Group selection of early replicators and the origin of life

J. theor. Biol. (1987) 128, 463-486

Group Selection of Early Replicators and the Origin of Life E6RS SZATH M,/~RY

Department of Plant Taxonomy and Ecology, Roland E6tv6s University, Budapest, Kun Bdla t~r 2., H-1083, Hungary AND

L,~SZLO DEMETER

Department of Biochemistry and Food Technology, Budapest University of Technology, Budapest, M~egyetem rakpart 3., H-1521, Hungary (Received 6 February 1987, and in revised form 8 May 1987) A major problem of the origin of life has been that of information integration. As Eigen (197 l) has shown, a mutant distribution of RNAs replicating without the aid of a replicase cannot integrate sufficient information for the functioning of a higher-level unit utilizing several types of encoded enzymes. He proposed the hypercycle model to bridge this gap in prebiology. It can be shown by a nonlinear game model, incorporating mutation of a hypercycle, that the selection properties of hypercycles make them inefficient information integrators as they cannot compete favourably with all kinds of less efficient information carriers or mutationally coupled hypercycles. The stochastic corrector model is presented as an alternative resolution of Eigen's paradox. It assumes that replicative templates are competing within replicative compartments, whose selective values depend on the internal template composition via a catalytic aid in replication and "metabolism". The dynamics of template replication are analyzed by numerical simulation of master equations. Due to the stochasticity in replication and compartment fission the best compartment types recur. An Eigen equation at the compartment level is set up and calculated. Even selfish template mutants cannot destroy the system though they make it less efficient. The genetic information of templates is evaluated at both levels, and the higher (compartment) level successfully constrains the lower (template) one. Compartmentation together with stochastic effects is sufficient to integrate information dispersed in competitive replicators. Compartment selection is considered to be group selection of replicators. Implications for the origin of life are discussed. I. Introduction

Our main goal in this p a p e r is to suggest a sufficient alternative to hypercycles as information integrators. In his seminal p a p e r Eigen (1971) has demonstrated that a distribution (quasispecies) o f macromolecular ( R N A ) sequences consisting o f a master copy and its mutants is incapable of ensuring coexistence o f templates sufficiently different from each other to code for a set of proteins of an early replicating machinery (of. Swetina & Schuster, 1982). Thus it seems that selection of the quasispecies leads evolution into a dead end. A higher level system o f t e m p l a t e s functionally linked to each other is needed in order to integrate the necessary a m o u n t o f information according to the criteria set by Eigen & Schuster (1978a, p. 26): 463

0022-5193187/200463 + 24 $03.00]0

© 1987 Academic Press Ltd.

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(1) Selective stability of each component due to favorable competition with error copies. (2) Cooperative behaviour of the components integraled into the new functional unit. (3) Favorable competition of the functional unit with other less efficient systems. Eigen (1971) proposed the hypercycle as an information integrator (a term introduced by Kiippers, 1983) and Eigen & Schuster (1977, 1978a) thought to have demonstrated that it indeed does satisfy these criteria. However, in a critical paper Bresch et al. (1980) argued that, since any one member of the hypercycle has triple functionality (target of replicase, messenger, and adaptor), of which only the target function is selective, the hypercycle is seriously threatened by parasites or cheaters (in a sociobiological sense) which do not do anything for the good of the hypercycle. In their reply Eigen et aL (1980) postulated that target function optimizes quickly enough to get the hypercycle started and it is only after this that the hypercycle must somehow be compartmentalized to allow for further evolution, e.g., the selection for better replicases or adaptor function. They also argued that a simple "package" of genes cannot solve the information integration problem. In the "package" model template replication rates were assumed to be equal (Bresch et aL, 1980; Niesert et aL, 1981), which is a pseudo-solution to the problem. Eigen et aL (1980, p. 410) have emphasized: "Compartments may assist--but do not replace--hypercycles". In a paper dealing explicitly with the cost of protein production Michod (1983) demonstrated that for the successful nucleation of a hypercycle some kind of a population structure providing at least partial localisation is compulsory. Therefore, the quoted statement needs at least some modification stating that "compartments" must assist hypercycles; i.e. hypercycles need compartments. As Wicken (1985, p. 549) in a recent critique writes: "Although one can represent a hypercycle schematically, it possesses no physical principle of closure to fix identity as a unit of competition in nature." A next logical question is whether compartments need hypercycles. Our answer is no. This result has been anticipated and demonstrated in another context by one of us (Szathm~iry, 1984, 1985, 1986a, b). In section 2 we clarify the problems raised in association with hypercycle evolution by a sort of minimum model. This is followed by section 3 where a mathematical model, potentially an alternative to hypercycles, is set up and some qualitative features are outlined. In section 4 we present some simulation results indicating that the expectations were justified. In section 5 the relation of our model to other ones is outlined. Finally, in section 6 we comment on some related aspects of the origin of life.

2. Involution of Hypercycles Two kinds of hypercycle mutations have been distinguished (Eigen et al., 1981; Eigen & Schuster, 1982). "Phenotypic" mutations affect the target function of RNAs (whether they are good substrates of replicases) and "genotypic'" mutations influence the efficiency of the encoded replicase proteins (Fig. 1). This terminology is unfortu-

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FIG. 1. Mutations in an elementary hypercycle. Replicators I~ and 12 constitute the original hypercycle, I' is a mutant. Arrows indicate catalytic activities, their widths the catalytic strengths. (a) "Phenotypic" mutation. I ' favours itself by its increased target efficiency so it can be selected for. (b) "Genotypic" mutation. I' acts as a better replicase but this benefit is experienced by 12 as well. The mutant cannot be selected for.

nate since both types of mutations arise in the sequence, i.e. the genotype, of the RNA molecules. Phenotypic mutants can be selected for but genotypic mutants cannot. As discussed above, the selection o f phenotypic mutants was thought to optimize target function quickly. Yet a problem, not addressed so far, arises here, because phenotypic and genotypic mutations affects one a n d the s a m e process; i.e., enzyme-catalyzed replication o f RNA molecules. Optimization of the target function assumes an environment to which this function has to be optimized. In the hypercycle, this environment is the replicases themselves, whose sequences are allowed to vary in a more or less random manner since as genotypic traits they cannot be selected for! A better clarification of this point is possible by the analysis of a model in which two alternative hypercycles share a c o m m o n member and the other members arise through mutations (Fig. 2). In this scheme I0 and I~ make up the original hypercycle and 12 is a mutant of I~ forming an alternative hypercycle with Io. Replicases are

i.. . . . . . . . . . . .! FtG. 2. Replications and mutations in two coupled hypercycles, k-s are replication rate constants, p.-s are mutation rate constants. Dashed lines indicate the process o f mutation.

