Lethal mutagenesis and the transient within-host dynamics of viral adaptation

Abstract

Beneficial mutations drive the within-host adaptation of viral populations and can prolong the duration of host infection. Yet, most mutations are not adaptive and the increase of the mean fitness of viral populations is hampered by deleterious and lethal mutations. Because of this ambivalent role of mutations, it is unclear if a higher mutation rate boosts or slows down viral adaptation. Here, we study the interplay between selection, mutation, genetic drift and within-host dynamics of viral populations. We obtain good approximations for the transient evolutionary epidemiology of viral adaptation under the assumption that the mutation rate is high and the effects of nonlethal mutations remain small. We use measures of fitness effects of mutations for a range of viruses to predict the critical mutation rate required to drive viral extinction. This analysis questions the feasibility of lethal mutagenesis because the fold increase of viral mutation rates induced by available mutagenic drugs is not high enough to reach the critical mutation rate predicted by our model.

Introduction

The within-host dynamics of viral infections depends both on the availability of susceptible host cells and the ability of the virus to infect and exploit these cells. Mathematical epidemiology provides a theoretical framework to model how these life-history traits affect the dynamics of viral populations (Anderson and May 1992, Nowak and May 2000, Diekmann et al. 2013). Yet these traits are not constant but may evolve and change during the course of the host infection. Indeed, many viruses have high rates of mutation (Sanjuán et al. 2010), yielding substantial genetic and phenotypic diversity of within-host viral populations. This influx of mutations challenges the simplicity of classical models of viral dynamics and has led to the concept of quasispecies to describe the dynamics of viruses with high mutation rates (Andino and Domingo 2015, Domingo and Perales 2019, Sardanyés et al. 2024). The effects of high mutation rates can also be captured within the classical population genetics framework (Wilke 2005, Bull et al. 2007, Martin and Gandon 2010). As most mutations have deleterious effects, the constant influx of mutations generates a ‘mutation load’, which measures the difference between fitness of the fittest strain and the mean fitness of the population (Crow 1989). In fact, some mutations can prevent viral replication and can be considered as ‘lethal mutations’ (Sanjuán et al. 2004). The massive impact of deleterious mutations on viral fitness led to the ‘lethal mutagenesis hypothesis,’ which states that there is a critical mutation rate above which a viral population cannot grow and is driven to extinction (Bull et al. 2007, 2013). Drugs increasing mutation rates may thus constitute a broadly applicable therapeutic strategy against many viruses (Loeb and Mullins 2000, Shiraki and Daikoku 2020), including SARS-CoV-2 (Kaptein et al. 2020, Driouich et al. 2021, Hadj Hassine et al. 2022, Masyeni et al. 2022, Swanstrom and Schinazi 2022). A better evaluation of the balance between the therapeutic potential and the risk associated with these drugs relies on a better understanding of the within-host viral dynamics with increased mutation rates.

First, it is important to realize that the mutation process is a double-edged sword: even if most mutations are deleterious, some may be beneficial and could speed up within-host evolution. Some concern has emerged regarding the potential risk of ‘sublethal mutagenesis’ that may result in immune escape or higher infectivity and transmission (Pillai et al. 2008, Sadler et al. 2010, Nelson and Otto 2021, Bank et al. 2022, Lobinska et al. 2023, Lobinska et al. 2024). Experimental evolution of bacteriophage T7 under high mutation rates showed that the exposure to a mutagen may boost adaptation and increase the mean fitness of the viral population (Bull et al. 2013, Paff et al. 2014). Several theoretical models explored the balance between the effects of deleterious and beneficial mutations (Martin and Gandon 2010, Keller et al. 2012, Gerrish et al. 2013, Anciaux et al. 2018). Yet, another aspect that is often overlooked in previous models is the population size dynamics taking place within the infected host. The drop in the mean fitness of the virus consequent to the accumulation of deleterious mutations is expected to reduce the within-host infectivity of the virus. This is expected to reduce the number of infected cells and to trigger a rebound in the density of susceptible host cells. This effect on the within-host population dynamics may also feed back on the intensity of selection on the viral population because selection for infectivity increases with the availability of susceptible cells. Hence, the critical mutation rate above which the virus population is expected to go extinct needs to account for this demographic feedback (Martin and Gandon 2010).

Second, even if lethal mutagenesis is a deterministic process occurring when the mutation load becomes overwhelmingly high, the effect of higher mutation rates may be amplified by demographic stochasticity. In finite populations, there is an irreversible accumulation of deleterious mutations because the most fit, least-loaded genotypes are repeatedly lost as genetic drift overwhelms the effect of natural selection, a process referred to as ‘Muller’s ratchet’ (Muller 1964, Felsenstein 1974). Accounting for population dynamics is expected to amplify this effect and leads to a mutation meltdown (Lynch and Gabriel 1990, Lynch et al. 1993, Matuszewski et al. 2017, Bank et al. 2022). Indeed, with each click of Muller’s ratchet, the mean fitness and the population size drops, which increases the magnitude of genetic drift and the speed of the ratchet. However, even a small influx of compensatory mutations can halt this runaway process and prevent the mutation meltdown and the extinction of populations (Poon and Otto 2000). Yet, demographic stochasticity could still increase the risk of extinction when the viral population becomes transiently very low after the start of a mutagenic treatment (Anciaux et al. 2018).

