Quantitative Traits in Populations Genes in populations Gene and genotype frequencies Hardy-Weinberg equilibrium Linkage disequilibrium Descriptive statistics for quantitative traits Genetic variance Heritability Relationships among relatives Inbreeding Kinship Selection alpine meadow of the Albion basin in the Wasatch Mountains Photo by Teresa Prendusi, USFS

Genetics So far… Molecular genetics Mendelian genetics Molecular genetics Possible questions… How do we conserve a plant species in a germplasm collection? How do we determine the mating system of a natural population of plants? How do we know if the variation that we see in a trait is due to genetics? How can we understand the genetic basis for expression of a quantitative trait? Additional tools are often needed in applied genetics

Genetics Population genetics Quantitative genetics Genomics Describes allele and genotype frequencies in populations over space and time Models evolutionary forces (survival, mutation, migration, etc.) Models are generally limited to one or a few loci Quantitative genetics Uses statistical approaches to describe variation in quantitative traits Does not tell us much about individual genes that control a trait Genomics Interdisciplinary Investigates the combined influence of all genes to determine their influence on growth and development of the organism

Population genetics – descriptive statistics Allele frequencies 2N = 100 Genotype frequencies N = 50 A a Allele A a Total # individuals 60 40 100 p q p + q Frequency 0.6 0.4 1.0 Genotype AA Aa aa Total # individuals 18 24 8 50 PAA PAa Paa PAA+PAa+Paa Frequency 0.36 0.48 0.16 1.0 Other statistics Number of alleles = 2 Ho = observed heterozygosity = 0.48

The multiplication rule of probability The probability of two or more independent events occurring together can be obtained by multiplying the probabilities of the individual events. Where have you seen this rule applied? Predict the progeny of a dihybrid cross: AaBb x AaBb Chance of an AA offspring: Chance of an AB gamete: Chance of an AABB offspring: Probability of rolling a six = 1/6 Probability of rolling another six = 1/6 Probability of rolling two sixes = Based on Mendel’s Laws Segregation Independent Assortment

The Hardy-Weinberg principle 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 ― example Genes in parents Genotypes in progeny A1 A2 A1A1 A1A2 A2A2 Frequencies p q P11 = p2 P12 = 2pq P22 = q2 Example 0.7 0.3 0.49 0.42 0.09 Expected genotype frequencies are obtained by expanding the binomial (p + q)2 = p2 + 2pq + q2 = 1 Can be extended for multiple alleles (p + q + r)2 = 1 A1 A2 Population from the previous slide is in HWE, but that would not necessarily be the case in a sample from an actual population. A1 p2=0.49 pq=0.21 p = 0.7 A2 pq=0.21 q2=0.09 q = 0.3 p = 0.7 q = 0.3

Linkage Disequilibrium Locus 1 Locus 2 Locus 3 Locus 4 Locus 5 Locus 1 Locus 2 Locus 3 Locus 4 Locus 5 Random association of alleles Slide courtesy of Alfonso Cuesta-Marcos

Linkage Disequilibrium (LD) Gametic phase disequilibrium is a better term (but it’s too late now) Refers to associations of alleles in gametes Loci don’t have to be linked or on the same chromosome Linkage Equilibrium Random association of alleles at different loci Loci are independent A a B b PAB PAb PaB Pab pA pa pB pb B b A a .38 .12 .50 PAB = pA pB DAB = 0 Disequilibrium DAB = PAB – pA pB DAB = 0.38 – 0.50*0.50 = 0.13

Origins of Linkage Disequilibrium Mutation Selection (particularly with linkage or epistasis) Genetic drift (sampling effects in small populations) Rare recombinants may be lost due to chance Certain allelic combinations favored or lost Bottlenecks in population size Migration Admixtures of different populations

Mutation creates LD Locus 1 Locus 2 Selection for Mutation at Locus 2 Selection for New allele occurs in gametes with

