Do selection coefficients add or multiply? And why it matters for resistance evolution

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. 

The addi­tion and mul­ti­plic­a­tion of inter­locus selec­tion coef­fi­cients, where a hap­loid indi­vidu­al with allele A and B has a selec­tion coef­fi­cient sA and sB, lead­ing to a com­bined fit­ness value (ωAB). 

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? 

For a mix­ture of pesti­cides, the level of con­trol attrib­ut­able to each pesti­cide inde­pend­ently and their mix­ture, and the selec­tion coef­fi­cient for a muta­tion for com­plete res­ist­ance to that pesticide.

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. 

For a mix­ture of syn­er­gist­ic pesti­cides, the level of con­trol attrib­ut­able to each pesti­cide inde­pend­ently and their mix­ture, and the selec­tion coef­fi­cient for a muta­tion for com­plete res­ist­ance to that pesti­cide (assum­ing that the res­ist­ance first evolves to the first pesticide). 

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.

For a mix­ture of pesti­cides under cross-res­ist­ance, the level of con­trol attrib­ut­able to each pesti­cide inde­pend­ently and their mix­ture, and the selec­tion coef­fi­cient for a muta­tion for com­plete res­ist­ance to that pesti­cide (assum­ing that the res­ist­ance muta­tion to the first pesti­cide has 50% pos­it­ive cross resistance). 

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.

For a mix­ture of pesti­cides under dose optim­isa­tion, the level of con­trol attrib­ut­able to each pesti­cide inde­pend­ently and their mix­ture, and the selec­tion coef­fi­cient for a muta­tion for com­plete res­ist­ance to that pesticide.

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. 

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