In the present work, we aim to account for these different effects and to study the joint evolutionary and within-host dynamics of a viral population exposed to high mutation rates. To model the effects of mutations on viral fitness, we use Fisher’s Geometric Model (FGM) of adaptation to link the phenotype of the virus to its within-host infectivity. We use the FGM because it allows us to generate realistic distributions of fitness effects (with beneficial, neutral, and deleterious mutations) and to account for pervasive epistasis in fitness among mutations (Orr 2000, Martin and Lenormand 2006a, Tenaillon 2014). Hence, the FGM provides a good starting point to model within-host adaptation. In addition, we account for a distinct type of strictly lethal mutations. The analysis of the long-term equilibrium of the population size and evolution of viral dynamics allows us to identify the critical mutation rate U_c, leading to the within-host eradication of the virus (Martin and Gandon 2010). We use estimates of growth rates, genomic mutation rates and fitness effects of mutations from previous experimental studies on a range of viruses to predict their critical mutation rate. This allows us to evaluate the feasibility of lethal mutagenesis given the efficacy of current mutagenic drugs (i.e. the fold increase in the genomic mutation rate induced by these drugs). Crucially, our model can also be used to describe the transient evolutionary and within-host dynamics of a viral population (i.e. away from a mutation–selection equilibrium). We use this theoretical framework to explore more realistic scenarios where the virus may be initially maladapted or when the immune response may gradually increase with time. These scenarios allow us to go beyond the analysis of the critical mutation rate U_c and to new risks and benefits associated with the use of mutagenic drugs. Finally, we compare our deterministic predictions with stochastic simulations to evaluate the influence of finite population size on our predictions.

Model

We assume that different virus strains may circulate and compete within the host. To describe this diversity, we assume that each strain is defined by a set of underlying continuous ‘phenotypic traits’ that are under stabilizing selection towards a single optimum (see Table 1 for the description of the notations) (Fisher 1999, Martin and Lenormand 2006b). More specifically, we consider that the within-host infectivity is governed by , a -dimensional vector, where each phenotypic dimension refers to an independent continuous trait. The infectivity depends on the Euclidian distance of the vector to the point of maximum infectivity (at the origin, i.e. at ) such that:

Table 1.

Model parameters and main notations

Symbol	Description
	Maximal value of infectivity
	Increased mortality rate induced by the infection
	Constant influx of susceptible cells
	Death rate of susceptible cells
	Mutation rate per genome and per generation
	Fraction of mutations that are lethal
	Variance of the mutational effect on each phenotypic trait
	Number of phenotypic traits (dimension of the landscape)
	Vector of phenotypic traits
	Density of cells infected by strains with phenotype
	Density of infected cells
	Density of uninfected cells
	Mean phenotype
	Phenotypic variance
	Critical mutation rate

(1)

We use this quadratic form to approximate a Gaussian shape of the fitness landscape (Martin and Gandon 2010) and the division by 2 is for consistency with earlier work (Bürger 2000). This approximation is good when the phenotype is not very far from the optimum, but it breaks down when the phenotype is very far from the optimum as infectivity can become negative (but we checked the validity of this approximation with simulations, see Supplementary Information section S4.1). Figure 1 illustrates how the variation on the underlying phenotypes affects the infectivity of the pathogen, and, as discussed below, how this translates into viral within-host fitness.

Figure 1.

The fitness landscape links the underlying phenotypic traits with life-history traits and malthusian fitness. We represent our phenotype-to-fitness landscape in two dimensions (). A phenotype is associated with the infectivity using equation (1). This infectivity is associated with the fitness value with equation (4), which depends on time through the number of susceptible cells . Note how the projection of a black circle around in the phenotypic landscape, results in a dashed circle in the infectivity landscape. This geometric distortion has major implications on the dynamics of adaptation. For instance, this distortion explains why the variance of the distribution of random mutants from increases with the distance to the optimum.

To model the joint evolutionary and within-host population dynamics of the virus population, we assume that the viral pathogen has access to a density of susceptible host cells. We focus on the dynamics of the density of infected cells, and we do not explicitly model the dynamics of the free virus stage because we assume that the lifetime of an infected cell is much longer than a free virus particle (Nowak and May 2000, Martin and Gandon 2010). Host cells are produced at a constant rate and die at a constant rate . refers to the density of host cells infected by strain . These cells infect susceptible cells at a rate , and they die at rate where is the virus-induced death rate. We assume that each host cell can only be infected with a single viral strain (i.e. no multiple infections). Next, to account for the effect of the mutation process, we assume that the virus mutates at the constant rate . With probability , the mutation is lethal and noninfectious, which implies that the cells infected by viruses with such a mutation cannot infect new cells. Because the influx of new susceptible cells is assumed to be independent of the total number of cells, the cells infected by viruses with lethal mutations do not affect the dynamics of the system. Therefore, we can simply treat lethal mutations as an additional death term . With probability , the mutation is nonlethal and the new phenotype becomes , where the mutation effect is sampled in an isotropic multivariate normal distribution with mean 0 and variance . The mutational variance is easily interpreted as it directly relates to the mean effect of random mutations on infectivity . Finally, combining the within-host dynamics with the mutation process yields the following integro-differential equation (the upper dot represents time derivation):

(2)

where the first three terms (first line) refer to ‘birth’ and ‘death’ rates of infected cells, while the final term (second line) refers to the influx and outflux of infected cells induced by the phenotypic change after a viable mutation. The above life cycle yields the following system of ordinary differential equations:

(3)

where is the total density of infected cells (i.e. cells infected by a transmissible virus), is the mean infectivity, and is the frequency of the phenotype in the infected population. We can now introduce, , the per capita growth rate of the phenotype (which we refer to as ‘fitness’) and the mean growth rate of the pathogen (i.e. the ‘mean fitness’):

(4)

The sign of the mean fitness indicates whether the viral population grows (when )or if it drops and eventually goes extinct (when ). The mean fitness is driven by the dynamics of uninfected cells and the mean infectivity , which can also be written in the following way [see equation (S.2) in the Supplementary Information]:

(5)

where the ‘lag load’ on infectivity depends on the distance to the optimum, while the ‘mutation load’ on infectivity depends on the phenotypic variance of the viral population (Lande and Shannon 1996).

