Transcript Quantitative genetics and breeding theory

Quantitative genetics
and breeding theory
Mini-course by Dag Lindgren
[email protected]
Acknowledgements to Darius Danusevicius for assistance in the lay out
Message from a senior and old
professor of Forest Genetics
At least one week attention on the concepts
behind TBT is needed each five years:
• For all who call themselves forest tree
breeders
• For all who get the doctors title in forest
genetics in the future
• For most professional forest geneticists
General website:
http://www.genfys.slu.se/staff/dagl/
In particular “Tree Breeding Tools” (TBT)
http://www.genfys.slu.se/staff/dagl/Breed_Home_Page/
The start of this course is almost identically given at
http://www.genfys.slu.se/staff/dagl/Breed_Home_Page/Tutorials
/Quant_Gen/Kurs01A_for_site.htm
This mini-course is much my personal view of the use of
quantitative genetics applied to forest tree breeding. Other
“schools” have other emphases.
Some concepts are established. Other I, or collaborators, have
coined. Most, but not all, stuff presented is published
somewhere,
Common assumptions
• One character – but may be
composite!
• Diploid zygotes and haploid
gametes
Meiosis
Haploid gametes
Diploid zygote
Mitosis
Diploid progeny
Semantics
• Many misunderstandings and conflicts are semantic (a
matter of definitions)
• Important to speak the same language – and use the
same symbols – at least within group. The second best
is to understand that people speak different languages.
This is
a tree
No it’s a
plant
The art of breeding is combining
a lot of things in a good way!
Gain(BV)
Environments
Genetic
parameters
Coancestr
y
Gene
diversity
Interaction
s
Inbreeding
Cost
Technique
To do that effectively, we must
have quantitative concepts and
measures
To optimize, a quantitative measure
must be defined and maximized!
Some concepts useful for quantitative genetics
Identical by descent (IBD) means that genes at the same locus are
copies of the same original gene in some ancestor.
The chance that both homologous genes in the same zygote are
identical by descent is called inbreeding (F) (or coefficient of
inbreeding).
Coancestry (, f) between pair of individuals is the probability
that genes, taken at random from each of the concerned
individuals, are identical by descent (=coefficient of coancestry).
A quantification of relatedness. We will widen that concept!
Self-coancestry: An individual's coancestry with itself is 0.5(1+F).
This can be realised e.g. by considering that coancestry in the
previous generation becomes inbreeding in next, and then
consider selfing.
If two individuals mate, their coancestry becomes the
inbreeding of their offspring.
Note that inbreeding and coancestry are relative to a
situation with no inbreeding or relatedness.
Founder population is the starting point of calculations. If all
inbreeding and coancestry of the founder population is known,
inbreeding and coancestry can be calculated from a pedigree. It
is usually practical and convenient to set inbreeding and
coancestry to zero in the "wild forest" (or source population)
and see the founders (plus trees) as a sample from the wild
forest.
Inbreeding and coancestry are relative to some real or imaginary
"base" or "reference" or "source" population. Most
conveniently this is the founder population or the wild forest.
Self-coancestry: An individual's coancestry with itself is 0.5(1+F).
This can be realised e.g. by considering that coancestry in the
previous generation becomes inbreeding in next, and then
consider selfing.
Gene pool means all genes in a population. It is convenient to
consider genes at one locus. The gene pool is independent on
how (or if) a population is organised in zygotes.
Gene pool
A population with N zygotes has 2N genes in the
gene pool
....2N
e.g 2 zygotes with 4 genes in the picture above
Each gene has the frequency 1/2N
....2N genes
Arrows = sampling with replacement (or infinite
copies of each gene)
Probability to sample the same gene twice is 1/2N
Probability that different genes will be sampled is
(1-0.5/N).
Genes can be IBD (identical by descent).
The probability is the coancestry (f).
f= coancestry
The gene pool is often structured in individuals
f = coancestry
Individual A
Individual B
The probability that the genes in two specific
individuals are IBD is the coancestry between
these two individuals.
The probability that the two different genes in the
same zygote are IBD is the coefficient of
inbreeding (F).
F= inbreeding
For self-coancestry, the genes need not to be
different. If they are different f=F; if the same
f=1; average f=(1+F)/2.
Different mechanisms genes sampled
from a population may be IBD:
1. The same gene sampled twice (drift);
2. The genes are homologous genes from
the same individual (inbreeding),
3. The genes originate from different
individuals (relatedness).
Coancestries arranged in a
coancestry matrix
Ind
1
2
3
1
0.5
0.25
0
2
0.25
0.5
0
3
0
0
1
We denote a certain value by f2,1=0.25
Symmetric, thus f2,1= f1,2
The values along the diagonal (self-coancestries) appear only once.
Coefficient of relationship are often arranged in such a matrix, (numerator
matrix), in absence of inbreeding these values are double as large.
Examples of coancestry
Coancestries are probabilities, thus 0 f 1.
Relative
Coancestry (f,)
Unrelated
0
Half sibs
0.125
Full sibs
0.25
Parent-offspring
0.25
Cousin
0.0625
Itself (selfcoancestry)
0.5
Group coancestry
mother
aunt
uncle
sister
cousin
What is the average
relatedness (group coancestry)
of this ”family”?
Group coancestry
Let's put all homologous genes in a big pool and select two (at
random with replacement). The probability that two are IBD we
define as group coancestry. (, this term was introduced by
Cockerham 1967).
f
To get overall probability; average over all individual probabilities,
f.
Group coancestry equals the average of all N2 coancestry values
among all combinations of the N individuals in a population (or the
average of all 4N2 combinations of individual genes). We could as
well define group coancestry as this average, the advantage of the
probabilistic definition appears in more complex situations.
Ind
1
2
3
1
0.5
0.25
0
2
0.25
0.5
0
3
0
0
1
Sum of the 9 values in matrix= 2.5;
Average = group coancestry = 2.5/9 = 0.278
Note that self-coancestries appear once, while
other coancestries appear twice (reciprocals).
If all individuals in a population are related in the same pattern,
it is enough to calculate the N coancestries for a single
individual.
Self-coancestry is the group coancestry for a population with a
single member.
All members in a full sib family have equal coancestries to all
other individuals. Thus it is enough to construct the coancestry
matrix for full sib families (and make some thinking).
Group coancestry depends on relatedness, not how uniting
gametes are arranged. A brother is equally related to his
brother as to his sister, in spite of that his gametes are able to
unite only with those of his sister.
Group coancestry for families
Family size = n, no inbreeding
Half sibs
3 n

