Lecture IV: Testing the syllogism 1
Last Time
1) Variation in parents = VAR(P)
2) Covariation between parents (mid-parent) & offspring = COV(O,P)
3) Heritability = COV(O,P) / VAR(P)
This Time
1) Our flow chart of relationships:
For interest only... Show that, for two points in x,y space, COV(x,y)/VAR(x) = slope of regression of y on x. Click here for answer.
In thinking of partitioning the total phenotypic variation for some trait within a population, it may be helpful to remember that variances are additive. In other words, you can think of the total phenotypic variation as a stick and break it into its components.
Heritability then is the proportion of the total phenotypic variation for some trait that is due to the additive effects of alleles.
VAR(P) is the total phenotypic variation [also = VAR(parents)]
VAR(G) is the total genetic variation,/ and is equal to VAR(A) + VAR(D) + VAR(to come)
VAR(A) is the "additive" genetic variance. Note that the additive genetic variance is some proportion of VAR(G).
Also, remember that VAR(A) is estimated as the COV(parents, offspring)
VAR(E) is the the variance due to random environmental perturbations.
And VAR(GxE) is the variance due to interactions between genotypes and environment.
2) Why is heritability usually less than 1?
A. Environmental variance
Consider a single genotype, Aa and first assume there is no environmental variance (aspects of the environment do not affect trait expression - here we'll use wing length).
As the above figure shows, there is no phenotypic variation in wing length for this genotype in our population.
Now let's assume there is some environmental variance (aspects of the environment do affect trait expression. Thus, at least part of the variation in wing length among individuals is due to variation in the environment experienced by each).
As the above figure shows, there is some phenotypic variation in wing length for this genotype in our population.
By adding the other two genotypes into the figure, we can see how the total phenotypic variation in the population is increased with the addition of an environmental variance term. (Using the variance stick above, before the total phenotypic variation was only due to the additive effects of alleles. Adding environmental variance must increase total phenotypic variance because variances are additive).
Because environmental variance increases the total phenotypic variation among parents and does not affect the covariance between offspring and parents (remember, this is random environmental variation), heritability MUST decrease. Graphically:
[HERE WE BROKE THE CLASS INTO GROUPS. THE ASSIGNMENT WAS TO DESIGN AN EXPERIMENT TO ESTIMATE VAR(E). One group designed experiments on clonal miniature cows. One group used clonal fish. One group used inbred cheetahs.]
B. Dominance variance
Consider a case of simple additivity. There is some "baseline" phenotype of 10 when no A alleles are present. Each A allele increases the phenotype by 10 units (so Aa = 20 & AA = 30). Graphically:
Now let allele A be dominant to allele a. Again, there is some "baseline" phenotype of 10 when no A alleles are present. Now, the addition of an A allele acts as though there are 2 A alleles present. Thus, Aa individuals have a phenotype of 30 and AA individuals have a phenotype of 30. Graphically:
We can construct a table that shows the average phenotype under strictly additive conditions or dominance conditions if we assume some population size (1 AA individual, 2 Aa individuals, & 1 aa individual). We can also determine the phenotypic variance in each population under strictly additive or strictly dominance conditions.
As you can see, there is greater phenotypic variance with dominance than there is under strictly additive conditions. Therefore, dominance variation will decrease measures of heritability relative to strictly additive scenarios. There are 2 reasons for this.
1) The phenotypic variation among parents is increased. We show this graphically, in a table format, and using the stick analogy (identical to the approach used above with environmental variance). The denominator of the heritability equation gets larger.
2) It is likely that the covariance between parents and offspring will decrease with dominance, relative to strictly additive scenarios. The numerator of the heritability equation gets smaller.
