Selection 1(2000)1–3, 135–145
Available online at http://www.akkrt.hu
Avoiding Catch-22 of Early Evolution by Stepwise Increase
in Copying Fidelity
I. SCHEURING
Department of Plant Taxonomy and Ecology, Research Group of Ecology and Theoretical Biology,
Eötvös University, Budapest, Hungary
(Received: 10 January 2000,
Accepted in revised form: 25 April 2000)
One of the most famous problems of early evolution is the reconstruction of the evolutionary processes creating a sophisticated, enzymatically aided replication apparatus from a system of primitive self-replicating oligomers. It is
known that the selectively maintained length, and thus the amount of information coded in replicators, are limited by
their copying fidelity, and that this length was too short in the primordial phase to code for specific replicase enzymes. However, as argued generally, copying accuracy could not be increased without the help of specific enzymes.
In this paper, we suggest a step-by-step solution to this paradoxical situation through studying both nonenzymatic or enzymatic replication. In the nonenzymatic case, longer templates may enhance replication fidelity
by their structural modification, while longer macromolecules could be a bit more precise replicase enzymes in the
enzymatic situation. If this increased copying accuracy is sufficiently high, and naturally if the longer replicators
outcompete the shorter ones, then the information content maintained in the winner template is increased. I use
simple strategic models to study the possibility of these coevolutionary processes. It is pointed out that the necessary per unit increase of copying fidelity decreases rapidly with increased template size, and it could be very small
even at shorter templates. Comparing the results to experimental data suggests that most probably it is the speed of
replication and decomposition of macromolecules rather than copying accuracy, that limits the length of
replicators.
Keywords: Error catastrophe, replicators, prebiotic evolution, RNA enzymes, copying fidelity
1. Introduction
Many scientists interested in the origin of life think
that there must have been a prebiotic phase in which
self-replicating oligomers were the only evolutionary units on Earth (Eigen, 1971; Gilbert, 1986; Maynard Smith and Szathmáry, 1995; Joyce and Orgel,
1999). The most adequate candidates for this role
are RNAs or alternative genetic systems that preceded the RNA world (Egholm et al., 1992; Eschenmoser, 1997; Joyce and Orgel, 1999). This view is
supported by the replicative and very diverse enzymatic capability of RNAs (Zaug and Cech, 1986;
Landweber, 1999 and ref. therein). Although nonenzymatic self-replicating RNAs are not discovered
yet, some artificial nucleic acid oligomers can repli-
Corresponding author: I. Scheuring, Department of Plant
Taxonomy and Ecology, Eötvös University, H-1083 Budapest,
Ludovika tér 2, Hungary, E-mail: [email protected]
cate without any enzyme (von Kiedrowski, 1986,
1993; Sievers and von Kiedrowski, 1994). Similarly, RNA-catalyzed RNA polymerization experiments (Ekland and Bartel, 1996) support the hypothesis of a nucleic acid-based prebiotic world.
Nearly thirty years ago, Eigen (1971) used the
mathematical apparatus of population genetics for
modeling the selection and mutation processes of
replicating macromolecules. He pointed out that the
length of selectively maintainable replicating molecules is limited by copying fidelity and by the selective superiority of the best replicator. Molecules longer than a limit should produce astronomical numbers of copies to yield a single error-free copy.
Considering nonenzymatic replication and assuming the copying fidelity per nucleotide per replication to be 0.99, the maximum length of the winner
macromolecule is about 100 nucleotides (Maynard
Smith, 1983). For further increasing this length the
copying fidelity has to be increased, which requires
the presence of specific enzymes. Prebiotic repli-
136
I. SCHEURING
cators smaller than 100 units were too short to code
for such enzymes, however. This is “Catch-22” of
prebiotic evolution (Maynard Smith, 1983) or
Eigen’s paradox (Szathmáry, 1989).
Earlier hypotheses attempting to solve this problem assume that, besides competition, there is some
kind of cooperation among the replicators, which
can maintain the coexistence of different replicators.
