# Genetic distance

Genetic distance map by Cavalli-Sforza et al. (1994) [1]

Genetic distance is a measure of the genetic divergence between species or between populations within a species, whether the distance measures time from common ancestor or degree of differentiation.[2] 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.[3] 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.[4]

## Biological foundation

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.[5]

## Measures

Although it is simple to define genetic distance as a measure of genetic divergence, there are several different statistical measures that have been proposed. This has happened because different authors considered different evolutionary models. The most commonly used are Nei's genetic distance,[5] Cavalli-Sforza and Edwards measure,[6] and Reynolds, Weir and Cockerham's genetic distance,[7] listed below.

In all the formulae in this section, ${\displaystyle X}$ and ${\displaystyle Y}$ represent two different populations for which ${\displaystyle L}$ loci have been studied. Let ${\displaystyle X_{u}}$ represent the ${\displaystyle u}$th allele frequency at the ${\displaystyle l}$th locus.

### 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.[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]

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

### 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[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.

### 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:[7]

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

### Other measures

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

#### Nei's DA 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.[9][10]

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

#### Euclidean distance

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]

#### 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.[12]

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

#### Nei's minimum genetic distance 1973

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

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

#### Roger's distance 1972

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

#### Fixation index

A commonly used measure of genetic distance is the fixation index (FST) 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. FST is a convenient measure of this differentiation, and as a result FST and related statistics are among the most widely used descriptive statistics in population and evolutionary genetics. But FST is more than a descriptive statistic and measure of genetic differentiation. FST is directly related to the Variance in allele frequency among populations and conversely to the degree of resemblance among individuals within populations. If FST 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.

## Software

• 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 DA distance between populations 1983

## References

1. ^ Cavalli-Sforza, L.L., Menozzi, P. & Piazza, A. (1994). The History and Geography of Human Genes. New Jersey: Princeton University Press.
2. ^ a b c Nei, M. (1987). "Chapter 9". Molecular Evolutionary Genetics. New York: Columbia University Press.
3. ^ Ramachandran S, Deshpande O, Roseman CC, Rosenberg NA, Feldman MW, Cavalli-Sforza LL (November 2005). "Support from the relationship of genetic and geographic distance in human populations for a serial founder effect originating in Africa". Proc Natl Acad Sci U S A. 102 (44): 15942–7. Bibcode:2005PNAS..10215942R. doi:10.1073/pnas.0507611102. PMC 1276087. PMID 16243969.
4. ^ Ruane J (1999). "A critical review of the value of genetic distance studies in conservation of animal genetic resources". Journal of Animal Breeding and Genetics. 116 (5): 317–323. doi:10.1046/j.1439-0388.1999.00205.x.
5. ^ a b c d Nei, M. (1972). "Genetic distance between populations". Am. Nat. 106 (949): 283–292. doi:10.1086/282771. S2CID 55212907.
6. ^ a b L.L. Cavalli-Sforza; A.W.F. Edwards (1967). "Phylogenetic Analysis – Models and Estimation Procedures". The American Journal of Human Genetics. 19 (3 Part I (May)): 233–257. PMC 1706274. PMID 6026583.
7. ^ a b John Reynolds; B.S. Weir; C. Clark Cockerham (November 1983). "Estimation of the coancestry coefficient: Basis for a short-term genetic distance". Genetics. 105 (3): 767–779. doi:10.1093/genetics/105.3.767. PMC 1202185. PMID 17246175.
8. ^ Nei, M. (1987) Genetic distance and molecular phylogeny. In: Population Genetics and Fishery Management (N. Ryman and F. Utter, eds.), University of Washington Press, Seattle, WA, pp. 193–223.
9. ^ Nei M., Tajima F., Tateno Y. (1983). "Accuracy of estimated phylogenetic trees from molecular data. II. Gene frequency data". J. Mol. Evol. 19 (2): 153–170. doi:10.1007/bf02300753. PMID 6571220. S2CID 19567426.{{cite journal}}: CS1 maint: multiple names: authors list (link)
10. ^ Takezaki N. (1996). "Genetic distances and reconstruction of phylogenetic trees from microsatellite DNA". Genetics. 144 (1): 389–399. doi:10.1093/genetics/144.1.389. PMC 1207511. PMID 8878702.
11. ^ Magalhães TR, Casey JP, Conroy J, Regan R, Fitzpatrick DJ, Shah N; et al. (2012). "HGDP and HapMap analysis by Ancestry Mapper reveals local and global population relationships". PLOS ONE. 7 (11): e49438. Bibcode:2012PLoSO...749438M. doi:10.1371/journal.pone.0049438. PMC 3506643. PMID 23189146.{{cite journal}}: CS1 maint: multiple names: authors list (link)
12. ^ Gillian Cooper; William Amos; Richard Bellamy; Mahveen Ruby Siddiqui; Angela Frodsham; Adrian V. S. Hill; David C. Rubinsztein (1999). "An Empirical Exploration of the ${\displaystyle (\delta \mu )^{2}}$ Genetic Distance for 213 Human Microsatellite Markers". The American Journal of Human Genetics. 65 (4): 1125–1133. doi:10.1086/302574. PMC 1288246. PMID 10486332.
13. ^ Nei M, Roychoudhury AK (February 1974). "Sampling variances of heterozygosity and genetic distance". Genetics. 76 (2): 379–90. doi:10.1093/genetics/76.2.379. PMC 1213072. PMID 4822472.
14. ^ Rogers, J. S. (1972). Measures of similarity and genetic distance. In Studies in Genetics VII. pp. 145−153. University of Texas Publication 7213. Austin, Texas.