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Error propagation in the hypercycle

P. R. A. Campos and J. F. Fontanari Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo Caixa Postal 369, 13560-970 S˜ao Carlos SP, Brazil

P. F. Stadler Institut f¨ur Theorestische Chemie, Universit¨at Wien W¨ahringerstraße 17, A-1090 Wien, Austria The Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe NM 87501, USA

some pathological, discontinuous replication landscapes We study analytically the steady-state regime of a net- [4], the coexistence problem seems to be more pervasive, work of n error-prone self-replicating templates forming an as it is associated to the form of the growth functions in asymmetric hypercycle and its error tail. We show that the the chemical kinetics equations [5,6]. existence of a master template with a higher non-catalyzed In order to circumvent the aforementioned limitations self-replicative productivity, a, than the error tail ensures the of the quasispecies model, Eigen and Schuster proposed stability of chains in which mribozymes) [8,9]. Interestingly, though the error ≤ threshold phenomenon has traditionally been considered p the main motivation for the proposal of the hypercycle PACS: 87.10.+e, 87.23.Kg (see [10], for instance), most of the seminal works in this field have dealt with the coexistence issue only, as they assume perfect replication accuracy for the hypercycle I. INTRODUCTION elements [7,11]. In this case an arbitrary number of tem- plates permanently coexist in a dynamical equilibrium The limitation of the length of a genome by the repli- state; if n>4, however, the template concentrations cation accuracy per nucleotide (q)hasledtoadeadlock vary with time [7], periodically decreasing to very small in the theories of the origin of life based on the evolution values. In practice, large hypercycles are therefore sus- of competing self-replicating polynucleotides. According ceptible to extinction via fluctuations, see e.g. [12], hence to Eigen’s quasispecies model [1,2], which may serve as the information gain due to the coexistence of different a paradigm here, polynucleotides have to replicate with templates in the hypercycle may not be very impressive high accuracy in order to reach a certain length, a re- after all. Furthermore, we will argue in this paper that quirement that is impossible to fulfill without the aid coexistence in the absence of a stable equilibrium can of specialized catalysts. However, to build those cata- also be achieved by a simpler arrangement, namely, the lysts a blueprint is necessary that amounts to a large free chains, in which the cyclic order of the catalysts is genome (the nucleotide sequence), which itself cannot interrupted. be maintained without the catalysts. In particular, for The effect of error-prone replication (mutation) in the polynucleotides of fixed length L, the quasispecies model hypercyclic organization was investigated by introducing predicts the existence of a minimal replication accuracy a mutation field as a perturbation of the error-free ki- L per genome Qc = qc , below which the genetic informa- netic equations [13]. This approach, however, is not very tion is irreversibly lost. This information crisis has been appropriate to study the error threshold phenomenon, termed error threshold transition. Above Qc the popula- since the results obtained cannot be easily compared with tion is composed of a master copy together with a cloud of those of the quasispecies model. In this sense, a better structurally similar mutants (quasispecies) [1,2]. Equally approach is to assume the existence of a special class of important is the finding that, except in a trivially degen- templates with no catalytic activity, so-called error-tail, erate case, two or more quasispecies cannot coexist [3], that appear as a consequence of the replication errors of thus precluding the coexistence of templates (i.e. polynu- the hypercycle elements [14–16]. However, the particu- cleotides) sufficiently different from each other to code for lar catalytic network investigated extensively within that any useful set of catalysts. Although it has been claimed framework was not the hypercycle, except for a short dis- that the information crisis is not really a fundamental cussion in [17], but the fully connected network in which issue, since the error threshold transition appears only in each element helps the replication of all other elements