In the following, we use different approaches to predict the evolutionary dynamics of the within-host population of the virus. First, we use a Weak Selection Strong Mutation (WSSM) approximation (i.e. valid when ) where adaptation is the result of many mutations of small effects and the distribution of phenotypes remains Gaussian (Martin and Roques 2016). Second, we study how mutations of larger effects can influence the evolutionary dynamics of the virus using additional approximations based on moment closure or the House of Cards (HC) approximation (Bürger 2000). Third, we check the robustness of our approximations using numerical simulations, where we study the dynamics of discrete phenotypes. We discuss the feasibility and the risks associated with lethal mutagenesis therapy under different scenarios. In particular, we use stochastic simulations to explore the effect of finite host and viral population sizes on the effect of mutagenic drugs on the within-host extinction of a viral population (i.e. a successful mutagenesis treatment).

Results

We use equation (2) to obtain dynamical equations for the cumulants of the distribution of the phenotypes . Under the WSSM assumptions, we can neglect higher-order terms of the ratio between the mutational variance and the genomic mutation rate (i.e. , , etc.), the phenotypic distribution is Gaussian, and the evolutionary dynamics can be described with the first two moments of the phenotypic distribution (Bürger 2000) (Supplementary Information section S2.5). Because of the isometry of the fitness landscape and of the model of mutation, the phenotypic distribution is the same along all dimensions. Therefore, all the components of vector behave the same, and we can fully capture the evolutionary dynamics using single dimension. We thus follow a single phenotypic trait , which yields (Supplementary Information S2.5):

(6)

where is the mean phenotype over the population defined as . The first equation in equation (6) shows how the mean phenotype moves towards the optimum at a speed governed by (i) the amount of susceptible cells , (ii) the phenotype variance , and (iii) the mean distance to the optimum . The dynamics of results from the balance between the effect of natural selection that consumes this variation and the effect of mutation that introduces more genetic variation. Interestingly, the number of dimensions only appears in the population size equations through the load terms (see S.36):

(7)

Combining equations (6) and (7) yields the dynamics of the mean infectivity:

(8)

where is the variance in infectivity. Note how the population size dynamics in equation (3) feeds back on the evolution of the phenotypic distribution. The mean infectivity is increased by natural selection with a speed controlled by the variance in infectivity, scaled by the density of susceptible cells. In contrast, the direct effect of the mutation process is negative and is equal to the rate of nonlethal mutations times , the mean effect of random mutations on infectivity.

Evolutionary equilibrium and critical mutation rate

At the evolutionary equilibrium, the mean phenotype is at the optimum and (where the indicates that the system is at equilibrium) and thus there is no lag load . The variance of the phenotypic distribution at the WSSM equilibrium is:

(9)

where is the equilibrium density of the susceptible cells. This expression shows how a more infectious virus that reduces is expected to lead to an increase phenotypic variance of the virus population at equilibrium.

There is a critical mutation rate , above which natural selection is overwhelmed by the accumulation of deleterious mutations, and the virus population is driven to extinction. This critical mutation rate is reached when the virus population cannot grow when the density of susceptible cells is (i.e. the equilibrium density of the host cells in the absence of the virus). At this point, the ‘birth’ and ‘death’ rates of new infections balance each other and the critical mutation rate verifies the following condition:

(10)

where . Using equation (9) for the equilibrium phenotypic variance yields the WSSM approximation for the critical mutation rate (Supplementary Information section S2.8):

(11)

with and . As expected, the fraction of lethal mutations has a massive influence on the critical mutation rate:

(12)

Figure 2 illustrates the combined effect of the fraction of lethal mutations and the number of phenotypic dimensions of the fitness landscape on . As expected, higher values of increase the cost of complexity for the virus and the critical mutation rate drops.

Figure 2.

Effect of the proportion of lethal mutations and the phenotypic dimension of the fitness landscape on the critical mutation rate and the fitness. Plot (a) shows the effect of the fraction of lethal mutations and the number of dimensions on the critical mutation rate in the WSSM regime. Arrows indicate the values of the critical mutation rate when . Plot (b) shows that the critical mutation rate corresponds to the point where the ‘birth’ and ‘death’ rates of new infections balance each other when (see equation (10) and section S2.8 in the supplementary information). Parameter values: .

Critical mutation rate when selection is stronger

Under the above WSSM assumption, we neglected higher order terms of the ratio . Next, if we relax the assumption that the effect of mutation is weak relative to genomic mutation rate, the phenotypic distribution is no longer multivariate Gaussian and we have to account for the effects of the third and fourth cumulants of the phenotypic distribution at equilibrium ( and , respectively). The fourth cumulant builds up with the second moment of the distribution of mutational effects and is always positive (see Supplementary Information S2.5 and Fig. S1). Hence, the equilibrium variance is smaller than expected in the WSSM approximation (Fig. S2). Consequently, when the effects of the mutation are stronger, we need to account for to compute the critical mutation rate (orange dashed line in Fig. 3). The critical mutation rate drops with higher values of as it increases equilibrium phenotypic variance and the mutation load associated with this variance [equations (7) and (9) and see Fig. S3], but the above analysis breaks down when becomes too high relative to the standing variance (see Supplementary Information section S2.6).

