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Conceived and designed the experiments: EBO COW. Performed the experiments: EBO. Analyzed the data: EBO. Contributed reagents/materials/analysis tools: EBO TEK. Wrote the paper: EBO COW.

The authors have declared that no competing interests exist.

Lethal mutagenesis is a promising new antiviral therapy that kills a virus by raising its mutation rate. One potential shortcoming of lethal mutagenesis is that viruses may resist the treatment by evolving genomes with increased robustness to mutations. Here, we investigate to what extent mutational robustness can inhibit extinction by lethal mutagenesis in viruses, using both simple toy models and more biophysically realistic models based on RNA secondary-structure folding. We show that although the evolution of greater robustness may be promoted by increasing the mutation rate of a viral population, such evolution is unlikely to greatly increase the mutation rate required for certain extinction. Using an analytic multi-type branching process model, we investigate whether the evolution of robustness can be relevant on the time scales on which extinction takes place. We find that the evolution of robustness matters only when initial viral population sizes are small and deleterious mutation rates are only slightly above the level at which extinction can occur. The stochastic calculations are in good agreement with simulations of self-replicating RNA sequences that have to fold into a specific secondary structure to reproduce. We conclude that the evolution of mutational robustness is in most cases unlikely to prevent the extinction of viruses by lethal mutagenesis.

The high mutation rate of RNA viruses, such as HIV, allows them to rapidly evolve resistance to host defenses and antiviral drugs. A new approach to treating these viruses—lethal mutagenesis—turns the mutation rate of these viruses against them. It uses mutagens to increase the viruses' mutation rates so much that the accumulation of harmful mutations drives viral populations to extinction. Is there any way that a virus could adapt to a drug that increases its mutation rate? One way is that the virus could evolve so that mutations tend to be less harmful. In previous experimental work, there have been reports that virus populations can differ in robustness. Yet, the evolution of mutational robustness did not seem to inhibit extinction by lethal mutagenesis. In this work, we model viral populations under lethal mutagenesis in order to see when viruses might escape extinction by evolving robustness to mutations. We find that viruses can benefit from robustness only at relatively low mutation rates because the extent to which robustness increases fitness is rapidly drowned out by the extent to which higher mutation rates decrease fitness. The implication is that the evolution of mutational robustness is not a fundamental impediment to lethal mutagenesis therapy.

Lethal mutagenesis is a proposed therapy for patients with viral infections. The general approach is to increase the deleterious viral mutation rate enough so that the viral population will go extinct

Research on lethal mutagenesis and the question of how much mutational robustness can affect mutagenesis are of practical importance. In support of the promise of lethal mutagenesis as a treatment for many human and agricultural viruses, there are reports of the addition of a mutagen severely reducing or extinguishing populations of coxsackievirus B3

An important limitation to any pathogen treatment is the ability of the pathogen to develop resistance. Since lethal mutagenesis introduces deleterious mutations throughout the genome of viruses, it seems that there are only two types of effective resistance mechanisms. First, the virus could evolve a mechanism to reduce the number of mutations that the therapeutic mutagen introduces. Ref.

Empirical studies of lethal mutagenesis appear to yield conflicting results. While Ref.

The organization of this paper parallels our line of inquiry. First we ask, when will a population at equilibrium go extinct? We find with a deterministic model that an approximation for the critical mutation rate, i.e. the mutation rate beyond which the population goes extinct, is the log of reproductive capacity divided by the non-neutrality of the population at equilibrium. The implication is that small increase in the mutation rate can compensate for relatively large increases in neutrality. Next, we ask, how will elevating the mutation rate increase the rate at which populations move to areas of a neutral network with higher equilibrium neutrality? We find with a semi-deterministic model that the time it takes for a population undergoing mutagenesis to find the optimal area of the network grows exponentially with the size of the barrier to it. The implication is that we can usually disregard these shifts of the virus population, since the population will quickly shift to the optimal area if the barrier is small and the population will stay where it begins if the barrier is large. Finally, we ask, when will a population that is not at equilibrium go extinct? We show with a stochastic analytical model and simulations based on RNA-secondary structure networks both the critical mutation rate in these more complex models and the probability of stochastic extinction at mutation rates below the critical mutation rate. The implication is that the initial robustness of the population can be important in some cases, but not when the mutation rate exceeds the critical mutation rate.

