PBG 650 Advanced Plant Breeding Module 1: Introduction Population Genetics – Hardy Weinberg Equilibrium – Linkage Disequilibrium

“The science, art, and business of improving plants for human benefit” Considerations: –Crop(s) –Production practices –End-use(s) –Target environments –Type of cultivar(s) –Traits to improve –Breeding methods –Source germplasm –Time frame –Varietal release and intellectual property rights Plant Breeding Bernardo, Chapter 1

Plant Breeding A common mistake that breeders make is to improve productivity without sufficient regard for other characteristics that are important to producers, processors and consumers.  Well-defined Objectives  Good Parents  Genetic Variation  Good Breeding Methods  Functional Seed System  Adoption of Cultivars by Farmers

Quantitative Traits Continuum of phenotypes (metric traits) Often many genes with small effects Environmental influence is greater than for qualitative traits Specific genes and their mode of inheritance may be unknown Analysis of quantitative traits –population parameters means variances –molecular markers linked to QTL

Populations In the genetic sense, a population is a breeding group –individuals with different genetic constitutions –sharing time and space In animals, mating occurs between individuals –‘Mendelian population’ –genes are transmitted from one generation to the next In plants, there are additional ways for a population to survive –self-fertilization –vegetative propagation Definition of ‘population’ may be slightly broader for plants –e.g., lines from a germplasm collection Falconer, Chapt. 1; Lynch and Walsh, Chapt. 4

Study genes in populations –Frequency and interaction of alleles –Mating patterns, genotype frequencies –Gene flow –Selection and adaptation vs random genetic drift –Genetic diversity and relationship –Population structure Related Fields –Evolutionary Biology – e.g., crop domestication –Landscape Genetics What do population geneticists do?

Gene and genotype frequencies AllelesGenotypes A1A1 A2A2 A1A1A1A1 A1A2A1A2 A2A2A2A2 FrequenciespqP 11 P 12 P 22 # Individuals Proportions For a population of diploid organisms: p + q = 1 P 11 + P 12 + P 22 = 1 Bernardo, Chapter 2

Gene frequencies (another way) AllelesGenotypes A1A1 A2A2 A1A1A1A1 A1A2A1A2 A2A2A2A2 FrequenciespqP 11 P 12 P 22 # Individuals Proportions Number of individuals = N = N 11 + N 12 + N 22 = 100 Number of alleles = 2N = N 1 + N 2 = 200

Allele frequencies in crosses Inbred x inbred Alleles are unknown, but allele frequencies at segregating loci are known F 1 and F 2 : p = q = 0.5 pq BC BC BC BC Value of q is reduced by ½ in each backcross generation

Factors that may change gene frequencies Population size –changes may occur due to sampling  assume ‘large’ population Differences in fertility and viability –parents may differ in fertility –gametes may differ in viability –progeny may differ in survival rate  assume no selection Migration and mutation  assume no migration and no mutation

Factors that may change genotype frequencies Changes in genotype frequency (not gene frequency) Mating system –assortative or disassortative mating –selfing –geographic isolation  assume that mating occurs at random (panmixia)

Hardy-Weinberg Equilibrium Assumptions –large, random-mating population –no selection, mutation, migration –normal segregation –equal gene frequencies in males and females –no overlap of generations (no age structure) Note that assumptions only need to be true for the locus in question  Gene and genotype frequencies remain constant from one generation to the next  Genotype frequencies in progeny can be predicted from gene frequencies of the parents  Equilibrium attained after one generation of random mating

Hardy-Weinberg Equilibrium Genes in parentsGenotypes in progeny A1A1 A2A2 A1A1A1A1 A1A2A1A2 A2A2A2A2 FrequenciespqP 11 = p 2 P 12 = 2pqP 22 = q 2 Example Expected genotype frequencies are obtained by expanding the binomial (p + q) 2 = p 2 + 2pq + q 2 = 1 A1A1 A2A2 A1A1 A2A2 p 2 =.16pq=.24p = 0.4 q = 0.6 q 2 =.36 pq=.24

Equilibrium with multiple alleles For multiple alleles, expected genotype frequencies can be found by expanding the multinomial (p 1 + p 2 + ….+ p n ) 2 For example, for three alleles: Lynch and Walsh (pg 57) describe equilibrium for autopolyploids Corresponding genotypes: A 1 A 1 A 1 A 2 A 1 A 3 A 2 A 2 A 2 A 3 A 3 A 3