Figure 3.

Lethal mutagenesis is feasible for lower mutation rates with steeper fitness landscapes. The critical mutation rate is shown as a function of the mutational variance . The results are shown under three approximations: The WSSM approximation (orange), the WSSM+K4 approximation accounting for the cumulant (dashed orange), the house of cards (HC) approximation (cyan). Data from numerical simulations are shown as filled circles when the infected population survives, and empty circles when the infected population goes to extinction. The shaded orange and cyan areas indicate the validity conditions of the WSSM and the house of cards regimes, respectively (see equation (14)). The vertical dashed line shows the maximum value of for which the WSSM+K4 approximation is valid. Parameter values: . For the stochastic simulations, we used

For even larger values of , we can use another approximation to describe the viral dynamics under a regime of mutation where the variance of mutation overwhelms the effect of the parental strain: the classical HC approximation, which is based on the assumption that the variance of the mutation distribution is much larger than the equilibrium variance. This approximation has been used to derive the equilibrium mutation load for haploids (Bürger 2000, Martin and Roques 2016). After incorporating the influence of population sizes feedbacks for pathogens, we obtain (see Supplementary Information S2.7) the following expectation for an upper bound of the phenotypic variance at the HC equilibrium:

(13)

where is the equilibrium density of susceptible cells under the HC approximation. Note that the mutation load on infectivity given by equation (7) becomes and the mutation load on ‘fitness’ is simply equal to , the nonlethal mutation rate (see (Martin and Roques 2016)). Another striking feature of the HC approximation is the independence of equilibrium phenotypic variance to the mutational variance . Note, however, that higher order HC approximations for the equilibrium phenotypic variance do depend on (Bürger 2000).

Numerical computations of the equilibrium phenotypic variance confirm the validity of the above approximations (Fig. S2). As expected, the equilibrium variance predicted with the WSSM approximation is very accurate when the ratio is low, but this approximation breaks down when the effect of mutations becomes high. There is a threshold value of the mutation variance below (above) which the WSSM (HC) approximation is more accurate. This threshold value is obtained by equating the equilibrium variance under the WSSM and the HC approximations [equations (9) and (13)], which yields:

(14)

Figure 3 shows that the critical mutation rate predicted by the WSSM approximation is accurate until the condition (14) is verified. For higher values of , the mutation rate required to achieve lethal mutagenesis is higher under the HC approximation than under the WSSM approximation (Fig. 3). Interestingly, the critical mutation rate predicted under the HC approximation is the same as the one obtained under the WSSM approximation if all mutations were lethal [see equation (12)]:

(15)

where is the maximal growth rate of the optimal phenotype in a fully susceptible population. In other words, the critical mutation rate under the HC approximation depends only on the demographic parameters but not on parameters governing the fitness landscape (Fig. 3).

Predicting the critical mutation rate of viruses

The above analysis allows us to predict the critical mutation rate using three types of estimates from a range of viruses (Supplementary Information S3): (i) we need to characterize the within-host demography of the viral population and in particular the maximal growth rate and the mean generation time of the virus, (ii) we need to characterize the fitness landscape of the virus and in particular the number of phenotypic dimensions, (iii) finally, we need to characterize the mutation process (the genomic mutation rate , the fraction of lethal mutations, and the mutational variance ). We use data from previous studies (Sanjuán 2010, Visher et al. 2016) to predict the critical mutation rates for five different viruses (Table 2). We explore the robustness of these predictions to variations of the parameters and in Tables S2 and S3. We find that the critical mutation rates are always higher for the WSSM estimates (Table S1), which implies that we should be using the WSSM approximation to predict (see Fig. 3). Interestingly, we find that the fold increase in the viral mutation rate required to reach the critical mutation rate of six different viruses ranges from 11 to 2700.

Table 2.

Critical mutation rates for five different viruses. For each considered virus, its genome size is given in kb. The following lines present parameters of demography and mutation rates: is the growth rate per hour of a wildtype virus (here we assume that the wildtype is at the optimum), is the generation time (we assume to be equal to , the duration of a cell infection), and mutation rates both per site per generation (s/n/c) and per generation (s/c). The next four lines are parameters related to the effects of mutation under our FGM framework: is the mean effect of non-lethal mutations scaled by the growth rate, is the proportion of lethal mutations, is the mutational variance and is the number of phenotypic dimensions. We show in supplementary information S3 how we estimate , , and in fine the critical mutation rate from previously published experimental data. We then show estimates of the critical mutation rates under the WSSM approximation for these viruses, expressed both per hour and per generation (s/c). Finally, we compute the fold-increase in mutation rate required to achieve viral extinction under the WSSM approximation. For the critical mutation rates and fold-increase in under the HC approximation, see Table S1. The HC estimates are lower than the WSSM estimates, meaning that the WSSM regime is more relevant in conditions of lethal mutagenesis for these viruses (see Fig. 3). We show in Table S2 the estimates of required fold-increase for different values of number of phenotypic dimensions and in Table S3 the sensitivity to the value of the proportion of lethal mutations . Grey cells refer to data from published studies (Baccam et al. 2006, Domingo-Calap et al. 2009, Sanjuán et al. 2010, Smith et al. 2010, Visher et al. 2016) and are described in the supplementary information S3. White cells refer to predictions derived from our model and from the data in the grey cells.