First, we consider the effects of mutational robustness in a deterministic model of lethal mutagenesis. In general, virus extinction is guaranteed if

We can write

In general, we can write the deleterious mutation rate as

Throughout the remainder of this paper, we consider populations evolving on neutral networks. All sequences on the neutral network have the same reproductive capacity

In the case where neutral sequences are distributed at equal density throughout the mutational network,

Under the assumption that

The set of mutation rates and reproductive capacities that allow the virus population to survive according to Equation (3) are shaded. This set is smaller in the absence of mutational robustness (

Of course, the critical mutation rate may be far above unity and the assumption that

In general, a neutral network may be broken into separate areas of differing neutrality and separated by entropic barriers. (The term

Depending on how great a barrier is in comparison to the mutation rate, the evolution of greater neutrality during lethal mutagenesis will be either inevitable or extremely unlikely. The barriers between areas of the neutral network at high mutation rates will often be so small that they can be neglected. In this case, the separate areas form one large, connected neutral network. Alternatively, the barriers will be so large that we may disregard the undiscovered areas of the neutral network. We next illustrate this concept with a specific example.

We consider the

Sequences in the neutral-staircase landscape consist of zeros and ones (bits). The bits are organized into

The neutral-staircase landscape can be solved analytically, and the full derivation can be found in Ref.

If we sum

In the neutral-staircase fitness landscape, the maximum neutrality increases as the number of active blocks increases. The expected time, in generations, for the number of active blocks

The prospect of the equilibrium neutrality increasing raises the question of how much increases in equilibrium neutrality may increase

Critical mutations rates derived from Equation (5) are plotted as a function of the total number of blocks

When barriers are small, we can expect that the area of the neutral network with the greatest connection density can be found in a reasonable number of generations. In this case, the main question is whether the population can find areas with high connection density before it goes extinct under mutagenesis. In the following subsections, we will address this question using fully stochastic models.

According to Equation (1), extinction is guaranteed if the mutation rate is so high that the equilibrium mean fitness of the population is less than 1. But lethal mutagenesis is not an equilibrium process. Therefore, we next explore how extinction occurs in a population out of equilibrium, using the mathematical framework of multi-type branching processes. Because this approach is a stochastic one, we calculate not only the mutation rate at which extinction is guaranteed but more generally the probability that extinction happens at any given mutation rate. Our main question here is how the extinction probability changes if the population resides initially in regions of the neutral network with particularly low or high connection density.

The mathematical framework we use to calculate the extinction probability under lethal mutagenesis is that of multi-type branching processes. This framework has been used previously to calculate the fixation probability of a rapidly mutating virus on a neutral network

Consider a population where all offspring are identical to their parents. A sequence produces a random number of offspring in the next generation. All these offspring sequences produce their own random number of offspring according to the same probability distribution. The number of progeny that a sequence has in two generations, then, is the sum of these random variables. The use of a probability generating function (p.g.f.) allows for convenient expression of these sums. We use

When there is a finite number

Extinction probabilities can easily be found numerically from Equation (12), but we next present two approximations to illuminate how extinction probabilities follow from offspring distributions.

First, we need an explicit expression for the multivariate p.g.f.s in the fixed-point equation. If the number of offspring of type

When extinction probabilities

When extinction probabilities

The previous subsection developed the general theory of stochastic extinction under lethal mutagenesis. We will now apply this theory to the special case of a neutral network of RNA sequences. To this end, we will first describe a model that links a sequence's location in a neutral network with the sequence's neutrality. This model yields the rates

Consider how the probability-density function of the offspring distribution

We define

Our approach is inspired by Ref.

Equations (19) and (20) say that, since there are only so many neutral sequences, if a sequence is in an area of the neutral network with a high connection density, then the connection density of neutral sequences must generally decline as we move away from it, and vice versa. This reasoning implies that

We can use this framework to determine the

Putting everything together, the probability that any one offspring of a parent of type

The matrix

The dominant eigenvalue decays exponentially with the genomic mutation rate

First, we present results based on the assumption that populations initially consist of a single sequence. This case is relevant to a scenario in which a patient is inoculated with a small dose of virus while on lethal mutagenesis therapy or a virus is establishing itself in a new tissue of a patient's body. With this assumption, we found that the probability of extinction declined with the initial sequence's neutrality, but also that the gradient in extinction probabilities rapidly leveled as the mutation rate increased (

Panel A displays results from simulations where sequence neutrality was determined by RNA folding. Panel B displays results from a branching process model derived from the correlation between sequence neutrality and epistasis. Only in a band of intermediate mutation rates does the extinction probability depend on initial neutrality

Next we used the analytic calculations to study the effect of the size of the neutral network. When going from a smaller neutral network to a larger neutral network, the extinction threshold

