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CSS 650 Advanced Plant Breeding. Module 1: Introduction Population Genetics Hardy Weinberg Equilibrium Linkage Disequilibrium. Plant Breeding. “The science, art, and business of improving plants for human benefit” Considerations: Crop(s) Production practices End-use(s)

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Css 650 advanced plant breeding

CSS 650 Advanced Plant Breeding

Module 1:


Population Genetics

Hardy Weinberg Equilibrium

Linkage Disequilibrium

Plant breeding
Plant Breeding

“The science, art, and business of improving plants for human benefit”


  • 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

Bernardo, Chapter 1

Plant breeding1
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
    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


    • 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

    Css 650 advanced plant breeding

    What do population geneticists do?

    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

    Gene and genotype frequencies
    Gene and genotype frequencies

    p + q = 1

    For a population of diploid organisms:

    P11 + P12 + P22 = 1

    Bernardo, Chapter 2

    Gene frequencies another way
    Gene frequencies (another way)

    Number of individuals = N = N11+ N12+ N22 = 100

    Number of alleles = 2N = N1 + N2 = 200

    Allele frequencies in crosses

    Inbred x inbred

    Alleles are unknown, but allele frequencies at segregating loci are known

    F1 and F2: p = q = 0.5

    Allele frequencies in crosses

    Value of q is reduced by ½ in each backcross generation

    Factors that may change gene frequencies
    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
    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
    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 equilibrium1
    Hardy-Weinberg Equilibrium






    p = 0.4


    q = 0.6

    Expected genotype frequencies are obtained by expanding the binomial

    (p + q)2 = p2 + 2pq + q2 = 1



    Equilibrium with multiple alleles
    Equilibrium with multiple alleles

    For multiple alleles, expected genotype frequencies can be found by expanding the multinomial (p1 + p2 + ….+ pn)2

    For example, for three alleles:

    Corresponding genotypes:

    A1A1 A1A2 A1A3 A2A2 A2A3 A3A3

    Lynch and Walsh (pg 57) describe equilibrium for autopolyploids

    Relationship between gene and genotype frequencies




    Relationship between gene and genotype frequencies

    • f(A1A2) has a maximum of 0.5, which occurs when p=q=0.5

    • Most rare alleles occur in heterozygotes

    • Implications for

      • F1?

      • F2?

      • Any BC?

    Applications of the hardy weinberg law
    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.

        q2 = 1/11,000q 0.0095

    • Estimate frequency of individuals that are carriers for a recessive allele

      p = 1 - 0.0095 = 0.9905 2pq = 0.0188  2%

    Testing for hardy weinberg equilibrium
    Testing for Hardy-Weinberg Equilibrium

    All genotypes must be distinguishable

    N = N11+ N12+ N22= 233 + 385 + 129 = 747

    Chi square test for hardy weinberg equilibrium
    Chi-square test for Hardy-Weinberg Equilibrium

    • Accept H0: 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 H0, 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
    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
    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
    Likelihood Ratio Test for HWE

    where is the expected number

    and is the observed number of the ijth genotype

    Calculations in Excel

    Gametic phase equilibrium

























    Gametic phase equilibrium

    Random association of alleles at different loci (independence)



    DAB = PAB – pApB

    DAB = PABPab – PAbPaB

    DAB = 0.40 – 0.5*0.5 = 0.15

    DAB = 0.4*0.4 – 0.1*0.1 = 0.15

    Lynch and Walsh, pg 94-100; Falconer, pg 15-19

    Linkage disequilibrium
    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 disequilibrium1
    Linkage Disequilibrium

    Excess of coupling phase gametes  +D

    Excess of repulsion phase gametes  -D

    Sources of linkage disequilibrium
    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









    0.5 AB

    0.5 Ab

    0.25 AB 0.25 aB

    0.25 Ab 0.25 ab

    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

    Decay of linkage disequilibrium
    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
    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
    Implications for breeding

    • Gametic Phase Disequilibrium that is not due to linkage is eliminated by making the F1 cross

    • Recombination occurs during selfing

    • There would be greater recombination with additional random mating, but it may not be worth the time and resources


    A1A1B1B1 x A2A2B2B2

    F1 A1A2B1B2


    A1B1 0.5*(1-c)

    A1B2 0.5*c

    A2B1 0.5*c

    A2B2 0.5*(1-c)

    Effect of mating system on ld decay

    no linkage

    Effect of mating system on LD decay

    c = effective recombination rate

    s = the fraction of selfing

    99% selfing


    Alternative measures of ld
    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 r2 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

    • r2indicates the degree of association between alleles at different loci due to various causes (linkage, mutation, migration)

    D minimum and maximum values
    D – minimum and maximum values

    If D>0 Look for the maximum value D can have

    PAb = pApb - D  0  D  pApb

    PaB = papB - D  0  D  papB

    D  min(pApb, papB)

    If D<0 Look for the minimum value D can have

    PAB = pApB + D  0  D  -pApB

    Pab = papb + D  0  D  -papb

    D  max(-pApB, -papb)

    Alternative measures of ld1
    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 r2 is more widely utilized to measure LD

    When DAB > 0

    When DAB < 0

    Testing for gametic phase disequilibrium
    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 112-133 for more information

    Depiction of linkage disequilibrium
    Depiction of Linkage Disequilibrium

    Disequilibrium matrix for polymorphic sites within sh1 in maize

    Prob valueFisher’s Exact Test


    Flint-Garcia et al., 2003. Annual Review of Plant Biology 54: 357-374.

    Extent of ld in maize
    Extent of LD in Maize

    Average LD decay distance is 5–10 kb


    Linkage disequillibrium across the 10 maize chromosomes measured with 914 SNPs in a global collection of 632 maize inbred lines.

    Yanet al. 2009. PLoS ONE 4(12): e8451

    Extent of ld in barley
    Extent of LD in Barley

    Elite North American Barley

    No adjustment for population structure 

    Average LD decay distance is ~5 cM


    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

    Waugh et al., 2009, Current Opinion in Plant Biology 12:218-222

    References on linkage disequilibrium
    References on linkage disequilibrium

    Flint-Garcia et al., 2003. Structure of linkage disequilibrium in plants. Annual Review of Plant Biology 54: 357-374.

    Gupta et al., 2005. Linkage disequilibrium and association studies in higher plants: present status and future prospects. Plant Molecular Biology 57:461–485.

    Slatkin, M. 2008. 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. 2009. 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. 2009. Genetic characterization and linkage disequilibrium estimation of a global maize collection using SNP Markers. PLoS ONE 4(12): e8451.

    Zhu et al., 2008. Status and prospects of association mapping in plants. The Plant Genome 1: 5–20.