1 of the network [15,16]. (Clearly, in the case of n = 2 ele- ments, these two networks become identical [14].) Such a network is more robust than the hypercycle since the mal- I 1 functioning or extinction of one of its elements does not compromise the whole network. Nevertheless, besides its aesthetic appeal, the cyclic coupling of the hypercycle seems to be more realistic [9]. The goal of this paper is to investigate analytically the steady-states of a deterministic system comprised of two I e parts, namely, a hypercycle made up of n self-replicating templates I1,I2,...,In and its error tail Ie. These parts are coupled such that any erroneous copy of the hyper- cycle elements will belong to the error tail. The focus of the present analysis is on the location in the parameters space of the model (i.e. replication accuracy per tem- plate, non-catalyzed and catalyzed productivity values, and hypercycle size) of the regions of stability of the di- verse possibilities of coexistence between the templates composing the hypercycle. In particular, we give empha- I 3 I2 sis to the characterization of the critical parameters at which the hypercycle becomes unstable against the error FIG. 1. The system composed of a hypercycle of size tail. n = 3 and its error tail Ie. The thin arrows represent The remainder of the paper is organized as follows. In the non-catalyzed self-replication reactions and the thick ar- Sec. II we present the chemical kinetics equations that rows represent the self-replication catalytically assisted by the govern the time evolution of the system and motivate neighbor template the specific choice of the parameters used throughout the paper. The fixed-points of the kinetic equations are ob- tained analytically in Sec. III and their stability discussed where x0 xn and 0≡ 0.5 1 in Sec. IV. The phase-diagrams showing the regions of n stability of the diverse coexistence states are presented Φ= xi (Ai + Kixi 1)+Aexe (3) − and analyzed in Sec. V. Finally, some concluding re- i=1 marks are presented in Sec. VI. X is a dilution flux that keeps the total concentration con- n stant, i.e., i=1 x˙ i +˙xe = 0. As usual, the dot denotes a II. THE MODEL time derivative. Henceforth we will assume that [15,16] P n We consider a system composed of a hypercycle made xi + xe =1. (4) up of elements and its error tail , as il- n I1,...,In Ie i=1 lustrated in Fig. 1. In contrast to the so-called ele- X mentary hypercycle [7], we assume that the templates Clearly, this formulation is equivalent to considering are capable of self-replication with productivity values polynucleotides of length L whose replication accu- →∞ Ai (i =1,...,n)andAe. Moreover, as usual, the growth racy per nucleotide q goes to 1 such that the replication promotion of template I as a result of the catalysis from accuracy per genome is finite, i.e. qL Q.Inthis i → template Ii 1 is measured by the kinetic constants Ki. limit, the back-mutations from the error-tail elements to The key ingredient− in the modeling is that in both pro- the templates that compose the hypercycle, as well as cesses of growth of template Ii the probability of success the mutations between those templates, can be safely ne- is given by the parameter Q [0, 1], so that an erro- glected. Hence, mutations can only increase the concen- neous copy, which will then belong∈ to the error tail, is tration of the elements in the error-tail. The advantage produced with probability 1 Q. Hence the concentra- of working in this limit is that the error threshold transi- − tions xi (i =1,...,n) of the hypercycle elements and the tion can be precisely located by determining the value of concentration xe of the error-tail evolve in time according Q at which the concentration of a relevant template van- to the kinetic equations ishes. For finite L, as well as for finite population sizes, the characterization of this transition is more involved, x˙ i = xi (AiQ + Kixi 1Q Φ) i =1, ..., n (1) − − being achieved through the use of finite-size scaling tech- and niques [18]. n In this work we consider the single-sharp-peak replica- tion landscape [1,2], in which we ascribe the productiv- x˙ e = xe (Ae Φ) + (1 Q) xi (Ai + Kixi 1)(2) − − − i=1 ity value A1 = a>1 to the so-called master template X

