We have shown that: To see what this means in the long run let α=.001 and graph p:

We see that in this case as t becomes large,, in fact this is the case for any α < 1 From this fact the probability of the position showing any one of the other three nucleotides is (1/3)*(1/4) = 1/12 = From these two facts we can gain some insight on the derivation of the Jukes- Cantor scoring matrix. Recall that the score in position (a,b) of the matrix at time, t, is: If we assume the nucleotides are essentially equally distributed throughout the sequences then q b = ¼ =.25 we have that

Multiplying s 1 by 10 and rounding, we get the score for a position having the same nucleotide as it initially had is 5. Noting that the score obtained for it having a different value is 9.5 units away from the score obtained when the value is unchanged, it is reasonable to assign it a value of -4. NOTE: One could argue that -5 makes more sense, except that then the scoring model would be no different than the simple counting model which assigns 1 to a match and -1 to a mismatch. Since 5 – 9.5 = –4.5 it makes sense to attribute a value of –4 as the score for a mismatch. The end result is that we have the familiar Jukes–Cantor scoring matrix

Returning to Phylogenetic distances. We saw that the distance between two genes, pseudogenes or conserved regions represented by nucleotide sequences S 1 and S 2 is given by the expression: d JC (S 1,S 2 ) = where is the fraction of sites that disagree when comparing S 1 to S 2. The exact same type of analysis can give us the formula for the distance based on the Kimura two parameter model that is shown in K&R on page 67: where p is the fraction of transitions and q is the fraction of transversions. In fact, we could consider a three parameter model and the same type of analysis would reveal that where, p, q, r are the three frequencies of the changes considered by the model.. And so the game goes on.

Phylogenetic Trees

An old and controversial question: What is our relationship to the modern species of apes? Consider the following species: gorilla, chimpanzee, orangatang, and gibbon Which is our closest evolutionary kin? On the other hand are these species more closely related to each other than they are to us? An examination of sequences for the HindIII Restriction Enzyme in these and 7 other primates revealed agreement of between 67% and 93% of the positions in the 898bp long sequences. Human Chimpanzee Gorilla Orangutan Gibbon Human Chimpanzee Gorilla Gibbon Orangutan

We can ask and answer some interesting questions based on the construction of the phylogenetic tree. If the following tree is correct, what can we say about our relationship with the apes? Human Chimpanzee Gorilla Orangutan Gibbon If we accept the other tree from the pair and use the fact that gorillas and chimpanzees are African in origin, while Orangutans and Gibbons are Asian, what can we deduce as the most likely place for the first appearance of Humans? Human Chimpanzee Gorilla Gibbon Orangutan

Unfortunately, knowing the Phylogenetic Distances, does not infer the shape of the tree. For example consider the following unrooted tree below A B C D E Unfortunately, we can not on this evidence alone construct a unique rooted version of the tree. For example, both of the following could be deduced. E A B D C A B D C E These trees are topologically as well as biologically different. However, both are possible without further evidence on which to base the construction.