Error threshold replication

Figure 11. The error threshold of replication and mutation in genotype space. Asexually reproducing populations with sufficiently accurate replication and mutation, approach stationary mutant distributions which cover some region in sequence space. The condition of stationarity leads to a (genotypic) error threshold. In order to sustain a stable population the error rate has to be below an upper limit above which the population starts to drift randomly through sequence space. In case of selective neutrality, i.e. the case of equal replication rate constants, the superiority becomes unity, Om = 1, and then stationarity is bound to zero error rate, pmax = 0. Polynucleotide replication in nature is confined also by a lower physical limit which is the maximum accuracy which can be achieved with the given molecular machinery. As shown in the illustration, the fraction of mutants increases with increasing error rate. More mutants and hence more diversity in the population imply more variability in optimization. The choice of an optimal mutation rate depends on the environment. In constant environments populations with lower mutation rates do better, and hence they will approach the lower limit. In highly variable environments those populations which approach the error threshold as closely as possible have an advantage. This is observed for example with viruses, which have to cope with an immune system or other defence mechanisms of the host.

Figure 12. The error threshold of replication and mutation in phenotype space. The genotypic error threshold approaches zero in the case of selective neutrality. Despite changing genotypes a phenotype may be conserved in evolution whenever it has higher fitness than the other phenotypes in the population. The concept of error threshold can easily be extended to competition between phenotypes. The distribution of phenotypes is stationary provided the error rate does not exceed the maximum value pmax which is a function of the mean fraction of nearest neighbors, X, and the superiority of the master phenotype, a. The illustration shows the position of the phenotypic error threshold in the X, p plane. Selective neutrality allows more errors to be tolerated and pmax increases accordingly with increasing X. If X approaches the inverse superiority, X — a-1, the tolerated error may grow to pmax = 1, and this means the phenotype will never be lost, no matter how many errors are made in replication.

Fig. 2.5. A quasi-species-type mutant distribution around a master sequence. The quasi-species is an ordered distribution of polynucleotide sequences (RNA or DNA) in sequence space. A fittest genotype or master sequence /m, which is commonly present at highest frequency, is surrounded in sequence space by a cloud of closely related sequences. Relatedness of sequences is expressed (in terms of error classes) by the number of mutations which are required to produce them as mutants of the master sequence. In case of point mutations the distance between sequences is the Hamming distance. In precise terms, the quasi-species is defined as the stable stationary solution of Eq. (2) [16,19, 20], In reality, such a stationary solution exists only if the error rate of replication lies below a maximal value called the error threshold. In this region, i.e. below...

It is worth considering Eq. (4) from a different point of view. The replication accuracy q is assumed to be determined by the replication machinery and therefore cannot be varied. Then, the error threshold restricts the chain length and defines an upper value for sufficiently faithful replication ... 

Eigen eventually found that it was replication that governed the optimization for molecules. He came out with this theory in Naturwissenschaften, and showed that Darwin was valid even for molecules if they were reproducing molecules. In this case it was possible to describe the process with a mathematical theory. He showed the necessity of a certain error threshold if the mutation rate is too high, information is lost, and if the mutation rate is too low, the progress rate is insufficient, and so on. This is the point where Eigen s studies of fast reactions and the molecular evolutionary theory are connected. 

The mutagenic ribonucleotide rPTP (87) has been used in an in vitro retroviral replication model. After four rounds of replication the mutation frequency was raised to 3.8 x 10 per nucleotide, with C U and U C mutations observed. It was suggested that such an analogue could induce mutations in a retroviral target beyond its error threshold. The tricyclic cytosine analogues phenoxazine and 9-(2-aminoethoxy)-phenoxazine (g-clamp, see (141)) have been incorporated into ODNs to study their effect with exonuclease. It was found that a single substitution at the 3 -terminus afforded complete protection of the ODN with snake venom phosphodiesterase. ... 

It should be briefly pointed out that simply decreasing the replication rates of other sequences to increase the superiority of the master does not indefinitely postpone the error threshold. Once the replication rates for mutant sequences fall below the destruction rate for the master sequence, the domain of viable existence for the quasi-species is limited by A Q > D . In Section III.3 a more fundamental statistical argument restricts the effective superiority to a finite value of order 1. 

The evaluations of the error threshold and effective superiority discussed so far [Eqn. (III.IO)] assume specific knowledge of the wild-type net rate of exact replication It is clear, however, that the wild type is distinguished from the other mutants only insofar as it has the maximum selective value, and so it is sensible to regard this value as the random extreme of n trials from the probability distribution f W) [29]. We may then speak of the general... 

In order to search for an interpretation of the error threshold relation by analogy to the spin system, we shall be more specific and consider binary sequences replicating with uniform error rates. Individual sequences are identified with the rows of a two-dimensional spin lattice. A genealogy corresponds to an entire, two-dimensional array of spins. We assign spin values (s= +1) to the digits (0,1). A sequence of v digits is identified with a string of spin values ... 

