PBG 650 Advanced Plant Breeding

Module 4: Quantitative Genetics– Components of phenotypes– Genotypic values– Average effect of a gene– Breeding values

Definition:

“Statistical branch of genetics based upon fundamental Mendelian principles extended to polygenic characters”

Primary goal:

To provide us with a mechanistic understanding of the evolutionary process

What is Quantitative Genetics?

Lynch and Walsh, Chapter 1

• How much of the observed phenotypic variation is due to genetic vs environmental factors?

• How much of the genetic variation is additive (can be passed on from parent to offspring)?

• What is the breeding value of the available germplasm?

• Are there genotype by environment interactions?

• What are the consequences of inbreeding and outcrossing? What are the underlying causes?

• Are there genetic correlations among traits?

Questions of relevance to breeders

• Answers to these questions will influence

– response to selection

– choice of breeding methods

– choice of parents

– optimal type of variety (pureline, hybrid, synthetic, etc.)

– strategies for developing varieties adapted to target environments

Questions of relevance to breeders

Phenotypic Value

P = phenotypic value

G = genotypic value

E = environmental deviation

Bernardo, Chapt. 3; Falconer & Mackay, Chapt. 7; Lynch & Walsh, Chapt. 4

P = G + E

For the population as a whole:E(E) = 0

= E(P) = E(G)

Cov(G, E) = 0

Components of an individual’s Phenotypic Value

For individual k withgenotype AiAj

P(ij)k = + gij + e(ij)k

Single locus model

A1A1A2A2 A1A2

-a 0 d a

degree of dominance = a

d

no dominance d = 0partial dominance 0 < d < +a or 0 > d > –a complete dominance d = +a or –aoverdominance d > +a or d < –a

z z+a+d z+2a GenotypicValue

CodedGenotypic Value

The origin ( ) is midway between the two homozygotesP

a-P dP aP P

Single locus model

Different scales have been used in the literature

0 (1+k)a 2a

A1A1A2A2 A1A2

Lynch & Walsh

-a 0 d a

Conversions can be readily made

Falconer

0 au a

0 a 2a+d

Comstock and Robinson (1948)

Hill (1971)

Population mean

FrequencyGenotypic

valueFrequency

x value

A1A1 p2 a p2a

A1A2 2pq d 2pqd

A2A2 q2 –a –q2a

M = p2a + 2pqd – q2a= a(p2 – q2) + 2pqd= a(p + q)(p - q) + 2pqd= a(p - q) + 2pqd

contribution from homozygotes and heterozygotes

Mean on coded scale(centered around zero)

This is a weighted average

Mean on original scale

Population mean

M = a(p - q) + 2pqd

When there is no dominance a(p - q) When A1 is fixed aWhen A2 is fixed -aPotential range 2a

If the effects at different loci are additive (independent), then

M = Σa(p - q) + 2Σpqd

= P + a(p - q) + 2pqd

Means of breeding populations

= P + a(p - q) + 2pqd

In an F2 population, p = q = 0.5

F2 = P + (1/2)d

In a BC1 crossed to the favorable parent, p = 0.75,so after random mating

BC1(A1A1) = P + (1/2)a + (3/8)d

In a BC1 crossed to the unfavorable parent, p = 0.25,so after random mating

BC1(A2A2) = P - (1/2)a + (3/8)d

For ½ A1A1, ½ A1A2

= P + ½(a + d)

For ½ A1A2, ½ A2A2

= P + ½(d - a)

Average effects

• We have defined the mean in terms of genotypic values

• Genes (alleles), not genotypes, are passed from parent to offspring

• Average effect of a gene (i)

– mean deviation from the population mean of individuals who received that gene from their parents (the other gene taken at random from the population)

Gamete

A1A1

aA1A2

dA2A2

-a Freq x valueAverage effect

of a gene

A1 p q pa + qd 1=q[a+d(q-p)]

A2 p q pd - qa 2=-p[a+d(q-p)]

subtract M = a(p - q) + 2pqd

Average effect of a gene substitution

• a and d are intrinsic properties of genotypes

1, 2, and are joint properties of alleles and the populations in which they occur (they vary with gene frequencies)

Average effect of changing from A2 to A1

= 1 - 2

q[a+d(q-p)] – (-p)[a+d(q-p)]

= a+d(q-p)Average effect of changing from A1 to A2 = -

Relating this to the average effects of alleles:

1 = q 2 = -p

Breeding Value

Breeding value of individual Aij = i + j

Breeding Value

GenotypeAverage effect of a

geneAverage effect of a gene substitution

A1A1 21 2qA1A2 1 + 2 (q - p)A2A2 22 -2p

• For a population in H-W equilibrium, the mean breeding value = 0

• The expected breeding value of an individual is the average of the breeding value of its two parents

• For an individual mated at random to a number of individuals in a population, its breeding value is 2 x the mean deviation of its progeny from the population mean.

Regression of breeding value on genotype

Breeding values

• can be measured

• provide information about genetic values

• lead to predictions about genotypic and phenotypic values of progeny

Additive genetic variance• variance in breeding values

• variance due to regression of genotypic values on genotype (number of alleles)

● genotypic value○ breeding value

Genotypic values

• Genotypic values have been expressed as deviations from a midparent

• To calculate genetic variances and covariances, they must be expressed as a deviation from the population mean, which depends on gene frequencies

Genotypic values Genotype Scaled Adjusted for mean

A1A1 a 2q(a-pd) 2q(-qd)A1A2 d a(q-p)+d(1-2pq) (q-p)+2pqdA2A2 -a -2p(a+qd) -2p(+pd)

subtract M = a(p - q) + 2pqd

Remember = a + d(q - p) Substitute a = - d(q - p)

Dominance deviation

Components of an individual’s Phenotypic Value P = G + E

G = A + D

• In terms of statistics, D represents – within-locus interactions– deviations from additive effects of genes

• Arises from dominance between alleles at a locus– dependent on gene frequencies– not solely a function of degree of dominance– (a locus with completely dominant gene action contributes

substantially to additive genetic variance)

Gij = + i + j + ij

Partitioning Genotypic Value

GenotypeGenotypic Value (adj. for mean)

Breeding Value(additive effects)

Dominance Deviation

A1A1 2q(-qd) 2q -2q2dA1A2 (q-p)+2pqd (q - p) +2pqdA2A2 -2p(+pd) -2p -2p2d

When p = q = 0.5 (as in a biparental cross between inbred lines)

Genotype Genotypic Value Breeding Value Dominance

A1A1 -(1/2)d -(1/2)dA1A2 (1/2)d 0 (1/2)dA2A2 --(1/2)d - -(1/2)d

Dominance deviations from regression

-2p2d

2pqd

-2q2dGenotypic Value

A1A1 2q - 2q2d

A1A2 (q-p)+2pqd

A2A2 -2p - 2p2d

Interaction deviation

• Components of an individual’s Phenotypic Value

P = G + E

P = A + D + E• When more than one locus is considered, there may also be

interactions between loci (epistasis)

G = A + D + I

P = A + D + I + E• ‘I’ is expressed as a deviation from the population mean and

depends on gene frequencies

• For a population in H-W equilibrium, the mean ‘I’ = 0