Background
The nucleotide substitution rate matrix, conventionally denoted Q, holds the rate of change from each type of nucleotide to each other nucleotide in the Markov model of molecular evolution
1
(see
2
for review). Q is often treated as a nuisance parameter in phylogenetic reconstruction, and is often assumed to be constant over a phylogenetic tree. However, if Q were really constant, all genomes would have identical composition, assuming that sufficient time had passed for the process to reach equilibrium. Neutrally-evolving sites reach equilibrium rapidly (e.g. within individual genera of nematodes
3
). However, genomes range in GC content (the fraction of nucleotides that are G or C, as opposed to A or T) from about 25% to 75%
4
5
, and members of all three domains of life have a wide range of GC content
6
. Consequently, discovering how Q varies is key to building better phylogenies and to understanding fundamental processes of molecular evolution. However, there are many methods for providing an estimate of Q, i.e.
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
, that may differ substantially in accuracy and performance, and lack of information about these performance characteristics has hindered discovery of patterns of Q on a large scale within and between genomes. Several factors are known to influence the proportions of the four nucleotides in the genome (hereafter, genome composition), and hence presumably Q, under specific conditions. For example, in the spirochete Borrelia burgdorferi, the strand-specific compositional biases induced by deamination are so great that the strand on which each gene is coded can be determined simply by examining its composition
7
8
. Other examples include the influence of CpG doublets on mammalian gene composition
9
, deamination biases in mitochondria
10
, and the origin of mammalian isochores through variation in mutation patterns
11
. Within vertebrate genomes, the balance between oxidation and deamination mutations can also differ radically
12
. However, despite the potential for relating changes in Q to specific mutational processes, little attention has been paid to obtaining empirical
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
in specific lineages
13
14
.
That Q does vary, and that variations in Q can tell us something interesting about the underlying processes in molecular evolution, has thus drawn substantial attention only in the last 10–15 years. Most interest in obtaining
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
has come from studies of phylogenetic reconstruction: for example, likelihood methods for tree-building typically involve optimizing a single
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
matrix for a phylogenetic tree
15
, and testing for the symmetries in
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
that best explain the sequence data
16
is a standard part of modern phylogenetic inference workflows
17
. Although some work has been done on non-stationary models
18
19
20
21
22
23
, and recent evidence suggests that non-stationarity may have a large influence on the accuracy of phylogenetic reconstruction
24
25
, no comparison of the different methods for obtaining
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
has yet been performed. Available methods for obtaining
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
are based on different underlying assumptions, and are hence expected to differ both in accuracy and in speed of calculation. Most available methods obtain
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
, where the entries of
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
are the estimated probabilities of changing from each base to each other base after a defined time interval of duration t.
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
is related to
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
by the equation
P
^
=
e
Q
^
t
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaacqGH9aqpcqWGLbqzdaahaaWcbeqaaiqbdgfarzaajaGaemiDaqhaaaaa@3240@
. Therefore,
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
t can be obtained from
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
in two ways: either by simply taking the matrix logarithm of
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
, or by using constrained optimization to find a valid
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
t without negative off-diagonal elements that, when exponentiated, produces
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
with minimal error. The time factor can be removed by scaling
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
t so that the trace is (-1), leaving
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
on a standard scale (but with arbitrary units). For equilibrium processes, the weighted trace can be scaled such that
−
∑
i
p
i
i
Q
^
i
i
=
1
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeyOeI0YaaabeaeaacqWGWbaCdaWgaaWcbaGaemyAaKMaemyAaKgabeaakiqbdgfarzaajaWaaSbaaSqaaiabdMgaPjabdMgaPbqabaaabaGaemyAaKgabeqdcqGHris5aOGaeyypa0JaeGymaedaaa@3A67@
