Quantitative genetics

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Quantitative genetics

• Many traits that are important in agriculture,
  biology and biomedicine are continuous in their
  phenotypes. For example,
• Crop Yield
• Stemwood Volume
• Plant Disease Resistances
• Body Weight in Animals
• Fat Content of Meat
• Time to First Flower
• IQ
• Blood Pressure

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The following image demonstrates the variation
for flower diameter, number of flower parts and
the color of the flower Gaillaridia pilchella
(McClean 1997). Each trait is controlled by a
number of genes each interacting with each other
and an array of environmental factors.
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• Number of Genes Number of Genotypes
• 1 3
• 2 9
• 5 243
• 10 59,049

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Consider two genes, A with two alleles A and a,
and B with two alleles B and b.- Each of the
alleles will be assigned metric values- We give
the A allele 4 units and the a allele 2 units-
At the other locus, the B allele will be given 2
units and the b allele 1 unit

• Genotype Ratio Metric value
• AABB 1 12
• AABb 2 11
• AAbb 1 10
• AaBB 2 10
• AaBb 4 9
• Aabb 2 8
• aaBB 1 8
• aaBb 2 7
• aabb 1 6

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A grapical format is used to present the above
results
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Normal distribution of a quantitative trait may
be due to

• Many genes
• Environmental effects
• The traditional view polygenes each with small
  effect and being sensitive to environments
• The new view A few major gene and many
  polygenes (oligogenic control), interacting with
  environments

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Traditional quantitative genetics research
Variance component partitioning

• The phenotypic variance of a quantitative trait
  can be partitioned into genetic and environmental
  variance components.
• To understand the inheritance of the trait, we
  need to estimate the relative contribution of
  these two components.
• We define the proportion of the genetic variance
  to the total phenotypic variance as the
  heritability (H2).
• - If H2 1.0, then the trait is 100 controlled
  by genetics
• - If H2 0, then the trait is purely affected
  by environmental factors.

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• Fisher (1918) proposed a theory for partitioning
  genetic variance into additive, dominant and
  epistatic components
• Cockerham (1954) explained these genetic variance
  components in terms of experimental variances
  (from ANOVA), which makes it possible to estimate
  additive and dominant components (but not the
  epistatic component)
• I proposed a clonal design to estimate additive,
  dominant and part-of-epistatic variance
  components
• Wu, R., 1996 Detecting epistatic genetic
  variance with a clonally replicated design
  Models for low- vs. high-order nonallelic
  interaction. Theoretical and Applied Genetics 93
  102-109.

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Genetic Parameters Means and (Co)variances

• One-gene model
• Genotype aa Aa AA
• Genotypic value G0 G1 G2
• Net genotypic value -a
  0 d
  a
• 
  origin(G0G1)/2
• a additive genotypic value
• d dominant genotypic value
• Environmental deviation E0 E1 E2
• Phenotype or
• Phenotypic value Y0G0E0 Y1G1E1 Y2G2E2
• Genotype frequency P0 P1 P2
• at HWE q2 2pq p2
• Deviation from population mean ? -a - ? d -
  ? a - ?
• -2pa(q-p)d (q-p)a(q-p)d
  2qa(q-p)d

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• Population mean ? q2(-a) 2pqd p2a
  (p-q)a2pqd
• Genetic variance ?2g q2(-2p?-2p2d)2
  2pq(q-p)?2pqd2 p2(2q?-2q2d)2
• 2pq?2 (2pqd)2
• ?2a (or VA) ?2d (or VD)
  
• Additive genetic variance, Dominant genetic
  variance,
• depending on both on a and d depending only on
  d
• Phenotypic variance ?2P q2Y02 2pqY12 p2Y22
  (q2Y0 2pqY1 p2Y2)2
• Define
• H2 ?2g /?2P as the broad-sense heritability
• h2 ?2a / ?2P as the narrow-sense heritability
• These two heritabilities are important in
  understanding the relative contribution of
  genetic and environmental factors to the overall
  phenotypic variance.

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What is ? a(q-p)d?