Thus, the information necessary to code for a replicase enzyme is stored in smaller cooperative fragments. Eigen and Schuster (1979) assumed the
replicators to be complete molecular mutualists, that
is, every molecule type catalyzes both its own replication and that of another member of the system. In
this so-called hypercycle, the coexistence of competitor-mutualist molecules is typical, but the system is unstable in both evolutionary and dynamical
terms (Szathmáry, 1989). Eigen’s model assumed
“point-like” macromolecules living in a perfectly
mixed medium. This idealisation can be offset by
considering macromolecules as discrete entities
moving on a surface by a diffusion for example. Coexistence of different replicators is typical in these
nonperfectly mixed spatially explicit models if some
kind of cooperation is assumed among the macromolecules (Boerlijst and Hogeweg, 1991; Czárán
and Szathmáry, 1998). In the meantime, experimentalists pointed out that template-directed selfreplicating oligomers display “parabolic” population growth at higher template concentrations (von
Kiedrowski, 1986, 1993; Sievert and von Kiedrowski, 1994; Lee et al., 1996). The word “parabolic”
means that template concentration increases with
the square of time. This behavior follows from the
self-inhibition of templates due to dimerization. Interestingly, the characteristic behavior of parabolic
replicating systems is not competitive exclusion but
the coexistence of different replicators (Szathmáry
and Gladkih, 1989). Thus, the necessary information was maintained partially in different shorter
templates by parabolic replication. Apparently we
do not fall into the catch now, but in reality we could
not invalidate the Eigen paradox this way. First, selfinhibition is effective only at template concentrations too high to be likely in prebiotic situations
(Maynard Smith and Szathmáry, 1995). At low concentrations, however, templates evolve according to
the Eigen equations, thus we arrive at the paradoxical situation again (Wills et al., 1998). Second, if
replication is limited by a common resource and
only single strands decompose spontaneously, then
types with the lowest fitness are selected out even in
parabolic systems (Lifson and Lifson, 1999). Third,
if we assume a mechanism that creates high template
concentrations, then the survival of everybody excludes the possibility of Darwinian selection.
The explanation of the problem starts from the
hypothesis that, at this prebiotic phase, replication
was non enzymatic. However, most of the previous
solutions of this problem consider more or less elaborated catalytic connections among the macromolecules. Thus, we investigate the possibility of a stepby-step increase in copying fidelity considering two
different mechanisms: nonenzymatic template directed and enzymatic replication.
Our basic assumption is either that the copying
accuracy of template-directed replicators depends
on the length of the replicating macromolecules or
the replicase function of the macromolecules could
increase with length. Similarly to protein-based enzymes, chemical reactions occur in the active center
of the RNA-enzymes. This center contains approximately 7 to 40 nucleotides (Landweber, 1999). The
other nucleotides are responsible for the stabilization, orientation and “tuning” of the active center
and for the stability of the entire molecule. So, adding new monomers to small RNA-enzymes or replicators can increase their effectiveness and stability.
The shorter the molecule the more meaningful is this
effect.
Then, the following process can increase the selectively maintained information: at the beginning,
the length of the winner nucleotide chain is below
the threshold. Then, a longer and fitter mutant
emerges the different structure of which allows for
more precise template-directed self-replication. Alternatively this mutant and/or its nearest neighbors
in the sequence space act as a more accurate replicase enzyme. If this longer mutant replicates with
an increased precision, then it has a higher chance to
avoid the error catastrophe. In this case, the winner
of the selection will be a longer information carrier
macromolecule until an even longer and more precise replicator defeats it, and so on. A really long
replicator molecule and a precise replication process
would be the end of these recursive evolutionary
steps. Recently, a similar hypothesis has been put
forward by Poole et al. (1999), but they do not explain the mechanism and the plausibility of this positive feedback in detail.
ERROR CATASTROPHE AND COPYING FIDELITY
It is shown in the following sections that the
above scenario is highly possible if either nonenzymatic template-directed self-replication or catalytically aided replication is considered. First, I determine the conditions in which longer master sequences are selected for. Then, I give the necessary
increase of copying fidelity by replicator length to
avoid the error catastrophe in the longer system. The
possibility of this scenario is discussed in the light of
existing experimental data.
2. Selection for longer templates
2.1. The template-directed case
Let us consider Eigen’s (1971) equation, which describes the mutation- selection process of replicating
macromolecules:
.
x i = (Ai Q − D i ) x i +
n
∑w
j ≠ i , j =1
ij
xj −
(1)
– F (x 1 , x 2 ,..., x n ) x i , (i = 1,..., n )
where xi is the concentration of sequence i, Ai is the
total rate of replication, Q is the copying fidelity per
replication, Di is the spontaneous decay rate, wij-s
are the mutation rates (the probability that a copy of
i will be type j) and Φ is a function introduced to
maintain constant total concentration in the chemostate. For convenience, the complementary nature of
replication is disregarded. As mentioned above,
template-directed self-replication leads to these
equations at low concentrations. Consequently,
equation (1) is the mathematical model of template-directed or (nonenzymatic) self-replication in
the biologically adequate limit case (Wills et al.,
1998).
