Course outline

Evolution:When violations in H-W assumptions cause changes in the genetic composition of a population

Population Structure:When violations in H-W assumptions cause changes in the distribution of alleles within/across populations

Unit 2: Evolution and Pop. Structure

(a.k.a. violations in H-W assumptions)

Unit 2.1: genetic driftUnit 2.2: natural selectionUnit 2.3: mutation

Unit 2.4: migrationUnit 2.5: assortative matingUnit 2.6: inbreeding

Consider adding some PCA plots, etc. to show population structure.

Course outline

Evolution:When violations in H-W assumptions cause changes in the genetic composition of a population

Population Structure:When violations in H-W assumptions cause changes in the distribution of alleles within/across populations

Unit 2: Evolution and Pop. Structure

(a.k.a. violations in H-W assumptions)

Unit 2.1: genetic driftUnit 2.2: natural selectionUnit 2.3: mutation

Unit 2.5: assortative matingUnit 2.6: inbreeding

Unit 2.4: migration

Migration

Feb 16, 2015

HUGEN 2022: Population Genetics

J. ShafferDept. Human GeneticsUniversity of Pittsburgh

Objectives

• At the end of the lecture you should be able to

1. identify whether scenarios constitute genetic migration

2. recognize the qualitative effects of migration

3. solve and interpret problems under various migration models

Hardy-Weinberg assumptions

• diploid organism• sexual reproduction• nonoverlapping generations• random mating• large population size• equal allele frequencies in the sexes• no migration• no mutation• no selection

Hardy-Weinberg assumptions

• diploid organism• sexual reproduction• nonoverlapping generations• random mating• large population size• equal allele frequencies in the sexes• no migration• no mutation• no selection

Big Picture: Population Structure

• Definition: – person-perspective: individuals in a population fall into genetically-

distinct groups– allele perspective: alleles are distributed across the population in some

way other than expected due to chance alone

• Population Structure vs. Hardy-Weinberg– The H-W Law tells us that if assumptions are met, alleles will be

distributed across genotype groups with the following frequencies: p2, 2pq, q2

– For population structure to occur, H-W assumption must be violated– Which processes cause population structure?

• non-random mating based on broad range of cultural and phenotypic characteristics (i.e. ethnicity)

• migration bringing two or more genetically distinct populations together

Migration

• Definition:– the movement of alleles among subpopulations

• Two viewpoints for population structure:– a meta-population may be divided into subpopulations

• geographical regions• ethnic groups

– people more often mate within their subpopulation, but there is some mixing

– from the point of view of the meta-population, the population structure is an example of non-random mating

– from the point of view of the subpopulation, the population structure is an example of migration

Key point: genetic migration• For the purpose of studying population genetics, migration does

NOT necessarily involve people migrating from one geographical place to another

• Migration from the genetic standpoint is only interested in movement of alleles among subpopulations

• Examples:– Sailors land on island, mate with natives, and then sail on– European settlers mate with native Americans– Mixing of people of African ancestry and European ancestry in North

America

• Counter example:– Entire island population moves from a volcanically active island to an

uninhabited nearby island (i.e. NO genetic migration)

Effects of migration

• Qualitative– Migration among populations is “homogenizing”

• allele frequencies of each population move toward the average

• do not confuse homogenous with homozygous – Affects the entire genome simultaneously– Speed of homogenization dependent on rate(s) of

migration among populations

• Quantitative– difficult to accurately model; we will do math under

some very simple models of migration

general migration model

• to understand how allele frequencies change in population i, we need to know:

1. migration rates, mij (i.e., mto from), among all populations• mij is the P(next generation allele comes from pop. j into pop. i)

2. allele frequencies, P(A) = pj, for all populations

general migration model

• allele frequency for population i after one generation of migration is:

3 4

1 2 1 2 3 4

1 m11 m12 m13 m14

2 m21 m22 m23 m24

3 m31 m32 m33 m34

4 m41 m42 m43 m44

from

to

pops

1jijji mpp

p1

p3

p2

p4

example: general model• four populations, 1-4, with allele frequencies 0.1, 0.2, 0.3, 0.4, respectively

• what is the allele frequency of population #2 after one generation of migration?

Exam

ple

1 2 3 4

1 0.9 0.00 0.03 0.07

2 0.01 0.85 0.09 0.05

3 0.02 0.06 0.9 0.02

4 0.15 0.00 0.05 0.8

from

to

= (p1)(m21) + (p2)(m22) + (p3)(m23) + (p4)(m24)

= (0.1)(0.01) + (0.2)(0.85) + (0.3)(0.09) + (0.4)(0.05)

= 0.001 + 0.17 + 0.027 + 0.02

= 0.218

pops

1jijji mpp

must be known

general migration model

• What if we have a lot of populations?

– migration matrix becomes bigger– we probably can’t accurately know all of the migration rates– migration rates among the network probably change from one generation to

another

• Make some simplifications– trade off between accuracy and feasibility

general migration model

• What if we have a lot of populations?

– migration matrix becomes bigger– we probably can’t accurately know all of the migration rates– migration rates among the network probably change from one generation to

another

• Make some simplifications– trade off between accuracy and feasibility

… quickly becomes a real mess!

general migration model

• What if we have a lot of populations?

– migration matrix becomes bigger– we probably can’t accurately know all of the migration rates– migration rates among the network probably change from one generation to

another

• Make some simplifications– trade off between accuracy and feasibility

… quickly becomes a real mess!

island model of migration

• many populations; migration rates among all– population of interest is an “island”– collectively call all of the other populations together a “continent”

• P(A) for each population, p

• p = average p over all populations (p of meta-population)– does not change over time (think of whole-world allele freq.)