• It is the average effect due to the substitution
  of gene from one allele (A say) to the other (a).
• Event A a contains two possibilities
• From Aa to aa From AA to Aa
• Frequency q p
• Value change d-(-a) a-d
• ? qd-(-a)p(a-d)
• a(q-p)d

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Midparent-offspring correlation

• __________________________________________________
  __________________
• Progeny
• Genotype Freq. of Midparent AA Aa aa Mean
  value
• of parents matings value a d -a of progeny
• __________________________________________________
  __________________
• AA AA p4 a 1 - - a
• AA Aa 4p3q ½(ad) ½ ½ - ½(ad)
• AA aa 2p2q2 0 - 1 - d
• Aa Aa 4p2q2 d ¼ ½ ¼ ½d
• Aa aa 4pq3 ½(-ad) - ½ ½ ½(-ad)
• aa aa q4 -a - - 1 -a
• ________________________________________________

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• Covariance between midparent and offspring
• Cov(OP)
• E(OP) E(O)E(P)
• p4a a 4p3q ½(ad) ½(ad) q4 (-a)(-a)
  (p-q)a2pqd2
• pq?2
• ½?2a
•  
• The regression of offspring on midparent values
  is
• b Cov(OP)/?2(P)
• ½?2a / ½?2P
• ?2a /?2P
• h2
• where ?2(P)½?2P is the variance of midparent
  value.

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• IMPORTANT
• The regression of offspring on midparent values
  can be used to measure the heritability!
• This is a fundamental contribution by R. A.
  Fisher.

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You can derive other relationships

• Degree of relationship Covariance
• __________________________________________________
  __
• Offspring and one parent Cov(OP) ?2a/2
• Half siblings Cov(FS) ?2a/4
• Full siblings Cov(FS) ?2a/2 ?2a/4
• Monozygotic twins Cov(MT) ?2a ?2d
• Nephew and uncle Cov(NU) ?2a/4
• First cousins Cov(FC) ?2a /8
• Double first cousins Cov(DFC) ?2a/4 ?2d/16
• Offspring and midparent Cov(O) ?2a/2
• __________________________________________________
  __
•  

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Cockerhams experimental and mating designs

• By estimating the covariances between relatives,
  we can estimate the additive (or mixed additive
  and dominant) variance and, therefore, the
  heritability.
• Next, I will introduce mating and experimental
  designs used to estimate the covariances between
  relatives.

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Mating design

• Mating design is used to generate genetic
  pedigrees, genetic information and materials that
  can be used in a breeding program
• Mating design provides genetic materials, whereas
  experimental design is utilized to obtain and
  analyze the data from these materials

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Objectives of mating designs

• Provide information for evaluating parents
• 2) Provide estimates of genetic parameters
• 3) Provide estimates of genetic gains
• 4) Provide a base population for selection

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Commonly used mating designs

• 1) Open-pollinated
• 2) Polycross
• 3) Single-pair mating
• 4) Nested mating
• 5) Factorial mating tester design
• 6) Diallel mating (full, half, partial
  disconnected)
•  

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Nested mating (NC Design I)

• Each of male parents is mated to a subset of
  different female parents

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• Cov(HSM)1/4VA
• V(female/male) Cov(FS) Cov(HSM)
• 1/2VA1/4VD 1/4VA
• 1/4VA 1/4VD
•  
• - Provide information for parents and full-sib
  families
• - Provide estimates of both additive and
  dominance effects
• - Provide estimates of genetic gains from both
  VA and VD
• - Not efficient for selection
• - Low cost for controlled mating

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Example Date structure for NC Design I

• Sample Male Female Full-sib family Individual Phen
  otype
• 1 1 A 1 1 y1A1
• 2 1 A 1 2 y1A2
• 3 1 B 2 1 y1B1
• 4 1 B 2 2 y1B2
• 5 1 C 3 1 y1C2
• 6 1 C 3 2 y1C2
• 7 2 D 4 1 y2D1
• 8 2 D 4 2 y2D2
• 9 2 E 5 1 y2E1
• 10 2 E 5 2 y2E2
• 11 2 F 6 1 y2F1
• 12 2 F 6 2 y2F2
• 13 3 G 7 1 y3G1
• 14 3 G 7 2 y3G2
• 15 3 H 8 1 y3H1
• 16 3 H 8 2 y3H2
• 17 3 I 9 1 y3I1
• 18 3 I 9 2 y3I2

