| fdnainvar

## Function

Nucleic acid sequence Invariants method

## Description

For nucleic acid sequence data on four species, computes Lake's and Cavender's phylogenetic invariants, which test alternative tree topologies. The program also tabulates the frequencies of occurrence of the different nucleotide patterns. Lake's invariants are the method which he calls "evolutionary parsimony".

## Algorithm

This program reads in nucleotide sequences for four species and computes the phylogenetic invariants discovered by James Cavender (Cavender and Felsenstein, 1987) and James Lake (1987). Lake's method is also called by him "evolutionary parsimony". I prefer Cavender's more mathematically precise term "invariants", as the method bears somewhat more relationship to likelihood methods than to parsimony. The invariants are mathematical formulas (in the present case linear or quadratic) in the EXPECTED frequencies of site patterns which are zero for all trees of a given tree topology, irrespective of branch lengths. We can consider at a given site that if there are no ambiguities, we could have for four species the nucleotide patterns (considering the same site across all four species) AAAA, AAAC, AAAG, ... through TTTT, 256 patterns in all.

The invariants are formulas in the expected pattern frequencies, not the observed pattern frequencies. When they are computed using the observed pattern frequencies, we will usually find that they are not precisely zero even when the model is correct and we have the correct tree topology. Only as the number of nucleotides scored becomes infinite will the observed pattern frequencies approach their expectations; otherwise, we must do a statistical test of the invariants.

Some explanation of invariants will be found in the above papers, and also in my recent review article on statistical aspects of inferring phylogenies (Felsenstein, 1988b). Although invariants have some important advantages, their validity also depends on symmetry assumptions that may not be satisfied. In the discussion below suppose that the possible unrooted phylogenies are I: ((A,B),(C,D)), II: ((A,C),(B,D)), and III: ((A,D),(B,C)).

### Lake's Invariants, Their Testing and Assumptions

Lake's invariants are fairly simple to describe: the patterns involved are only those in which there are two purines and two pyrimidines at a site. Thus a site with AACT would affect the invariants, but a site with AAGG would not. Let us use (as Lake does) the symbols 1, 2, 3, and 4, with the proviso that 1 and 2 are either both of the purines or both of the pyrimidines; 3 and 4 are the other two nucleotides. Thus 1 and 2 always differ by a transition; so do 3 and 4. Lake's invariants, expressed in terms of expected frequencies, are the three quantities:

    (1)      P(1133) + P(1234) - P(1134) - P(1233),

    (2)      P(1313) + P(1324) - P(1314) - P(1323),

    (3)      P(1331) + P(1342) - P(1341) - P(1332),

He showed that invariants (2) and (3) are zero under Topology I, (1) and (3) are zero under topology II, and (1) and (2) are zero under Topology III. If, for example, we see a site with pattern ACGC, we can start by setting 1=A. Then 2 must be G. We can then set 3=C (so that 4 is T). Thus its pattern type, making those substitutions, is 1323. P(1323) is the expected probability of the type of pattern which includes ACGC, TGAG, GTAT, etc.

Lake's invariants are easily tested with observed frequencies. For example, the first of them is a test of whether there are as many sites of types 1133 and 1234 as there are of types 1134 and 1233; this is easily tested with a chi-square test or, as in this program, with an exact binomial test. Note that with several invariants to test, we risk overestimating the significance of results if we simply accept the nominal 95% levels of significance (Li and Guoy, 1990).

Lake's invariants assume that each site is evolving independently, and that starting from any base a transversion is equally likely to end up at each of the two possible bases (thus, an A undergoing a transversion is equally likely to end up as a C or a T, and similarly for the other four bases from which one could start. Interestingly, Lake's results do not assume that rates of evolution are the same at all sites. The result that the total of 1133 and 1234 is expected to be the same as the total of 1134 and 1233 is unaffected by the fact that we may have aggregated the counts over classes of sites evolving at different rates.

