The evolutionary rate of duplicated genes under concerted evolution

Abstract

The effect of directional selection on the fixation process of a single mutation that spreads in a multigene family by gene conversion is investigated. A simple two-locus model with two alleles, A and a, is first considered in a random-mating diploid population with size N. There are four haplotypes, AA, Aa, aA, and aa, and selection works on the number of alleles A in a diplod (i = 0, 1, 2, 3, 4). Because gene conversion is allowed between the two loci, when the mutation rate is very low, either AA or aa will fix in the population eventually. We consider a situation where a single mutant, A, arises in one locus when a is fixed in both loci. Then, we derive the fixation probability analytically, and the fixation time is investigated by simulations. It is found that gene conversion has an effect to increase the "effective" population size, so that weak selection works more efficiently in a multigene family. With these results, we discuss the effect of gene conversion on the rate of molecular evolution in a multigene family undergoing concerted evolution. We also argue about the applicability of the theoretical results to models of multigene families with more than two loci.

Figure 1.—

Gene tree of a pair duplicated genes from two species. The two boxes represents the duplicated gene pair. The gene pairs are present in the two species, because the duplication event predates the speciation event. The duplicated gene pairs are still undergoing concerted evolution, so that the resultant gene tree of the four genes should look as illustrated: the paralogous gene pair in each species is more closely related than the orthologous pairs. The circles represent the mutations that occurred along the evolution of the four genes since the speciation event. The open circles are those fixed in each species, while the shaded ones are the variation between duplicated copies.

Figure 2.—

(A–D) Typical trajectories of the frequencies of the four haplotypes, AA, Aa, aA, and aa in a two-locus model, represented by the blue, green, yellow, and red parts. The trajectories were obtained by simulations with a random-mating diploid population with size 100, where s = 0.01 is assumed. (A) c = 0.01 and r = 0. (B) c = 0.01 and r = 0.1. (C) c = 0.0001 and r = 0. (D and E) Typical trajectories of haplotype frequencies in a three-locus model. (D) c = 0.01 and r = 0. (E) c = 0.0001 and r = 0.

Figure 3.—

Selection coefficients of the five gametes when the dominance effect is incorporated by Equation 22. The solid line shows the selection coefficients when the effect is additive. (A) h = 0.2, representing a case where A is recessive. (B) h = 0.5; that is, the selection effect is completely additive. (C) h = 0.8, representing a case where A is dominant.

Figure 4.—

Fixation probability and fixation time in the two-locus model. (A and B) Simulation results for the fixation probability when the recombination rate is 0 and 0.1, respectively. The shaded lines represents the theoretical result from Equation 37. (C and D) Simulation results for the fixation time when the recombination rate is 0 and 0.1, respectively. The results are based on simulations with 10⁷ replications for each parameter set.

Figure 5.—

Fixation probability and fixation time in the three-locus model. (A and B) Simulation results for the fixation probability when the recombination rate is 0 and 0.1, respectively. The number of replications is 10⁷ for each parameter set. The shaded lines represent the theoretical result from Equation 35 with n = 3. (C and D) Simulation results for the fixation time when the recombination rate is 0. Data for the mean fixation time when s = −0.01 are missing because the fixation event is so rare that we were not able to obtain reliable mean values.

Figure 6.—

Fixation probability for n = {1, 2, 3, 4, 5, 6} predicted by Equation 36 with simulation results for c = 0.00001, 0.0001, 0.001, and 0.01. The number of replications is 10⁷ for each parameter set.

Figure 7.—

(A) Fixation probability for n = {1, 2, 3, 4} predicted by Equation 36. N = 100 is assumed so that the probability of the neutral case is given by 1/(2nN) = 1/(200n). (B) Substitution rate for n = {1, 2, 3, 4} from Equation 38. The substitution rate is measured such that the rate is 1 under neutrality (note that the substitution rate per site per generation is identical to the mutation rate for any n).

Figure 8.—

(A and B) Effect of biased gene conversion on the fixation probability in the two-locus model. A total of 10⁷ replications of simulations were carried out for each parameter set.