Maynard Smith (1983) analyzed a simplified version of Eigen’s model. Because his more tractable
equations lead to the same qualitative results as the
original ones, I start from Maynard Smith’s model.
He divided the population of different replicators
into two parts: the fittest master (denoted by m) and
all the rest (denoted by j). j stands for a fictitious sequence, each parameter of which is the average of
the corresponding parameter values in the quasispecies. It is assumed that there are no back mutations to the master, because it is much more likely
that mutant sequences mutate to other mutants than
137
to the master. Consequently, Q = 1 in the mutant
class. Then, equation (1) becomes:
½m = (AmQ – Dm) xm – xm Φ (xm, xj),
(2a)
½j = (Aj – Dj) xj + Am(1 – Q) xm – xj Φ (xm, xj)
(2b)
where xm + xj = 1 and Φ (xm, xj) = (Am – Dm) xm +
+ (Aj – Dj) xj keeps the total concentration constant
during replication. Equation (2b) indicates that the
concentrations are scaled to sum up to one.
Now, following Maynard Smith (1983), I show
that the length of the selectively maintained information is limited by copying fidelity. The master sequence is selectively maintained if the concentration
of the master is greater than zero at dynamical equilibrium, that is, when ½m = 0 and ½j = 0. Solving equations (2a–b) we conclude that the equilibrium concentration of the master is positive if
Q>
Aj − D j + D m
Am
1
= ,
s
(3)
where s is the selective superiority of the master sequence. Relation (3) can be simplified to Q > Aj/Am
by exploiting the fact that the difference of the decay
rates must be much smaller than the replication
rates. This assumption enables us to neglect decay
rates Dj and Dm in (2a–b) (Maynard Smith, 1983).
Let q be the average copying fidelity of one nucleotide base per replication and N the number of bases
in the sequences. Then, the copying fidelity of the
master is Q = qN. It is known, however, that
−N 1 − q
q N ≈ e ( ) , so that from relation (3) we obtain
N<
ln s
,
1−q
(4)
which means that the selectively maintained quantity of information (N) is limited by the per unit
copying fidelity (q) (Eigen, 1971). The upper limit
for N, mentioned in the Introduction, is estimated
from this relation, and the related problem of
prebiotic evolution is called Eigen’s paradox (Szathmáry, 1989).
If increased copying fidelity belongs to longer
templates, then q must be some increasing function
of N. From condition (3) we have that
Q (N ) = q (N )N > A j / Am .
(5)
138
I. SCHEURING
in the A j >> D j − D m limit. Let us imagine now that
a new M mutant sequence emerges in the system.
We are interested in the possibility of increasing the
coded information, so the mutant is assumed to a nucleotide longer unit, and q is some increasing function of N. Then, using the previously introduced notations and denoting the new master and its mutants
with index M and J, respectively, the selection process can be described by:
½m = [Amq(N)N – Φ(xM, xm, xj, xJ)] xm,
(6a)
½M = [AMq(N + 1)N+1 – Φ(xM, xm, xj, xJ)] xM,
(6b)
ôj = Ajxj + Am(1 – q(N)N)xm – xjΦ(xM, xm, xj, xJ), (6c)
ôJ = AJxJ + Am(1 – q(N + 1)N + 1)xM –
– xJ Φ(xM, xm, xj, xJ),
(6d)
where xM + xm + xj + xJ = 1, Φ(xM, xm, xj, xJ) = AMxM +
+ Amxm + Ajxj + AJxJ .
The invading mutant M outcompetes the resident
m if its net growth rate is greater than that of m, that
is, if
AMq(N + 1)N+1 – Φ(xM, xm, xj, xJ) > Amq(N)N –
– Φ(xM, xm, xj, xJ),
or equivalently if
AM
q (N )N
.
>
Am q (N + 1 )N +1
(7)
The above condition changes somewhat if the decay
rates are not neglected; I shall discuss that case later.
