Theory of Lethal Mutagenesis for Viruses

Mutation is the basis of adaptation. Yet, most mutations aredetrimental, and elevating mutation rates will impair a population'sfitness in the short term. The latter realization has led tothe 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, althoughreference is commonly made to Eigen's error catastrophe theory.Here, we propose a theory of lethal mutagenesis. With an obviousparallel to the epidemiological threshold for eradication ofa disease, a sufficient condition for lethal mutagenesis isthat each viral genotype produces, on average, less than oneprogeny virus that goes on to infect a new cell. The extinctionthreshold involves an evolutionary component based on the mutationrate, but it also includes an ecological component, so the thresholdcannot be calculated from the mutation rate alone. The geneticevolution of a large population undergoing mutagenesis is independentof whether the population is declining or stable, so there isno runaway accumulation of mutations or genetic signature forlethal mutagenesis that distinguishes it from a level of mutagenesisunder which the population is maintained. To detect lethal mutagenesis,accurate measurements of the genome-wide mutation rate and thenumber of progeny per infected cell that go on to infect newcells are needed. We discuss three methods for estimating theformer. Estimating the latter is more challenging, but broadlimits to this estimate may be feasible.

INTRODUCTION

Top

Introduction

Some viral infections for which vaccines are unavailable orineffective can be treated with antiviral drugs. One of themore interesting mechanisms suspected for some antiviral drugsis lethal mutagenesis, pushing a within-host population of virusesto extinction by overwhelming it with an elevated mutation rate.Lethal mutagenesis has emerged as a concept and practice withoutmuch theoretical underpinning. Consequently, it is not evenclear how to make the measurements to ascertain whether lethalmutagenesis is operating. Much of the viral literature equateslethal mutagenesis with the error catastrophe originally proposedby Eigen in the context of quasispecies (1, 13, 27). Indeed,the very idea of lethal mutagenesis seems to have been inspiredby theories of the error catastrophe. Ironically, the two conceptsare not the same: an error catastrophe is an evolutionary shiftin genotype space, whereas extinction is a demographic process,a drop in the absolute abundance of individuals in the population.An error catastrophe can delay or even prevent extinction byshifting the population to genotypes that are robust to mutation(see below), while lethal mutagenesis is by definition a processthat pushes the population to extinction.

Despite the confusion over error catastrophes and their relationto extinction, empirical evidence broadly supports the principleof lethal mutagenesis. Chemical mutagens have been used to artificiallyincrease error rates in a variety of RNA viruses, includingvesicular stomatitis virus (VSV) (33, 39), human immunodeficiencyvirus type 1 (HIV-1) (40), poliovirus type 1 (12, 33), foot-and-mouthdisease virus (55), lymphocytic choriomeningitis virus (30),Hantaan virus (54), and hepatitis C virus (66). The drugs severelyreduced viral titers and in some cases achieved extinction.Thus, lethal mutagenesis appears to have merit in principleand also to be biochemically feasible with various drugs.

Whereas the theory of error catastrophe has been developed andexpounded for decades, the theory of lethal mutagenesis remainsto be developed, which is our purpose here. More specifically,our intent is to synthesize existing empirical and theoreticalwork to explain the quantities relevant to lethal mutagenesis.None of the theory offered here is specifically original; rather,it is the application of simple models and the interpretationof those results in the context of empirical methods that makesthis paper original.

Understanding the genetic and demographic bases of populationextinction has been the goal of many papers in the evolutionaryand the ecological literature. The ecological literature hasaddressed population size and inability to adapt as key featuresof extinction (21, 37, 38). In the evolutionary literature,a major focus has been to discover the reason why parthogeneticplants and animals do not persist (4, 45). Processes such asmutational meltdown via Muller's ratchet and fixation of deleteriousgenes in small populations have been entertained as mechanismsof extinction (31, 42, 47). Muller's ratchet is the progressiveaccumulation of deleterious mutations in finite, asexual populations.If back mutations cannot occur, then any finite asexual populationwill eventually reach the point at which each genome carriesat least one deleterious mutation. The mutation-free wild-typegenome is forever lost from the population at this point. Bythe same mechanism, eventually each genome in the populationwill carry at least two mutations and then at least three, andso on, and all genomes with fewer mutations are forever lostand cannot be reconstructed without recombination.

Lethal mutagenesis is distinct from those processes becausethe latter require small populations. Lethal mutagenesis isa deterministic process that will overwhelm the largest of populations.Furthermore, the time scale over which lethal mutagenesis operatesis potentially much shorter than the time scale usually attributedto processes such as Muller's ratchet, one of the few otherextinction mechanisms that can operate in relatively large populations(50). Viral extinction may occur at two levels: (i) a clearanceof the infection within one host or (ii) extinction of the virusacross the entire population of hosts. There are mathematicalsimilarities between the two cases but profound biological differences.Historically, the domain of lethal mutagenesis has been extinctionwithin a host, which is what we consider here. To eradicatea virus by lethal mutagenesis across the entire population ofhosts would require treatment of virtually every infected hostthroughout the time of its infection, which is not practical.Below, therefore, the use of the word "population" in referenceto viruses refers to the viruses within one host.