	Q	x174	F1/M13	VSV	Influenza
Genome size	4.22 kb	5.39 kb	6.4 kb	11.2 kb	13.6 kb
	3.6	10	4.3	0.9	0.32
Generation time	2.07	0.46	1	7.68	6–31.6
Mutation rate (s/n/c)	1.4	1.1	7.8	3.5	2.3
Mutation rate (s/c)	5.91	5.93	5.00	3.92	3.13
	0.10	0.13	0.10	0.13	0.12
	29%	18%	20%	39%	30%
	0.69/h	2.7/h	0.88/h	0.21/h	0.056/h
	1	1	1	1	2
	8.82	32.1	13.7	1.50	0.569
	18.3	14.8	13.7	11.5	3.4–18
Fold-increase	31	2 500	2 700	29	11–57

Transient dynamics of adaptation

Next, we jointly use equations (3), (5), (6), and (8) to study the interplay between the transient within-host dynamics of the virus and the adaptation of the viral populations in the WSSM regime. In Fig. 4, we explore the dynamics of the mean infectivity for different initial conditions. We vary the initial distance of the mean phenotype to the optimal phenotype and the standing variance. We contrast a scenario where the virus population is monomorphic and a scenario with some standing genetic variance. As expected, regardless of these initial conditions, the dynamics converge to the same equilibrium, which is given by equations (7) and (9). However, the initial conditions govern the speed at which this equilibrium is reached. First, the standing genetic variance induces a mutation load in infectivity, which explains the lower value of at in Fig. 4. Second, the absence of a genetic variation in the clonal population implies that the speed of adaptation is initially very slow. In fact, the mean infectivity initially drops because of the effect of deleterious mutations [see equation (8)]. Genetic variation first needs to build up before selection can act on the mean infectivity. In contrast, the speed of adaptation is faster with standing genetic variance. This faster adaptation allows the population to rapidly overcome the initial mutation load, and the mean infectivity quickly becomes higher than in initially clonal populations. Finally, the rate of change increases with the initial distance to the optimum (or initial lag load in infectivity). This faster rate of adaptation is due to the increase in the phenotypic variance with the distance to the optimum [see equation (8) and Fig. 1 for a geometric interpretation of this effect].

Figure 4.

The transient evolutionary dynamics depends on the distance to the optimum and the initial phenotypic variance. We show the dynamics of mean infectivity through time starting from a clonal population at different initial values of . The solid black lines refer to an initially clonal population () while the dashed lines the population is initially polymorphic (). We also vary the initial distance value of the mean phenotypic trait: and . and indicate the initial values of the lag load and the mutation load on infectivity for the scenario . is the mutation load on infectivity at equilibrium: The difference between and the equilibrium mean infectivity . Parameter values: . Initial conditions were

The transient dynamics is generally well described by equation (6), but when the mutational variance is large relative to the genomic mutation rate , the phenotypic distribution does not remain Gaussian during adaptation. We show in the Supplementary Information (section S2.5) that and are expected to build up during the transient phase of adaptation. Crucially, these higher moments increase the phenotypic variance transiently, which speeds up the adaptation of the virus population (see Fig. S1). In the long-term, however, we expect and . Consequently, when these higher moments of the distribution are accounted for, the phenotypic variance is lower at the endemic equilibrium (Fig. S2). Hence, as pointed out in the previous section, the critical mutation rate is higher when is accounted for (Fig. S3).

Yet the dynamics of the population size of the virus population is driven by and not by . To better understand the dynamics of viral adaptation, it is useful to decompose the dynamics of viral mean fitness into separate effects following the framework of Gandon & Day (Gandon and Day 2009) (Fig. S4):

(16)

where , , and refer to the changes in mean fitness due to natural selection, mutation, and environmental change, respectively. In our model, these different components of the dynamics of adaptation can also be expressed as:

(17)

First, as expected from Fisher’s fundamental theorem, the change of mean fitness from natural selection is always positive and equal to the variance in fitness (Fig. S4). This variance increases with the lag load (the farther a phenotype is from the optimum, the larger the strength of selection towards this optimum) and the mutation load (even if this load has a negative impact on mean fitness, it has a positive influence on the speed of adaptation). Selection is also fuelled by the density of susceptible cells and the phenotypic variance .

Second, the effect of mutations on mean fitness is simply equal to the influx of nonlethal mutation multiplied by the mean effect of mutations on fitness . Since , the direct effect of the mutation process is always negative on the dynamics of mean fitness (Fig. S4). This drop in fitness is exactly equal to the expected drop in mean fitness in mutation accumulation experiments where a radical bottlenecking at each passage ensures that natural selection does not operate (because the variance in fitness ). As expected, lethal mutagenesis occurs when the positive effect of natural selection is overwhelmed by the negative effect of deleterious mutation (Bull et al. 2013, Gerrish et al. 2013). But lethal mutagenesis may also be driven by a degradation of the within-host environment of the pathogen, which we discuss next.

The third term of equation (16) accounts for the environmental change consecutive to a drop in the density of susceptible cells. This final term can be either positive or negative, depending on the change in the density of susceptible host cells (Fig. S4). During the initial phase of an infection, the density of susceptible cells is expected to drop and to have a negative impact on the growth rate of the epidemic (density-dependent regulation). In contrast, during the initial phase of therapy, drugs are expected to reduce the density of infected cells and, consequently, the density of susceptible cells may increase. We illustrate below how these within-host dynamics may affect the feasibility of lethal mutagenesis to treat viral infections.