2.2. The enzymatic case
The other possibility is that sequences or some parts
of them behave as primitive replicase enzymes,
named as ribozymes. These ribozymes, being in a
well-mixed chemostate, catalyze the replication according to their relative concentrations. We assume
that ribozymes are obligatory for replication. If molecules would replicate in template-directed and enzymatic way simultaneously, then the second order
enzymatic process can be neglected because of the
very small absolute template concentrations (Wills
et al., 1998). Thus, the above defined situation can
be modeled by the following phenomenological
equations:
ôm = amCm(xm, xj)xm – xmΦ(xm, xj),
(8a)
ôj = ajxj + am(1 – Cm(xm, xj)) xm – xjΦ(xm, xj),
(8b)
where Cm(xm, xj) = q(N)Nxm + Õ(N)Nxj and xm + xj = 1,
Φ(xm, xj) = amxm + ajxj. Cm(xm, xj) estimates the catalytically aided copying fidelity, where q(N) is the
per base copying accuracy of master replication if it
is aided by other masters. Similarly, Õ(N) is the per
base copying fidelity if masters replicated by the catalytic help of mutants. Because all mutants are collected into a single class in this model, Õ(N) is an average value which characterizes the mutants. If the
system is not saturated by the substrate, then reaction speed is proportional to enzyme concentration,
which is assumed in the function Cm(xm, xj). Enzymes can modify replication speed as well, thus the
total enzyme function is Em(xm, xj) = hmq(N)Nxm +
+ hjÕ(N)Nxj, where hm, hj are kinetic constants. One
of the speed constants is greater than the other, say
let hm > hj , then Em(xm, xj) > hj(q(N)Nxm + hjÕ(N)Nxj)
= hjCm(xm, xj). Denoting hjAm by am, we estimate the
replication speed of the master from lower bound in
(8a–b). It seems to be strange that the enzymatic interactions do not emerge in (8b), but it is the result of
a similar estimation. Since mutants replicate into the
mutant class, the enzymatic effect is Ej(xm, xj) = εmxm
+ εjxj with the kinetic constants εm and εj. If, for example εm > εj then Ej(xm, xj) < εm(xm + xj) = εm. Thus,
the enzyme function can be estimated from upper
bound by εmAj = aj replication rate. This “worst
case” assumption helps us give a fair and simpler estimations to be used later.
We can determine the stable nonzero equilibrium
density of master sequence from (8b) assuming that
ôj = 0 and substituting, 1–xm into xj. After some rearrangement, we conclude that
x *m =
a m q (N )N − a j
a m (1 − q (N )N ) + a m q (N )N − a j
.
(9)
Like in the nonenzymatic situation, the master
avoids the error catastrophe if its concentration is
positive at equilibrium. There are two basically different cases when the x *m > 0 condition is satisfied,
if either
aj
(10)
q (N )N >
,
am
or
ERROR CATASTROPHE AND COPYING FIDELITY
aj
1
(11)
=1 − .
am
s
Condition (10) suits the scenario when the mutants
give – on average – a precise and effective catalytic
aid to the master. It has to be stressed that the actual
value of q(N) is not important here, it could even be
zero. The second case requirest Õ(N)N to be very
small, and q(N)N to be greater than a fixed value. We
have to investigate the stability of the fixed points as
well. Substituting 1–xm into xj and analyzing equation (8a), we find that condition (10) always leads to
a stable fixed point, while relation (11) defines an
unstable fixed point at x *m . In the latter case, the
x **
m = 0 fixed point is stable indicating that the master is selected out (see Appendix A). Thus, I do not
discuss further.
Now a master M, being one unit longer than master m, is introduced in the system. Because enzyme
fidelity depends on the length of the replicators, the
mutants of M (denoted by J) form a new mutant
class. With this notation, the system is evolved by
the dynamics
q (N )N << 1 and q (N )N > 1 −
ôm =
[amCm(xM, xm, xj, xJ) –
– Φ(xM, xm, xj, xJ)] xm
ôM = [aMCM(xM, xm, xj, xJ)
=
(q (N )N − q (N )N ) x *m + q (N )N
(q (N )N +1 − q (N )N +1 ) x *m + q (N )N +1
.
(13)
(12b)
(9)). Thus, q(N)N x *m >> Õ(N)N (1 − x *m ) in (13).
Then condition (13) simplifies to
(12c)
ôJ = aJxJ + aM (1 – CM (xM, xm, xj, xJ)) xm –
– xJ Φ(xM, xm, xj, xJ),
aM
q (N )N x *m + q (N )N (1 − x *m )
=
>
a m q (N )N +1 x *m + q (N )N +1 (1 − x *m )
(12a)
ôj = ajxj + am (1 – Cm(xM, xm, xj, xJ)) xm –
– xj Φ(xM, xm, xj, xJ)
speed is built into am and aM, and the enzymatic effects in the mutant classes are estimated from above
by aj and aJ. To avoid unnecessary complications,
we assume that mutants are identical with their masters in length, and both mutant classes have the same
average catalytic activity at a given length (that is
Õj(N) = ÕJ(N) = Õ(N)).