THEORY

Top

Theory

Approach and models. Lethal mutagenesis is a form of extinction. Most basically,it requires that deleterious mutations are happening often enoughthat the population cannot maintain itself, but it is otherwiseno different from any other extinction process in which fitnessis not great enough for one generation of individuals to fullyreplace themselves in the next. (We consider viral fitness tobe the average number of progeny capable of infecting new cellsproduced by a specific viral genome. Thus, the more viable progenya virus produces, the higher its fitness.) Although lethal mutagenesisis a genetic (evolutionary) phenomenon because it is drivenby mutations, it is not an exclusively genetic one because italso depends on absolute reproductive rates. Consequently, therewill be no universal mutation rate that signals extinction forall viruses or even for the same virus under different conditions.Consider the following example. For a virus to establish aninfection in the body, it must produce an excess number of progeny,whereby one infected cell gives rise, on average, to more thanone new infected cell; otherwise, the infection will not spread.Lethal mutagenesis is a mechanism by which that excess is curtailedand rendered negative. If the excess is 49, whereby one infectedcell creates 50 new infected cells, mutagenesis will need tobe high enough to harm 98% of the progeny; if the excess isonly one, mutagenesis will need to be high enough to harm only50% of the progeny. Thus, the ecology or natural history ofthe infection will determine how much mutagenesis is requiredfor extinction, and that dependence on ecology means that thereis no universal genetic law for lethal mutagenesis.

The dependence of lethal mutagenesis on fitness means that itindeed contains an evolutionary component in addition to theecological one. This evolutionary component requires a detailedunderstanding of the relationship between mutation rate andfitness, and that relationship is affected by how fitness changeswith increasing numbers of mutations in the genome. These evolutionaryproperties defy intuition, so we resort to mathematical models.The models will proceed in three steps. First, we specify howfitness declines with increasing numbers of mutations in thegenome. Here, we consider three simple models. Second, we considerthe relationship between mutation rate and average fitness whenthe population has reached genetic equilibrium. As has alreadybeen established in the population genetics literature, thisrelationship is a simple one that applies across broad classesof models. It is also an important part of the lethal mutagenesisthreshold. Third, we add the ecological component to the lethalmutagenesis model to achieve the threshold.

General assumptions. Our models make several simplifying assumptions about the viralmutation process and mutational effects on viral fitness. Ourapproach best suits an infection that, except for treatment,is maintained indefinitely. Foremost, the viral population sizeis very large and the target cell population is even larger.We assume discrete generations in an ongoing infection processin which virus in each infected cell is subjected to a genomic-mutationrate of U mutations per genome per replication. Viral progenyare released and go on to infect new cells, where they are againsubjected to a mutation rate of U per genome.

For all models, mutations occur at random and are equally likelyto affect any site in the genome. Under this assumption, thenumber of mutations in a genome follows a Poisson distribution.The Poisson distribution assigns a probability of occurrenceto each possible number of mutations that may arise in a genome(i.e., 0, 1, 2,..., ${infty}$ ) and is characterized by a single parameter,U, which gives the mean number of mutations per genome. In whatfollows below, we neglect finite population effects on mutationfrequencies and assume that recombination is absent and thatall mutations are unique and either deleterious or neutral;the possibilities of beneficial mutations, compensatory evolution,parallel evolution, and reversion are consequently absent. Sincewe assume that all mutations are either neutral or deleterious,we can subdivide the mutation rate U into component U_n, comprisingpurely neutral mutations, and component U_d, comprising mutationswith a (deleterious) fitness effect, and write U = U_n + U_d.

All models assume that the fitness of individuals with j deleteriousmutations is independent of the identity of those mutations.For convenience and without loss of generality, the relativefitness of mutation-free genotypes is set at unity (w₀ = 1);models that require absolute fitnesses are indicated where neededand parameterized accordingly.

Three models. We consider three simple models (Fig. 1). None of these modelsis considered biologically realistic, but they collectivelyspan the spectrum of possibilities usually addressed. (i) Inthe multiplicative fitness landscape, each additional deleteriousmutation reduces viral fitness by a fraction s, independentlyof the number of mutations already present, and the fitnessof a genotype carrying j nonneutral mutations is w_j = (1 –s)^j. (ii) In the Eigen two-class fitness landscape, the wild-typegenotype has one fitness (arbitrarily set to 1) and all genotypeswith one or more nonneutral mutations have the same (lower)fitness (18, 19). We denote the fitness of the non-wild-typesequences by w_{j > 0} = 1 – s. Note that mutations inthe Eigen model are conditionally neutral: each mutation individuallyhas a fitness effect of 1 – s, but multiple nonneutralmutations have the same fitness effect as a single nonneutralmutation. (iii) In the truncation landscape, a small numberof mutations can be tolerated without effect, but any genotypecarrying too many mutations is inviable. More specifically,genotypes carrying 0 to k nonneutral mutations have fitnesslevels of 1, and genotypes with k + 1 or more mutations aredead (w_j = 1 for j $≤$ k and w_j = 0 for j > k). Note that mutationsin the truncation landscape are conditionally deleterious: eachmutation individually has no fitness effect, but k + 1 or moremutations are lethal.

View larger version (9K):
[in this window]
[in a new window]

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.

