This type of increase with ?
The following argument shows how a negative relation between ?na and synonymous diversity could arise. First, because ?na is the component of KA/Kna and hence KA (SI Appendix, Fig. S3, shows that genes with lower KA have lower ?na). Second, stronger selection can also reduce the strength of BGS for a single gene, leading to higher synonymous diversity (SI Appendix, Eq. S5). This pattern results from the fact that weakly deleterious mutations achieve higher equilibrium frequencies than more strongly selected mutations, so that a closely linked neutral mutation has a higher chance of association with a mutation that is destined to be eliminated from the population (5).
To make this analysis quantitative, we used both an exact summation formula and a more tractable, but approximate, integral method; each of these included BGS effects of both NS and UTR sites, and are described by Eq. 1 of Materials and Methods, and SI Appendix, Eqs. S10b and S12, respectively. The equations take both gene conversion and crossing over into account and determine the mean value of E over all synonymous sites in a gene, where E is the negative of the natural logarithm of the ratio of the predicted value ?S at a site to its value in the absence of BGS, ?0. The larger E, the greater the reduction in diversity due to BGS. A subsidiary question is the extent to which the two methods for determining E agree. Continue reading “The relationship anywhere between imply E and ?”