Transcript 2002-11-19: Quantitative Traits V

Lecture 25: Quantitative Traits V
Date: 11/19/02
 Environmental variation
 Resemblance of relatives
 Parent-offspring regression
Types of Environmental Effects
 General environmental effects are trait-affecting
factors that are shared by groups of individuals.


local habitat effects
maternal effects
 Special environmental effects are the residual
deviations from the expected phenotype given
genetic and general environmental effects.



microenvironmental variation
developmental noise
measurement error
Genotype x Environment Effect
 If different genotypes respond to environmental
change in different ways (e.g. trait increases for one
genotype, but decreases for another when moved
from environment A to B), then there are genotype
by environment effects.
 Here, we develop our general linear model for all
genetic and environmental effects to give us a final
model for PHENOTYPE.
Linear Model for Phenotypes
 E : general environmental effect
 e : specific environmental effect
 I : genotype by environment interaction
effect.
 zijk : phenotype of kth individual of the ith
genotype in the jth general environment.
zijk  Gi  I ij  E j  eijk
Properties of Full Linear Model
 Genotypic value Gi are linear functions of additive,
dominance, and epistatic effects.
 Terms are mean 0 deviations from lower order expectations.
 m G  zijk is mean phenotype in population.
 Gi is the expected phenotype of genotype i averaged over all
environmental conditions.
 mG + Ej is mean phenotype of all genotypes if they are
assayed in jth general environment.
 Gi + Iij + Ej is expected phenotype of genotype i in the jth
general environment.
Example – Thrips
Daisy
Line
1
2
3
G
I
Thrip Line
II
III
E
77
61
40
59.33
34
159
71
88.00
47
51
107
68.33
-19.22
18.44
0.78
mG = 71.89
- Average density of thrips reported from replicated experiments.
- Specific environmental effect is small and ignored in the analysis.
Example – Thrips
Daisy
Line
1
2
3
G
I
Thrip
Linephenotype of
Expected
Mean
trait value
obtained
genotype
1, calculated
II
III
bybyassuming
averagingequal
over
weight
for each cell
and
environments
(equal
34averaging
weights)47
77
61Average 159
51
environmental
across
40effect is average
71
107
genotypes in this
59.33
88.00
68.33
environment
(plant line),
E
-19.22
18.44
0.78
mG = 71.89
mean phenotype.
- Average densitysubtracting
of thrips reported
from replicated experiments.
- Error is small and ignored in the analysis.
Example – Thrip Interactions
 Because we are assuming measurement error
and microenvironmental effects are small
(eijk = 0), we can estimate interaction effects.
I ij  zij  Gi  E j
II,1 = 36.89
III,1 = -34.78
IIII,1 = -2.11
II,2 = -16.78
III,2 = 52.56
IIII,2 = -35.78
II,3 = -20.11
III,3 = -17.78
IIII,3 = 37.89
Variance
 I and e are uncorrelated with the other variables:
s  s  s  2s G,E  s  s
2
P
2
G
2
I
2
E
2
e
 sG,E is genotype-environment covariance. It
measures the physical association of genotypes and
environments. If individuals are randomly
distributed by environment, then it is 0.
 sI is a measure of the variation in phenotypic
response to specific environments.
Genotype-Environment
Covariance
 There is no way to estimate sG,E.
 It is confounded with genetic variance and results in
biased estimates of genetic variance.
 Experimental design can reduce, but you cannot
always control for:



