Transcript Modeling Populations

One-way migration
Migration



There are two populations (x and y), each with a
different frequency of A alleles (px and py).
Assume migrants are from population x, and residents
are population x; unidirectional).
After migration, m is the migrant portion of the
population y, and (1-m) is the resident portion of the
population y. py’ is the p after migration:





py’ = m x px + (1-m) x py
dpy = m x px + (1-m) x py – py
dpy = m x px + py – m x py – py
dpy = m x px + m x py
dpy = m(px-py)
Change in allele frequency with
one-way migration (m = 0.01)
Natural Selection

The interaction between alleles and
environment shapes the direction of the
change in allele frequencies resulting in
evolution of adaptable traits.
Fitness and coefficient of
selection (s)


Darwinian fitness is defined as the relative
reproductive ability of a genotype.
The genotype that produces the most
offspring is assigned a fitness (W) value of 1.
Selection coefficient (s) equals (1-W)



AA produces on average 8 offspring
Aa produces on average 4 offspring
aa produces on average 2 offspring.



WAA = 1.0; sAA = 1-1 = 0
WAa = 0.5; sAa = 1-0.5 = 0.5
Waa = 0.25; saa = 1-0.25 = 0.75
How to calculate change in
allele frequency after selection
AA
Aa
aa
Initial
genotypic
frequencies
p2
2pq
q2
Fitness
WAA
WAa
Waa
Frequency after
selection
p2 WAA
2pq WAa
q2 Waa
Relative
frequency after
selection
p2 WAA/WMEAN
2pq WAa /WMEAN
q2 Waa /WMEAN
Wmean =
p2 WAA
+
2pq WAa
+
q2 Waa
Possibilities
1.
WAA = WAa = Waa: no natural selection
2.
WAA = WAa < 1.0 and Waa = 1.0: natural selection and complete
dominance operate against a dominant allele.
3.
WAA = WAa = 1.0 and Waa < 1.0: natural selection and complete
dominance operate against a recessive allele.
4.
WAA < WAa < 1.0 and Waa = 1.0: heterozygote shows
intermediate fitness; natural selection operates without effects
of complete dominance.
5.
WAA and Waa < 1.0 and WAa = 1.0: heterozygote has the highest
fitness; natural selection/codominance favor the heterozygote
(also called overdominance).
6.
WAa < WAA and Waa = 1.0: heterozygote has lowest fitness;
natural selection favors either homozygote.
Selection against a recessive
lethal phenotype



Recessive trait result
in reduced fitness.
Frequency of the
recessive allele
decreases over time.
Not completely
eliminated since
present in
heterozygotes.
Heterozygote superiority

Distribution of
malaria and
frequency of Hb-s
allele leading to
sickle cell disease in
homozygotes.
Balance between mutation
and selection



When an allele becomes rare, changes in
frequency due to natural selection are small.
Mutation occurs at the same time and
produces new rare alleles.
For a complete recessive allele at equilibrium:


q = õ/s
If homozygote recessive is lethal (s = 1) then q =
õ
Model 1

Simulate the change in allele frequencies directly
by mathematical modeling of the forces that act on
them.





Set initial values for p and q;
Set initial sample size (effective population size);
Set the HWE as the null model (p2 + 2pq + q2 = 1);
Allow for forces such as mutation rate, migration,
genetic drift, and selection to act on the null model.
Estimate the change in allele frequencies over time using
iterations (i.e., the program loops over for a number of
generations as given by the arguments).
Model 2

Simulate individuals of a population(s) having DNA
sequence polymorphisms, and allow them to evolve
randomly or under certain forces.






Set initial number of individuals (N at t = 0, equals to the effective
size of the population, Ne);
Generate a null matrix for N x K x G, where K = 2 (diploid), and G
equals to the number of genes considered (start with a single gene,
if else assume genes are not linked for simplicity).
Set the total number of alleles (Nk, start with Nk = 2) for each G.
Set the initial number of homozygotes, heterozygotes for G.
Allow for the individuals mate randomly to produce offspring, iterate
to simulate generations; for simplicity assume that all individuals die
after reproduction. E.g., annual plants where Nt+1 = bNt + 0 Nt
Allow for forces to act on the null model, and test their effects on the
allelic evolution.