Again, we are interested in the conditions which
allow M to outcompete the resident master sequence. The rare M mutant can spread in the resident
system if ôM > 0 when xM ≈ 0, xJ = 0 and ôm > 0.
Then, we obtain the condition for the invasion of M
from equations (12a) and (12b):
For a more convenient analysis of the above relation, we study it in three characteristic cases. In the
first case, the master has a more accurate enzymatic
function than the mutant, that is, q(N)N >> Õ(N)N.
Then, x *m ≈ 1 − x *m or x *m >> 1 − x *m (see equation
–
– Φ(xM, xm, xj, xJ)] xM
139
(12d)
where
xM + xm + xj + xJ = 1, Φ(xM, xm, xj, xJ) = aMxM + amxm +
+ ajxj + aJxJ,
Cm (xM, xm, xj, xJ) = q(N + 1)NxM + q(N)Nxm +
+ Õ(N)Nxj + Õ(N + 1)NxJ,
CM (xM, xm, xj, xJ) = q(N + 1)N + 1xM +
+ q(N)N + 1xm + Õ(N)N + 1xj + Õ(N + 1)N + 1xJ.
Functions Cm (xM, xm, xj, xJ) and CM (xM, xm, xj, xJ)
estimate the average copying fidelity of replicators
m and M. As before, the enzyme-modified copying
aM
1
>
.
a m q (N )
(14)
We refer to this limit as the “dominant” situation. In
the second possible case the master and the mutants
possess approximately the same enzymatic function,
which means that q(N)N ≈ Õ(N)N. Then, independently of the actual value of x *m , the condition of invasion becomes aM /am > 1/q(N)N ≈ 1/Õ(N). Mutants
and master help one another’s replication effectively, so this is the “cooperative” case. In the third
characteristic situation, the master’s enzymatic
function is insignificant, while it exploits the mutant
enzymes thoroughly, that is, Õ(N)N >> q(N)N. Since
mutants help the replication of the master, we may
call this case the “altruistic” or “parasitic” limit.
Then, Õ(N)N x *m >> q(N)N x *m in (13), and M can invade if
aM
1
>
.
a m q (N )
(15)
140
I. SCHEURING
On the other hand, M could be the winner of the selection if a rare m could not invade into a resident
population M. It means that ôm < 0 when xm ≈ 0, xj = 0
and ôM = 0. Following the previous analysis and approximations, we conclude that m does not invade
the resident master M if
It is reasonable to assume that increasing AM (aM) entails the increase of AJ (aJ) in the mutant class as
well, consequently selective superiority does not
change meaningfully, that is, S ≈ s. Then, comparing
inequality (5) with (18) we conclude that condition
(18) is valid if
aM
1
>
a m q (N + 1 )
Q(N + 1) ≥ Q(N).
(16)
when q(N)N >> Õ(N)N at the dynamic equilibrium,
or
aM
1
>
,
a m q (N + 1 )
(17)
when Õ(N)N ≈ q(N)N or Õ(N)N >> q(N)N.
It can be seen from the structure of (12a) and
(12b) and from conditions (14–17) that m and M
could coexist in none of the dominant, cooperative
and altruistic cases. Thus, we may assume that if M
can invade in the resident quasispecies, then m generally will be outcompeted at the dynamic equilibrium (for mathematical details, see Appendix B).
Since Õ(N) and q(N) must be monotone increasing
functions, conditions (14) and (15) are stricter than
(16) and (17). If a rare M can invade the resident m,
then it is stable against the invasion of rare m-s. Consequently, inequilibrium pairs of (10) and (14) or
(10) and (15) give biologically relevant conditions
for a sequence to be a master, and to be a successful
invader in the enzymatic model.
(20)
Similarly, it follows from (10) and (19) that Û(N + 1)
≥ Û(N) is a sufficient condition for M to be an information carrying master sequence in the enzymatic
model. Since the above conditions are identical for
Q(N) and Û(N), we concentrate only on Q(N) below.
Q(N) is originally defined for positive integer
values of N, but this function could be extended for
every positive real z value. We define Q(z) to be a
continuous and differentiable extension of Q(N).
Thus, condition (20) is supported if Q′(z) ≥ 0, that is,
if (q (z ) z )′ ≥ 0. After derivation it means that
z (ln q (z ))′ + ln q (z ) ≥ 0.
