## Philip Madgwick and Ricardo Kanitz

**Our new paper ‘Evolution of resistance under alternative models of selective interference’ was recently published in JEB. The paper addresses a fundamental problem in evolutionary biology that has consequences for an important application of evolutionary theory. The introduction and discussion of the paper is focused on the application in the evolution of resistance, but here instead we focus on clearly explaining the fundamental problem to contextualize the key findings of our paper in a different way. **

To describe the change of a trait over time, the basic setup of a population genetic model uses a selection coefficient that describes the mean change in relative fitness that is associated with an individual carrying a mutant allele at a trait-encoding locus. By convention, when selection coefficients come from alleles at the same locus, selection coefficients are added together to calculate an individual’s relative fitness; when selection coefficients come from alleles at different loci, selection coefficients are multiplied together. The convention of adding intralocus selection coefficients and multiplying interlocus selection coefficients gives the basic setup of an evolutionary model a convenient set of mathematical properties. The addition of intralocus selection coefficients means that evolutionary change can be accurately described assuming haploidy (*i.e.* one allele per locus) and the multiplication of interlocus selection coefficients means that the evolutionary change at a locus can be calculated independently (*i.e.* irrespective of the evolutionary change at other loci). Such convenient mathematical properties can make modelling easier, but this does not justify whether addition or multiplication is more appropriate. Moreover, by implicating a special explanation for deviation from this basic setup (of dominance for the non-addition of intralocus selection coefficients and epistasis for the non-multiplication of interlocus selection coefficients), this way of modelling comes with a theoretical framework that establishes a ‘default’ scenario. Yet this does beg the question: should selection coefficients add or multiply? This is not really a question about whether addition or multiplication is more appropriate, but is a question about what factors can lead to a scenario that differs from the basic setup.

Consider the evolution of resistance to a mixture of two pesticides with different modes of action that each afford 90% mortality when used solo to control a haploid pest. The different modes of action mean that we should expect resistance to each pesticide to evolve at separate loci, and let us assume that resistance to each pesticide involves a mutation that completely prevents pesticidal mortality (*i.e.* to 0%). In the basic setup with the multiplication of interlocus selection coefficients, the mixture would be expected to lead to 99% mortality, such that having one resistance mutation would reduce mortality to 90% and having two resistance mutations would reduce mortality to 0%. This would give each resistance mutation a large selection coefficient of (0.1/0.01= or 1/0.1=) 10. So, what factors would cause the non-multiplication of the selection coefficients?

First, the pesticides could have synergism or antagonism when used together in a mixture, which can cause a deviation from the expected mortality of an individual without resistance mutations. As a hypothetical example, a chloride channel blocker that causes hyperexcitation could have synergism with an uncoupler of oxidative phosphorylation that inhibits respiration to have higher-than-expected mortality through increasing the demand for respiration. Alternatively (also hypothetically), a ryanodine receptor modulator that causes paralysis could have antagonism with an uncoupler of oxidative phosphorylation to have lower-than-expected mortality through decreasing the demand for respiration. Although synergism and antagonism are properties of pesticides (and not their resistance mutations), they would lead the first resistance mutation that appears in the population to have a different selection coefficient. For example, if a pesticide mixture synergistically has 99.9% mortality, the first resistance mutation that reduces the pesticide mixture’s mortality to 90% would have a selection coefficient of (0.1/0.001=) 100.

Second, the resistance mutations could have cross-resistance, which can cause a deviation from the expected mortality of an individual with one resistance mutation (or both). Although behavioural or physiological changes could also produce cross-resistance, a major form of positive cross-resistance occurs through increases in the metabolic rate of detoxification that can reduce the amount of pesticide that reaches the site of action. As a hypothetical example, increased metabolism could provide resistance to a ryanodine receptor modulator by removing toxins from the cytoplasm before they reach the endoplasmic reticulum, which could also provide some level of positive cross-resistance to an uncoupler of oxidative phosphorylation by decreasing the amount of pesticide that reaches the mitochondria (that are also within the cytoplasm). Negative cross-resistance is also conceivable, if resistance to one pesticide can enhance the susceptibility to the other. Positive or negative cross-resistance would afford both resistance mutations different selection coefficients. For example, if a positive cross-resistance mutation provides a 100% reduction to one pesticide’s mortality and a 50% reduction to the other’s, (only if it spreads first) the positive cross-resistance mutation would have a selection coefficient of (0.55/0.01=) 55 whilst the other resistance mutation would have a selection coefficient of (1/0.55=) 1.8.

Third, pesticides may be used at a reduced dose in a mixture. To achieve the same level of mortality as using either pesticide solo, a mixture can reduce the dose of one or both pesticides. This may be done to optimise the mixture product to balance factors like its economic cost, level of control relative to market competitors and sustainability of an acceptable level of control (reflecting the time to resistance). Although this does not technically lead to non-multiplication, the reduced dose can decrease the selection coefficient for resistance to the pesticide mixture relative to the solo use of each pesticide. For example, to achieve 90% mortality from their mixture, the dose of each pesticide could be reduced until they produce ~68% mortality when used solo, which would reduce the selection coefficient to each pesticide in the mixture from 10 to (0.32/0.1 or 1/0.32=) ~3.2.

Therefore, factors like synergism, cross-resistance and dose optimisation can lead to a scenario that differs from the basic setup of the multiplication of interlocus selection coefficients, but we are still framing the question in terms of ‘what explains a deviation from the basic setup’. This should not imply that the basic setup is the ‘general’ scenario, as it only holds the position as the ‘default’ scenario for evolutionary models because of its convenient mathematical properties. In the context of pesticide mixtures, its generality depends on how often factors like synergism, cross-resistance or dose optimisation are important for the evolution of resistance, which is a question for experimental research.

Why does this matter? Pesticides are often used in mixtures as a management strategy to delay the evolution of resistance. In our paper, we show that the addition and multiplication of interlocus selection coefficients leads to substantially different outcomes for the probability and time it takes for a new mutation to spread through a population during the spread of another (already existing) mutation. In short, addition leads to substantial decreases in the probability of spreading and increases in the time it takes to spread, whereas multiplication does not. Practically, this means that addition is highly likely to lead to the sequential spread of resistance mutations, which would delay resistance evolution, whereas multiplication leads to their more rapid simultaneous spread. As such, the assumption of addition or multiplication can cause large differences in the predictions about the benefits of using mixtures for resistance-management. Whilst the importance of this result requires experimental verification, our paper demonstrates that the fundamental question of whether selection coefficients should add or multiply has very tangible implications for how we manage the evolution of resistance.