Genetic distance

In 1972, Masatoshi Nei published what came to be known as Nei's standard genetic distance. This distance has the nice property that if the rate of genetic change (amino acid substitution) is constant per year or generation then Nei's standard genetic distance (D) increases in proportion to divergence time. This measure assumes that genetic differences are caused by mutation and genetic drift.[5]

${\displaystyle D=-\ln {\frac {\sum \limits _{\ell }\sum \limits _{u}X_{u}Y_{u}}{\sqrt {\left(\sum \limits _{u}X_{u}^{2}\right)\left(\sum \limits _{u}Y_{u}^{2}\right)}}}}$

This distance can also be expressed in terms of the arithmetic mean of gene identity. Let ${\displaystyle j_{X}}$ be the probability for the two members of population ${\displaystyle X}$ having the same allele at a particular locus and ${\displaystyle j_{Y}}$ be the corresponding probability in population ${\displaystyle Y}$. Also, let ${\displaystyle j_{XY}}$ be the probability for a member of ${\displaystyle X}$ and a member of ${\displaystyle Y}$ having the same allele. Now let ${\displaystyle J_{X}}$, ${\displaystyle J_{Y}}$ and ${\displaystyle J_{XY}}$ represent the arithmetic mean of ${\displaystyle j_{X}}$, ${\displaystyle j_{Y}}$ and ${\displaystyle j_{XY}}$ over all loci, respectively. In other words,

${\displaystyle J_{X}=\sum _{u}{\frac {{X_{u}}^{2}}{L}}}$

${\displaystyle J_{Y}=\sum _{u}{\frac {{Y_{u}}^{2}}{L}}}$

${\displaystyle J_{XY}=\sum _{\ell }\sum _{u}{\frac {X_{u}Y_{u}}{L}}}$

where ${\displaystyle L}$ is the total number of loci examined.[8]

Nei's standard distance can then be written as[5]

${\displaystyle D=-\ln {\frac {J_{XY}}{\sqrt {J_{X}J_{Y}}}}}$

In 1967 Luigi Luca Cavalli-Sforza and A. W. F. Edwards published this measure. It assumes that genetic differences arise due to genetic drift only. One major advantage of this measure is that the populations are represented in a hypersphere, the scale of which is one unit per gene substitution. The chord distance in the hyperdimensional sphere is given by[2][6]

${\displaystyle D_{\text{CH}}={\frac {2}{\pi }}{\sqrt {2\left(1-\sum _{\ell }\sum _{u}{\sqrt {X_{u}Y_{u}}}\right)}}}$

Some authors drop the factor ${\displaystyle {\frac {2}{\pi }}}$ to simplify the formula at the cost of losing the property that the scale is one unit per gene substitution.

In 1983, this measure was published by John Reynolds, Bruce Weir and C. Clark Cockerham. This measure assumes that genetic differentiation occurs only by genetic drift without mutations. It estimates the coancestry coefficient ${\displaystyle \Theta }$ which provides a measure of the genetic divergence by:[7]

${\displaystyle \Theta _{w}={\sqrt {\frac {\sum \limits _{\ell }\sum \limits _{u}(X_{u}-Y_{u})^{2}}{2\sum \limits _{\ell }\left(1-\sum \limits _{u}X_{u}Y_{u}\right)}}}}$

Many other measures of genetic distance have been proposed with varying success.

This distance assumes that genetic differences arise due to mutation and genetic drift, but this distance measure is known to give more reliable population trees than other distances particularly for microsatellite DNA data.[9][10]

${\displaystyle D_{A}=1-\sum _{\ell }\sum _{u}{\sqrt {X_{u}Y_{u}}}/{L}}$

Euclidean genetic distance between 51 worldwide human populations, calculated using 289,160 SNPs.[11] Dark red is the most similar pair and dark blue is the most distant pair.

${\displaystyle D_{EU}={\sqrt {\sum _{u}(X_{u}-Y_{u})^{2}}}}$[2]

It was specifically developed for microsatellite markers and is based on the stepwise-mutation model (SMM). ${\displaystyle \mu _{X}}$ and ${\displaystyle \mu _{Y}}$ are the means of the allele sizes in population X and Y.[12]

${\displaystyle (\delta \mu )^{2}=\sum _{\ell }{\frac {(\mu _{X}-\mu _{Y})^{2}}{L}}}$

This measure assumes that genetic differences arise due to mutation and genetic drift.[13]

${\displaystyle D_{m}={\frac {J_{X}+J_{Y}}{2}}-J_{XY}}$

${\displaystyle D_{R}={\frac {1}{L}}{\sqrt {\frac {\sum \limits _{u}(X_{u}-Y_{u})^{2}}{2}}}}$[14]

A commonly used measure of genetic distance is the fixation index (F_ST) which varies between 0 and 1. A value of 0 indicates that two populations are genetically identical (minimal or no genetic diversity between the two populations) whereas a value of 1 indicates that two populations are genetically different (maximum genetic diversity between the two populations). No mutation is assumed. Large populations between which there is much migration, for example, tend to be little differentiated whereas small populations between which there is little migration tend to be greatly differentiated. F_ST is a convenient measure of this differentiation, and as a result F_ST and related statistics are among the most widely used descriptive statistics in population and evolutionary genetics. But F_ST is more than a descriptive statistic and measure of genetic differentiation. F_ST is directly related to the Variance in allele frequency among populations and conversely to the degree of resemblance among individuals within populations. If F_ST is small, it means that allele frequencies within each population are very similar; if it is large, it means that allele frequencies are very different.