, taking into account the equilibrium base frequencies
26
to standardize to a fixed number of substitutions per site, but for nonstationary processes this procedure is not justified. The simplest method for obtaining
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
is to count the frequencies that a base i ∈ {A, C, G, T} in one sequence is aligned with base j ∈ {A, C, G, T} in the other sequence, and denote this by n
ij
. To isolate the effects of mutation, regions of the sequence that are thought to be under no selection or very weak selection, e.g. introns or 4-fold degenerate synonymous sites, are used for this purpose. If the substitution process is time-reversible, n
ij
= n
ji
in expectation. It suffices to consider the special case where the observed counts n is symmetric. If n is not symmetric, we replace n by the average of n and the transpose n'
26
. Then the entries of
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
are given by
P
^
i
j
=
n
i
j
∑
k
n
i
k
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabg2da9KqbaoaalaaabaGaemOBa42aaSbaaeaacqWGPbqAcqWGQbGAaeqaaaqaamaaqababaGaemOBa42aaSbaaeaacqWGPbqAcqWGRbWAaeqaaaqaaiabdUgaRbqabiabggHiLdaaaaaa@3D48@
, which can then be used to obtain
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
as described above
2
26
27
. Alternatively, since this method of obtaining
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
assumes the divergence between sequences is small,
P
^
~
I
+
Q
^
t
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaacqGG+bGFcqWGjbqscqGHRaWkcuWGrbqugaqcaiabdsha0baa@333B@
. Then an estimate for the rate of change of i to j is given by
(
Q
^
t
)
i
j
=
n
i
j
∑
k
n
i
k
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIafmyuaeLbaKaacqWG0baDcqGGPaqkdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabg2da9KqbaoaalaaabaGaemOBa42aaSbaaeaacqWGPbqAcqWGQbGAaeqaaaqaamaaqababaGaemOBa42aaSbaaeaacqWGPbqAcqWGRbWAaeqaaaqaaiabdUgaRbqabiabggHiLdaaaaaa@406D@
for i ≠ j, and the diagonal elements are set such that each row of
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
t sums to 0
28
29
. We refer to this method as the 'Goldman' method
29
. Again the time factor can be removed by scaling
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
t. However, both of these methods are based on the assumption that the evolutionary process is time-reversible. A method that can better recapture non-time-reversible substitution processes uses rooted triples of sequences where X and Y are the sister taxa, and Z is the outgroup
30
. If X has the same base at that position as Z, the assumption is that the change most likely occurred between Y and its common ancestor with X. This method thus leads to a directional and not necessarily time-reversible
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
, which can then be used to obtain
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
(see
31
for additional details, and for a comparison of directional and undirectional estimates of Q). We refer to this three-sequence method as the 'Gojobori' method of otaining
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
30
, and use the implementation in the PyCogent toolkit
32
.
More elaborate methods include maximum likelihood methods and Markov triple analysis
13
33
34
. Each of these methods assume nucleotide sites are independent and identically distributed, and do not require the evolutionary process to be stationary or reversible (recall that the sequence composition produced by evolution according to a Markov process over time need not match the composition of the starting sequence, which might be modeled differently). To explain the methods, we introduce the following notation. Consider a rooted triple of sequences from taxa X, Y, Z. Define R to be the root and let
V
^
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOvayLbaKaadaWgaaWcbaGaemOuaifabeaaaaa@2E73@
denote the estimated vector of nucleotide probabilities for the sequence at R. Let
P
^
U
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemyvaufabeaaaaa@2E6D@
denote the estimated nucleotide probability transition matrix from R to taxa U, for U ∈ {X, Y, Z}, i.e.,
P
^
U
(
i
,
j
)
=
P
r
(
U
=
j
|
R
=
i
)
,
for
i
,
j
∈
{
A
,
C
,
G
,
T
}
,
U
∈
{
X
,
Y
,
Z
}
.
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@64B8@
The transition probabilities in each row of
P
^
U
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemyvaufabeaaaaa@2E6D@
sum to 1, therefore each matrix
P
^
U
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemyvaufabeaaaaa@2E6D@
consists of 12 unknown parameters and similarly the vector
V
^
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOvayLbaKaadaWgaaWcbaGaemOuaifabeaaaaa@2E73@
consists of 3 unknown parameters. To determine the unknown parameters, the methods each make use of the observed joint probability distribution between the three taxa. Let
J
^
X
,
Y
,
Z
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemiwaGLaeiilaWIaemywaKLaeiilaWIaemOwaOfabeaaaaa@329F@
denote this distribution. That is,
J
^
X
,
Y
,
Z
(
i
,
j
,
k
)
=
P
r
(
X
=
i
,
Y
=
j
,
Z
=
k
)
,
for
i
,
j
,
k
∈
{
A
,
C
,
G
,
T
}
.
