Page 241 - Applied Probability

10. Molecular Phylogeny
227
this revelation, then use formula (10.17) to find the equilibrium dis-
tribution of the nucleotide substitution chain of Section 10.5 when
detailed balance is not assumed. Your answer should match the dis-
tribution appearing in Problem 15.
17. In his method of evolutionary parsimony, Lake [14] has highlighted
the balanced transversion assumption. This assumption implies the
constraints λ AC = λ AT , λ GC = λ GT , λ CA = λ CG , and λ TA = λ TG
in the nucleotide substitution model with general transition rates.
Without further restrictions, infinitesimal balanced transversions do
not imply finite-time balanced transversions. For example, the iden-
tity p AC (t)= p AT (t) may not hold. Prove that finite-time balanced
transversions follow if the additional closure assumptions
λ AG − λ GA = λ G − λ A
λ CT − λ TC = λ T − λ C
k
are made [1]. (Hint: Show by induction that the matrices Λ have the
balanced transversion pattern for equality of entries.)
18. In Lake’s balanced transversion model of the last problem, show that

λ AG λ GT = λ AT λ GA
λ CT λ TA = λ TC λ CA
are necessary and sufficient conditions for the corresponding Markov
chain to be reversible.
19. In the Ising model, one can explicitly calculate the partition function
H(c)
e of Section 10.10. To simplify matters, we impose circular
c
symmetry and write
m m

H(c) = θ 0 c i + θ 1 1 {c i=c i+1 } ,
i=1 i=1
where c m+1 = c 1 . Show that

1 1 m
H(c)
e
e = ··· e θ 0c i θ 1 1 {c i =c i+1 }
c c 1 =0 c m =0 i=1
m
t
= ··· u Zu i+1 . (10.18)
i
u 1 u m i=1
t
t
Here each u i ranges over the set of two vectors (1, 0) and (0, 1) , and
Z is the 2 × 2 matrix

e θ 1 e θ 0
Z = .
e θ 0 e 2θ 0 +θ 1