The multiplicative model assumes that mutations have independenteffects; hence, there is no genetic interaction (epistasis).The Eigen model represents an extreme case of antagonistic epistasis(in which mutations have reduced impact levels as they accumulate),and the truncation model represents an extreme case of synergisticepistasis (in which mutations have increased impact levels asthey accumulate). The Eigen model is well known because it exhibitsan error threshold, as discussed below.

RESULTS

Top

Results

Mean fitness level at equilibrium declines exponentially with the deleterious-mutation rate. When a population is first subjected to a sustained, increasedlevel of mutation, fitness will typically decline over severalgenerations, until it eventually levels off and reaches a newequilibrium. The reason for this gradual approach to the newequilibrium is that mutations accumulate with each succeedinggeneration. The ultimate drop in fitness is due to the totalburden of mutations from many preceding generations. At equilibrium,further accumulation of deleterious mutations is fully counteractedby selection against those mutations already present. Whethermutagenesis is lethal (ultimately causes population extinction)will depend in part on this equilibrium fitness.

For each of the three models considered here, with an exceptionnoted below, the mean fitness at equilibrium () is simply the Poisson fraction of mutation-freegenotypes (not counting neutral mutations), = e^–U_d (35). It is remarkable that this equilibriumis independent of the selective effects of those mutations andindependent of epistasis (in the absence of recombination).It is important to use the deleterious-mutation rate U_d ratherthan the overall genome-wide mutation rate U in this expression(61). Recall that beneficial mutations are not allowed in ourmodels.

This relationship between mutation rate and mean fitness leveldoes not hold at high mutation rates in the Eigen error catastrophemodel: an error catastrophe actually maintains fitness abovee^–U_d (Fig. 2). An error catastrophe is an evolutionaryphenomenon in which high-fitness genotypes are lost from thepopulation because they are sensitive to mutations, and thepopulation evolves to genotypes that are low in fitness butrobust to the effects of mutations. Thus, mean fitness in theEigen model behaves as a hybrid of two processes. At low mutationrates, fitness declines exponentially with increasing mutationrate, as in the other models. This relationship applies up tothe error threshold. At mutation rates above the error threshold,fitness stops declining: there is no change in mean fitnesslevel because all genotypes are insensitive to mutation. Therefore,contrary to common perceptions, in the Eigen two-class fitnesslandscape an error catastrophe actually retards the extinctionof the population.

View larger version (10K):
[in this window]
[in a new window]

FIG. 2. Equilibrium mean fitness level as a function of deleterious-mutation rate U_d. The solid curve is Formula 4 , 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.

In conclusion, except at the error catastrophe, mean fitnesslevel depends entirely on the deleterious-mutation rate, noton the fitness effects of mutations or on the way mutationsinteract. The higher the mutation rate, the lower the population'sfitness. This result holds for infinite, asexual populations;for sexual populations, it holds only if there is no epistasis(35). This result also relies on the assumption of no back mutation.When this assumption is relaxed, mean fitness is determinedby fitness effects (53, 61, 63), and low population sizes amplifythis effect (36). Nevertheless, = e^–U_d is a good first-order approximation even in thesecases (8). Note also that none of the results that we presentin the following sections are strictly dependent on the assumptionof no back mutations. We employ this assumption mainly for mathematicalsimplicity and clarity of presentation.

An extinction criterion. The results described above provide the mean fitness level ona relative scale, in which the best (wild-type) genotype hasa fitness level of 1. Unfortunately, models of relative fitness(which include all of Eigen's error threshold models) do notenable direct calculation of extinction rates, because actualnumbers of offspring cancel out in those types of models (seethe supplemental material). The difference between extinctionand survival depends on actual birth rates or numbers of offspringand thus depends on absolute fitness (6, 60, 62). Extinctionis a demographic phenomenon. Although the relationship betweenmutation rate and population mean fitness level is central tothe calculation of extinction conditions, knowledge of onlymean relative fitness level is insufficient to determine extinction,as we show now.

Deterministically, a decline in the population size will occurwhen the average number of offspring per parent is less than1 for all genotypes:

Formula 1 (1)
Ifthe inability of parents to replace themselves continues indefinitely,population extinction will ultimately result (neglecting limitingcases of infinitesimal declines). To extend this result to lethalmutagenesis, it is necessary to separate the evolutionary (mutational)and demographic components. If we define R as the number ofsuccessful viral offspring released per cell in the absenceof mutation, the general condition for a decline in the populationsize is then

Formula 2 (2)
(seethe supplemental material). This formula multiplies relativefitness level () by the maximumnumber of offspring per parent to create a measure of absolutefitness, i.e., number of progeny. The term represents the evolutionary component, andthe R term represents the demographic or ecological component;our use of R is in fact borrowed from the demography literature.

Formally, R applies to the wild-type, mutation-free genotypeand is the number of progeny released from one infected cellthat go on to establish infections in other cells. The reasonfor basing R on the best or mutation-free genotype is so thatall effects of mutation may be subsumed into . This dependence on the best genotype poses someempirical challenges when these quantities are being measured.These difficulties will be addressed below, although one ofthose dependencies will be overcome by a modification introducednext.

