Lecture 3: Allele Frequencies and Hardy-Weinberg

Equilibrium August 27, 2012

Last Time  Review of genetic variation and

Mendelian Genetics

 Methods for detecting variation  Morphology  Allozymes  DNA Markers

 Anonymous  Sequence-tagged

Today  Sequence probability calculation

 Molecular markers: DNA sequencing

 Introduction to statistical distributions

 Estimating allele frequencies

 Introduction to Hardy-Weinberg Equilibrium

 Using Hardy-Weinberg: Estimating allele frequencies for dominant loci

If nucleotides occur randomly in a genome, which sequence should occur more

frequently? AGTTCAGAGT

AGTTCAGAGTAACTGATGCT

What is the expected probability of each sequence to occur once?

How many times would each sequence be expected to occur by chance in a 100 Mb

genome?

AGTTCAGAGT

What is the expected probability of each sequence to occur once?

What is the sample space for the first position? A

T

G C

Probability of “A” at that position? 41

Probability of “A” at position 1, “G” at position 2, “T” at position 3, etc.?

710 1054.925.041

41

41

41

41

41

41

41

41

41 −== xxxxxxxxxx

AGTTCAGAGTAACTGATGCT 1320 1009.925.0 −= x

AGTTCAGAGT

How many times would each sequence be expected to occur in a 100 Mb genome?

( )( ) 4.95101054.9 87 =−x

AGTTCAGAGTAACTGATGCT

( )( ) 5813 101.9101009.9 −− = xx

Why is this calculation wrong?

),()|()( BPBAPBAP =∩),()()()( BAPBPAPBAP ∩−+=∪

A B

AGTTCAGAGTAACTGATGCT AGT TCA GAG TAA CTG ATG CT

UCA AGU CUC AUU GAC UAC GA

Ser Cys Phe Ile Asp Tyr

UGA AGU CUC AUU GAC UAG GA Stop Cys Phe Ile Asp Stop

DNA Sequencing   Direct determination of

sequence of bases at a location in the genome

  Shotgun versus PCR sequencing

  Dye terminators (Sanger) and capillaries revolutionized DNA sequencing

  Modern sequencing methods (sequencing by synthesis, pyrosequencing) have catapulted sequencing into realm of population genetics

  Human genome took 10 years to sequence originally, and hundreds of millions of dollars

  Now we can do it in a week for <$2,000

SNPs   A Single Nucleotide Polymorphism

(SNP) is a single base mutation in DNA.

  The most common source of genetic polymorphism (e.g., 90% of all human DNA polymorphisms).

  Identify SNP by screening a sample of individuals from study population: usually 16 to 48

  Once identified, SNP are assayed in populations using high-throughput methods

Genotyping by Sequencing   New sequencing methods generate 10’s of millions of short sequences

per run

  Combine restriction digests with sequencing and pooling to genotype thousands of markers covering genome at very high density

http://www.maizegenetics.net/images/stories/GBS_CSSA_101102sem.pdf

Generate 10’s of thousands of markers for <$100 per sample

Presence-Absence Polymorphism

SNP

Genotyping by Sequencing Cost Example

http://www.maizegenetics.net/gbs-overview

Statistical Distributions: Normal Distribution

  Many types of estimates follow normal distribution

  Can be visualized as a frequency distribution (histogram)   Can interpret as a probability density function

Variance (Vx): A measure of the dispersion around the mean:

∑=

−−

=n

iix xx

nV

1

2)(11

Expected Value (Mean): ∑=

=n

iixn

x1

1

where n is the number of samples

Standard Deviation (sd): A measure of dispersion around the mean that is on same scale as mean

xVsd =

1 sd

2 sd

Standard Error of Mean

  Standard Deviation is a measure of how individual points differ from the mean estimates in a single sample

  Standard Error is a measure of how much the estimate differs from the true parameter value (in the case of means, µ)

  If you repeated the experiment, how close would you expect the mean estimate to be to your previous estimate?

Standard Error of the Mean (se): nVse x=

95% Confidence Interval: )(96.1 sex ±

Estimating Allele Frequencies, Codominant Loci   Measured allele frequency is maximum likelihood estimator

of the true frequency of the allele in the population (See Hedrick, pp 82-83 for derivation)

N

NNp

1211 21

+=

  Expected number of observations of allele A1: E(Y)=np

  Where n is number of samples   For diploid organisms, n = 2N , where N is number of

individuals sampled

  Expected number of observations of allele A1 is analogous to the mean of a sample from a normal distribution

  Allele frequency can also be interpreted as an estimate of the mean

  Assume a population of Mountain Laurel (Kalmia latifolia) at Cooper’s Rock, WV

Allele Frequency Example

Red buds: 5000 Pink buds: 3000 White buds: 2000

  Phenotype is determined by a single, codominant locus: Anthocyanin

  What is frequency of “red” alleles (A1), and “white” alleles (A2)?