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabeWaaaqaaiqbdQeakzaajaWaaSbaaSqaaiabdIfayjabcYcaSiabdMfazjabcYcaSiabdQfaAbqabaGccqGGOaakcqWGPbqAcqGGSaalcqWGQbGAcqGGSaalcqWGRbWAcqGGPaqkcqGH9aqpcqWGqbaucqWGYbGCcqGGOaakcqWGybawcqGH9aqpcqWGPbqAcqGGSaalcqWGzbqwcqGH9aqpcqWGQbGAcqGGSaalcqWGAbGwcqGH9aqpcqWGRbWAcqGGPaqkcqGGSaalaeaacqqGMbGzcqqGVbWBcqqGYbGCaeaacqWGPbqAcqGGSaalcqWGQbGAcqGGSaalcqWGRbWAcqGHiiIZcqGG7bWEcqWGbbqqcqGGSaalcqWGdbWqcqGGSaalcqWGhbWrcqGGSaalcqWGubavcqGG9bqFcqGGUaGlaaaaaa@63BB@
Each state of
J
^
X
,
Y
,
Z
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemiwaGLaeiilaWIaemywaKLaeiilaWIaemOwaOfabeaaaaa@329F@
can be written in terms of
P
^
X
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemiwaGfabeaaaaa@2E73@
,
P
^
Y
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemywaKfabeaaaaa@2E75@
,
P
^
Z
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemOwaOfabeaaaaa@2E77@
and
V
^
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOvayLbaKaadaWgaaWcbaGaemOuaifabeaaaaa@2E73@
. For example,
J
^
X
,
Y
,
Z
(
A
,
C
,
T
)
=
∑
i
∈
{
A
,
C
,
G
,
T
}
P
r
(
R
=
i
)
P
r
(
X
=
A
|
R
=
i
)
P
r
(
Y
=
C
|
R
=
i
)
P
r
(
Z
=
T
|
R
=
i
)
=
∑
i
∈
{
A
,
C
,
G
,
T
}
V
^
R
(
i
)
P
^
X
(
i
,
A
)
P
^
Y
(
i
,
C
)
P
^
Z
(
i
,
T
)
.
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@A574@
Maximum likelihood methods obtain
P
^
X
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemiwaGfabeaaaaa@2E73@
,
P
^
Y
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemywaKfabeaaaaa@2E75@
and
P
^
Z
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemOwaOfabeaaaaa@2E77@
by finding the values for the parameters that maximize the likelihood of the observed joint frequencies
33
34
. That is, align the X, Y, and Z sequences and let n
XYZ
(i, j, k) denote the number of times X = i, Y = j, Z = k occurs in the alignment. The likelihood function is proportional to the conditional probability, given by
P
r
(
n
X
Y
Z
|
P
X
,
P
Y
,
P
Z
,
V
R
)
=
∏
i
,
j
,
k
∈
{
A
,
C
,
G
,
T
}
J
^
X
,
Y
,
Z
(
i
,
j
,
k
)
n
(
i
,
j
,
k
)
.
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7145@
One method for finding the 39 unknown parameters (12 unknowns in each of three
P
^
i
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemyAaKgabeaaaaa@2E95@
and three unknowns in
V
^
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOvayLbaKaadaWgaaWcbaGaemOuaifabeaaaaa@2E73@
) is to use a global optimization algorithm to search parameter space and find
P
^
X
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemiwaGfabeaaaaa@2E73@
,
P
^
Y
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemywaKfabeaaaaa@2E75@
,
P
^
Z
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemOwaOfabeaaaaa@2E77@
and
V
^
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOvayLbaKaadaWgaaWcbaGaemOuaifabeaaaaa@2E73@
that maximize the likelihood. We refer to this method as the 'MW' method below. Since the chosen ordering of the nucleotides is arbitrary, there are 4! permutations of the elements of
V
^
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOvayLbaKaadaWgaaWcbaGaemOuaifabeaaaaa@2E73@
(while keeping the nucleotide ordering fixed), and 4! corresponding permutations of the rows of the
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
matrices, that will result in the same
J
^
X
,
Y
,
Z
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemiwaGLaeiilaWIaemywaKLaeiilaWIaemOwaOfabeaaaaa@329F@
and therefore the same likelihood. Thus, once the global optimization algorithm converges to one of the maximum likelihood solutions, a unique solution can be determined by choosing the permutation that maximizes the sum of the traces of the
P
^
X
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemiwaGfabeaaaaa@2E73@
,
P
^
Y
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemywaKfabeaaaaa@2E75@
and
P
^
Z
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemOwaOfabeaaaaa@2E77@
matrices
34
, which provides the solution that implies least change in the overall sequence (the diagonal elements represent the probability that a nucleotide remains unchanged after t time units). If the underlying process is not one that maximizes the diagonal entries, the process is unidentifiable using this method (i.e. cannot be recovered even with infinite sequence data). One solution might be to restrict the class of processes to those with diagonal-dominant P
35
, although we did not perform such restrictions in this study.
