Evolutionary safety of lethal mutagenesis driven by antiviral treatment

Abstract

Nucleoside analogs are a major class of antiviral drugs. Some act by increasing the viral mutation rate causing lethal mutagenesis of the virus. Their mutagenic capacity, however, may lead to an evolutionary safety concern. We define evolutionary safety as a probabilistic assurance that the treatment will not generate an increased number of mutants. We develop a mathematical framework to estimate the total mutant load produced with and without mutagenic treatment. We predict rates of appearance of such virus mutants as a function of the timing of treatment and the immune competence of patients, employing realistic assumptions about the vulnerability of the viral genome and its potential to generate viable mutants. We focus on the case study of Molnupiravir, which is an FDA-approved treatment against Coronavirus Disease-2019 (COVID-19). We estimate that Molnupiravir is narrowly evolutionarily safe, subject to the current estimate of parameters. Evolutionary safety can be improved by restricting treatment with this drug to individuals with a low immunological clearance rate and, in future, by designing treatments that lead to a greater increase in mutation rate. We report a simple mathematical rule to determine the fold increase in mutation rate required to obtain evolutionary safety that is also applicable to other pathogen-treatment combinations.

Fig 1

(A) Mechanism of action of molnupiravir. SARS‑CoV‑2 has a positive-sense single-stranded RNA genome, represented schematically in (1). Its replication proceeds in 2 steps: first, the synthesis of a negative-sense template strand (2), which is then used to synthesize a positive-sense progeny genome (3). Molnupiravir (M) is incorporated against of A or G during the synthesis of the negative-sense template strand (2). When the template strand is replicated, M can be base-paired with either G or A. Hence, all A and G in the parent genome become ambiguous and can appear as A or G in the newly synthetized positive-strand genome. C and T are not affected by molnupiravir during the synthesis of the template strand, (1) to (2), but can be substituted to M during the synthesis of the progeny genome from the template strand, (2) to (3). M can then base-pair with A or G when used as a template; see (3) to (4), which can cause A->U and U->A transitions in the final progeny genome (5). (B) Virus dynamics within an infected person. Wild type (x) and mutant (y) replicate at rate b and quality q = 1−u. The per base mutation rate, u, is increased by treatment with molnupiravir. Both wild type and mutant need to maintain m positions to remain viable. Mutating any of n positions in the wild type results in a mutant. In the beginning of the infection, the adaptive immune response is weak, and virus is cleared at a rate a₀ which is less than b. After some time, T, the adaptive immunity is strong, and virus is cleared at the rate a₁ which is greater than b. (C) Graphical summary of the influence of mutagenic drugs on virus mutants. White circles represent wild type, beige circles viable mutant, and black circles dead virus. When the mutation rate is low, few viable mutants and few lethal mutants are produced. Most mutations occur when the virus load is already high; hence, they have little influence on subsequent generations. For intermediate mutation rate, the total virus load declines but the amount of viable mutant increases. When the mutation rate is high, both the virus load and the amount of viable mutant decline. SARS‑CoV‑2, Severe Acute Respiratory Syndrome Coronavirus 2.

Fig 2. Cumulative mutant virus load versus mutation rate, u₁, during treatment.

The cumulative mutant virus load increases with mutation rate u₁ before reaching a peak and then decreases to low values. If the peak is reached at a mutation rate that is less than the natural mutation rate, u₀ (red dotted line), then any increase in mutation rate reduces the cumulative mutant load. If the peak is reached for a mutation rate greater than u₀, then the increase in mutation rate caused by mutagenic treatment must exceed a threshold value (blue dotted line) to reduce the cumulative mutant virus load. We also consider mutants with a 1% advantage in the birth rate. As expected, we observe a higher cumulative mutant load for the advantageous mutant (green line) compared to the neutral mutant (blue line). But the minimum mutation rate under treatment that is required for evolutionary safety is slightly lower for the advantageous mutant. (A) Treatment starts at peak virus load. (B) Treatment starts at infection. The red arrow indicates the mutation rate at the error threshold of the growth phase. Parameters: b = 7.61 per day, a₀ = 3 per day, n = 1 position, T = 5 days, m and a₁ as shown. The code used to generate this figure can be found at DOI: 10.5281/zenodo.8017992.

Fig 3. Evolutionary safety of mutagenic treatment.

In the green parameter region, any increase in mutation rate reduces the cumulative mutant virus load and is therefore evolutionarily safe. In the red shaded region, we indicate the minimum fold increase in mutation rate that is required to reduce the cumulative mutant load. Contour lines for 3-fold and 10-fold increase are shown. (A) Treatment starts at peak virus load. (B) Treatment starts at infection. Parameters: b = 7.61 per day, a₀ = 3 per day, n = 1, T = 5 days, u₀ = 10⁻⁶ per bp. The code used to generate this figure can be found at DOI: 10.5281/zenodo.8017992.

Fig 4. ERF for a grid of parameters.

For each pair of parameters, we numerically compute the ERF for a range of values, while all other parameters are fixed. We observe that the value of n has little effect on the ERF. Evolutionary risk factors above 1 are only observed for low values of the number of lethal positions, m. The ERF decreases with early treatment, high viral mutation rate under treatment, and large number of lethal positions. Initial condition: x₀ = 1 and y₀ = 0. The code used to generate this figure can be found at DOI: 10.5281/zenodo.8017992. ERF, evolutionary risk factor.

Fig 5. The ERF versus the number, n, of positions in the viral genome giving rise to concerning (or viable) mutations.

The ERF of mutagenic treatment is the ratio of the cumulative mutant virus load with and without treatment. We explore all values of n subject to the constraint that m+n remains below the length of the SARS‑CoV‑2 genome. We observe that the ERF decreases as function of n. (A) Treatment starts at peak virus load. (B) Treatment starts at infection. Parameters: a₀ = 3 per day, b = 7.61 per day, u₀ = 10⁻⁶ per bp, u₁ = 3∙10⁻⁶ per bp, T = 5 days. Initial condition: x₀ = 1 and y₀ = 0. The code used to generate this figure can be found at DOI: 10.5281/zenodo.8017992. ERF, evolutionary risk factor; SARS‑CoV‑2, Severe Acute Respiratory Syndrome Coronavirus 2.