No, there's nothing to bluff about, since anyone familiar with
molecular systematics is aware of how common it is for researchers in
that field to try to calculate divergence times of groups based on
rough estimates of mutation rates, and the fact that these estimates
don't give ridiculous numbers like a minimum of 300 trillion years for
the time of divergence of mammals and reptiles with the maximum
conceivable rate of mutation (and these rates are indeed
well-constrained by empirical data) implies that mutations do have
something do with the genetic differences between living things.
In cases where we don't have direct biochemical measurements of
mutation rates, this "molecular clock" is calibrated by geological
data. Superficially this is circular reasoning, but that would only be
true if you used the rate of mutation derived from the divergence time
of those two groups to show that mutations were responsible for the
genetic divergence of exactly those two groups; if you apply that
calibration method to *other* groups, one would wonder what process is
forcing the other groups to match up with their time of appearance in
the fossil record, as predicted by the mutation rate calculated using
the calibration of the molecular clock using the first two groups.
And that's just the beginning of the involvement of mutations with the
genetic differences between major groups of living things. Genes
which code for functionally less constrained proteins can be expected
on theoretical grounds to evolve more quickly, and when we compare two
groups, they tend to be more dissimilar in the parts of DNA that code
for exactly those types of proteins, indicating again that it's the
mutation process that's leading to the DNA differences between the
major groups. More generally, it's common for molecular systematists
to talk about how "such and such a protein is known to evolve at a
high rate, or mitochondrial DNA evolves at a higher rate than nuclear
DNA, and that's why we find this large divergence between groups when
we use those high-rate evolvers as a measure of genetic divergence."
So that's pretty much the big picture on the reasons why we think
mutations are responsible for the major genetic differences between
living things (if we ever had to wonder, really; I mean seriously,
since mutations happen in the genome, and nothing much else, and there
are differences in the genome, you would really have to suspect that
it was the mutations in there causing the differences and not Super
Yahweh of the Bible or whatever fantasy Fundamentalists want to think
up as an alternative was responsible).
And the fact that mutations are responsible for genetic divergences
among living things is just a subset of the more general evidence that
the genome in general has been modified by physical processes, and not
The Divine Command of Yahweh; other examples of this set of processes
include evidence that many genomes have undergone the physical
processes of gene duplication, genome duplication, and DNA viral
insertion.
Now, you wanted some math, as in you were in your mode where you're
pretending to want to learn rather than your usual blustering attempts
to bolster your religious prejudices with attempts at criticizing the
scientific data. The following Web site outlines the basics of
calculating the chimp-human divergence time from known measured
mutation rates among humans:
http://sandwalk.blogspot.com/2013/03/estimating-human-mutation-rate.html
You can use that, or you can use Wikipedia's numbers to do the
calculations yourself. I'll take the latter route, and you can follow
either Larry Moran's reasoning there or mine here.
The human genome is about 3200 mega-bases long (or 3.2*10^9 bases;
we'll be using scientific notation to simply and clarify the
calculation). The human mutation rate is about a couple of hundred
millionths of a mutation per base pair per generation, according to
Wikipedia, i.e., 10^-9 yr-1, if you want to use fancy hertz notation
for the mutation rate, but written out in long-hand this would be
10^-9 mutations per base pair per year, where I've converted from
generations to years by assuming a generation time of 20 years). This
idea of mutations per base pair per year seems too complex to allow us
to use our familiar formula of
distance = rate * time
but it's not really. The quantity "mutations per base pair per year,"
with all those per's in it is just a regular rate of change if we
refer to mutations per base pair as the quantity whose rate of change
we calculate by dividing by years. So let's plug it into the distance
formula
rate = 10^-9 yr -1 (i.e., that many something or others per year,
where the something or other is the concatenation of words: mutations
per base pair). Okay.
time = ?
Well, we can estimate this from our knowledge of hominid evolution.
The first australopithecines showed up around 4 million years ago.
These primitive ape-human intermediates are similar enough to
chimpanzees that a round number of 5 million years for the time of
divergence between chimps and humans is not unreasonable.
so time = 5 million years = 5*10^6
Plug in the numbers to get:
(10^-9) *( 5*10^6)
Now, because we're using scientific notation, you don't even have to
use your calculator to do this multiplication (remember that with
scientific notation, you just add or subtract the exponents and leave
the other numbers alone). We get
5*10^-3 (since we added the -9 exponent to the 6 exponent
to get -3, and put the 5 right next to it)
Because our changing quantity was (mutations per base pair) per
(year), when we multiplied by years, we got mutations per base pair.
Now let's use the known size of the human genome, 3.2*10^9 bp, to
calculate how many mutations in one genome that would be (since
(mutations/base pair)*(base pair/genome) should equal mutations per
genome (using the type of dimensional analysis they taught us in high
school chemistry, or the laws of fraction cancellation we learned in
high school algebra). That's
(3.2*10^9) * (5*10^-3)
or 16*10^6
(with scientific notation, you actually have to multiply the
non-exponent numbers if there's more than one of them; you can't just
put the five there to get the answer like we did in the first
calculation; but your calculator will do this for you, so nothing to
worry about there). 16*10^6 is of course 16 million, since 10^6 is
a million.
That means in 5 million years, 16 million mutations would have
occurred in a mutating human or chimp genome. So if the human and
chimp genomes physically separated from each other 5 million years ago
and began to become different from each other due to the mutation
process, 16 million mutations would have occurred in the human genome
and 16 million mutations would have occurred in the human genome. The
human and chimp genomes are approximately the same size, so that we
can consider this to be one big genome of 3.2 giga bases (i.e., 3200
mega bases) that has accumulated 32 million mutations in it, 16
million from the human part of it that was mutating for 5 million
years, and 16 million from the chimp part of it that was mutating for
16 million years. What percentage of this genome has mutations in it?
To get that, we'll just divide the number of mutations by the size of
the genome.
The rules of scientific notation become a bit too unwieldy for this
last step in the calculation, so we'll just use regular numbers:
32 million/3.2 billion = 0.01
That means 1% of this big human-and-chimpanzee genome entity should
have mutations in it, which is pretty much the same thing as saying
that there should be a 1% difference between chimps and humans in
their genomes. Wikipedia lists the chimp-human difference as less
than 2%, so it looks like we got the right answer using the idea that
mutations (and not the Power of Jesus at Creation) are the source of
the genetic differences between chimps and humans.
Now, that being said, I have one more thing to say. You seized on the
last statement in my post about the origin of nested hierarchies, as
if that nested-hierarchy evidence for evolution was somehow weaker
than the math-based one you claimed you were interested in. But I'm
not going to buy the idea that you evaluate the evidence in such an
idiotic proof-text based manner.
You don't need a mathematical disproof of your views; the smoking-gun
association of the biological nested hierarchy with common descent
(due to its lack of functional or logical structure) should in itself
have been enough to show you that you evolution is the correct
explanation of the origin of life's diversity -- unless you want to be
an irrational person, in which case there's no hope for you.