Lecture 1 Introduction I: Pedigrees, genetics and ... · Lecture 1 Introduction I: Pedigrees,...
Transcript of Lecture 1 Introduction I: Pedigrees, genetics and ... · Lecture 1 Introduction I: Pedigrees,...
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Lecture 1
Introduction I: Pedigrees, genetics and probabilities
Magnus Dehli Vigeland
Statistical methods in genetic relatedness and pedigree analysis
NORBIS course, 6th – 10th of January 2020, Oslo
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Outline
• Part I: Pedigrees
– Pedigree symbols and terminology
– Some common relationships
• Part II: Genetics
– Terminology
Locus, allele, genotype, marker
– Mendelian inheritance
Autosomal, X, Y
• Part III: Pedigree likelihoods
– Motivation: Real-life problems
– Ingredients:
Hardy-Weinberg equilibrium
Mendelian transition probabilities
– Likelihoods by hand
– Computer algorithms
Statistical methods in genetic relatedness and pedigree analysis
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Outline
• Part I: Pedigrees
– Pedigree symbols and terminology
– Some common relationships
• Part II: Genetics
– Terminology
Locus, allele, genotype, marker
– Mendelian inheritance
Autosomal, X, Y
• Part III: Pedigree likelihoods
– Motivation: Real-life problems
– Ingredients:
Hardy-Weinberg equilibrium
Mendelian transition probabilities
– Likelihoods by hand
– Computer algorithms
Statistical methods in genetic relatedness and pedigree analysis
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Pedigrees: Symbols and terminology
Statistical methods in genetic relatedness and pedigree analysis
= male
= female
Founders: No parents included
in the pedigree
Nonfounders
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Pedigrees: Symbols and terminology
Statistical methods in genetic relatedness and pedigree analysis
= male
= female
Consanguineous marriage
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Pedigrees: Symbols and terminology
Statistical methods in genetic relatedness and pedigree analysis
● ●
●
●
= affected
= unaffected
= carrier of disease allele●
Medical pedigrees:
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Alternative ways of drawing pedigrees
Statistical methods in genetic relatedness and pedigree analysis
Standard
1
3
5
Simplified Directed acyclic graph
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Statistical methods in genetic relatedness and pedigree analysis
Some common relationships
(and some less common...)
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Cousin relationships
Statistical methods in genetic relatedness and pedigree analysis
Full siblings
First cousins
Second cousins
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Cousin relationships
Statistical methods in genetic relatedness and pedigree analysis
First cousins once removed
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Cousin relationships
Statistical methods in genetic relatedness and pedigree analysis
Aunt - nephew
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Cousin relationships
Statistical methods in genetic relatedness and pedigree analysis
Grandaunt
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Half cousin relationships
Statistical methods in genetic relatedness and pedigree analysis
Half siblings (paternal)
Half first cousins
Half second cousins
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Half cousin relationships
Statistical methods in genetic relatedness and pedigree analysis
Half aunt /half nephew
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Half cousin relationships
Statistical methods in genetic relatedness and pedigree analysis
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More complicated relationships
Statistical methods in genetic relatedness and pedigree analysis
3/4 siblings
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What about this?
Double first cousins
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Quadruple half first cousins!
The connoisseur's favourite!
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Outline
• Part I: Pedigrees
– Pedigree symbols and terminology
– Some common relationships
• Part II: Genetics
– Terminology
Locus, allele, genotype, marker
– Mendelian inheritance
Autosomal, X, Y
• Part III: Pedigree likelihoods
– Motivation: Real-life problems
– Ingredients:
Hardy-Weinberg equilibrium
Mendelian transition probabilities
– Likelihoods by hand
– Computer algorithms
Statistical methods in genetic relatedness and pedigree analysis
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Genetics
• Human genome:
– Diploid
– 22 pairs of autosomes
– Sex chroms: X and Y
• Some important terms
– Locus
– Allele
– Genotype
– Genetic markers
SNPs
microsatellites
Statistical methods in genetic relatedness and pedigree analysis
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Locus, allele, genotype
Statistical methods in genetic relatedness and pedigree analysis
M F
BA locus
alleles
Homologous chromosomes
• LOCUS = a specific place in the genome, e.g. a base pair, a gene or a region
• ALLELE = any of the alternative forms of a locus
• GENOTYPE = the set of alleles carried by an individual at a given locus
genotype: A/B
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Genetic markers
• Small parts of the genome which ...– have known position
– vary in the population
– are easy to genotype
• SNPs (single nucleotide polymorphisms)– two alleles
– usual requirement: MAF > 1%
– very common in the genome (millions!)
