version 3.69

with molecular clock

© Copyright 1986-2008 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 indirectly related to Dnaml. Details of the algorithm are not yet published, but many aspects of it are similar to Dnaml, and these are published in the paper by Felsenstein and Churchill (1996). The model of base substitution allows the expected frequencies of the four bases to be unequal, allows the expected frequencies of transitions and transversions to be unequal, and has several ways of allowing different rates of evolution at different sites.

The assumptions of the model are:

- Each site in the sequence evolves independently.
- Different lineages evolve independently.
- There is a molecular clock.
- Each site undergoes substitution at an expected rate which is chosen from a series of rates (each with a probability of occurrence) which we specify.
- All relevant sites are included in the sequence, not just those that have changed or those that are "phylogenetically informative".
- 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 transition.
- (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:
- Determine whether the existing base is a purine or a pyrimidine.
- Draw from the proper pool:
Purine pool: Pyrimidine pool: | | | | | 0.4706 A | | 0.5714 C | | 0.5294 G | | 0.4286 T | | (ratio is | | (ratio is | | 0.24 : 0.27) | | 0.28 : 0.21) | |_______________| |_______________|

- Second kind of event:
Draw from the overall pool:

| | | 0.24 A | | 0.28 C | | 0.27 G | | 0.21 T | |__________________|

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). Another explanation is available in the paper by Felsenstein and Churchill (1996).

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.

This program uses a Hidden Markov Model (HMM) method of inferring different rates of evolution at different sites. This was described in a paper by me and Gary Churchill (1996). 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 is. The rates can also be chosen by the program to approximate a Gamma distribution of rates, or a Gamma distribution plus a class of invariant sites. The program computes the 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 A 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 give 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.

The Hidden Markov Model framework for rate variation among sites was independently developed by Yang (1993, 1994, 1995). We have implemented a general scheme for a Hidden Markov Model of rates; we allow the rates and their prior probabilities to be specified arbitrarily by the user, or by a discrete approximation to a Gamma distribution of rates (Yang, 1995), or by a mixture of a Gamma distribution and a class of invariant sites.

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.

Another layer of rate variation also is available. The user can assign categories of rates to each site (for example, we might want first, second, and third codon positions in a protein coding sequence to be three different categories). This is done with the categories input file and the C option. We then specify (using the menu) the relative rates of evolution of sites in the different categories. For example, we might specify that first, second, and third positions evolve at relative rates of 1.0, 0.8, and 2.7.

If both user-assigned rate categories and Hidden Markov Model rates are allowed, the program assumes that the actual rate at a site is the product of the user-assigned category rate and the Hidden Markov Model regional rate. (This may not always make perfect biological sense: it would be more natural to assume some upper bound to the rate, as we have discussed in the Felsenstein and Churchill paper). Nevertheless you may want to use both types of rate variation.

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.

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.

The options are selected using an interactive menu. The menu looks like this:

Nucleic acid sequence Maximum Likelihood method with molecular clock, version 3.69 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 R Rate variation among sites? constant rate G Global rearrangements? No W Sites weighted? 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, ANSI, none)? 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 5 Reconstruct hypothetical sequences? No 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, W, 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 sequence programs.

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 (they are 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 R (Hidden Markov Model rates) option allows the user to approximate a Gamma distribution of rates among sites, or a Gamma distribution plus a class of invariant sites, or to specify how many categories of substitution rates there will be in a Hidden Markov Model of rate variation, and what are the rates and probabilities for each. By repeatedly selecting the R option one toggles among no rate variation, the Gamma, Gamma+I, and general HMM possibilities.

If you choose Gamma or Gamma+I the program will ask how many rate categories you want. If you have chosen Gamma+I, keep in mind that one rate category will be set aside for the invariant class and only the remaining ones used to approximate the Gamma distribution. For the approximation we do not use the quantile method of Yang (1995) but instead use a quadrature method using generalized Laguerre polynomials. This should give a good approximation to the Gamma distribution with as few as 5 or 6 categories.

In the Gamma and Gamma+I cases, the user will be
asked to supply the coefficient of variation of the rate of substitution
among sites. This is different from the parameters used by Nei and Jin
(1990) but
related to them: their parameter *a* is also known as "alpha",
the shape parameter of the Gamma distribution. It is
related to the coefficient of variation by

CV = 1 / a^{1/2}

or

a = 1 / (CV)^{2}

(their parameter *b* is absorbed here by the requirement that time is scaled so
that the mean rate of evolution is 1 per unit time, which means that *a = b*).
As we consider cases in which the rates are less variable we should set *a*
larger and larger, as *CV* gets smaller and smaller.

