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.
In a finite size population, after some generations, all individuals will have a common ancestor :
f
→
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
≫
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
=
k
−
1
k
N
+
k
(
N
−
1
)
k
N
f
n
−
1
≈
1
N
+
(
1
−
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
−
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
.
This computation trivially extends to the inbreeding coefficients of alleles in a sexual population by changing
N
to
2
N
(the number of gametes).
