1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

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

12. 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

13. 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

14. 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

15. Inbreeding Coefficient FA 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)

16. 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

17. Single locus model -a 0 d a A2 A2 A1 A2 A1 A1 Genotypic Value
Coded Genotypic Value
-a 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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 Plant Cell 17: