DNAMLK -- DNA Maximum Likelihood program with molecular clock
(c) Copyright 1986-1993 by Joseph Felsenstein and by the University of
Washington. Written by Joseph Felsenstein. Permission is granted to copy this
document provided that no fee is charged for it and that this copyright notice
is not removed.
This program implements the maximum likelihood method for DNA sequences
under the constraint that the trees estimated must be consistent with a
molecular clock. The molecular clock is the assumption that the tips of the
tree are all equidistant, in branch length, from its root. This program is
fairly slow, and can be expensive to run. The program is indirectly related to
DNAML. Its algorithmic details are not yet published, but details of the
algorithm of a previous version of DNAML are described in my paper in Journal
of Molecular Evolution (1981a). For some uses of this program and further
developments, the papers of Masami Hasegawa are particularly instructive
(Hasegawa and Yano, 1984a, 1984b; Hasegawa et. al. 1985a, 1985b). As with
DNAML, the model of base substitutions used in this program is more general
than the model described in the 1981a paper, as it allows transition and
transversion rates to be expected to be unequal.
The assumptions of the model are:
1. Each site in the sequence evolves independently.
2. Different lineages evolve independently.
3. There is a molecular clock.
4. Each site undergoes substitution at an expected rate which is chosen
from a series of rates (each with a probability of occurrence) which we
5. All relevant sites are included in the sequence, not just those that
have changed or those that are "phylogenetically informative".
6. A substitution consists of one of two sorts of events:
a. The first kind of event consists of the replacement of the
existing base by a base drawn from a pool of purines or a pool of
pyrimidines (depending on whether the base being replaced was a
purine or a pyrimidine). It can lead either to no change or to a
b. The second kind of event consists of the replacement of the
existing base by a base drawn at random from a pool of bases at
known frequencies, independently of the identity of the base
which is being replaced. This could lead either to a no change,
to a transition or to a transversion.
The ratio of the two purines in the purine replacement pool is the same as
their ratio in the overall pool, and similarly for the pyrimidines.
The ratios of transitions to transversions can be set by the user. The
substitution process can be diagrammed as follows: Suppose that you specified
A, C, G, and T base frequencies of 0.24, 0.28, 0.27, and 0.21.
First kind of event:
1. Determine whether the existing base is a purine or a pyrimidine.
2. Draw from the proper pool:
Purine pool: Pyrimidine pool:
I I I I
I 0.4706 A I I 0.5714 C I
I 0.5294 G I I 0.4286 T I
I (ratio is I I (ratio is I
I 0.24 : 0.27) I I 0.28 : 0.21) I
Second kind of event:
Draw from the overall pool:
I 0.24 A I
I 0.28 C I
I 0.27 G I
I 0.21 T I
Note that if the existing base is, say, an A, the first kind of event has a
0.4706 probability of "replacing" it by another A. The second kind of event
has a 0.24 chance of replacing it by another A. This rather disconcerting
model is used because it has nice mathematical properties that make likelihood
calculations far easier. A closely similar, but not precisely identical model
having different rates of transitions and transversions has been used by
Hasegawa et. al. (1985b). The transition probability formulas for the current
model were given (with my permission) by Kishino and Hasegawa (1989).
Note the assumption that we are looking at all sites, including those that
have not changed at all. It is important not to restrict attention to some
sites based on whether or not they have changed; doing that would bias branch
lengths by making them too long, and that in turn would cause the method to
misinterpret the meaning of those sites that had changed.
An important new development in the 3.5 release is the Hidden Markov Chain
method of inferring different rates of evolution at different sites. This wil
l be described in a future paper by me and Gary Churchill, of Cornell
University. It allows us to specify to the program that there will be a number
of different possible evolutionary rates, what the prior probabilities of
occurrence of each is, and what the average length of a patch of sites all
having the same rate. The program then computes the likelihood by summing it
over all possible assignments of rates to sites, weighting each by its prior
probability of occurrence.
For example, if we have used the C and R options (described below) to
specify that there are three possible rates of evolution, 1.0, 2.4, and 0.0,
that the prior probabilities of a site having these rates are 0.4, 0.3, and
0.3, and that the average patch length (number of consecutive sites with the
same rate) is 2.0, the program will sum the likelihood over all possibilities,
but giving less weight to those that (say) assign all sites to rate 2.4, or
that fail to have consecutive sites that have the same rate.
This feature effectively removes the artificial assumption that all sites
have the same rate, and also means that we need not know in advance the
identities of the sites that have a particular rate of evolution.
INPUT FORMAT AND OPTIONS
Subject to these assumptions, the program is a correct maximum likelihood
method. The input is fairly standard, with one addition. As usual the first
line of the file gives the number of species and the number of sites. There
follow the character W if the Weights options is being used.
Next come the species data. Each sequence starts on a new line, has a
ten-character species name that must be blank-filled to be of that length,
followed immediately by the species data in the one-letter code. The sequences
must either be in the "interleaved" or "sequential" formats described in the
Molecular Sequence Programs document. The I option selects between them. The
sequences can have internal blanks in the sequence but there must be no extra
blanks at the end of the terminated line. Note that a blank is not a valid
symbol for a deletion.
After that are the lines (if any) containing the information for the C,
and W options, as described below.
The options are selected using an interactive menu. The menu looks like
Nucleic acid sequence
Maximum Likelihood method with molecular clock, version 3.5c
Settings for this run:
U Search for best tree? Yes
T Transition/transversion ratio: 2.0
F Use empirical base frequencies? Yes
C One category of substitution rates? Yes
G Global rearrangements? No
J Randomize input order of sequences? No. Use input order
M Analyze multiple data sets? No
I Input sequences interleaved? Yes
0 Terminal type (IBM PC, VT52, ANSI)? ANSI
1 Print out the data at start of run No
2 Print indications of progress of run Yes
3 Print out tree Yes
4 Write out trees onto tree file? Yes
Are these settings correct? (type Y or the letter for one to change)
The user either types "Y" (followed, of course, by a carriage-return) if the
settings shown are to be accepted, or the letter or digit corresponding to an
option that is to be changed.
The options U, J, O, M, and 0 are the usual ones. They are described in
the main documentation file of this package. Option I is the same as in other
molecular sequence programs and is described in the documentation file for the
The T option in this program does not stand for Threshold, but instead is
the Transition/transversion option. The user is prompted for a real number
greater than 0.0, as the expected ratio of transitions to transversions. Note
that this is not the ratio of the first to the second kinds of events, but the
resulting expected ratio of transitions to transversions. The exact
relationship between these two quantities depends on the frequencies in the
base pools. The default value of the T parameter if you do not use the T
option is 2.0.
The F (Frequencies) option is one which may save users much time. If you
want to use the empirical frequencies of the bases, observed in the input
sequences, as the base frequencies, you simply use the default setting of the F
option. These empirical frequencies are not really the maximum likelihood
estimates of the base frequencies, but they will often be close to those values
(what they are is maximum likelihood estimates under a "star" or "explosion"
phylogeny). If you change the setting of the F option you will be prompted for
the frequencies of the four bases. These must add to 1 and are to be typed on
one line separated by blanks, not commas.
The C (categories) option allows the user to specify how many categories
of substitution rates there will be, and what are the rates and probabilities
for each. The user first is asked how many categories there will be (for the
moment there is an upper limit of 5, which should not be very restrictive).
Then the program asks for the rates for each category. These rates are only
meaningful relative to each other, so that rates 1.0, 2.0, and 2.4 have the
exact same effect as rates 2.0, 4.0, and 4.8. Note that a category can have
rate of change 0, so that this allows us to take into account that there may be
a category of sites that are invariant. Note that the run time of the program
will be proportional to the number of rate categories: twice as many categories
means twice as long a run. Finally the program will ask for the probabilities
of a random site falling into each of these categories. These probabilities
must be nonnegative and sum to 1. Default for the program is one category,
with rate 1.0 and probability 1.0 (actually the rate does not matter in that
If more than one category is specified, then another option, R, becomes
visible in the menu. This allows us to specify that we want to assume that
sites that have the same rate category are expected to be clustered. The
program asks for the value of the average patch length. This is an expected
length of patches that have the same rate. If it is 1, the rates of successive
sites will be independent. If it is, say, 10.25, then the chance of change to
a new rate will be 1/10.25 after every site. However the "new rate" is
randomly drawn from the mix of rates, and hence could even be the same. So the
actual observed length of patches with the same rate will be somewhat larger
than 10.25. Note below that if you choose multiple patches, there will be an
estimate in the output file as to which combination of rate categories
contributed most to the likelihood.
With the current options C and R the program has gained greatly in its
ability to infer different rates at different sites and estimate phylogenies
under a more realistic model. Note that Likelihood Ratio Tests can be used to
test whether one combination of rates is significantly better than another.
The G (global search) option causes, after the last species is added to
the tree, each possible group to be removed and re-added. This improves the
result, since the position of every species is reconsidered. It approximately
triples the run-time of the program.
If the U (user tree) option is chosen another option appears in the menu,
the L option. If it is selected, it signals the program that it should take
any branch lengths that are in the user tree and simply evaluate the likelihood
of that tree, without further altering those branch lengths. This means that
if some branches have lengths and others do not, the program will estimate the
lengths of those that do not have lengths given in the user tree. Note that
the program RETREE can be used to add and remove lengths from a tree. This
means that we can test hypothesis such as that a certain branch has zero
length, and by doing a series of runs with different specified lengths for a
branch we can plot a likelihood curve for its branch length while allowing all
other branches to adjust their lengths to it. If all branches have lengths
specified, none of them will be iterated. This is useful to allow a tree
produced by another method to have its likelihood evaluated. The L option has
no effect and does not appear in the menu if the U option is not used.