2 I1 and Ai = Ae = 1 to the n 1 other elements of the 0, i.e., xi = 0. So, there is no fixed point corresponding hypercycle as well as to the error-tail.− Also, for the sake to such a chain. Any chain of survivors therefore must of simplicity we set Ki = K for all i. The motivation start with template n or 1. In the first case we get Φ = Q for this particular choice of parameters is the observa- fromx ˙ n =0andaQ + KQxn = Q fromx ˙ 1 = 0, yielding tion that the emergence of the hypercycle requires both xn =(1 a)/K < 0, which rules out this possibility. spatial and temporal coexistence of the templates form- The equilibria− of interest for our study thus are either ing the network, and this can be achieved by a quasis- the interior equilibrium in which all templates survive, pecies distribution, which guarantees the coexistence of or a fixed point that corresponds to a chain of survivors the master template and its close mutants, despite the beginning with I1. purely competitive character of the quasispecies model Accordingly, we define a m-coexistence state as the n- [19]. Once the coexistence is established, the appear- component template vector x =(x1,x2,...,xn)inwhich ance of catalytic couplings between the templates is not the first m components are strictly positive and the rest a very unlike event. Of course, as soon as those cooper- are equal to zero. Clearly, given the template vector x ative couplings become sufficiently strong to balance the the concentration of the error tail xe is determined by the competition imposed by the constant concentration con- constraint (4). In the following we solve analytically the straint, the mutants will certainly depart from the master kinetic equations in the steady-state regimex ˙ i =0 i. template due to the relentless pressure of the mutations, The simplest fixed point is the zero-coexistence state∀ so that no trace will remain of the original quasispecies (m = 0) which corresponds to the solution x1 = ... = distribution. xn =0andxe = 1, existing for the complete range of parameter values. In the case of chains, i.e. 0 x˙ 1 = 0 which then yields 0 and extinct templates x = 0. A survivor I is said i j a 1 isolated if xj 1 = xj+1 = 0. Hence, x1 = x2 = ...=xm 1 = − . (12) − − K x˙ = x (Q Φ) j>1(5) j j − Next we insert this result in Eq. (3) to obtain and Qa 1 a 1 x = − (m 1) − . (13) m a 1 − − K − x˙ e = xe(1 Φ)+(1 Q) x1 (a 1)+1+K xixi 1 . − −  − −  However, since xi (0, 1) i, this solution is physically i=j,j+1 ∈ ∀ 6 X meaningful in the region K>a 1andQ>Qm where  (6) − 1 1 Q = + (m 1) (a 1)2 . (14) In the steady-state regime, Eq. (5) yields Φ = Q which, m a Ka − − for Q<1, is incompatible with Eq. (6), since the term within brackets in this equation is positive. Therefore, all We note that the 1-coexistence state (quasispecies) is isolated survivors with the exception of the master tem- obtained by setting m = 1 in Eq. (13) and its region plate are unstable against the error tail. Next consider of existence is simply Q>1/a, since the other condi- the following chain of surviving templates: tion, namely K>a 1, is derived by considering the other templates in the− chain. In fact, this very simple result quantifies nicely the notion that the cooperative x˙ i = xi(Q Φ) (7) − couplings must reach a certain minimum strength so as to balance the competition between templates. x˙ i+1 = xi+1(Q + KQxi Φ) (8) The analysis of the hypercycle, i.e. m = n, is a little − more involved. ¿Fromx ˙ =0wegetΦ Q=KQx 2 − 1 which, inserted in the equationsx ˙ 3 = ... =˙xn =0, x˙ i+2 = xi+2(Q + KQxi+1 Φ) (9) yields x1 = x2 = ...=xn 1 and − − a 1 . x = x − . (15) . (10) n 1 − K

Finally, using these results in Eq. (3) we find that x1 is x˙ k = xk(Q + KQxk 1 Φ) (11) given by the roots of the quadratic equation − − which does not contain x . Again, in the steady-state nKx2 (KQ + a 1) x +1 Q=0. (16) 1 1 − − 1 − regime the first equation yields Φ = Q, implying KQxi =

