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 modelFor what organisms is this model appropriate?

Initial frequencies, fitnessf(A) = p(t)

f(a) = q(t)

WA = relative fitness of A

Wa = relative fitness of a

One-locus haploid model

p[t]q[t]

WA

Wa

f'(A) = __WAp(t)___ WAp(t) + Waq(t) 

 f'(a) = __Waq(t)___ WAp(t) + Waq(t)

 Example:p(t) = 0.5; q(t) = 0.5WA = 1; Wa = 0.8

One-locus haploid model

p(t)WA

p(t)WA + q(t)Wa

Relative, not absolute, fitness determines changes in allele

frequencies

a

Aa

A

A

A A

Aa

a

a

a

a

Aa

A

A

A A

Aa

a

a

a

a

A

A

A

A A

Aa

a

a

a

a

A

A A

a

6 A, 6 a 6 A, 6 a

Survival of A = 1, of a = 2/3

Survival of A = 1/2, of a = 1/3

f’(A) = 0.6 f’(A) = 0.6

Haploid selection: rest of life cycle

Adults mate at random

Undergo meiosis

One-locus haploid model

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

• We can use a simple trick to answer this question. If we divide p[t+1] by q[t+1]:

What will happen over periods of time longer than one generation?

The ratio of p[t] to q[t] changes by W

A/W

a every generation.

p(t+1) p(t)WA

q(t+1) q(t)Wa

=

Predicting allele frequencies

q(t) = 1- p(t), so

p(0)WAt

p(0)WAt + q(0)Wa

t

p(t) =

Now, for any generation t:

p(t) p(0)WAt

q(t) q(0)Wat=

hint: keep right side together, divide by fraction

Using the model I

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?

p(0)WAt

p(0)WAt + q(0)Wa

t

p(t) =

Using the model IIIf A changes in frequency from

0.001 to 0.01 in 10 generations, by how much must it be favored?

p(t) p(0)WAt

q(t) q(0)Wat=

Selection coefficients

Selection coefficient exampleHow 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?

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

Some general principles

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 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

The Fundamental Theorem of Natural Selection

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)?

Example: Dykhuizen and Dean (1990)

References and readings

ReferencesHeidler, 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

ReadingsChapter 6.1 – 6.3 (5.1 – 5.3), question 3.

More questionsWould 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 30What is the selection co-efficient?Assume that the mixture starts out with f(A) = 0.5. What will the frequency be after 20 generations?