8n
Full sibs
n 1

4n
Self sibs
2n  1

4n
Group Coancestry may be expressed
N

N

i 1 j 1
2
N
ij
Pair-coancestry and Self-coancestry
The term pair-coancestry is used here for the average of all
coancestry-values among different individuals excepting selfcoancestry. Using “Coancestry” for “pair-coancestry” invites
to misunderstandings.
Group-coancestry can be separated in two types: Selfcoancestry and pair-coancestry.
The term pair-coancestry (or avearage pair coancestry) is my
own construct, I am not sure it is best or has not been better
coined by someone else. I have used cross-coancestry earlier
and Ola used pair-wise-coancestry, but I did not see what
”wise” was good for.
Ind
1
2
3
1
0.5
0.25
0
2
0.25
0.5
0
3
0
0
1
Pair-coancestry for this matrix is 2*0.25/6=0.083
n

n
n
  
i 1
ii
i 1 j  i
2
n
ij

0.5  0.5  1  0.25  0.25

3
2
Pair-coancestry, Inbreeding and
Group Coancestry relations
A population can be described by:
• Inbreeding (or average self-coancestry)
• Group-coancestry
• Pair-coancestry
If two are known, the third can be derived
Using the following relationships, group coancestry and
average pair-coancestry can be derived
1  F  (2 N  2) f

