148 Modeling Molecular Evolution
transversion are equal so β = γ , then this model includes the Jukes-Cantor
one as a special case with α = 3β = 3γ .
An even more general model is the Kimura 3-parameter model, which
assumes a transition matrix of the form
M =
∗ βγ δ
β ∗ δγ
γδ∗ β
δγβ∗
.
By appropriate choice of the parameters, this includes both the Jukes-Cantor
and Kimura 2-parameter models as special cases.
Part of the Kimura models is the assumption that the initial base distribution
vector is p
0
=
1
4
,
1
4
,
1
4
,
1
4
. Because this vector is an eigenvector with eigen-
value 1 for both the Kimura 2- and 3-parameter matrices, sequences evolving
according to these models have this uniform base distribution at all times. As
you will see in the exercises, all the work done above for the Jukes-Cantor
model can be performed for the Kimura 3-parameter model as well.
The general Markov model may well provide the most accurate description
of the base substitutions that actually occur in evolution, because it assumes
nothing special about the entries in the Markov matrix. It does not require any
particular relationship between the various conditional probabilities. There
are 12 parameters in picking a matrix for this model, since of the 16 entries we
may freely pick 3 in each column, with the fourth determined by the condition
that the columns sum to 1. If we also allow any initial base composition vector
p
0
, then there are 3 additional parameters.
Why are there only 3 parameters for p
0
, even though it has 4 entries?
Unless we have specific parameter values in mind for the general Markov
model, it is hard to derive detailed results for it of the sort we found for the
Jukes-Cantor model. However, as long as all entries of the matrix are positive,
the two theorems stated above do tell us that there must be an equilibrium
base distribution. Furthermore, by applying the Strong Ergodic Theorem of
Chapter 2, we know that, over time, the general Markov model will result in p
t
approaching this equilibrium distribution, even if the initial base distribution
is something else.
Problems
4.4.1. Review the forest succession model in the text of Chapter 2 to interpret
it as a Markov model of a single plot in the forest.
a. What are the “states” for this model?