J. J Bull; Lauren Ancel Meyers; Michael Lachmann

“ It is essential to distinguish the process of evolution toward the mutation–selection balance from an error catastrophe, as the two are completely different. We thus begin by briefly illustrating the dynamic approach to mutation–selection balance. For any combination of mutation rates and fitnesses in our model, a population will evolve until it reaches a unique equilibrium distribution of A1 and A2. Figure 2 illustrates the evolution of two populations—one that initially consisted entirely of A1 and another that consisted almost entirely of A2. ”

“ For this scenario, we assume that the population size is sufficiently small relative to the mutational distance between the two neutral networks, so that it is essentially impossible for the population to reach one network from the other via mutation (as modeled in some of the “survival of the flattest” work cited above) . If one of the networks is more mutationally robust but of lower fitness than the other, an error catastrophe can cause the loss of the higher-fitness network. ”

“ In summary, the error threshold decreases as the fitness landscape around the inferior phenotype broadens, either via the extension of the neutral network to include additional genotypes or via a net decrease in mutation off the network. The error threshold above which a particular wild-type is lost will depend on its fitness relative to the fitness and breadth of the inferior network. The broader and more mutationally connected the inferior neutral network, the greater its tolerance for mutation and the more likely an error catastrophe that tips the population in its favor. ”

“ These models show that a single fitness landscape may contain multiple, hierarchically organized error thresholds and that an error threshold is affected by the extent of back mutation and redundancy in the genotype-to-phenotype map. Importantly, an error threshold is distinct from an extinction threshold, which is the complete loss of the population through lethal mutations. Based on this framework, we argue that the lethal mutagenesis of a viral infection by mutation-inducing drugs is not a true error catastophe, but is an extinction catastrophe. ”

“ Our model assumes deterministic dynamics and ignores the stochastic effects of finite population sizes. We have argued that populations will equilibrate either to a combination of A1 and A2 or to A2 exclusively. This equilibration strictly requires an infinite amount of time in an infinite population. In a finite population, as long as the effective population size times the mutation rate is significantly bigger than one (Nμ ≫ 1) , the finite population behaves similarly to the infinite population, except that the error threshold is not as sharp. ”

“ Sella and Hirsh [18] showed that population size rather than mutation rate is the critical randomizing factor: the smaller the population size, the less the population is able to maintain genotypes with high fitness.
We ignored the effect of recombination, as does most of the literature on the error catastrophe. The effect of recombination on the quasispecies can be very complicated. A surprising result is that recombination can make a genotype more vulnerable to an error catastrophe, causing the error threshold to occur at a lower mutation rate [19] . ”

“ No error catastrophe is possible because no other genotypes exist, but extinction is obviously possible. Our model is one genotype more complicated: mutant offspring of A1 survive as A2, but mutants of A2 are inviable. Thus, there is not an additional error threshold beyond A2, because the model does not allow a third inferior genotype. An A2 population, however, will go extinct if its net growth rate drops below one (the horizontal axis in Figure 3) . ”

“ Yet two components of the quasispecies framework suggest that a permanently elevated mutation rate may be deleterious to the virus even when extinction is not achieved. First, a higher mutation rate reduces mean fitness around any local fitness peak. Second, a high mutation rate “flattens” the fitness landscape, so that some narrow peaks cannot be attained. Thus, a sustained high mutation rate that does not drive the virus to extinction may enforce a low mean fitness. ”

“ Again, we emphasize that an error catastrophe is not a process per se; rather it is a change in the equilibrium distribution of genotypes arising under a relatively high mutation rate. If the mutation rate of a population is increased from below to above an error threshold, however, then there will be a dynamical loss of A1 as the population approaches the new equilibrium.
Three implications of this analysis are significant. First, an error catastrophe is not equivalent to a population extinction. Second, fitness landscapes may contain multiple error thresholds. ”

“ Abstract Quasispecies are clouds of genotypes that appear in a population at mutation–selection balance. This concept has recently attracted the attention of virologists, because many RNA viruses appear to generate high levels of genetic variation that may enhance the evolution of drug resistance and immune escape. ”

“ One would then test for characteristics of an error catastrophe, such as a population shift in genotype space or increased mutational robustness of the genotypes evolved at high mutation rate. The key difficulty in demonstrating an error catastrophe without a priori knowledge of alternative networks is that there are population genetic processes not considered in the simple quasispecies models that give rise to superficially similar outcomes. ”

“ Identification of an error threshold is certainly facilitated by use of a previously characterized genotype–fitness map, but that kind of information is not usually available. In ignorance of the alternative neutral networks for a phenotype, the obvious protocol to use in blindly testing for an error catastrophe is to impose stabilizing selection for the wild-type trait while progressively increasing the mutation rate. ”

“ A2 reproduces its genotype with fidelity 1 − μ2, and all of its mutants die.
Quasispecies concepts address equilibia, that is, the final distributions of genotypes in populations that have evolved to a stable state. In the quasispecies model, mutation and natural selection steer the population toward the equilibrium distribution, regardless of the initial distribution of genotypes. If the population does not start at the equilibrium, then mutation and natural selection steer it toward equilibrium in the quasispecies model. ”

“ Figure 10 illustrates a succession of error catastrophes in a single landscape with varying widths and heights of neutral networks, for a genome with 40 loci. In general, the narrower networks are skipped as the error catastrophes move to the flatter networks. In this particular model, there is no back mutation. Error thresholds exist because the genomic mutation rate is proportional to the number of unmutated loci, hence the phenotypes of lower fitness experience lower overall mutation rates. ”

“ While an error threshold depends on the relative replacement rates of the genotypes, the extinction threshold depends on the absolute replacement rates of the genotypes.
There are several possible relationships between error thresholds and extinction thresholds. The genotype with highest fitness in this model (A1) can disappear through either an error catastrophe or a population extinction, but A2 can disappear only through extinction. ”

“ Other parameters are the same as for the right side of Figure 3: w1 = 3.0, w2 = 2, and k = 2. When Nμ ≪ 1, the above description does not hold. Mutations occur so rarely, that the population is very likely to fix or lose each mutation before the next one arises. In this case, it no longer makes sense to talk about a quasispecies—there is no mutation–selection balance, but instead a fixed type, the wild-type, with possibly some recent mutants. The wild-type essentially undergoes a random walk in genotype space, in which it tends to “hang out” more often in areas with high fitness [18] . ”

“ The ligase ribozymes will permit an easier demonstration of an error threshold than most other empirical systems, because the alternative functional networks have been identified and can be seeded into the experiment. ”

“ Figures Model of Two Genotypes with Forward MutationEach genotype Ai has its own fitness wi and mutational loss μi. Mutation is asymmetric, so that A1 gives rise to A2, but not vice versa.Dynamic Approach to the Quasispecies Equilibrium, for the Genotypes Illustrated in Figure 1Drawn for w1 = 1.5, w2 = 1.0, μ1 = 0.7, and μ2 = 0.6. ”