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Pieter J. Wijngaarden1Ã, Frank van den Bosch2, Michael J. Jeger3 1Laboratory of Genetics, Wageningen University, Arboretumlaan 4, NL-6703 BD Wageningen, The Netherlands 2Biomathematics Unit, Rothamsted Research, Harpenden, Hertfordshire AL5 2JQ, UK 3Department of Agricultural Sciences, Imperial College London, Wye Campus, Wye, Ashford, Kent TN25 5AH, UK Populations of pathogenic organisms often evolve resistance in response to the use of pesticides or anti-biotics. This rise of resistance may be followed by a fall when chemical control is suspended and resistancealleles carry a fitness cost. Another possibility is that mutations at secondary loci compensate for the cost,usually without loss of resistance. This enables resistant types to withstand invasion by the susceptible wild-type; resistance then persists in the population, which reduces the efficacy of future pesticide or antibioticuse. We examined a two-locus model of a haploid organism that adapts to the cost of resistance by a singlecompensatory mutation. We addressed the question how different combinations of cost and compensationand different levels of recombination affect the consequences of a single pesticide application. Resistancewill become fixed in the population when the fraction of the population exposed to pesticide exceeds the costof resistance. Compensatory mutations reduce the cost of resistance and therefore this threshold level of pes-ticide use. In the absence of pesticide, recombination promotes stability of equilibria. In the presence of pes-ticide, recombination accelerates the fixation of resistance and compensating alleles; recombination mayalso enable the persistence of compensated resistant types after pesticide use.
Keywords: cost of resistance; compensatory evolution; pesticides; epistasis; resistance Mutations that modify the cost can facilitate the evolution The use of chemicals to control populations of pathogenic of both attack and defence mechanisms. Compensatory organisms has often led to the rapid spread of alleles con- adaptation not only applies to resistance mutations, how- ever, but also to deleterious mutations in general. Resistance frequently comes at a cost, however, so that the showed that deleterious effects of random inser- susceptible wild-type has a higher fitness than resistant tion mutations in evolving lines of E. coli were compensated mutants when antibiotics or pesticides are absent. On the for while retaining these mutations (see also positive side, this fitness difference may cause susceptible for a short review of the experimental literature).
types to return to their dominant position in pathogen Compensatory mutations are not only of practical populations when chemical control is suspended importance but also of theoretical interest: they provide an opportunity to study the origin and maintenance of gene the negative side, a cost of resistance selects for mutations interactions (epistasis) in adaptive evolution. Compensat- that reduce this cost, usually without loss of resistance.
ing second site mutations make a return to the susceptible Recent experiments on micro-organisms show the ease state unlikely, as back mutations at either the resistance with which the cost of resistance is reduced by mutations locus or the compensatory locus result in genotypes with a at secondary loci: the cost of resistance was rapidly com- lower fitness than both the uncompensated susceptible and pensated for in streptomycin-resistant Escherichia coli the compensated resistant genotypes. Hence, it shows how two loci can become parts of a group of coadapted genes resistant E. coli and Helicobacter pylori that effectively prevent reverse evolution Pesticide and antibiotic resistance have been modelled extensively (reviewed by and a cost of resistance Compensatory mutations may also have occurred in is often taken into account. Subsequent modification of the organisms where cells are attacked from within, e.g. by cost is rarely dealt with, however. As compensatory muta- restriction enzymes or toxins in bacteria tions may reduce the efficacy of chemical control by slowing In these systems, cells without protection destroy them- the decline of resistance, understanding their dynamics is selves while protection without toxicity is probably costly.
critical. Here we present a simple model of a haploid organ-ism, e.g. a pathogenic fungus, that adapts to the cost ofresistance by a single compensatory mutation. We use both à Author for correspondence (pieter_wijngaarden@yahoo.com).
analytical solutions (for continuous pesticide use) and Table 1. Fitness values of the susceptible-uncompensated (SU ), susceptible-compensated (SC ), resistant-uncompensated (RU )and resistant-compensated (RC ) genotypes in unexposed (Wu) and exposed (We) environments.
(These environments occur with frequencies 1Àf and f, respectively. Wi is the mean fitness across environments of genotype i.
The selection coefficients sr, src and sc represent the cost of resistance without compensation, the cost of resistance with (or after)compensation, and the cost of compensation, respectively.) simulations (for temporary pesticide use) to examine howselection and recombination affect populations dominated by the wild-type (susceptible and uncompensated) or the double mutant (resistant and compensated) genotypes.