The sizes of the neutral networks in panels A, B, and C are

Since lethal mutagenesis is intended to eliminate virus populations that have grown to high levels, we also considered the effect of the initial population size. We considered an initial population that was uniformly composed of sequences with a given initial robustness

The initial population sizes in panels A, B, and C are 1, 100, and 100,000, respectively. The dependence of the extinction probability on the initial robustness is greatest in panel A, where the initial population size is small. These results are from a branching process model derived from the correlation between sequence neutrality and epistasis. Parameters: sequence length

We verified our branching-process model by carrying out simulations with individual RNA sequences (see

We have studied how the evolution of mutational robustness affects lethal mutagenesis. Using a simple deterministic theory, we found that extinction was guaranteed past a critical mutation rate

In our model of replicating RNA sequences, we found that the critical mutation rate

We found that the stochastic model behaved nearly deterministically when the initial population size was 100,000, which is not a large population for viruses. This result assumed a completely homogeneous initial population. If the initial population were heterogeneous, we would likely see nearly deterministic behavior at even lower initial population sizes. At high heterogeneity, the population might contain a single individual with high neutrality. This individual would have a low extinction probability unless

What are reasonable values for the fraction of deleterious mutations

Yet mutational robustness can only increase to the extent to which it is not already present. Theory predicts that populations evolve robustness if the product of mutation rate and population size exceeds one, and that the level of robustness achieved is largely independent of the actual mutation rate

The reproductive capacity

The sequence lengths of 40 and 400 used in the stochastic models are short in comparison to the genomes of RNA viruses, which are about 10,000 base pairs long. Since the relationship between

For our model of replicating RNA sequences under mutagenesis, we found that the critical mutation rate

The bulk of our results implies that the evolution of mutational robustness during lethal mutagenesis is not a serious threat to the efficacy of lethal mutagenesis. As long as lethal-mutagenesis treatment aims to increase

Additionally, our results are not a contradiction to the report that a mutationally robust strain of vesicular stomatitis virus (VSV) prevailed in competition against a strain that was more fit in the absence of a mutagen when 5-FU doses were 20, 40, 60, and 80

While our models do show that the initial neutrality of a population can affect its probability of extinction, this relationship may be overshadowed in practice. For example, the models neglect the effect of defective interfering particles, which may contribute to extinction by lethal mutagenesis

This work has provided quantitative support for the statement that the evolution of mutational robustness will have only a minor effect on lethal mutagenesis. In an extreme case, half of all non-beneficial mutations could evolve to become neutral. In this case, doubling the mutation rate will be sufficient to cause extinction (

We evaluated the convergence times given by Equation (8), numerically derived

We obtained the fixed point

Sequences that folded into a target shape were considered neutral, and all others were considered inviable. The neutrality of a sequence was the fraction of neighbors at a Hamming distance of one that also had the target phenotype. The RNAfold function in the Vienna package

((((....))))............................

.(((..........(((((.....))))))))........

Here, positions that form base pairs are indicated with matching parentheses, and unpaired positions are indicated with dots. For the first target, which was used to generate the results in

The extinction probability of a sequence was determined by simulation of a branching process on the RNA secondary-structure neutral network. Simulations began with a single neutral sequence. These sequences were selected from the sample of sequences used to estimate the size of the neutral network so as to get the full range of initial neutralities. At each iteration, each sequence in the population had a Poisson distributed number of offspring. Each letter of the sequence changed to any of the other three possible letters with a probability equal to the genomic mutation rate divided by the sequence length. Mutation rates ranged from zero to fifteen. Each sequence was tested to see if it folded into the target, and sequences that did not were removed. Simulation was continued until the population size reached zero or 10,000. Simulations were replicated 100 times for each of 500 initial sequences and the extinction probability was the number of simulations in which extinction occurred divided by the total number of simulations. A local polynomial fitting function (the loess function in R

The code written for these analyses is in

Negative correlation between neutrality and epistasis. Equations (19) and (20) predict that as the parameter α of a sequence increases, the epistasis parameter β decreases. The panels are labeled with the sequence lengths of 40, 5000, and 10000. For each sequence length, the lines, from highest to lowest, are numerical solutions where log_{4}(

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Extinction probability as a function of initial neutrality and deleterious genomic mutation rate. Panel A displays results from simulations where sequence neutrality was determined by RNA folding. Panel B displays results from a branching process model derived from the correlation between sequence neutrality and epistasis. Only in a band of intermediate mutation rates does the extinction probability depend on initial neutrality 1-^{29}, reproductive capacity

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Raw data and computer code necessary to reproduce all results reported in this paper.

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Supplementary text.

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