1) Quantitative genetics focuses on inheritance of quantitative traits controlled by multiple genes and influenced by the environment. 2) A basic single-gene model is used to explain quantitative genetic theory, including calculations of population mean, genetic effects, and variance components. 3) More complex multi-gene models and analyses like ANOVA and heritability are then introduced to better capture quantitative traits controlled by numerous genes and environmental influences.

1. QUANTITATIVE
GENETICS
Avjinder Singh Kaler

2. Introduction
• Quantitative Genetics: Focus on the inheritance of quantitative trait
• Number of genes controlling a trait increases and importance of the
environmental effect on phenotype of trait increases

3. Single-Gene Model
• Quantitative Genetic theory start with single gene model
• Single locus A with two alleleA and a and have three possible genotypes; AA, Aa, aa in a
population and three assigned genotypic values; a, d and –a
• Two alleles have frequency: p and 1-p =q
• Population mean 𝜇 in terms of allelic frequencies and genotypic values
• 𝜇 = 𝑝2a + 2pqd − 𝑞2a
• Deviation of genotypic value of AA from the population mean
• AA = 𝑎 − 𝜇 = 𝑎 − 𝑝2a + 2pqd − 𝑞2a = 2q(a − pd)
• aa = 𝑑 − 𝜇 = 𝑑 − 𝑝2a + 2pqd − 𝑞2a = −2p(a + qd)
• Aa = 𝑑 − 𝜇 = 𝑑 − 𝑝2a + 2pqd − 𝑞2a = a q − p + d(1 − 2pq)

4. Average Effect of Gene Substitution
• Average Effect of Gene Substitution α : average effect on the trait of one allele
being replaced by another allele.
• Gamete containing allele A: results in progeny with genotype AA and Aa
• Gamete containing allele a: results in progeny with genotype Aa and aa
• Mean value of genotype produced for two gametes:
• A = pa + qd and a = pd-qa
• α = A-a = (pa + qd) – (pd-qa)= a + (q-p)d

5. BreedingValue
• Average genotypic value of its progeny
• α =Genetic effect of gametes transferred to progeny
• AA receive two copies of alleleA and genetic effect 2 α
• Aa receive one copy of alleleA and genetic effect α
• Average = 2p α
• Breeding values of three Genotypes:
• AA =2p α
• Aa=(q-p) α
• Aa=-2p α
• Large breeding value important for genetic improvement

6. BreedingValue
• When dominance =0, breeding value of AA and Aa have linear relationship with
the frequency of A allele increase
• Gene with rare favorable allele has more potential breeding significance than gene
with a favorable allele already at median or high frequency in the population.
• Rare favorable alleles contribute less than median and high frequency favorable
alleles to population mean and additive variance

7. Dominance Deviation (DD)
• Breeding values for a single locus are additive effects of genotypic values
• Dominance deviation is portion of genotypic values which cannot be explained by
breeding values
• Obtained by subtraction of breeding value from the genotypic value
• DD for AA: 2q(a-pd) -2p[a+ (q-p)d]=-2𝑞2
a, for Aa= 2pqd, and for aa=-2𝑝2
a

8. Variance
• Total genetic variance in a population is the variance of the genotypic values
• Genetic variance 𝜎 𝐺
2
= 2𝑝𝑞𝛼2
+ 4𝑝2
𝑞2
𝑑2
• Additive genetic variance 𝜎𝐴
2
= 2𝑝𝑞𝛼2
• Dominance variance 𝜎 𝐷
2
= 4𝑝2 𝑞2 𝑑2
• Genetic variance 𝜎 𝐺
2
=𝜎𝐴
2
+ 𝜎 𝐷
2
• When p=q=0.5, the additive variance has no relation to the degree of dominance
and dominance variance reaches its maximum
• Under complete dominance d=1, additive variance reaches its maximum at p =1/3
• Genetic variance is small when allelic frequency of A is less than 5%

9. Trait Model
• Under quantitative genetic assumptions, a trait may be controlled by a number of genes
• However, in classical quantitative analysis, number of genes and their genotypic effect are usually
unknown
• A simple model for continuous trait: 𝑦𝑖𝑗 = 𝜇 + 𝐺𝑖 + 𝜀𝑖𝑗
• 𝑦𝑖𝑗 = trait value of genotype i in replication j, μ = population mean, 𝐺𝑖 = genetic effect for genotype i, 𝜀𝑖𝑗 =
error term associated with genotype i in replication j
• All component in model are distributed as normal variables
• y ~ N(𝜇, 𝜎 𝑝
2
), G ~ N(0, 𝜎𝑔
2
), 𝜀 ~ N(0, 𝜎𝑒
2
)
• Covariance between genetic effect and experimental error is zero, then
• 𝜎 𝑝
2
= 𝜎𝑔
2
+ 𝜎𝑒
2
• Same genotype is replicated in b times in an experiment and phenotypic means are used, then relation
becomes: 𝜎 𝑝
2
= 𝜎𝑔
2
+ 1/𝑏𝜎𝑒
2

10. ANOVA
• If same genotype are tested in several environments such as locations or years, then simple model of equation becomes
• 𝑦𝑖𝑗𝑘 = 𝜇 + 𝐺𝑖 + 𝐸𝑗 + (𝐺𝐸)𝑖𝑗+ 𝜀𝑖𝑗
• Here is 𝐸𝑗 = environmental effect and (𝐺𝐸)𝑖𝑗 = genetic by environmental interaction effect
• ANOVA is used to estimate variance components associated with model of equation
Source DF EMS
ENV e -1
BLOCKS (b-1)e
Genotypes g -1
G x E (g-1)(e-1)
Error (b-1)(g-1)e

11. Heritability
• Ratio of genotypic to phenotypic variance H =
𝜎 𝑔
2
𝜎 𝑝
2 =
𝜎 𝑔
2
𝜎 𝑔
2 + 𝜎 𝑒
2
• Broad sense heritability = total genetic variance (additive, dominance, and
epistatic interaction)/ phenotypic variance
• Narrow sense heritability = additive variance/phenotypic variance

12. Genetic Correlation
• Two related traits with models
• 𝑦1𝑗 = 𝜇 + 𝐺1𝑖 + 𝜀1𝑗
• 𝑦2𝑗 = 𝜇 + 𝐺2𝑖 + 𝜀2𝑗
• Relationship between two traits quantified by
• 𝜌 𝑝
=
𝜎 𝑝12
𝜎 𝑝1
2 𝜎 𝑝2
2
, 𝜌 𝑔
=
𝜎 𝑔12
𝜎 𝑔1
2 𝜎 𝑔2
2
, 𝜌 𝑒
=
𝜎 𝑒12
𝜎 𝑒1
2 𝜎 𝑒2
2
• Genetic correlation between two traits may be caused by linkage of genes or same
gene controlling both traits (pleiotropy)
• Pleiotropic effect could be explained by a physiological relationship between traits