2. MODELConsider a large, random-mating population of a haploid organism with discrete generations. A fraction, f, of this population is exposed to a pesticide, leaving a fraction, 1Àf, unexposed. At the first (or resistance) locus the wild-type S allele renders its bearer susceptible to this pesticide; the mutant R allele confers resistance but this resistance comes at a cost (sr > 0). At the second (or compensatory) locus initial frequency non-compensating allele the mutant C allele compensates for (part of) the cost of 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 r to src, which may result in src < 0) whereas the wild-type U allele has no effect on the cost.
Figure 1. The effect of different initial conditions after 25 000 The C allele may be costly, however, when associated with generations. The grey area corresponds to initial allele an S allele. lists the genotypes and their fitnesses.
frequencies that result in fixation of the SU genotype; initial The expected frequency of genotype i after selection allele frequencies in the white area result in fixation of the RC genotype. Parameter values: f ¼ 0, sr ¼ 0:25, src ¼ 0:05, sc ¼ and RC ) or À1 (for i ¼ SC and RU ), r is the recombi- We ignore stochastic effects, migration and mutation so We will now consider how the conditions for stability are that we can focus exclusively on the effects of selection and affected by different values of sr, sc, src and r in both the absence and presence of pesticide use.
The different outcomes of our model can be explored by considering the behaviour around the boundary equilibria of the four genotypes. The matrix of partial derivatives (the Jacobian matrix) describes the linearized dynamics of the The first thing to note is that equilibria 2 and 3 will never be system around these equilibria. The local stability of equi- stable when, as we assume, sc > 0 and src < sr. Without librium points can be determined by inspection of the recombination (r ¼ 0), the first equilibrium is stable when eigenvalues of the Jacobian matrix evaluated at equilib- sr > 0, src > 0 and sc > 0; equilibrium 4 is unstable in this rium; stability is indicated when the absolute values of all case. With r > 0, equilibrium 1 is stable when sr > 0, sc > four eigenvalues are smaller than unity.
0 and src > 1 À 1 (note that because 0 6 r 6 0:5, non-zero values for r allow src < 0); equilibrium 4 is stable when x1 ¼ 1, x2 ¼ 0, x3 ¼ 0 and x4 ¼ 0.
src < sc and src < r. The conditions for stability of these The eigenvalues of the Jacobian matrix are 0, 1 À sc, two equilibria are not mutually exclusive and which equi- librium will be attained depends on the initial allele fre- x1 ¼ 0, x2 ¼ 1, x3 ¼ 0 and x4 ¼ 0.
quencies With recombination the numerators of The eigenvalues of the Jacobian matrix are eigenvalues containing r (these eigenvalues are associated with genotypes containing alleles not present in the fixed genotype) become smaller and the conditions for stability x1 ¼ 0, x2 ¼ 0, x3 ¼ 1 and x4 ¼ 0.
will therefore be more readily fulfilled than without recom- The eigenvalues of the Jacobian matrix are 1Àf , 1 ¼ 0, x2 ¼ 0, x3 ¼ 0 and x4 ¼ 1.
Applying pesticide does not inevitably lead to a stable The eigenvalues of the Jacobian matrix are (1Àf )(1Àr), equilibrium of one of the resistant genotypes. With r ¼ 0, equilibrium 1 remains stable as long as f < s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 fraction of the population exposed to pesticide Figure 3. Changes in allele frequency under continuous The effect of recombination (r) and the level of R allele (solid line) and the C allele (dashed f ) on the outcome of pesticide use after 25 000 lines). Each dashed line shows a different amount of generations. Grey area: fixation of SU genotype; white area: compensation decreasing from left to right. Parameter values: 0:05 and sc ¼ 0:1. The initial frequency of both the S and the :9, r ¼ 0, sr ¼ 0:1, sc ¼ 0:025. The initial frequencies of both the R and the C alleles are 0.0001.
sc > 0; when f > sr and f > src, equilibrium 4 is stable. Wetherefore have a threshold level of spraying f > sr that marks the transition from equilibrium 1 to equilibrium 4 The evolution of sets of interacting genes (coadapted gene (we assume src < sr so the condition f > src is always met complexes) is a long-standing problem in evolutionary when f > sr). With r > 0 and src < f < sr the outcome is determined by the amount of recombination Wright’s shifting balance theory of evolution where they rium 4 whereas higher values for r yield equilibrium 1.
