Preferential entailment

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

$f$ 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), $f$ can be calculated by examining detailed pedigree records. Modernly, $f$ 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 : $f\rightarrow 1$. Consider a non-sexual population of fixed size $N$, and call $f_{i}$ the inbreeding coefficient of generation $i$. Here, $f$ means the probability that two individuals picked at random will have a common ancestor. At each generation, each individual produces a large number $k\gg 1$ of descendants, from the pool of which $N$ individual will be chosen at random to form the new generation.

At generation $n$, 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" :

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

$\approx {\frac {1}{N}}+(1-{\frac {1}{N}})f_{n-1}.$

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

$f_{0}=0$, we get

$f_{n}=1-(1-{\frac {1}{N}})^{n}.$

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

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

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

References

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