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) W A = relative fitness of A W a = relative fitness of a

One-locus haploid model p[t] q[t] WAWaWAWa f'(A) = __W A p(t)___ W A p(t) + W a q(t) f'(a) = __W a q(t)___ W A p(t) + W a q(t) Example: p(t) = 0.5; q(t) = 0.5 W A = 1; W a = 0.8

One-locus haploid model p(t)W A p(t)W A + q(t)W a

Relative, not absolute, fitness determines changes in allele frequencies a A a A A A A A a a a a a A a A A A A A a a a a a A A A A A A a a a a a A A A 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

Haploid selection: rest of life cycle

Adults mate at random Undergo meiosis One-locus haploid model

p(t+1) = p(t)W A p(t)W A + q(t)W a  p = p(t+1) – p(t) = (W A – W a )p(t)q(t) W(t) W(t) = p(t)W A + q(t)W a

One-locus haploid model  p = p(t+1) – p(t) = (W A – W a )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)W A q(t+1) q(t)W a =

Predicting allele frequencies q(t) = 1- p(t), so p(0)W A t p(0)W A t + q(0)W a t p(t) = Now, for any generation t: p(t) p(0)W A t q(t) q(0)W a t = 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)W A t p(0)W A t + q(0)W a t p(t) =

Using the model II If A changes in frequency from to 0.01 in 10 generations, by how much must it be favored? p(t) p(0)W A t q(t) q(0)W a t =

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?

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)(W A - W(t)) 2 + q(t)(W a -W(t)) 2 = p(t)q(t)(W A - W a ) 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 References Heidler, J. et al Partitioning, Persistence, and Accumulation in Digested Sludge of the Topical Antiseptic Triclocarban during Wastewater Treatment. Environ. Sci. Technol.; 40(11); 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:Aa starting count (before selection) ending count (after selection) 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?