Bioinformatics, Volume I Data, Sequence Analysis and Evolution

262 Liò and Bishop
4. Codon Models
data substantially better than a series of independent pairwise
nucleotide distributions (a zero-order Markov chain). They also
used a Markov model on a phylogenetic tree, parameterized by
a di-nucleotide rate matrix and an independent-site equilibrium
sequence distribution, and estimated substitution parameters
using an expectation maximization (EM) procedure (23).
Whelan and Goldman (24) have developed the singlet-double-triplet
(SDT) model, which incorporated events that change
one, two, or three adjacent nucleotides. This model allows for
neighbor- or context-dependent models of base substitutions,
which consider the N-bases preceding each base and are capable
of capturing the dependence of substitution patterns on neighboring
bases. They found that the inclusion of doublet and triplet
mutations in the model gives statistically significant improvements
in fit of model to data, indicating that larger-scale mutation events
do occur. There are indications that higher-order states, autocorrected
rates, and multiple functional categories all improve the
fit of the model and that the improvements are roughly additive.
The effect of higher-order states (context dependence) is particularly
pronounced.
In an attempt to introduce greater biological reality through
knowledge of the genetic code and the consequent effect of nucleotide
substitutions in protein coding sequences on the encoded
amino acid sequences, Goldman and Yang (25, 26) described a
codon mutation model. They considered the 61 sense codons
i consisting of nucleotides i 1 i 2 i 3 . The rate matrix Q consisted of
elements Q ij describing the rate of change of codon i = i 1 i 2 i 3 to
j = j 1 j 2 j 3 (i ≠ j) depending on the number and type of differences
between i 1 and j 1 , i 2 and j 2 , and i 3 and j 3 as follows:
Q
ij
0 if 2 or 3 of the pairs ik, jk
are different
−daai
, aaj
/ V
= mp je
if one pair differ by a transversion
daai , aaj
/ V
mkp je
i
−
⎧
⎪
⎨
⎪
⎩⎪
f one pair differ by a transition
where d aai,aaj is the distance between the amino acid coded by the
codon i (aa i ) and the amino acid coded by the codon j (aa j ) as
calculated by Grantham (27) on the basis of the physicochemical
properties of the amino acids. This model takes account of codon
frequencies (through p j ), transition/transversion bias (through
k), differences in amino acid properties between different codons