Supporting Information
Measuring population differentiation using GST or D? A
simulation study with microsatellite DNA markers under a finite
island model and non-equilibrium conditions
Liang Leng and De-Xing Zhang
When is the equilibrium value of GST larger than that of D?
We are interested in finding out the conditions under which the equilibrium
value of GST is larger than, equal to or smaller than that of D. Then we defined a
function DMG( ,m ,N )  D(equilibrium )  G ST(equilibrium ), where D(equilibrium) and
GST(equilibrium) were calculated by equations 10 and 11, respectively. Here μ
is the mutation rate, m the migration rate and N the subpopulation size. The
number of subpopulations (n) was fixed to 10. It is obvious that D is greater
than GST when DMG > 0, equal to GST when DMG = 0, and less than GST when
DMG < 0. It is impossible to list all combinations that make DMG larger than,
equal to or less than zero, so we concentrated on conditions that make DMG =
0.
We calculated DMG under different combinations of μ, m and N, and recorded
the condition when DMG = 0, using a C script we wrote. We set μ increasing
from 0.000001 to 0.0001, m from 0 to 0.01 and N from 1 to 10,000. In these
ranges, all combinations of μ, m and N were traversed. At each step the
-1-
program judged that whether the absolute value of DMG(μ,m,N) is smaller than ε
(ε is a small positive number, e.g. 0.000001). If so, it recorded the combination
of μ, m and N. Such combinations were named as zero points of DMG. Then
we drew the zero points in the 3-dimensional space of μ, m and N (Fig. S4), and
projected them on the 2-dimensional planes of μ and m (Fig. S5), m and N (Fig.
S6), μ and N (Fig. S7), respectively.
Theoretically, it seems that at any points in the planes shown in Figures S5 and
S6, DMG can be equal to zero. This implies that DMG can be larger than, equal
to or less than zero for any combination of m and N or μ and m, if given an
appropriate value of μ or N, respectively. However, it is quite different in Figure
S7. There we see a curve dividing the plane into two regions. In region II, DMG
is always larger than zero, no matter how great or small m is. When we fixed m
to different values and drew the zero points in the plane of μ and N, we
obtained a series of representative curves (Fig. S8). Invariably, in the upper
region of a curve (which is corresponding to region II in Fig. S8) DMG is larger
than zero. When migration rate is greater the curve shifts somewhat upper-right,
but when migration rate becomes larger than 0.001 there is only a tiny shift.
Moreover, even when migration rate is very great (m = 0.1) the upper region of
the curve is still overwhelmingly larger than that of the lower one, which means
that DMG is usually larger than zero, that is, D(equilibrium) is usually larger
than GST(equilibrium). In other words, no matter how great the migration rate is,
DMG becomes less than zero, that is, the equilibrium value of GST is larger than
that of D, only when mutation rate or population size is very small.
-2-
Figure S1 Effects of mutation on GST, D, HT and HS in populations with different
initial heterozygosities when population size (N) is 1,000 and migration rate (m)
0.001. A: initial heterozygosity (I.H.) is about 0.3 and mutation rate (µ) is 0.01; B:
I.H. is about 0.3 and µ is 0.001; C: I.H. is about 0.3 and µ is 0.0001; D: I.H. is
about 0.65 and µ is 0.01; E: I.H. is about 0.65 and µ is 0.001; F: I.H. is about
0.65 and µ is 0.0001.
-3-
Figure S2 Effects of population size (N) on GST, D, HT and HS in populations
with different initial heterozygosities when mutation rate (µ) is 0.001 and
migration rate (m) 0.001. A: initial heterozygosity (I.H.) is about 0.3 and N is
100; B: I.H. is about 0.3 and N is 1,000; C: I.H. is about 0.3 and N is 10,000; D:
I.H. is about 0.65 and N is 100; E: I.H. is about 0.65 and N is 1,000; F: I.H. is
about 0.65 and N is 10,000.
-4-
Figure S3 Effects of migration (m) on GST, D, HT and HS in populations with
different initial heterozygosities when population size (N) is 1,000 and mutation
rate (µ) 0.001. A: initial heterozygosity (I.H.) is about 0.3 and m is 0.01; B: I.H.
is about 0.3 and m is 0.001; C: I.H. is about 0.3 and m is 0; D: I.H. is about 0.65
and m is 0.01; E: I.H. is about 0.65 and m is 0.001; F: I.H. is about 0.65 and m
is 0.
-5-
Figure S4 Zero points of DMG in the 3-dimensional space of μ, m and N. The
initial heterozygosity is zero.
-6-
Figure S5 Projection of the zero points of DMG on the 2-dimensional plane of m
and N. The initial heterozygosity is zero.
-7-
Figure S6 Projection of the zero points of DMG on the 2-dimensional plane of μ
and m. The initial heterozygosity is zero.
-8-
Figure S7 Projection of the zero points of DMG on the 2-dimensional plane of μ
and N. The initial heterozygosity is zero.
-9-
Figure S8 Zero points of DMG on the 2-dimensional plane of μ and N, under
different m. The initial heterozygosity is zero.
- 10 -
Table S1 Observed numbers of allelic states in simulation.
Possible allelic states set to 50
μ
m
Possible allelic states set to 999
Maximum allelic number
Maximum allelic number
within 100,000 generations
within 100,000 generations
N
0.01
1 000
0.001
42.05
67.7
0.001
1 000
0.001
22
22.65
0.0001
1 000
0.001
8.2
8.3
0.001
1 000
0
30
34.4
11
0.001
1 000
0.01
18
21.6
0.001
100
0.001
8.5
9.2
43.15
48.1
0.001
10 000 0.001
In most simulations under SSM, we set the maximum possible allelic states to 50. In order to examine whether 50
possible allelic states is enough, we also set it to 999 and then compared the two results. It is clear that the setting of 50
allelic states conforms to the expectation of SSM model (i.e. the observed allele number is always much smaller than 50)
unless mutation rate is very high (μ ≥ 0.01). The initial heterozygosity is zero.
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Table S2 Values of GST and D from simulation and theoretical derivation when the initial heterozygosity is about 0.65.
μ
m
N
GST (t)
GST*
T(GST)
D (t)
D
D’
0.01
0.001
1 000
0.0197
0.03771
~2 000
0.99
0.5288
0.6018
0.001
0.001
1 000
0.1009
0.09927
~3 000
0.9089
0.4612
0.5028
0.0001
0.001
1 000
0.1696
0.15545
~3 000
0.4996
0.2766
0.2892
0.001
0.001
100
0.5288
0.5104
~3 000
0.9089
0.7055
0.7120
0.001
0.001
10 000
0.0111
0.01122
~2 000
0.9089
0.1137
0.2450
0.001
0.01
1 000
0.0197
0.01821
<1 000
0.4961
0.1235
0.1436
0.001
0
1 000
0.1834
0.2489
>10 000
1
0.7387
0.9122
GST (t): theoretically deduced equilibrium value of GST, derived from equations 10; GST*: stationary value of GST in
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simulation; T(GST): time (in generations) to reach the stationary value of GST in simulation; D (t): theoretical equilibrium
value of D, derived from equation 11; D: value of D at the 10,000th generation in simulation; D ’: value of D at the
100,000th generation in simulation.
* Here we recorded the stationary value of GST assuming that it reached the stationary value within 10,000 generations. It
is worth noting that when gene flow is highly constrained (e.g. migration rate is 0), GST still increases after 10,000
generations, albeit slowly.
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