3 2 For K<(a 1) /4n, this equation has real roots for all IV. STABILITY ANALYSIS Q 0, otherwise− it has real roots for Q Q where Q ≥ ≥ h h is the unique positive root of the equation In order to perform a standard linear stability analysis of the fixed points obtained in the previous section, it is K2Q2 +2K(a 1+2n)Q +(a 1)2 4nK =0. h − h − − convenient to rewrite the kinetic equations (1) and (2) as (17) follows

x˙ i = xiFi (x) i =1,...,n (19) In particular, for large K we find Qh 2 n/K.Further- more, it can be easily seen from Eqns.≈ (15) and (16) that p where xn vanishes at Q = Qn with Qn given in Eq. (14). To un- derstand the role of Qh and Qn (wenotethatQn Qh) ≥ Fi(x)=AiQ+KQxi 1 Ae xj (Aj Ae+Kxj 1) in delimiting the region of existence of the n-coexistence − − − − − j state we must look at the behavior of the two real roots X + (20) of Eq. (16). Let us denote them by x1 and x1− with + x1 x1−, which, according to Eq. (15), correspond to + ≥ and we have used the constraint (4) to eliminate .The x and x−, respectively. Of course, these roots become xe n n stability of a fixed point is ensured provided that the identical at Q = Qh and so the two solutions for xn will real parts of all eigenvalues of the n n Jacobian are vanish simultaneously only at the value of K = Kh at negative. In our case the elements× of the JacobianJ are which Qh equals Qn. Explicitly, we obtain given by

Kh =(a 1) [n (a +1) 1] . (18) − − ∂Fi Jij = δij Fi + xi i, j =1,...,n. (21) by inserting Eq. (14) into Eq. (17). Although both roots ∂xj + x1 and x1− are in the simplex (0, 1),thisisnotsofor + The evaluation of the eigenvalues is simple only for the x and x−. In particular, for K However, while x becomes positive for Q>Qn (it 11 ii n 1. Therefore this steady state− becomes unstable− for vanishes at Qn), xn− remains always negative. Since 2 1 , which coincides with the lowest replication K > (a 1) /4n the same conclusion holds in the Q> /a h − accuracy required for the existence of the 1-coexistence range K<(a 1)2 /4n as well, provided we define state. However, for a general m-coexistence state we have = 0 in this− region. The situation is reversed for Qh to resort to a numerical evaluation of the Jacobian eigen- : both concentrations are positive within the K>Kh values. range Q QQn xn alternative way to look at the stability of the fixed points, always positive. Despite the small region in the parame- as hinted by the stability analysis of the zero-coexistence ters space where the root yields concentrations inside x1− state, which becomes unstable due to the emergence of the simplex, the linear stability analysis discussed in the the 1-coexistence state. In fact, it can be easily seen sequel indicates that this solution is always unstable, so that any perturbation of the m-coexistence fixed point we only need to consider the root x+. Thus the range of 1 which makes the concentration x non-zero will be existence of the hypercycle fixed point m = n is Q Q m+1 n amplified if A Q + KQx Φ is positive. For m>0 if K K and Q Q if K>K . ≥ m+1 m h h h we use Φ = aQ and A = 1− together with the value of In≤ models without≥ error-tail, i.e., pure replicator equa- m+1 x given in Eq. (13) to obtain the following (necessary) tions a much stronger statement on coexistence is possi- m condition for the stability of the m-coexistence state, ble. The “time average theorem” [20] states that if there is a trajectory along which a certain subset of templates Q0, (22) J survives, then there is a fixed point with exactly the