2N
f 
N  0.5(1  F )
N 1
where:
 = group coancestry;
N = individuals;
f = average pair-coancestry;
F = average inbreeding.
Linking generations
Group coancestry changes at generation shifts can be calculated
retrospectively from a known pedigree linking to the founders.
Future group coancestry can be calculated with knowledge or
assumptions about future pedigrees. For other cases predictions
may be made, but this is often far from trivial.
Note that there may be doubt if assumptions are realistic (neutral
selection, many genes with infinitesimal action etc.)
The link between generations is the gametes.
parents
offspring
The gene pool of the offspring is identical
to the gene pool of the successful gametes
of the parents.
Consider a pair of genes, which may equivalently be regarded as
in offspring zygotes or in parental successful gametes!
A pair of genes in offspring may be IBD as they are copies of the
same gene in the parent population. This may happen if a parent
has more than one offspring.
2Nparents parents
2N
offspring
2N
offspring offspring
offspring
A pair of genes may originate from homogenous genes of the same
parental zygote in the parental generation, if that was inbred, the
considered genes may be IBD.
F
2Nparents parents
2Noffspring offspring
Different gametes from a parent get coancestry (1+Fparent)/2
Sibs sharing that parent (half sib) get coancestry (1+Fparent)/8.
If the considered gene pair originates from different parents, the
coancestry will be fparent.
f
2Nparents parents
2Noffspring offspring
IBD may occur by the following
mechanisms:
1. The same gene in the current generation is
sampled twice,
2. The genes are copies of the same gene in the
parental generation,
3. The genes origin from homologous genes in
the same inbred parent,
4. The genes come from different, but related,
parents.
Gene diversity!
Group coancestry and gene diversity
• Group coancestry is the
probability that two genes are IBD;
• Diversity means that things
are different;
• Gene Diversity means that
genes are different.
• Evidently 1 - group coancestry is
the probability that the genes are
non-identical, thus diverse.
GD = 1 - group coancestry
is the probability that the genes are
non-identical, thus diverse.
GD  1  
GD is Gene Diversity!
Group coancestry is a measure of
gene diversity lost!
That seems to be something worth
knowing!
This way of thinking sees all genes in the source (reference)
populations as unique (“tagged”).
GD is similar to expected average heterozygosity (the chance
that two genes are different).
Group coancestry based measures are (like inbreeding)
relative to some reference population. For forest tree
breeding the wild forest usually constitutes a good
reference. The gene diversity of the wild forest is 1, and
the group coancestry is the share of the initial gene
diversity lost.
Monitor group coancestry in tree improvement
operations! That says how much gene diversity has
been lost since the initiation of the breeding program!
Deriving coancestry and group
coancestry
An algorithm for calculation of coancestry and group coancestry
(example from Lindgren et al 1997).
Tabulate pedigree for the population, points (.) for founders.
Parents always defined before used as parents.
Task: Calculate group coancestry of reds!
1
2
5
6
10
3
4
7
9
8
11
12
1,2,3,4,9 and one parent to 13 can
be considered founders.
13
Ind
1
2
3
4
5
6
7
8
9
10+
11+
12+
13+
Parent A
.
.
.
.
1
2
2
3
.
5
7
8
9
Parent B
.
.
.
.
1
3
3
4
.
6
8
9
.
Calculation of the coancestry matrix.
Pedigree for population in the example.
Fill the matrix (thus the coancestry of all pair of the 13
individuals) using the pedigree information.
This can be done step by step.

Fill rows from left to right
Ind
Parent A

Start with the diagonal element
1
.

Proceed leftwards to the row’s end
Ind
1
2
3
4
5
1
0.5
0
0
0
0.5
6
7
8
9
Parent B
.
2
.
.
3
.
.
4
.
.
5
1
1
10+
11+
12+
13+

