable by them. Avarietyof solutions havebeen proposed to overcomethese problems. Wecan have evolution within the hypercycleif the hypercycleitself acts asacorefor growth, incorpo- rating more and more quasi-species.But, the most interesting prospects maylie in the incorporation of spatial dependence to the model.Preliminary results show that, in this way,stability of hypercyclesagainst parasites can beachieved[47]. However, the spatially explicitmodel is stable against parasites onlyif the number of coupled quasi-speciesis greater than 4.But the spatiallyexplicit system isshown to beresistant against onlythe so-called\sel¯ sh parasites", whereas the \short-cut" parasites can invadeit, decreasing the number ofcoexistingquasi-species in the hypercycle.The error threshold was avery serious problem for the Eigen model,which justi¯ed the introduction of the hypercycle concept, because at the timethe general understanding was that onlyproteins can be- haveas acatalyst.A protein synthesis requires cooperation of roughly 100 genes,and the minimalorganism with catalysiswas considered quitecomplicated. The current viewis, that RNAmoleculescan store information and playthe roleof enzymes,so the minimal organism requires onlya singlegene, which isable to reproduce itself.In this sense,the hypothesis ofRNAcatalysts reformulates the problem completely.See[48] and references therein, for an up-to-date discussion. Anyway,at this moment this isan activearea of research [49]. However,the hypercycleis not the onlypossible resolution of the error-catastrophe problem, there are numerous alternative hypotheses. First of allthere are other exper- imentallytested mechanisms (the product inhibition) which could maintain the coexis- tenceof di®erent replicators [50],and non-perfect mixingcan havethe samee® ect in open chaotic° ows [51].The other possibility isthat co-evolutionof replicator ¯delity and template function can leadto the overcomingerror catastrophe [48,52]. 538 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555

9A¯nalremark

In 1994 E.SchrÄodinger wrote alittlebook called:\What IsLife?" [53]This essayposed basicand precisequestions: How isthe information to build and maintain an organism stored, read and executedat amolecularlevel? How isit copiedand transmitted from generation to generation? These questions determined research in the ¯eldin the next decades.But subtly SchrÄodinger laidaside questions di±cult to answer in the near future. Tobespeci¯c, he did not speakabout metabolism. Without reallysaying it,he replaced the complexproblem of the origin ofLifeby the more simplequestion ofhow information isreplicated. As mentioned before inthe second paragraph ofthis work: \we can rephrase the old question about the origin of lifeas: What isthe origin ofthe information stored in livingsystems, and why isit stable?" Sixtyyears later, metabolism isstillaside so we ignore how thousands of moleculesare organized within acell.T oday we ¯nd ourselves in the post-genomic era having sequencedseveral organisms, including ahuman being. Metabolism isour next big theoretical problem to understand the origin of life.

Acknowledgments

Wewould liketo thank Antonio Ferrera, Fernando Ballesteros,Jordi Bascompte, Aida Agea,Ricardo Garc¶³a-Pelayoand two anonymous referees for their valuablecontributions and opinions. This work has been supported by CICYT BFM2002-01812.

References

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Fig. 1 Experimentof evolution in vitro of phago Q- ­ .Atube collectionwith Q- ­ replicasa alongwith monomerslike A TP,GTP,UTP orCTP isassembled. W ebegininserting phago Q- ­ inthe ­rsttube. After afew hourswe transferan aliquot of the ­rsttube intoa secondtube. And repeatsuccessively to order of seventeentwice. The resultis a RNA virusthat has a length ofonly17% its original one, and was reproducing at a rate15 times faster. 544 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555

I 1, I 2, ... I s

valve 3

I 1

I s m*

valve 1 m

I 2 valve 2

(a)

1

E=0.8

0.75

E=0.2

x(t) 0.5

0.25 E=­ 0.1 E=­ 0.5

0 0 2 4 6 8 10 time

(b)

Fig. 2 (a):Schematic ® ux reactorfor molecular evolution. The self-replicatingmolecules are representedas ball chains. Rich energetic monomers are introduced by valve1 andresidual productsare eliminated by valve2. W ecanmaintain the molecularpopulation constant by valve3. (b): Segregation process between survivaland extinct molecular species. In the model without restrictionsonly the moleculeswith positiveproductivity survive andgrow without limit. B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 545

1

E=0.8 0.75

x(t) 0.5

E=0.6 0.25

E=0.3 E=0.1 0 0 10 20 30 time

(a)

1

E=0.6 0.75

x(t) 0.5

0.25 E=0.8 E=0.3

E=0.1 0 010 20 30 40 50 60 70 80 90 100 time

(b)

Fig. 3 (a):Simulation for four specieswith constantpopulation conditions. The bestreplicator dominatesthe system.The selectiveequilibrium is reached in a few steps.(b): W erepeat the samenumerical experiment as in ­ gure(a) but without the bestreplicator. The selective equilibriumis reached for the molecularspecies with productivity E = 0:6,the \­ttest"of the threeinitial species. At time t =25we introducean in­ nitesimal amount of a specieswith E = 0:8.A new processof selectiontakes place and a new selectiveequilibrium is reached. 546 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555