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not explicitly considered so we have elementary hypercycles (Eigen & Schuster, 1978a). The corresponding dynamic equations are formulated as follows: X~o = Xo( ko, x , + ko2x 2 - ~ ) x', = XoX2tX, + x , ( k , x o - ~

( 1) )

(2)

x ; = XoX,tZ 2 + x2( k2xo - ~ )

(3)

where xi (i = 1,2, 3) stand for the concentrations o f the templates with c o r r e s p o n d i n g indices; primes indicate time derivation. The k0-s are replication rate constants and p.,-s indicate mutation rates, qb serves as a constraint keeping total c o n c e n t r a t i o n constant in time (of. Eigen, 1971): dp = XoXt(koK + k~ + # 2 ) + XoX2(ko2 + k2 + t z , )

(4)

xo + x t + x2 = 1.

(5)

Note that the mutation terms are second o r d e r in c o n s e q u e n c e o f the s e c o n d order replication processes; mutants arise from unfaithful replications. C o n d i t i o n (5) does not violate generality as the growth functions in (1-3) are h o m o g e n e o u s (Eigen & Schuster, 1978a). System (1-5) without mutations has been analyzed by H o f b a u e r et al. (1980). They have f o u n d that except for the degenerate case k~ = k2 the system is competitive and the template with better target function (k; = k,,~.,) o u t c o m p e t e s the other, so it is the p h e n o t y p i c trait which matters. In order to get an insight into the d y n a m i c s o f the mutative system we sought for fixed-points and determined stability properties by local stability analysis (cf. Stucki, 1978). The system is c o m p l i c a t e d e n o u g h to make calculation o f s o m e o f its fixed-point coordinates and the associated stabilities c u m b e r s o m e by hand. The formal processing software R E D U C E was applied to obtain them (Table 1). The qualitative results are depicted in Fig. 3. Without back TABLE 1 Coordinates

and stability properties of the fixed-points

of equations

(1-5), / f / ~ 2 > 0,

/~ = 0 Coordinates Number

( Xo, xt , x 2)

Eigenvalues

Type centre centre

1.

O, O, 1

O~ > O, O~= 0

2.

O, 1, 0

O~> O, 02 = 0

3.

1 - x2, O, k2/(k2 + k~2 )

if k, > k i 01 < O, 0 2 < 0 ifk2
node

O~> O, 02< 0 G > O, O, > 0

saddle source

O~ < O, 02 < 0

node

4. 5.

t, O,0 xo = l - ( x l + x 2) xt = ( kl - ( k2 + ko2)X2)/ ( kl +#2 + k.a )

x, = tz2kl/ (tz2ko2 - ( k I + kol )k 2 + kt( kl + I-t2+km )) exists only if k I > k 2

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(c]

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FIG. 3. Fixed-points of the system in Fig. 2. (a) k~ > k2, /z~ =g2=0; cf. Hofbauer el aL (1980); competition. (b) k I > k2, /.t~= 0, P-2> 0; coexistence. (c) k~ <- k> g~ = 0, g2 > 0; competition. (d) Fixedpoints on the boundaries of the simplex with #~, #2 > 0. The system is noncompetitive. (e) A numerical example for the degenerate case k~ = k2 =/z~ = P-2= 0.1, ko~ = ko.~= 1. I. Assumption of equal replication and mutation rate constants is not plausible, unless copying fidelity is very low (template length very high). In certain cases the nature of the fixed-points depend on the kinetic rate constants; for these cases every possibility is shown. Symbols: • node, © source, 0) saddle, (D ~ centres; see Eigen & Schuster (1978a).

m u t a t i o n ( / ~ t - - 0 ) it is a g a i n the r e l a t i o n b e t w e e n kl a n d k2 that d e t e r m i n e s the o u t c o m e . It can be seen that the h y p e r c y c l e with b e t t e r target f u n c t i o n c a n n o t get rid o f its m u t a n t . T h e s y s t e m is definitely n o n c o m p e t i t i v e if b a c k m u t a t i o n s a r e allowed. H e r e the p r o b l e m e m e r g e s that the m e a n i n g o f the term "inefficient s y s t e m " in the i n f o r m a t i o n i n t e g r a t i o n criteria (section 1) is o b s c u r e . H o w is efficiency m e a s u r e d ? O n e c o u l d t h i n k that the h y p e r c y c l e m a x i m i z e s total p r o d u c t i v i t y , r e a c h ing a c e r t a i n ~max. AS it c a n e a s i l y be d e m o n s t r a t e d for a t w o - m e m b e r e d h y p e r c y c l e this e x p e c t a t i o n is not fulfilled. A t h e o r e m in e v o l u t i o n a r y g a m e t h e o r y ( M a y n a r d Smith, 1982) p r o v i d e s us with a g e n e r a l answer. T h e h y p e r c y c l e e q u a t i o n a n d the l i n e a r e v o l u t i o n a r y g a m e e q u a t i o n are j u s t s p e c i a l cases o f a m o r e g e n e r a l r e p l i c a t o r e q u a t i o n ( S c h u s t e r & S i g m u n d , 1983). S t a b l e e q u i l i b r i a o f e l e m e n t a r y h y p e r c y c l e s with less t h a n 4 m e m b e r s c o r r e s p o n d to e v o l u t i o n a r i l y stable s t r a t e g i e s , ESSs ( I s h i d a , 1984). P o h l e y & T h o m a s (1983) e m p h a size that l i n e a r a s y m m e t r i c g a m e s are g e n e r a l l y n o t m a x i m a l in the a b o v e s u g g e s t e d m a n n e r (they do not r e a c h qbma.0. A n e l e m e n t a r y h y p e r c y c l e with u n e q u a l rate c o n s t a n t s (kol # kl in o u r case) c o r r e s p o n d s to an a s y m m e t r i c l i n e a r e v o l u t i o n a r y g a m e , a n d is t h e r e f o r e n o t m a x i m a l (cf. Eigen & S c h u s t e r , 1978a). T h e s i t u a t i o n is no b e t t e r for o u r m u t a t i v e system either. It is a n a l o g o u s to a n o n l i n e a r g a m e with

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frequency-dependent payoffs:

Io

Ii

12

Io

0

ko,

ko2

I,

kl

0

tZtXo/ Xl

12

k2

I.t2xo/ x2

O.