As the matrix is symmetric, column values can be filled
from the row

Start with next diagonal
Ind
1
2
3
4
5
6
7
8
9
10+
11+
12+
13+
1
0.5
0
0
0
0.5
0
0
0
0
0.25
0
0
0
2
0
0.5
The matrix below has been filled to element (6,6). Individual 6 has parents 2
and 3, it is demonstrated how diagonal element (6,6) is filled.
Ind
1
2
3
4
5
6
7
8
9
10+
11+
12+
13+
1
0.5
0
0
0
0.5
0
0
0
0
0.25
0
0
0
2
0
0.5
0
0
0
0.25
0.25
0
0
0.125
0.125
0
0
3
0
0
0.5
0
0
0.25
0.25
0.25
0
0.125
0.25
0.125
0
4
0
0
0
0.5
0
0
0
0.25
0
0
0.125
0.125
0
5
0.5
0
0
0
0.75
0
0
0
0
0.375
0
0
0
0
0.25
0.25
0
0
0.5
6
The diagonal (6,6)=0.5+(3,2)=0.5+0
Self-coancestry = (1+F)/2 = average
of 0.5 and coancestry for parents 2
and 3.
Ind
Parent A
Parent B
1
.
.
2
.
.
3
.
.
4
.
.
5
1
1
6
2
3
The matrix below has been filled to element (6,7). Individual 8 has parents 3
and 4, it is demonstrated how off-diagonal element (6,8) is filled.
Ind
1
2
3
4
5
6
7
8
9
10+
11+
12+
13+
1
0.5
0
0
0
0.5
0
0
0
0
0.25
0
0
0
2
0
0.5
0
0
0
0.25
0.25
0
0
0.125
0.125
0
0
3
0
0
0.5
0
0
0.25
0.25
0.25
0
0.125
0.25
0.125
0
4
0
0
0
0.5
0
0
0
0.25
0
0
0.125
0.125
0
5
0.5
0
0
0
0.75
0
0
0
0
0.375
0
0
0
6
0
0.25
0.25
0
0
0.5
0.25
0.125
3
.
.
4
.
.
5
1
1
6
2
3
7
2
3
8
3
4
The off diagonal
the average of coancestry with 6 and the
parents to eight (3 and 4)
(6,8)=0.5[(6,3)+(6,4)]=0.5[0.25+0]=0.125
The average of the parents to 7’s coancestry
with 6.
The full coancestry matrix. Group coancestry is wanted for 10-13
1
2
3
4
5
6
7
8
9
10+
11+
12+
13+
0.5
0
0
0
0.5
0
0
0
0
0.25
0
0
0
2
0
0.5
0
0
0
0.25
0.25
0
0
0.125
0.125
0
0
3
0
0
0.5
0
0
0.25
0.25
0.25
0
0.125
0.25
0.125
0
4
0
0
0
0.5
0
0
0
0.25
0
0
0.125
0.125
0
5
0.5
0
0
0
0.75
0
0
0
0
0.375
0
0
0
6
0
0.25
0.25
0
0
0.5
0.25
0.125
0
0.25
0.188
0.063
0
7
0
0.25
0.25
0
0
0.25
0.5
0.125
0
0.125
0.313
0.063
0
8
0
0
0.25
0.25
0
0.125
0.125
0.5
0
0.063
0.313
0.25
0
9
0
0
0
0
0
0
0
0
0.5
0
0
0.25
0.25
10+
0.25
0.125
0.125
0
0.375
0.25
0.125
0.063
0
0.5
0.094
0.031
0
11+
0
0.125
0.25
0.125
0
0.188
0.313
0.313
0
0.094
0.563
0.156
0
12+
0
0
0.125
0.125
0
0.063
0.063
0.25
0.25
0.031
0.156
0.5
0.125
13+
0
0
0
0
0
0
0
0
0.25
0
0
0.125
0.5
Ind
1
The red population get the red coancestry values, the group
coancestry for the population 10-13 is the average of the red values
(= 2.875/16=0.1797).
Status number
• Status number is half the inverse of
group coancestry
1
NS 
2
Or, equivalently
• Status number is half the inverse of the
probability that two genes drawn at
random are IBD.
1
NS 
2
Status Number
An attractive property of the status number is
that it is the same as the census number for a
population of unrelated, non-inbred trees.
Status number is an intuitively appealing way
of presenting group coancestry, as it connects
to the familiar concept of number (population
size).
Status number is an effective number. It relates
a real population to an ideal population.
The ideal population consists of unrelated, noninbred trees with the same probability of IBD.
Gene diversity as a function of
status number
1
GD  1    1 
2NS
Note that 1/2N is familiar in genetics!
The status number says that the probability to draw two genes IBD
is the same as if it were so many unrelated non-inbred individuals
contributing to the gene pool. Therefore we can call it an effective
number.
The ratio of the status number and the census number is useful,
thus Nr=Ns/N. I call this the relative status number.
An example of the predicted drop of status number over time in a
breeding program
POPSIM simulation; BP=100; four controlled matings made for each member of
the breeding population, the family size was 40, the next generation was recruited
from the previous by phenotypic selection, the initial heritability was 0.2.
(Lindgren et al 1997).
100
90
Status number
80
70
60
50
40
30
20
10
0
0
2
4
6
Generation
8
10
The drop of Gene Diversity
The same data looks less drastic when considering gene diversity!
This is to exemplify what may happen to Gene Diversity during breeding (from Lindgren et al
1997). Data from a simulated breeding program. POPSIM simulation; Breeding Population=100;
four controlled matings made for each member of the breeding population, the family size was 40,
the next generation was recruited from the previous by phenotypic selection (selecting the best 100