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Estimates by statistical software

• VTotal 40
• VFS Cov(FS) 10
• VM Cov(HSM) 4
• VE VTotal VFS 40 10 30
• V(female/male) Cov(FS) Cov(HSM)
• 10 4 6
• VA 4Cov(HSM) 4 4 16 h2 16/40
  0.x
• V(female/male) 1/4VA 1/4VD 4 1/4VD 6
• VD 8, VG VA VD 16 6 22
• H2 22/40 0.x

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Factorial mating (NC Design II)

• Each member of a group of males is mated to each
  member of group of females

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• Cov(HSM) 1/4 VA
• Cov(HSF) 1/4 VA
•  V(female ? male) Cov(FS)Cov(HSM)Cov(HSF)
• 1/4 VD
•  
• - Provide good information for parents and
  full-sib families
• - Provide estimates of both additive and
  dominance effects
• - Provide estimates of genetic gains from both
  VA and VD
• - Limited selection intensity
• - High cost

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Tester mating design (Factorial)

• Each parent in a population is mated to each
  member of the testers that are chosen for a
  particular reason

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• Cov(HSM)1/4VA
• Cov(HSF)1/4VA
• V(female ? male) Cov(FS)COV(HSM)-COV(HSF)
• 1/4VD
•  
• - Provide good information for parents and
  full-sib families
• - Provide estimates of both additive and
  dominance effects
• - Provide estimates of genetic gains from both
  VA and VD
• - Limited selection intensity
• - High cost

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Diallel mating design

• Full diallel each parent is mated with every
  other parent in the population, including selfs
  and reciprocal
•  

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• Half diallel each parent is mated with every
  other parent in the population, excluding selfs
  and reciprocal

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• Partial Diallel selected subsets of full
  diallels
•  

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• Disconnected half diallel selected subsets of
  full diallels

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• Diallel analysis
•  
• Cov(HS) 1/4VA
• Cov(FS) 1/2VA 1/4VD
• Cov(FS) Cov(FS) 2Cov(HS) 1/4VD
•  
• - Provide good evaluation of parents and
  full-sib families
• - Provide estimates of both additive and
  dominance effects
• - Provide estimates of genetic gains from both
  VA and VD
• - High cost

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Genomic Imprinting or parent-of-origin effectThe
same allele is expressed differently, depending
on its parental origin

• Consider a gene A with two alleles A (in a
  frequency p) and a (in a frequency q)
• Genotype Frequency Value
• AA p2 a Average effect
• Aa pq di No imprinting ? a
  d(q-p)
• aA qp d-i Imprinting ?M a
  i d(q-p) A ? a
• aa q2 -a ?P a i d(q-p)
  A ? a
• Mean a(p-q)2pqd
• No imprinting ?g2 2pq?2 (2pqd)2
• Imprinting ?gi2 2pq?2 (2pqd)2 2pqi2
• Imprinting leads to increased genetic variance
  for a quantitative trait and, therefore, is
  evolutionarily favorable.

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Genomic Imprinting
The callipygous animals 1 and 3 compared to
normal animals 2 and 4 (Cockett et al. Science
273 236-238, 1996)
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Predicting Response to Selection
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Population Mean, Xp - phenotypic mean of the
animals or plants of interest and expressed in
measurable units. Selection Mean, Xs - phenotypic
mean of those animals or plants chosen to be
parents for the next generation and expressed in
measurable units. Selection Differential, SD -
difference between the phenotypic means of the
entire population and its selected mean.
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Genetic Gain the amount that the phenotypic
mean in the next generation change by selection.
- that change can be or -
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Selection Differential
G h2 SD
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How to Calculate Genetic Gain
M2 M h2 (M1 - M) M2 resulting mean
phenotype M mean of parental population M1
mean of selected population h2 heritability of
the trait ? M2 - M h2 (M1
- M) ? G h2 SD (SD/?p)h2?p ih2?p i
selection intensity h2 narrow-sense
heritability ?p standard phenotypic deviation
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• Factors that influence
• the Genetic Gain
• Magnitude of selection differential
• Selection intensity
• Broad-sense heritability heritability
• Phenotypic variation