An alternate approach for finding the maximum likelihood parameters that does not involve a global optimization routine was introduced by Barry and Hartigan and is referred to as the 'BH' method
33
34
below (we ran this method using default parameters as supplied). BH rewrites the likelihood function in terms of joint probability matrices along each edge of the tree. Solving for the maximum value of the likelihood leads to a system of iterative equations for the joint probability matrices along each edge. Thus, given initial starting guesses for the joint distributions along each edge, the system can be iterated until convergence is reached
33
34
. The joint probability matrices along each edge can then be used to obtain their corresponding
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
matrices.
BH is valid on unrooted trees of any size, but for simplicity we briefly outline it here using the unrooted version of the rooted triple defined above. Let u
i
denote the ith element in the sequence from taxon U, with U ∈ X, Y, Z. Similarly, let r
i
denote the ith element in the sequence at node R. Then
J
^
X
,
Y
,
Z
(
x
1
,
y
1
,
z
1
)
=
∑
r
1
∈
{
A
,
C
,
G
,
T
}
J
^
X
,
R
(
x
1
,
r
1
)
P
^
Y
(
r
1
,
y
1
)
P
^
Z
(
r
1
,
z
1
)
.
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@71CF@
Let N denote the length of the sequences. Then the conditional probability can be written as
P
r
(
x
,
y
,
z
|
J
^
X
,
R
,
P
Y
,
P
Z
)
=
∏
i
=
1
N
∑
r
∈
{
A
,
C
,
G
,
T
}
J
^
X
,
R
(
x
i
,
r
)
P
^
Y
(
r
,
y
i
)
P
^
Z
(
r
,
z
i
)
.
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaemOCaiNaeiikaGIaemiEaGNaeiilaWIaemyEaKNaeiilaWIaemOEaONaeiiFaWNafmOsaOKbaKaadaWgaaWcbaGaemiwaGLaeiilaWIaemOuaifabeaakiabcYcaSiabdcfaqnaaBaaaleaacqWGzbqwaeqaaOGaeiilaWIaemiuaa1aaSbaaSqaaiabdQfaAbqabaGccqGGPaqkcqGH9aqpdaqeWbqaamaaqafabaGafmOsaOKbaKaadaWgaaWcbaGaemiwaGLaeiilaWIaemOuaifabeaakiabcIcaOiabdIha4naaBaaaleaacqWGPbqAaeqaaOGaeiilaWIaemOCaiNaeiykaKIafmiuaaLbaKaadaWgaaWcbaGaemywaKfabeaakiabcIcaOiabdkhaYjabcYcaSiabdMha5naaBaaaleaacqWGPbqAaeqaaOGaeiykaKIafmiuaaLbaKaadaWgaaWcbaGaemOwaOfabeaakiabcIcaOiabdkhaYjabcYcaSiabdQha6naaBaaaleaacqWGPbqAaeqaaOGaeiykaKcaleaacqWGYbGCcqGHiiIZcqGG7bWEcqWGbbqqcqGGSaalcqWGdbWqcqGGSaalcqWGhbWrcqGGSaalcqWGubavcqGG9bqFaeqaniabggHiLdaaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGobGta0Gaey4dIunakiabc6caUaaa@7A8C@
The maximum likelihood solutions are therefore the values of
J
^
X
,
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemiwaGLaeiilaWIaemOuaifabeaaaaa@3074@
,
P
^
Y
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemywaKfabeaaaaa@2E75@
and
P
^
Z
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemOwaOfabeaaaaa@2E77@
that maximize the likelihood subject to the constraint that the entries in
J
^
X
,
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemiwaGLaeiilaWIaemOuaifabeaaaaa@3074@
sum to 1, as it is a joint probability matrix. In other words, BH maximizes
log
(
P
r
(
x
,
y
,
z
|
J
^
X
,
R
,
P
^
Y
,
P
^
Z
)
)
+
λ
(
1
−
∑
x
,
r
∈
{
A
,
C
,
G
,
T
}
J
^
X
,
R
(
x
,
r
)
)
.
