## EMBOSS: PHYLIP

# Original PHYLIP dnamlk documentation.

This is the original PHYLIP documentation and includes information on the old menu driven program which is now replaced with the EMBOSS acd file.

                                                                       version 3.5c

             DNAMLK -- DNA Maximum Likelihood program with molecular clock

    (c) Copyright  1986-1993  by  Joseph  Felsenstein  and  by  the  University  of
    Washington.  Written by Joseph Felsenstein.  Permission is granted to copy this
    document provided that no fee is charged for it and that this copyright  notice
    is not removed.

         This program implements the maximum likelihood method  for  DNA  sequences
    under  the  constraint  that  the  trees  estimated  must  be consistent with a
    molecular clock.  The molecular clock is the assumption that the  tips  of  the
    tree  are  all  equidistant,  in branch length, from its root.  This program is
    fairly slow, and can be expensive to run.  The program is indirectly related to
    DNAML.   Its  algorithmic  details  are  not  yet published, but details of the
    algorithm of a previous version of DNAML are described in my paper  in  Journal
    of  Molecular  Evolution  (1981a).   For  some uses of this program and further
    developments, the  papers  of  Masami  Hasegawa  are  particularly  instructive
    (Hasegawa  and  Yano,  1984a,  1984b; Hasegawa et. al.  1985a, 1985b).  As with
    DNAML, the model of base substitutions used in this  program  is  more  general
    than  the  model  described  in  the  1981a  paper, as it allows transition and
    transversion rates to be expected to be unequal.

         The assumptions of the model are:

         1.  Each site in the sequence evolves independently.

         2.  Different lineages evolve independently.

         3.  There is a molecular clock.

         4.  Each site undergoes substitution at an expected rate which  is  chosen
    from  a  series  of  rates  (each  with  a  probability of occurrence) which we
    specify.

         5.  All relevant sites are included in the sequence, not just  those  that
    have changed or those that are "phylogenetically informative".

         6.  A substitution consists of one of two sorts of events:

               a.  The first kind of event  consists  of  the  replacement  of  the
                  existing base by a base drawn from a pool of purines or a pool of
                  pyrimidines (depending on whether the base being replaced  was  a
                  purine or a pyrimidine).  It can lead either to no change or to a
                  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:

       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 wil
    l be described in  a  future  paper  by  me  and  Gary  Churchill,  of  Cornell
    University.  It allows us to specify to the program that there will be a number
    of different possible evolutionary  rates,  what  the  prior  probabilities  of
    occurrence  of  each  is,  and  what the average length of a patch of sites all
    having the same rate.  The program then computes the likelihood by  summing  it
    over  all  possible  assignments of rates to sites, weighting each by its prior
    probability of occurrence.

         For example, if we have used the C and  R  options  (described  below)  to
    specify  that  there are three possible rates of evolution,  1.0, 2.4, and 0.0,
    that the prior probabilities of a site having these rates  are  0.4,  0.3,  and
    0.3,  and  that  the average patch length (number of consecutive sites with the
    same rate) is 2.0, the program will sum the likelihood over all  possibilities,
    but  giving  less  weight  to those that (say) assign all sites to rate 2.4, or
    that fail to have consecutive sites that have the same rate.

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

                               INPUT FORMAT AND OPTIONS

         Subject to these assumptions, the program is a correct maximum  likelihood
    method.   The  input is fairly standard, with one addition.  As usual the first
    line of the file gives the number of species and the number  of  sites.   There
    follow the character W if the Weights options is being used.

         Next come the species data.  Each sequence starts on a  new  line,  has  a
    ten-character  species  name  that  must  be blank-filled to be of that length,
    followed immediately by the species data in the one-letter code.  The sequences
    must  either  be  in the "interleaved" or "sequential" formats described in the
    Molecular Sequence Programs document.  The I option selects between them.   The
    sequences  can  have internal blanks in the sequence but there must be no extra
    blanks at the end of the terminated line.  Note that a blank  is  not  a  valid
    symbol for a deletion.

         After that are the lines (if any) containing the information  for  the  C,
    and W options, as described below.

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

    Nucleic acid sequence
       Maximum Likelihood method with molecular clock, version 3.5c

    Settings for this run:
      U                 Search for best tree?  Yes
      T        Transition/transversion ratio:  2.0
      F       Use empirical base frequencies?  Yes
      C   One category of substitution rates?  Yes
      G                Global rearrangements?  No
      J   Randomize input order of sequences?  No. Use input order
      M           Analyze multiple data sets?  No
      I          Input sequences interleaved?  Yes
      0   Terminal type (IBM PC, VT52, ANSI)?  ANSI
      1    Print out the data at start of run  No
      2  Print indications of progress of run  Yes
      3                        Print out tree  Yes
      4       Write out trees onto tree file?  Yes

    Are these settings correct? (type Y or the letter for one to change)

    The user either types "Y" (followed, of course, by a  carriage-return)  if  the
    settings  shown  are to be accepted, or the letter or digit corresponding to an
    option that is to be changed.

