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 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 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 Founders: Pedigrees: Symbols and terminology No parents included in the pedigree = male = female Nonfounders Statistical methods in genetic relatedness and pedigree analysis Pedigrees: Symbols and terminology = male = female Consanguineous marriage Statistical methods in genetic relatedness and pedigree analysis Pedigrees: Symbols and terminology ● ● Medical pedigrees: ● ● = affected = unaffected ● = carrier of disease allele Statistical methods in genetic relatedness and pedigree analysis Alternative ways of drawing pedigrees 1 3 5 Standard Simplified Directed acyclic graph Statistical methods in genetic relatedness and pedigree analysis Some common relationships (and some less common...) Statistical methods in genetic relatedness and pedigree analysis Cousin relationships Full siblings First cousins Second cousins Statistical methods in genetic relatedness and pedigree analysis Cousin relationships First cousins once removed Statistical methods in genetic relatedness and pedigree analysis Cousin relationships Aunt - nephew Statistical methods in genetic relatedness and pedigree analysis Cousin relationships Grandaunt Statistical methods in genetic relatedness and pedigree analysis Half cousin relationships Half siblings (paternal) Half first cousins Half second cousins Statistical methods in genetic relatedness and pedigree analysis Half cousin relationships Half aunt / half nephew Statistical methods in genetic relatedness and pedigree analysis Half cousin relationships Statistical methods in genetic relatedness and pedigree analysis More complicated relationships 3/4 siblings Statistical methods in genetic relatedness and pedigree analysis What about this? Double first cousins The connoisseur's favourite! Quadruple half first cousins! 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 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 Locus, allele, genotype M F alleles A B locus genotype: A/B 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 Statistical methods in genetic relatedness and pedigree analysis Genetic markers • Small parts of the genome which ... – have known position – vary in the population – are easy to genotype • SNPs (single nucleotide polymorphisms) ...CCGTTATATGGGC... – two alleles = minor allele frequency ...CCGTTAGATGGGC... – usual requirement: MAF > 1% ...CCGTTATATGGGC... – very common in the genome (millions!) ...CCGTTATATGGGC... – used in medical genetics +++ ...CCGTTAGATGGGC... • STRs (short tandem repeats) = microsatellites – consecutive repeats of 2-5 bases ...ACG TTAG TTAG TTAG TTAG AAC.. – multiallelic: 5 - 50 alleles ...ACG TTAG TTAG AAC.. – allele names: # repeats ...ACG TTAG TTAG TTAG TTAG TTAG AAC.. – used in forensics Statistical methods in genetic relatedness and pedigree analysis 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 Mendelian inheritance: Autosomal (chromosomes 1-22) Example: autosomal marker with 3 alleles: A, B, C homozygous A/A B/C heterozygous A/B A/C A/B B/C Probability of transmitting either allele: always 50% Statistical methods in genetic relatedness and pedigree analysis Mendelian inheritance: X-linked Example: X-linked marker with 3 alleles: A, B, C males are A B/C hemizygous A/C A/B C A no transmisison forced transmission from father to son from father to daughter Statistical methods in genetic relatedness and pedigree analysis Mendelian inheritance: Y-linked Example: Y-linked marker with 2 alleles: A, B A B • no transmission involving females B • father-son forced Statistical methods in genetic relatedness and pedigree analysis Assumptions throughout (most of) this course • Diploid species • No cytogenetic abnormalitites • No de novo mutations COFFEE BREAK! Statistical methods in genetic relatedness and pedigree analysis 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 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 Questions related to pedigrees with genotypes D/N Disease locus: ? D alleles D and N Will my child have the disease? Statistical methods in genetic relatedness and pedigree analysis Questions related to pedigrees with genotypes 1 2 11/13 -/- 13/18 11/18 Suppose: • 11 is common • 18 is rare Who is the true father? Statistical methods in genetic relatedness and pedigree analysis Questions related to pedigrees with genotypes Brothers or half brothers? Statistical methods in genetic relatedness and pedigree analysis Questions related to pedigrees with genotypes 12/14 32/40 7/11 6/21 11/14 32/40 13/13 6/25 12/16 34/40 7/7 12/21 11/16 32/41 7/13 6/25 Is this woman related to the family? Statistical methods in genetic relatedness and pedigree analysis Questions related to pedigrees with genotypes A/B A/A A/A ?/? A/B A/B Can we predict the missing genotype? Statistical methods in genetic relatedness and pedigree analysis • Common to all of these: The need to calculate probabilities P( genotypes | pedigree, marker info, allele freqs, .. ) • Called the likelihood of the pedigree. Statistical methods in genetic relatedness and pedigree analysis Ingredients for likelihood computations founder probabilities A/B A/A A/A -/- A/A transition probabilities A/B untyped individuals Statistical methods in genetic relatedness and pedigree analysis Ingredient 1: Founder probabilities • Suppose the allele frequencies are: 푃 퐴 = 푝 푃 퐵 = 푞 • What are the frequencies of the genotypes AA, AB, BB? • Under certain assumptions, the alleles can be treated as independent: 푃(퐴퐴) = 푃 퐴 ∗ 푃 퐴 = 푝2 푃 퐵퐵 = 푃 퐵 ∗ 푃 퐵 = 푞2 푃 퐴퐵 = 푃 퐴퐵 표푟 퐵퐴 = 푝푞 + 푞푝 = 2푝푞 two possible orderings! Statistical methods in genetic relatedness and pedigree analysis The Hardy-Weinberg principle Assumptions: A 푝 B 푞 • infinite population • random mating ∗ ∗ ∗ • no selection AA AB BB • no mutations • no migration A 푝 B 푞 Hardy (1908): Shows 2 2 «... using a little mathemathics of the multiplication table kind»: 푝 2푝푞 푞 AA AB BB • allele freqs are unchanged from generation to generation • after 1 generation the genotype freqs stay unchanged A 푝 B 푞 2 2 푃 퐴퐴 = 푝2 푝 2푝푞 푞 푃 퐴퐵 = 2푝푞 HW equilibrium AA AB BB 푃(퐵퐵) = 푞2 Statistical methods in genetic relatedness and pedigree analysis 2 푝퐴퐴 = 푝 푝퐴퐵 = 2푝푞 2 푝퐵퐵 = 푞 assuming HWE Allele Genotype frequencies frequencies always 푝 = 푝퐴퐴 + 0.5 푝퐴퐵 푞 = 푝퐵퐵 + 0.5 푝퐴퐵 Statistical methods in genetic relatedness and pedigree analysis Ingredient 2: Transition probabilities P(gchild | gparents) • Easy - follows directly from Mendel's laws! child A/A AB BB parents AA×AA 1 0 0 AA×AB 0.5 0.5 0 A/A A/B 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 Statistical methods in genetic relatedness and pedigree analysis Example 1 2 A/A A/B 3 A/B L = 푃 푔1, 푔2, 푔3 = 푃 푔1) ∙ 푃(푔2)