• m = migration rate– P(next-gen. allele in island comes from the continent)– 1 – m = P(next-gen. allele is from the island)

island model of migration

• simplifying assumptions:

– continental population very large compared to island• migration FROM continent TO island may meaningfully impact genetic

composition of island• migration FROM island TO continent has negligible impact on genetic

composition of continent • island population has negligible effect on p

– migrant allele frequency = p• ignores population substructure within the continent

island model as general model

p0

p

island continent

island 1-m m

continent 0.0 1.0to

from

= (p0)(1-m) + (p)(m)

m

pops

1jijji mpp

from island from migrants

in the next generation:

island model over time• for the next generation:

p1 = p0(1 – m) + pm

• For t generations:

pt = p + (p0 – p)(1 – m)t

• After many, many generations

(1 – m)∞ = 0

p∞ = p

island model example

• specifics the continental population has p = 0.6

1 island population has p0 = 0m = 0.01

i.e. 1% of alleles from continent, 99% from island

• what is p10? p100? p1000?• how many generations until p = 0.4?

Exam

ple

island model example

p0 = 0.0 m = 0.01p = 0.6

pt = p + (p0 – p)(1 – m)t p10 = (0.6) + (0 – 0.6)(1 – 0.01)10

p10 = 0.057p100 = 0.380p1000 = 0.59997

Exam

ple

Migration example

p0 = 0.0 m = 0.01p = 0.6

How many generations until pt = 0.4 in the subpopulation of interest?

pt = p + (p0 – p)(1 – m)t

(0.4) = (0.6) + (0 – 0.6) (1 – 0.01)t

(0.4) = (0.6) + (-0.6) (0.99)t

(0.6 – 0.4) / (0.6) = (0.99)t

0.3333 = 0.99t t = log(0.3333) / log(0.99)

t = 109.3

Exam

ple

More models for humans

• one-way “racism” model (variant of island model)– offspring of mixed parentage are all members of one of the

parental groups– example: African Americans

• two-way racism model (example of general model)– child of mixed parentage are distinct and from a new population

of their own– example: Anglo-Indians

migrantpop.

admixedpop.

parentalpop. 1

admixedpop.

parentalpop. 2

one-way racism as general model

admixed migrant

admixed 1-m m

migrant 0.0 1.0to

from

= pA1 = (1-m)(pA) + (m)(pM)

pops

1jijji mpp

from admixed from migrants

in the next generation:

migrantpop.

admixedpop.

pApM

m

for t generations: pAt = pM + (pA0 – pM)(1-m)t

• comments– pM not dependent on t because migration is one way– what happens to pAt in the long run? It approaches pM

Example: one-way racism model

• How much admixture is there in U.S. African American population?– clinical and historical interest

• We need to know: pAt, pM, pA0, t and solve for mpAt = pM + (pA0 – pM)(1-m)t

• We will consider a study of populations from Claxton, GA

CaucasianAfrican

American

Exam

ple

Example: one-way racism model

• Where do we get values for our variables?pA0 = original allele frequency in African Americans prior to admixturepAt = current allele frequency in the admixed populationpM = allele frequency in the migrating populationt = number of generations since admixture

• These values must be estimated from what we can measure now

• Because of all this uncertainty, we cannot be confident in the results of a single locus.

• Instead, test many loci to see if the results are consistent– remember, migration effects all loci simultaneously!

Exam

ple

Example: one-way racism model

• the data for an example locus:

pA0 = 0.474 (measured from West Africa)

pAt = 0.484 (current Claxton African Americans)pM = 0.507 (current Claxton Caucasians)t = 15 (historical record of time of max. slave influx to Claxton area)

pAt = pM + (pA0 – pM)(1-m)t

0.484 = 0.507 + (0.474 – 0.507)(1-m)15

m = 1 – (0.696969)1/15

m = 0.024Interpretation:On average, over 15 generations, 2.4% of the alleles in each generation of African Americans came from Caucasians

Exam

ple

two-way racism as general model

admixed migrant 1 migrant 2

admixed 1-mA1-mA2 mA1 mA2

migrant 1 0.0 1.0 0.0

migrant 2 0.0 0.0 1.0

to

from

= pA1 = (1-mA1-mA2)(pA) + (mA1)(pM1)+ (mA2)(pM2)

pops

1jijji mpp

from admixed from migrants 1

in the next generation:

pA pM2

mA1

for t generations: pAt = pM + (pA0 – pM)(1-m)t

• where:

parentalpop. 1

admixedpop.

parentalpop. 2

mA2

pM1

from migrants 2

2A1A

2A2M1A1MM mm

mpmpp

and m = mA1 + mA2

Wahlund’s effect• reduction in heterozygosity in the meta-population due to

population substructure

– suppose each subpopulation is in HWE• p2, 2pq, q2

– suppose allele frequencies differ among subpopulations

– meta-population is not in HWE• it has an excess of homozygotes

Sidebar

Wahlund’s effect• reduction in heterozygosity in the meta-population due to

population substructure

– suppose each subpopulation is in HWE• p2, 2pq, q2

– suppose allele frequencies differ among subpopulations

– meta-population is not in HWE• it has an excess of homozygotes

Sidebar

extreme case:

subpop. 1 in HWE:p1 = 1.0

p2 = 100%2pq = 0q2 = 0

subpop. 2 in HWE:p2 = 0

p2 = 02pq = 0q2 = 100%

meta-pop.not in HWE

Summary

• Migration– from meta-population viewpoint:

homogenizing – from subpopulation viewpoint: source of

variation• general model

– island model– one- and two-way racism models