version 3.69

# Dnadist -- Program to compute distance matrix
from nucleotide sequences

© 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 uses nucleotide sequences to compute a distance matrix, under four different models of nucleotide substitution. It can also compute a table of similarity between the nucleotide sequences. The distance for each pair of species estimates the total branch length between the two species, and can be used in the distance matrix programs Fitch, Kitsch or Neighbor. This is an alternative to using the sequence data itself in the maximum likelihood program Dnaml or the parsimony program Dnapars.

The program reads in nucleotide sequences and writes an output file containing the distance matrix, or else a table of similarity between sequences. The four models of nucleotide substitution are those of Jukes and Cantor (1969), Kimura (1980), the F84 model (Kishino and Hasegawa, 1989; Felsenstein and Churchill, 1996), and the model underlying the LogDet distance (Barry and Hartigan, 1987; Lake, 1994; Steel, 1994; Lockhart et. al., 1994). All except the LogDet distance can be made to allow for for unequal rates of substitution at different sites, as Jin and Nei (1990) did for the Jukes-Cantor model. The program correctly takes into account a variety of sequence ambiguities, although in cases where they exist it can be slow.

Jukes and Cantor's (1969) model assumes that there is independent change at all sites, with equal probability. Whether a base changes is independent of its identity, and when it changes there is an equal probability of ending up with each of the other three bases. Thus the transition probability matrix (this is a technical term from probability theory and has nothing to do with transitions as opposed to transversions) for a short period of time dt is:

                  To:    A        G        C        T
                       ---------------------------------
                   A  | 1-3a      a         a       a
           From:   G  |  a       1-3a       a       a
                   C  |  a        a        1-3a     a
                   T  |  a        a         a      1-3a

where a is u dt, the product of the rate of substitution per unit time (u) and the length dt of the time interval. For longer periods of time this implies that the probability that two sequences will differ at a given site is:

p = 3/4 ( 1 - e- 4/3 u t)

and hence that if we observe p, we can compute an estimate of the branch length ut by inverting this to get

ut = - 3/4 loge ( 1 - 4/3 p )

The Kimura "2-parameter" model is almost as symmetric as this, but allows for a difference between transition and transversion rates. Its transition probability matrix for a short interval of time is:

                  To:     A        G        C        T
                       ---------------------------------
                   A  | 1-a-2b     a         b       b
           From:   G  |   a      1-a-2b      b       b
                   C  |   b        b       1-a-2b    a
                   T  |   b        b         a     1-a-2b

where a is u dt, the product of the rate of transitions per unit time and dt is the length dt of the time interval, and b is v dt, the product of half the rate of transversions (i.e., the rate of a specific transversion) and the length dt of the time interval.

The F84 model incorporates different rates of transition and transversion, but also allows for different frequencies of the four nucleotides. It is the model which is used in Dnaml, the maximum likelihood nucelotide sequence phylogenies program in this package. You will find the model described in the document for that program. The transition probabilities for this model are given by Kishino and Hasegawa (1989), and further explained in a paper by me and Gary Churchill (1996).

The LogDet distance allows a fairly general model of substitution. It computes the distance from the determinant of the empirically observed matrix of joint probabilities of nucleotides in the two species. An explanation of it is available in the chapter by Swofford et al. (1996).

The first three models are closely related. The Dnaml model reduces to Kimura's two-parameter model if we assume that the equilibrium frequencies of the four bases are equal. The Jukes-Cantor model in turn is a special case of the Kimura 2-parameter model where a = b. Thus each model is a special case of the ones that follow it, Jukes-Cantor being a special case of both of the others.

The Jin and Nei (1990) correction for variation in rate of evolution from site to site can be adapted to all of the first three models. It assumes that the rate of substitution varies from site to site according to a gamma distribution, with a coefficient of variation that is specified by the user. The user is asked for it when choosing this option in the menu.