– used in medical genetics +++
• STRs (short tandem repeats) = microsatellites
– consecutive repeats of 2-5 bases
– multiallelic: 5 - 50 alleles
– allele names: # repeats
– used in forensics
Statistical methods in genetic relatedness and pedigree analysis
...CCGTTATATGGGC...
...CCGTTAGATGGGC...
...CCGTTATATGGGC...
...CCGTTATATGGGC...
...CCGTTAGATGGGC...
= minor allele frequency
...ACG TTAG TTAG TTAG TTAG AAC..
...ACG TTAG TTAG AAC..
...ACG TTAG TTAG TTAG TTAG TTAG AAC..
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Outline
• Part I: Brief introductions
– Pedigrees symbols and terminology
– Some common relationships
– Genetics
Locus, allele, genotype, marker
– Mendelian inheritance
Autosomal
X, Y
• Part II: Pedigree likelihoods
– Motivation: Real-life problems
– Ingredients:
Hardy-Weinberg equilibrium
Mendelian transition probabilities
– Likelihoods by hand
– Computer algorithms
Statistical methods in genetic relatedness and pedigree analysis
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Mendelian inheritance: Autosomal (chromosomes 1-22)
Statistical methods in genetic relatedness and pedigree analysis
A/A B/C
A/B A/B
B/C
A/C
Example: autosomal marker with 3 alleles: A, B, C
homozygous heterozygous
Probability of transmitting either allele: always 50%
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Mendelian inheritance: X-linked
Statistical methods in genetic relatedness and pedigree analysis
A B/C
A/C C
A
A/B
Example: X-linked marker with 3 alleles: A, B, C
no transmisison from father to son
males are hemizygous
forced transmission from father to daughter
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Mendelian inheritance: Y-linked
Statistical methods in genetic relatedness and pedigree analysis
A
B
B
Example: Y-linked marker with 2 alleles: A, B
• no transmission involving females
• father-son forced
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Assumptions throughout (most of) this course
• Diploid species
• No cytogenetic abnormalitites
• No de novo mutations
Statistical methods in genetic relatedness and pedigree analysis
COFFEE BREAK!
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Outline
• Part I: Pedigrees
– Pedigree symbols and terminology
– Some common relationships
• Part II: Genetics
– Terminology
Locus, allele, genotype, marker
– Mendelian inheritance
Autosomal, X, Y
• Part III: Pedigree likelihoods
– Motivation: Real-life problems
– Ingredients:
Hardy-Weinberg equilibrium
Mendelian transition probabilities
– Likelihoods by hand
– Computer algorithms
Statistical methods in genetic relatedness and pedigree analysis
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Questions related to pedigrees with genotypes
• Will my child have the disease?
• Is NN the true father?
• Brothers or half brothers?
• Is NN related to this family? How?
• Predict the missing genotype?
Statistical methods in genetic relatedness and pedigree analysis
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Questions related to pedigrees with genotypes
Statistical methods in genetic relatedness and pedigree analysis
? D
Disease locus:alleles D and N
Will my child have the disease?
D/N
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Questions related to pedigrees with genotypes
Statistical methods in genetic relatedness and pedigree analysis
Who is the true father?
13/18
1 2
11/18
11/13 -/-
Suppose:• 11 is common• 18 is rare
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Questions related to pedigrees with genotypes
Statistical methods in genetic relatedness and pedigree analysis
Brothers or half brothers?