Relationship between gene and genotype frequencies f(A 1 A 2 ) has a maximum of 0.5, which occurs when p=q=0.5 Most rare alleles occur in heterozygotes Implications for –F 1 ? –F 2 ? –Any BC? A2A2A2A2 A1A1A1A1 A1A2A1A2

Applications of the Hardy-Weinberg Law Predict genotype frequencies in random-mating populations Use frequency of recessive genotypes to estimate the frequency of a recessive allele in a population –Example: assume that the incidence of individuals homozygous for a recessive allele is about 1/11,000. q 2 = 1/11,000 q  Estimate frequency of individuals that are carriers for a recessive allele p = = pq =  2%

Testing for Hardy-Weinberg Equilibrium All genotypes must be distinguishable GenotypesGene frequencies A1A1A1A1 A1A2A1A2 A2A2A2A2 A1A1 A2A2 Observed Expected N = N 11 + N 12 + N 22 = = 747

Chi-square test for Hardy-Weinberg Equilibrium Accept H 0 : no reason to think that assumptions for Hardy- Weinberg equilibrium have been violated –does not tell you anything about the fertility of the parents When you reject H 0, there is an indication that one or more of the assumptions is not valid –does not tell you which assumption is not valid Example in Excel only 1 df because gene frequencies are estimated from the progeny data

Exact Test for Hardy-Weinberg Equilibrium Chi-square is only appropriate for large sample sizes If sample sizes are small or some alleles are rare, Fisher’s Exact test is a better alternative –Calculate the probability of all possible arrays of genotypes for the observed numbers of alleles –Rank outcomes in order of increasing probability –Reject those that constitute a cumulative probability of <5% Example in Excel Weir (1996) Chapt. 3

Likelihood Ratio Test Maximum of the likelihood function given the data (z) when some parameters are assigned hypothesized values Maximum of the likelihood function given the data (z) when there are no restrictions When the hypothesis is true:   2 df=#parameters assigned values Likelihood ratio tests for multinomial proportions are often called G-tests (for goodness of fit) Lynch and Walsh Appendix 4

Likelihood Ratio Test for HWE where is the expected number and is the observed number of the ij th genotype Calculations in Excel

Gametic phase equilibrium Lynch and Walsh, pg ; Falconer, pg A a Bb P AB P Ab P aB P ab pApA papa pBpB pbpb Random association of alleles at different loci (independence) P AB =p A p B Disequilibrium D AB = P AB – p A p B D AB = P AB P ab – P Ab P aB D AB = 0.40 – 0.5*0.5 = 0.15 D AB = 0.4*0.4 – 0.1*0.1 = 0.15 Bb A a

Linkage Disequilibrium Nonrandom association of alleles at different loci –the covariance in frequencies of alleles between the loci Refers to frequencies of alleles in gametes (haplotypes) May be due to various causes in addition to linkage –‘gametic phase disequilibrium’ is a more accurate term –‘linkage disequilibrium’ (LD) is widely used to describe associations of alleles in the same or in different linkage groups

Linkage Disequilibrium Gametic typesABAbaBab Observed P AB P Ab P aB P ab Expected pA pBpA pB pA pbpA pb pa pBpa pB pa pbpa pb Disequilibrium+D-D +D Excess of coupling phase gametes  +D Excess of repulsion phase gametes  -D

Sources of linkage disequilibrium Linkage Multilocus selection (particularly with epistasis) Assortative mating Random drift in small populations Bottlenecks in population size Migration or admixtures of different populations Founder effects Mutation

Two locus equilibrium For two loci, it may take many generations to reach equilibrium even when there is independent assortment and all other conditions for equilibrium are met –New gamete types can only be produced when the parent is a double heterozygote A A B b 0.5 AB 0.5 Ab A a B b 0.25 AB0.25 aB 0.25 Ab0.25 ab

Decay of linkage disequilibrium In the absence of linkage, LD decays by one-half with each generation of random mating c = recombination frequency

Factors that delay approach to equilibrium Linkage Selfing – because it decreases the frequency of double heterozygotes Small population size – because it reduces the likelihood of obtaining rare recombinants