Within-host dynamics during and after drug therapy

A drug may act in at least two different ways in our model. First, a drug may have a mutagenic effect and act via an increase of the mutation load of the virus population. Second, a drug may move the fitness optimum away from the virus population and increase the phenotypic distance . In the following, we explore how these two effects act on the transient within-host dynamics of the virus in the WSSM regime. For simplicity, we assume that the other parameters of the model are unaffected by the use of the drug.

Figure 5 shows virus dynamics during and after the start of drug therapy using the approximation, which accounts for the moments and [see equations (S36)–(S39)]. When, the mutation rate induced by the drug is sufficiently high above , the drug can drive the virus population to extinction before the end of treatment. But when the mutagenic effect is lower, the initial stress level is low and/or the duration of the treatment is short, the drug may fail to clear the infection and could even lead to higher rates of viral replication and higher viral loads after the end of the treatment. This rebound is due to the increased rate of viral adaptation induced by the mutagen and to the increase of the density of the susceptible host cells during treatment. It would thus be particularly important to monitor the patients before deciding to end the treatment to avoid the risk of recrudescence. The risk of viral evolutionary rescue is less likely if we account for the build-up of an effective immune response of the host against the virus as in the case of an acute infection where the death rate of the infected cells will increase after some time. For instance, we show in Fig. S5 how host immunity may reduce the risk of recrudescence. In this scenario, a mutagen may be used as a way to buy some time and reduce the viral load before the immune response kicks in and effectively controls the infection.

Figure 5.

Effect of a mutagenic treatment on chronic infections. The mean infectivity (a,B) and infected density (C,D) are shown through time depending on an initial distance to the optimum of the pathogen of (a,C) or (B,D) infection, for the WSSM+cumulant model. Simulations are initialized with clones (null variance) and the densities of susceptible and infected cells are initialized with the equilibrium values obtained with . During the treatment (red shaded window), the mutation rate is increased from 1 to the value of specified on the right of the plots. The computed deterministic critical mutation rate is . The treatment stops at time where the mutation rate goes back to 1. When the infected density drops below an arbitrary value of , the corresponding section of the infected density and mean infectivity curves become dashed. This highlights how the changes in mean infectivity observed in these time frames are highly dependent on escaping stochastic extinction when density is low. The horizontal red line in (a,B) shows the maximum infectivity . Parameters values: .

The above numerical exploration of the effect of a mutagenic drug on within-host viral dynamics ignored the influence of demographic stochasticity. Demographic stochasticity is expected to be particularly high when the viral density drops to very low levels. We explore the influence of demographic stochasticity in Fig. 6 by computing the probability of evolutionary rescue after the start of the mutagenic treatment as a function of the viral mutation rate induced by the drug and , the initial stress. Note that we use higher values of compared to previous figures to obtain cases where initial fitness is negative and thus observe potential evolutionary rescue. As expected, viral extinction occurs when and larger values of always promote viral extinction. Note, however, that when is sufficiently large, the virus population may only survive when the mutation rate is above a minimal mutation rate. Indeed, intermediate mutation rates can speed up viral adaptation and allow the viral population to avoid extinction at the beginning of the treatment without inducing an unbearable mutation load at equilibrium. The influence of the mutation rate on the probability of extinction has been analysed in Anciaux et al. (2019) without demographic feedback. Our simulation model accounts for the within-host populations’ size dynamics and shows that a larger fraction of lethal mutants can have a massive impact on the interaction between the effect of and the initial stress level . As expected, the effects of demographic stochasticity are amplified by smaller host and pathogen population size (see Fig. S6). To appreciate the influence of the demographic feedback, we show in Fig. S7 scenarios of evolutionary rescue with the same parameters but a constant susceptible density of (i.e. without a demographic feedback) that show increased probabilities of rescue overall.

Figure 6.

Probability of evolutionary rescue with (A) and without (B) lethal mutations. The simulations are initialized with the equilibrium values of , and variance for a mutation rate of . The colour scale represents the proportion of simulations in which the infected populations survived. There are 10 simulations per parameter combination. The horizontal dashed white line represents the value of critical mutation rate above which the infected population goes to extinction in the deterministic model. Parameter values: .

Discussion

We develop a mathematical framework to explore the feasibility of the lethal mutagenesis hypothesis, which states that the use of some mutagenic drugs could help clear infections by viral pathogens (Loeb and Mullins 2000, Bull et al. 2007, 2013, Shiraki and Daikoku 2020). To study the effects of these drugs, we track the within-host dynamics of the density of a viral population exposed to a high rate of mutation. New infections are driven by the availability of uninfected cells, as well as the infectivity of the virus and our model accounts for the fact that both these quantities are dynamical variables. First, the density of susceptible cells varies with the within-host influx of new cells and drops with the within-host spread of the virus population. Second, the infectivity of the virus population is also expected to vary with time because infectivity is assumed to be an evolving trait. The mutation process fuels this evolution through the recurrent introduction of phenotypic variation. This variation may be beneficial (higher infectivity) but could also be deleterious (lower infectivity) or even lethal (no infectivity) for the virus. Deleterious mutations bring down the mean infectivity of the virus and may eventually lead to viral extinction. Hence, a key ingredient of this model is the way we account for the effects of mutation on virus infectivity and thus on viral fitness.

