Could anyone give me the direction or hints on 'how to derive the F81
model distance formula' from its probability formula ?? The F81 distance is
given as follows :
D = -(1-([P_A]^2 + [P_C]^2 + [P_G]^2 + [P_T]^2))ln(1-(P/([P_A]^2 + [P_C]^2 +
[P_G]^2 + [P_T]^2)).
The probability formula for F81 is given as:
P(ij) = P_j + (1-P_j)*Exp[-at] for all i == j
P_j*(1-Exp[-at]) for all i != J
where P_j is the equilibrium value for the frequency of each nucleotide.
Exp is the exponential function.
at is the expected rate of change per unit time.
I use the idea given by Li's book [Molecular Evolution - The method to
derive Juke-Cantor model distance based on sequence similiarity (p.69 and
p.80] for F81 model, but it doesn't come out that what i expected.