Error threshold rates

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.

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...

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. 

Appendix 6. Brillouin-Wigner Perturbation Theory of the Quasi-species. Appendix 7. Renormalization of the Perturbation Theory Appendix 8. Statistical Convergence of Perturbation Theory Appendix 9. Variables, Mean Rate Constants, and Mean Selective Values for the Relaxed Error Threshold... 

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 ... 

Figure 29. RNA sequences, obtained by de novo synthesis using Q -replicase, can be optimally adapted to presence of inhibitors [76]. Growth rate of such variant as function of ethidium bromide concentration indicates not only much higher degree of adaptation (as to be expected from broader quasi-species-like mutant spectrum obtained under de novo synthesis conditions, i.e., near error threshold) but also a drug addiction, i.e., higher synthesis rate at finite ethidium bromide concentration.

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... 

According to Head [Trans. Am. Soc. Mech. Eng., 78, 1471-1479 (1956)], a pulsation-intensity limit of F = 0.1 is recommended as a practical pulsation threshold below which the performance of all types of flowmeters will differ negligibly from steady-flow performance (an error of less than 1 percent in flow due to pulsation). F is the peak-to-trough flow variation expressed as a fraction of the average flow rate. According to the ASME Research Committee on Fluid Meters Report (op. cit., pp. 34—35), the fractional metering error E for liquid flow i he "... 

---