are necessary (but not sufficient) for the occurrence of fit- ness peaks and valleys. A difficulty is how to cross fitness r and f ) src the time to fixation of the RC genotype depends on the reduction of the cost of resistance valleys: a high-fitness double mutant is unlikely to arise when intermediate genotypes are rare because of theirlower fitness. It is therefore not surprising that solutions tothis problem (e.g. usually involve an important role for genetic drift so that So far we have assumed the continuous presence of pes- intermediate types can rise in frequency by chance. The ticide. With increasing values of f, however, the number of evolution of the cost of resistance points to a way of cir- generations of pesticide use (s) that suffices to cause fix- cumventing this problem. When selection temporarily ation of both R and C alleles decreases; this number can befound by simulation. Obviously, s decreases rapidly when favours an intermediate genotype it allows this intermedi- ate to reach a high frequency; the emergence of the double rc approaches zero but recombination also reduces s ). This effect of recombination only mutant is then much more likely. If such an episode of occurs for low levels of pesticide use; for high values of f selection leaves no traces (e.g. it did not affect other loci) there is even a slight increase (not visible in the figure) in s then, in hindsight, the widespread occurrence of the double with increasing amounts of recombination.
When pesticides are used for only a short period of time Adaptation to the cost of resistance makes it easier for resistance to invade a predominantly susceptible popu- lation during pesticide use and to persist in that population may occur via equilibria with x1 ¼ 0, x2 > once the pesticide has disappeared from the environment.
3 ¼ 0 and x4 > 0. This is illustrated in start- ing with a mainly susceptible and uncompensated popu- The conditions for the RC genotype to persist in the lation less than 39 generations spraying yields equilibrium absence of chemical control are still stringent, however, 1 while more than 49 generations spraying results in equi- and require recombination when compensation is not com- librium 4; 39–49 generations of pesticide use, however, plete. Fungi are predominantly haploid; however, recombi- causes fixation of the C allele but not of the R allele.
nation is important in outcrossing and the parasexual cycle.
shows how this result is arrived at for 40 genera- In strictly asexual organisms, however, other mechanisms tions of pesticide use: the R allele goes to near-fixation are needed to maintain RC types in a population. quickly, but its frequency then declines and remains ‘fro- suggest that bottleneck effects can account for zen’ at a frequency of 0.92. The C allele, however, rises only slowly in frequency. The timing and duration compensatory mutations occur more often than reversals (in terms of number of generations of pesticide use) of the they are more likely to pass through bottlenecks; they can internal equilibria depend on f: the higher the value of f, the therefore remain in the population despite their lower fit- earlier and shorter this transitional spraying period.
ness. This mechanism will especially work for mutations Figure 4. The minimum number of generations spraying (s) that results in fixation of the RC genotype for different levels ofpesticide use ( f ); and (a) different reduced costs of resistance (src) and (b) different recombination rates (r). Parameter values: (a) sr ¼ 0:2, sc ¼ 0:1 and r ¼ 0; (b) sr ¼ 0:3, src ¼0:2 and sc ¼ 0:1. The initial frequency of both the S and the U Figure 5. The result of temporary pesticide use. (a) The outcome after 25 000 generations with pesticide use in the first30–50 generations. Open circles, SU genotype; filled circles, with small effect as these are expected to arise more often RC genotype; squares, SC genotype. Parameter values: r ¼ 0:125, f ¼ 0:5, sr ¼ 0:15, src ¼ 0:05, sc ¼ 0:05. The initial Some of the results we presented apply to evolutionary frequency of both the S and the U allele is 0.9999. (b) Changesin allele frequencies after 40 generations of pesticide use. Solid rather than ecological time-scales. When only a small part line, R allele; dashed line, C allele. Parameter values: of the pathogen population is exposed to pesticide it may r ¼ 0:125, f ¼ 0:5, sr ¼ 0:15, src ¼ 0:05, sc ¼ 0:05. The initial take thousands of generations until the RC genotype frequencies of both the R and the C allele are 0.0001.
becomes fixed. However, when most of the population isexposed, fixation occurs within tens of generations,especially when there is recombination. Important here is breaking up the combinations RC (at the SU boundary) or our assumption that compensating alleles are already SU (at the RC boundary). At the SU boundary, recombi- present in the population. An additional ‘waiting time’ nation even allows src to take negative values so that RC would have to be taken into account when compensating types have a higher fitness than the wild-type SU. With alleles have to be introduced into the population by constant pesticide levels recombination affects the equili- mutation or migration. looked at the time bria when src < f < sr: in this range little or no recombi- to fixation of compensatory mutations in a situation com- nation (and therefore little or no decoupling of R and C parable to our unsprayed environment (i.e. both resistance alleles) benefits the RC genotype whereas high recombi- and compensatory alleles are deleterious individually). He nation rates cancel the RC genotype’s fitness advantage in showed that the average waiting time can be extremely long the sprayed part of the population. A similar effect is seen in both small populations (where mutations are rare) and with temporary pesticide use where recombination has the large populations (where selection is very effective). In our effect of considerably shortening the time to fixation.