Implications for breeding Gametic Phase Disequilibrium that is not due to linkage is eliminated by making the F 1 cross Recombination occurs during selfing There would be greater recombination with additional random mating, but it may not be worth the time and resources P1 P2 A 1 A 1 B 1 B 1 x A 2 A 2 B 2 B 2 F 1 A 1 A 2 B 1 B 2 gamete frequency A 1 B 1 0.5*(1-c) A 1 B 2 0.5*c A 2 B 1 0.5*c A 2 B 2 0.5*(1-c)

Effect of mating system on LD decay c = effective recombination rate s = the fraction of selfing no linkage 99% selfing outcrossing

Alternative measures of LD D is the covariance between alleles at different loci Maximum values of D depend on allele frequencies It is convenient to consider r 2 to be the square of the correlation coefficient, but it can only obtain a value of 1 when allele frequences at the two loci are the same r 2 indicates the degree of association between alleles at different loci due to various causes (linkage, mutation, migration)

D – minimum and maximum values Bb AP AB = p A p B + DP Ab = p A p b - DpApA aP aB = p a p B - DP ab = p a p b + Dpapa pBpB pbpb If D>0 Look for the maximum value D can have P Ab = p A p b - D  0  D  p A p b P aB = p a p B - D  0  D  p a p B D  min(p A p b, p a p B ) If D<0 Look for the minimum value D can have P AB = p A p B + D  0  D  -p A p B P ab = p a p b + D  0  D  -p a p b D  max(-p A p B, -p a p b ) fyi

Alternative measures of LD D’ is scaled to have a minimum of 0 and a maximum of 1 D’ indicates the degree to which gametes exhibit the maximum potential disequilbrium for a given array of allele frequencies D’=1 indicates that one of the haplotypes is missing D’ is very unstable for small sample sizes, so r 2 is more widely utilized to measure LD When D AB > 0 When D AB < 0 fyi

Testing for gametic phase disequilibrium Best when you can determine haplotypes –inbred lines or doubled haploids –haplotypes of double heterozygotes inferred from progeny tests Use a Goodness of Fit test if the sample size is large –Chi-square –G-test (likelihood ratio) Use Fisher’s exact test for smaller sample sizes Use a permutation test for multiple alleles Need a fairly large sample to have reasonable power for LD (~200 individuals or more) See Weir (1996) pg for more information

Depiction of Linkage Disequilibrium Flint-Garcia et al., Annual Review of Plant Biology 54: Disequilibrium matrix for polymorphic sites within sh1 in maize r2r2 Prob value Fisher’s Exact Test

Extent of LD in Maize Linkage disequillibrium across the 10 maize chromosomes measured with 914 SNPs in a global collection of 632 maize inbred lines. Yan et al PLoS ONE 4(12): e8451 r2r2 Average LD decay distance is 5–10 kb

Extent of LD in Barley Average LD decay distance is ~5 cM Waugh et al., 2009, Current Opinion in Plant Biology 12: r2r2 No adjustment for population structure  Adjusted for population structure  Other studies Wild barley – LD decays within a gene Landraces ~ 90 kb European germplasm - significant LD: mean 3.9 cM, median 1.16 cM Elite North American Barley

References on linkage disequilibrium Flint-Garcia et al., Structure of linkage disequilibrium in plants. Annual Review of Plant Biology 54: 357–374. Gupta et al., Linkage disequilibrium and association studies in higher plants: present status and future prospects. Plant Molecular Biology 57: 461–485. Mangin et al., Novel measures of linkage disequilibrium that correct the bias due to population structure and relatedness. Heredity 108: 285–291. Slatkin, M Linkage disequilibrium – understanding the evolutionary past and mapping the medical future. Nature Reviews Genetics 9: 477–485. Waugh, R., Jean-Luc Jannink, G.J. Muehlbauer, L. Ramsay The emergence of whole genome association scans in barley. Current Opinion in Plant Biology 12(2): 218–222. Yan, J., T. Shah, M.L Warburton, E.S. Buckler, M.D. McMullen, et al Genetic characterization and linkage disequilibrium estimation of a global maize collection using SNP Markers. PLoS ONE 4(12): e8451. Zhu et al., Status and prospects of association mapping in plants. The Plant Genome 1: 5–20.