A1A1 A1A2 A2A2

,2

221

12111211

NNN

N

NNp +

=+

=

Frequency of A1 = p

,2

221

12221222

NNN

N

NNq +

=+

=

Frequency of A2 = q

Allele Frequencies are Distributed as Binomials

  Binomials are variables that can be interpreted as the number of successes and failures in a series of trials

  Based on samples from a population

  For two-allele system, each sample is like a “trial”   Does the individual contain Allele A1?   Remember, q=1-p, so only one parameter is estimated

Number of ways of observing y positive results in n trials

Probability of observing y positive results in n trials once

,)( yny fsyn

yYP −⎟⎟⎠

⎞⎜⎜⎝

⎛==

)!(!!yny

nCyn n

y −==⎟⎟

⎠

⎞⎜⎜⎝

⎛

where s is the probability of a success, and f is the probability of a failure

Given the allele frequencies that you calculated earlier for Cooper’s Rock

Kalmia latifolia, what is the probability of observing two “white” alleles in a

sample of two plants?

Variation in Allele Frequencies, Codominant Loci   Binomial variance is pq or p(1-p)

  Variance in number of observations of A1: V(Y) = np(1-p)

  Variance in allele frequency estimates (codominant, diploid):

NppVp 2)1( −

=

  Standard Error of allele frequency estimates:

NppSEp 2)1( −

=

  Notice that estimates get better as sample size increases

  Notice also that variance is maximum at intermediate allele frequencies

Maximum variance as a function of allele frequency for a codominant locus

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p

p(1-p)

Why is variance highest at intermediate allele frequencies?

p = 0.5

If this were a target, how variable would your outcome be in each case (red versus white hits)?

Variance is constrained when value approaches limits (0 or 1)

p = 0.125

What if there are more than 2 alleles?   General formula for calculating allele frequencies in

multiallelic system with codominant alleles:

NppV ii

pi 2)1( −

=

  Variance and Standard Error of allele frequency estimates remain:

NppSE ii

pi 2)1( −

=

ijN

NNp

n

jijii

i ≠

+

=∑= ,

21

1

How do we estimate allele frequencies for dominant loci?

A2A2

Codominant locus Dominant locus

A1A1 A1A2 A2A2 -

+

A1A1 A1A2

Codominant locus Dominant locus -

+

Hardy-Weinberg Law  After one generation of random mating,

single-locus genotype frequencies can be represented by a binomial (with 2 alleles) or a multinomial function of allele frequencies

222 2)( qpqpqp ++=+

Frequency of A2A2 (Q) Frequency of A1A1 (P) Frequency of A1A2 (H)

How does Hardy-Weinberg Work?   Reproduction is a sampling process

  Example: Mountain Laurel at Cooper’s Rock Red Flowers: 5000 Pink Flowers: 3000 White Flowers: 2000

A1A1 A1A2 A2A2

Frequency of A1 = p = 0.65

Frequency of A2 = q = 0.35

: A2=14 : A1=26 Alleles:

: 4 : 10 Genotypes:

: 6 Phenotypes:

: 4 : 10 : 6

What are expected numbers of phenotypes and genotypes in a sample of 20 trees?

What are expected frequencies of alleles in pollen and ovules?

What will be the genotype and phenotype frequencies in the next

generation?

What assumptions must we make?

Hardy-Weinberg Equilibrium

 After one generation of random mating, genotype frequencies remain constant, as long as allele frequencies remain constant

 Provides a convenient Neutral Model to test for departures from assumptions

 Allows genotype frequencies to be represented by allele frequencies: simplification of calculations

Hardy-Weinberg Assumptions  Diploid

 Large population

 Random Mating: equal probability of mating among genotypes

 No mutation

 No gene flow

 Equal allele frequencies between sexes

 Nonoverlapping generations

Graphical Representation of Hardy-Weinberg Law (p+q)2 = p2 + 2pq + q2 = 1

Relationship Between Allele Frequencies and Genotype Frequencies

under Hardy-Weinberg