case, however, resistance and compensatory alleles may So far we have not dealt with the nature of the cost of already be at a high frequency as a result of recent pesticide resistance nor have we been explicit about the mechan- ism(s) by which compensation is achieved. However, by In the absence of pesticide use two boundary equilibria assigning fitness values 1Àsr and 1Àsrc, respectively, to the are possible, depending on the initial allele frequencies and RU and RC genotypes in both the presence and absence of recombination rate. The initial allele frequencies will pesticide we implicitly assumed that the cost is not con- mainly depend on the history of selection (i.e. pesticide ditional on the presence or absence of pesticide. When use). The effect of recombination is a consequence of resistance is costly only in the absence of pesticide (and the RU and RC genotypes have fitness 1 instead of 1Àsr Christiansen, F. B., Otto, S., Bergman, A. & Feldman, M. W.
1998 Waiting with and without recombination: the time to pro- rc in the sprayed part of the environment) both resistance and compensation will spread even faster than duction of a double mutant. Theor. Popul. Biol. 53, 199–215.
Cowen, L. E., Kohn, L. M. & Anderson, J. B. 2001 Diver- More realism can be added to our model in several ways.
gence in fitness and evolution of drug resistance in experi- We considered a single compensatory mutation, but as mental populations of Candida albicans. J. Bacteriol. 183, long as compensation is not complete there will be selec- Fenster, C. B., Galloway, L. F. & Chao, L. 1997 Epistasis and tion for any additional mutation that reduces the remaining its consequences for the evolution of natural populations.
fitness cost. These second (third, etc.) compensatory mutations may be conditional on the first (second, etc.) Hayes, F. 2003 Toxins-antitoxins: plasmid maintenance, pro- mutation and thereby add to the complexity of the model.
grammed cell death, and cell cycle arrest. Science 301, 1496– Furthermore, we assumed that sc > 0, i.e. the compensa- tory mutation is slightly deleterious on a susceptible back- Levin, B. R., Lipsitch, M., Perrot, V., Schrag, S., Antia, R., ground, to account for its low frequency in the population Simonson, L., Moore Walker, N. & Stewart, F. M. 1997 before spraying. However, the C allele will still be com- The population genetics of antibiotic resistance. Clin. Infect.
pensatory (and epistatic) as long as its fitness enhancing effect is stronger when associated with an R allele than with Levin, B. R., Perrot, V. & Walker, N. 2000 Compensatory mutation, antiobiotic resistance and the population genetics and the conditions for stability of equilibria 3 and 4 become of adaptive evolution in bacteria. Genetics 154, 985–997.
more stringent. A stable equilibrium 2 becomes a possi- Lipsitch, M. 2001 The rise and fall of antimicrobial resistance.
cM < 0, however, resulting in the continued Michalakis, Y. & Slatkin, M. 1996 Interaction of selection and presence of compensatory alleles in a susceptible popu- recombination in the fixation of negative-epistatic genes.
Extension of the model to a diploid pest organism Moore, F. B.-G., Rozen, D. E. & Lenski, R. E. 2000 Pervasive requires that we consider the degree of dominance of both compensatory adaptation in Escherichia coli. Proc. R. Soc.
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refuge (i.e. an untreated area) for susceptible genotypes Palumbi, S. R. 2001 Humans as the world’s greatest evol- that can keep most of the resistance alleles in heterozygous utionary force. Science 293, 1786–1790.
condition. With a high dose heterozygotes are killed; resist- Peck, S. L. 2001 Antibiotic and insecticide resistance model- ance is then effectively recessive which slows its spread ing: is it time to start talking? Trends Microbiol. 9, 286–292.
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however, and the HRD strategy is therefore unlikely to Poon, A. & Otto, S. P. 2000 Compensating for our load of mutations: freezing the meltdown of small populations.
model shows that leaving part of the environment (1Àf ) untreated is an important factor in determining the RC Rausher, M. D. 2001 Co-evolution and plant resistance to genotype’s success. Clearly then, there is the need to know natural enemies. Nature 411, 857–864.
if compensatory evolution can interfere with the HRD Reynolds, M. G. 2000 Compensatory evolution in Rifampin- strategy in diploid organisms such as insects.
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