Fig. 4 Schematicrepresentation of aquasi-species.The gridrepresents the spaceof sequences. The verticallines show the ­tnessor relativeconcentration of the correspondingsequence in the quasi-speciesdistribution. The mastersequence in this case is the mostrepresented. The cloud aroundthe mastersequence in the sequencespace indicates the regionof the sequencespace occupiedby the quasi-species. B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 547

1

4

0.75 1

x(t) 0.5

0.25 3

2

0 0 5 10 15 time

(a)

1

0.75 1

x(t) 0.5

4 3 0.25 2

0 0 5 10 15 time

(b)

Fig. 5 (a):Simulation of the quasi-speciesmodel with four di¬erent species. The mutational values Wij aresmall in this case. Thus the systembehaves as in the previousmodel without mutationand CP constraint:when the selectiveequilibrium is reached, the mastersequence dominatesthe population.The mutantcloud is minimally represented. (b): W erepeatthe simulationof ­ gure5(a) but now the mutationalmatrix has non-negligible elements. W esee now thatthe relativeconcentrations of the di¬erent species (master sequence and mutants) are ofequalorder. 548 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555

8 Fig. 6 Pathevolution of asimulationin Tierra with amutationalrate of q = 0:5 10¡ per bit. The timein horizontal axis is in units of 10 9 executed instructions.Shown onthe £ verticalaxis isthe mostcommon ­ tnessof the existingcreatures, the mostpopular genotype (F rom[24]). B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 549

Fig. 7 Powerspectrum of ­ gure6 (From[24]). 550 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555

Fig. 8 Integrateddistribution of dominanttimes N(t)ofcuasispecies.(From[24]) B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 551

Fig. 9 Above:v ariationof relativepopulations xd (equilibriumconcentrations) as afunction of the errorrate 1 q.The errorthreshold 1 q = 0:046is marked with averticaldashed line. Beyond itthe distribution¡ of relative populations ¡ becomes a randomdistribution. This isthe errorcatastrophe. Below: The samegraph but with alogarithmscale in the verticalaxis to clearlyappreciate the sharpdrop of the mastersequence to zero concentration (F rom[29]). 552 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555

I1 ......

I8 ......

......

time Ij ......

Ii ......

......

...... I4 ...... I9 1 2 3 4 5 6 7 8 n-1 n nucleotide number

Fig. 10 Successivereproductions in time of ainitialspin chain I9 subject tospin ® ips(muta- tions). B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 553

Q 3 Q Q Q n 1 1 ...

i k1 k Q i Q i ... k3 Q Q 2 Q n 2 Q 3 k2

(a) (b)

kn Q Q n 1 ...

i k1 k Q i ... k3 Q 2 Q 3 k2

(c)

Fig. 11 (a):A setof independent quasi-species.(b): An acycliccoupled quasi-species by mutualcatalysis. (c): A cyclicchain of coupledquasi-species by mutualcatalysis, more aptly ,a hypercycle. 554 B.Luque /CentralEuropean Journal of Physics3 (2003)516{555

1 1

q 1 n=2 0.75 0.75 n=2

q(t) 0.5 0.5 q 1

0.25 0.25

q 2

0 0 0 5 10 15 20 0 0.25 0.5 0.75 1 time q 2

(a) (b)

1 1

q1 0.75 n=3 0.75 n=3

q(t) 0.5 q 0.5 2

0.25 0.25 q 2

q3 0 0 0510 15 20 25 30 0 0.25 0.5 0.75 1 time q 1

(c) (d)

1 1

q 1 0.75 0.75 n=4 n=4 q 2

q(t) 0.5 0.5 q2

0.25 0.25

0 0 0 100 200 300 0 0.25 0.5 0.75 1 time q 1

(e) (f)

Fig. 12 Dynamicalbehavior of the elementalhypercycles. Concentrations in timeof ahypercycle with: (a): n =2quasi-species,(c): n = 3 and (d): n =4.Phase portrait, plane q1(t) q2(t), forhypercycle with: (b): n =2,a directfocus, (d): n =3afocusin spiral and (f ): n¡ = 4, a limitcycle. B.Luque /CentralEuropean Journal of Physics 3 (2003)516{555 555

1 1

n=5 q1 0.75 0.75 n=5

q2

q(t) 0.5 q 0.5 2

0.25 0.25

0 0 0 100 200 300 0 0.25 0.5 0.75 1 time q 1

(a) (b)

1 1

q 1 0.75 0.75 n=10 n=10

q(t) 0.5 q 0.5 2

q 2 0.25 0.25

0 0 0100 200 300 400 500 600 700 800 900 1000 0 0.25 0.5 0.75 1 time q 1

(c) (d)

1 1

q n=4 3 0.75 0.75

q(t) 0.5 q 0.5 2 n=4 q 1 0.25 0.25

q q 4 2

0 0 0100 200 300 400 500 0 0.25 0.5 0.75 1 time q 1

(e) (f)

Fig. 13 The samenumerical experiments as in ­ gure12 but with n =5in(a), (b) and n = 10 in(c), (d). In both casesthe behavioris the limitcycle. In the lastcase (e), (f )we represent ahypercyclewith n =4but now with non-symmetricconstants coupled. Again the behavioris the limitcyclic.