Predicting the Evolution of Sex on Complex Fitness Landscapes
Sexual reproduction is widespread among multi-cellular organisms 1 ., However , the ubiquity of sex in the natural world is in stark contrast to its perceived costs , such as the recombination load 2 or the two-fold cost of producing males 3 , 4 ., Given these disadvantages it is puzzling that sexual reproduction has evolved and is maintained so commonly in nature ., The “paradox of sex” has been one of the central questions in evolutionary biology and a large number of theories have been proposed to explain the evolution and maintenance of sexual reproduction 5 ., Currently , the most prominent theories include, ( i ) the Hill-Robertson effect 6–8 ,, ( ii ) Mullers ratchet 9 ,, ( iii ) the Red Queen hypothesis 10 , 11 , and, ( iv ) the Mutational Deterministic hypothesis 12 , 13 ., While originally described in various different ways , the underlying benefit of sex can always be related to the role of recombination in breaking up detrimental statistical associations between alleles at different loci in the genome ., What fundamentally differentiates the theories is the proposed cause of these statistical associations , assigned to either the interactions between drift and selection ( Fisher-Muller effect , Mullers ratchet , and Hill-Robertson effect ) or gene interactions and epistatic effects ( Red Queen hypothesis and Mutational Deterministic hypothesis ) ., The present list of hypotheses is certainly not exhaustive , with new ones continuously being proposed , complementing or replacing the existing ones 14 ., However , it is not new hypotheses that are most needed , but the real-world evidence that allows us to distinguish between them ., The major question that still remains is whether the assumptions and requirements of different theories are fulfilled in the natural world ., Accordingly , there has been considerable effort to experimentally test these assumptions , mainly for the epitasis-based theories ( reviewed in 15–17 ) ., However , an even more basic and crucial problem underlies all work on evolution of sex: how does one choose , measure , and interpret appropriate population properties that relate to different theories 17–19 ., The difficulty stems from the often large divide between the theoretical and experimental research: theories are frequently formulated as mathematical models and rely on simplistic fitness landscapes or small genome size ( e . g . two locus , two allele models ) 13 , 20–25 ., As a result , it may be unclear how a property established based on these simplified assumptions relates to actual properties of natural populations ., In this study we attempt to bridge the gap between the theoretical and experimental work and to identify which population measures are predictive of the evolution of sexual reproduction by simulating the evolution of both sexual and asexual populations on fitness landscapes with different degrees of complexity and epistasis ., The measures we use are the change of mean fitness , of additive genetic variance , or of variance in Hamming distance as well as four epistasis-based measures , physiological , population , mean pairwise , and weighted mean pairwise epistasis ., While this certainly is not an exhaustive list , we took care to include major quantities previously considered in theoretical and experimental literature ( e . g . 26–28 ) ., With some exceptions 29–32 , earlier work generally focused on the smooth , single peaked landscapes , while here we also use random landscapes and NK landscapes ( random landscapes with tunable ruggedness ) ., Some studies of more complex rugged landscapes tested whether they would select for sex but have not found a simple and unique answer , even in models with only two-dimensional epistasis 33 , 34 ., A recent paper , which uniquely combines experimental and theoretical approaches and simulates evolution of sex on empirical landscapes , also finds that landscape properties greatly affect the outcome of evolution , sometimes selecting for but more often against sex 35 ., However , what specifically distinguishes our study is the goal of not only determining when sex evolves but also of quantifying our ability to detect and predict such outcome in scenarios where we know how the evolution proceeds ., Whether the more complex landscapes we are using here are indeed also more biologically realistic is open to debate as currently little is known about the shape and the properties of real fitness landscapes ( for an exception see for example 35 , 36 ) ., Our goal is to move the research focus away from the simple landscapes mostly investigated so far to landscapes with various higher degrees of complexity and epistasis , and to probe our general understanding of the evolution of sexual reproduction on more complex fitness landscapes ., Notably , we find that some of the measures routinely used in the evolution of sex literature perform poorly at predicting whether sex evolves on complex landscapes ., Moreover , we find that genetic neutrality lowers the predictive power of those measures that are typically robust across different landscapes types , but not of those measures that perform well only on simple landscapes ., The difficulty of predicting sex even under the ideal conditions of computer simulations , where in principle any detail of a population can be measured with perfect accuracy , may be somewhat sobering for experimentalists working on the evolution of sex ., We hope , however , that this study will evoke interest among theoreticians to tackle the challenge and develop more reliable predictors of sex that experimentalists can use to study the evolution of sex in natural populations ., We investigated the evolution of sex in simulations on three types of fitness landscapes with varying complexity ( smooth , random and NK landscapes ) and used seven population genetic quantities ( ΔVarHD , ΔVaradd , ΔMeanfit , Ephys , Epop , EMP , and EWP , Table, 1 ) as predictors of change in frequency of the recombination allele ( see Methods for more details ) ., We calculated predictor accuracy ( the sum of true positives and true negatives divided by the total number of tests ) and used it to assess their quality on 110 smooth landscapes with varying selection coefficients and epistasis , 100 random landscapes , and 100 NK landscapes each for K\u200a=\u200a0 , … , 5 ., All landscapes are based on 6 biallelic loci and they were generated such that an equal number of landscapes of each type select for versus against sex in deterministic simulations with infinite population size ., Hence , random prediction by coin flipping is expected to have an accuracy of 0 . 