Stationary mutant population distribution

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 conditions under which a population approaches a stationary, i.e. time independent, mutant distribution were derived from the kinetic differential equations. In this stationary distribution called quasispecies, the most frequent genotype of highest fitness, the master sequence, is surrounded by closely related mutants1 (Figure 10). 

The result is true for most fitness landscapes and seems to hold for all realistic landscapes in molecular evolution. There are, however, very smooth distributions of fitness values sometimes used in population genetics for which the transition between stationary quasispecies and drifting populations is smooth. A simple landscape showing a sharp transition is the single-peak fitness landscape that assigns a higher fitness value to the master sequence and the same lower fitness value to all mutants. It has some similarity to mean field approximations often applied in physics. 

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