(6)

Nonlinear evolutionary games are not always maximal (Pohley & Thomas, 1983) and, as the fixed-point analysis shows it, (6) is not maximal either. Let us consider a hypercycle in which back mutations are neglected. How can a possible evolution of successive mutants be visualized? Using the results in Fig. 3 this is depicted in Fig. 4(a). The sole criterion of selection is the increase in target efficiency o f the mutant, while the replicase efficiency may arbitrarily decrease. Indeed, a parasite with ko2 = 0 can outcompete the hypercycle (Eigen & Schuster, 1978a). At least it can be said that the target function develops despite the fact that total productivity may approach zero (without the hypercycle the parasite dies). The scenario in Fig. 4(a) is special in that the new mutant template always arises as a mutant of the former mutant. Another possibility, shown in Fig. 4(b), is that alternative target functions are affected by successive mutations. As a general trend every rate constant (including target function) is allowed to decrease. There is absolutely no selective difference between these possible scenarios; survival of the hypercycle is determined by chance mutations alone. We emphasize that the above case is not general but it need not be; its mere possibility demonstrates how easily the hypercycle can be deteriorating. We conclude that the hypercycle violates the (1) and (3) criteria of the integration of information set up by Eigen & Schuster (1978a) since none of its members has to

(o)

125( i2" "

(

(b) FIG. 4. (a) Evolution of -phenotypic- mutants. One catalytic activity increases considerably. (b) Involution of a hypercycle. Both catalytic activities decrease. Back mutations are neglected.

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be selectively stable if its error copies are able to form an alternative hypercycle with the rest of the original cycle and, consequently, the hypercycle is unable to compete favorably with all systems less efficient in either total productivity or even target efficiency o f its members. Through a series of mutations, target function can decrease as well. A compartmentalized hypercycle could be an efficient information integrator (cf. Maynard Smith, 1979; Feistel et al., 1980; Michod, 1983), but do compartments have to enclose hypercycles?

3. The Stochastic Corrector Model

We consider compartments in which templates replicate, degrade, and contribute to the metabolic function of the whole compartment. As a result of their catalytic aid the whole compartment undergoes fission into two. The plausibility of such a system is discussed in section 6. An essential feature of the system is that templates are allowed to compete against each other. We consider two templates, X and Y whose concentrations x and y, respectively, change according to the following equations x' = ax(xy)~/4 _ dx - x ( x + y ) / K.

(7)

y ' = by(xy) ' / 4 - dy - y ( x + y ) / K.

(8)

a and b are replication rate constants; d is the (aspecific) degradation rate constant and K influences the degree of competition. Competition is expressed by a > b.

(9)

The power 1/4 makes the nonlinearity of growth decrease for calculation purposes. The deterministic fate of the system is unequivocal: both templates die out. Starting from sufficiently low concentrations both x and y increase, with x increasing faster. At a certain point y starts to decrease because of increasing competition, x is still growing, however, until the decrease of y makes the first term in (7) small enough. It is easy to prove that given condition (9) the point (0, 0) is globally asymptotically stable (see Appendix A). The equations express, by the multiplicative growth term, that templates compete against each other, but neither can replicate alone (at zero concentration of the other). Note that the mutual coupling is unspecific; we do not re-smuggle hypercylic organization (Appendix B). This can be best illustrated by a minimum model of Eigen & Schuster (1978b, p. 342) describing primitive translation, in which the translation "complex" (Ft~) is made up by four enzymes encoded by competitive templates. The effect of Ft~ is aspecific. It results that "the couplings present are not sufficient to guarantee a mutual stabilization of the different genotypic constituents L. The general replicase function exerted by E0 and the general translation function Ft~ are represented in all differential equations by the same term. The equations then reduce to those for uncoupled competitors, multiplied by a common time function f ( t ) . The system, which initially functions quite well, is predestined to deteriorate, owing to internal competition." Eigen & Schuster (1978b) have

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considered this as an example of non-hypercyclic, coupled systems; and our system is in this regard analogous to theirs. Another equation describes the production and spontaneous degradation of a "metabolic substrate" S whose concentration s is used to measure the system's preparedness to undergo fission by an unspecified mechanism (but see section 6). The dynamics of s reads s ' = s( ( c , x + c2y)(xy)'/2)'/'-/ ( M + s + ( x + y ) 2 / iz ) - as.

(10)

Here c, and c_, express the specific catalytic contribution of X and Y to the growth of s; M is an inhibition constant (of a generalized Michaelis-Menten type); tz is another constant influencing inhibition; and 8 is a degradation constant. (10) assumes that catalysis by X and Y involve additive as well as multiplicative effects; i.e., X or Y alone cannot catalyze the production of S. On the other hand, too many templates inhibit metabolism. (Plausible effects include energetic load, increasing viscosity, etc.) The compartment is unable to divide before reaching a certain smax starting from s , , a J 2 . As described later, the time necessary for this increase will be the compartment's "generation time". Changes in compartment volume are neglected. (9-10) can nicely be displayed as surfaces. Figure 5 illustrates (10) at certain values of the parameters. All edges of the surface are below zero indicating degradation and inhibition. Figure 6 illustrates the dynamic (8).

FIG. 5. The metabolic surface according to eqn (10) for system I. s = M = 2 2 , c~ = c, =3500. The maximum point has a value of 1477.7.

tz =0.23, fi=20,

The dynamic (7-8, 10) is deterministic. Given the fact that we are dealing with macromolecules inside small compartments it should be replaced by stochastic equations. We have chosen one-step master equations (cf. Gardiner, 1983). Although not exactly expressing chemical reality of bimolecular reactions, one-step master equations have already been applied to problems of macromolecular competition

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FIG, 6, T h e r e p l i c a t i o n surface a c c o r d i n g to eqn (7) for s y s t e m I. a = 90, K = 0.1, d = 0,05. M a x i m u m : 1023-1.

(Jones & Leung, 1981). We describe the time evolution o f the probabilities P(n, m) of the different numbers n and m of X and Y, respectively

P(n, m ) ' = a ( n - 1 ) ( ( n - 1)m)l/4P(n- 1, m) + ( n + 1)(d + ( n + 1 + m ) / K ) P ( n + 1, m)

-(d(n+m)+(n+m)2/K + (an + bm)(nm)l/4)p(n, m) + b ( m - 1 ) ( n ( m - 1))l/4P(n, m - 1) + ( m + 1 ) ( d + ( n + m+ 1)/K)P(n, m + 1).

(11)

The boundary n =0, m = 0 acts as a natural absorbing one (cf. Gardiner, 1983). Analytical solutions to (8-11) are impossible to obtain. The numerical simulation had the following steps. (1) We integrated (8-10) by the fourth-order Runge-Kutta algorithm. Compartments are distinguished by their initial template concentrations. The time elapsed from the initial state with s = Smax/2 till s = Sm~x equals the generation time T~ of the i-th compartment. From T~ the amplification factor Ai (Eigen, 1971) can be calculated A, = In (2)/T~.