         The options U, J, O, M, and 0 are the usual ones.  They are  described  in
    the  main documentation file of this package.  Option I is the same as in other
    molecular sequence programs and is described in the documentation file for  the
    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 5, which should not be very restrictive).
    Then the program asks for the rates for each category.  These  rates  are  only
    meaningful  relative  to  each  other, so that rates 1.0, 2.0, and 2.4 have the
    exact same effect as rates 2.0, 4.0, and 4.8.  Note that a  category  can  have
    rate of change 0, so that this allows us to take into account that there may be
    a category of sites that are invariant.  Note that the run time of the  program
    will be proportional to the number of rate categories: twice as many categories
    means twice as long a run.  Finally the program will ask for the  probabilities
    of  a  random  site falling into each of these categories.  These probabilities
    must be nonnegative and sum to 1.  Default for the  program  is  one  category,
    with  rate  1.0  and probability 1.0 (actually the rate does not matter in that
    case).

         If more than one category is specified, then another  option,  R,  becomes
    visible  in  the  menu.   This allows us to specify that we want to assume that
    sites that have the same rate category  are  expected  to  be  clustered.   The
    program  asks  for  the value of the average patch length.  This is an expected
    length of patches that have the same rate.  If it is 1, the rates of successive
    sites  will be independent.  If it is, say, 10.25, then the chance of change to
    a new rate will be 1/10.25  after  every  site.   However  the  "new  rate"  is
    randomly drawn from the mix of rates, and hence could even be the same.  So the
    actual observed length of patches with the same rate will  be  somewhat  larger
    than  10.25.   Note below that if you choose multiple patches, there will be an
    estimate in the  output  file  as  to  which  combination  of  rate  categories
    contributed most to the likelihood.

         With the current options C and R the program has  gained  greatly  in  its
    ability  to  infer  different rates at different sites and estimate phylogenies
    under a more realistic model.  Note that Likelihood Ratio Tests can be used  to
    test whether one combination of rates is significantly better than another.

         The G (global search) option causes, after the last species  is  added  to
    the  tree,  each  possible group to be removed and re-added.  This improves the
    result, since the position of every species is reconsidered.  It  approximately
    triples the run-time of the program.

         If the U (user tree) option is chosen another option appears in the  menu,
    the  L  option.   If it is selected, it signals the program that it should take
    any branch lengths that are in the user tree and simply evaluate the likelihood
    of  that tree, without further altering those branch lengths.   This means that
    if some branches have lengths and others do not, the program will estimate  the
    lengths  of  those  that do not have lengths given in the user tree.  Note that
    the program RETREE can be used to add and remove lengths  from  a  tree.   This
    means  that  we  can  test  hypothesis  such  as that a certain branch has zero
    length, and by doing a series of runs with different specified  lengths  for  a
    branch  we can plot a likelihood curve for its branch length while allowing all
    other branches to adjust their lengths to it.  If  all  branches  have  lengths
    specified,  none  of  them  will  be  iterated.  This is useful to allow a tree
    produced by another method to have its likelihood evaluated.  The L option  has
    no effect and does not appear in the menu if the U option is not used.

         The W (Weights) option is invoked in the usual way, with  only  weights  0
    and  1 allowed.  It selects a set of sites to be analyzed, ignoring the others.
    The sites selected are those with weight 1.  If the W option  is  not  invoked,
    all sites are analyzed.

                                     OUTPUT FORMAT

         The output starts by giving the number of species, the  number  of  sites,
    and  the base frequencies for A, C, G, and T that have been specified.  It then
    prints out the transition/transversion ratio that  was  specified  or  used  by
    default.    It   also   uses   the  base  frequencies  to  compute  the  actual
    transition/transversion ratio implied by the parameter.

         If the C (Categories) option is used a table  of  the  relative  rates  of
    expected  substitution  at  each  category  of sites is printed, as well as the
    probabilities of each of those rates.