(21)
If the inequality is replaced by an equation sign, then
the resulting differential equation gives a sufficient
function for q(z) that enables the information integration through the step-by-step mechanism. Let us
substitute ln q(z) with L(z), so that we have to solve
1
L ′ (z ) + L (z ) = 0.
z
(22)
α
, where a is
z
the constant of integration. Consequently, if the per
unit copying fidelity increases at least according to
Integrating this equation yields L (z ) =
α
3. Avoiding the error catastrophe
q (N ) = e N ,
The mutant M could be a hopeful invader only if it is
resistant against the error catastrophe as well. According to equation (5), it requires in the nonenzymatic case that
then the longer invading mutant remains resistant
against the error catastrophe. Since q(N) is an increasing function of N, a < 0. (An identical expression is valid for Õ(N).) Substituting this form of q(N)
into (7) we conclude that M can be fitter than m if AM
> Am in the nonenzymatic case. That is, if copying
accuracy increases with the increased coded information according to (23), then the longer mutant is
the winner of the selection, provided it replicates
faster than the resident master. However, if decay
rates are not neglected in (6a–c), then
Q (N + 1 ) = q (N + 1 )N +1 > AJ / AM = 1 / S.
(18)
Relation (10) reflects the enzymatic situation, that
is, M avoids the error catastrophe if
Q (N + 1 ) = q (N + 1 )N +1 > a J / a M = 1 / S.
(19)
(23)
ERROR CATASTROPHE AND COPYING FIDELITY
(24)
is the condition for M to be the winner of selection.
That is, M can overcome m even if it replicates
slower, provided it is more resistant against spontaneous decomposition.
We studied the enzymatic model for some special
cases. In the “dominant” case, the invasion of M is
determined by q(N) (14), which is much greater than
Õ(N). The master is the best template of the replication and it has an outstanding enzymatic capability
as well. As these functions are in negative trade-off,
this situation is improbable and is not treated in
more detail. In the cooperative and in the altruistic
situations, M is a successful invader if aM > am /
/Õ(N) = am / ea/N.
Like in the nonenzymatic situation, if death rates
of replicators differ in (12), the invasion condition
for M modifies to
a M (q (N )N x *m + q (N )N (1 − x *m )) − D M >
> a m (q (N )N +1 x *m + q (N )N +1 (1 − x *m )) − D m ,
(25)
thus M can invade easier if it decays slower than m.
4. Discussion
Let us see the biological relevance of these results. It
is an important property of the function exp(a/N)
that it increases very fast for small N-s, but much
slower if N is high above a (Fig. 1). The longer the
ribozymes and templates the smaller copying fidelity enhancing suffices for increasing the length of
replicators.
Eigen and Schuster (1979) thought the copying
fidelity q to be about 0.99 and the length of the selectively maintained master sequence to be about 100
nucleotides in nonenzymatic replication. According
to recent experiments, 0.99 seems to be an optimistic
value for q. Ekland and Bartel (1996), using specialized RNA polymerizing ribozyme, found that q is
between 0.76 and 0.96 depending on the AU/GC ratio. This ribozyme is about 100 units long and with
the help of an external template it can polymerize
oligomers up to six nucleotides.
Assume that our master replicates about e to e3
times faster than the mutants, that is, s is between
e ≈ 2.7 and e3 ≈ 20. Substituting s and the values of q
q(N)
AM q (N + 1 )N +1 − D M > Am q (N )N − D m
141
N
FIG. 1. The necessary per unit copying fidelity in function of unit
number. The parameters are a = –1, (solid line) and a = –3,
(dashed line)
measured by Ekland and Bartel into (4), we conclude that N, the selectively maintained replicator
length, is between 4 and 75. The lowest value applies if ln s = 1 and the RNAs consist of A and U nucleotides only (q = 0.76), the highest one when
ln s = 3 and RNAs are pure GC sequences (q = 0.96).
Knowing N and q, we can estimate a from (23),
which is approximately between –1 (ln s = 1) and –3
(ln s = 3). These functions are plotted in Fig. 1. We
are interested in the necessary increase in per base
replication fidelity when the length increases from N
to N + 1, which is estimated by q(N + 1) – q(N). Figure 2 shows this difference as a function of N. The
necessary increase of copying fidelity is 4 × 10–2 in
the worst, and 5 × 10–4 in the best case in our estimation (see Fig. 2).
The ribozyme in Ekland and Bartel’s experiment
is probably too long to leave room under the error
threshold. However, Joyce and Orgel (1999) argue
that the minimum size of a replicase ribozyme is
about 40–60 nucleotides, because they must form a
“dumbbell” or a pseudoknot structure to have enzymatic activity. They do not exclude completely that
a single stem-loop, containing 12–17 nucleotides,
has some replicase activity as well. This later view is
supported by the fact that very short oligomers with
7 nucleotides have been demonstrated to show catalytic activity (Landweber, 1999). These values are in
concordance with our estimated size for a selectively maintained replicator-replicase system.