(12)

It is obvious that Ai = 0 whenever a compartment never reaches s . . . . (2) The stochastic dynamic (11) was also calculated by Runge-Kutta. The difficulty here is that in principle one is faced with an infinite number of states from n = 0 , m = 0 to infinity. The problem has been circumvented by introducing an artificial reflecting boundary (cf. Gardiner, 1983) sufficiently distant from the origin. This approach can be justified a posteriori (section 4). The simulation time for the different compartment types was always T~ long. At time T~ one obtains a probability

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distribution corresponding to the state just before fission takes place. As the volume of the compartment is assumed to be constant, concentrations are always simply proportional to the number of templates within the compartment. Dimensions are taken to be such that x - - n and y = rn numerically. O f course, entire consistency would require the stochastic analogue to (10) be calculated as well. This would make the computation task formidable; in fact intractable. For each compartment type one would obtain a continuous distribution of generation times (representatives of the same compartment type would have various generation times). This distribution is replaced by 7",.. As it will become clear this does not mean a serious restriction in the qualitative sense. (3) Having the probability distribution P;(n, m) for each compartment type i at time T, the distribution of the templates during fission can be calculated. Distribution between offspring compartments is taken to be completely random and is expressed by the hypergeometric distribution (sampling without replacement) giving the probability rr(n', m'; n, m) of drawing n' and m' pieces of templates out of a compartment ("urn") containing exactly n and m pieces, respectively, of these templates; provided the total number of templates drawn equals n ' + m' rr(n', m'; n, m ) = (" ,,.)(" ,,,.)/( ...... ,,..... .)

(13)

where the brackets enclose the binomial coefficients. (Through eqns (13-17) primes do not indicate time derivatives.) In order to obtain Prob~ (n', m'; n, m), the probability of having n' and m' organelles in an offspring compartment and n and m in the parent compartment (originating from an /-type compartment) at time ~ , (13) has to be multiplied by the probabilities rr'(n'+ m'; n + m) of drawing exactly ( n ' + m') templates and by P,(n, m) at time T,;

Probi(n',m';n,m)=rr(n',m';n,m)Tr'(n'+m';n+m)Pi(n,m)

(14)

where ~"(n'+m';n+m)=(

..... ,,..... .)/~,~.( ...... ~,)

(15)

and the brackets contain again the binomial coefficients. In the following calculations all terms less than a certain probability (10 -s or 10 -9) have been neglected. From (14) the so-called quality factor (Q~), expressing the probability of exact reproduction of a compartment (cf. Eigen, 1971), and the mutation rates wo can be calculated

Q~=Ek.X-rProb~(k,l;k',l'); w,j=AjEk-~,r. Probi(k,l;k',l');

k'>-k,l'>-I

(16)

i#j,k'>-k,l'>-I

(17)

where k, l correspond to the initial template composition of an /-type compartment (as already stated compartments are distinguished by their initial template compositions). (16) and (17) contain P, and Pj at times T~ and Tj, respectively. Summation is terminated whenever the probability decreases below the given limit. Of course, (13) implies that during fission one offspring compartment inherits (n', m') templates and another (n - n', m - m') templates; necessarily with the same probability. But, according to our rules, they qualify as representatives of different compartment types, and are dealt with separately in the following. Both offspring compartments are considered in the whole system.

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(4) From the Ai, Qi, and w0 values thus obtained an appropriate Eigen equation (Eigen, 1971; Eigen & Schuster, 1977; Kiippers, 1983) is constructed X'i = (A,Q,- E)X, +XjwijXj

(18)

where Xi is the density of the ith compartment type (with prime for time derivation) and E = Z~(AiQ~X, + 2jw~jXj)

(19)

the overall excess productivity (Eigen, 1971), is incorporated so as to ensure that XiX~ = 1

(20)

thus the selection constraint of constant total concentration (cf. Eigen, 1971) is realized. System (18-20) is different from the usual form in two respects; firstly death rates at the compartment level are not incorporated and, secondly, nonviable compartment types are not counted with. These differences do not, however, influence the conclusions obtainable by the analysis of this system. While being continuously produced by viable compartments nonviable compartments (never reaching Smax) cannot produce any offspring either correctly or incorrectly, hence they are unable to mold the relative frequencies within the viable sub-population. Their omission is legitimate. Equations (18-20) can be written in the usual form X'= ( W - E ) X

(21)

where X is the vector of X~- s; W is the fitness matrix with entries w~j (w, = A~Q~); and E is a diagonal matrix with aspecific entries E. Analytical solutions to (21) are known (Thompson & McBride, 1974; Jones et al., 1976; Ebeling & Feistel, 1982; Bessho & Kuroda, 1983). A dominant quasispecies (Eigen & Schuster, 1977) emerges associated with a positive (dominant) eigenvector meaning coexistence of all types; the only condition being the irreducibility of W. This is guaranteed by the fact that the off-diagonal elements are necessarily greater than zero in our case since each cell type can "mutate" to all others. Originally the quasispecies was defined for macromolecular populations. Here we face a quasispecies of compartments with internal competition. The macromolecular quasispecies is not a satisfactory information integrator, because sequences sufficiently different from each other are generally represented in such small concentrations that individual sequences show up only in a stochastic manner (Swetina & Schuster, 1982), i.e., the system is underoccupied (Ebeling & Sonntag, 1986). In a set of sequences necessary for the functioning of a moderately complicated system (e.g., a hypercycle) this random presence has fatal consequences. Templates coding for different functions can be expected to have fairly different sequences. Hence they cannot be closely related members of the dominant sub-population. Loss or non-appearance of any important sequence tends to have deleterious qualitative consequences. On the contrary, absence of any type from the quasispecies of compartments makes only a quantitative difference. (Numbers of templates are quantitative traits.)

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4. Simulated Behaviour of the Stochastic Corrector As we have shown, in the deterministic case o u r templates w o u l d b e h a v e competitively and involution o f internal template c o m p o s i t i o n o f c o m p a r t m e n t s would b e c o m e worse and worse. Stochasticity is able to correct this malignant trend: hence we use the term " c o r r e c t o r " to indicate this fact. In Figs 5 and 6 we have already shown the " m e t a b o l i c " and template replication surfaces in case o f a concrete numerical e x a m p l e (System I). The distribution Pi(n, m; T~) is s h o w n for one c o m p a r t m e n t (Fig. 7). It is pleasing to see that probability has not a c c u m u l a t e d at the artificially introduced reflecting b o u n d a r y . This indicates that the choice o f its position is good. An e x a m p l e shows the distribution o f mutation rates for one c o m p a r t m e n t type (Fig. 8). It is a p p a r e n t that the d o m i n a n t quasispecies (Fig. 9) involves its m e m b e r s in a p p r o x i m a t e p r o p o r t i o n to the AiQi fitness values (Fig. 10). A n o t h e r system's ( I I ) stationary quasispecies is displayed in Fig. 11. The only difference between this System I is that c~ was increased from 1 to 1-2. The effect is remarkable. There are more viable cell types and the " a v e r a g e " fitness ( E )

0

30

FIG. 7. Simulation result of the master equation ( 11)just before fission for compartment with P( 1, 2) = 1 at t = 0 (system I, b = 45). Because of the short generation time the distribution is still narrow. Maximum = 0.19.