If both terms in equation 2 are constant over time, then condition2 ensures population extinction. However, R may not be constant.As is recognized in the literature on demography (2), the valueof R will often be density dependent, largest when the populationis at its lowest density. If, by the same argument, R increasesas the viral population nears extinction, inequality 2 may besatisfied initially but later reverse and lead to a stable populationreduced in size. A more stringent condition is thus the replacementof R with R_max, representing the maximum reproductive rate ofthe mutation-free genotype across all viral population densities.Using the results for the equilibrium mean fitness level atmutation rate U_d, equation 2 is easily modified to provide asufficient condition for lethal mutagenesis in the absence ofan error catastrophe:

Formula 3 (3)
Thisextinction criterion can be justified from two perspectives.As suggested above, this condition merely states that the averagenumber of successful offspring per infected cell is less than1 once the mutation-selection equilibrium has been attained(successful offspring are those that go on to infect new cells).The population size must then decline. A different justificationfor this result comes from the theory of branching processes(32). e^–U_d is both the mean fitness level and also thefraction of offspring with no nonneutral mutations. If thisnumber is so low that the best genotype produces (on average)less than 1 viable offspring that successfully initiates a newinfection, then the population will go extinct eventually.

Recall that e^–U_d is the mean fitness level in all modelslacking an error threshold and in the Eigen model before theerror catastrophe. It may be impossible to satisfy condition3 in the Eigen model if the error threshold occurs before themean fitness level drops to the requisite value e^–U_d.Extinction is impossible in this specific case because all mutantgenotypes have absolute fitness levels high enough to replacethemselves, and fitness cannot drop lower. Of course, this extremecase is unrealistic, but it serves to illustrate the possibilitythat some realistic models may not obey criterion 3.

Dynamics of extinction: population decline may not happen immediately. It is instructive to consider some numerical examples to developa sense of the overall process of lethal mutagenesis. Figure3 shows the decline in fitness over the first 10 generationsof exposure to mutation rate U_d = 2 for each of our three models.The equilibrium mean fitness level is 0.135 for this mutationrate in all three models. There are five curves in each graphbecause two of the models are illustrated with two values eachfor the selection coefficient s. The calculations assumed astarting genotype lacking mutations, and the curves representthe initial changes in fitness as the viral population evolvestoward equilibrium. The data shown in Fig. 3A are the same asthose in Fig. 3B, except that the mean absolute fitness levelon the vertical scale in Fig. 3B is exactly twice what it isin Fig. 3A: the mutation-free genotype was assumed to have fouroffspring in Fig. 3A and eight offspring in Fig. 3B. Gene frequencyevolution is thus the same for the corresponding curves betweenFig. 3A and B, but the two graphs differ in whether the outcomeis ultimately extinction or survival. In both graphs, the extinctionthreshold is an absolute fitness level of 1, indicated withthe horizontal black line. Extinction (lethal mutagenesis) wouldnever occur for any of the curves in Fig. 3B but will eventuallyoccur for all the curves in Fig. 3A. The reason for the differencein extinction is evident from our criterion, e^–U_dR_max< 1. Thus, 0.135 x 4 = 0.54 < 1 implies extinction inFig. 3A but 0.135 x 8 = 1.08 > 1 implies survival in Fig.3B. All but one of the curves in Fig. 3A have crossed the extinctionthreshold by generation 10; hence, their populations would bedeclining in generation 10; one curve in Fig. 3A has not crossedthe threshold by generation 10 but would by generation 12. Evenafter crossing the extinction threshold, the population maypersist for hundreds or thousands of generations, dependingon its initial size and how close to 1.0 the absolute fitnesslevel remains at the mutation-selection equilibrium.

View larger version (12K):
[in this window]
[in a new window]

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.

These graphs emphasize two points. First, the same gene frequencydynamics operate regardless of whether the outcome is extinctionor survival (while population size remains large). Thus, thereis no genetic signature of lethal mutagenesis to distinguishit from the same mutagenesis and relative fitness effects observedwhen the population survives. In particular, lethal mutagenesisoperating in large populations is not a runaway mutation accumulationin which mutations increase indefinitely (although stochasticeffects will augment mutation accumulation once the populationbecomes small). The ecology of the infection determines whetherthe outcome is extinction or survival via its effect on offspringnumber, R_max, but the genetics behaves the same. Second, lethalmutagenesis is usually a progressive decline. Unless mutationrates are elevated to extreme levels, absolute fitness may stayabove the extinction threshold for several generations, untilmutations have accumulated. Even after the threshold is breached,reproduction continues, just not enough to maintain populationsize. This prediction is compatible with results from mutagenesisexperiments indicating that viral extinction is not sudden.For example, the presence of the base analog 5-hydroxydeoxycytidineresulted in the loss of HIV-1 infectivity after 9 to 24 serialpassages in human cells (40). Similarly, 5-fluorouracil or 5-azacytidinecaused occasional extinction of foot-and-mouth disease viruspopulations after 11 to 21 serial passages (55).

Estimating parameters of lethal mutagenesis. The extinction threshold for lethal mutagenesis involves twocomponents. One is evolutionary and depends only on the deleterious-mutationrate. The other is demographic, an absolute fecundity specificto the infection, and applies to the best genotype. Both presentdifficulties in estimation, especially in vivo. However, thedeleterious-mutation rate is potentially the most importantand most empirically tractable of the two.