J-coordinates non-zero. While we have not been able to with Qm given in Eq. (14). Hence the maximum value prove the “time average theorem” in full generality for of Q allowed for the stability of the m-coexistence state Eqns. (1) and (2), it is easily verified for free chains. coincides with the minimum Q required for the existence Hence, if there is no m-coexistence equilibrium, then of the (m + 1)-coexistence state. Interestingly, though there is no trajectory at all along which the templates for m =0wehaveΦ=1,A1 =aand x0 = 0, condition I1 through Im survive. (22) holds true in this case too. Hitherto we have determined the ranges of the pa- At this point two caveats are in order. First, the entire rameter Q where the m-coexistence states are physically argument leading to the stability condition (22) is flawed meaningful, in the sense that xi (0, 1) i. The next if the ( + 1)-coexistence state happens to be unstable. ∈ ∀ m step is to find the regions where these states are locally Therefore, we must guarantee via the numerical evalu- stable. ation of the Jacobian eigenvalues that the l-coexistence

4 state is stable before using that condition to study the 0.3 stability of chains with m

(iii) for n 6, it is always unstable. (1)−(4) ≥ (3)−(4) Second, the derivation of the stability condition (22) is Q (3) (2)−(4) based on the analysis of a single eigenvalue of the Jaco- (2) bian and so it does not yield a sufficient condition for (0)−(4) the stability of the fixed points. Nevertheless, we have (1) verified through the numerical evaluation of all n eigen- 0.1 values that, provided the (m+1)-coexistence state is sta- ble, the eigenvalue associated to fluctuations leading to an increase of the chain length is the first one to become positive. (0)

V. DISCUSSION 0 0K 1000 2000 h Kc Combining the existence and the stability results de- K rived in the previous sections we can draw the phase- FIG. 2. Phase-diagram in the space (K, Q)forn=4and diagrams in the plane (K, Q)forfixedaand n. In par- a= 10 showing the regions of stability of the diverse coexis- ticular, for n 4them-coexistence state is stable within tence states. The numbers between parentheses indicate the the interval ≤ number of coexisting templates. Regions of bistability ap- pear for K>Kh. The thin lines are (from bottom to top) Qm a 1. Interestingly, for fixed Q, Eq. (23) shows that the incre-− which Qh equals 1/a, the minimal replication accuracy ment δK in the catalytic coupling needed to incorporate of the quasispecies. It is given by a new template into the chain is K = a (a 1) 2n 1+ n(n 1) . (25) c − − − 2 (a 1) h p i δK = − , (24) Beyond this value the error threshold of the hypercycle aQ 1 Q is smaller than that of the quasispecies. Moreover, as − h mentioned before, for large K, it vanishes like 1/√K.A regardless the number of elements in the chain. The case rather frustrating aspect of K and K is that both are of 1, for which no chains with 1 are allowed, h c K order a2, indicating then that the productivity of catalyt- does not− require any special consideration. In fact, we ically assisted self-replication is much larger than that find that the only stable states are the zero-coexistence of non-catalyzed self-replication. While this is obviously (Q<1/a) and the 1-coexistence (1/a Q 1) states. ≤ ≤ true for biochemical catalysis, it is difficult to argue for However, since Q2 1onlyforK a 1, this result is the existence of such efficient catalysts in prebiotic con- consistent with Eq.≤ (23). ≥ − ditions. On the other hand, we can take a different, more The n-coexistence state (i.e., the hypercycle solution) optimistic viewpoint and argue that modern biochemical is stable for Q>Q if K K and for Q>Q,oth- n ≤ h h catalysts (enzymes) are so efficient because their precur- erwise, where Kh and Qh are given by Eqns. (18) and sors had to satisfy the stringent conditions imposed by (17), respectively. We define the error threshold of the surpassing K . hypercycle as the value of the replication accuracy that h Q In Fig. 3 we present the phase-diagram for n =5.The delimits the region of stability of the -coexistence state. n main difference from the previous figure is that the 5- The phase diagram for a =10andn=4showninFig.2 coexistence state is stable only within the thin region be- illustrates the major role played by K in the hypercyclic h tween Q and the dashed curve, obtained through the nu- organization: only for the hypercycle becomes 5 K>Kh merical evaluation of the Jacobian eigenvalues. As these more stable than a chain of the same length. Another curves intersect at some K K , the 5-membered hy- important quantity is the value of , denoted by ,at h K Kc percycle is not very interesting,≤ since it has the same