Each distance that is calculated is an estimate, from that particular pair of species, of the divergence time between those two species. For the Jukes- Cantor model, the estimate is computed using the formula for ut given above, as long as the nucleotide symbols in the two sequences are all either A, C, G, T, U, N, X, ?, or - (the latter four indicate a deletion or an unknown nucleotide). This estimate is a maximum likelihood estimate for that model. For the Kimura 2-parameter model, with only these nucleotide symbols, formulas special to that estimate are also computed. These are also, in effect, computing the maximum likelihood estimate for that model. In the Kimura case it depends on the observed sequences only through the sequence length and the observed number of transition and transversion differences between those two sequences. The calculation in that case is a maximum likelihood estimate and will differ somewhat from the estimate obtained from the formulas in Kimura's original paper. That formula was also a maximum likelihood estimate, but with the transition/transversion ratio estimated empirically, separately for each pair of sequences. In the present case, one overall preset transition/transversion ratio is used which makes the computations harder but achieves greater consistency between different comparisons.

For the F84 model, or for any of the models where one or both sequences contain at least one of the other ambiguity codons such as Y, R, etc., a maximum likelihood calculation is also done using code which was originally written for Dnaml. Its disadvantage is that it is slow. The resulting distance is in effect a maximum likelihood estimate of the divergence time (total branch length between) the two sequences. However the present program will be much faster than versions earlier than 3.5, because I have speeded up the iterations.

The LogDet model computes the distance from the determinant of the matrix of co-occurrence of nucleotides in the two species, according to the formula

       D  = - 1/4(loge(|F|) - 1/2loge(fA1fC1fG1fT1fA2fC2fG2fT2))

Where F is a matrix whose (i,j) element is the fraction of sites at which base i occurs in one species and base j occurs in the other. fji is the fraction of sites at which species i has base j. The LogDet distance cannot cope with ambiguity codes. It must have completely defined sequences. One limitation of the LogDet distance is that it may be infinite sometimes, if there are too many changes between certain pairs of nucleotides. This can be particularly noticeable with distances computed from bootstrapped sequences.

Note that there is an 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 the distances by making them too large, and that in turn would cause the distances to misinterpret the meaning of those sites that had changed.

For all of these distance methods, the program allows us to specify that "third position" bases have a different rate of substitution than first and second positions, that introns have a different rate than exons, and so on. The Categories option which does this allows us to make up to 9 categories of sites and specify different rates of change for them.

In addition to the four distance calculations, the program can also compute a table of similarities between nucleotide sequences. These values are the fractions of sites identical between the sequences. The diagonal values are 1.0000. No attempt is made to count similarity of nonidentical nucleotides, so that no credit is given for having (for example) different purines at corresponding sites in the two sequences. This option has been requested by many users, who need it for descriptive purposes. It is not intended that the table be used for inferring the tree.

## INPUT FORMAT AND OPTIONS

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 -- neither is dot (".").

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

    Nucleic acid sequence Distance Matrix program, version 3.69

    Settings for this run:
      D  Distance (F84, Kimura, Jukes-Cantor, LogDet)?  F84
      G          Gamma distributed rates across sites?  No
      T                 Transition/transversion ratio?  2.0
      C            One category of substitution rates?  Yes
      W                         Use weights for sites?  No
      F                Use empirical base frequencies?  Yes
      L                       Form of distance matrix?  Square
      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

      Y to accept these or type 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 D option selects one of the four distance methods, or the similarity table. It toggles among the five methods. The default method, if none is specified, is the F84 model.

If the G (Gamma distribution) option is selected, 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 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 / a1/2

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.

The F (Frequencies) option appears when the Maximum Likelihood distance is selected. This distance requires that the program be provided with the equilibrium frequencies of the four bases A, C, G, and T (or U). Its default setting 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 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 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

If both user-assigned rate categories and Gamma-distributed rates are allowed, the program assumes that the actual rate at a site is the product of the user-assigned category rate and the Gamma-distributed rate. This allows you to specify that certain sites have higher or lower rates of change while also allowing the program to allow variation of rates in addition to that. (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.

The L option specifies that the output file is to have the distance matrix in lower triangular form.

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 option 0 is the usual one. It is 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.

## OUTPUT FORMAT

As the distances are computed, the program prints on your screen or terminal the names of the species in turn, followed by one dot (".") for each other species for which the distance to that species has been computed. Thus if there are ten species, the first species name is printed out, followed by nine dots, then on the next line the next species name is printed out followed by eight dots, then the next followed by seven dots, and so on. The pattern of dots should form a triangle. When the distance matrix has been written out to the output file, the user is notified of that.