Measuring the deleterious-mutation rate, U_d. The meaning of U_d is straightforward: it is the genomic rateof deleterious mutations per generation. Using the tools ofmolecular biology, this value is perhaps most easily soughtas the product of two numbers, the genome-wide mutation rateU times the fraction of mutations that are deleterious, ${alpha}$ = U_d/U.The proportion of mutations that are deleterious, ${alpha}$ , has beenestimated as 70% in VSV for randomly generated point mutations(40% lethal, 30% viable but deleterious [51]). Direct estimatesof ${alpha}$ for other viruses are not available. Interestingly, however,our models can provide an indirect estimation of a componentof ${alpha}$ , the fraction of mutations that are nonviable mutations.We obtain this estimate by reanalyzing data from a study thataddressed the impact of the mutagen ribavirin on poliovirustype 1 infectivity (12). Mutations were counted by sequencingof biological clones obtained from isolated PFU, and infectivitywas measured as the number of PFU per standard amount of genomicRNA. These two variables are plotted in Fig. 4b of reference12. To carry out our analysis, we have to assume that the observednumbers of mutations equal the mutation rates (U). Exact equalityexists only in the first generation after mutation (Table 1),but equating numbers with rates seems a reasonable approximationin this case because viruses were sequenced only a few replicationrounds after mutagenesis. Since the infectivity assay distinguishesonly viable and nonviable mutants, we must take into accountonly neutral (s = 0) and lethal (s = 1) mutations. EquationS8 in the supplemental material takes the following form here:infectivity = e^–^m, where ${gamma}$ is the fraction of lethal mutationsand m is the mutation count. (Note that ${alpha}$ measures the fractionof deleterious mutations, which is a superset of the lethalmutations. We always have ${gamma}$ $≤$ ${alpha}$ .) A least-squares regression yieldsan estimate of ${gamma}$ = 0.33 ± 0.13, which means that approximatelyone-third of the mutations produce noninfectious virions. Thisresult is similar to that obtained for VSV in the absence ofdrugs. Hence, even though the effects of individual mutationsdepend on the environment, the overall fraction of lethal mutationsmight be roughly constant for different RNA viruses in differentenvironments. It would obviously be most useful to lethal mutagenesisapplications if the value of ${alpha}$ were relatively constant.

View larger version (11K):
[in this window]
[in a new window]

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.

View this table:
[in this window]
[in a new window]

TABLE 1. Number of mutations per genome equals the mutation rate only in the first generation before selection

Whereas ${alpha}$ , the fraction of all mutations that are deleterious,might be independent of context and thus be estimable from experimentsdone outside the context of specific drugs, mutation rate estimationsmust be carried out in the presence of the mutagenic drug. Threeapproaches to mutation rate estimation have been commonly used.Each has its own strengths, but none is free of difficulties,as summarized in Table 2.

View this table:
[in this window]
[in a new window]

TABLE 2. Summary of methods for estimating mutation rates and their respective advantages and drawbacks

The classic method for estimating mutation rates is the Luria-Delbrückfluctuation test (41; see reference 65 for a review): an easilyselected phenotype is scored in replicate cultures inoculatedwith phenotype-negative genotypes. For example, a small inoculumof an antibody-sensitive virus would be used to infect a numberof identical cultures without antibody, grown for a period oftime and then plated in the presence of the antibody to testfor the appearance of resistant mutants. In its simplest version,the proportion of phenotype-negative cultures is used to estimatethe mutation rate from the formula P₀ = e^–µN; P₀is the proportion of mutant-free cultures, ${Delta}$ N is the change inpopulation size during the growth of the cultures, and µis the rate per replication event at which the phenotype convertsinto the selectable state. Under some additional assumptions,it is possible to estimate µ from the entire observeddistribution of the number of phenotype-positive individualsper culture (24, 56). To extrapolate phenotypic mutation ratesto genome-wide mutation rates (U), it is further necessary toknow the number of different mutations that give rise to theselectable phenotype. This method has been used to estimatemutation rates for influenza A virus (57), measles virus (52),bacteriophage ${Phi}$ 6 (11), and VSV (24). Extrapolation to genomicerror rates was done for the last three viruses, yielding U= 1.4, U = 0.03, and U = 0.07, respectively. Note that thisextrapolation may be subject to a potentially large and difficult-to-quantifyerror. If there are few ways of mutating to the phenotype, thenthe estimates obtained from the Luria-Delbrück assay maydeviate considerably from the genomic average because site-to-sitevariation can be large. One should also consider that biaseswith Luria-Delbrück estimates may exist when a substantialfraction of mutations are lethal, but we have not explored thispossibility.

A second approach to estimating the mutation rate is to measurethe number of mutations in sequences (mutation count), as hasbeen done in several mutagenesis experiments (12, 29, 30, 33,40, 54, 55, 66). The difficulty is that there is no feasiblemethod for converting numbers of mutations into a mutation rate.In particular, when mutation-free templates are used initially,the observed numbers of mutations increase with each succeedinggeneration, counteracted by selection against the mutations.The net accumulation thus depends on the fitness effects ofthe mutations, mutation rate, and time; the observed accumulationwill not generally allow a unique determination of mutationrate without independent knowledge of fitness effects and numberof generations. If all mutations are neutral, the number ofmutations increases by the neutral mutation rate each generation,whereas if all mutations are lethal, then mutations do not accumulate,because mutated genomes die. In between these two extremes,any observed rate of mutation accumulation could stem from ahigh rate of highly deleterious mutations or a low rate of weaklydeleterious mutations. Table 1 shows that the number of mutationsper genome equals U only in the first generation. After thispoint, the number of mutations per genome depends on the deleterious-mutationrate, the generation number, and the selective effect of mutations.We are unaware of any empirical study in which these considerationshave been applied when estimating mutation rates from mutationfrequencies.

