This program implements the maximum likelihood method for DNA sequences. This program is fairly slow, and can be expensive to run. The present version is, however, faster than earlier versions of DNAML. 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 and Hirohisa Kishino are particularly instructive (Hasegawa and Yano, 1984a, 1984b; Hasegawa et. al. 1985a, 1985b; Kishino and Hasegawa, 1989). My 1981 algorithm is similar to the present one, except that it did not allow for different rates of transitions and transversions and for different rates of evolution at different sites. The assumptions of the present model are:

- Each site in the sequence evolves independently.
- Different lineages evolve independently.
- 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:
- 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.
- 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 I I 0.24 A I I 0.28 C I I 0.27 G I I 0.21 T I I 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 will be described in a future paper by me and Gary Churchill, of Cornell University. It allowsus 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.

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 follows 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 this:

Nucleic acid sequence Maximum Likelihood method, version 3.5c Settings for this run: U Search for best tree? Yes T Transition/transversion ratio: 2.0000 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 O Outgroup root? No, use as outgroup species 1 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 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 (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 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 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 case).

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 an unrooted tree topology (possibly rooted by outgroup if so requested). 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. Note that the trees printed out have a trifurcation at the base. 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 (in units of expected nucleotide substitutions per site), as well as (very) rough confidence limits on their lengths. As with CONTML, if a confidence limit is negative, this indicates that rearrangement of the tree in that region is not excluded, while if both limits are positive, rearrangement is still not necessarily excluded because the variance calculation on which the confidence limits are based results in an underestimate, which makes the confidence limits too narrow.

In addition to the confidence limits, the program performs a crude Likelihood Ratio Test (LRT) for each branch of the tree. The program computes the ratio of likelihoods with and without this branch length forced to zero length. This done by comparing the likelihoods changing only that branch length. A truly correct LRT would force that branch length to zero and also allow the other branch lengths to adjust to that. The result would be a likelihood ratio closer to 1. Therefore the present LRT will err on the side of being too significant. YOU ARE WARNED AGAINST TAKING IT TOO SERIOUSLY. If you want to get a better likelihood curve for a branch length you can do multiple runs with different prespecified lengths for that branch, as discussed above in the discussion of the L option.

One should also realize that if you are looking not at a previously-chosen branch but at all branches, that you are seeing the results of multiple tests. With 20 tests, one is expected to reach significance at the P = .05 level purely by chance. You should therefore use a much more conservative significance level, such as .05 divided by the number of tests. The significance of these tests is shown by printing asterisks next to the confidence interval on each branch length. It is important to keep in mind that both the confidence limits and the tests are very rough and approximate, and probably indicate more significance than they should. Nevertheless, maximum likelihood is one of the few methods that can give you any indication of its own error; most other methods simply fail to warn the user that there is any error! (In fact, whole philosophical schools of taxonomists exist whose main point seems to be that there isn't any error, that the "most parsimonious" tree is the best tree by definition and that's that).

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.

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.

Confidence limits on the branch lengths are also given. Of course a negative value of the branch length is meaningless, and a confidence limit overlapping zero simply means that the branch length is not necessarily significantly different from zero. 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. Note that versions 2.7 and earlier of this program printed out the branch lengths in terms of expected probability of change, so that they were scaled differently.

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.

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 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.

This program, which in version 2.6 replaced the old version of DNAML, is not derived directly from it but instead was developed by modifying CONTML, with which it shares many of its data structures and much of its strategy. 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-------------------------- 5 13 Alpha AACGTGGCCAAAT Beta AAGGTCGCCAAAC Gamma CATTTCGTCACAA Delta GGTATTTCGGCCT Epsilon GGGATCTCGGCCC (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) ------- Nucleic acid sequence Maximum Likelihood method, version 3.5c Site category Rate of change Probability 1 1.000 0.400 2 3.200 0.600 Expected length of a patch of sites having the same rate = 1.500 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 0.24615 C 0.29231 G 0.24615 T(U) 0.21538 Transition/transversion ratio = 2.000000 (Transition/transversion parameter = 1.523077) +Epsilon +------------------------------------------3 +--2 +----Delta ! ! ! +Beta ! --1---------------------Gamma ! +---Alpha remember: this is an unrooted tree! Ln Likelihood = -72.22208 Examined 15 trees Between And Length Approx. Confidence Limits ------- --- ------ ------- ---------- ------ 1 2 0.06639 ( zero, 0.46657) ** 2 3 3.32192 ( zero, infinity) 3 Epsilon 0.00006 ( zero, 0.30554) ** 3 Delta 0.31809 ( zero, 0.78094) ** 2 Beta 0.00003 ( zero, 0.41027) ** 1 Gamma 1.47499 ( zero, 3.68232) 1 Alpha 0.27845 ( zero, 0.72666) ** * = significantly positive, P < 0.05 ** = significantly positive, P < 0.01 Combination of categories that contributes the most to the likelihood: 2222111111 222