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Knowing the Selection Differential, and the
response to selection, an estimate of the traits
heritability can be calculated G / SD Realized
Heritability
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Realized heritability can also be calculated
as M2 M h2 (M1 - M) rearranged,
(M2 - M) (M1 - M)
h2
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• Maximizing Genetic Gain
• Examples

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N48, Population Mean 109.7
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Goal Improve the Mean Select those in red, N
6, Mean of Selected 119.5 SD 9.8 G h2 SD
0.7 x 9.8 6.86
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Goal Reduce the Mean Select those in blue, N
8, Mean of Selected 100.4
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Nature 432, 630 - 635 (02 December 2004)The
role of barren stalk1 in the architecture of maize

• ANDREA GALLAVOTTI1,2, QIONG ZHAO3,
  JUNKO KYOZUKA4, ROBERT B. MEELEY5,
  MATTHEW K. RITTER1,, JOHN F. DOEBLEY3,
  M. ENRICO PÈ2 ROBERT J. SCHMIDT1
• 1 Section of Cell and Developmental Biology,
  University of California, San Diego, La Jolla,
  California 92093-0116, USA2 Dipartimento di
  Scienze Biomolecolari e Biotecnologie, Università
  degli Studi di Milano, 20133 Milan,
  Italy3 Laboratory of Genetics, University of
  Wisconsin, Madison, Wisconsin 53706,
  USA4 Graduate School of Agriculture and Life
  Science, The University of Tokyo, Tokyo 113-8657,
  Japan5 Crop Genetics Research, Pioneer-A DuPont
  Company, Johnston, Iowa 50131, USA Present
  address Biological Sciences Department,
  California Polytechnic State University, San Luis
  Obispo, California 93407, USA

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Mapping Quantitative Trait Loci (QTL) in the F2
hybrids between maize and teosinte
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Maize Teosinte tb-1/tb-1 mutant maize
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Effects of ba1 mutations on maize development
Mutant Wild type No tassel
Tassel
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Data format for a backcross

• Sample Height Marker 1 Marker 2 QTL
• (cm, y)
• 1 184 Mm (1) Nn (1) ?
• 2 185 Mm (1) Nn (1) ?
• 3 180 Mm (1) Nn (1) ?
• 4 182 Mm (1) nn (0) ?
• 5 167 mm (0) nn (0) ?
• 6 169 mm (0) nn (0) ?
• 7 165 mm (0) nn (0) ?
• 8 166 mm (0) Nn (1) ?

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• Heights classified by markers (say marker 1)
• Marker Sample Sample Sample
• group size mean variance
• Mm n1 4 m1182.75 s21
• mm n0 4 m0166.75 s20

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The hypothesis for the association between the
marker and QTL

• H0 m1 m0
• H1 m1 ? m0
• Calculate the test statistic
• t (m1m0)/?s2(1/n11/n0),
• where s2 (n1-1)s21(n0-1)s20/(n1n02)
• Compare t with the critical value
  tdfn1n2-2(0.05) from the t-table.
• If t gt tdfn1n2-2(0.05), we reject H0 at the
  significance level 0.05 ? there is a QTL
• If t lt tdfn1n2-2(0.05), we accept H0 at the
  significance level 0.05 ? there is no QTL

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Why can the t-test probe a QTL?

