# Asymptotic MKT

Messer and Petrov (Messer & Petrov 2013) proposed a simple asymptotic extension of the MK test that yields accurate estimates of α, as it also takes slightly deleterious mutations presence into account. This is referred as the asymptotic MKT method. Briefly, this approximation first estimates α for each Derived Allele Frequency (DAF) category using its specific *P _{i}* and

*P*counts and then fits an exponential function to this values, of the form: α

_{0}*. Finally, the asymptotic α estimate is obtained by extrapolating the value of this function to 1:*

_{fit(x)}=a+b×e^{-cx}*α*.

_{asymptotic}=α_{fit}(x=1)For further information about this methodology, please see Messer and Petrov work (Messer & Petrov 2013).

The asymptotic MKT has been extended to also estimate the fraction of substitutions which are non-beneficial. Polymorphism and divergence data can be used in an extended MKT framework which allows estimating the fraction of sites that are under negative selection regimens (*d*: strongly deleterious, *b*: slightly deleterious, *f*: neutral), for each gene and class of site.

The fraction of strongly deleterious mutations (*d*) is estimated as the difference between neutral (*0*) and selected (*i*) polymorphic sites relative to the number of analyzed sites: *d=1-P _{0}/m_{0}/P_{i}/m_{i}*. The fraction of weakly deleterious mutations (

*b*) corresponds to the relative proportion of selected polymorphic sites that cause the underestimation of α at low DAF categories. In detail, if

*α*is lower than the low CI estimate of α

_{(x)}*model fitting, we considered the presence of slightly deleterious mutations in*

_{asymptotic}*DAF*category. The weakly deleterious fraction among the segregating sites is estimated as

_{(x)}*w*minα

_{d}=(α_{asymptotic}-α_{(x)}×(P_{ix}/∑P_{i}}))/(α_{asymptotic}-_{(x)}). Then, the proportion of weakly deleterious mutations is

*b=w*. Finally, the fraction of neutral sites (

_{d}/(P_{0}/m_{0}/P_{i}/m_{i})*f*) is estimated as:

*f=1-d-b*.

Note that this approach only fits the data with the exponential model, and not the linear one.

**Figure 1**. **Example iMKT with negative fractions**.