POPULATION GENOMICS
The objective of this exercise is to conclude what is the relationship between adaptive
evolution and population size, through the comparison of differing in size populations which
contain mutations in different proportion and with different strength of selection.
The first step was to fix the fitness coefficient (s) and the proportion of each mutation. A negative
s, indicates a negative strength of selection, so a lower fixation of this mutation after many
generations is expected. Therefore, a positive s leads to a higher fixation of the mutations after
many generations. But this fixation not only depends on the s value, but also in the proportion
of each mutation in the population, i.e. even though a negative s, the fixation of a mutation that
exists in a high proportion could be higher than another mutation with a positive s, if its
proportion is too small. To see what happens in all of these cases, I have selected different
combinations of s and proportions. The graphic of the proportions for each fitness class is
indicated in Figure 1.
Fitness distribution of new mutations
14
Proportion
12
10
8
6
4
2
0
Fitness class
Figure 1. Proportion of each mutation in the population.
Note that for s with values -0.00006 and 0,001, I have fixed different proportions of the mutation
in the population, to can see the relation between different cases.
However, this is not what exactly happens in real populations. Actually, as genetic code is
degenerated about one third of mutations are neutral, so the proportion of mutations should
be more or less concentrated in the middle of the graph. The majority of the rest of mutations
are expected to have a really small positive or negative effect, and those that have a huge effect
tend to be deleterious, so they are not expected to be fixed in the population in a high
percentage (Figure 2).
Now we will see how the fixation probability of each mutation changes after many generations
depending on the values indicated above and the size of population. As we predicted, we can
see in the Figure 2 that those mutations for which fitness coefficient is negative, the proportion
of fixed mutations after many generations is lower. The opposite happens when s is positive, in
this case we can see that the proportion of mutations that are finally fixed is higher. Also we
can conclude that as explained before, for the same s the percentage of fixed mutation is higher
in those ones in which the proportion was bigger. The reason of this is that as I will analyze later,
the fixation probability of each mutation is influenced but these parameters.
Proportion
% Fixed mutations
0.5
0.46219361
0.45
0.4
0.35
0.131906492
0.3
0.052604365
0.25
0.2
0.075681864
0.053448696
0.015942898
0.15
0.053458317
0.00534006
0.026752151
0.031891536
0.1
0.021340375
0.050.005292605
0.06414703
0
Fitness class
Figure 2. Proportion of fixed mutations after 10^6 generations for each fitness class, for an
effective population size of 50 and a census of 500.
In the row number 3, we had to calculate the number of mutations arising per generation. I have
calculated this by using next formula:
2*N*(10^5)*Proportion of mutations
As we can see, this value is directly proportional to the initially fixed proportion and to the
population size, so that as the population size or proportion of the mutation is incremented, the
number of mutation arising per generation is higher. For example, for a proportion of mutations
of 13% and a s of 0,001:
-
If N=500 -> The number of mutations arising per generation is 0,13.
If N=1000 -> The number of mutations arising per generation is 2,6.
If N=500000 -> The number of mutations arising per generation is 100.
If N=10000000-> The number of mutations arising per generation is 2600.
The next step was to calculate the fixation probability of each mutation (row 5) and the rate of
evolution (row 6). It has been calculated with next formulas:
-
Fixation probability: u (p) = (1 – e‐4Nes p)/(1 – e‐4Nes)
Rate of evolution: 2*N*10^-5*Fixation probability
As results indicate, fixation probability increases with s and with the proportion of the
mutations. Thereby, those mutations occurring in a positive fitness class, has a higher fixation
probability than those occurring for a negative fitness class. And as rate of evolution is
proportional to the fixation probability, the same occurs with this parameter. Another important
aspect is that rate of evolution is also influenced by the effective population size (Ne) – cause
fixation probability depends on this- and the cense (N), so that as N increases, this increases as
well. For example, for the previous case (proportion of mutations of 13% and s of 0,001):
-
N=500, Ne=50 -> Rate of evolution= 0,00141584
N=10000, Ne=1000 -> Rate of evolution= 0,20372527
N=500000, Ne=5000 -> Rate of evolution= 10
N=10000000, Ne=1000000-> Rate of evolution= 200
These results show that for the same s and proportion of mutations, populations with a high
census has a higher rate of evolution. This supports the theoretical idea that the amount of
adaptive evolution has a positive correlation with the population size.
The graphs of the percentage of fixed mutations for the rest of examples are the followings:
% Fixed mutations
0.18
0.16
0.08096355
Proportion
0.14
0.12
0.1
0.08
0.06
0.131864108
0.131337705
0.066977806
0.157699648
0.067025808
0.06580019
0.06580019
0.037572113
0.052215698
0.06580019
0.06580019
0.04
0.011142801
0.02
0
Fitness class
Figure 3. Number of mutation fixed after 10^6 of generations for each fitness class, for an
effective population size of 1000 and a census of 10000.
% Fixed mutations
0.18
Proportion
0.16
0.139369423
0.161183197
0.14
0.139369423
0.139369423
0.141969689
0.139369423
0.12
0.1
0.08
0.139369423
0.06
0.04
0.02
0
0
0
0
0
0
0
Fitness class
Figure 4. Number of mutation fixed after 10^6 of generations for each fitness class, for an
effective population size of 1000000 and a census of 10000000.
Note how the Figure 4 represents the power of the population size increasing adaptive
evolution. This occurs since as the population size is bigger, natural selection becomes more
powerful than genetic drift, which becomes statistically less effective at changing allele
frequencies. Therefore, as Ne increases, a greater number of advantageous mutations appear
in the population so that a greater proportion of these are effectively selected. Then, species
with larger Ne are expected to undergo more adaptive evolution than species with small Ne.
Summarizing, these results correlate with the theoretical predictions which affirms thatguessing that adaptation is limited by the supply of mutations- large effective population sizes
undergo a higher rate of adaptive evolution than those with a low effective population size. This
is promoted cause as explained, in large populations a higher proportion of mutations are
expected to have a Ne*s>1, so that a higher proportion are expected to be effectively selected.
UNEXPECTED RESULTS:
1.2
1
% fixed mutations
Proportion
1
0.8
5.4508E-30
2.25532E-24
1.24827E-31
0.6
0.4
0.2
9.47895E-24 2.31952E-16
0
6.88594E-31
1.24179E-26
1.24827E-31
8.76664E-30
3.94788E-29
1.24827E-31
1.24827E-31
Fitness class
Figure 5. Number of mutation fixed after 10^6 of generations for each fitness class, for an
effective population size of 1000000 and a census of 10000000.
In these graph the data is not correlated with the expected results, since the population
size is high but there is no more fixation for the mutations with a high s even when the
proportions of the initial mutations are also great in some cases. I do not find the reason
why this graph does not follow the same way as the others since the calculation
procedure that has been followed has been the same for all of them.
Lur Alonso Cotchico
NIU: 1354974