The W (Weights) option is invoked in the usual way, with only weights 0
and 1 allowed. It selects a set of sites to be analyzed, ignoring the others.
The sites selected are those with weight 1. If the W option is not invoked,
all sites are analyzed.
The output starts by giving the number of species, the number of sites,
and the base frequencies for A, C, G, and T that have been specified. It then
prints out the transition/transversion ratio that was specified or used by
default. It also uses the base frequencies to compute the actual
transition/transversion ratio implied by the parameter.
If the C (Categories) option is used a table of the relative rates of
expected substitution at each category of sites is printed, as well as the
probabilities of each of those rates.
There then follow the data sequences, with the base sequences printed in
groups of ten bases along the lines of the Genbank and EMBL formats. The trees
found are printed as a rooted tree topology. The internal nodes are numbered
arbitrarily for the sake of identification. The number of trees evaluated so
far and the log likelihood of the tree are also given. The branch lengths in
the diagram are roughly proportional to the estimated branch lengths, except
that very short branches are printed out at least three characters in length so
that the connections can be seen.
A table is printed showing the length of each tree segment, and the time
(in units of expected nucleotide substitutions per site) of each fork in the
tree, measured from the root of the tree. I have not attempted in include code
for approximate confidence limits on branch points, as I have done for branch
lengths in DNAML, both because of the extreme crudeness of that test, and
because the variation of time for different forks would be highly correlated.
The log likelihood printed out with the final tree can be used to perform
various likelihood ratio tests. One can, for example, compare runs with
different values of the expected transition/transversion ratio to determine
which value is the maximum likelihood estimate, and what is the allowable range
of values (using a likelihood ratio test, which you will find described in
mathematical statistics books). One could also estimate the base frequencies
in the same way. Both of these, particularly the latter, require multiple runs
of the program to evaluate different possible values, and this might get
This program makes possible for the first time a (reasonably) legitimate
statistical test of the molecular clock. To do such a test, run DNAML and
DNAMLK on the same data. If the trees obtained are of the same topology (when
considered as unrooted), it is legitimate to compare their likelihoods by the
likelihood ratio test. In DNAML the likelihood has been computed by estimating
2n-3 branch lengths, if their are n tips on the tree. In DNAMLK it has been
computed by estimating n-1 branching times (in effect, n-1 branch lengths).
The difference in the number of parameters is (2n-3)-(n-1) = n-2. To perform
the test take the difference in log likelihoods between the two runs (DNAML
should be the higher of the two, barring numerical iteration difficulties) and
double it. Look this up on a chi-square distribution with n-2 degrees of
freedom. If the result is significant, the log likelihood has been
significantly increased by allowing all 2n-3 branch lengths to be estimated
instead of just n-1, and molecular clock may be rejected.
If the U (User Tree) option is used and more than one tree is supplied,
and the program is not told to assume autocorrelation between the rates at
different sites, the program also performs a statistical test of each of these
trees against the one with highest likelihood. This test, due to Kishino and
Hasegawa (1989), uses the mean and variance of log-likelihood differences
between trees, taken across sites. If the mean is more than 1.96 standard
deviations different then the trees are declared significantly different. This
use of the empirical variance of log-likelihood differences is more robust and
nonparametric than the classical likelihood ratio test, and may to some extent
compensate for the any lack of realism in the model underlying this program.
The program prints out a table of the log-likelihoods of each tree, the
differences of each from the highest one, the variance of that quantity as
determined by the log-likelihood differences at individual sites, and a
conclusion as to whether that tree is or is not significantly worse than the
best one. However the Kishino-Hasegawa-Templeton test is not available if we
assume that there is autocorrelation of rates at neighboring sites (option R)
and is not done in those cases.
The branch lengths printed out are scaled in terms of expected numbers of
substitutions, counting both transitions and transversions but not replacements
of a base by itself, and scaled so that the average rate of change, averaged
over all sites analyzed, is set to 1.0 if there are multiple categories of
sites. This means that whether or not there are multiple categories of sites,
the expected fraction of change for very small branches is equal to the branch
length. Of course, when a branch is twice as long this does not mean that
there will be twice as much net change expected along it, since some of the
changes occur in the same site and overlie or even reverse each other. The
branch lengths estimates here are in terms of the expected underlying numbers
of changes. That means that a branch of length 0.26 is 26 times as long as one
which would show a 1% difference between the nucleotide sequences at the
beginning and end of the branch. But we would not expect the sequences at the
beginning and end of the branch to be 26% different, as there would be some
overlaying of changes.