Although there are many complications in estimating mutationrates from counts, we see from Table 1 that one simple methodmay be feasible: expose viral genomes to a single generationof mutagenesis and measure the counts before selection. Forexample, cells could be infected at a multiplicity near 1 inthe presence of a drug. The drug may prolong the infectiouscycle, but as long as nearly all cells are infected initially,there will be few cells to be infected in second cycles. Eitherthe virus should kill the cell or the infected cell should resistsuperinfection for this method to give meaningful results. Individualvirions from the resulting culture are then sequenced directly,in the absence of any subsequent infection or other biologicalamplification process that would cause a bias against deleteriousmutations.

Yet, even this simple protocol, which is technically feasiblewith many viruses, gives a direct estimate of mutation rateonly if viral replication within a cell does not select againstmutations that arise within that cell (as when the infectinggenome is the template for all copies). If the nature of replicationis unknown or differs from the single-template mechanism, onesolution may be to measure mutation rates in parts of the genomethat would not be subject to selection within the cell. Anotherlimitation of this method is that mutations may have accumulatedprior to the beginning of the experiment or may be introducedby reverse transcription during the synthesis of the cDNA. Sophisticatedtechnology has been developed for measuring mutation rates duringa single infection cycle and in the absence of selection inretroviruses (see reference 44 for a review). This technology,based on the use of genetic constructions that carry nonviralgenes, offers a valuable tool for lethal mutagenesis experiments.These genes are partially released from selection and hencecan be used to estimate mutation rates more accurately. A similarapproach was undertaken using a cognate mutational target ina study with tobacco mosaic virus in which the viral gene thatencodes the movement protein was complemented by a plant transgene(43).

The two methods described above provide estimates of mutationrates but require independent estimates of the fraction of deleteriousmutations. A third method, mutation accumulation, partly overcomesthis problem. Mutation accumulation experiments offer a methodfor directly estimating deleterious-mutation rates, but onlyfor viable mutations. For VSV (20) and bacteriophage ${phi}$ 6 (7),this method has yielded estimates of U_d = 1.2 and U_d = 0.07,respectively. (Note that these values are not directly comparableto the Luria-Delbrück estimates of U for the same virusesbecause of differences in the methodologies.) Starting froma single clone, several lineages are founded and propagatedat the lowest possible population size, which is typically doneas plaque-to-plaque transfers. Small population size facilitatesthe accumulation of all nonlethal mutations, even deleteriousones, through genetic drift. Together, the average rate of fitnessdecline and the variance between lines enable estimates of thedeleterious-mutation rate and the average deleterious effect;the Bateman-Mukai (3, 46), maximum-likelihood (34), and minimum-distance(25) methods are different statistical approaches to these estimates.A benefit of estimating mutation rates by this method is thatneutral mutations do not affect the estimates, and of course,neutral mutations are also not relevant to lethal mutagenesis.On the other hand, unlike for experiments with higher organisms,it is impossible to maintain population size to a single individual,so some degree of selection is inevitable (e.g., during plaqueoutgrowth), introducing a bias against strongly deleteriousmutations. As a consequence, lethal mutations will be completelymissed. Another limitation of this design for viruses is thatplaques may become undetectable at very low fitness levels,a situation that is particularly likely in mutagenized populations.

Measuring maximum fecundity, R_max. The second parameter in the extinction threshold is a type offecundity, R_max. R is the average number of offspring per cellinfected by the mutation-free genotype that would go on to establishnew infected cells. This parameter specifically applies in vivo,so its measurement is not trivial. Some histological observationson plants inoculated with tobacco mosaic virus suggest thatR may be as low as 3 to 6 particles per cell (43).

It is easiest to contemplate the fecundity value R as the productof two parameters, S and b (R = Sb). The parameter S is thesuccess rate, corresponding to the survival of mutation-freeprogeny in establishing infections in new cells, and b is theburst size, i.e., the number of viable viral offspring releasedfrom a cell infected by the mutation-free genome. In general,b is a number much larger than 1, but S is always strictly smallerthan 1, being reduced by senescent decay, immune clearance,and other properties specific to the infection in vivo. Thus,R may be much less than the number of offspring (burst size)per infected cell, because the success rate, or survival ofprogeny virus, may be low. Additionally, the values of S (andpossibly b) may be density dependent, larger when the viralload is low and smaller when the viral load is high. Indeed,for a viral infection to reach an equilibrium density, one orboth of these quantities must decline as the infection grows.Therefore, as mentioned previously, the R value sufficient tosatisfy our extinction criterion must be the maximum acrossall stages of the infection, R_max. Otherwise, mutagenesis mightreduce viral load down to the point that inequality 3 is reversed.

The mutagenic drug may impair b or S directly, contributingto achievement of the lethal threshold by nonmutagenic processes.For example, ribavirin negatively impacts viral fitness by possiblyfour molecular mechanisms besides mutagenesis (28). Those effectsfacilitate satisfaction of the extinction criterion of equation3 by reducing R_max. Indeed, if a mutagenic drug is so harmfulto viral reproduction that R_max is <1, extinction will occurregardless of the mutagenic effect. (In general, Fig. 4 canbe used to calculate the combination of effects on U_d and R_maxthat will cause extinction together.)

