\magnification=\magstep1
\leftline{Sarah and Isabella,}
\bigskip
Your paper and presentation on the Jukes-Cantor model and relations
with algebraic statistics were good although there were some (relatively 
small) inaccuracies
and also some things I would have liked to see explained in more detail to 
show you really understood what was going on.  Two of the difficult things about
projects like this are
\bigskip
\item{1)}  Learning new things as necessary (and I guess you 
needed to do a lot of that here).
\item{2)}  Reconciling the information you get from different
sources and understanding the terminology and notation they use. 
\bigskip
\noindent 
I think there were several times when similar notation used for different 
things in different sources confused you.  For instance on page 6, 
the $\pi_A, \pi_C, \pi_G, \pi_T$
should not be transition probabilities.  The quantities that are
all equal are (by assumption), the distribution of the nucleotides
over the whole genome.  In other words, if you pick a random location
in the genome, the probabilities of seeing $A,C,G,T$ at that location
are all the same.  

It would have been good to reorganize what you say and include
some more details about the exact relation 
between the JC rate matrix from page 7 is related to the transition
matrix from page 6.  They are equivalent in a sense, but I think
it's probably better to start from the rate matrix $Q$.  The idea is
that $\theta(t)$ has to be a solution of the differential 
equation $\theta'(t) = Q\cdot \theta(t)$ with $\theta(0) = I$ (identity matrix).
Then, as you said, $\theta(t) = e^{Qt}$, and $\theta(t)$ is given as in
the equation on your page 7.  The point though, is that the entries
in $\theta(t)$ are the transition probabilities per unit time $\mu_i$
and $\pi_i$ from before (you need the $i$'s if you're looking at more
than one site in the genome, but otherwise, they could be omitted as 
below).  Hence
$$\mu = {1\over 4}\left(1 + 3 e^{-4\alpha}\right)$$
and 
$$\pi = {1\over 4}\left(1 - e^{-4\alpha}\right)$$
from setting $t = 1$ in the formula for $\theta(t)$.  (Note that 
there's a small error in the formulas at the top of page 8.)

The expected number of mutations over a time period $t$ is $3\alpha t$
as you say.  But since you have set up all the equations, it would also
have been good to say more about the actual derivation of the 
JC distance (or ``correction'').  The idea is that starting from any
mother (ancestor) species, the probability of a change after time $t$
is the sum of the three off-diagonal terms in the corresponding 
row of the $\theta(t)$ matrix:  ${3\over 4}(1 - e^{-4\alpha t})$.
So if you see that $k/n$ of the positions in a sequence of the 
genome have changed, then the maximum likelihood estimate for
the branch length is obtained just by solving the equation 
$${k\over n} = {3\over 4}(1 - e^{-4\alpha t})$$
for $-4\alpha t$, then multiplying that by ${-3\over 4}$ to get 
the $3\alpha t$:
$${4k\over 3n} = 1 - e^{-4\alpha t},$$
so 
$$-4\alpha t = \ln\left(1 - {4k\over 3n}\right)$$
and then multiply both sides by ${-3\over 4}$.  

It would have been good to say more about where the $p_{123}, p_{dis},
p_{ij}$ formulas come from in the paper (as you did in the talk).

Also see the project paper hardcopy for some more specific comments.

\bigskip
Final Project Presentation: 92 (A-)

\bigskip
Final Project Paper:  92 (A-)
\bye









 Some specific comments:
\bigskip
