Virus Evolution, 2017, 3(1): vex004

doi: 10.1093/ve/vex004 Reflections

Two sides of the same coin: A population genetics perspective on lethal mutagenesis and mutational meltdown Sebastian Matuszewski,1 Louise Ormond,1 Claudia Bank,2 and Jeffrey D. Jensen1,3,*

1School of Life Sciences, Ecole Polytechnique Fe´de´rale de Lausanne, Lausanne 1015, Switzerland, 2Instituto Gulbenkian de Ciencia,^ Oeiras 2780-156, Portugal and 3Center for Evolution & Medicine, School of Life Sciences, Arizona State University, Tempe, AZ 85287, USA

*Corresponding author: E-mail: [email protected]

Abstract The of RNA virus populations upon application of a mutagenic drug is frequently referred to as evidence for the existence of an error threshold, above which the population cannot sustain the mutational load. To explain the extinction process after reaching this threshold, models of lethal mutagenesis have been proposed, in which extinction is described as a deterministic (and thus population size-independent) process. As a separate body of literature, the population genetics community has developed models of mutational meltdown, which focus on the stochastic (and thus population-size de- pendent) processes governing extinction. However, recent extensions of both models have blurred these boundaries. Here, we first clarify definitions in terms of assumptions, expectations, and relevant parameter spaces, and then assess similar- ities and differences. As concepts from both fields converge, we argue for a unified theoretical framework that is focused on the evolutionary processes at play, rather than dispute over terminology.

Key words: lethal mutagenesis; mutational meltdown; Hill–Robertson interference; Muller’s Ratchet.

1. Introduction proposed, invoking previously developed evolutionary proc- Fisher (1930) argued that an intermediate rate is likely esses including Muller’s ratchet (Muller 1964; Felsenstein 1974) optimal for populations to survive, as it ensures a constant in- and Hill–Robertson interference (Hill and Robertson 1966; put of beneficial providing the ‘fuel’ for adaptation, McVean and Charlesworth 2000). Relatedly, from the study of while limiting the impact of accumulating deleterious muta- virus evolution, the model of lethal mutagenesis (Bull, Sanjuan, tions. Subsequently, the effects of high population mutation and Wilke 2007; Wylie and Shakhnovich 2012) has emerged. rates and the related risk of population extinction have been Despite the different fields in which these ideas were de- explored thoroughly in the theoretical literature, spanning the veloped, there is a considerable amount of parallelism between fields of both population genetics and virology. these notions. Though mutational meltdown is generally dis- This literature has spawned a number of concepts describing cussed within the context of ‘small’ population sizes in which the extinction of populations owing to the excessive accumula- stochastic effects play an important role, and lethal mutagen- tion of deleterious mutations. From population genetics, the esis is generally discussed within the context of ‘large’ popula- mutational meltdown model (Lynch and Gabriel 1990) has been tion sizes driven by deterministic factors—recent extensions of

