Theory of lethal mutagenesis for viruses

Abstract

Mutation is the basis of adaptation. Yet, most mutations are detrimental, and elevating mutation rates will impair a population's fitness in the short term. The latter realization has led to the concept of lethal mutagenesis for curing viral infections, and work with drugs such as ribavirin has supported this perspective. As yet, there is no formal theory of lethal mutagenesis, although reference is commonly made to Eigen's error catastrophe theory. Here, we propose a theory of lethal mutagenesis. With an obvious parallel to the epidemiological threshold for eradication of a disease, a sufficient condition for lethal mutagenesis is that each viral genotype produces, on average, less than one progeny virus that goes on to infect a new cell. The extinction threshold involves an evolutionary component based on the mutation rate, but it also includes an ecological component, so the threshold cannot be calculated from the mutation rate alone. The genetic evolution of a large population undergoing mutagenesis is independent of whether the population is declining or stable, so there is no runaway accumulation of mutations or genetic signature for lethal mutagenesis that distinguishes it from a level of mutagenesis under which the population is maintained. To detect lethal mutagenesis, accurate measurements of the genome-wide mutation rate and the number of progeny per infected cell that go on to infect new cells are needed. We discuss three methods for estimating the former. Estimating the latter is more challenging, but broad limits to this estimate may be feasible.

FIG. 1.

Fitness models considered in this work. The multiplicative model [w_j = (1 − s)^j, shown for s = 0.5], the Eigen model (w₀ = 1, w_{j > 0} = 1 − s, shown for s = 0.5), and the truncation model (w_j = 1 for j ≤ k, w_j = 0 for j > k, shown for k = 2) are shown. The mutation number j counts nonneutral mutations only.

FIG. 2.

Equilibrium mean fitness level as a function of deleterious-mutation rate U_d. The solid curve is , which is the equilibrium for all models in which an error catastrophe is absent or has not occurred. The dashed line is the mean fitness level for the simple Eigen model beyond the error catastrophe, in which the best genotype has a fitness level of 1.0 and all mutants have fitness levels of 0.1 (no back mutations are allowed). In models without error thresholds, the mean fitness levels decay to arbitrarily small values for high mutation rates, whereas an error catastrophe slows down this decay and, in the simple Eigen model, sets a lower bound on the mean fitness level of the population.

FIG. 3.

Decay in average fitness level over the initial 10 generations of mutagenesis with mutation rate U_d = 2. All models illustrated have the same equilibrium mean relative fitness level of e⁻² = 0.135. Multiplicative models are indicated with circles (filled, s = 0.1; open, s = 0.5), truncation models by squares (open, k = 1; filled, k = 3), and the Eigen model by filled diamonds (s = 0.9). Graphs A and B represent the same changes in relative fitness level but different absolute fitness levels (relative fitness levels have been multiplied by R = 4.0 in A and by R = 8.0 in B for conversion to absolute fitness levels). The extinction threshold is shown as a thick black line at the absolute fitness level of 1. All populations in panel A will eventually go extinct, although one of the multiplicative models does not cross the extinction threshold until generation 12. None of the populations in panel B will go extinct, because their intrinsic fecundities (R) are high enough to offset the deleterious effects of a mutation rate of U_d = 2. Once a curve drops below the extinction threshold, the population size begins declining, but the time until complete loss of the population depends on initial population size and may take many generations. Gene frequency dynamics are the same in both graphs despite the different outcomes in extinction.

FIG. 4.

Lethal mutagenesis threshold according to mutation rate U_d and maximum fecundity R_max, from inequality 3. The relationship is log linear, so that changes in mutation rate have a much larger effect on extinction than changes in fecundity. In turn, modest increases in mutation rate, especially for RNA viruses, may be especially amenable to achievement of extinction.