Genetic Distance

Populations with many similar alleles have small genetic distances. This indicates that they are closely related and have a recent common ancestor.

Genetic distance is useful for reconstructing the history of populations, such as the multiple human expansions out of Africa. It is also used for understanding the origin of biodiversity. For example, the genetic distances between different breeds of domesticated animals are often investigated in order to determine which breeds should be protected to maintain genetic diversity.

In the genome of an organism, each gene is located at a specific place called the locus for that gene. Allelic variations at these loci cause phenotypic variation within species (e.g. hair colour, eye colour). However, most alleles do not have an observable impact on the phenotype. Within a population new alleles generated by mutation either die out or spread throughout the population. When a population is split into different isolated populations (by either geographical or ecological factors), mutations that occur after the split will be present only in the isolated population. Random fluctuation of allele frequencies also produces genetic differentiation between populations. This process is known as genetic drift. By examining the differences between allele frequencies between the populations and computing genetic distance, we can estimate how long ago the two populations were separated.

Different statistical measures exist that aim to quantify genetic deviation between populations or species. By utilizing assumptions gained from experimental analysis of evolutionary forces, a model that more accurately suits a given experiment can be selected to study a genetic group. Additionally, comparing how well different metrics model certain population features such as isolation can identify metrics that are more suited for understanding newly studied groups The most commonly used genetic distance metrics are Nei's genetic distance, Cavalli-Sforza and Edwards measure, and Reynolds, Weir and Cockerham's genetic distance, listed below.

Nei's standard 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.

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

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.

Nei's standard distance can then be written as

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

Cavalli-Sforza chord distance

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

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

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.

Reynolds, Weir, and Cockerham's genetic distance

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:

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

Other measures

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

Nei's D_A distance 1983

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. This method is not ideal in cases where natural selection plays a significant role in a populations genetics.

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

${\displaystyle D_{A}}$ : Nei's DA distance, the genetic distance between populations X and Y

${\displaystyle \ell }$ : A locus or gene studied with ${\displaystyle \sum _{\ell }}$  being the sum of loci or genes

${\displaystyle X_{u}}$  and ${\displaystyle Y_{u}}$ : The frequencies of allele u in populations X and Y, respectively

L: The total number of loci examined

Euclidean distance

 

Euclidean distance is a formula brought about from Euclid's Elements which is used to convey, as simply as possible, the genetic dissimilarity between populations with a larger distance indicating greater dissimilarity. The work of René Descartes brought about the cartesian coordinate system which can be used to visually convey the results of euclidean distance calculations.

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

${\displaystyle D_{EU}}$ : Euclidean genetic distance between populations X and Y

${\displaystyle X_{u}}$  and ${\displaystyle Y_{u}}$ : Allele frequencies at locus u in populations X and Y, respectively

Goldstein distance 1995

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.

${\displaystyle (\delta \mu )^{2}=\sum _{\ell }{\frac {(\mu _{X}-\mu _{Y})^{2}}{L}}}$ 
${\displaystyle \delta \mu }$ : Goldstein genetic distance between populations X and Y
${\displaystyle \mu _{x}}$  and ${\displaystyle \mu _{y}}$ : Mean allele sizes in populations X and Y
L: Total number of microsatallite loci examined

Nei's minimum genetic distance 1973

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

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

Roger's distance 1972

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

Fixation index

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.

• PHYLIP uses GENDIST
  • Nei's standard genetic distance 1972
  • Cavalli-Sforza and Edwards 1967
  • Reynolds, Weir, and Cockerham's 1983
• TFPGA
  • Nei's standard genetic distance (original and unbiased)
  • Nei's minimum genetic distance (original and unbiased)
  • Wright's (1978) modification of Roger's (1972) distance
  • Reynolds, Weir, and Cockerham's 1983
• GDA
• POPGENE
• POPTREE2 Takezaki, Nei, and Tamura (2010, 2014)
  • Commonly used genetic distances and gene diversity analysis
• DISPAN
  • Nei's standard genetic distance 1972
  • Nei's D_A distance between populations 1983

• Coefficient of relationship
• Degree of consanguinity
• Human genetic variation
• Phylogenetics
• Allele frequency