Lab 6: Genetic Drift andEffective Population Size

Goals1. To calculate the probability of fixation or loss of an allele.

2. To estimate mean time until fixation of an allele.

3. To estimate effective population size affected by past cataclysms.

4. To learn how genetic drift and selection interact in populations of various Ne.

What is MC method?• The (pseudo-random) number

generator has certain characteristics (e.g., a long "period" before the sequence repeats)

• The (pseudo-random) number generator produces values that pass tests for randomness

• There are enough samples to ensure accurate results

• It simulates the phenomenon in question.

p = ? p = ?

Probability of fixation or loss

1. Genetic drift results from chance changes in allele frequencies that result from sampling of gametes from generation to generation in a finite population.

2. Exact probability of fixation of an allele is equal to the initial frequency of that allele in absence of selection.

3. Probability of fixation of an allele can be calculated empirically by using Monte Carlo simulations as implemented in Populus.

0)( ppu

• Population size is assumed to remain constant from generation to generation• Allelic frequency changes result only from the

random sampling process

Important assumption for Genetic Drift model in Populus

Problem 1 (15 minutes). The frequencies of alleles A1 and A2 arep = 0.7 and q = 0.3, respectively.

Use Populus to empirically estimate the probabilities of fixation and loss for each of these alleles.

What do you think are the exact probabilities of fixation and loss for each allele?

Do these probabilities depend on the population size?

a) Use Populus to empirically estimate the mean time (in number of generations) until fixation for allele A1 for each case.

b) Show the mean time until fixation of A1 calculated using the diffusion approximation (given above) for each case.

c) Discuss the reasons for the differences (if any) between the two types of estimates. What are some of the assumptions underlying each method?

Problem 2 (15 minutes). Consider a population with the following genotype counts:

Genotype counts

Case A1A1 A1A2 A2A2

1 18 4 3

2 0 7 21

.)1ln()1(4

)(p

ppNpT

Mean time until fixation of an allele depends on population size and initial frequency of that allele.

A1A1(N11)

A1A2(N12)

A2A2(N22) N p q T(p)

T(p) in terms of

N

16 2 2 20 0.85 0.15 26.78 1.34N

0 2 18 20 0.05 0.95 77.96 3.90N

.)1ln()1(4

)(

:ionapproximatdiffusion (1971) Ohta and Kimura

p

ppNpT

Table courtesy of K. Ritland

Effective Population Size: Effects of Different Numbers of Males and Females

Effective population size

Problem 3 (15 minutes). The census population size of an isolated population of finches on the Galapagos Islands is as follows. What is the effective population size in 2010?

Year Females Males

1930 120 2501950 15 2901970 250 3501990 1500 25002010 3500 5000

Dr. Robert Rothman, Rochester Institute of Technologyhttp://people.rit.edu/rhrsbi/GalapagosPages/DarwinFinch.html

When time is discontinuous, a transition matrix can be used to determine the probability of fixation in the next generation.

iNittij qp

iiN

NjYiYPx

2

1 ''!)!2(

)!2()|(

sppqhs

hspqspp

2

2

21

)1()1('

GenotypeA1A1 A1A2 A2A2

Fitness ω11 ω12 ω22

Fitness in terms of s and h(adaptive Darwinian selection)

1 + s 1 + hs 1

Fitness in terms of s and h(purifying selection) 1 1 − hs 1 − s

• Using traditional setup for adaptive Darwinian (positive) selection• These can be easily converted to terms for purifying selection

selection purifyingunder effect usheterozygo hLet

selectionDarwinian under effect usheterozygo hLet

selection purifyingunder t coefficienselection sLet

selectionDarwinian under t coefficienselection sLet

P

D

P

D

D

DP s

ss

1 Dp hh 1

Problem 4 (15 minutes). If adaptive Darwinian selection (characterized by h = 0.5 and s = 0.25) is operating on a locus and the frequency of allele A1 at that locus is p = 0.25, predict whether A1 is more likely to get lost or to become fixed:

(i) In a population with Ne = 10.

(ii) In a population with Ne = 30.

a) For each of the two cases, calculate the probability of fixation of A1 empirically (i.e., using Populus).

b) If Ne affects the probabilities of fixation and loss of A1, explain why. If not, explain why not.

Problem 5. GRADUATE STUDENTS ONLY: Starting with the conditions in Problem 4-a), calculate the probability that:

a) The frequency of A1 becomes 0.1 in the next generation.

b) A1 becomes fixed in the next generation.

c) If the two transition probabilities differ dramatically, explain why. If not, explain why not.