An experimental test on the probability of extinction of new genetic variants

Abstract

In 1927, J.B.S. Haldane reasoned that the probability of fixation of new beneficial alleles is twice their fitness effect. This result, later generalized by M. Kimura, has since become the cornerstone of modern population genetics. There is no experimental test of Haldane's insight that new beneficial alleles are lost with high probability. Here we demonstrate that extinction rates decrease with increasing initial numbers of beneficial alleles, as expected, by performing invasion experiments with inbred lines of the nematode Caenorhabditis elegans. We further show that the extinction rates of deleterious alleles are higher than those of beneficial alleles, also as expected. Interestingly, we also find that for these inbred lines, when at intermediate frequencies, the fate of invaders might not result in their ultimate fixation or loss but on their maintenance. Our study confirms the key results from classical population genetics and highlights that the nature of adaptation can be complex.

Figure 1. Extinction decreases with number of invading alleles.

(a) The dynamics of invasions. Observed number of GFP individuals, when two or five individuals initially invade a population with N=10³ wild-type individuals. Thirty-five replicates were used in the experiments starting with two GFP individuals, whereas 56 replicates were followed in the invasions with five GFP individuals. (b) Observed probability of extinction at G5 with 2 s.d. of 1,000 bootstrap replicates is shown as the error bars. (c,d) Selection coefficient estimates. For the two-individual (c) and five-individual (d) GFP invasion experiments, the likelihood values of selective coefficients (s) for the observed GFP allele counts at G3, G4 and G5 are presented. These were obtained by simulations of 10⁸ invasions. In the region close to the initial ML estimate, defined as ML-10log₁₀ (black circles), an extra set of 9 × 10⁸ simulations were performed. A quadratic function was then fitted to this interval (dashed line) to obtain the final ML estimate of s (grey line) with −2log₁₀ confidence limits (grey area).

Figure 2. Probability of extinction and fixation.

(a) Observed probability of extinction at G5 when five wild-type individuals invade a population with 10³ GFP individuals (grey line). Fifty replicate invasions were followed. The interquantile range of the median probability of extinction (dashed lines) was obtained by numerical simulations of the invasion experiment with selection coefficient between −0.4 and 0.1. The median selection coefficient that best reflects the observed number of line losses is of s=−0.138, with the red error bar indicating the expected interquantile range (n=10³ simulations). (b) Expected probability of fixation of invader alleles, using M. Kimura diffusion process approximation, obtained for the selective coefficients estimated from the three invasion experiments performed (symbols). Lines indicate the probability of fixation with N=10³ or 10² and constant s=0.16 or −0.138 (for probability of fixation with frequency-dependent selection, see Fig. 4). With strong selection on beneficial invaders, the probability of fixation is the same for the two population sizes (half-filled symbols). For strong selection, the approximation , where n is the number of beneficial invaders, is indistinguishable from Kimura’s approach employed here. The probability of fixation scale is truncated for clarity.

Figure 3. Frequency-dependent selection.

(a) The GFP allele is deleterious. Intermediate-frequency competitions between the GFP and the wild-type inbred lines show that the GFP allele declines in frequency. GFP allele frequency changes are shown for 11 replicate competitions (grey lines). The mean GFP frequency in adults through time is shown as a black line. The mean selection coefficient estimated from multiple generations, at the adult stage, is s=−0.12±0.03 s.d. (see Methods). The frequency declines in GFP allele observed over one generation both at the L1 stage (dashed grey lines) and at the adult stage (solid grey lines) are shown. (b) Estimated selection coefficients at adult and L1 stages for GFP allele frequency changes in a single generation, from a (symbols), show no effect of life-history stage (Welch’s t-test, t_14.7=−0.45, P=0.66). Error bars indicate 2 s.e.m. among the 10 replicates at each stage. (c) The form of frequency-dependent selection. Estimated selection coefficients from one-generation competition experiments at several intermediate initial GFP frequencies are shown as triangles. Measurements were done at the L1 stage. Error bars indicate 2 s.e.m. among the 10 replicates for each starting frequency. Selection coefficient estimates from the invasion experiments with two (square) or five (circle) starting GFP individuals are also shown as empty symbols (from the data in Figs 1c and 2a). Note that the selection coefficient estimated from the invasion experiment where the GFP allele was at an initial proportion of 0.995 is the absolute selection coefficient of the wild-type invader allele. Frequency-dependent selection is illustrated by a quadratic regression on all data points (R²=0.91; inset). Neutrality is indicated as a grey line.

Figure 4. Selection may maintain diversity and increase extinction.

(a) GFP allele frequency dynamics in simulated populations when starting with five GFP invaders. Simulations were run under frequency-dependent (black) or frequency-independent selection (grey), where s=0.12–2.32 p+2.36 p² or s=0.108 (assumed constant and calculated with P=0.005), respectively (from Fig. 3c). The stable polymorphic equilibrium is rapidly reached at around 5% under frequency-dependent selection, while most fixations occurred by generation 100 under frequency-independent selection. (b) Enlargement of the dashed box in a for frequency-dependent selection. Several replicate populations face extinction even when their sizes were simulated with N=10³. (c) Expected probabilities of extinction and fixation estimated from the 1,000 simulated populations of a and b, under frequency-dependent selection or frequency-independent selection (black or grey lines). Frequency-dependent selection can increase extinctions when compared with frequency-independent selection after generation 50 (dashed lines). The triangle indicates the ultimate probability of extinction (infinite time) when using the approximation P_ext=e^−2s for frequency independence with large selection coefficients.