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


"Malcolm" <regniztar@xxxxxxxxxxxxxx> wrote in message news:d9k7t3$1rlc$1@xxxxxxxxxxxxxxxxxxxxxx
>
> "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.

Thanks for the correction, Malcolm. Yes, I definitely was wrong
in my blanket statement that the non-linear aspects of epistasis
are "not heritible". I still don't completely understand the math,
but I wonder whether the following attempted revision of my
statement is closer to the truth:

The non-linear aspects of epistasis don't respond to selection well.
In the short term, their response is quite similar to the response
associated with the non-heritable component of fitness variance - v[e].
That is, there is no response.

Over the long term, there is a response, but one way of looking at
this is to treat fitness at a locus as dependent upon the gene
frequencies at all other loci. That is, we treat the variance in
what alleles are present at those other loci as if it were variance
in the environment. For example, if A/a is the focal locus, and
if the fitness of A (over a) depends on whether you have B or b at
some other locus, and if the frequency of B is 50%, then we can
calculate the average fitness of A as the average of the fitnesses
of AB and Ab. The variance (arising from the fitness difference
between AB and Ab) can be treated as part of v[e]. It is non-
heritible in this sense.


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