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n

F1G. 8. Fitness o f a n d m u t a t i o n s rates to the c o m p a r t m e n t with P(3, 1 ) = 1 M a x i m u m = 2.08.

I¸

at 1 = 0

{system I).

3 t7

FIG. 9. D o m i n a n t q u a s i s p e c i e s d i s t r i b u t i o n ( s y s t e m I) as the s t a t i o n a r y n u m e r i c a l s o l u t i o n to e q n (21). M a x i m u m = 0,42.

increased from 13-81 to 15.22 ( E is known to approach the largest dominant eigenvalue; Kiippers, 1983). This demonstrates clearly that the stochastic corrector can, as expected, utilize "'genotypic'" mutations in the sense that templates catalyzing metabolism better can give rise to fitter quasispecies.

A test for the fate of a selfish template (a complete parasite) has also been carried out. Instead of eqns (6-8, 10) one has now x' = axx ~/2 _ dx - x ( x + y ) / K

(22)

y ' = byx I / 2 - d y - y ( x + y ) / K

(23)

s' = c s x / ( M + s + ( x + y)2/Iz ) - 8s

(24)

where c is a constant and the condition b > a makes Y an efficient parasite (see

476

E. S Z A T H M , , ~ R Y

AND

L. D E M E T E R

n FIG. 10. F i m e s s e s (w,, = A,Q~) o f s y s t e m 1. M a x i m u m = 7.96.

I

..5 n

i

FIG. 11. D o m i n a n t q u a s i s p e c i e s d i s t r i b u t i o n o f s y s t e m I1 with i n c r e a s e d c a t a l y t i c activity c t = 4 2 0 0 . O t h e r p a r a m e t e r s are i d e n t i c a l with t h o s e o f s y s t e m 1. M a x i m u m : 0.38.

Appendix C). The appropriate master equation reads: P(n, m)'= a ( n - 1 ) ( n - 1)l/2P(n- 1, m ) + b ( m - 1)nl/2P(n, m - 1) - (d(n + rn) + (n + rn)2/K + (an + bm)n I/2)p(n, m) + ( n + 1 ) ( d + ( n + 1 + m ) / K ) P ( n + 1, rn) +(rn+l)(d+(n+rn+l)/K)P(n,

rn + 1).

(25)

The metabolic surface according to (24) is displayed in Fig. 12. It shows clearly that template Y as a complete parasite only inhibits metabolism. The stationary quasispecies distribution in Fig. 13 is interesting. No compartment with y ( 0 ) > 2 is viable in accord with the selfishness of Y. Yet the whole population, due to the correcting effect of stochasticity, does not go extinct because of internal competition within the compartments. Remarkably, the equilibrium density of the first compart-

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301

FIG. 12. Metabolic surface according to eqn (24) indicating selfishness of Y (system III). s = M =22, c = 1000, ,u =0.5, ~5= 10. Maximum: 950.21.

I

n

II

FIG. 13. Dominant quasispecies distribution of system 111. There are only three viable compartments with initial m = 2 due to the selfishness of Y. Maximum: 0.37. Parameters of system III not listed in legends to Fig. 12: a = 10, b = 20, K =20, d =0-05.

ment type (n(0) = m(0) = 1) is much larger than one would expect from the fitnesses (Fig. 14). This is attributable to the large mutation rates w~j (Fig. 15). 5. Discussion of the Model

The simulation results presented in the preceding section show that the stochastic corrector model offers an alternative resolution for Eigen's paradox; "the 'Catch-22' of the origin of life: no large genome without enzymes, and no enzymes without a large genome." (Maynard Smith, 1983, p. 317). The hypercycle, according to our argument in section 2, is insufficient to resolve this problem owing to its selective instability. Ishida (1986) analyzed fluctuations in the hypercycle by a master

478

E. S Z A T H M A R Y

AND

L. D E M E T E R

l

I

FaG. 14. Fitnesses (w,) t = 0 . Maximum: 1-79.

compartment with P(I, 1) = 1 at

FIG. 15. Fitness of and mutation rates to the compartment P(1, 1)= 1 at t = 0 . Maximum =3.21.

equation. He argued that given a two-membered hypercycle H2 "the fluctuations in H2 are anomalously enhanced through the interaction between/-/2 and its fluctuating environment, and so it becomes unstable" (p. 8). He then assumed the appearance of a mutant (somehow facilitated by the fluctuation) which could interact with/-/2 in various manners. Whether H2 expands itself to H2 incorporating the new mutant is a matter of chance (i.e., it "depends on the relative magnitude of the coupling factors", p. 8). The stochastic corrector model behaves as a darwinian population in which the intrinsic adaptiveness is not masked by nonlinearities of growth (cf. Michod, 1984). We emphasize that competition within the compartments, contrary to other models (Gabriel, 1960; Niesert et al., 1981; Koch, 1984), has in each case been allowed. In their pioneering works Ebeling & Feistel (1982) and Feistel (1983a, b) imagined a

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quasispecies of replicative compartments. In particular, Feistel (1983a) has set up a complicated master equation for the probability distribution of "coacervates" (microreactors with templates inside) accounting for chemical reactions inside the coacervates, reactions in the environment, particle exchange between coacervates and the former, spontaneous creation, decay, fission, and association of coacervates. From this, he has managed to deduce a linear partial integro-differential equation assuming rare association, a stationary environment, and not too small particle numbers inside (re-allocation of templates during fission was taken to be Gaussian). In all applications, fast selective equilibration inside the coacervates has been assumed (Ebeling & Feistel, 1982; Feistel, 1983b). This would mean zero concentration of both templates in our case. Our compartments undergo fission in selective inequlibrium. The two sources of stochasticity: (1) replication and degradation of templates ("demographic stochasticity"), and (2) their chance allocation into the offspring compartments during fission, re-create the best compartment types. To put it clearly, stochasticity "generates variability, on which selection (between cells) can act" (Maynard Smith, J., pers. comm.). All this justifies Kimura's (1983) conjecture that stochastic theory may significantly contribute to the understanding of the origin of life. Based on the foregoing analytic and simulation results, we think that the stochastic corrector model is an efficient information integrator and is a sufficient alternative of (compartmentalized) hypercycles (cf. Szathmfiry, 1986b). One could argue that the basic equations (7-8, 10, 22-24) are too complicated and one might as well use simpler models such as the Lotka-Volterra equations. While it is true that the beneficial effect of the stochastic corrector should apply to that case as well, yet the plausibility of the model would be considerably reduced and interesting effects (those of increased catalytic activity or selfishness) could not be investigated at all. As is apparent from the replication (Fig. 6) and metabolic (Figs 5 and 12) surfaces, the nature of the interactions (mutualism or competition; catalysis or inhibition, respectively) depend on the template concentrations. In ecology, an analogous case is called density dependence forcing modification of the Lotka-Volterra equations. It should be noted that the powers 1/2, 1/4, and 3/2 in the deterministic dynamics do not express chemical reality: they were introduced to reduce the nonlinearity, but their configuration in the equations ensures the required qualitative behaviour. We conjecture that any mathematical model with the same qualitative foundations: (1) aspecifically coupled, competitive templates with no deterministic coexistence, (2) compartment fission rate depending on internal template composition with (3) some composition being optimal, would have the same outcome, i.e. coexistence of replicative compartments in a quasispecies. The chosen simulation technique of the stochastic problem (numerical integration of master equations) is time-consuming. One could ask if some analytic approximation technique exists. Van Kampen's (1983) system-size elimination procedure leads to a corresponding Fokker-Planck equation which can usually be solved easily. Recently, Leung (1985) applied this approach to the Eigen equation with faithful replications. We deal here with small particle numbers and the presence of the absorbing boundary (n = 0, m = 0) cannot be neglected. This immediately leads to