5 1 0.5

Ι1 (a) Ι2

Ι3 0.4 Ι4

Ι5

0.8 complex dynamics 0.3

xi

0.2

0.6 0.1

(5) Q 0 01020

0.8 Ι 0.4 1 Ι2 (b) Ι (4) 3 Ι4 (3) Ι 0.6 5 (2)

0.2 (1) xi 0.4

(0) 0.2

0 0 100 200 300 K 0 01020 FIG. 3. Same as Fig. 2 but for n = 5. There are no stable time fixed points above the dashed curve. The solid curves are FIG. 4. Time evolution of the five concentrations of the (from bottom to top) Q1,Q2,Q3,Q4, and Q5. templates composing a hypercycle of size n =5fora= 10, Q =1,and(a)K= 37 (inside the region of stability) and (b) K = 40 (outside the region of stability). The initial state is characteristics of a chain of length m =5.Toconfirm xi(0) = 0.2 i. this result we have carried out the numerical integra- ∀ tion of the kinetic equations using the ninetieth-order Runge-Kutta method. The results are shown in Fig. 4, in Fig. 3 for the 5-coexistence state does not appear which illustrates the time evolution of the concentrations in the symmetric case a =1,soitmustbeaconse- xi (i =1,...,5) inside and outside the region of stability. quence of the asymmetry in the productivity values of Although the behavior pattern in the region of instabil- the non-catalyzed self-replication reaction. We note that, ity seems periodic, we have not explored completely the differently from the asymmetric case (a>1), the zero- space of parameters to discard the existence of chaotic coexistence state is always stable. behavior. Hence we use the term complex dynamics to For the sake of completeness, we present some results label this region in Fig. 3. We note that the phase- on the elementary hypercycle (Ai =0,i=1,...,n)cou- diagram shown in this figure describes also the regions pled to an error tail (Ae = 1) via the imperfect catalyti- of stability of hypercycles and chains of size n 5, since cally assisted self-replication. Inserting these parameters ≥ m-coexistence states with m>5 are always unstable. into Eq. (20) and setting Fi =0 iyields x1 = ... =xn ∀ An interesting limiting case which deserves special at- with x1 given by the larger root of the quadratic equation tention is the symmetric hypercycle (a = 1). According Knx2 (n + KQ)x +1=0, (27) to the argument put forward in the beginning of Sec. III, 1 − 1 the only fixed points in this case are the zero-coexistence state and the hypercycle, i.e., chains are not allowed. since we have verified that the smaller root is always un- stable. As in the symmetric case discussed above, for Moreover, Eq. (15) yields x = x = ... = x where x 1 2 n 1 n 4 the stability condition coincides with the condi- is given by Eq. (16) with a replaced by 1. The analysis ≤ of the roots of that quadratic equation and the numeri- tion for real x1,namely, cal evaluation of the Jacobian eigenvalues yield that the n n symmetric hypercycle is stable for Q 2 . (28) ≥ K−K r 4n K> (1 Q) , (26) Thus the term in the right hand side of this inequality Q2 − yields the error threshold of the elementary hypercycle. provided that n 4. The region of stability observed ≤

