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assumed values for three parameters of the two phase mutation model ( = 4N
e
, p
s
= mean
percentage of mutations that add or delete only one repeat unit, and
g
= mean size of larger
mutations). Once M is obtained, it is ranked relative to the equilibrium distribution. Using
conventional criteria, there is evidence of a significant reduction in population size if less than
5% of the replicates are below the observed value. The average values employed in this
analysis were obtained from the MISAT program by Nielsen (1997), where p
s
and
g
were
0.0325 (overall Amazon) and 0.1318 (Bolivia) for the first parameter and 3.5 for the second
one, respectively. This analysis was carried out with the M-P-Val and Critical-M programs
from Garza & Williamson (2001).
The last microsatellite demographic analysis applied to the
Inia sp.
populations reported
here were the within locus kurtosis (k) test and the interlocus (g) test proposed by Reich &
Goldstein (1998) and Reich et al. (1999). Both tests estimate if there is population expansion
in a given species, or population.
The first test is based on the following principles. A population with constant size has
gene genealogies, which tend to have a single ancient bifurcation. Therefore, the allele length
distributions have multiple discrete peaks. On the contrary, in a growing population, most of
the gene genealogy bifurcations tend to date back to the time expansion and as a result the
allele length distribution is clearly more smoothly peaked. To measure these differences
between the multipeaked allele distribution of a constant size population and the smooth
single-peaked allele distribution of a population in expansion, the k statistic was calculated as
follows:
k = 2.5 Sig
4
+ 0.28 Var - (0.95/n) - Gam
4
,
where Sig
4
is the unbiased variance squared of allele length, Var is the sample variance of the
same concept and Gam
4
is the unbiased fourth central moment of the allele, respectively. The
equations to estimate Sig
4
and Gam
4
are as follows
2
(
n
3
n
)3
1
4
2
2
4
Sig
(
(
Xi
X
)
)
(
Xi
X
)
n
(
n
1
)(
n
2
)(
n
)3
(
n
2
)(
n
)3
2
(
n
2
n
)3
(
6
n
)9
4
2
2
Gam
(
Xi
X
)
(
(
Xi
X
)
)
4
(
n
1
)(
n
2
)(
n
)3
(
n
2
)(
n
)3
where
n
is the number of chromosomes analyzed, X
i
is the number of repeats of each allele
found and X is the average repeat number of all the alleles found in a given population for a
determined microsatellite. In order to assess significance levels, a binomial distribution is
used with the number of trials equal to the number of loci based on the expectation of an
almost (P = 0.515) equal probability of negative and positive k-values for the set of loci
N
N
!
r
N
r
r
N
r
p
(
p
)
p
(
p
)
analyzed (
)
r
r
!
N
r
)!
When there is a smaller loci number associated with positive k values than would be
expected for a constant-sized population, there is evidence of a population expansion. The
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