5 ., Figure 1 shows the accuracy of the predictors for the different landscape types ., Increasing levels of blue indicate greater accuracy of prediction ., For the simulations with infinite population size ( deterministic simulations ) we ran a single competition between sexual and asexual populations to assess whether sex was selected for ., For simulations with finite population size ( stochastic simulations ) , we ran 100 simulations of the competition phase and assessed whether the predictor accurately predicts the evolution of sex in the majority of these simulations ., Focusing on the top left panel we find that for deterministic simulations most predictors are only highly accurate in predicting evolutionary outcomes for the smooth landscapes ., The exception is the poor performance of ΔMeanfit , which is not surprising , as theory has shown that for populations in mutation-selection balance ΔMeanfit is typically negative 2 ., According to our use of ΔMeanfit as a predictor , it always predicts no selection for sex when negative and thus is correct in 50% of cases , due to the way the landscapes were constructed ., For the NK0 landscapes , all predictors perform poorly , because such NK landscapes have no epistasis by definition ( see Methods ) ., For infinite population size , theory has established that in absence of epistasis there is no selection for or against sex ., Indeed , in our simulations the increase or decrease in the frequency of sexual individuals is generally so small ( of order 10−15 and smaller ) that any change in frequency can be attributed to issues of numerical precision ., Generally , the accuracy of most predictors is much weaker for complex landscapes ( NK and random landscapes ) than for the simpler , smooth landscapes ., The predictors that have highest accuracy across different landscape types are ΔVarHD and Epop ., To test whether combinations of the predictors could increase the accuracy of prediction of the evolution of sex we plot for each landscape the value of the predictors ΔVarHD , ΔVaradd and ΔMeanfit against each other and color code whether the number of sexual individuals increased ( red ) or decreased ( blue ) during deterministic competition phase ( see Figure 2 ) ., If the blue and red points are best separated by a vertical or a horizontal line , then we conclude that little can be gained by combining two predictors ., If , however , the points can be separated by a different linear ( or more complex ) function of the two predictors , then combining these predictors would indeed lead to an improved prediction ., Figure 2 shows the corresponding plots for the smooth , the random , and the NK2 landscapes ., For the smooth landscapes the criterion ΔVarHD>0 or ΔVaradd>0 are both equally good in separating cases where sex evolved from those where it did not ., As already shown in Figure 1 , ΔVarHD is generally a more reliable predictor of the evolution of sex than ΔVaradd in the more complex random or NK landscapes ., Epistasis-based theories suggest that the selection for sex is related to a detrimental short-term effect ( reduction in mean fitness ) and a possibly beneficial long-term effect ( increase in additive genetic variance ) 28 ., The plots of ΔVaradd against ΔMeanfit , however , do not indicate that combining them would allow a more reliable prediction of the evolution of sex ., Generally , the plots show that blue and red points either tend to overlap ( in the more complex landscapes ) or can be well separated using horizontal or vertical lines ( in the smooth landscapes ) such that combining predictors will not allow to substantially increase the accuracy of prediction ., This is also the case for all other landscapes and all other pairwise combinations of predictors ( data not shown ) ., It is possible that some of the effect described in 28 and expected here are too small to be detected with the level of replication in our study ., However , as the level of replication used in this computational study goes way beyond what can be realistically achieved in experimental settings we expect that these effects would also not be detected in experimental studies ., We also used a linear and quadratic discriminant analysis to construct functions to predict the outcome of competitions between the two modes of reproduction ., For these purposes , half of the data set was used for training and the other half for testing of the discriminant functions , and the procedure was repeated separately for each of the three population sizes ( 1 , 000 , 10 , 000 , and 100 , 000 ) and the deterministic case ., In no case did these methods improve the accuracy of predictions ( data not shown ) ., While there certainly are other , potentially more sophisticated techniques that could be used here , our analysis indicates that there may not be much additional information in our metrics that could be extracted and used to increase the accuracy of the predictions ., All predictors performed much worse for simulations with finite population size ( Figure 1 ) , most likely because the selection coefficient for