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6BFC@
Differentiating with respect to
J
^
X
,
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemiwaGLaeiilaWIaemOuaifabeaaaaa@3074@
and λ and setting equal to zero leads to the following iterative equation for
J
^
X
,
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemiwaGLaeiilaWIaemOuaifabeaaaaa@3074@
,
J
^
X
,
R
(
x
,
r
)
=
1
N
∑
i
=
1
N
I
(
x
∈
{
x
i
}
)
J
^
X
,
R
(
x
,
r
)
P
^
Y
(
r
,
y
i
)
P
^
Z
(
r
,
z
i
)
∑
r
∈
{
A
,
C
,
G
,
T
}
J
^
X
,
R
(
x
,
r
)
P
^
Y
(
r
,
y
i
)
P
^
Z
(
r
,
z
i
)
,
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@9710@
where I(x ∈ {x
i
}) is an indicator function that takes the value 1 if x = x
i
and 0 otherwise
33
34
. This iterative equation can be written entirely in terms of joint distributions along each edge by using the relation
P
^
U
(
r
,
u
i
)
=
J
^
U
,
R
(
u
i
,
r
)
∑
u
i
∈
{
A
,
C
,
G
,
T
}
J
^
U
,
R
(
u
i
,
r
)
,
for
U
∈
{
X
,
Y
,
Z
}
.
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6FCC@
These steps are repeated to derive iterative equations for
J
^
Y
,
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemywaKLaeiilaWIaemOuaifabeaaaaa@3076@
and
J
^
Z
,
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemOwaOLaeiilaWIaemOuaifabeaaaaa@3078@
, leading to an iterative system for
J
^
X
,
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemiwaGLaeiilaWIaemOuaifabeaaaaa@3074@
,
J
^
Y
,
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemywaKLaeiilaWIaemOuaifabeaaaaa@3076@
and
J
^
Z
,
R
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemOwaOLaeiilaWIaemOuaifabeaaaaa@3078@
. Suitable initial values are chosen and the system is iterated until it converges
33
.
The
J
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaaaaa@2D02@
matrices can then be converted to
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
matrices along each edge using the relationship in (3). Finally, the
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
matrices can then be used to obtain
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
matrices using the relationship
P
^
=
e
Q
^
t
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaacqGH9aqpcqWGLbqzdaahaaWcbeqaaiqbdgfarzaajaGaemiDaqhaaaaa@3240@
.
Both the global optimization and BH methods maximize the same likelihood function, but they differ in the computational methods used to find the maximum and in whether they write the unknown parameters in terms of joint probability matrices or probability transition matrices. Maximizing the likelihood does not result in a unique solution without adding further restrictions. MW chooses the
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
matrices with the largest sum of the traces, while in the BH method, the initial starting values determine which maximum is chosen. The BH method's default initial
J
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaaaaa@2D02@
matrices, are diagonally dominant. Thus we expect the iteration to converge to a diagonally dominant solution (if one exists). Thus, in theory the BH and MW methods will result in the same
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
matrices. In practice, their answers can differ if either converge instead to a local maximum, or if the initial starting points used in either algorithm lead to too slow of a rate of convergence, or if the original P matrices used to generate the sequences are not diagonally dominant.