If the user instead chooses the general Hidden Markov Model option, they are first asked how many HMM rate categories there will be (for the moment there is an upper limit of 9, which should not be 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 an HMM rate 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 HMM 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 regional rate categories. These probabilities must be nonnegative and sum to 1. The default for the program is one category, with rate 1.0 and probability 1.0 (actually the rate does not matter in that case).

If more than one category is specified, then another option, A, becomes visible in the menu. This allows us to specify that we want to assume that sites that have the same HMM rate category are expected to be clustered so that there is autocorrelation of rates. 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 a bit 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.

Note that the autocorrelation scheme we use is somewhat different from Yang's (1995) autocorrelated Gamma distribution. I am unsure whether this difference is of any importance -- our scheme is chosen for the ease with which it can be implemented.

The C option allows user-defined rate categories. The user is prompted for the number of user-defined rates, and for the rates themselves, which cannot be negative but can be zero. These numbers, which must be nonnegative (some could be 0), are defined relative to each other, so that if rates for three categories are set to 1 : 3 : 2.5 this would have the same meaning as setting them to 2 : 6 : 5. The assignment of rates to sites is then made by reading a file whose default name is "categories". It should contain a string of digits 1 through 9. A new line or a blank can occur after any character in this string. Thus the categories file might look like this:

122231111122411155 1155333333444

With the current options R, A, and C 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, provided one rate scheme represents a restriction of another with fewer parameters. The number of parameters needed for rate variation is the number of regional rate categories, plus the number of user-defined rate categories less 2, plus one if the regional rate categories have a nonzero autocorrelation.

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.

The User tree (option U) is read from a file whose default name is
`intree`.
The trees can be multifurcating. This allows us to test the
hypothesis that a given branch has zero length.

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. In the case of a clock, if some branches have lengths and others do not, the program does not estimate the lengths of those that do not have lengths given in the user tree. If any of the branches do not have lengths, the program re-estimates the lengths of all of them. This is done because estimating some and not others is hard in the case of a clock.

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 Weights (W) option takes the weights from a file whose default name is "weights". The weights follow the format described in the main documentation file.

The M (multiple data sets) option will ask you whether you want to use multiple sets of weights (from the weights file) or multiple data sets from the input file. The ability to use a single data set with multiple weights means that much less disk space will be used for this input data. The bootstrapping and jackknifing tool Seqboot has the ability to create a weights file with multiple weights. Note also that when we use multiple weights for bootstrapping we can also then maintain different rate categories for different sites in a meaningful way. If you use the multiple data sets option rather than multiple weights, you should not at the same time use the user-defined rate categories option (option C), because the user-defined rate categories could then be associated with the wrong sites. This is not a concern when the M option is used by using multiple weights.

The algorithm used for searching among trees is faster than it was in version 3.5, thanks to using a technique invented by David Swofford and J. S. Rogers. This involves not iterating most branch lengths on most trees when searching among tree topologies, This is of necessity a "quick-and-dirty" search but it saves much time.

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 R (HMM rates) 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, if the user has selected the menu option to print them out, 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 to 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 times 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 expensive.

This program makes possible 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 there 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. If there are two user trees, the test done is one which is due to Kishino and Hasegawa (1989), a version of a test originally introduced by Templeton (1983). In this implementation it uses the mean and variance of log-likelihood differences between trees, taken across sites. If the two trees' means are 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.

If there are more than two trees, the test done is an extension of the KHT test, due to Shimodaira and Hasegawa (1999). They pointed out that a correction for the number of trees was necessary, and they introduced a resampling method to make this correction. In the version used here the variances and covariances of the sum of log likelihoods across sites are computed for all pairs of trees. To test whether the difference between each tree and the best one is larger than could have been expected if they all had the same expected log-likelihood, log-likelihoods for all trees are sampled with these covariances and equal means (Shimodaira and Hasegawa's "least favorable hypothesis"), and a P value is computed from the fraction of times the difference between the tree's value and the highest log-likelihood exceeds that actually observed. Note that this sampling needs random numbers, and so the program will prompt the user for a random number seed if one has not already been supplied. With the two-tree KHT test no random numbers are used.

In either the KHT or the SH test 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 test is not available if we assume that there is autocorrelation of rates at neighboring sites (option A) 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 length 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 R option is in effect with multiple HMM rates, the program will print a list of what site categories contributed the most to the final likelihood. This combination of HMM 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 calculation 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.

