QUANTITATIVE GENETICS

Avjinder Singh Kaler

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

Single-Gene Model

• Quantitative Genetic theory start with single gene model

• Single locus A with two allele A 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)

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

Breeding Value

• Average genotypic value of its progeny

• α =Genetic effect of gametes transferred to progeny

• AA receive two copies of allele A and genetic effect 2 α

• Aa receive one copy of allele A 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

Breeding Value

• 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

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]=-2𝑞2a, for Aa= 2pqd, and for aa=-2𝑝2a

Variance

• Total genetic variance in a population is the variance of the genotypic values

• Genetic variance 𝜎𝐺2 = 2𝑝𝑞𝛼2 + 4𝑝2𝑞2𝑑2

• Additive genetic variance 𝜎𝐴2 = 2𝑝𝑞𝛼2

• Dominance variance 𝜎𝐷2 = 4𝑝2𝑞2𝑑2

• Genetic variance 𝜎𝐺2 =𝜎𝐴

2 + 𝜎𝐷2

• 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%

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, 𝜎𝑝2), G ~ N(0, 𝜎𝑔

2), 𝜀 ~ N(0, 𝜎𝑒2)

• Covariance between genetic effect and experimental error is zero, then

• 𝜎𝑝2 = 𝜎𝑔

2 + 𝜎𝑒2

• Same genotype is replicated in b times in an experiment and phenotypic means are used, then relation becomes: 𝜎𝑝

2 = 𝜎𝑔2 + 1/𝑏𝜎𝑒

2

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

Heritability

• Ratio of genotypic to phenotypic variance H = 𝜎𝑔2

𝜎𝑝2 =

𝜎𝑔2

𝜎𝑔2 + 𝜎𝑒

2

• Broad sense heritability = total genetic variance (additive, dominance, and epistatic interaction)/ phenotypic variance

• Narrow sense heritability = additive variance/phenotypic variance

Genetic Correlation

• Two related traits with models• 𝑦1𝑗 = 𝜇 + 𝐺1𝑖 + 𝜀1𝑗

• 𝑦2𝑗 = 𝜇 + 𝐺2𝑖 + 𝜀2𝑗

• Relationship between two traits quantified by• 𝜌 𝑝 =

𝜎𝑝12

𝜎𝑝12 𝜎𝑝2

2, 𝜌 𝑔 =

𝜎𝑔12

𝜎𝑔12 𝜎𝑔2

2, 𝜌 𝑒 =

𝜎𝑒12

𝜎𝑒12 𝜎𝑒2

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