# Coefficient of relationship

Coefficient of relationship

In population genetics, Sewall Wright's coefficient of relationship or coefficient of relatedness or relatedness or r is defined as 2 times the Coefficient of Inbreeding.[1] The 'Coefficient of Relatedness' (or: coefficient of kinship) is defined as the probability that the alleles at a particular locus chosen at random from two individuals are identical by descent.

## Calculation

The coefficient of relationship between parent and offspring is 0.5, meaning half of the offspring's genome comes from that parent.[2] Or, more precisely said, half the genes are identical by descent (i.e. not accounting for genomic imprinting and other non-Mendelian inheritance).

There is a notable difference between exact and average relatedness. The former describes the inheritance of an exact and consistent fraction of identical alleles whereas the latter is an average of variable relatedness. For example, one might share exactly half of one's alleles from a parent. On the other hand, a sibling does not receive exactly half of the same alleles as oneself, but can theoretically receive any fraction from all to none of the same alleles. In this case, genetic relatedness merely dictates the expected fraction of shared alleles.

Some other relatedness examples: (assuming no consanguinity of the parents, for diploid species unless noted otherwise)

r relationship exact or average
0.5 (½) parent-offspring exact
0.25 (¼) grandparent-grandchild average
0.125 (⅛) great grandparent-great grandchild average
1 identical twins; clones exact
0.5 (½) full siblings average
0.25 (¼) half siblings average
0.125 (⅛) first cousins average
0.03125 (1/32) second cousins[3] average
0.75 (¾) full hymenopteran sisters[4] average

Some of these definitions of r are tricky to calculate. For example, consider that you are one of a family of half brothers with the same mother but different fathers. Call rm the coefficient of matrilineal relatedness and call rp the coefficient of patrilineal relatedness. What is the coefficient of relatedness to each of your half brothers?

rm=0.5 as it does for normal siblings, since there is a 1 in 2 chance that you and your half brother developed from eggs harboring the same haploid genome selected during maternal meiosis. But rp=0 since there is no chance that any two half-brothers inherited the same allele from different fathers by descent. The average of these rs equals 0.25, as we see above. You and your half brother are one quarter genetically identical by descent.

The following images demonstrates the effect of gene distribution through 16 generations.

## Coefficient is relative to a base population

It might be objected that all copies of all genes are identical by descent, in that they are all descended from a single ancestral gene. Shouldn't all these numbers be 1? However these calculations implicitly rely on a base population, in which all copies are assumed to not be identical by descent.[5] For example, for a pair of siblings the base population is the one from which their parents are drawn, and only identity by descent that occurred since then is counted. This gives the above numbers.

## Effect of imprinting

In the example above, the differentiation between maternal and paternal coefficients of relatedness was made in order to introduce the phenomenon of imprinting, or parent-specific gene expression into the calculation of r.

To illustrate how an imprinting turns normal siblings into half siblings, consider that you are one of a cadre of full siblings. What is your coefficient of relatedness to one of your siblings at an imprinted locus in a maternally inherited allele? (Maternally inherited alleles are called "madumnal," and paternally inherited alleles are called "padumnal".)

The value of r at this imprinted locus is equal to the value of r for the entire genome of the half brothers. For the brothers above, rm=0.5 and rp=0. For the madumnal locus in the family of full siblings rm=0.5 too, since all expression is from the mother's allele. Normally, rp=0.5 for full siblings with the same father, but the effect of imprinting is to negate paternal expression, hence rp=0. Therefore you and your full siblings resemble half-siblings at imprinted loci.

## References

1. ^ Malécot, G. (1948) Les mathématiques de l’hérédité, Masson et Cie, Paris. Lange, K. (1997) Mathematical and statistical methods for genetic analysis, Springer-Verlag, New-York.
2. ^ "Kin Selection". Benjamin/Cummings. Retrieved 2007-11-25.
3. ^ "Relatedness Part 2". Retrieved 2007-11-25.  (excerpted from The Selfish Gene, Richard Dawkins)
4. ^ "Effects of Insemination Quantity on Honey Bee Queen Physiology". NIH PLoS ONE. Retrieved 2007-11-25.  (In haplodiploid honeybees, supersisters are daughters of the same queen who has been inseminated by a single drone.)
5. ^ Oliehoek, Pieter; Jack J. Windig, Johan A. M. van Arendonk and Piter Bijma (May 2006). "Estimating Relatedness Between Individuals in General Populations With a Focus on Their Use in Conservation Programs". Genetics 173: 483–496. doi:10.1534/genetics.105.049940. PMC 1461426. PMID 16510792. Retrieved 17 April 2011.

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