Error threshold

We can now revise Algorithm 1 with the definition of the error threshold. 

The selection to minimize absolute error [Eq. (6)] calls for optimization algorithms different from those of the standard least-squares problem. Both problems have simple and extensively documented solutions. A slight advantage of the LP solution is that it does not need to be solved for the points for which the approximation error is less than the selected error threshold. In contrast, the least squares problem has to be solved with every newly acquired piece of data. The LP problem can effectively be solved with the dual simplex algorithm, which allows the solution to proceed recursively with the gradual introduction of constraints corresponding to the new data points. 

As expected, a response to the hypercycle criticisms appeared, in fact in the same issue of the Journal of Theoretical Biology (Eigen et al., 1980). According to this, the Freiburg investigations refer to one particular evolution model, in which the occurrence of mutants with different, selective values is ignored. In such realistic models, the error threshold loses its importance for the stability of the wild type. If the latter reaches a finite fitness value, it can always be the subject of selection, as no rivals are present. 

The capacity of the wild-type for adaptation is greatest close to the error threshold. The quantity of information compatible with a stable distribution is then in optimal relation to the variety in its spectrum of mutants. Such a system responds very flexibly to changes in its environment. The wild-type is the predominant individual sequence, but it makes up only a small fraction of the complete mutant spectrum. 

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.

The phenotype of highest fitness is called the master phenotype in complete analogy to the fittest genotype, the master sequence. A derivation that is closely related to those for the conventional (genotypic) error threshold yields the conditions for the existence of stationary phenotype distributions which is tantamount to a phenotypic error threshold occurring at a minimum accuracy (Figure 12 Schuster, 1997b) ... 

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.

The quasispecies model defines an optimal mutation rate for evolving populations (Eigen et al., 1988). At the critical mutation rate pmml (referred to as the error threshold), the distribution becomes too broad for selection to withstand the dispersion and it wanders stochastically on the fitness landscape. The optimal mutation rate for evolvability should be as close to pm Crit as possible without exceeding it. Indeed, it was found that viral mutation rates are very close to pm m,. By assuming that the mutation probability is the same at each residue, the error threshold in terms of mutation rate pm ai, was derived as... 

For quasispecies, the error threshold is lower for finite populations because fitness information can be lost through fluctuations in the population as well as a high mutation rate (Nowak and Shuster, 1989 Bonnaz and Koch, 1998). Under the limit of a finite population M, the error threshold has to be modified... 

Simulations of RNA secondary structure landscapes provide insight into the necessary mutation rate to drive adaptation. Huynen etal. (1996) found that the ability of a population to adapt is determined by the error threshold of the fitness and not the sequence. Indeed, they found that any mutation rate greater than zero will cause the population to drift on the neutral network [The error threshold on landscapes with high neutrality approaches zero (Derrida and Peliti, 1991).] A second, higher mutation threshold causes the fitness information to be lost. To accelerate the diffusion of the population on the neutral network, it is necessary to be above the sequence error threshold and as close to the fitness error threshold as possible. Under these criteria, the population will diffuse rapidly without losing fitness information. On a flat landscape, the diffusion constant D0 for a population of M sequences of length N can be approximated by Eq. (37). 

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. 

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