The output file contains on its first line the number of species. The distance matrix is then printed in standard form, with each species starting on a new line with the species name, followed by the distances to the species in order. These continue onto a new line after every nine distances. If the L option is used, the matrix of distances is in lower triangular form, so that only the distances to the other species that precede each species are printed. Otherwise the distance matrix is square with zero distances on the diagonal. In general the format of the distance matrix is such that it can serve as input to any of the distance matrix programs.

If the option to print out the data is selected, the output file will precede the data by more complete information on the input and the menu selections. The output file begins by giving the number of species and the number of characters, and the identity of the distance measure that is being used.

If the C (Categories) option is used a table of the relative rates of expected substitution at each category of sites is printed, and a listing of the categories each site is in.

There will then follow the equilibrium frequencies of the four bases. If the Jukes-Cantor or Kimura distances are used, these will necessarily be 0.25 : 0.25 : 0.25 : 0.25. The output then shows the transition/transversion ratio that was specified or used by default. In the case of the Jukes-Cantor distance this will always be 0.5. The transition-transversion parameter (as opposed to the ratio) is also printed out: this is used within the program and can be ignored. 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 distances 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 may 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.

One problem that can arise is that two or more of the species can be so dissimilar that the distance between them would have to be infinite, as the likelihood rises indefinitely as the estimated divergence time increases. For example, with the Jukes-Cantor model, if the two sequences differ in 75% or more of their positions then the estimate of divergence time would be infinite. Since there is no way to represent an infinite distance in the output file, the program regards this as an error, issues an error message indicating which pair of species are causing the problem, and stops. It might be that, had it continued running, it would have also run into the same problem with other pairs of species. If the Kimura distance is being used there may be no error message; the program may simply give a large distance value (it is iterating towards infinity and the value is just where the iteration stopped). Likewise some maximum likelihood estimates may also become large for the same reason (the sequences showing more divergence than is expected even with infinite branch length). I hope in the future to add more warning messages that would alert the user the this.

If the similarity table is selected, the table that is produced is not in a format that can be used as input to the distance matrix programs. It has a heading, and the species names are also put at the tops of the columns of the table (or rather, the first 8 characters of each species name is there, the other two characters omitted to save space). There is not an option to put the table into a format that can be read by the distance matrix programs, nor is there one to make it into a table of fractions of difference by subtracting the similarity values from 1. This is done deliberately to make it more difficult for to use these values to construct trees. The similarity values are not corrected for multiple changes, and their use to construct trees (even after converting them to fractions of difference) would be wrong, as it would lead to severe conflict between the distant pairs of sequences and the close pairs of sequences.

## PROGRAM CONSTANTS

The constants that are available to be changed by the user at the beginning of the program include "maxcategories", the maximum number of site categories, "iterations", which controls the number of times the program iterates the EM algorithm that is used to do the maximum likelihood distance, "namelength", the length of species names in characters, and "epsilon", a parameter which controls the accuracy of the results of the iterations which estimate the distances. Making "epsilon" smaller will increase run times but result in more decimal places of accuracy. This should not be necessary.

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.

### TEST DATA SET

       5   13
    Alpha     AACGTGGCCACAT
    Beta      AAGGTCGCCACAC
    Gamma     CAGTTCGCCACAA
    Delta     GAGATTTCCGCCT
    Epsilon   GAGATCTCCGCCC

### CONTENTS OF OUTPUT FILE (with all numerical options on)

(Note that when the options for displaying the input data are turned off, the output is in a form suitable for use as an input file in the distance matrix programs).

    Nucleic acid sequence Distance Matrix program, version 3.69

     5 species,  13  sites

      F84 Distance

    Transition/transversion ratio =   2.000000

    Name            Sequences
    ----            ---------

    Alpha        AACGTGGCCA CAT
    Beta         ..G..C.... ..C
    Gamma        C.GT.C.... ..A
    Delta        G.GA.TT..G .C.
    Epsilon      G.GA.CT..G .CC

    Empirical Base Frequencies:

       A       0.24615
       C       0.36923
       G       0.21538
      T(U)     0.16923

        5
    Alpha       0.000000  0.303900  0.857544  1.158927  1.542899
    Beta        0.303900  0.000000  0.339727  0.913522  0.619671
    Gamma       0.857544  0.339727  0.000000  1.631729  1.293713
    Delta       1.158927  0.913522  1.631729  0.000000  0.165882
    Epsilon     1.542899  0.619671  1.293713  0.165882  0.000000