limited seed/pollen dispersal
genetically-based social structure
maternal/paternal effects
Example – Two-Way ANOVA
Trait
Var(G)
Var(E)
Var(I)
Var(e)
Longest
leaf length
Longest
leaf width
26.7
29.5
6.1
37.7
27.5
7.8
3.6
61.3
Grow various plant clones (each clone is different
genotype) in three different sites (sand dunes,
grassland, marsh) .
Example – Two-Way ANOVA
Trait
Var(G)
Var(E)
Var(I)
Longest
26.7
29.5
6.1
Variance among
leaf length
Clone by site
Variance within
Var(e)
37.7
three sites
Among-clone
variance
is
an within
clones
sites
Longest estimates
27.5
7.8
3.6
61.3
variance
estimates
estimate ofestimates
the
specific
leaf width generalgenotype total
genetic
byenvironmental
environmental
variance.
environment
effects.
Two-way ANOVA
with
factors
clone
(genotype) and
effects. variance.
environment.
Estimating Variance
Components
 We have partitioned phenotypic variance of trait
into various components.
 We still need methods to estimate the components.
 Genetic and environmental sources of variance
contribute differently to different types of relatives.
 Use this fact to estimate components.
 To show that there are differences between relative
types, we derive expressions for phenotypic
covariance between relatives.
Covariance of Relatives
 Consider two relatives x and y.
Covariance of one
 z x = G x + Ex + e x
genotypic value with
environment of other.
 zy = Gy + Ey + ey
 ex and ey do not contribute to resemblance.
s x, y   s Gx  Ex  ex , Gy  E y  ey 
 s G x, y   s G , E x, y   s G , E  y, x   s E x, y 
Assume No GenotypeEnvironment Covariance
 Experiment can be designed to remove such crossed
genotype by environment covariance.
 One can assume these covariance terms are nonzero.
 If they are not, they are absorbed in the genetic
variance.
s x, y   s G x, y   s E x, y 
Relatedness
 Relatedness is defined with respect to an
identified frame of reference. In the global
frame of reference, we are all related.
 Usually assume that founders of the
observed pedigree are unrelated.
 Then, relatedness is determined by IBD
status of alleles.
Coefficient of Identity
 Individuals are inbred if they contain pairs of
alleles which are IBD.
 We know there are 9 identity states (next slide).
With each identity state is associated a probability
Di, a condensed coefficient of identity.
 All 9 coefficients define the complete identity state
probability distribution of single loci in two
individuals.
Possible IBD Status
S1
S2
S3
S4
S5
S6
S7
S8
S9
Identity State Distributions
Relative
Type
1
2
3
4
5
6
7
8
9
Parent/
Offspring
0
0
0
0
0
0
0
1
0
Full Sibs
0
0
0
0
0
0
¼
½
¼
Coefficient of Coancestry
 AKA: coefficient of consanguinity / kinship
/ de parente
 It is the probability that two alleles selected
randomly from two individuals, one from
each individual, are IBD.
1
1
  D1  D 3  D 5  D 7   D 8
2
4
P(2 alleles are IBD | IBD state 8)
Inbreeding Coefficient
 The inbreeding coefficient of an individual
is the probability that the two alleles they
have at a locus are IBD.
 This is merely the coefficient of ancestry of
the individual’s parents.
f z   xy
Coefficient of Coancestry for
Self
S7
1
 xx  PIBD IBD state 7  
2
Coefficient of Coancestry of
Self with Inbreeding
 xx  Psame selected and IBD S 7   Pdifferent selected and IBD S 7 
1
1
 1   f x
2
2
1
 1  f x 
2
Coefficient of Ancestry for
Parent/Offspring
 Assume non-inbred parent and offspring:
 po
1

4
 Assume parent inbred with inbreeding
coefficient fp:
 po  Pdraw passed gene from both and IBD S8 
 Pdraw passed gene from child and non - passed from parent and IBD S8 
1
1
 1   f p
4
4
Coefficient of Ancestry for
Parent/Offspring
 Suppose offspring is inbred with inbreeding
coefficient fo:
 po  Pdraw passed gene in parent and offspring and IBD S8 
 Pdraw other parent gene from offspring and IBD S8 

1
1
1   f o
4
2
 Most general equation:
 po
1
 1  f p  2 f o 
4
Coefficient of Ancestry for Full
Sibs
1
 xy  2  f m  f f  4 mf 
8
Inbreeding
coefficient
of mother
Inbreeding
coefficient
of father
Coefficient of
coancestry of
mother and father
Coefficient of Fraternity
 The probability that the single-locus
genotype of two individuals are identical by
descent is the coefficient of fraternity.
Coefficient of Fraternity
 px my
 mx m y
px
mx
 mx p y
py
mx
 px p y
x
y
D xy  mx my  px p y  mx p y  px my
Example - Dxy
 Consider full sibs: Then they share a mother
and father (mx = my = m and px = py = p).
D xy  mm pp  
2
mp
 What if the parents are unrelated?
 What if the parents are not inbred?
Example - Dxy
 Consider half sibs sharing a father:
px = py = p
D xy  mx my  pp  mx p  pmy
 What if the parents are unrelated?
 What if the parents are not inbred?
Genetic Covariance






Assume autosomal loci
Assume random mating
Assume unlinked loci
Assume linkage equilibrium
Assume no maternal effects
Assume genotype-environment covariance and
interaction are ~0.
 Assume no sexual dimorphism
 Assume no selection
Genotypic Values of Two
Individuals

 

Gijkl  x   m G   ix   xj   kx   lx     ijx   klx  

  

   
  ik   il    jk    jl  
x
x
ikl

x
x
x
   jkl   kij   lij
x
x
x
 

x
ijkl

Gijkl  y   m G   iy   jy   ky   ly     ijy   kly  

  

   
  ik   il    jk    jl  
y
y
ikl
y
y
y
   jkl   kij   lij
y
y
y
Gx  Ax  Dx  AAx  ADx  DDx
y
ijkl