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Questions related to pedigrees with genotypes
Statistical methods in genetic relatedness and pedigree analysis
Is this woman related to the family?
12/1432/407/116/21
12/1634/40
7/712/21 11/16
32/417/136/25
11/1432/4013/136/25
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Questions related to pedigrees with genotypes
Statistical methods in genetic relatedness and pedigree analysis
Can we predict the missing genotype?
A/B A/A
A/A A/B
A/B
?/?
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• Common to all of these: The need to calculate probabilities
• Called the likelihood of the pedigree.
Statistical methods in genetic relatedness and pedigree analysis
P( genotypes | pedigree, marker info, allele freqs, .. )
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Ingredients for likelihood computations
Statistical methods in genetic relatedness and pedigree analysis
A/B A/A
A/A A/A
A/B
-/-
founder probabilities
untypedindividuals
transition probabilities
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Ingredient 1: Founder probabilities
• Suppose the allele frequencies are:
𝑃 𝐴 = 𝑝𝑃 𝐵 = 𝑞
• What are the frequencies of the genotypes AA, AB, BB?
• Under certain assumptions, the alleles can be treated as independent:
Statistical methods in genetic relatedness and pedigree analysis
𝑃(𝐴𝐴) = 𝑃 𝐴 ∗ 𝑃 𝐴 = 𝑝2
𝑃 𝐵𝐵 = 𝑃 𝐵 ∗ 𝑃 𝐵 = 𝑞2
𝑃 𝐴𝐵 = 𝑃 𝐴𝐵 𝑜𝑟 𝐵𝐴 = 𝑝𝑞 + 𝑞𝑝 = 2𝑝𝑞
two possible orderings!
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The Hardy-Weinberg principle
Assumptions:
• infinite population
• random mating
• no selection
• no mutations
• no migration
Hardy (1908): Shows
«... using a little mathemathics of the multiplication table kind»:
• allele freqs are unchanged from generation to generation
• after 1 generation the genotype freqs stay unchanged
Statistical methods in genetic relatedness and pedigree analysis
𝑝2
AA
2𝑝𝑞
AB
𝑞2
BB
𝑝A 𝑞B
𝑝2
AA
2𝑝𝑞
AB
𝑞2
BB
𝑝A 𝑞B
∗
AA
∗
AB
∗
BB
𝑝A 𝑞B
HW equilibrium𝑃 𝐴𝐴 = 𝑝2
𝑃 𝐴𝐵 = 2𝑝𝑞𝑃(𝐵𝐵) = 𝑞2
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Statistical methods in genetic relatedness and pedigree analysis
Allele frequencies
Genotype frequencies
assuming HWE
𝑝 = 𝑝𝐴𝐴 + 0.5 𝑝𝐴𝐵𝑞 = 𝑝𝐵𝐵 + 0.5 𝑝𝐴𝐵
𝑝𝐴𝐴 = 𝑝2
𝑝𝐴𝐵 = 2𝑝𝑞𝑝𝐵𝐵 = 𝑞2
always
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Ingredient 2: Transition probabilities
• Easy - follows directly from Mendel's laws!
Statistical methods in genetic relatedness and pedigree analysis
childparents
A/A AB BB
AA×AA 1 0 0
AA×AB 0.5 0.5 0
AA×BB 0 1 0
AB×AA 0.5 0.5 0
AB×AB 0.25 0.5 0.25
AB×BB 0 0.5 0.5
BB×AA 0 1 0
BB×AB 0 0.5 0.5
BB×BB 0 0 1
A/B
-/-
A/A
P(gchild | gparents)
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Example
Statistical methods in genetic relatedness and pedigree analysis
1 2
3
A/B
A/B
A/A
L = 𝑃 𝑔1, 𝑔2, 𝑔3
= 𝑃 𝑔1) ∙ 𝑃(𝑔2) ∙ 𝑃(𝑔3 | 𝑔1, 𝑔2
= 𝑃 𝐴𝐴 ∙ 𝑃 𝐴𝐵 ∙ 𝑃 𝐴𝐵 parents = 𝐴𝐴 × 𝐴𝐵)
= 𝑝2 ∙ 2𝑝𝑞 ∙ 0.5
= 𝑝3𝑞
assuming HWE!