480

E. S Z A T H M , ~ R Y

AND

L. D E M E T E R

a Fokker-Planck equation with moving boundary conditions. Such problems are immensely difficult to solve (cf. Cranck, 1984); we failed to succeed in our case. Generation times were borrowed from the deterministic dynamic throughout the simulations. If we applied the stochastic version of equations (10) and (24) then a continuum of generation times would emerge for each compartment type, making the simulation infinitely more complicated. Although not exactly analogous to our case, McCaskill's (1984a) investigation of a quasispecies with continuously distributed replication rates may be referred to. It seems that the overall behaviour of the system remains similar to the discrete distribution case. Given the stochastic dynamics within compartments the application of deterministic eqn (18) at the compartment level may look awkward. One could argue that the total density of compartments (EiX~) may be so large that the relative frequencies practically equal the probabilities. A more adequate interpretation may be offered by the result in Demetrius et al. (1985) according to which the Eigen equation applies even to the stochastic case as far as the relative frequencies of the expectation values are considered, provided the total population does not go extinct. (In the cited paper the authors assumed constant copying times of polynucleotides. Our fixed T~ values are in agreement with this simplification.) McCaskill (1984b) has shown that, in a stochastically treated population analogous to ours, metastable "'metaspecies'" arise which follow each other slowly in a succession towards the deterministic quasispecies. Compare these with the stochastic behaviour of elementary hypercycles, in which case hypercycles with more than four members quickly die out (Rodriguez-Vargas & Schuster, 1984). We referred to selfishness in connection with a template not possessing catalytic activity but consuming the common metabolic pool for replication within a compartment. Since Dawkins' (1976, 1982) selfish gene concept the idea of molecular selfishness has received considerable attention (e.g., Orgel & Crick, 1980; Doolittle & Sapienza, 1980; Orgel et al., 1980). A theoretical outgrowth of this line is the concept of molecular drive (Dover, 1982) of which novel mathematical models are also available (Ohta & Dover, 1984). Doolittle (1982) pointed out that a replicating RNA molecule in a prebiotic evolution would per definitionem be selfish. The need for a trick to overcome selfishness arises immediately. The trick is to package the selfish replicators into a next higher level of replicators obtaining thus a two level selection process where the higher level selection can overcome that on the lower level. This is analogous to not only the selfish gene versus individual selection case but that of individual versus group selection as well. It is known that various forms of group selection can maintain more or less altruistic individuals in the population (see Wilson (1983) for a review of group selection). Our replicative compartments are groups of replicating templates. Maynard Smith (1983, p. 319) prefers to confine the term group selection "to cases in which the entities with the properties of multiplication, heredity, and variation, whose evolution is being described, are in fact the groups, and no individuals". According to this the stochastic corrector model describes group selection of early replicators. Compartmentation provides the necessary context for the evolution of group-beneficial traits (cf. Wicken, 1985). But as compartments obey eqn (18), a sort of replicator equation (Schuster & Sigmund, 1983), they are true replicators on the next higher level of organization.

GROUP SELECTION OF EARLY REPLICATORS

481

6. The Stochastic Corrector and the Origin of Life In the presented model the action of metabolism and a functional membrane subsystem were explicitly assumed. The relevance of this assumption has to be discussed. G~inti (1975, 1979) has put forward the chemoton model (Fig. 16) to resolve some unclear issues in prebiology. He pointed out that the three subsystems (genetic material made of RNA, an autocatalytic cycle of small molecules as metabolism, and a two-dimensional fluid membrane) could function as a whole. The chemoton's behaviour was numerically simulated (B6k6s, 1975; Csendes, 1984). A review of relevant experimental evidence (G~inti, 1979) showed that the spontaneous generation and coupling of the three sub-systems could have been plausible processes.

FIG. 16. Schematic drawing of a chemoton. The system is bounded by a membrane encapsulating an autocatalytic metabolic network (M) whose reactions are catalysed by self-replicating templates I,. Arrows indicate catalytic actions, dashed lines stand for couplings through mass-action kinetics.

Originally, the coupling among the subsystems was imagined to be based solely on mass action. Later, following earlier speculations (Crick, 1968, White, 1976), G~inti (1983) assumed that the genetic material, in concreto RNA, could function not only autocatalytically in its own replication but heterocatalytically in metabolism as "enzyme-RNA (eRNA)', too. This assumption seems to be more and more justified (Cech, 1985; Gilbert, 1986). Resting on a functional eRNA (or ribozyme) background, the origin and evolution of protein enzymes and the genetic code seems somewhat easier to conceive. Furthermore, the origin of a catalytic system consisting entirely of RNAs should have higher probability than that of a protein-RNA set with the appropriate genetic code (cf. Kaplan, 1981). The division of the chemoton was thought to be a result of the growth of its membrane by adding constituents to it from inside synthesized by the metabolic subsystem, and the simultaneous modification of the inner osmotic pressure (G~inti, 1979). Recently, Koch (1985) has outlined a plausible model of the growth and fission of phospholipid vesicles. His mechanism can be borrowed for the fission process of the chemoton, too. He writes: "Assuming that the genetic material could

