Biology:Malecot's method of coancestry
Malecot's coancestry coefficient, [math]\displaystyle{ f }[/math], refers to an indirect measure of genetic similarity of two individuals which was initially devised by the France mathematician Gustave Malécot.
[math]\displaystyle{ f }[/math] 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), [math]\displaystyle{ f }[/math] can be calculated by examining detailed pedigree records. Modernly, [math]\displaystyle{ f }[/math] 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 : [math]\displaystyle{ f \rightarrow 1 }[/math]. Consider a non-sexual population of fixed size [math]\displaystyle{ N }[/math], and call [math]\displaystyle{ f_i }[/math] the inbreeding coefficient of generation [math]\displaystyle{ i }[/math]. Here, [math]\displaystyle{ f }[/math] means the probability that two individuals picked at random will have a common ancestor. At each generation, each individual produces a large number [math]\displaystyle{ k \gg 1 }[/math] of descendants, from the pool of which [math]\displaystyle{ N }[/math] individual will be chosen at random to form the new generation.
At generation [math]\displaystyle{ n }[/math], 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" :
- [math]\displaystyle{ f_n = \frac{k-1}{kN} + \frac{k(N-1)}{kN}f_{n-1} }[/math]
What is the source of the above formula? Is it in a later paper than the 1948 Reference.
- [math]\displaystyle{ \approx \frac{1}{N}+ (1-\frac{1}{N})f_{n-1}. }[/math]
This is a recurrence relation easily solved. Considering the worst case where at generation zero, no two individuals have a common ancestor,
- [math]\displaystyle{ f_0=0 }[/math], we get
- [math]\displaystyle{ f_n = 1 - (1- \frac{1}{N})^n. }[/math]
The scale of the fixation time (average number of generation it takes to homogenize the population) is therefore
- [math]\displaystyle{ \bar{n}= -1/\log(1-1/N) \approx N. }[/math]
This computation trivially extends to the inbreeding coefficients of alleles in a sexual population by changing [math]\displaystyle{ N }[/math] to [math]\displaystyle{ 2N }[/math] (the number of gametes).
See also
References
Bibliography
- Malécot, G. (1948). Les mathématiques de l'hérédité. Paris: Masson & Cie.
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