Because of limitations of the numerical algorithm, branch length estimates
of zero will often print out as small numbers such as 0.00001. If you see a
branch length that small, it is really estimated to be of zero length.
Another possible source of confusion is the existence of negative values
for the log likelihood. This is not really a problem; the log likelihood is
not a probability but the logarithm of a probability. When it is negative it
simply means that the corresponding probability is less than one (since we are
seeing its logarithm). The log likelihood is maximized by being made more
positive: -30.23 is worse than -29.14.
At the end of the output, if the C option is in effect with multiple rate
categories, the program will print a list of what site categories contributed
the most to the final likelihood. This combination of rate categories need not
have contributed a majority of the likelihood, just a plurality. Still, it
will be helpful as a view of where the program infers that the higher and lower
rates are. Note that the use in this calculations of the prior probabilities
of different rates, and the average patch length, gives this inference a
"smoothed" appearance: some other combination of rates might make a greater
contribution to the likelihood, but be discounted because it conflicts with
this prior information. See the example output below to see what this printout
of rate categories looks like.
If the 3 (write out tree) option is used, then the tree with branch
lengths will be written out on the tree file. The program DRAWGRAM can then be
used to draw the tree more precisely.
The constants defined at the beginning of the program include "maxtrees",
the maximum number of user trees that can be processed. It is small (10) at
present to same some further memory but the cost of increasing it is not very
great. Other constants include "maxcategories", the maximum number of site
categories, "namelength", the length of species names in characters, and three
others, "smoothings", "iterations", and "epsilon", that
The other CONSTants include maxcategories, the maximum number of site
categories, namelength, the length of species names in characters, and three
CONSTants, smoothings, iterations, and epsilon, that help "tune" the algorithm
and define the compromise between execution speed and the quality of the branch
lengths found by iteratively maximizing the likelihood. Reducing iterations
and smoothings, and increasing epsilon, will result in faster execution but a
worse result. These values will not usually have to be changed.
The program spends most of its time doing real arithmetic. Any software
or hardware changes that speed up that arithmetic will speed it up by a nearly
proportional amount. For example, microcomputers having a numeric co-processor
will run the program much faster, if the compiled program uses it. The
algorithm, with separate and independent computations occurring for each
pattern, lends itself readily to parallel processing.
PAST AND FUTURE OF THE PROGRAM
This program was developed in 1989 by combining code from DNAPARS and from
DNAML. It was speeded up by two major developments, the use of aliasing of
nucleotide sites (version 3.1) and pretabulation of some exponentials (added by
Akiko Fuseki in version 3.4). In version 3.5 the Hidden Markov Chain code was
added and the method of iterating branch lengths was changed from an EM
algorithm to direct search. The Hidden Markov Chain code slows things down,
especially if there is autocorrelation between sites, so this version is slower
than version 3.4. Nevertheless we hope that the sacrifice is worth it.
Two changes that are needed in the future are to put in some way of
allowing for base composition of nucleotide sequences in different parts of the
phylogeny, and to re-introduce the code that allowed us to assign relative
rates of change to individual sites. By allowing prespecified rates and also
allowing the Hidden Markov Chain method of inferring rates at different sites,
we could then deal with cases in which (say) first, second, and third positions
in a coding sequence had rates of (say) 1.0 : 0.8 : 2.7 but the Hidden Markov
Chain also inferred rates that varied along the length of the coding sequence.
This would be much more realistic.
----------------------------TEST DATA SET--------------------------
(it was run with two categories with rates 1.0 and 3.2, and with
probabilities 0.4 and 0.6 for these rates, and with patch length
parameter = 1.5)
----- CONTENTS OF OUTPUT FILE (with all numerical options on) ------
5 Species, 13 Sites
Nucleic acid sequence
Maximum Likelihood method with molecular clock, version 3.5c
Site category Rate of change
Alpha AACGTGGCCA AAT
Beta ..G..C.... ..C
Gamma C.TT.C.T.. C.A
Delta GGTA.TT.GG CC.
Epsilon GGGA.CT.GG CCC
Empirical Base Frequencies:
Transition/transversion ratio = 2.000000
(Transition/transversion parameter = 1.523077)
Ln Likelihood = -72.40499
Ancestor Node Node Time Length
-------- ---- ---- ---- ------
3 4 3.40037 3.40037
4 Delta 3.55871 0.15834
4 Epsilon 3.55871 0.15834
3 2 2.75248 2.75248
2 Gamma 3.55871 0.80623
2 1 3.38653 0.63405
1 Alpha 3.55871 0.17219
1 Beta 3.55871 0.17219
Combination of categories that contributes the most to the likelihood: