I'll try to make some more progress with my understanding your 59 allele
argument.
You wrote:
>With that as a back ground, a generation is, well, a generation-parent to
>child.
Ok, now I can ask more focused questions. (Some of what I asked in my last
post was due to my misunderstanding of what you meant by generation).
You wrote:
>If the logic of the situation requires that allele 2 be derived by
>random substitution from allele 1 then what you have is one of those puzzles
>I can never do, in which you are asked to go from the word "wore" to "bass" by
>making a one letter substitution at each step always having a real word at
>each step. In other words you can't have "wors" because that is not a word.
>wore-ware-wart-dart-dare-bare-bars-bass
>This is how allele 2 must be derived from allele 1.
>We know the probability of a substituion at each location each generation, so
>it is mere probability to look at allele 1 and tell how long (on average) it
>must have taken to derive allele 2. With six substituions, the odds are that
>it took 60,000 generations to make the six substituions. Since human
>generations are 20 years or so, then 20 x 60,000 = 1,200,000 years.
This calculation is way too optimistic. You assume all substitutions occur
at a rate of 10^-4, but really only the first one does. To pile up
substitutions, you quickly have to revert to the rate of 10^-7.
Actually, the math is a bit more complicated than this. I just worked out
the problem you specified above. Statistical Optics is my field, so I'm not
afraid to jump into a statistics problem - though mutations occur in my
derivations at a rate higher than 10^-7 :-). Here's what I get.
Assuming a probability *p* of having a mutation for each offspring at a
particular location, and further assuming the mutations never go backwards
(it would slow things down even more), we can calculate the relative
frequency of the alleles after N generations. I assumed an exponential
growth rate in the population, though I think I can relax that stipulation
since it canceled out in the end. This calculation also applies to a
specific location, (it won't change things too much to consider a multitude
of locations since we are looking for substitutions that pile up; it would
make less difference than stepping k down by 1). I label the alleles A0 (the
unmutated allele), A1, A2, A3.....Ak where k represents the kth mutation.
The expected relative frequency of Ak/A0 in the Nth generation is (1/k!)(pN)^k
Basically, this says that you'll never find anything until pN approaches 1.
Thus we need 10 million generations if p = 10^-7. We're up to 200 million
years to make an allele with 6 substitutions, and have it reasonably common
in the human population; i.e. A6/A0 = 1/720
>I do not know how common the alleles are in the population but that is not
>relevant to the problem at all.
By using your assumptions, I calculate that the total number of people in
the population who would have allele 2 is **very very very very** small. So
small, that I doubt the writers of the Scientific American article would
know they exist. How common the alleles are is an issue, because if they are
too rare, we won't know they exist. Many rare ones can exist after a short
time, but it takes 100s of millions of years to generate a large number of
common ones.
I'm not sure where my musings are leading me, but my point seems to be the
opposite of what Walter Remine was arguing. I think I'm saying that there's
no way after only a few million years 49 new alleles could be generated
**and be in sufficient quantity in the population** that the writers of the
SA article could find them. There must be some other explanation for their
existence - like the mutation rates are much higher, or something else.
>I am a little unclear what you are asking here but why should that stop me
from >answering. :-)
Well, then go ahead and answer this post, too.
Jim
Jim Blake
Associate Professor
Department of Electrical Engineering
Texas A&M University
College Station, TX 77843