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Fluctuating Selection in the Moran Genetics: Early Online, published on January 20, 2017 as 10.1534/genetics.116.192914 1 2 3 Fluctuating selection in the Moran 4 5 Antony M. Dean*,†,‡,1, Clarence Lehman† and Xiao Yi†,‡ 6 7 8 *State Key Laboratory of Biocontrol, School of Life Sciences, Sun Yat-sen University, 9 Guangzhou 510275, PR China 10 †Department of Ecology, Evolution, and Behavior, University of Minnesota, St. Paul, MN 55108 11 ‡ BioTechnology Institute, University of Minnesota, St. Paul, MN 55108 12 13 14 15 16 17 18 19 20 1 Corresponding author: BioTechnology Institute, University of Minnesota, St. Paul, MN 55108. 21 E-mail: [email protected] Copyright 2017. 22 Abstract 23 Contrary to classical population genetics theory, experiments demonstrate that fluctuating 24 selection can protect a haploid polymorphism in the absence of frequency dependent effects on 25 fitness. Using forward simulations with the Moran model we confirm our analytical results 26 showing that a fluctuating selection regime, with a mean selection coefficient of zero, promotes 27 polymorphism. We find that increases in heterozygosity over neutral expectations are especially 28 pronounced when fluctuations are rapid, mutation is weak, the population size is large and the 29 variance in selection is big. Lowering the frequency of fluctuations makes selection more 30 directional and so heterozygosity declines. We also show that fluctuating selection raises dn/ds 31 ratios for polymorphism, not only by sweeping selected alleles into the population, but also by 32 purging the neutral variants of selected alleles as they undergo repeated bottlenecks. Our analysis 33 shows that randomly fluctuating selection increases the rate of evolution by increasing the 34 probability of fixation. The impact is especially noticeable when the selection is strong and 35 mutation is weak. Simulations show the increase in the rate of evolution declines as the rate of 36 new mutations entering the population increases, an effect attributable to clonal interference. 37 Intriguingly, fluctuating selection increases the dn/ds ratios for divergence more than for 38 polymorphism, a pattern commonly seen in comparative genomics. Our model, which extends 39 the classical neutral model of molecular evolution by incorporating random fluctuations in 40 selection, accommodates a wide variety of observations, both neutral and selected, with 41 economy. 42 43 SUMMARY All environments vary. Yet molecular sequence analyses interpret patterns of 44 polymorphism and divergence assuming that selection is constant and directional. We extend the 45 classic neutral model of molecular evolution to incorporate fluctuating selection. Contrary to 46 classical selection theory fluctuating selection can promote polymorphism in haploids in the 47 absence of frequency dependent fitness effects. The conditions for neutral fixation are broadened 48 so that alleles temporarily subject to directional selection might fix as if neutral. Fluctuating 49 selection raises the dn/ds ratio for divergence more than for polymorphism, a pattern commonly 50 seen in genomic comparisons. 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 Introduction 71 Available field evidence suggests selection fluctuates in direction over time (Dobzhansky 1943; 72 Fisher and Ford 1947; Cook and Jones 1996; Cain et al. 1990; Lynch 1987; Saccheri et al. 2008). 73 However, the importance of fluctuating selection in patterning polymorphisms, probabilities of 74 fixation, and evolutionary divergence remains poorly understood. Classical population genetics 75 theory suggests that fluctuating selection promotes polymorphism whenever the geometric mean 76 fitness of the heterozygote is greater than both homozygotes (Kimura 1954; Dempster 1955; 77 Haldane and Jayakar 1963; Gillespie 1972, 1973; Jensen 1973; Karlin and Levikson 1974; Karlin 78 and Liberman 1975; Felsenstein 1976; Maynard Smith 1998; Bell 2008). Fluctuating selection 79 alone cannot promote polymorphisms in haploids simply because the clone with the largest 80 geometric mean fitness inevitably wins the competition. Only if fitness is frequency dependent 81 (Felsenstein 1976; Bell 2008; Bell 2010), as in the lottery model (Chesson and Warner 1981), or 82 if recombination between selected loci occurs (Kirzhner et al. 1994), can selection promote 83 polymorphism in haploids. 