There are several difficulties in estimating R_max and thus band S. First, the model assigns these values to the wild-type,mutation-free genotype. It may not be possible to confine assaysto mutation-free genotypes or even to know which genomes aremutation free. This difficulty could lead to underestimationof both parameters, as genomes with accumulated mutations willlikely have lower values of b and S than those that are mutationfree. Another complication is that mutagenesis may confoundthe estimates of b and S. For example, the estimate of b, thenumber of viral progeny produced per infected cell, might seemto be obtained easily, but mutations arising in progeny thatkill or otherwise harm them may interfere with progeny counts(which are typically done by plaque assays) and thus lead tounderestimation of b. Finally, the estimate of b will likelydepend on whether b was measured at a low or high multiplicityof infection.

The difficulties in obtaining direct estimates of R_max may requireworking with crude upper limits and indirect estimates. Fortunately,great accuracy in the estimate of R_max is not essential, becausethe mutation rate satisfying equation 3 appears as an exponent,so small changes in U_d can overwhelm large differences in R_max(Fig. 4). A gross upper limit to R_max might be obtained by settingS to 1 and measuring b in cell cultures. For example, for bacteriophage ${Phi}$ 6, the burst size was estimated as b = 76 PFU per cell (11),whereas in an animal virus, such as VSV, a single infected cellcan often produce several thousand particles (22, 23). Theseestimates for b are not fully independent of the mutation rate,as mentioned above, but they give an idea of the order of magnitudeof b for cellular cultures in the absence of mutagens. If wereplace R_max by b in equation 3, we have

Formula 4 (4)
This is a crude but, most probably, conservativecondition for lethal mutagenesis. For example, for b = 100,a deleterious-mutation rate of U_d ${approx}$ 4.6 would suffice to fulfillthis condition, whereas U_d ${approx}$ 6.9 would be necessary for b = 1,000.If basal mutation rates in RNA viruses were around 1 (14, 15),given that the majority of mutations are deleterious (17, 51),an approximately five- to sevenfold increase in mutation rateswould be sufficient to achieve lethal mutagenesis. Since somemutagens are known to reduce the replicative capacities of virusesby mechanisms independent of mutagenesis (28) and b is a crudeupper limit to R_max, more-modest increases in U_d will probablysuffice to induce lethal mutagenesis. A series of experimentswith RNA viruses or retroviruses replicating in the presenceof base analogs have shown that reductions of several ordersof magnitude in viral titers and even extinction can be achievedwith modest increases in mutation counts, ranging from lessthan twofold to sixfold relative to those for the untreatedcontrols (12, 30, 33, 40, 55).

DISCUSSION

Top

Discussion

Lethal mutagenesis is an elevation of mutation rate to the pointthat a population is so overwhelmed by deleterious mutationsthat it cannot maintain itself. This method has been suggestedas the basis of successful treatments of viral infections byuse of drugs known to elevate mutation rates. This paper hasdeveloped a simple theoretical condition for the operation oflethal mutagenesis in the viral infection of a culture or host: Formula 4

, where U_d is the genomicrate of deleterious mutation and R_max is the maximum averagenumber of viral progeny per cell infected with wild-type virusthat go on to establish new infected cells. Although the Eigenerror catastrophe theory is often invoked as the theoreticalbasis of lethal mutagenesis, that process is different fromlethal mutagenesis and may actually retard lethal mutagenesis.

The goal of treatment could be to reduce viremia during an acuteinfection or to end a persistent infection. With an acute infection,it is likely that any decrease in mean fitness due to mutagenesiswill slow the ascent of the viremia and thereby augment recoveryby the immune system. For this case, mutagenesis need not surpassthe extinction threshold to have a beneficial effect: any reductionin viral fitness will reduce the rate at which the within-hostviral population expands, potentially enabling the immune systemto clear the infection earlier. Our theory applies also to casesof persistent infection, which is associated with the usualapplication of lethal mutagenesis.

Extinction threshold equation 3 is sufficient to cause viraldecline but is possibly conservative and may specify a highermutation rate than necessary. For example, the within-host growthrate of the viral population could be important to the outcomeof the infection, and a slowing of viral growth rate will beachieved even if Formula 4 is not satisfied. The exact mutation-extinction threshold lies in theecology of each type of infection, and a specific model of thesedynamics, as well as of the impact of mutation on the differentinfection parameters, is required. What generalities will befound by studying specific models remains to be seen, however.

As a second example of how our extinction criterion is possiblyconservative, stochastic effects may contribute to the fixationof deleterious mutations in finite populations through processessuch as mutational meltdown mediated by Muller's ratchet (42).For a population of size N with deleterious-mutation rate U_d,the expected number of individuals without any nonneutral mutationsat equilibrium is Formula 4 . For N₀< 1, there is a high probability that these mutation-freeindividuals are lost by chance or never exist, and the losswill be irreversible without back mutation (9, 26, 31). Thisstochastic mutation accumulation process, or Muller's ratchet,can ultimately lead to the extinction of the population (42).Small populations are prone to fixing deleterious mutations,as has been amply demonstrated in several experiments (10, 16,64). The emphasis in those studies has been small populationsize, but as the important quantity in the above formula isN₀, increasing the mutation rate through the use of chemicalmutagens is an alternative to reducing the total populationsize. For the multiplicative model, beyond a mutation rate ofU_d = s ln N, the population risks extinction even without deterministiclethal mutagenesis. Not surprisingly, the combination of populationbottlenecking and chemical mutagenesis has proven to be themost efficient way to achieve viral extinction (55).