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the latter model have relaxed this assumption to incorporate high), or because RmaxðÞe drops below unity as a consequence of stochastic effects (Wylie and Shakhnovich 2012). Hence, the ex- a change in the environment (e.g., due to the application of a tent to which these models are overlapping descriptions of novel drug treatment). related processes or events, or are even subsets of one another, In this general formulation, viral populations are assumed to is unclear. This has resulted in an inherent confusion when be large initially such that is not affecting invoking these models to describe biological observations. mutation-selection balance, recombination is thought to be ab- We here seek firstly to clearly define and review these mod- sent, and beneficial mutations are disregarded (see below for a els in terms of parameter requirements, predicted effects, and discussion of recent relaxations of these model assumptions). biological relevance, and then compare them side-by-side in This simplified model allows the calculation of critical mutation light of their similarities and differences. Fundamentally, we rates Uc, beyond which absolute mean drops below unity propose that the notion of lethal mutagenesis largely describes and the population will become extinct. In particular, under a the deterministic subset of the dynamics of mutational melt- multiplicative genotype-to-fitness map xðÞ¼ði 1 sÞi, where the down, and thus call for a more integrated focus on the underly- fitness of a genotype is reduced by a constant s per deleterious ing processes driving extinction. mutation, the mean relative fitness is w ¼ eU (Kimura and Maruyama 1966). Thus, the population would go extinct if 2. Lethal mutagenesis U > Uc ¼ Log½RmaxðeÞ . Notably, due to environmental depend- ence of RmaxðeÞ, there is no universal critical mutation rate Originally introduced and coined to describe a therapeutic strat- across viral populations—not even for a single species. egy for curing vaccine-lacking viral diseases, the concept of As shown by the above back-of-the envelope calculation, le- lethal mutagenesis refers to the drug-induced increase in thal mutagenesis is independent of population size—a result viral mutation rate (achieved through incorporating non- made explicit in the original model in noting that it is fundamen- complementary nucleotides or nucleotide analogs during the tally a deterministic process that will operate even in very large DNA/RNA replication process), which reduces population mean populations (Bull, Sanjuan, and Wilke 2007). Importantly, al- fitness (and increases mutation load) and in turn leads to a de- though the outcome of lethal mutagenesis is deterministic, popu- cline in viral population size and eventual extinction (Loeb et al. lation dynamics, extinction times, and individual trajectories of 1999; Anderson, Daifuku, and Loeb 2004; Bull, Sanjuan, and mean absolute fitness are not. Thus, demography is nonetheless Wilke 2007). Although the concept itself has been applied to important here (Nowak and May 2000), where finite population both RNA and DNA viruses—including human immunodefi- sizes will always induce an additional (drift) load that is not ac- ciency virus (HIV), influenza A virus (IAV) and hepatitis B virus counted for in these models, but which can have a strong effect (HBV) (see Anderson, Daifuku, and Loeb 2004)—results were ini- on population dynamics owing to the fixation of deleterious mu- tially discussed in light of Eigen’s error catastrophe model tations further decreasing population mean fitness. (Eigen 1971) owing to the lack of a separate, formal theoretical framework of lethal mutagenesis. In an attempt to synthesize existing empirical and theoret- 3. Mutational meltdown ical work, Bull, Sanjuan, and Wilke (2007) proposed a general Muller’s ratchet (Muller 1964; Felsenstein 1974) describes the theory of lethal mutagenesis, which, as the authors noted, stepwise loss of the fittest class of individuals in a population “offered [nothing] ... specifically original”, but is “rather [an] ... and the associated reduction in absolute fitness due to the accu- application of simple models and the interpretation of those re- mulation of deleterious alleles, or drift load (Whitlock and ... sults in the context of empirical methods that made this Bourguet 2000). In a finite population, Muller’s ratchet eventu- [theory] original”. This model is composed of three basic fea- ally results in the extinction of the population if it is not x tures: First, a genotype-to-fitness map ðÞg characterizes how opposed by compensatory or beneficial mutations. Lynch and the number of (deleterious) mutations affects the genotype’s fit- Gabriel (1990) were the first to combine the study of population ness; second, a mutation-rate-to-fitness map addresses how dynamics (i.e., size changes and absolute growth rates) and the population mean fitness at mutation-selection balance population genetics (i.e., allele frequency distributions and rela- changes with the rate of mutations U (Haldane 1932; Bu¨ rger tive fitnesses) in order to describe this extinction process, which 1998); finally, a demographic model that links the two above they termed “mutational meltdown”. In a series of papers, the components with an ecological component R ðeÞ that quanti- max authors proposed and analyzed various related models and dis- fies the (environmentally dependent) maximal absolute popula- cussed the properties and implications of mutational meltdown tion growth rate (sometimes also called the maximal fecundity). in both asexual and sexual populations (Lynch and Gabriel 1990; Therefore, the population absolute mean fitness W is given by Gabriel, Lynch, and Bu¨ rger 1993; Lynch et al. 1993; Lynch, Conery, and Burger 1995a, b; Gabriel and Bu¨ rger 1994). W ¼ R ðeÞw ðxðgÞ; UÞ: max This process is fundamentally described with respect to the accumulation of mutations over time. The mean number of mu- Depending on the choice of the underlying demographic tations, n(t) , depends on the carrying capacity K, the absolute model, the number of viral particles in the next (discrete) gener- growth rate R, the deleterious effect of a mutation s, and the ation N is then simply a function of the current population tþ1 (deleterious) mutation rate l, and can be expressed as size Nt and the absolute mean fitness (e.g., a simple exponential " # growth model of the form Ntþ1 ¼ NtW or a more complex density-dependent model; see Gabriel and Bu¨ rger 1992). 2 1 nðÞt þ 1 ffi n ðÞt þ l s l þ rnðtÞ 1 ; However, independent of the choice of demographic model, the K population will eventually become extinct if the mean absolute 2 fitness W < 1, such that the population can no longer maintain where rnðtÞ describes the variance in n over time, which is gen- itself. This can happen either because the mean relative fitness erated by mutation and reduced by selection. The dynamics of w drops below a critical value (i.e., mutation load becomes too mutation accumulation, beginning with an isogenic population, S. Matuszewski et al. | 3