84 85 In contrast to classical theory, experiments demonstrate that fluctuating selection can maintain a 86 haploid/clonal polymorphism in the absence of frequency-dependent effects on fitness (Yi and 87 Dean 2013). Key is in recognizing two sources of variability when using the growth rates rA and rAt rat 88 ra over a period t to define the absolute fitnesses WA = e and Wa = e . One (well-recognized) 89 is variability in the relative growth rate, rA/ra = LogeWA/LogeWa. The other (previously 90 overlooked) is variability in the time available for growth, t. 91 92 Consider a serial transfer experiment. Following Yi and Dean (2013), let p and q = 1 – p be the 93 frequencies of A and a immediately following dilution into fresh medium, and let eD.i be the fold 94 increase in population density after growth to carrying capacity in environment i. Then the r t r t D 95 growth of the mixed culture at carrying capacity is described by pe A.i i + qe a.i i = e i . This 96 model (experimentally verified for strains of Escherichia coli competing for limiting glucose (Yi 97 and Dean 2013)) describes exponential growth that ceases abruptly once the carrying capacity is * 98 reached. When A is rare the time taken to reach carrying capacity is ti = Di / ra.i . When a is rare ** 99 the time taken is ti = Di / rA.i . Placing a bound on population size (i.e. restricting the fold- D 100 increase to e i ) means that the time spent growing to carrying capacity is a dependent variable, 101 one that varies with allele frequency. 102 103 This frequency-dependent slippage in "time-for-growth" promotes diversity (Figure 1). Alleles 104 experience more doublings when the least fit is most common. This favors the rare fitter allele. 105 Alleles experience fewer doublings when the fittest is most common. This mitigates selection 106 against the rare less fit allele. Fluctuating selection is passively biased in favor of rare alleles. 5 time in tetracycline 4 3 2 1 0 -1 -2 -3 -4 time in chloramphenicol -5 0 20 40 60 80 100 120 140 160 107 108 Figure 1. This serial transfer experiment with fluctuating selection shows two E. coli strains (one 109 resistant to chloramphenicol, Clmr.T5R, and the other resistant to tetracycline, Tetr.T5S) 110 coexisting in a stable oscillation. Selection is not frequency dependent because the selection 111 coefficients (the slopes) are constant and the lines are either parallel up or parallel down. On 112 reaching carrying capacity (at the peaks and in the valleys) the culture is immediately diluted 113 into fresh medium and the direction of selection switched. Coexistence is possible because the 114 time spent growing in each antibiotic (horizontal bars) is frequency dependent. It takes longer to 115 reach carrying capacity in tetracycline when Clmr.T5R is common near the start of the 116 experiment than in the stable oscillation at the end. Similarly, it takes less time to reach carrying 117 capacity in chloramphenicol when Clmr.T5R is common near the start of the experiment than in 118 the stable oscillation at the end. Coexistence is not possible in the alternative transfer regime 119 where the culture is diluted at a fixed time interval before carrying capacity is reached. After Yi 120 and Dean (2013). 121 122 Motivated by the experimental results in Figure 1, we re-examined the impact of randomly 123 fluctuating selection on polymorphism and evolution using two continuous time overlapping 124 generation models. The first, of an infinite (though bounded) population, sets the stage for the 125 second, of a finite population undergoing continuous mutation and allelic fixations. Results show 126 that fluctuating selection indeed promotes polymorphism, increases rates of evolution, and raises 127 the ratio of non-synonymous to synonymous substitutions more for divergence than for 128 polymorphism. Kimura’s neutral model of molecular evolution is the limiting case where 129 selection is zeroed. 130 131 132 133 Infinite Populations 134 Basic model. The conditions needed for fluctuating selection to promote polymorphism in a 135 population growing continuously at carrying capacity are identical to those in serial transfer 136 (Dean 2005; Yi and Dean 2013). Consider two clonal (i.e. non-recombining) populations 137 (densities NA and Na) competing in a succession of n arbitrary environments (each of time 138 interval ti) for a continuously replenished growth-limiting nutrient (concentration Ri). Death rates 139 are fixed and birth rates vary with resource availability. To make matters more concrete, and 140 without loss of generality, let the populations inhabit a chemostat, a continuous culture device. 141 The differential equations describing the competition are during interval i are: 142 dN 143 A = r !! N [1] dt ( A.i i ) A dN 144 a = r !! N [2] dt ( a.i i ) a dR r r 145 i = ! R ! R ! A.i N ! a.i N [3] dt i ( i.0 i ) Y A Y a 146 147 where the per capita growth rates rA.i = fA(Ri) and ra.i = fa(Ri) are increasing monotonic functions 148 of Ri, the concentration of resource in the growth chamber. Ri.0 is the concentration of the 149 limiting nutrient in the fresh medium entering the growth chamber. δi is the chemostat dilution 150 rate (the fractional rate at which fresh medium enters the growth chamber and spent medium and 151 cells are washed out of it). Y is the yield coefficient (the number of organisms produced per 152 amount of limiting resource consumed). 153 154 During growth at quasi-steady state, dRi/dt ≈ 0 and so the mean population growth rate equals the 155 dilution rate, prA.i + qra.i = δi.
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