An interesting outcome of the theory presented here is thatthere is no genetic signature of lethal mutagenesis that distinguishesit from nonlethal mutagenesis. Mutagenesis itself obviouslyhas a genetic signature, but whether extinction will resultdoes not. The same elevated mutation rate may or may not causepopulation extinction, and at least while the population isstill large, the genetic evolution of deleterious mutationsis the same whether the population is stable or declining. Thereis no mutational runaway accumulation of mutations accompanyinglethal mutagenesis. The reason for this genetic independenceof population survival versus extinction is that genetic evolutiondepends on relative fitness, whereas population survival dependson absolute fitness, i.e., total numbers of offspring.

There are demonstrations, both in vivo and in vitro, that theaddition of mutagens can lead to the extinction of the viralpopulation (29, 30, 40, 55). Whether these results constitutea clear demonstration of lethal mutagenesis depends on the otherpotential effects of the mutagen. The most thorough empiricalstudy of this problem measured mutation counts (as approximationsof rates) in poliovirus subjected to ribavirin treatment invitro (12). Even if those estimates of mutation counts are acceptedas rates, it is further necessary to estimate the number ofviruses produced by one cell that go on to infect other cells.At the highest dose in that study, mutation counts were indeedquite high per genome (15.5, a high rate even if due to an accumulationover a few generations), and it seems likely a priori that mutagenesiswould have been high enough to ensure extinction (Fig. 4). However,it is also possible that other effects of the drug would havebeen high enough to eradicate the virus without mutagenesis.

Every viral infection that could potentially be treated witha mutagen falls into one of three categories: (i) Formula 4 , where U₀ is the deleterious-mutation rate inthe absence of mutagenesis, in which the virus cannot successfullyestablish an infection, and mutagenesis is not necessary forextinction but might shorten the total duration of infection;(ii) , in which mutagenesis is necessary for extinction; and (iii) Formula 4 , in which extinction does not occur despite mutagenesis.The last case is potentially a worry because the elevated mutationrate might facilitate evolution to a part of the fitness landscapethat was otherwise not likely to be accessed. In general, ifmutagenesis increases the mutation rate closer to the mutationrate optimum for the virus, then mutagenesis will presumablybe counterproductive for treatment. This possibility seems unlikelyfor RNA viruses, as their intrinsic mutation rates are so high.However, the relevant parameters are not adequately known toexclude this possibility, so the caution seems warranted.

The assumption of a constant mutation rate across all infectedcells is possibly valid for in vitro systems but may be violatedin vivo. In a multicellular host, refugia might exist with lowdrug concentrations, as has been observed for HIV-1 patientsunder antiretroviral therapy. Mutagenesis levels may rise andfall with drug concentrations and cause genetic differentiationof viruses replicating in different compartments (5). Any decreasein mutation rate, whether spatially or temporally, will obviouslywork against lethal mutagenesis.

Our assumption that all mutations are deleterious or neutralis unrealistic. Beneficial mutations invariably exist. Furthermore,the spectrum of beneficial effects may vary during the courseof mutagenesis, such that more beneficial mutations become availableas mean fitness level declines (49, 58). A low rate of beneficialmutations should not preclude lethal mutagenesis per se, althoughit will raise the threshold for extinction, so that a higherdose of mutagen will be required to achieve the same effect.There are two mechanisms by which beneficial mutations can work.First, some genuine beneficial mutations may increase the intrinsicreplicatory ability of the virus, increasing b and thus R. Second,mutations can confer partial or complete resistance to the mutagen(48). While partial resistance might be possible to overcomewith an increased mutagen dosage, complete resistance will preventlethal mutagenesis. A treatment strategy for preventing theevolution of significant or complete resistance could be combinationtherapy with several mutagens or with a mutagen in combinationwith other antiviral drugs.

ACKNOWLEDGMENTS

J.B. was supported by NIH grant GM57756 and the Miescher RegentsProfessorship at the University of Texas. R.S. was supportedby fellowship ASTF 37900-05 from the EMBO and grant GV06/031from the Generalitat Valenciana. C.O.W. was supported by NIHgrant AI 065960.

We thank Holly Wichman, Isabel Novella, and two anonymous reviewersfor many helpful comments and suggestions.

FOOTNOTES

* Corresponding author. Mailing address: Section of Integrative Biology, The University of Texas at Austin, Austin, TX 78712. Phone: (512) 471-6028. Fax: (512) 471-3878. E-mail: cwilke{at}mail.utexas.edu.

Published ahead of print on 3 January 2007.

Supplemental material for this article may be found at http://jvi.asm.org/.

Present address: Instituto de Biología Molecular y Celularde Plantas (CSIC-UPV), Avenida de los naranjos s/n, 46022 Valencia,Spain.

REFERENCES

Top