SLIDE 1
Recap of the additive model
Trait value = genetic
Z
+ non-genetic
- E
The infinitesimal with dominance CIRM, February 2020 Recap of the - - PowerPoint PPT Presentation
The infinitesimal with dominance CIRM, February 2020 Recap of the - - PowerPoint PPT Presentation
The infinitesimal with dominance CIRM, February 2020 Recap of the additive model Trait value = genetic + non-genetic Z E I am going to ignore the environmental component E . Parental trait values z 1 , z 2 .
SLIDE 2
SLIDE 3
The infinitesimal model
Let
- 1. P(t) denote the pedigree relationships between all individuals
up to and including generation t;
- 2. Z(t) denote the traits of all individuals in the pedigree up to
and including the tth generation. The distribution of trait values in generation t, conditional on knowing P(t) and Z(t−1), is multivariate normal. THIS IS A STATEMENT ABOUT DISTRIBUTION WITHIN FAMILIES, NOT ACROSS THE WHOLE POPULATION The pedigree can be quite general. It can, for example, capture population structure.
SLIDE 4
Why does it work in the additive case?
For simplicity, consider haploid model. In Amandine’s notation, Zj = ¯ z0 +
M
- l=1
1 √ M ηl(χj
l ).
Simplest imaginable model: suppose ηl are i.i.d. with ηl = ±1 with equal probability, ¯ z0 = 0. Because many different combinations of alleles can lead to the same allelic state, ‘typically’, knowing the trait value of an individual tells us very little about the allelic state at any given locus.
SLIDE 5
Put more mathematically
P[η1 = 1|Z = k/ √ M] = P M
l=1 ηl = k
- η1 = 1
- P
M
l=1 ηl = k
- P [η1 = 1]
= P M
l=2 ηl = (k − 1)
- P
M
l=1 ηl = k
- P [η1 = 1]
=
1 2M−1 1 2M
- M−1
(M+k−2)/2
- M
(M+k)/2
P [η1 = 1] =
- 1 + k
M
- P [η1 = 1] .
SLIDE 6
Toy example continued
If scaled allelic effects are i.i.d. Bernoulli, P
- η1 = 1
- Z =
k √ M
- =
- 1 + k
M
- P [η1 = 1] .
For a ‘typical’ trait value, k/M = O(1/ √ M). For extreme values (k = ±M), the trait gives complete information about the allelic effect at each locus. For ‘typical’ k, the distribution of η1 is almost unchanged because there are so many different configurations of allelic effects that correspond to the same trait value.
SLIDE 7
Dominance?
Trait value: Z = ¯ z0 + 1 √ M
M
- l=1
- ηl(χ1
l ) + ηl(χ2 l ) + φl(χ1 l , χ2 l )
- .
For an individual in the ancestral population, χ1
l ,
χ2
l , two
independent draws from νl. Conventions: E[ηl( χl)] = 0, E[φl( χ1
l ,
χ2
l )] = 0,
for any value x′ of the allelic state at locus l, E[φl( χl, x′)] = 0 = E[φl(x′, χl)].
SLIDE 8
Why does this not result in a loss in generality?
Replace φl( χ1
l ,
χ2
l ) by
φl( χ1
l ,
χ2
l ) − E[φl(
χ1
l ,
χ2
l )|
χ1
l ] − E[φl(
χ1
l ,
χ2
l )|
χ2
l ] + E[φl(
χ1
l ,
χ2
l )].
Second and third terms (functions of χ1
l ,
χ2
l ) can be subsumed
into ηl( χl). For any value x′ of the allelic state at locus l: E[φl( χl, x′)] =
- φl(x, x′)
νl(dx) = 0 = E[φl(x′, χl)]. With this modification, E[φl( χ1
l ,
χ2
l )] = 0. Moreover, absorbing the
mean into ¯ z0, we may assume that E[ηl( χl)] = 0.
SLIDE 9
Decomposing the trait value
Write i[1], i[2] for parents of individual i Decompose trait value into two parts,
◮ component shared by all offspring of Zi[1] and Zi[2]; ◮ component which is independent for each individual within
the family. trait value
- Zi
= shared
Ai+Di
+ residual
Ri+Si
Ai + Ri additive component; Di + Si dominance deviation. Residuals Ri, Si determined by Mendelian inheritance
SLIDE 10
Inheritance
Label chromosomes in individual i, 1 and 2, according to whether inherited from i[1] or i[2]. Xi
l , Y i l , independent Ber(1/2) r.v.’s ◮ Xi l = 1 if type at locus l on chromosome 1 in individual i,
inherited from chromosome 1 in i[1];
◮ Y i l = 1 if type at locus l on chromosome 2 in individual i
inherited from chromosome 1 in i[2]. Zi = ¯ z0 + Ai + Di + Ri + Si. Mirrors our approach in additive case: Ai + Di centre on the means of Xi
l , Y i l ; residuals Ri + Si capture deviation of Xi l , Y i l
from their means, i.e. the randomness of Mendelian inheritance.
SLIDE 11
Terms shared by all descendants of i[1], i[2]
Ai = 1 2 √ M
M
- l=1
- ηl(χi[1],1
l
) + ηl(χi[1],2
l
) + ηl(χi[2],1
l
) + ηl(χi[2],2
l
)
- and
Di = 1 4 √ M
M
- l=1
- φl(χi[1],1
l
, χi[2],1
l
) + φl(χi[1],1
l
, χi[2],2
l
) + φl(χi[1],2
l
, χi[2],1
l
) + φl(χi[1],2
l
, χi[2],2
l
)
- .
