Malecot's coancestry coefficient, f {\displaystyle f} , refers to an indirect measure of genetic similarity of two individuals which was initially devised by the French mathematician Gustave Malécot.
f {\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), f {\displaystyle f} can be calculated by examining detailed pedigree records. Modernly, f {\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 : f → 1 {\displaystyle f\rightarrow 1} . Consider a non-sexual population of fixed size N {\displaystyle N} , and call f i {\displaystyle f_{i}} the inbreeding coefficient of generation i {\displaystyle i} . Here, f {\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 k ≫ 1 {\displaystyle k\gg 1} of descendants, from the pool of which N {\displaystyle N} individual will be chosen at random to form the new generation. At generation n {\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" :
f n = k − 1 k N + k ( N − 1 ) k N f n − 1 {\displaystyle f_{n}={\frac {k-1}{kN}}+{\frac {k(N-1)}{kN}}f_{n-1}}
What is the source of the above formula? Is it in a later paper than the 1948 Reference.
≈ 1 N + ( 1 − 1 N ) 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,
f 0 = 0 {\displaystyle f_{0}=0} , we get
f n = 1 − ( 1 − 1 N ) n . {\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
n ¯ = − 1 / log ( 1 − 1 / N ) ≈ N . {\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 N {\displaystyle N} to 2 N {\displaystyle 2N} (the number of gametes).
See also Coefficient of relationship Consanguinity Genetic distance
References
Bibliography Malécot, G. (1948). Les mathématiques de l'hérédité. Paris: Masson & Cie.