• Assume a backcross with two genes, one marker
  (alleles M and m) and one QTL (allele Q and q).
• These two genes are linked with the recombination
  fraction of r.
• MmQq Mmqq mmQq mmqq
• Frequency (1-r)/2 r/2 r/2 (1-r)/2
• Mean effect ma m ma m
• Mean of marker genotype Mm
• m1 (1-r)/2 (ma) r/2 m m (1-r)a
• Mean of marker genotype mm
• m0 r/2 (ma) (1-r)/2 m m ra
• The difference
• m1 m0 m (1-r)a m ra (1-2r)a

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• The difference of marker genotypes can reflect
  the size of the QTL,
• This reflection is confounded by the
  recombination fraction
• Based on the t-test, we cannot distinguish
  between the two cases,
• - Large QTL genetic effect but loose linkage with
  the marker
• - Small QTL effect but tight linkage with the
  marker

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Example marker analysis for body weight in a
backcross of mice

• __________________________________________________
  ___________________
• Marker class 1 Marker class 0
• ______________________ _____________________
• Marker n1 m1 s21 n1 m1 s21 t P value
• __________________________________________________
  ___________________________
• 1 Hmg1-rs13 41 54.20 111.81 62 47.32 63.67 3.754
  lt0.01
• 2 DXMit57 42 55.21 104.12 61 46.51 56.12 4.99
  lt0.01
• 3 Rps17-rs11 43 55.30 101.98 60 46.30 54.38 5.231
  lt0.000001
• __________________________________________________
  ___________________

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Marker analysis for the F2

• In the F2 there are three marker genotypes, MM,
  Mm and mm, which allow for the test of additive
  and dominant genetic effects.
• Genotype Mean Variance
• MM m2 s22
• Mm m1 s21
• mm m0 s20

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Testing for the additive effect

• H0 m2 m0
• H1 m2 ? m0
• t1 (m2m0)/?s2(1/n21/n0),
• where s2 (n2-1)s22(n0-1)s20/(n1n02)
• Compare it with tdfn2n0-2(0.05)

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Testing for the dominant effect

• H0 m1 (m2 m0)/2
• H1 m1 ? (m2 m0)/2
• t2 m1(m2 m0)/2/?s21/n11/(4n2)1/(4n0)
  ,
• where s2 (n2-1)s22(n1-1)s21(n0-1)s20/(n2n1
  n03)
• Compare it with tdfn2n1n0-3(0.05)

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Example Marker analysis in an F2 of maize

• __________________________________________________
  ____________________________________________
• Marker class 2 Marker class 1 Marker class
  0 Additive Dominant
• ____________ ______________ ______________
• M n2 m2 s22 n1 m1 s21 n0 m0
  s20 t1 P t2 P
• __________________________________________________
  _____________________________________________
• 43 5.24 2.44 86 4.27 2.93 42
  3.11 2.76 6.10 lt0.001 0.38 0.70
• 2 48 4.82 3.15 89 4.17 3.26 34
  3.54 2.84 3.28 0.001
  -0.05 0.96
• 3 42 5.01 3.23 92 4.14 3.18 37
  3.57 2.68 3.71 0.0002
  -0.57 0.57
• __________________________________________________
  _____________________________________________

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Testing for the dominant effect

• H0 m1 (m2 m0)/2
• H1 m1 ? (m2 m0)/2
• t2 m1(m2 m0)/2/?s21/n11/(4n2)1/(4n0)
  ,
• where s2 (n2-1)s22(n1-1)s21(n0-1)s20/(n2n1
  n03)
• Compare it with tdfn2n1n0-3(0.05)

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Example Marker analysis in an F2 of maize

• __________________________________________________
  ____________________________________________
• Marker class 2 Marker class 1 Marker class
  0 Additive Dominant
• ____________ ______________ ______________
• M n2 m2 s22 n1 m1 s21 n0 m0
  s20 t1 P t2 P
• __________________________________________________
  _____________________________________________
• 43 5.24 2.44 86 4.27 2.93 42
  3.11 2.76 6.10 lt0.001 0.38 0.70
• 2 48 4.82 3.15 89 4.17 3.26 34
  3.54 2.84 3.28 0.001
  -0.05 0.96
• 3 42 5.01 3.23 92 4.14 3.18 37
  3.57 2.68 3.71 0.0002
  -0.57 0.57
• __________________________________________________
  _____________________________________________