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Example on X
Statistical methods in genetic relatedness and pedigree analysis
𝐿 = 𝑃 𝑔𝑒𝑛𝑜𝑡𝑦𝑝𝑒𝑠 | 𝑝𝑒𝑑𝑖𝑔𝑟𝑒𝑒, 𝑝, 𝑞
= 𝑝 ∙ 2𝑝𝑞 ∙ 0.5 ∙ 0.5 ∙ 𝑞2 ∙ 1
= 0.5 𝑝2 𝑞3
21
4
A/B
A/B
A
3
6
5
B
B/B
B/B
1 2 3 4 5 6 contribution from each individual
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Ingredient 3: How to deal with untyped individuals
Solution: Sum of all possible genotypes for the untyped
Statistical methods in genetic relatedness and pedigree analysis
1 2
3
-/-
A/B
A/A
𝑃 𝑔1, 𝑔3 =
𝑔2
𝑃(𝑔1, 𝑔2, 𝑔3) =
𝑔2
𝑃 𝑔1) ∙ 𝑃(𝑔2) ∙ 𝑃(𝑔3|𝑔1, 𝑔2
= 𝑃(𝐴𝐴) ∙ 𝑃(𝐴𝐴) ∙ 𝑃 𝐴𝐵 𝐴𝐴 × 𝐴𝐴) + 𝑃(𝐴𝐴) ∙ 𝑃(𝐴𝐵) ∙ 𝑃 𝐴𝐵 𝐴𝐴 × 𝐴𝐵) + 𝑃(𝐴𝐴) ∙ 𝑃(𝐵𝐵) ∙ 𝑃 𝐴𝐵 𝐴𝐴 × 𝐵𝐵)
= 𝑝2 ∙ 𝑝2 ∙ 0 + 𝑝2 ∙ 2𝑝𝑞 ∙ 0.5 + 𝑝2∙ 𝑞2 ∙ 1
= 𝑝3𝑞 + 𝑝2𝑞2 = 𝑝2𝑞 𝑝 + 𝑞 = 𝑝2𝑞
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Pedigree likelihood: General formula
• Given:
– pedigree with n individuals
– k members are genotyped: g1, g2, …, gk
• Then:
𝑃 𝑔1, ..., 𝑔𝑘 =
𝐺1
𝐺2
…
𝐺𝑛
𝑃 𝑔1 ⋯𝑃 𝑔𝑗 ∙ 𝑃 𝑔𝑗+1 𝑝𝑎𝑟 ⋯𝑃 𝑔𝑛 𝑝𝑎𝑟
• If everyone is typed: Only one term → easy
• Number of terms grows exponentially in #(untyped)
– but clever algorithms exist!
Statistical methods in genetic relatedness and pedigree analysis
founders non-founders
Gi = set of possible genotypes for individual i
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Computer algorithms for pedigree likelihoods
• Elston-Stewart algorithm
– a peeling algorithm
– linear in pedigree size!
• Lander-Green
– based on inheritance vectors
– hidden Markov model
– best choice with many linked markers
– small/medium pedigrees only
Statistical methods in genetic relatedness and pedigree analysis
A/B A/A
A/A A/A
A/B
-/-
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Software
• R/pedprobr
– Part of the ped suite
– Elston-Stewart
– general likelihoods, inbreeding, genotype distributions ++
• Familias
– GUI for forensic applications
– Elston-Stewart
– handles mutations, HW deviations, ++
• MERLIN
– command line program
– Lander-Green
– medical applications: multipoint linkage
Statistical methods in genetic relatedness and pedigree analysis