among the offspring considering only the phenotype), the initial heritability was 0.2.
Breeding population size 50, differently managed over generations
50
Mating Offspring/parent
Selfing
2 (1=2)
Full sibbing 2
random
2
random
random
45
Status number
40
35
30
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9
From Lindgren et al 1996, Silvae Genetica
Generation
10
Some properties of status number
• NS can never be higher than the census number (N);
q NS can never be lower than 0.5 (NS of a gamete);
q NS considers relatedness and inbreeding;
q NS may be derived for any hypothetical population (with known relatedness
patterns to a known source population). It is irrelevant if "population members"
belong to the same generation or the same “subpopulation”;
q NS cannot exceed the minimum N in any of the preceding generations, if all
ancestors are confined to a range of discrete generations;
q NS does not care about the gender of the population members;
q NS after a generation shift depends only on the number of offspring for each
parent;
q NS is independent on the mating patterns of the parents it is derived from;
q NS describes a gene pool, not how it is organised;
q NS usually declines at generation shifts, but it can rise if the initial genomes
may get a more equal representation after a generation shift than before.
Mating patterns matters for development of NS in later generations, and they are constraining for
possible values of NS, thus they are a relevant matter, even if not formally.
NS is closely associated to inbreeding, but the associations become cleared with the concept group
coancestry, they are better developed in connection to that concept.
Status number and group coancestry measure gene dispersion!
Cockerham (1969) concluded that the variance of the gene frequency (that’s the mean of the
occurrence of a gene) is
 2p  p(1  p)
this can equivalently be expressed
p(1  p)
p 
2Ns
2
This is the binomial expression of the variance for the gene frequency in a population with Ns noninbred non-related members!
Status number is the size of unrelated non inbred trees sampled
from the reference population, which have the same drift as the
accumulated drift of the population under study (compared to
the reference population).
Gene frequency
Generations
Gene frequency over generations
Status number
Effective number
An effective number (size) is an effort to characterise a complicated
system by the number of individuals in a simpler and more ideal
system, which have the same characteristic value or behaviour from
some important respect.
Effective population size in the inbreeding or variance sense
To understand how these concepts are usually used in genetics one
has to understand that they compare population dynamics to that of
an "ideal population". (Caballero, 1994 p 658 from Fisher 1930)
“An idealized population consists of infinite, randomly mated base populations subdivided
into infinitely many subpopulations, each with a constant number, N, of breeding
individuals per generation. In each subpopulation, parents produce an infinite number of
male and female gametes into a large pool from which only 2N are sampled and united to
produce the N zygotes of the following generation...Both the sampling of the gametes and
their union (including self-fertilization) are random, so that all parents have an equal
chance of producing offspring.... Generations do not overlap.”
(there are some less important omissions)
Based on this “The effective size of a population is defined as the size
of an idealized population which would give rise to the variance of
change in gene frequency or the rate of inbreeding observed in the
actual population under consideration”
Thus, effective population size says how a studied population
develops over generations compared to the development of the ideal
population. Note that the number is not associated to any particular
generation.
Usually two variants of effective population sizes are recognised, the
inbreeding sense and the variance sense (there are more).
The classical effective population size is the size of an ideal
population, which accumulates inbreeding or widen variance at the
same rate as the ideal population. The status number does not do