482

E. S Z A T H M . / t R Y

AND

L. D E M E T E R

be replicated in a semiconservative fashion, the thesis of this paper is that such liposomelike structures could have the three necessary functions: (1) for segregation of genetic units, (2) as units for selection of phenotypes, and (3) for energy development." (p. 275). Taken together the plausibility of the chemoton's eRNAs and its spontaneous fission, it seems to be the model required by Schwemmler (1985) to describe the origin of living systems. What is more, it is a detailed and plausible illustration of Varela's (1979) "autopoietic system" concept. Cs~inyi & Kampis (1985) have argued that autogenesis is not a mysterious process. The chemoton hypothesis shows the same in a more concrete manner. Despite all these merits, the integration of information by the chemoton has been an open question. Our stochastic corrector model shows that the chemoton remains a promising model for prebiology without the incorporation of hypercycles. On the other hand, the stochastic corrector model can also be applied to problems of organellar competition and coexistence in eukaryotic cells (Szathm~ry, 1986a) as well. The a u t h o r s are i n d e b t e d for discussions a n d s u p p o r t to Profs J. Corliss, T. G:~nti, P. Juh~isz-Nagy, S. Koch, T. Simon, a n d G. Vida a n d Dr. M. Zim~inyi. Special t h a n k s are due to Prof. J. M a y n a r d Smith for e n c o u r a g e m e n t a n d criticism. Dr L Sasv~ri m a d e useful comments on the stochastic simulation problem. Dr Irina G l a d k i h ( C e n t r a l Research Institute for Physics, Budapest) obtained the coordinates of fixed-points in the R E D U C E system. Drs F. B6k6s a n d I. Moln~ir s h o w e d also active interest. O n e o f us (E. Sz.) t h a n k s his brother for a Sinclair Q L pc. REFERENCES B~KI~S, F. (1975). BioSystems 7, 189. BESSHO, C. & KURODA, N. (1983). Bull. math. BioL 45, 143. BRANDON, R. N. & BURIAN, R. M. (eds) (1984). Genes, Organisms, Populations. Cambridge: MIT Press. BRESCH, C., NIESERT, U. & HARNASCH, D. (1980). J. theor. Biol. 85, 399. CECH, T. R. (1985). Int. Rev. CytoL 93, 3. CRANCK, J. (1984). Free and Moving Boundary Problems. Oxford: Clarendon Press. CRICK, F. H. C. (1968). 3". tool. Biol. 38, 467. CSANYI, V. & KAMPIS, G. (1985). J. theor. Biol. 114, 303. CSENDES, T. (t984). Kybernetes 13, 79. DAWKINS, R. (1976). The Selfish Gene. Oxford: Oxford University Press. DAWKINS, R. (1982). The Extended Pbenot)Te. Oxford and San Francisco: W. H. Freeman & Co. DEMETRIUS, L., SCHUSTER, P. & SIGMUND, K. (1985). Bull. math. BioL 47, 239. DOOLIXTLE, W. F. (1982). In: Genome Evolution. (Dover, G. A. & Flavell, R. B., eds), pp. 3-28. London: Academic Press. DOOLrr-rLE, W. F. & SAPIENZA, C. (1980). Nature 284, 601. DOVER, G. (1982). Nature 299, 111. EBELING, R. W. & FEISTEL, R. (1982). Physik der Selbstorganisation und Evolution. Berlin: AkademieVerlag. EBELING, R. W. & SONNTAG, I. (1986). BioSystems 19, 91. EIGEN, M. (1971). Naturwissenschaften 58, 465. EIGEN, M., GARDINER, W. C. JR & SCHUSTER, P. (1980). J. theor Biol. 85, 407. EIGEN, M. & SCHUSTER, P. (1977). Naturwissenschaften 64, 541. EIGEN, M. & SCHUSTER, P. (1978a). Naturwissenschaften 65, 7. EIGEN, M. & SCHUSTER, P. (1978b). Naturwissenschaften 65, 341. EIGEN, M. & SCHUSTER, P. (1982). J. tool. EvoL 19, 47. EIGEN, M., SCHUSTER, P., GARDINER, W. ~,~ WINKLER-OSWATITSCH, R. (1981). Sci. Am. 244(4), 78.

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FEISTEL, R. (1983a). studia biophysica 93, 113. FEISTEL, R. (1983b). studia biophysica 93, 121. FEISTEL, R., ROMANOVSKY, YU. M. & VASILIEV, V. A. (1980). Biofizika 25, 882. GABRIEL, M. L. (1960). Am. Nat. 44, 257. GA.NTI, T. (1975). BioSystems 7, 15. GANTI, T. (1979). A Theory of Biochemical Superxystems and its Application to Problems of Natural and Artificial Biogenesis. Budapest: Akad6miai Kiad6 & Baltimore: University Park Press. G.~NT1, T. (1983). Biol6gia 31, 47. GARD1NER, C. W. (1983). Handbook of Stochastic Methods. Heidelberg: Springer-Verlag. GILBERT, W. (1986). Nature 319, 6t8. HOFBAUER, J., SCHUSTER, P., SIGMUND, K. & WOLFF, R. (1980). SIAM.I. Appl. Math. 38, 282. 1SHIDA, K. (1984). J. theor. Biol. 108, 469. ]SHIDA, K. (1986). J. theor. Biol. !18, 3. JONES, B. L., ENNS, R. H. & RANGNEKAR, S. S. (1976). Bull. math. Biol. 38, 15. JONES, B. L. & LEUNG, H. K. (1981). Bull. math. Biol. 43, 665. KAPLAN, R. W. (1981). Biol. Zbl. 100, 23. KI MURA, M. (1983). The Neutral Theory of Molecular Evolution. Cambridge: Cambridge University Press. KocH, A. k (1984). J. mot. Evol. 20, 71. KocH, A. L. (1985). J. tool. Evol. 21,270. KOPPERS, B.-O. (1983). Molecular Theory o f Evolution. Heidelberg: Springer-Verlag. LEUNG, H. K. (1985). Bull. math. Biol. 47, 231. MAYNARD SMITH, J. (1979). Nature 280, 445. MAYNARD SMITH, J. (1982). Evolution and the Theoo, of Games. Cambridge: Cambridge University Press. "MAYNARD SMITH, J. (1983). Proc. R. Soc. Lond. B 219, 315. McCASKILL, J. S. (1984a). J. Chem. Phys. 80, 5194. McCASKILL, J. S. (1984b). Biol. Cybern. 50, 63. MICHOD, R. E. (1983). Amer. Zool. 23, 5. MICHOD, R. E. (1984). In: A New Ecology. (Price, P. W., Slobodchikoff, C. N. & Gaud, W. S., eds), pp. 253-278. New York: John Wiley & Sons. NIESERT, U., HARNASCH, D. & BRESCH, C. (1981). J. tool. Evol. 17, 348. OHTA, T. & DOVER, G. (1984). Genetics 108, 501. ORGEt_, L. & CRICK, F. H. C. (1980). Nature 284, 604. ORGEL, L., CRICK, F. H. C. & SAPIENZA, C. (1980). Nature 288, 645. POHLEY, H.-J., THOMAS, B. (1983). BioSystems 16, 87. RODRIGUEZ-VARGAS, A. M. & SCHUSTER, P. (1984). In: Stochastic Phenomena and Chaotic Behaviour in Complex Systems. (Schuster, P., ed), pp. 208-219. Heidelberg: Springer-Verlag. SCHUSTER, P. & SIGMUND, K. (1983). J. theor. Biol. 100, 533. SCHWEMMLER, W. (1985). J. theor. Biol. i17, 187. STUCKI, J. W. (1978). Prog. Biophys. Molec. Biol. 33, 99. SWETINA, J. & SCHUSTER, P. (1982). Biophys. Chem. 16, 329. SZATHMARY, E. (1984). In: Evolution IV. Frontiers of Evolutionary Research (in Hungarian). (Vida, G., ed), pp. 37-157. Budapest: Natura. SZATHMARY, E. (1985). ICSEB 111.3rd Int. Congr. Syst. Evo/. Biol. Brighton. Abstracts 188. SZATHMARY, E. (1986a). Endocyt. C. Res. 3, 113. SZATHMARY, E. (1986b). Endocyt. C. Res. 3, 337. THOMPSON, C. J. & MCBRIDE, J. L. (1974). Math. Biosci. 21, 127. VAN KAMPEN, N. G. (1983). Stochastic Processes in Physics and Chemistry. Amsterdam: North-Holland Publ. Co. VARELA, F. J. (1979). Principles of Biological Autonomy. New York: North-Holland. WHITE, H. B. III. (1976). J. tool Evol. 7, 101. WICKEN, J. S. (1985). J. theor. Biol. 117, 545. WILSON, D. S. (1983). Ann. Rev. EcoL Syst. 14, 159. APPENDIX