Another method, which we refer to as the 'Lake' method, uses Markov triple analysis (MTA) as a different approach to finding the
P
^
U
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemyvaufabeaaaaa@2E6D@
matrices for rooted triples of sequences
13
. This method uses the fact that the conditional joint probabilities
J
^
X
,
Y
|
Z
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOsaOKbaKaadaWgaaWcbaGaemiwaGLaeiilaWIaemywaKLaeiiFaWNaemOwaOfabeaaaaa@333F@
can be written in terms of
P
^
X
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemiwaGfabeaaaaa@2E73@
,
P
^
Y
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemywaKfabeaaaaa@2E75@
, and
P
^
Z
*
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaqhaaWcbaGaemOwaOfabaGaeiOkaOcaaaaa@2F54@
, where
P
^
Z
*
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaqhaaWcbaGaemOwaOfabaGaeiOkaOcaaaaa@2F54@
is the probability transition matrix from taxon Z to R. (Note that because the evolutionary process is not required to be reversible,
P
^
Z
*
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaqhaaWcbaGaemOwaOfabaGaeiOkaOcaaaaa@2F54@
differs from
P
^
Z
*
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaqhaaWcbaGaemOwaOfabaGaeiOkaOcaaaaa@2F54@
.) For example,
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^
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,
Y
,
|
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(
A
,
C
|
T
)
=
∑
r
∈
{
A
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}
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(
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)
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,
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)
.
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@65E1@
MTA uses the observed conditional joint probabilities as estimates of the true conditional joint probabilities. This results in a system of 48 nonlinear equations in terms of the 36 unknowns in
P
^
X
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemiwaGfabeaaaaa@2E73@
,
P
^
Y
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemywaKfabeaaaaa@2E75@
and
P
^
Z
*
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaqhaaWcbaGaemOwaOfabaGaeiOkaOcaaaaa@2F54@
. MTA solves this system by first using the properties of determinants to set up a simpler system in terms of
P
^
Z
*
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaqhaaWcbaGaemOwaOfabaGaeiOkaOcaaaaa@2F54@
. Specifically, determinants of the conditional joint probabilities are used to derive a system of 6 quartic polynomials.
P
^
Z
*
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaqhaaWcbaGaemOwaOfabaGaeiOkaOcaaaaa@2F54@
is determined by first finding the roots of the polynomials and then searching through 245 possible orderings of the roots to find the ordering that results in a consistent system. In the case that the original system of 48 equations is inconsistent, the Lake method chooses the ordering that minimizes an inconsistency function defined by the author
13
. These ordered roots are coefficients in a system of equations that is linear in the unknown parameters of
P
^
Z
*
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaqhaaWcbaGaemOwaOfabaGaeiOkaOcaaaaa@2F54@
. Thus once the ordered roots are found,
P
^
Z
*
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaqhaaWcbaGaemOwaOfabaGaeiOkaOcaaaaa@2F54@
can be determined by solving the linear system. Again, any of the 4! permutations of the columns of
P
^
Z
*
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaqhaaWcbaGaemOwaOfabaGaeiOkaOcaaaaa@2F54@
will also be a valid solution. Thus the Lake method chooses the ordering with a positive determinant that maximizes the trace. Once a unique
P
^
Z
*
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaqhaaWcbaGaemOwaOfabaGaeiOkaOcaaaaa@2F54@
has been determined, 3 systems of equations that are linear in the parameters of
P
^
X
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemiwaGfabeaaaaa@2E73@
,
P
^
Y
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemywaKfabeaaaaa@2E75@
,
P
^
Z
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemOwaOfabeaaaaa@2E77@
can be solved to determine these matrices. Again, the
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
matrices estimated using the Lake method can then be used to obtain
Q
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
matrices.
If the observed conditional frequencies do in fact accurately estimate the true conditional frequencies and result in a consistent system of 48 equations, the Lake method provides a computationally feasible way to find the solutions of the system. However, since this method estimates the joint probability distribution from observed frequencies, the accuracy of this estimation will be dependent on sequence length and sequence alignment. Therefore, the system of 48 equations can often be inconsistent (i.e. values for the 36 parameters that satisfy all of the equations and that are between 0 and 1 do not exist). In this case, it is possible that the Lake method will be unable to find valid estimations of the probability transition matrices, as the linear systems in the method may also be inconsistent or may result in
P
^
X
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemiwaGfabeaaaaa@2E73@
,
P
^
Y
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemywaKfabeaaaaa@2E75@
or
P
^
Z
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaadaWgaaWcbaGaemOwaOfabeaaaaa@2E77@
having negative elements.
In this paper, we present a comparison of the speed and accuracy of the different methods described above, using sequences of different lengths and divergences. We also compare two methods for obtaining
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^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D10@
matrices from the resulting
P
^
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@
matrices: using the matrix log
36
and using constrained optimization.