As outlined in previous sections, much depends on the properties of physically realistic sets of replication rate constants, despite the comparative insensitivity of error thresholds to details in their distributions. At present it is not feasible to measure or estimate real-valued landscapes empirically, not even in the most simple experimental systems like RNA replication in the Qj ... 

Several attempts to describe replication-mutation networks by stochastic techniques were made in the past. We cannot discuss them in detail here, but we shall brieffy review some general ideas that are relevant for the quasispecies model. The approach that is related closest to our model has been mentioned already [51] the evolutionary process is viewed as a sequence of stepwise increases in the populations mean fitness. Fairly long, quasi-stationary phases are interrupted by short periods of active selection during which the mean fitness increases. The approach towards optimal adaptation to the environment is resolved in a manner that is hierarchical in time. Evolution taking place on the slow time scale represents optimization in the whole of the sequence space. It is broken up into short periods of time within which the quasi-species model applies only locally. During a single evolutionary step only a small part of sequence space is explored by the population. There, the actual distributions of sequences resemble local quasispecies confined to well-defined regions. Error thresholds can be defined locally as well. 

Figure 30. Error threshold as function of population size. Stochastic replication-mutation dynamics in ensemble of polynucleotide sequences with chain length v = 20 simulated by Gillespie s algorithm [95]. Critical single-digit accuracy of replication (q in) at which ordered quasi-species is converted into changing population of sequences with finite lifetimes is plotted as function of 1/N, reciprocal population size (lower curve). For further details see ref. 96. Upper curve is theoretical prediction of Eqn. (V.l) based on ref. 51.

We have recently developed a quasi-species approach for analyzing mutation and selection in catalytic reactions of varying order. We discussed how the error catastrophe that reflects the transition from localized to delocalized quasi-species population is affected by catalytic replication of different reaction orders. Specifically, the second-order mechanisms lead to a discontinuity in the mean fimess of the popnlation at the error threshold. This is in contrast to the behavior of the first-order, antocatalytic replication mechanism, considered in the standard quasi-species model. This suggested that quasi-species models with higher order of replication mechanisms produce discontinuities in the mean fitness and, hence, in the viable population fraction, at the error threshold, while lower-order replication mechanisms yield a continuous mean fitness fnnction. 

Low copying fidelities can lead to an error catastrophe where a replicative process cannot maintain the sequence information [41—45]. The error threshold marks the onset of this catastrophe it refers to a critical value in the ratio n s of the mutation rate to the selection strength s. Beyond the threshold, the mutants with compromised function dominate over the original master sequence, because the... 

It is important to note, however, that the concept of an error threshold can acquire a somewhat different meaning in the prebiotic context. In particular, if we consider the molecules that form the core of a primitive replication process, then the most fundamental maintenance requirement is that the process generates at least one functional copy per core replication molecule before the template is destroyed. This situation differs from the standard error threshold scenario In the latter case, both master and mutants replicate and an error catastrophe results from their competition, whereas in the former case the catastrophe is no replication at all. However, this replication breakdown can also be induced by a low copying fidelity. 

Nowak MA, Schuster P (1989) Error thresholds of replication in finite populations. Mutation frequencies and the onset of Muller s ratchet. J Theor Biol 137 375-395. doi 10.1016/S0022-5193(89)80036-0... 

The environmental conditions of the primitive Earth were surely different from those of Spiegelman s and Eigen s test-tubes, but this can be regarded as a secondary complication, and in a first approximation it can be ignored. What we cannot ignore, however, is the fact that any replication process is inevitably affected by errors, and it is therefore imperative to understand the consequences that such errors have for the very survival of a replicating system. This is a crucial problem for all replication-first theories, because it has been proved that any self-replicating system can tolerate replication errors only below a critical threshold. Above such a threshold, the system is overwhelmed by a runaway error catastrophe, and is inexorably condemned to collapse. This is a fundamental problem, and in order to address it we need first to quantify the critical threshold. 

Equation (III.2) may be rewritten to isolate the dependence on the copying fidelity q in order to demonstrate that for a given set of replication parameters there is an error-rate-dependent threshold sequence length for quasi-species instability. To this end the selective advantage or superiority parameter a was introduced ... 

Here, ln(ot) is typically 0( 1), while the error rate in the replication of monomer is estimated to be around 0.01-0.1, in the usual polymer replication process. Then the above condition gives N < 100 or so. In other words, information using a polymer with a sequence longer than this threshold N is hardly sustained. This problem was first posed by Eigen and is called error catastrophe [7]. On the other hand, information for the replication for a minimal life system must require much more information. Of course, the error rate could be reduced once some machinery for faithful replication as in the present life emerges. However, such machinery requires much more information to be transmitted by the polymer. 

The model is simulated as follows At each step, a pair of molecules, say, i and j, is chosen randomly. If there is a reaction path between species i and j, and i (j) catalyzes j (i), one molecule of the species j ( ) is added with probability c, (cj), respectively. The molecule is then changed to another randomly chosen species with the probability of the replication error rate p. When the total number of molecules exceeds a given threshold (denoted as N), the cell divides into two such that each daughter cell inherits half (N/2) of the molecules of the mother cell, chosen randomly [2],... 

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