can then be split into three phases (see Fig. 1 of Lynch et al. The notion of lethal mutagenesis arose to provide a compre- 1993): first, mutations accumulate rapidly, until mutation- hensive framework to describe the deterministic (and thus selection-balance is reached. This is followed by a slower population size-independent) processes that lead to population constant-rate accumulation of mutations at constant popula- extinction via the crossing of a distinct error threshold. In con- tion size (i.e., when the population is at its carrying capacity). trast, the notion of mutational meltdown was fundamentally Once the mean viability drops below 1/R, carrying capacity can- concerned with the stochastic nature of this extinction process, not be maintained and population size starts to decline, thus invoking classical population genetic models describing small increasing the susceptibility to further accumulate deleterious population size dynamics. However, as discussed, the model of mutations (which in turn again reduces mean viability), ultim- mutational meltdown has also been examined with regards to ately resulting in rapid population extinction. large population sizes. Further, recent extensions of the model One important difference in the dynamics of the meltdown of lethal mutagenesis have begun to consider stochastic effects. model is the dependence on the order of events in the life cycle For example, Wylie and Shakhnovich (2012) studied the role of (i.e., whether selection acts before or after population size regu- population size and mutation rate on extinction times, finding, lation). In the first case, the carrying capacity K can be main- as expected, that small populations may go extinct very quickly, tained over a long period, resulting in a constant population whereas large populations survive almost indefinitely. size and linear accumulation of mutations, followed by a rapid Thus, in some ways the model of lethal mutagenesis has extinction phase (Lynch and Gabriel 1990; Lynch et al. 1993). In converged with that of mutational meltdown, certainly in terms the second case, each click of the ratchet (i.e., loss of the least- of appreciating the importance of stochastic processes in driv- loaded class of individuals) results in fewer offspring; thus the ing extinction events. Indeed, the genetic processes underlying population size declines gradually and the speed of the ratchet population extinction are governed by the effective population increases over time (Gabriel, Lynch, and Bu¨ rger 1993). size, Ne (Wright 1931; Crow 1954; Charlesworth 2009). Estimates Independent of the type of model, the conclusions from these of Ne=N taken from over 100 species (excluding viruses), demon- papers were that extinction times of a few hundreds of gener- strated that census population size is on average an order of ations are expected for small populations, and that the process magnitude larger owing to fluctuating population sizes, unequal is slowed by roughly an order of magnitude in sexual popula- sex-ratios, and/or variance in reproductive success (Frankham tions (Lynch, Conery, and Burger 1995b). A simple rule deter- 1995). In particular, the latter has been argued to significantly mines the beginning of the final meltdown phase, which was affect viral populations (Neher and Hallatschek 2013; Irwin subsequently used in Lande’s treatment of the same problem in et al. 2016). The ladder-like genealogy of the influenza a quantitative genetics framework (Lande 1994, 1998), and in A virus hemagglutinin segment, for example, suggests that only models of lethal mutagenesis (Bull, Sanjuan, and Wilke 2007; a few viruses seed the entire next generation (Grenfell et al. and see below): the population is doomed to extinction when 2004). Along the same lines, estimates of effective population the mean viability decreases below the reciprocal of the abso- size in HIV range from 103 to 106, but generally show an extraor- lute growth rate (i.e., the number of offspring an individual can dinarily low Ne=N ratio (Pennings, Kryazhimskiy, and Wakeley produce; Lynch et al. 1993). An interesting finding is that an 2014). intermediate magnitude of the deleterious selection coefficient Another similarity common to both models is the need to in- minimizes the time to extinction through mutational melt- corporate the