# Malecot's method of coancestry

Malecot's coancestry coefficient, ${\displaystyle f}$, refers to an indirect measure of genetic similarity of two individuals which was initially devised by the French mathematician Gustave Malécot.

${\displaystyle f}$ is defined as the probability that any two alleles, sampled at random (one from each individual), are identical copies of an ancestral allele. In species with well-known lineages (such as domesticated crops), ${\displaystyle f}$ can be calculated by examining detailed pedigree records. Modernly, ${\displaystyle f}$ can be estimated using genetic marker data.

## Evolution of inbreeding coefficient in finite size populations

In a finite size population, after some generations, all individuals will have a common ancestor : ${\displaystyle f\rightarrow 1}$. Consider a non-sexual population of fixed size ${\displaystyle N}$, and call ${\displaystyle f_{i}}$ the inbreeding coefficient of generation ${\displaystyle i}$. Here, ${\displaystyle f}$ means the probability that two individuals picked at random will have a common ancestor. At each generation, each individual produces a large number ${\displaystyle k\gg 1}$ of descendants, from the pool of which ${\displaystyle N}$ individual will be chosen at random to form the new generation.

At generation ${\displaystyle n}$, the probability that two individuals have a common ancestor is "they have a common parent" OR "they descend from two distinct individuals which have a common ancestor" :

${\displaystyle f_{n}={\frac {k-1}{kN}}+{\frac {k(N-1)}{kN}}f_{n-1}}$
${\displaystyle \approx {\frac {1}{N}}+(1-{\frac {1}{N}})f_{n-1}.}$

This is a recurrence relation easily solved. Considering the worst case where at generation zero, no two individuals have a common ancestor,

${\displaystyle f_{0}=0}$, we get
${\displaystyle f_{n}=1-(1-{\frac {1}{N}})^{n}.}$

The scale of the fixation time (average number of generation it takes to homogenize the population) is therefore

${\displaystyle {\bar {n}}=-1/\log(1-1/N)\approx N.}$

This computation trivially extends to the inbreeding coefficients of alleles in a sexual population by changing ${\displaystyle N}$ to ${\displaystyle 2N}$ (the number of gametes).

## References

• Malécot G. Les mathématiques de l'hérédité. Paris: Masson & Cie, 1948.