sex is weak 19 , 20 ., To further examine the effect of finite population size on the evolution of sex on different landscape types we analyzed 100 independent simulations of the competition phase starting from the genotype frequencies obtained from the burn-in phase on each landscape ., Figure 3 shows the fraction of cases in which the frequency of sexual individuals increased for three population sizes ( 1 , 000 , 10 , 000 , and 100 , 000 ) , plotted separately for those landscapes in which frequency of the recombination modifier increased or decreased in deterministic simulations ., For almost all landscapes the fraction of cases in which sex evolves is close to 50% , indicating that selection for sexual reproduction is indeed extremely weak , and can thus easily be overwhelmed by stochastic effects ( in contrast to simulations with infinite populations where selection coefficients of any size will always produce a consistent observable effect ) ., As a consequence , even for relatively large population sizes the outcome of the competition between sexual and asexual populations is largely determined by drift ., Such weak selection may in part due to the small number of loci used for these simulations and stochastic simulations with larger genomes have indeed been shown to result in stronger selection for or against sex 37 , 38 ., However , accurate deterministic simulations are computationally not feasible for large genome sizes , because of the need to account for the frequency of all possible genotypes in deterministic simulations ( see Supporting Information ( Text S1 ) for more details ) ., According to the Hill-Robertson effect ( HRE ) 8 , 21 selection for recombination or sex may be stronger in populations of limited size , because in such populations the interplay between drift and selection can generate negative linkage disequilibria , which in turn select for increased sexual reproduction ., The strength of HRE vanishes for very small populations and for populations of infinite size 21 ., In an intermediate range of population sizes , the HRE increases with increasing number of loci ( as does the range of population sizes in which the effect can be observed ) 38 and for large genome size it can be strong enough to override the effect of weak epistasis 37 ., In our simulations , however , HRE is weak , as is evidenced by the fact that , in the NK0 landscape , which by definition have no epistasis , the fraction of runs in which sex evolves is only very marginally above 50% ( Figure 3 ) ., Our results indicate that for finite population size the predictors generally perform poorly ., Of course this does not imply that they could not be better than a simple coin toss ., However , the results suggest that these predictors will likely be of limited use , as any experiment will have difficulties to reach even the replicate number that we have used to generate Figure 1 ., We also examined additional fitness landscapes , characterized by increased neutrality ( for full details and figures see Text S1 ) ., We found that the allelic diversity at neutral loci both decreases the accuracy and generates a systematic bias in the previously best performing predictors , Epop and ΔVarHD ., In contrast , other predictors investigated here , ΔVaradd , ΔMeanfit , Ephys , EMP , and EWP are not affected by including neutral loci , but still have poor accuracy of prediction on more complex fitness landscapes ., The central message of our study is that the prediction of the evolution of sex is difficult for complex fitness landscapes , even in the idealized world of computer simulations where in principle one can measure any detail of a given population and fitness landscape ., Here we put the emphasis on predictors that are experimentally measurable and are based on conditions for the evolution of sex established in the population genetic literature using simple fitness landscapes ., We have however included EMP and EWP , two predictors which would be more difficult to measure experimentally , but are based on the most fundamental and general theoretical treatment of the evolution of sex 28 ., Of course , while our choice of predictors , landscapes and selection regimes is comprehensive , we are aware that it can never be exhaustive or complete – there will always be other options to try out and test ., Future work will have to focus on identifying more reliable predictors of the evolution of sex that can be used in conjunction with experimental data ., Additionally , a better characterization of properties of natural fitness landscapes is badly needed to improve our understanding of the forces selecting for the evolution of sex ., As it stands , ΔVarHD , our best candidate for a predictor of the evolution of sex , has nevertheless important shortcomings ., In particular , it never reaches high levels of accuracy on many of the landscapes ., Still , ΔVarHD at least suggests a potential direction for future research: a focus on predictors that would take advantage of the rapidly increasing number of fully or partially sequenced genomes and allow us to determine the advantage of sex in large numbers of taxa , bringing us closer to fully understanding the evolution of sex ., All simulations of the evolution of a haploid population on a given fitness landscape are divided into a “burn-in” and a “competition” phase ., In the burn-in phase an asexually reproducing population is allowed to equilibrate on the landscape starting from random initial genotype frequencies ., In the competition phase we determine whether the frequency of an allele coding for increased recombination increases in the population ., The burn-in phase consists of repeated cycles of mutation and selection ., Genotype frequencies after selection are given by the product of their frequency and relative fitness before selection ., In all simulations mutations occur independently at each locus with a mutation rate μ\u200a=\u200a0 . 