Fitness landscapes

Earlier models of lethal mutagenesis often assume that the effects of mutations are fixed, always deleterious, and independent of the genetic background (i.e. no epistasis in fitness) (Swetina and Schuster 1982, Bull et al. 2005). Yet, recent empirical studies measuring the effects of combinations of mutations on viral fitness have found that the effect of mutations on fitness is distributed and that epistasis in fitness is widespread (reviewed in, e.g. Bank 2022). Hence, our attempt to explore quantitatively the feasibility of lethal mutagenesis needs to account for these nonadditive effects in fitness. Two main classes of epistatic fitness landscape models have been used (Blanquart et al. 2014, Bank 2022). First, probabilistic genotype fitness landscape models like the rough Mount Fuji model (RMFM) (Aita et al. 2000). This model allows for moving from a fully additive fitness landscape (no epistasis) to a model with maximal epistasis where each genotype is assigned a given fitness independent of its component alleles (Kauffman and Levin 1987). However, fitting these models to empirical patterns of epistasis requires a very large number of parameters to describe a specific network of interactions (see, e.g. Kemble et al. 2020). Hence, the use of genotype-fitness landscape models may limit our ability to explore the feasibility of lethal mutagenesis over a broad range of viruses. Second, an alternative way to build a genotype-fitness landscape is to combine a genotype–phenotype mapping (with some distribution of additive mutation effects on phenotype) with a phenotype-fitness mapping. This is the basis of the various forms of FGM, a central model of evolutionary ecology and quantitative genetics (Tenaillon 2014). Here, we use the most classic version of the FGM with a multivariate and isotropic Gaussian distribution of additive mutation effects on phenotypes and a quadratic phenotype-fitness map (where fitness is driven by the infectivity of the virus). This simple model requires a few parameters, which can be estimated from the effects of single mutants (see Supplementary Information S3). Crucially, the FGM has been shown to quantitatively reproduce various empirical distributions of mutation fitness effects like (i) the shape of nonlethal single step mutations (Martin and Lenormand 2006b), (ii) the distribution of epistasis among random and/or beneficial mutations (Martin et al. 2007, Schoustra et al. 2016) (but see Blanquart et al. 2014), (iii) the quantitative pattern of diminishing returns epistasis (i.e. the quantitative relation between initial maladaptation and compensatory evolution in response to this maladaptation; Perfeito et al. 2014). Besides, the Gaussian FGM that we use is also the limit that emerges from much less stringent assumptions if one considers mutations to be small perturbations of a highly integrated biological system connecting many highly pleiotropic underlying phenotypes with a lower-dimensional set of key traits (Martin 2014) (but see also Reddy and Desai 2021). Hence, we contend that the FGM offers a simple and solid first step to model the effects of mutations on viral within-host fitness. Future studies, however, should explore the robustness of our predictions for more realistic and more complex fitness landscapes with time-varying fitness optima, anisotropic mutational effects or with coinfections.

Critical mutation rate

Our model can be used to predict the long-term evolutionary equilibrium of the virus population. In particular, we can predict the critical mutation rate U_c above which the mutation load is so high that the virus population is driven to extinction. We derive approximations for U_c under different regimes of selection. First, when the effect of mutation λ is small relative to the mutation rate (WSSM regime), the equilibrium mutation load L_M [combine equations (7) and (9)] is driven by the effects of nonlethal mutations, which depend on the number n of phenotypic dimensions (i.e. complexity). Indeed when phenotypic complexity is high, the proportion of beneficial mutations decreases in favour of deleterious mutations. Because the mutation load increases with the variance of the mutation distribution λ, larger values of λ reduce U_c (Fig. S3). Second, when the effect of mutation is strong relative to the mutation rate, the phenotypic variance of the viral population is lower than in the WSSM regime and we can use other approximations to predict the critical mutation rate. We showed that we can account for the fourth cumulant of the phenotypic distribution to improve our predictions, but when selection is much stronger [condition (14)], we can use the classical HC approximation. In this regime, the effects of nonlethal mutations are so strong that they can be viewed as effectively lethal in the WSSM regime [compare equations (12) and (15)].

Empirical tests of lethal mutagenesis

Several attempts have been made in the past to study experimentally the feasibility of lethal mutagenesis. First, Bank et al. (2016) studied how the mutagen favipiravir may drive a population of influenza to extinction in vitro (Bank et al. 2016). In a treatment with constant mutagen concentration and a 2.1–3.9 fold-increase in genomic mutation rate, the virus did not become extinct. Yet, when the mutagen concentration was increased daily, reaching very high concentrations of up to 200 μM and leading to a 4.2–5.4 fold-increase in genomic mutation rate, the virus did become extinct. Our model, however, predicts that influenza extinction requires at least a 10-fold increase in the genomic mutation rate (Table 2). Second, evolution experiments in the presence of a mutagenic drug have also been conducted on bacteriophage T7 (Springman et al. 2010). Interestingly, Springman et al. (2010) used the NG mutagen to reach a 100–1000 fold-increase in the genomic mutation rate of the virus without achieving viral extinction. This result is consistent with the prediction of our model, which states that DNA bacteriophages require at least a 2500-fold increase in the genomic mutation rate to drive the population extinct (Table 2). Third, it is also useful to discuss recent attempts to use mutagenic drugs to treat SARS-CoV-2 infections. Some clinical trials with molnupiravir report an increase in viral clearance (Fischer et al. 2021) and a reduction in the number of hospitalizations or deaths (Jayk Bernal et al. 2022). However, a later meta-analysis showed that while the drug accelerated clinical improvement and RNA negativity by days 5 and 10, it did not significantly reduce rates of hospitalization or all-cause mortality compared to placebo or standard care (Tian et al. 2023). Although high-throughput deep mutational scanning (DMS) of SARS-CoV-2 spike provided detailed functional maps, including how thousands of individual spike mutations impact both antibody neutralization and receptor affinity (Starr et al. 2020, Greaney et al. 2021, Dadonaite et al. 2023), we do not have access to a genome-wide distribution of within-host fitness effects of mutations. Interestingly, some studies estimated the distribution of fitness effects from the expected and observed frequency across the genome of SARS-CoV-2 (Bloom and Neher 2023, Thadani et al. 2023) but these results are difficult to translate into within-host measures of fitness. This prevents us from properly fitting our within-host model to this data and thus to predict the critical mutation rate estimate of SARS-CoV-2. However, it is interesting to note that the inferred mutational parameters (λ and n) were remarkably consistent across the different viruses of Table 2, suggesting that SARS-CoV-2 might also share the same parameter values. Given that the estimated per-generation mutation rate of SARS-CoV-2 is seven times lower than that of influenza (Markov et al. 2023), our analysis implies that lethal mutagenesis requires a mutagenic drug seven times stronger than to treat influenza. More specifically, our model predicts that treating SARS-CoV-2 would require an 80–400 fold increase in mutation rate to reach U_c. Hence, lethal mutagenesis seems difficult to achieve since available estimates of the mutagenic effect of nucleoside analogues like ribavirin, favipiravir, and molnupiravir indicate that they increase viral mutation rate by a factor of 10 at most (Crotty et al. 2001; Zhou et al. 2021; Sanderson et al. 2023).