The mutant M can outcompete m if its net growth
rate is greater than that of the resident one (AM > Am)
in the nonenzymatic case, or if aM > am / eα/N in the
“cooperative” and “altruistic” situation. Since M is
142
I. SCHEURING
TABLE 1
The necessary conditions for invasion and the avoidance
of the error catastrophe
q(N+1)–q(N)
Maximum
length of
replicators
Necessary
increase of
copying fidelity
Necessary
increase of
replication speed
Nonenzymatic Enzymatic
case
case
N
N
FIG. 2. The difference of copying fidelity in function of N in a
log-log plot. Arrows indicate the estimated worst (diamonds, N
= 4, α = –1, q = 0.76,) and the best cases (crosses, N = 75, α = –3,
q = 0.96). The dashed line decreases with N–2, showing a strong
decreasing trend in the difference of copying fidelity
one unit longer than m, the replication rate per unit
has to be increased in the longer template. Table 1
collects the estimated values necessary for invasion
and to avoid the error catastrophe of the longer mutant in the nonenzymatic and in the enzymatic situations. For example, if the resident oligomers are four
units long, then the invader has to copy at least 5
units in the same time to invade. It means that AM/Am
has to be greater than 1.25. According to Table 1, we
can conclude that to replicate faster sets up a more
rigorous condition to increase the length of replicators than the error catastrophe itself. As I have
shown above, if the longer replicator is more stable
against spontaneous decay, then it could be the winner of the selection even it has a slower replication
rate (see (24, 25)). Similarly, if replication fidelity
increases even more than the sufficient exp(a/N)
function demands, then the longer replicator outcompetes the smaller one easier (see (7, 14, 15)).
Naturally, it can happen that the longer replicator
kills out the smaller one, but itself is not resistant
against the error catastrophe. Then, both the master
and its mutants will be present at a very low concentration in the original Eigen model (Eigen, 1971).
Because concentrations become very low, it is highly probable that sooner or later the master dies out by
external disturbance (Eigen and Schuster, 1979). In
this case a smaller master can invade and the system
q(N + 1) – q(N) > AM /Am >
aM/am/eα/N >
4
4 × 10–2
(α = –1, q = 0.76)
1.25
1.6
75
5 × 10–4
(α = –3, q = 0.96)
1.02
1.03
The estimation process is described in the main text
returns to the initial state. This cyclical process continues until a longer invading master increases the
copying accuracy in the necessary manner as well.
It is important to note that our enzymatic model is
identical to a two-membered hypercycle in which
one of the replicators (the master) can mutate into
the other one (mutants). It is well known that selfish
parasites, which do not give any enzymatic help to
the other members of the hypercycle but exploit
the others as ribozymes for their replication, infect
and kill the hypercycle (Szathmáry, 1989). This evolutionary problem does not emerge in this model
even if the master exploits the mutants totally
(hmq(N) = 0), because masters (including the selfish
ones) mutating to the mutant class, produce ribozymes for its reproduction continuously. To be stable against selfish parasites is an important question,
and this problem is under study in a spatially explicit
cellular automaton model, where backward mutations and trade-offs among the enzymatic and replicative functions are considered as well.
Acknowledgements
I am grateful Tamás Czárán, János Podani, Csaba Pál, Eörs
Szathmáry, and to an anonymous reviewer for comments and
helpful criticism. The financial support from the OTKA (Hungarian Scientific Found), T025793 and T029789 is gratefully
acknowledged. Special thanks are due to Inter Metal Ltd. for
supporting my traveling expenses.
ERROR CATASTROPHE AND COPYING FIDELITY
143
APPENDIX A
APPENDIX B
Substituting 1–xm into xj in (8a) and rearranging the
right-hand side of this equation we obtain
We show below that M and m masters could not coexist at the studied situations in the enzymatic system. That is, if M can invade when it is rare, then it
will outcompete the resident m.
Equations (12a–d) describe the dynamics of master (m), its mutants (j) and the invader (M) with its J
mutants as follows:
ôm = Ax m2 + Bx m ,
where
A = a m (q (N )N − 1 ) + a j − a m q (N )N ,
B = a m q (N )N − a j .
(A1)
ôm =
The fixed points of (A1) are
x *m =
a m q (N ) − a j
N
a m (1 − q (N )N ) + a m q (N )N − a j
and x **
m = 0.
