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| {{underlinked|date=December 2012}}
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| '''Malecot's coancestry coefficient''', '''<math>f</math>''', refers to an indirect [[measure]] of genetic [[similarity]] of two individuals which was initially devised by the [[France| French]] mathematician [[Gustave Malécot]].
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| <math>f</math> is defined as the probability that any two [[alleles]], [[sample (statistics)|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>f</math> can be calculated by examining detailed pedigree records. Modernly, <math>f</math> can be estimated using [[genetic marker]] data.
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| == Evolution of inbreeding coefficient in finite size populations ==
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| In a finite size [[population]], after some generations, all individuals will have a [[common ancestor]] : <math>f \rightarrow 1 </math>.
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| Consider a non-sexual population of fixed size <math>N</math>, and call <math>f_i</math> the inbreeding coefficient of generation <math>i</math>. Here, <math>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>k \gg 1</math> of descendants, from the pool of which <math>N</math> individual will be chosen at random to form the new generation.
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| At generation <math>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" :
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| :<math>f_n = \frac{k-1}{kN} + \frac{k(N-1)}{kN}f_{n-1}</math> | |
| :<math> \approx \frac{1}{N}+ (1-\frac{1}{N})f_{n-1}. </math>
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| This is a [[recurrence relation]] easily solved. Considering the worst case where at generation zero, no two individuals have a common ancestor,
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| :<math>f_0=0</math>, we get
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| :<math>f_n = 1 - (1- \frac{1}{N})^n.</math>
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| The scale of the fixation time (average number of generation it takes to homogenize the population) is therefore
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| :<math> \bar{n}= -1/\log(1-1/N) \approx N. </math> | |
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| This computation trivially extends to the inbreeding coefficients of alleles in a sexual population by changing <math>N</math> to <math>2N</math> (the number of [[gametes]]).
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| == References ==
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| *Malécot G. ''Les mathématiques de l'hérédité.'' Paris: Masson & Cie, 1948.
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| [[Category:Classical genetics]]
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