Decay of Linkage Disequilibrium For a single locus, Hardy-Weinberg equilibrium can be obtained after one generation of random mating: p2 AA + 2pq Aa + q2 aa For two loci, LD decays gradually over generations, even when there is independent assortment New gamete types can only be produced when the parent is a double heterozygote A B b 0.5 AB 0.5 Ab 0.25 AB 0.25 aB 0.25 Ab 0.25 ab A a B b

Decay of linkage disequilibrium over time In the absence of linkage, LD decays by one-half with each generation of random mating Factors that reduce recombination will slow down the rate of decay Selfing Linkage c = recombination frequency Disequilibrium in the next generation Disequilibrium at generation t

Inbreeding Increases homozygosity May lead to inbreeding depression Outcrossing species suffer more, in general Causes of inbreeding Due to small population size Due to mating between relatives and/or selfing

Inbreeding Coefficient F A C B D E For an individual, F is the probability that alleles at the same locus are identical by descent Varies from zero (noninbred) to one (fully inbred) Effect of population size Change in inbreeding in a single generation Inbreeding accumulates over generations Inbreeding can be estimated as the deficiency of observed heterozygotes in a population (Ho) relative to expectation (He) F-statistics are also used to describe population structure Distinguish within-population inbreeding (FIS) from divergence among populations (FST)

Quantitative Traits In general, quantitative traits… Many exceptions Show continuous variation (metric, not discrete) Are controlled by many genes (polygenic) Influenced by the environment Mean = 30 Variance = 36 sd = 6 Many exceptions Threshhold traits Binary traits (e.g., alive or dead) Counts Ratings or scores Skewed distributions A few major genes

Single locus model -a 0 d a A2 A2 A1 A2 A1 A1 Genotypic Value Coded Genotypic Value -a 0 d a Dominance effect Additive effect -a Additive effect +a The origin ( ) is midway between the two homozygotes no dominance d = 0 partial dominance 0 < d < +a or 0 > d > –a complete dominance d = +a or d = –a overdominance d > +a or d < –a

Genetic Variance P = G + E Components of an individual’s Phenotypic Value P = phenotypic value G = genotypic value E = environmental deviation Variance Components VP = VG + VE VG = VA + VD VG = Genetic variance VP = VA + VD + VE VA = Additive genetic variance VD = Dominance variance P = G + E

Covariance between relatives Measures degree of genetic relationship and resemblance Strategy for estimating genetic variances and heritability Determine theoretical relationships, and compare to experimental observations Need to account for genetic relationships when conducting association studies (GWAS) Relationships Additive Covariance (F=0) parent : offspring 1/2 full-sibs half-sibs 1/4 uncle (aunt) : nephew (niece) grandparent : grandchild 1/8 first cousins

Kinship matrix Shows genetic covariances for all pairs of individuals Can be calculated from pedigrees Can be estimated from molecular marker scans and GBS studies T Bae, Harold & Perls, Tom & Sebastiani, Paola. (2014). Frontiers in genetics. 5. 390.

Heritability Broad sense heritability Narrow sense heritability Proportion of phenotypic variation that is due to genetic differences Varies from 0 to 100% (0 to 1) Narrow sense heritability Extent to which the phenotypic variation is determined by genes transmitted from the parents

Calculating H2 VP = VG + VE Measure the variation among clones VG = 0 , so VP = VE Measure the variation among individuals that are not identical to get VP Calculate VG = VP – VE Estimate broad sense heritability

Parent-offspring regression Parent-offspring regression can be used to estimate narrow sense heritability. Example Flower diameter was measured on male and female parents and their offspring in the greenhouse (50 crosses, 5 progeny/cross). Regression on one parent h2 = 2b = 2*0.462 = 0.924 Regression on midparent h2 = b = 0.804

Response to selection R = h2 S Predict response S Selection differential Response to selection Realized heritability 60 70 80 90 100 110 120 130 140 150 S Select best 10% of C0 Intermate selections to form C1 60 70 80 90 100 110 120 130 140 150 R

Domestication and plant breeding Availability of high density genetic maps and high throughput marker genotyping platforms has created new opportunities for understanding the genetic basis of quantitative traits. Signatures of Selection Yamasaki et al. 2005. Plant Cell 17: 2859-2872