| fpromlk

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## Function

Protein phylogeny by maximum likelihood

## Description

Same as PROML but assumes a molecular clock. The use of the two programs together permits a likelihood ratio test of the molecular clock hypothesis to be made.

Estimates phylogenies from protein amino acid sequences by maximum likelihood. The PAM, JTT, or PMB models can be employed, and also use of a Hidden Markov model of rates, with the program inferring which sites have which rates. This also allows gamma-distribution and gamma-plus-invariant sites distributions of rates across sites. It also allows different rates of change at known sites.

## Algorithm

This program implements the maximum likelihood method for protein amino acid 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 PROML. It uses the Dayhoff probability model of change between amino acids. Its algorithmic details are not yet published, but many of them are similar to DNAMLK.

The assumptions of the model are:

1. Each position in the sequence evolves independently.
2. Different lineages evolve independently.
3. Each position undergoes substitution at an expected rate which is chosen from a series of rates (each with a probability of occurrence) which we specify.
4. All relevant positions are included in the sequence, not just those that have changed or those that are "phylogenetically informative".
5. The probabilities of change between amino acids are given by the model of Jones,
6. Taylor, and Thornton (1992), the PMB model of Veerassamy, Smith and Tillier (2004), or the PAM model of Dayhoff (Dayhoff and Eck, 1968; Dayhoff et. al., 1979).

Note the assumption that we are looking at all positions, including those that have not changed at all. It is important not to restrict attention to some positions 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 positions that had changed.

This program uses a Hidden Markov Model (HMM) method of inferring different rates of evolution at different amino acid positions. 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 positions all having the same rate. 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 positions. The program computes the likelihood by summing it over all possible assignments of rates to positions, 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 position having these rates are 0.4, 0.3, and 0.3, and that the average patch length (number of consecutive positions 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 positions to rate 2.4, or that fail to have consecutive positions that have the same rate.

The Hidden Markov Model framework for rate variation among positions 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 positions.

This feature effectively removes the artificial assumption that all positions have the same rate, and also means that we need not know in advance the identities of the positions that have a particular rate of evolution.

Another layer of rate variation also is available. The user can assign categories of rates to each positions (for example, we might want amino acid positions in the active site of a protein to change more slowly than other positions. This is done with the categories input file and the C option. We then specify (using the menu) the relative rates of evolution of amino acid positions in the different categories. For example, we might specify that positions in the active site evolve at relative rates of 0.2 compared to 1.0 at other positions. If we are assuming that a particular position maintains a cysteine bridge to another, we may want to put it in a category of positions (including perhaps the initial position of the protein sequence which maintains methionine) which changes at a rate of 0.0.

If both user-assigned rate categories and Hidden Markov Model rates are allowed, the program assumes that the actual rate at a position 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.

## Usage

Here is a sample session with fpromlk

    % fpromlk
    Protein phylogeny by maximum likelihood
    Input (aligned) protein sequence set(s): promlk.dat
    Phylip tree file (optional):
    Phylip promlk program output file [promlk.fpromlk]:

    Adding species:
       1. Alpha
       2. Beta
       3. Gamma
       4. Delta
       5. Epsilon

    Output written to file "promlk.fpromlk"

    Tree also written onto file "promlk.treefile"

    Done.

Go to the input files for this example
Go to the output files for this example

## Command line arguments

    Protein phylogeny by maximum likelihood
    Version: EMBOSS:6.5.0.0

       Standard (Mandatory) qualifiers:
      [-sequence]          seqsetall  File containing one or more sequence
                                      alignments
      [-intreefile]        tree       Phylip tree file (optional)
      [-outfile]           outfile    [*.fpromlk] Phylip promlk program output
                                      file

       Additional (Optional) qualifiers (* if not always prompted):
       -ncategories        integer    [1] Number of substitution rate categories
                                      (Integer from 1 to 9)
    *  -rate               array      Rate for each category
    *  -categories         properties File of substitution rate categories
       -weights            properties Weights file
    *  -lengths            boolean    [N] Use branch lengths from user trees
       -model              menu       [Jones-Taylor-Thornton] Probability model
                                      for amino acid change (Values: j
                                      (Jones-Taylor-Thornton); h (Henikoff/Tillier
                                      PMBs); d (Dayhoff PAM))
       -gammatype          menu       [n] Rate variation among sites (Values: g
                                      (Gamma distributed rates); i
                                      (Gamma+invariant sites); h (User defined HMM
                                      of rates); n (Constant rate))
    *  -gammacoefficient   float      [1] Coefficient of variation of substitution
                                      rate among sites (Number 0.001 or more)
    *  -ngammacat          integer    [1] Number of categories (1-9) (Integer from
                                      1 to 9)
    *  -invarcoefficient   float      [1] Coefficient of variation of substitution
                                      rate among sites (Number 0.001 or more)
    *  -ninvarcat          integer    [1] Number of categories (1-9) including one
                                      for invariant sites (Integer from 1 to 9)
    *  -invarfrac          float      [0.0] Fraction of invariant sites (Number
                                      from 0.000 to 1.000)
    *  -nhmmcategories     integer    [1] Number of HMM rate categories (Integer
                                      from 1 to 9)
    *  -hmmrates           array      [1.0] HMM category rates
    *  -hmmprobabilities   array      [1.0] Probability for each HMM category
    *  -adjsite            boolean    [N] Rates at adjacent sites correlated
    *  -lambda             float      [1.0] Mean block length of sites having the
                                      same rate (Number 1.000 or more)
    *  -njumble            integer    [0] Number of times to randomise (Integer 0
                                      or more)
    *  -seed               integer    [1] Random number seed between 1 and 32767
                                      (must be odd) (Integer from 1 to 32767)
    *  -global             boolean    [N] Global rearrangements
       -outgrno            integer    [0] Species number to use as outgroup
                                      (Integer 0 or more)
       -[no]trout          toggle     [Y] Write out trees to tree file
    *  -outtreefile        outfile    [*.fpromlk] Phylip tree output file
                                      (optional)
       -printdata          boolean    [N] Print data at start of run
       -[no]progress       boolean    [Y] Print indications of progress of run
       -[no]treeprint      boolean    [Y] Print out tree
       -hypstate           boolean    [N] Reconstruct hypothetical sequence

       Advanced (Unprompted) qualifiers: (none)
       Associated qualifiers:

       "-sequence" associated qualifiers
       -sbegin1            integer    Start of each sequence to be used
       -send1              integer    End of each sequence to be used
       -sreverse1          boolean    Reverse (if DNA)
       -sask1              boolean    Ask for begin/end/reverse
       -snucleotide1       boolean    Sequence is nucleotide
       -sprotein1          boolean    Sequence is protein
       -slower1            boolean    Make lower case
       -supper1            boolean    Make upper case
       -scircular1         boolean    Sequence is circular
       -sformat1           string     Input sequence format
       -iquery1            string     Input query fields or ID list
       -ioffset1           integer    Input start position offset
       -sdbname1           string     Database name
       -sid1               string     Entryname
       -ufo1               string     UFO features
       -fformat1           string     Features format
       -fopenfile1         string     Features file name

       "-outfile" associated qualifiers
       -odirectory3        string     Output directory

       "-outtreefile" associated qualifiers
       -odirectory         string     Output directory

       General qualifiers:
       -auto               boolean    Turn off prompts
       -stdout             boolean    Write first file to standard output
       -filter             boolean    Read first file from standard input, write
                                      first file to standard output
       -options            boolean    Prompt for standard and additional values
       -debug              boolean    Write debug output to program.dbg
       -verbose            boolean    Report some/full command line options
       -help               boolean    Report command line options and exit. More
                                      information on associated and general
                                      qualifiers can be found with -help -verbose
       -warning            boolean    Report warnings
       -error              boolean    Report errors
       -fatal              boolean    Report fatal errors
       -die                boolean    Report dying program messages
       -version            boolean    Report version number and exit

Qualifier | Type | Description | Allowed values | Default
--- | --- | --- | --- | ---
Standard (Mandatory) qualifiers
[-sequence] (Parameter 1) | seqsetall | File containing one or more sequence alignments | Readable sets of sequences | Required
[-intreefile] (Parameter 2) | tree | Phylip tree file (optional) | Phylogenetic tree |
[-outfile] (Parameter 3) | outfile | Phylip promlk program output file | Output file | <*>.fpromlk
Additional (Optional) qualifiers
-ncategories | integer | Number of substitution rate categories | Integer from 1 to 9 | 1
-rate | array | Rate for each category | List of floating point numbers |
-categories | properties | File of substitution rate categories | Property value(s) |
-weights | properties | Weights file | Property value(s) |
-lengths | boolean | Use branch lengths from user trees | Boolean value Yes/No | No

-model

list

Probability model for amino acid change

j | (Jones-Taylor-Thornton)
h | (Henikoff/Tillier PMBs)
d | (Dayhoff PAM)

Jones-Taylor-Thornton

-gammatype

list

Rate variation among sites

g | (Gamma distributed rates)
i | (Gamma+invariant sites)
h | (User defined HMM of rates)
n | (Constant rate)

n

-gammacoefficient | float | Coefficient of variation of substitution rate among sites | Number 0.001 or more | 1
-ngammacat | integer | Number of categories (1-9) | Integer from 1 to 9 | 1
-invarcoefficient | float | Coefficient of variation of substitution rate among sites | Number 0.001 or more | 1
-ninvarcat | integer | Number of categories (1-9) including one for invariant sites | Integer from 1 to 9 | 1
-invarfrac | float | Fraction of invariant sites | Number from 0.000 to 1.000 | 0.0
-nhmmcategories | integer | Number of HMM rate categories | Integer from 1 to 9 | 1
-hmmrates | array | HMM category rates | List of floating point numbers | 1.0
-hmmprobabilities | array | Probability for each HMM category | List of floating point numbers | 1.0
-adjsite | boolean | Rates at adjacent sites correlated | Boolean value Yes/No | No
-lambda | float | Mean block length of sites having the same rate | Number 1.000 or more | 1.0
-njumble | integer | Number of times to randomise | Integer 0 or more | 0
-seed | integer | Random number seed between 1 and 32767 (must be odd) | Integer from 1 to 32767 | 1
-global | boolean | Global rearrangements | Boolean value Yes/No | No
-outgrno | integer | Species number to use as outgroup | Integer 0 or more | 0
-[no]trout | toggle | Write out trees to tree file | Toggle value Yes/No | Yes
-outtreefile | outfile | Phylip tree output file (optional) | Output file | <*>.fpromlk
-printdata | boolean | Print data at start of run | Boolean value Yes/No | No
-[no]progress | boolean | Print indications of progress of run | Boolean value Yes/No | Yes
-[no]treeprint | boolean | Print out tree | Boolean value Yes/No | Yes
-hypstate | boolean | Reconstruct hypothetical sequence | Boolean value Yes/No | No
Advanced (Unprompted) qualifiers
(none)
Associated qualifiers
"-sequence" associated seqsetall qualifiers
-sbegin1 -sbegin_sequence | integer | Start of each sequence to be used | Any integer value | 0
-send1 -send_sequence | integer | End of each sequence to be used | Any integer value | 0
-sreverse1 -sreverse_sequence | boolean | Reverse (if DNA) | Boolean value Yes/No | N
-sask1 -sask_sequence | boolean | Ask for begin/end/reverse | Boolean value Yes/No | N
-snucleotide1 -snucleotide_sequence | boolean | Sequence is nucleotide | Boolean value Yes/No | N
-sprotein1 -sprotein_sequence | boolean | Sequence is protein | Boolean value Yes/No | N
-slower1 -slower_sequence | boolean | Make lower case | Boolean value Yes/No | N
-supper1 -supper_sequence | boolean | Make upper case | Boolean value Yes/No | N
-scircular1 -scircular_sequence | boolean | Sequence is circular | Boolean value Yes/No | N
-sformat1 -sformat_sequence | string | Input sequence format | Any string |
-iquery1 -iquery_sequence | string | Input query fields or ID list | Any string |
-ioffset1 -ioffset_sequence | integer | Input start position offset | Any integer value | 0
-sdbname1 -sdbname_sequence | string | Database name | Any string |
-sid1 -sid_sequence | string | Entryname | Any string |
-ufo1 -ufo_sequence | string | UFO features | Any string |
-fformat1 -fformat_sequence | string | Features format | Any string |
-fopenfile1 -fopenfile_sequence | string | Features file name | Any string |
"-outfile" associated outfile qualifiers
-odirectory3 -odirectory_outfile | string | Output directory | Any string |
"-outtreefile" associated outfile qualifiers
-odirectory | string | Output directory | Any string |
General qualifiers
-auto | boolean | Turn off prompts | Boolean value Yes/No | N
-stdout | boolean | Write first file to standard output | Boolean value Yes/No | N
-filter | boolean | Read first file from standard input, write first file to standard output | Boolean value Yes/No | N
-options | boolean | Prompt for standard and additional values | Boolean value Yes/No | N
-debug | boolean | Write debug output to program.dbg | Boolean value Yes/No | N
-verbose | boolean | Report some/full command line options | Boolean value Yes/No | Y
-help | boolean | Report command line options and exit. More information on associated and general qualifiers can be found with -help -verbose | Boolean value Yes/No | N
-warning | boolean | Report warnings | Boolean value Yes/No | Y
-error | boolean | Report errors | Boolean value Yes/No | Y
-fatal | boolean | Report fatal errors | Boolean value Yes/No | Y
-die | boolean | Report dying program messages | Boolean value Yes/No | Y
-version | boolean | Report version number and exit | Boolean value Yes/No | N

## Input file format

fpromlk reads any normal sequence USAs.

### Input files for usage example

### File: promlk.dat

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

## Output file format

fpromlk output starts by giving the number of species, the number of amino acid positions.

If the R (HMM rates) option is used a table of the relative rates of expected substitution at each category of positions 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 amino acids. 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. The unit of branch length is the expected fraction of amino acids changed (so that 1.0 is 100 PAMs).

A table is printed showing the length of each tree segment, and the time (in units of expected amino acid substitutions per position) 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 PROML, 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 relative rate of change in the active site and in the rest of the protein 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 possibl 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 PROML and PROMLK 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 PROML the likelihood has been computed by estimating 2n-3 branch lengths, if their are n tips on the tree. In PROMLK 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 (PROML 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 amino acid positions, 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 amino acid positions. 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 amino acid positions 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 positions (option A) and is not done in those cases.

The branch lengths printed out are scaled in terms of 100 times the expected numbers of amino acid substitutions, scaled so that the average rate of change, averaged over all the positions analyzed, is set to 100.0, if there are multiple categories of positions. This means that whether or not there are multiple categories of positions, the expected percentage 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 position and overlie or even reverse each other. underlying numbers of changes. That means that a branch of length 26 is 26 times as long as one which would show a 1% difference between the amino acid 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 amino acid position 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 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. A second list will also be printed out, showing for each position 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). The symbol printed out is for the amino acid which accounts for the largest fraction of the likelihood at that position. In that table, if a position has an amino acid which accounts for more than 95% of the likelihood, its symbol printed in capital letters (W rather than w). One limitation of the current version of the program is that when there are multiple HMM rates (option R) the reconstructed amino acids are based on only the single assignment of rates to positions 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.

### Output files for usage example

### File: promlk.fpromlk

    Amino acid sequence
       Maximum Likelihood method with molecular clock, version 3.69.650

    Jones-Taylor-Thornton model of amino acid change

                                              +-----------Epsilon
      +---------------------------------------4
      !                                       +-----------Delta
    --3
      !                      +----------------------------Gamma
      +----------------------2
                             !                   +--------Beta
                             +-------------------1
                                                 +--------Alpha

    Ln Likelihood =  -134.70332

     Ancestor      Node      Node Height     Length
     --------      ----      ---- ------     ------
     root            3
       3             4          0.66464      0.66464
       4          Epsilon       0.85971      0.19507
       4          Delta         0.85971      0.19507
       3             2          0.37420      0.37420
       2          Gamma         0.85971      0.48551
       2             1          0.70208      0.32788
       1          Beta          0.85971      0.15763
       1          Alpha         0.85971      0.15763

### File: promlk.treefile

    ((Epsilon:0.19507,Delta:0.19507):0.66464,(Gamma:0.48551,
    (Beta:0.15763,Alpha:0.15763):0.32788):0.37420);

## Data files

None

## Notes

None.

## References

None.

## Warnings

None.

## Diagnostic Error Messages

None.

## Exit status

It always exits with status 0.

## Known bugs

None.

## See also

Program name | Description
--- | ---
distmat | Create a distance matrix from a multiple sequence alignment
ednacomp | DNA compatibility algorithm
ednadist | Nucleic acid sequence distance matrix program
ednainvar | Nucleic acid sequence invariants method
ednaml | Phylogenies from nucleic acid maximum likelihood
ednamlk | Phylogenies from nucleic acid maximum likelihood with clock
ednapars | DNA parsimony algorithm
ednapenny | Penny algorithm for DNA
eprotdist | Protein distance algorithm
eprotpars | Protein parsimony algorithm
erestml | Restriction site maximum likelihood method
eseqboot | Bootstrapped sequences algorithm
fdiscboot | Bootstrapped discrete sites algorithm
fdnacomp | DNA compatibility algorithm
fdnadist | Nucleic acid sequence distance matrix program
fdnainvar | Nucleic acid sequence invariants method
fdnaml | Estimate nucleotide phylogeny by maximum likelihood
fdnamlk | Estimates nucleotide phylogeny by maximum likelihood
fdnamove | Interactive DNA parsimony
fdnapars | DNA parsimony algorithm
fdnapenny | Penny algorithm for DNA
fdolmove | Interactive Dollo or polymorphism parsimony
ffreqboot | Bootstrapped genetic frequencies algorithm
fproml | Protein phylogeny by maximum likelihood
fprotdist | Protein distance algorithm
fprotpars | Protein parsimony algorithm
frestboot | Bootstrapped restriction sites algorithm
frestdist | Calculate distance matrix from restriction sites or fragments
frestml | Restriction site maximum likelihood method
fseqboot | Bootstrapped sequences algorithm
fseqbootall | Bootstrapped sequences algorithm

## Author(s)

This program is an EMBOSS conversion of a program written by Joe Felsenstein as part of his PHYLIP package.

Please report all bugs to the EMBOSS bug team (emboss-bug © emboss.open-bio.org) not to the original author.

## History

Written (2004) - Joe Felsenstein, University of Washington.

Converted (August 2004) to an EMBASSY program by the EMBOSS team.

## Target users

This program is intended to be used by everyone and everything, from naive users to embedded scripts.

## Comments

    None