         There then follow the data sequences, with the base sequences  printed  in
    groups of ten bases along the lines of the Genbank and EMBL formats.  The trees
    found are printed as a rooted tree topology.  The internal nodes  are  numbered
    arbitrarily  for  the sake of identification.  The number of trees evaluated so
    far and the log likelihood of the tree are also given.  The branch  lengths  in
    the  diagram  are  roughly proportional to the estimated branch lengths, except
    that very short branches are printed out at least three characters in length so
    that the connections can be seen.

         A table is printed showing the length of each tree segment, and  the  time
    (in  units  of  expected nucleotide substitutions per site) of each fork in the
    tree, measured from the root of the tree.  I have not attempted in include code
    for  approximate  confidence limits on branch points, as I have done for branch
    lengths in DNAML, both because of the  extreme  crudeness  of  that  test,  and
    because the variation of time for different forks would be highly correlated.

         The log likelihood printed out with the final tree can be used to  perform
    various  likelihood  ratio  tests.   One  can,  for  example, compare runs with
    different values of the expected  transition/transversion  ratio  to  determine
    which value is the maximum likelihood estimate, and what is the allowable range
    of values (using a likelihood ratio test, which  you  will  find  described  in
    mathematical  statistics  books).  One could also estimate the base frequencies
    in the same way.  Both of these, particularly the latter, require multiple runs
    of  the  program  to  evaluate  different  possible  values, and this might get
    expensive.

         This program makes possible for the first time a  (reasonably)  legitimate
    statistical  test  of  the  molecular  clock.  To do such a test, run DNAML and
    DNAMLK on the same data.  If the trees obtained are of the same topology  (when
    considered  as  unrooted), it is legitimate to compare their likelihoods by the
    likelihood ratio test.  In DNAML the likelihood has been computed by estimating
    2n-3  branch  lengths,  if their are n tips on the tree.  In DNAMLK it has been
    computed by estimating n-1 branching times (in  effect,  n-1  branch  lengths).
    The  difference in the number of parameters is (2n-3)-(n-1) =  n-2.  To perform
    the test take the difference in log likelihoods between  the  two  runs  (DNAML
    should  be the higher of the two, barring numerical iteration difficulties) and
    double it.  Look this up on a  chi-square  distribution  with  n-2  degrees  of
    freedom.    If   the  result  is  significant,  the  log  likelihood  has  been
    significantly increased by allowing all 2n-3 branch  lengths  to  be  estimated
    instead of just n-1, and molecular clock may be rejected.

         If the U (User Tree) option is used and more than one  tree  is  supplied,
    and  the  program  is  not  told to assume autocorrelation between the rates at
    different sites, the program also performs a statistical test of each of  these
    trees  against  the one with highest likelihood.  This test, due to Kishino and
    Hasegawa (1989), uses the  mean  and  variance  of  log-likelihood  differences
    between  trees,  taken  across  sites.   If the mean is more than 1.96 standard
    deviations different then the trees are declared significantly different.  This
    use  of the empirical variance of log-likelihood differences is more robust and
    nonparametric than the classical likelihood ratio test, and may to some  extent
    compensate  for  the  any lack of realism in the model underlying this program.
    The program prints out a  table  of  the  log-likelihoods  of  each  tree,  the
    differences  of  each  from  the  highest one, the variance of that quantity as
    determined by  the  log-likelihood  differences  at  individual  sites,  and  a
    conclusion  as  to  whether that tree is or is not significantly worse than the
    best one.  However the Kishino-Hasegawa-Templeton test is not available  if  we
    assume  that  there is autocorrelation of rates at neighboring sites (option R)
    and is not done in those cases.

         The branch lengths printed out are scaled in terms of expected numbers  of
    substitutions, counting both transitions and transversions but not replacements
    of a base by itself, and scaled so that the average rate  of  change,  averaged
    over  all  sites  analyzed,  is  set to 1.0 if there are multiple categories of
    sites.  This means that whether or not there are multiple categories of  sites,
    the  expected fraction of change for very small branches is equal to the branch
    length.  Of course, when a branch is twice as long  this  does  not  mean  that
    there  will  be  twice  as much net change expected along it, since some of the
    changes occur in the same site and overlie or even  reverse  each  other.   The
    branch  lengths  estimates here are in terms of the expected underlying numbers
    of changes.  That means that a branch of length 0.26 is 26 times as long as one
    which  would  show  a  1%  difference  between  the nucleotide sequences at the
    beginning and end of the branch.  But we would not expect the sequences at  the
    beginning  and  end  of  the branch to be 26% different, as there would be some
    overlaying of changes.

         Because of limitations of the numerical algorithm, branch length estimates
    of  zero  will  often print out as small numbers such as 0.00001.  If you see a
    branch length that small, it is really estimated to be of zero length.

