Download Models of Selection

Models of Selection
Goal: to build models that can predict a
population’s response to natural selection
What are the key factors?
Today’s model: haploid, one locus
Outline: triclosan in biosolids
fitness
haploid life cycle
selection coefficients
long term predictions
When does selection act?
Triclosan and biosolids
Triclosan:
Biosolids:
Triclosan in biosolids??
Fitness: The sum total effect of
selection within a generation
Absolute Fitness =
Relative Fitness =
Key questions for model
One-locus haploid model
For what organisms is
this model appropriate?
Initial frequencies, fitness
f(A) = p(t)
f(a) = q(t)
WA = relative
fitness of A
Wa = relative
fitness of a
One-locus haploid model
f'(A) = __WAp(t)___
WAp(t) + Waq(t)
f'(a) = __Waq(t)___
WAp(t) + Waq(t)
Example:
p(t) = 0.5; q(t) = 0.5
WA = 1; Wa = 0.8
W
WA
a
p[t]
q[t]
One-locus haploid model
p(t)WA
p(t)WA + q(t)Wa
Relative, not absolute, fitness determines
changes in allele frequencies
6 A, 6 a
6 A, 6 a
a
A
a
a
A
a
A
a
A
A
a
A
a
A
A
A
A
a
a
A
a
Survival of A = 1, of a = 2/3
a
A
a
a
A
a
Survival of A = 1/2, of a = 1/3
a
A
a
A
A
A
A
A
a
a
A
f’(A) = 0.6
A
a
f’(A) = 0.6
Haploid selection: rest of life cycle
One-locus haploid model
Adults mate at random
Undergo meiosis
One-locus haploid model
p(t+1) =
p(t)WA
p(t)WA + q(t)Wa
p = p(t+1) – p(t) = (WA – Wa)p(t)q(t)
W(t)
W(t) = p(t)WA + q(t)Wa
One-locus haploid model
p = p(t+1) – p(t) = (WA – Wa)p(t)q(t)
W(t)
What does this tell us about selection?
A note about variance
What will happen over periods of time longer than
one generation?
• We can use a simple trick to answer
this question. If we divide p[t+1] by
q[t+1]:
p(t+1) p(t)WA
=
q(t+1) q(t)Wa
The ratio of p[t] to q[t] changes by W /W
A
a
every generation.
Predicting allele frequencies
Now, for any generation t:
p(t)
q(t)
=
p(0)WAt
q(0)Wat
q(t) = 1- p(t), so
p(t) =
p(0)WAt
p(0)WAt + q(0)Wat
hint: keep right side together, divide by fraction
Using the model I
p(t) =
t
p(0)WA
p(0)WAt + q(0)Wat
What would the frequency of allele A be after
100 generations of selection if A is 10% more fit
than allele a and if one in every hundred alleles
is initially A?
Using the model II
If A changes in frequency from 0.001 to
0.01 in 10 generations, by how much
must it be favored?
p(t)
q(t)
=
p(0)WAt
q(0)Wat
Selection coefficients
Selection coefficient example
How long would it take for 95% of the alleles
to be A if A is initially present in 5% of the
population and if the selection coefficient
favoring allele A is...
s = 0.1?
Some general principles
The time needed for an allele to go
from low frequency to high is the
inverse of the selection coefficient
s = 0.1 -> tens of generations
Does the mean fitness of a population
always increase over time?
Var(W(t)) = p(t)(WA - W(t))2 + q(t)(Wa-W(t))2
= p(t)q(t)(WA - Wa)2
ΔW = W(t+1) - W(t) = Var(W(t))
W(t)
The Fundamental Theorem of Natural
Selection
"The rate of increase in fitness of any
organism at any time is equal to its genetic
variance in fitness at that time."
R. A. Fisher (1930) The Genetical Theory of Natural
Selection
Example: Dykhuizen and Dean (1990)
Two strains of E. coli (TD9 and TD1)
• Had a genetic difference in the lactose pathway
• Competed in two environments:
• Glucose-limited (Open symbols)
• Lactose-limited (Closed symbols)
• What is the selection coefficient (s)?
References and readings
References
Heidler, J. et al. 2006. Partitioning, Persistence, and Accumulation in
Digested Sludge of the Topical Antiseptic Triclocarban during Wastewater
Treatment. Environ. Sci. Technol.; 40(11); 3634-3639
Readings
Chapter 6.1 – 6.3 (5.1 – 5.3), question 3.
More questions
Would a dominant or recessive allele change frequency faster in a haploid
organism? why?
Calculate the relative fitnesses for these two genotypes:
genotype:
A
a
starting count (before selection)
100 100
ending count (after selection)
90
30
What is the selection co-efficient?
Assume that the mixture starts out with f(A) = 0.5. What will the frequency
be after 20 generations?