A

Stability Analysis of (7-8) W e a n a l y z e ( 7 - 8 ) w i t h c o n d i t i o n (9). S e t t i n g t h e r h s s o f (7) a n d ( 8 ) e q u a l t o z e r o w e s e e t h a t p o i n t (0, 0) is a n o b v i o u s s o l u t i o n . A s s u m i n g

that x > 0 and y > 0 we

484

E. SZATHM,~RY AND L. D E M E T E R

divide (7) and (8) by x and y, respectively to yield a(xy) I / 4 - d - ( x

+ y)/ K =0

(A1)

b(xy) ~/4_ d - (x + y ) / K = 0

(A2)

a ( x y ) t / a = b ( x y ) '/4

(A3)

which would require that

which contradicts (9). Thus, there is no fixed-point in the positive orthant. System (7-8) cannot grow to infinity. To this end we take (7), divide it by x, and write x'
if a ( x v ) ~ / 4 - d - ( x + y ) / K


(A4)

It is sufficient to show that with growing x and y (A4) must become true. We require that a(xy) t'4 < (x + y ) / K

(A5)

)4/K4.

(A6)

or

a 4xy < (x + y

According to the binomial expansion of the rhs it contains 6 x 2 y 2 / K 4. Using this we write a4x)' < 6(.~V)2/ g 4

(A7)

xy > (aK)4/6.

(A8)

which is true if

If x and y grow (A8) will become true. In consequence, (A4) must hold also. The same procedure can be applied to (8) with b replacing a. Therefore, x and y cannot grow without limit. Trajectories are thus bound to terminate in the only fixed point (0, 0) as on the plane they may grow to infinity, converge to a fixed-point or to a limit cycle. Existence of the latter is excluded by the fact that there is no fixed-point in the positive orthant. APPENDIX B

Non-hypereyclic Nature of (7-8) We demonstrate that despite a crude resemblance to a t w o - m e m b e r e d hypercycle the central terms in (7-8) are definitely non-hypercyclic. In order to show this in the most orthodox way we set up the following system: x' = ax(.~v)" - x ( a x + b y ) ( x y y / C

(A9)

y' = by(x),)" - y ( a x + b y ) ( x y ) " / C

(A10)

n>0,

C =x+y

(All)

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485

thus we apply the constraint of "constant overall organization" (Eigen, 1971). (9) is valid for the above system. Two fixed-points (0, C) and (C, 0) are obvious. There can be none else, for it would require

a(xy)"=b(xy)';

x,y>O

(A12)

which contradicts (9). The absence of deterministic, stable coexistence is in marked contrast with the properties of two-membered elementary hypercycles (of. Eigen & Schuster, 1978a). It is easy to show that x will replace y in selection equilibrium, y can be eliminated by using the constraint (A11)

x ' = x"+l(C - x ) " ( a - b+ x(b - a ) / C ) .

(A13)

Due to (9), (A13) is always positive if 0 < x < C. Hence x = 0 is an unstable and x = C a stable fixed-point. The aspecific coupling between the replicators cannot prevent

the system from behaving, competitively. APPENDIX C

Stability Analysis of (22-23) System (22-23) with the condition b> a

(A14)

has a stable fixed-point (0, 0), all other fixed-points are unstable. First we show that x and y cannot exhibit unlimited growth. For x it is true that

ax3/2
ifaK
I/2

(A15)

which means that x' in (22) must become negative at sufficiently high x (y makes x' only decrease). For y one finds that

byxl/'- < y x / K

ifbK
I/2

(A16)

hence at sufficiently large concentrations y ' < 0 holds as well. From (22-23) it is apparent that point (0, 0) is a stationary one. If x = 0 y has to decrease until y = 0. There can be two fixed-points (x, 0), x > 0. Setting the rhs of (22) equal to zero, dividing by x and taking y = 0 we obtain a.~ I/2 =

d + .~/ K.

(A17)

Raising both sides to the second power gives the solution for x:

Xl.2 = (Ka)2/2 - d K + K2a(a "-- 4 d / K ) l / ' - / 2

(A18)

which is feasible (positive) if

a2K > 4d.

(A19)

486

E. S Z A T H M , , ~ R Y A N D

L. D E M E T E R

For instability it is sufficient to show that the fixed-points (x, 0) are unstable against invasion by y. To this end we calculate the partial derivatives associated with (22-23): 3x'/Ox = 3axl/2/2 3x'/Oy = -x/

(A21)

K

3y'/Ox = byx-t/2/23y'/ay

(A22)

y/ K

( A23 )

= b x t~ 2 _ d - x ~ K - 2 y / K .

We take the above elements of the Jacobian with £ > 0 characteristic equation (3a.~/2/2

(A20)

- d - 2x/ K - y/ K

and ~ = 0

to obtain the

P-2) = 0

(A24)

- d - 2 . ¢ : / K - i,z t )( b.~ t/a _ d - :~/ K -

where p.~ are the eigenvalues. For p.2 < 0 we require that b:~ ~/2 < d + :~/ K

(A25)

where the rhs can be replaced by (A17) b.~, ~/2 < a:~ I/~

(A26)

which explicitly contradicts (A14). Thus we have found that even if the two feasible fixed-points ( ~ . 2 , 0 ) exist they are unstable. Even if they exist, x ' < 0 if x < min (~.2), hence a small perturbation of (0, 0) will die out since as x = 0, y' < 0 and y = 0 will be asymptotically reached as well. The fixed-point (0, 0) is locally asymptotically stable, and this is the only stable point of system (22-23) with (A14).