Re: Fundamental theorems, dilemmas, fitness, and information.


"Perplexed in Peoria" <jimmenegay@xxxxxxxxxxxxx> wrote
>
> Not quite correct, John. Fisher says that there is an additive
> part and a non-additive part to the effect of any allele on fitness.
> Only the additive portion of the effect is heritable. At least that
> is how I understand it.
>
This is extremely confusing. There is broad-sense and narrow-sense
heritability.

We can partion the variance in the trait into variance due to genetic
factors and variance due to non-gentic factors. So broad sense heritability
would be

v[g] / (v[g] + v[e])

genetic variance over total variance.

However let's say that we have three loci, a, b and c. For each there is a
mutant allele which has a positive effect on out trait, tail length in
squirrels. For simplicity let's say that all are dominant (aA is the same as
AA).

In case 1, abc has a ten inch tail. Each allele contributes 1 inch, so abC
is 11 inches, Abc 11 inches, ABc 12 inches, ABC 13 inches, and so on.

In case 2, abc has a ten inch tail, but any one will extend the tail to 13
inches. Abc is 13 inches, ABc is also thirteen inches, etc.

In case 3, we have complex interactions. abc still has a ten inch tail.

Abc is 11 inches
aBc is 11 inches
abC is 11 inches
ABc is 13 inches
aBC is 12 inches
AbC is 11 inches
ABC is 10 inches

Now in all case we have set the environmental variance and the contribution
from other genes to zero. Tail length is wholly determined by there three
genes.Broad sense heitability is 1.

When we try to breed long-tailed squirrels, however, we find that only
population 1 responds as we had hoped. In population 1, children tend to be
intermediate between their parents (occasionally if you have a parent abC
and another ABc you will get a child who is ABC and thus longer tailed than
either, but the average child will be intermediate).

In case 2 we will find that we very rapdily get a population of long tailed
squirrels, but then "wild" short tailed squirrels keep on popping up,
because an Abc and aBc parent will frequently produce an abc bad recessive.

In case 3, results will be even more confusing. Once we fix ABc we have our
maximum length tails, but initial selection will tend to fix allele C, which
is advantageous in some circumstances but not others.

So only in the first case would narrow sense heritability be one. In the
other cases you would see some correlation between genotype and phenotype,
but not a nice straight line. Narrow sense heritability actually changes
according to the number of alleles in the gene pool. For instance, if allele
C in the third case is set to a very low frequency, then we have nearly an
additive situation, and narrow sense heritability is high. If C is very
common, narrow sense heritability will be low and the regression line all
over the place.




.

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