         Another possible source of confusion is the existence of  negative  values
    for  the  log  likelihood.  This is not really a problem; the log likelihood is
    not a probability but the logarithm of a probability.  When it is  negative  it
    simply  means that the corresponding probability is less than one (since we are
    seeing its logarithm).  The log likelihood is  maximized  by  being  made  more
    positive: -30.23 is worse than -29.14.

         At the end of the output, if the C option is in effect with multiple  rate
    categories,  the  program will print a list of what site categories contributed
    the most to the final likelihood.  This combination of rate categories need not
    have  contributed  a  majority  of the likelihood, just a plurality.  Still, it
    will be helpful as a view of where the program infers that the higher and lower
    rates  are.   Note that the use in this calculations of the prior probabilities
    of different rates, and the  average  patch  length,  gives  this  inference  a
    "smoothed"  appearance:  some  other  combination of rates might make a greater
    contribution to the likelihood, but be discounted  because  it  conflicts  with
    this prior information.  See the example output below to see what this printout
    of rate categories looks like.

         If the 3 (write out tree) option  is  used,  then  the  tree  with  branch
    lengths will be written out on the tree file.  The program DRAWGRAM can then be
    used to draw the tree more precisely.

                                   PROGRAM CONSTANTS

         The constants defined at the beginning of the program include  "maxtrees",
    the  maximum  number  of user trees that can be processed.  It is small (10) at
    present to same some further memory but the cost of increasing it is  not  very
    great.   Other  constants  include  "maxcategories", the maximum number of site
    categories, "namelength", the length of species names in characters, and  three
    others, "smoothings", "iterations", and "epsilon", that

         The other CONSTants include maxcategories,  the  maximum  number  of  site
    categories,  namelength,  the  length of species names in characters, and three
    CONSTants, smoothings, iterations, and epsilon, that help "tune" the  algorithm
    and define the compromise between execution speed and the quality of the branch
    lengths found by iteratively maximizing the  likelihood.   Reducing  iterations
    and  smoothings,  and increasing epsilon, will result in faster execution but a
    worse result.  These values will not usually have to be changed.

         The program spends most of its time doing real arithmetic.   Any  software
    or  hardware changes that speed up that arithmetic will speed it up by a nearly
    proportional amount.  For example, microcomputers having a numeric co-processor
    will  run  the  program  much  faster,  if  the  compiled program uses it.  The
    algorithm, with  separate  and  independent  computations  occurring  for  each
    pattern, lends itself readily to parallel processing.

                            PAST AND FUTURE OF THE PROGRAM

         This program was developed in 1989 by combining code from DNAPARS and from
    DNAML.   It  was  speeded  up by two major developments, the use of aliasing of
    nucleotide sites (version 3.1) and pretabulation of some exponentials (added by
    Akiko  Fuseki in version 3.4).  In version 3.5 the Hidden Markov Chain code was
    added and the method of  iterating  branch  lengths  was  changed  from  an  EM
    algorithm  to  direct  search.  The Hidden Markov Chain code slows things down,
    especially if there is autocorrelation between sites, so this version is slower
    than version 3.4.  Nevertheless we hope that the sacrifice is worth it.

         Two changes that are needed in the future  are  to  put  in  some  way  of
    allowing for base composition of nucleotide sequences in different parts of the
    phylogeny, and to re-introduce the code that  allowed  us  to  assign  relative
    rates  of  change to individual sites.  By allowing prespecified rates and also
    allowing the Hidden Markov Chain method of inferring rates at different  sites,
    we could then deal with cases in which (say) first, second, and third positions
    in a coding sequence had rates of (say) 1.0 : 0.8 : 2.7 but the  Hidden  Markov
    Chain  also inferred rates that varied along the length of the coding sequence.
    This would be much more realistic.

    ----------------------------TEST DATA SET--------------------------

       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) ------

       5 Species,   13 Sites

    Nucleic acid sequence
       Maximum Likelihood method with molecular clock, version 3.5c

    Site category   Rate of change

              1        1.000
              2        3.200

    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)

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

    Ln Likelihood =   -72.40499

     Ancestor      Node      Node Time     Length
     --------      ----      ---- ----     ------
     root            3
       3             4         3.40037     3.40037
       4          Delta        3.55871     0.15834
       4          Epsilon      3.55871     0.15834
       3             2         2.75248     2.75248
       2          Gamma        3.55871     0.80623
       2             1         3.38653     0.63405
       1          Alpha        3.55871     0.17219
       1          Beta         3.55871     0.17219

    Combination of categories that contributes the most to the likelihood:

                 2222221111 112