Even if we condition on Zi[1], Zi[2], with nontrivial dominance, these terms are random
SLIDE 12
Residuals
Ri = 1 √ M
M
- l=1
- Xi
l − 1
2
- ηl(χi[1],1
l
) + 1 2 − Xi
l
- ηl(χi[1],2
l
) +
- Yi − 1
2
- ηl(χi[2],1
l
) 1 2 − Yi
- ηl(χi[2],2
l
)
- ,
Si = 1 √ M
M
- l=1
- Xi
l Y i l − 1
4
- φl(χi[1],1
l
, χi[2],1
l
) +
- Xi
l (1 − Y i l ) − 1
4
- φl(χi[1],1
l
, χi[2],2
l
) +
- (1 − Xi
l )Y i l − 1
4
- φl(χi[1],2
l
, χi[2],1
l
) +
- (1 − Xi
l )(1 − Y i l ) − 1
4
- φl(χi[1],2
l
, χi[2],2
l
)
- .
SLIDE 13
When will everything be well defined?
By construcion, E[Ri + Si] = 0, and E[ηl( χl)] = 0 = ⇒ E[Ai] = 0. Di = 1 4 √ M
M
- l=1
- φl(χi[1],1
l
, χi[2],1
l
) + φl(χi[1],1
l
, χi[2],2
l
) +φl(χi[1],2
l
, χi[2],1
l
) + φl(χi[1],2
l
, χi[2],2
l
)
- .
E[φl( χ1
l ,
χ2
l )] = 0, but what if parents are related?
Define inbreeding depression ι = 1 √ M
M
- l=1
E[φl( χl, χl)] E[Di] = ιFi[1],i[2]. Need this to be finite
SLIDE 14
. . . and then it gets really messy
To calculate the variance of Di, we need to know quantities like E
- φl(χi[1],1
l
, χi[2],1
l
) + φl(χi[1],1
l
, χi[2],2
l
) + φl(χi[1],2
l
, χi[2],1
l
) + φl(χi[1],2
l
, χi[2],2
l
) 2 and this will depend upon 4-way identities.
SLIDE 15
Identities required in expressions for variances
P P P P P
1 1 2 2
F F F F F
1122 1122 112 122 1212
~ ~ = = = = =
Parent 1 Parent 2
- r
- r
1 2
- r
Gene 1 Gene 2
SLIDE 16
QG coefficients required to describe variance
Additive variance σ2
A = 2 M
M
l=1 E[ηl(
χl)2] Dominance variance σ2
D = 1 M
M
l=1 E[φl(
χ1
l ,
χ2
l )2]
Inbreeding depression ι =
1 √ M
M
l=1 E[φl(
χl, χl)] Sum of squared locus-specific ι∗ =
1 M
M
l=1 E[φl(
χl, χl)]2 inbreeding depressions Variance of dominance effects σ2
DI = 1 M
M
l=1
- E[φl(
χl, χl)2] in inbred individuals −E[φl( χl, χl)]2 Covariance of additive and σADI =
2 M
M
l=1 E [ηl(
χl)φl( χl, χl)] dominance effects in inbred individuals
SLIDE 17
But it can be done . . .
And, since the actual trait value depends only on the two alleles carried at each locus, Var(Zi) only depends on pairwise identities: Var(Zi) = σ2
A(1 + Fii) + σ2 D(1 − Fii) + (σ2 DI + ι∗)Fii
+ 2FiiσADI − F 2
iiι∗.
(We have assumed that the ancestral population is in linkage equilibrium, or there is another term.)
SLIDE 18
Conditioning on trait values of parents
Z = ¯ z0 + 1 √ M
M
- l=1
- ηl(χ1
l ) + ηl(χ2 l ) + φl(χ1 l , χ2 l )
- .
Suppose ηl, φl uniformly bounded, and inbreeding depression ι =
1 √ M
M
l=1 E[φl(
χl, χl)] < ∞. Before conditioning, normality from a CLT. Knowing the trait values of the parents, tells us little about allelic states at any pair of loci. Mendelian inheritance independent at unlinked loci, guarantees normality residuals after conditioning much as in additive case. For Ai + Di have to work harder; use a variant of Stein’s method
- f exchangeable pairs.
SLIDE 19
Main result
◮ Trait values across pedigree well approximated by multivariate
normal;
◮ Trait can be decomposed into two (normally distributed)
parts, (Ai + Di) (shared by all individuals in family) and (Ri + Si) (picked independently for each individual in the family);
◮ Conditioning on the parental trait values does not affect the
variance of the trait values among offspring, and it shifts mean trait values in a predictable way. The effect of conditioning can be calculated in the usual way: xA xB
- ∼ N
µA µB
- ,
ΣAA ΣAB ΣBA ΣBB
- .
Then xA|xB ∼ N
- µA + ΣABΣ−1
BB(xB − µB), ΣAA − ΣABΣ−1 BBΣBA
- .
SLIDE 20
Key conditions for the approximation
◮ Pedigree not too inbred. (Σt not too small); ◮ Parental trait values not too extreme. (∆t not too big); ◮ inbreeding depression, ιF12 not too big.
When we say that the infinitesimal with dominance holds, we mean that the trait values across the pedigree are a multivariate normal; the distribution of trait values across the whole population may be far from normal. Knowing trait values of the parents shifts the mean in a predictable way; the variance is independent of the parental trait values, and is determined by the pedigree and variance components in the ancestral population.
SLIDE 21