01 per replication cycle ., This high mutation rate was chosen in order to obtain sufficient levels of genetic diversity ., However , we also tested mutation rates up to 10 times lower and found no qualitative differences in the results ( data not shown ) ., In the competition phase the population undergoes recombination in addition to mutation and selection in each reproduction cycle ., To this end a recombination modifier locus is added to one end of the genome , with two alleles m and M , each present in exactly half of the population ., Recombination between two genotypes depends on the modifier allele in both genotypes , with the corresponding recombination rates denoted by rmm , rmM , and rMM ., For the simulations discussed in the main text we used rmm\u200a=\u200armM\u200a=\u200a0 and rMM\u200a=\u200a0 . 1 ., For this parameter choice individuals carrying distinct modifier alleles cannot exchange genetic material and thus any effect of increased recombination remains linked to the M allele ., Sexual and asexual individuals compete directly with each other , and we refer to this scenario as the evolution of sex ., In contrast , if rmm<rmM<rMM , then genetic material can be exchanged between all individuals ., We refer to this scenario as the evolution of recombination ., For the sake of simplicity , we primarily consider the evolution of sex in the main text , but analogous simulations of the evolution of recombination scenario led to qualitatively indistinguishable results ( Text S1 ) ., Moreover , for the evolution of sex scenario we also tested values of rMM ranging from 0 . 01 to 0 . 3 ( data not shown ) , which produced qualitatively indistinguishable results ., All recombination values refer to a probability of recombination happening between neighboring loci with one recombination event per genome ., The position of the crossover point is chosen randomly ., No mutations occur between m and M alleles at the modifier locus ., Recombination , mutation and selection as described above are deterministic and are calculated assuming infinite population size ., To examine stochastic effects , we also considered populations with 1 , 000 , 10 , 000 , and 100 , 000 individuals ., Those simulations included a step in which the frequencies of genotypes are sampled from a multinomial distribution according to their frequencies as calculated based on infinite population size ., The burn-in phase always consists of 2500 generations of mutation and selection ., We confirmed that 2500 generations were typically sufficient for the system to go into mutation-selection balance from random initial genotype frequencies ( data not shown ) ., The competition phase consists of 250 generations of recombination , mutation and selection ., For infinite population size we ran a single competition phase for each burn-in phase ., For finite-size populations , the outcome was estimated as the average of 100 simulations of the competition phase .
Most population genetic theories on the evolution of sex or recombination are based on fairly restrictive assumptions about the nature of the underlying fitness landscapes ., Here we use computer simulations to study the evolution of sex on fitness landscapes with different degrees of complexity and epistasis ., We evaluate predictors of the evolution of sex , which are derived from the conditions established in the population genetic literature for the evolution of sex on simpler fitness landscapes ., These predictors are based on quantities such as the variance of Hamming distance , mean fitness , additive genetic variance , and epistasis ., We show that for complex fitness landscapes all the predictors generally perform poorly ., Interestingly , while the simplest predictor , ΔVarHD , also suffers from a lack of accuracy , it turns out to be the most robust across different types of fitness landscapes ., ΔVarHD is based on the change in Hamming distance variance induced by recombination and thus does not require individual fitness measurements ., The presence of loci that are not under selection can , however , severely diminish predictor accuracy ., Our study thus highlights the difficulty of establishing reliable criteria for the evolution of sex on complex fitness landscapes and illustrates the challenge for both theoretical and experimental research on the origin and maintenance of sexual reproduction .
One of the biggest open questions in evolutionary biology is why sexual reproduction is so common despite its manifold costs ., Many hypotheses have been proposed that can potentially explain the emergence and maintenance of sexual reproduction in nature , and currently the biggest challenge in the field is assessing their plausibility ., Theoretical work has identified the conditions under which sexual reproduction is expected ., However , these conditions were typically derived , making strongly simplifying assumptions about the relationship between organisms genotype and fitness , known as the fitness landscape ., Building onto previous theoretical work , we here propose different population properties that can be used to predict when sex will be beneficial ., We then use simulations across a range of simple and complex fitness landscapes to test if such predictors generate accurate predictions of evolutionary outcomes ., We find that one of the simplest predictors , related to variation of genetic distance between sequences , is also the most accurate one across our simulations ., However , stochastic effects occurring in small populations compromise the accuracy of all predictors ., Our study both illustrates the limitations of various predictors and suggests directions in which to search for new , experimentally attainable predictors .
computational biology/evolutionary modeling, evolutionary biology, genetics and genomics/population genetics