that. The status number measures a state, the classical effective
population size measure a rate.
The status number and the "traditional" effective population size
are sometimes similar when studied after respectively over the first
generation turn over, in particular when large progenies are
considered. An analogy: Distance and a speed may appear the
same, if studied over a unit time from a common starting point.
E.g. many results concerning diversity from Lindgren and Wei (e.g.
Wei 1995) and others can be considered as status number results
even if different variants of effective numbers or diversity has been
used. But then families are limited there is no equivalence.
Currently I believe effective population size in the inbreeding
sense is a concept we have better to forget about in forest tree
improvement, it is much better just to try to predict the
inbreeding than stray around in never needed - but complex calculations of an abstract and often misleading entity.
I have easier to see the need and accept some intuitively odd
characteristics for the effective population size in the variance
sense, there may be a need for such calculations, and status
number may be viewed as a complement.
Status number, group coancestry and variance
effective number
These concepts may (in an over-simplified world) be linked
N S  0.5 /(1  (1  0.5 / NV ) )
t
lim N S
 NV / t
NV  
Where NS = status number, NV variance effective number and t
generations
Can also be expressed:
t  (1  (1  0.5 / NV ) )
t
Gene frequency
Generations
Gene frequency over generations
Variance effective number
Status number
The initial founders matters, so the formulas are more relevant for the
development over generations that the absolute values.
Different effective numbers for the same object
It may be of interest to see how different effective numbers compare
for the same object, this has been done by Kjaer and Wellendorf
(1998) for a Norway spruce seed orchard and its crop:
Entity
Value
Number of clones in seed orchard
100
Variance effective population size
236.7
Inbreeding effective population size
18.1
Status number of the crop
70.4
Note that the effective population size expresses changes between the
100 parental clones and their progeny, while the status number
expresses the relationship between the orchard seeds and the base
population with unrelated non inbred trees (the “wild forest”) the
seed orchard clones were drawn from.
Status number may be interpreted
Status number is the number of clones drawn from the wild forest
which has the same group coancestry and gene diversity as the seeds
harvested in the orchard.
Inbreeding interpretation
Status number is the number of clones drawn from the wild forest
which following random mating would produce as much inbreeding as
expected in the seed crop of the forest created with the seed orchard
crop.
Drift interpretation
Status number is the number of clones drawn from the wild forest,
which has the same expected drift in gene frequencies as the seeds
harvested in the orchard. Note that the variance effective population
size is a measure of the drift between the seed orchard and its seed,
Inbreeding follows group coancestry
Simulation of Swedish Norway spruce breeding program
Probability of identity by descent
0.08
by POPSIM, BP=48, DPM,
equal representation (2/parent)
Message: Group coancestry can often
be regarded as a potential inbreeding,
which becomes realized some generations later
0.06
f
0.04
0.02
0
0
2
4
6
Generations
8
10
Rosvall, Lindgren & Mullin 1999
Group coancestry and Wright's
F-statistics
What is called FIS is the difference between inbreeding and
cross-coancestry. If Hardy-Weinberg balance they are equal
(the same chance of IBD if the genes are in the same as in
different individuals).
I have developed the relations with  as follows:
F    0.5(1  F ) / N
FIS 
1    0.5(1  F ) / N
Forest tree breeding and status
number
The status number concept is more useful to forest tree breeders
than other breeders or geneticists. Forest tree breeders :
• are still very close to the founders
• thus close to the "wild forest“, a natural
reference point for evaluating impact of
breeding
• deal with few generations
• change strategy between generations
• structure population in sublines
• "own" and control the breeding population