6 the hypercycle (Qh) becomes smaller than that of the quasispecies (Q =1/a) for catalytic couplings (K)of Ι1 1 Ι 2 2 order of a ,whereais the productivity value of the mas- Ι3 0.6 ter template. In particular, Qh vanishes like n/K for Ι4 Ι 5 large K. Perhaps, even more important is our finding that the asymmetry in the non-catalyzed self-replicationp reaction (a>1) entails the existence of chains of size n 5. We note that these chains are unstable in the symmetric≤ hypercycle as well as in the fully connected network [15,16]. 0.4 Adding to the scenario for the emergence of the hyper- xi cycle described in Sec. II, which starts with an isolated quasispecies, our results indicate that the chains may well be the next step in this complex evolutionary process. In fact, according to Eq. (24) the strengths of the catalytic couplings needed to form a chain are of order a, while the 0.2 hypercycle only acquires its desirable stability character- istics for strengths of order a2 (see Eq. (18)). Although we realize that an evolutionary step leading from chains to hypercycles is still a major one, it is certainly much more plausible than a direct transition from quasispecies to hypercycle. In any event, we think that the emergence of the hypercycle can be explained as a series of plausible 0 smooth transitions, without need to postulating the hy- 01020 time percycle as an unique event in prebiotic evolution. In this FIG. 5. Time evolution of the five concentrations of the vein, this work represents a modest first step to tackle templates composing a free chain of size n = 5. The parame- this fundamental problem within a firm basis. ters and initial state are the same as for Fig. 4(b). To conclude we must mention an alternative resolution for the information crisis in prebiotic evolution which has received some attention recently [21], namely, the Before concluding this section, we must note that the stochastic corrector model [22]. This model builds on chains of size m considered hitherto are bonded in a hy- ideas of the classical group selection theory for the evo- percycle of size n>m. We could study free chains of lution of altruism [23], since it considers replicative tem- size n as well by simply setting x0 = 0 in the kinetic plates competing inside replicative compartments, whose equations (1) and (2). Not surprisingly, the results are selective values depend on their template composition. essentially the same as for bonded chains, with Qm play- However, the chemical kinetics equations governing the ing a similar fundamental role in delimiting the regions of dynamics of the templates inside the compartments dis- stability of the shorter chains (m4 are always unstable. Moreover, as il- must be relaxed, or at least justified, before it can be con- lustrated in Fig. 5, the oscillatory behavior pattern of the sidered an important alternative to the more traditional template concentrations, which ensues a dynamic coexis- approach based on the hypercycle and its variants. tence among all templates in the chain, is similar to that observed in the hypercycle (compare with Fig. 4). In this P.R.A.C. thanks Prof. P. Schuster and Prof. P. F. sense, the free chains seem as good as the hypercycle to Stadler for their kind hospitality at the Institut f¨ur The- attain that kind of coexistence. However, we must em- orestische Chemie, where part of his work was done, and phasize that, for sufficiently large K (i.e. K>Kh), the Prof. P. E. Phillipson for illuminating discussions. The fixed points describing the coexistence of n 4 templates work of J. F. F. was supported in part by Conselho ≤ in a hypercycle are much more robust against replication Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico errors than their counterparts in the free chains. (CNPq). The work of P.F.S. was supported in part by the Austrian Fonds zur F¨orderung der Wissenschaftlichen Forschung, Proj. No. 13093-GEN. P.R.A.C. is supported VI. CONCLUSION by FAPESP.

Our study of the steady states of an asymmetric hy- percycle composed of n error-prone self-replicating tem- plates indicates that, for n 4, the error threshold of ≤

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8 I 1

I e

I 3 I2

0 0.5 1 0.3

(4) 0.2

(1)−(4)

(3)−(4) Q (3) (2)−(4)

(2) (0)−(4) (1)

0.1

(0)

0 0K 1000 2000 h Kc K 1

0.8 complex dynamics

0.6

(5) Q

0.4 (4) (3)

(2)

0.2 (1)

(0)

0 0 100 200 300 K 0.5

Ι1 (a) Ι2

Ι3 0.4 Ι4

Ι5

0.3 xi

0.2

0.1

0 01020

0.8

Ι1 Ι (b) 2 Ι3

Ι4 Ι 0.6 5

xi 0.4

0.2

0 01020 time Ι1

Ι2 Ι 0.6 3 Ι4

Ι5

0.4

xi

0.2

0 01020 time