[amCm(xM, xm, xj, xJ) –
– Φ(xM, xm, xj, xJ)] xm
ôM = [aMCM(xM, xm, xj, xJ)
(B1a)
–
– Φ(xM, xm, xj, xJ)] xM
(B1b)
ôj = ajxj + am (1 – Cm(xM, xm, xj, xJ)) xm –
– xj Φ(xM, xm, xj, xJ)
(B1c)
ôJ = aJxJ + aM (1 – CM (xM, xm, xj, xJ)) xM –
– xJ Φ(xM, xm, xj, xJ)
(B1d)
where
xM + xm + xj + xJ = 1, Φ(xM, xm, xj, xJ) = aMxM + amxm +
+ ajxj + aJxJ,
Cm (xM, xm, xj, xJ) = q(N + 1)NxM + q(N)Nxm +
+ Õ(N)Nxj + Õ(N + 1)NxJ,
FIG. 3. The graphical representation of (A1) when A < 0. The
value of xm changes along the horizontal line according to
dxm/dt, which determines the velocity and direction of this
change. Arrows indicate the direction of change showing that x *m
is stable and x *m* is unstable
According to standard vector field analysis (see
Strogatz, 1994), the stability of the fixed points is
determined by A. If A < 0 then x *m is stable and x **
m
is unstable, while stability is reversed when A > 0
(Fig. 3).
According to (10), aj – amÕ(N)N < 0, and naturally
q(N)N – 1 < 0, consequently A < 0 and x *m is globally
stable. In the second studied situation q(N)N – 1 >
– aj / am (11). Then am(q(N)N – 1) + aj > 0 and –
amÕ(N)N can be neglected in A. Thus A > 0 and x *m is
unstable.
CM (xM, xm, xj, xJ) = q(N + 1)N + 1xM +
+ q(N)N + 1xm + Õ(N)N + 1xj + Õ(N + 1)N + 1xJ.
Templates m and M might be in coexistence if both
x *m and x *M fixed points are positive. Let us assume
indirectly that x *m and x *M are the positive solutions
of (B1). Then it follows from equations (B1a) and
(B1b) that
a m C m (x *M , x *m , x *j , x *J ) =
= a M C M (x *M , x *m , x *j , x *J ),
or in expanded form
am(q (N )N x *m + q (N + 1 )N x *M + q (N )N x *j +
+ q (N + 1 )N x *J ) =
(B2)
144
I. SCHEURING
= aM(q (N )N +1 x *m + q (N + 1 )N +1 x *M +
+q (N )N +1 x *j + q (N + 1 )N +1 x *J ).
(B3)
We study the conditions of invasion of M and its resistance against invasion of m in special cases in the
main text. In the “dominant” limit (q(N)N >> Õ(N)N),
the frequency of masters could not be much smaller
than the frequency of mutants (see equation (9)).
Thus, the second two terms can be omitted in both
sides of (B3):
a m (q (N )N x *m + q (N + 1 )N x *M ) =
= a M (q (N )N +1 x *m + q (N + 1 )N +1 x *M ).
(B4)
M is assumed to be a successful invader which
means that aM > am/q(N). Substituting this relation
into (B4) and knowing that q(N) increases monotonically we conclude that the right side of (B4) is
greater than the left side if the fixed points are positive. We arrived at a contradiction, so our indirect
assumption was invalid.
The cooperative situation assumes that q(N)N and
Õ(N)N are in the same order. Suppose that q(N)N <
Õ(N)N. Then, the right side of (B3) can be estimated
from below
aM (q (N )N +1 x *m + q (N + 1 )N +1 x *M +
+q (N )N +1 x *j + q (N + 1 )N +1 x *J ) >
> aMq (N )(q (N )N x *m + q (N + 1 )N x *M +
+q (N )N x *j + q (N + 1 )N x *J ).
Since aM > am/q(N), it is still greater than the left side
of the (B4), so coexistence is impossible. (Similar
estimation is valid if q (N )N > q (N )N .)
In the “altruistic” case, the enzymatic function of
mutants is dominant over masters, that is, Õ(N)N >>
q(N)N. According to (9), we conclude that the frequencies of masters could not be much higher than
the frequencies of mutants, thus (B3) simplifies to
a m (q (N )N x *j + q (N + 1 )N x *J ) =
= a M (q (N )N +1 x *j + q (N + 1 )N +1 x *J ).
(B6)
However, the right-hand side is greater than the lefthand side, i.e., M can spread if aM > am / Õ(N). This
result is again in contradiction with our indirect assumption.
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