Genetic Covariance
 The effects are uncorrelated within and
between individuals if the assumptions are
met.
s G x, y   s A x, y   s D x, y   s AA x, y   s AD x, y   s DD x, y   
 Now, write each term as a function of
variance components and coefficients of
relationship.
Additive Genetic Covariance
 Mean effects are 0 by definition.
s A x, y   Eix   xj   kx  lx iy   jy   ky  ly 

 

 E 
E   


E  ix iy  E  ix iy IBD PIBD   E  ix iy not IBD Pnot IBD 
2
i
xy
x
i
x
j
y
i

 
  jy  4E i2 xy
s A x, y   2xys A2
Dominance Genetic Variance

 

E  ijx ijy  E  ij2 IBD genotype PIBD genotype 


 E  ijx ijy not IBD genotype Pnot IBD genotype 
s D x, y   D xys
2
D
Additive x Additive Genetic
Covariance

 

E  ik  ik  E   IBD alleles selected at 1, IBD alleles selected at 2 PIBD at 1, IBD at 2
x
y

2

   not IBD at 1, not IBD at 2 Pnot IBD at 1, not IBD at 2
2

  2xy E  
2

s AA x, y   2 xy  s
2
2
AA
In General...
 Any covariance due to higher-order epistatic
effects is the product of:



probability of identity for each additive effect
probability of identity for each dominance effect
corresponding variance component.
2
s AD x, y   2 xy D xys AD
 additive x dominance:
2
2


s
x
,
y

D
s
 dominance x dominance: DD
xy DD
Overall Covariance
s G x, y    2 xy n Dmxys A2 D
n
m
2
 2 xys A2  D xys D2  2 xy  s AA
2
2
2
 2 xy D xys AD
 D2xys DD
 2 xy  s 2AAA  
3
Coefficients
Relationship
Parent-offspring
Grandparent-grandchild
A
½
¼
D
Great grandparent-great
grandchild
Half sibs
Full sibs (DZ)
1/8
1/64
¼
½
1/16
¼ ¼ 1/8
1/16
Uncle – nephew
First cousins
Monozygotic twins (MZ)
¼
1/8
1
1/16
1/64
1
1
1
AA AD
¼
1/16
1
DD
Heritability
 Though many components of variance
cannot be estimated, progress can be made in
estimating
additive genetic variance**
 dominance genetic variance
 environmental variance due to common families
2
s
 Narrow-sense heritability is the ratio h 2  A2
sz

Estimating Heritability
 One can use sets of relatives to estimate h2.
 If additive genetic variance is the dominant
source of phenotypic covariation, then
h 
2
because
Covz x , z y 
2 xy Var  z 
Covz x , z y   2 xys A2
Parent-Offspring Regression
 Heritabilities are often estimated by regressing
offspring phenotypes on parental phenotypes.




Parent-offspring relationships are often easiest to
identify.
Simple computation: least-squares
Dominance and linkage do not influence covariance
between parents and offspring.
Unbiased by selection on the parents.
The Role of Mama
 Often only the mother can be identified.
 Or the trait can only be observed on mothers (e.g.
birth weight).
 Maternal effects can confound the analysis.
 We assume no maternal effects in the following
analysis.
 Also, assume no significant genotype by
environment interaction nor covariance.
Balanced Design
 Assume single offspring and single parent
observed in each family.
zoi     op z pi  ei
offspring
phenotype for
ith family
parent
phenotype for
ith family
Regression Coefficient
Assumes no
environmental
causes of
resemblance
 op 
Covzo , z p 
Var z p 
1 2 1 2 1 2
s zo , z p  2 s A  4 s AA  8 s AAA  
E  op   2

s z p 
s z2
Estimate of Heritability
 Assuming no additive x additive interactions:
h  2bop
2
Midparent Values
 When both parents are known, a more
precise estimate is available.
 Let zmi be the phenotype of the mother in the
ith family. For father, zpi.
 zmi  z pi 
  ei
zoi     op 
2 

Estimating Heritability
 zm  z p 
Cov  zo ,

2


bop 
 zm  z p 
Var 
 in both sexes

2


 in both generations
1
Cov zo , z m   Covzo , z p 
Assume resemblance in  2
1
Var z   Var z 
relatives is independent
4
of sex.
2Covzo , z p 

Var  z 
 2bop
 Assume phenotypic
variance is same