potential effects of beneficial and compensatory down; this is of particular interest given the recently accumulat- mutations in modifying the rate of fitness decline and time to ing empirical evidence for a generally bimodal distribution of extinction (see Manrubia, Domingo, and Lazaro 2010; Bull et al. fitness effects (DFEs) of new mutations (e.g., Eyre-Walker and 2013). Specifically, small increases in mutation rate may im- Keightley 2007; Hietpas et al. 2011, 2013; Bank et al. 2014). prove the ability of populations to respond to novel environ- Beyond the “”, other elements of lethal mental challenges, and there indeed exist examples of selection mutagenesis were indeed first discussed in the seminal papers for hypermutator strains in bacteria under particular stressors on mutational meltdown (Lynch and Gabriel 1990; Lynch et al. (e.g., Sniegowski, Gerrish, and Lenski 1997; Gerrish, Colato, and 1993). First, Lynch et al. (1993) describe the conditions under Sniegowski 2013). However, owing to the effects of Hill– which mutational meltdown is driven by genetic drift versus by Robertson interference, as well as the far greater input of newly mutational pressure, and conclude that “when the mutation arising deleterious relative to beneficial mutations, this concern rate is on the order of 1 per individual per generation [...] the fundamentally suggests a need to quantify the extent to which [least-loaded] class will be lost due to mutation pressure alone”. mutation rates must be increased in order to ultimately result Second, the authors demonstrate that the extinction time is in population extinction. only relatively weakly (logarithmically) dependent on the popu- Recent empirical studies attempting to test the genetic mod- lation size. Finally, Lynch et al. (1993) compare the mean fitness els underlying lethal mutagenesis have largely failed to match reached after the first phase with that of an infinite-population its (qualitative) predictions (Springman et al. 2010; Bull et al. under mutation-selection-balance (i.e., the starting point for le- 2013). This is presumably owing to the evolutionary mechan- thal mutagenesis; Kimura and Maruyama 1966), and observe isms neglected by the (original) theory such as adaptive evolu- that it is indeed only slightly larger in the case of a finite popu- tion (i.e., beneficial/compensatory mutations), interactions lation. Thus, though generally associated with small- between mutations (i.e., epistasis) and non-constant mutational population size effects, mutational meltdown has been dis- effect sizes (i.e., the DFEs). While Antoneli et al. (2013) recently cussed both with regards to high-mutation-rate regimes and derived a generalization of the lethal mutagenesis extinction large population sizes (Lynch and Gabriel 1990). criterion that allows for a small fraction of (fixed effect) benefi- cial mutations, epistasis and the DFE are inherently connected to the genotype-to-fitness and/or the genotype-to-phenotype 4. Comparing and interpreting the models map. Two alternatives to the frequently used multiplicative fit- “Everything should be made as simple as possible, but not simpler.” ness model of Kimura and Maruyama (1966) have been pro- Albert Einstein posed: First, in biophysics-based fitness landscape approaches, 4|Virus Evolution, 2017, Vol. 3, No. 1

the DFE is derived from the mutational effects on protein fold- sizes and changing environmental pressures, and the effects of ing and its thermodynamic properties (e.g., Chen and tissue-specific drug permeability. Shakhnovich 2009; Stich, Lazaro, and Manrubia 2010; Wylie and Conflict of interest: None declared. Shakhnovich 2011, 2012). While these approaches may indeed account for an important class of mutational fitness effects in viruses (Wylie and Shakhnovich 2011), they have been criticized References for directly equating fitness with protein stability—thus neg- Achaz, G., Rodriguez-Verdugo, A., Gaut, B. S., and Tenaillon, O. lecting ecological aspects underlying fitness (Martin and (2014) ‘The Reproducibility of Adaptation in the Light of Gandon 2010). As an alternative, phenotypic landscape models Experimental Evolution With Whole Genome Sequencing’, in naturally accommodate epistasis, variation in mutational ef- Ecological Genomics, eds. Landry,C. 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