The above three empirical examples show that our model does provide testable predictions that may help understand why previous experimental attempts to reach the critical mutation rate have generally failed (i.e. our model indicates that the fold increase in the mutation rate by the mutagen is too low). However, these examples also illustrate the lack of good estimates of the parameters of the model used to describe the dynamics of T7 and SARS-CoV-2. For influenza, the estimation of U_c in Table 2 does not account for the increasing level of mutagen used in the experimental treatments that led to the extinction of the virus (Bank et al. 2016). Indeed, accounting for the additional “direct effects” of the drugs and/or the host immune response on the growth rate of the virus are expected to dramatically reduce the critical mutation rate needed to eradicate the virus population. It is also important to recall that the above discussion of the feasibility of lethal mutagenesis using the critical mutation rate is based on the assumption that the virus population has reached a mutation–selection equilibrium in the presence of a constant dose of the mutagen. However, the time needed to eradicate the virus population could be longer than the duration of the treatment. Hence, in practice, viral recrudescence may occur after the end of a treatment with a strong mutagen (Fig. 5). On the other hand, we also showed scenarios where the virus population can go extinct even when U < U_c (Fig. 6). In other words, the condition U > U_c is likely a higher bound to reach viral extinction. In the following section, we show how our analysis of the transient dynamics of the virus population may help explore these more realistic scenarios.

Transient within-host dynamics

We studied the transient within-host dynamics of the virus in the WSSM regime, where the phenotypic distribution remains Gaussian during the adaptation of the virus population. When selection is stronger, however, the phenotypic distribution of the virus can move away from the Gaussian distribution [equations (S36)–(S39)] (Barton and Turelli 1987, Burger 1991, Turelli and Barton 1994, Bürger 2000). The transient buildup of skewness increases the variance in fitness and tends to speed up the rate of adaptation (Fig. S1). This framework allowed us to explore the effect of a mutagenic drug on viral dynamics. The drug is expected to affect the population dynamics (both the density of infected and uninfected cells) and the evolutionary potential of the virus. We showed that if the treatment is stopped before the eradication of the virus, the infection could rebound and reach very high viral loads (even if the drug is an effective mutagen and U > U_c). The risk of this viral rebound is limited in hosts that are able to mount an effective immune response is eventually expected to kick in and help clear the infection (see Fig. S5). Yet, the risk of rebound could be high for immunodepressed hosts and chronic infections. Hence, it is particularly important to monitor the patients treated with a mutagen after the end of the treatment.

When we explored the robustness of our predictions in a finite population of host cells, we found that the virus could go extinct even when . These extinctions occur when the viral population is initially far away from its fitness optimum. The lower the initial level of adaptation, the more likely the virus population may be driven to extinction by the drug. This could occur if a mutagenic treatment is combined with a direct reduction in fitness by another drug. This effect is particularly strong when the fraction of lethal mutations is high because the evolutionary potential is limited in this case (Fig. 6).

Concluding remarks

Our model focuses on within-host dynamics, but if a patient treated with a mutagenic drug does not fully clear the infection, drug-mutated viruses can be transmitted to a new host. For example, the use of molnupiravir, an antiviral mutagen used against SARS-CoV-2 in some countries, is associated with the rise of specific G-to-A mutations in the virus in these countries (Sanderson et al. 2023). In principle, some of these mutations may affect the between-host fitness of the virus (e.g. higher between-host transmission, ability to evade host immunity and infect vaccinated hosts). These mutations may not necessarily affect within-host dynamics, but they could have dramatic consequences on the emergence of new variants. Hence, a more comprehensive measure of evolutionary safety of lethal mutagenesis therapy should also consider the cumulative number of mutants that may be transmitted to new hosts (Lobinska et al. 2023). An in-depth analysis of the within-host and between-host consequences of mutagenic drugs remains to be explored.

Conflict of interest: None declared.

Funding

None declared.

Data availability

No new data were generated or analysed in this study. All data used are publicly available from the original sources cited within the manuscript.

Code availability

The code produced for this work is available at: https://github.com/martingui/Lethal-mutagenesis.git