A second list will also be printed out, showing for each site which rate accounted for the highest fraction of the likelihood. If the fraction of the likelihood accounted for is less than 95%, a dot is printed instead.

Option 3 in the menu controls whether the tree is printed out into the output file. This is on by default, and usually you will want to leave it this way. However for runs with multiple data sets such as bootstrapping runs, you will primarily be interested in the trees which are written onto the output tree file, rather than the trees printed on the output file. To keep the output file from becoming too large, it may be wisest to use option 3 to prevent trees being printed onto the output file.

Option 4 in the menu controls whether the tree estimated by the program is written onto a tree file. The default name of this output tree file is "outtree". If the U option is in effect, all the user-defined trees are written to the output tree file.

Option 5 in the menu controls whether ancestral states are estimated at each node in the tree. If it is in effect, a table of ancestral sequences is printed out (including the sequences in the tip species which are the input sequences). In that table, if a site has a base which accounts for more than 95% of the likelihood, it is printed in capital letters (A rather than a). If the best nucleotide accounts for less than 50% of the likelihood, the program prints out an ambiguity code (such as M for "A or C") for the set of nucleotides which, taken together, account for more than half of the likelihood. The ambiguity codes are listed in the sequence programs documentation file. One limitation of the current version of the program is that when there are multiple HMM rates (option R) the reconstructed nucleotides are based on only the single assignment of rates to sites which accounts for the largest amount of the likelihood. Thus the assessment of 95% of the likelihood, in tabulating the ancestral states, refers to 95% of the likelihood that is accounted for by that particular combination of rates.

The constants defined at the beginning of the program include "maxtrees", the maximum number of user trees that can be processed. It is small (100) at present to save 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 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. The algorithm, with separate and independent computations occurring for each pattern, lends itself readily to parallel processing.

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 Model code was added and the method of iterating branch lengths was changed from an EM algorithm to direct search. The Hidden Markov Model 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.

One change that is needed in the future is to put in some way of allowing for base composition of nucleotide sequences in different parts of the phylogeny.

5 13 Alpha AACGTGGCCAAAT Beta AAGGTCGCCAAAC Gamma CATTTCGTCACAA Delta GGTATTTCGGCCT Epsilon GGGATCTCGGCCC |

(It was run with HMM rates having gamma-distributed rates approximated by 5 rate categories, with coefficient of variation of rates 1.0, and with patch length parameter = 1.5. Two user-defined rate categories were used, one for the first 6 sites, the other for the last 7, with rates 1.0 : 2.0. Weights were used, with sites 1 and 13 given weight 0, and all others weight 1.)

Nucleic acid sequence Maximum Likelihood method with molecular clock, version 3.69 5 species, 13 sites Site categories are: 1111112222 222 Sites are weighted as follows: 01111 11111 110 Name Sequences ---- --------- 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: A 0.23636 C 0.29091 G 0.25455 T(U) 0.21818 Transition/transversion ratio = 2.000000 Discrete approximation to gamma distributed rates Coefficient of variation of rates = 1.000000 (alpha = 1.000000) State in HMM Rate of change Probability 1 0.264 0.522 2 1.413 0.399 3 3.596 0.076 4 7.086 0.0036 5 12.641 0.000023 Expected length of a patch of sites having the same rate = 1.500 Site category Rate of change 1 1.000 2 2.000 +-Epsilon +---------------------------------------------------4 ! +-Delta --3 ! +-------Gamma +---------------------------------------------2 ! +-Beta +-----1 +-Alpha Ln Likelihood = -58.51728 Ancestor Node Node Height Length -------- ---- ---- ------ ------ root 3 3 4 4.14820 4.14820 4 Epsilon 4.29769 0.14949 4 Delta 4.29769 0.14949 3 2 3.67522 3.67522 2 Gamma 4.29769 0.62247 2 1 4.12429 0.44907 1 Beta 4.29769 0.17340 1 Alpha 4.29769 0.17340 Combination of categories that contributes the most to the likelihood: 1122121111 111 Most probable category at each site if > 0.95 probability ("." otherwise) .......... ... Probable sequences at interior nodes: node Reconstructed sequence (caps if > 0.95) 3 .ayrtykcsr cm. 4 .GkaTctCgg Cc. Epsilon GGGATCTCGG CCC Delta GGTATTTCGG CCT 2 .AykTcgtcA ca. Gamma CATTTCGTCA CAA 1 .AcgTcGCCA AA. Beta AAGGTCGCCA AAC Alpha AACGTGGCCA AAT |