### Cavender's Invariants, Their Testing and Assumptions

Cavender's invariants (Cavender and Felsenstein, 1987) are for the case of a character with two states. In the nucleic acid case we can classify nucleotides into two states, R and Y (Purine and Pyrimidine) and then use the two-state results. Cavender starts, as before, with the pattern frequencies. Coding purines as R and pyrimidines as Y, the patterns types are RRRR, RRRY, and so on until YYYY, a total of 16 types. Cavender found quadratic functions of the expected frequencies of these 16 types that were expected to be zero under a given phylogeny, irrespective of branch lengths. Two invariants (called K and L) were found for each tree topology. The L invariants are particularly easy to understand. If we have the tree topology ((A,B),(C,D)), then in the case of two symmetric states, the event that A and B have the same state should be independent of whether C and D have the same state, as the events determining these happen in different parts of the tree. We can set up a contingency table:

                                     C = D         C =/= D
                               ------------------------------
                              |
                       A = B  |   YYYY, YYRR,     YYYR, YYRY,
                              |   RRRR, RRYY      RRYR, RRRY
                              |
                     A =/= B  |   YRYY, YRRR,     YRYR, YRRY,
                              |   RYYY, RYRR      RYYR, RYRY

and we expect that the events C = D and A = B will be independent. Cavender's L invariant for this tree topology is simply the negative of the crossproduct difference,

          P(A=/=B and C=D) P(A=B and C=/=D) - P(A=B and C=D) P(A=/=B and C=/=D).

One of these L invariants is defined for each of the three tree topologies. They can obviously be tested simply by doing a chi-square test on the contingency table. The one corresponding to the correct topology should be statistically indistinguishable from zero. Again, there is a possible multiple tests problem if all three are tested at a nominal value of 95%.

The K invariants are differences between the L invariants. When one of the tables is expected to have crossproduct difference zero, the other two are expected to be nonzero, and also to be equal. So the difference of their crossproduct differences can be taken; this is the K invariant. It is not so easily tested.

The assumptions of Cavender's invariants are different from those of Lake's. One obviously need not assume anything about the frequencies of, or transitions among, the two different purines or the two different pyrimidines. However one does need to assume independent events at each site, and one needs to assume that the Y and R states are symmetric, that the probability per unit time that a Y changes into an R is the same as the probability that an R changes into a Y, so that we expect equal frequencies of the two states. There is also an assumption that all sites are changing between these two states at the same expected rate. This assumption is not needed for Lake's invariants, since expectations of sums are equal to sums of expectations, but for Cavender's it is, since products of expectations are not equal to expectations of products.

It is helpful to have both sorts of invariants available; with further work we may appreciate what other invaraints there are for various models of nucleic acid change.

## Usage

Here is a sample session with fdnainvar

    % fdnainvar -printdata
    Nucleic acid sequence Invariants method
    Input sequence: dnainvar.dat
    Input file:
    Output file [dnainvar.fdnainvar]:

    Output written to output file "dnainvar.fdnainvar"

    Done.

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

## Command line arguments

       Standard (Mandatory) qualifiers:
      [-sequence]          seqsetall  File containing one or more sequence
                                      alignments
       -weights            properties (no help text) properties value
      [-outfile]           outfile    Output file name

       Additional (Optional) qualifiers (* if not always prompted):
       -printdata          boolean    Print data at start of run
    *  -[no]dotdiff        boolean    Use dot-differencing to display results
       -[no]printpattern   boolean    Print counts of patterns
       -[no]printinvariant boolean    Print invariants
       -[no]progress       boolean    Print indications of progress of run

       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
       -sformat1           string     Input sequence format
       -sdbname1           string     Database name
       -sid1               string     Entryname
       -ufo1               string     UFO features
       -fformat1           string     Features format
       -fopenfile1         string     Features file name

       "-outfile" associated qualifiers
       -odirectory2        string     Output directory

       General qualifiers:
       -auto               boolean    Turn off prompts
       -stdout             boolean    Write standard output
       -filter             boolean    Read standard input, write 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. 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 deaths

Standard (Mandatory) qualifiers | Allowed values | Default
--- | --- | ---
[-sequence] (Parameter 1) | File containing one or more sequence alignments | Readable sets of sequences |
-weights | (no help text) properties value | Property value(s) |
[-outfile] (Parameter 2) | Output file name | Output file | <sequence>.fdnainvar
Additional (Optional) qualifiers | Allowed values | Default
-printdata | Print data at start of run | Boolean value Yes/No | No
-[no]dotdiff | Use dot-differencing to display results | Boolean value Yes/No | Yes
-[no]printpattern | Print counts of patterns | Boolean value Yes/No | Yes
-[no]printinvariant | Print invariants | Boolean value Yes/No | Yes
-[no]progress | Print indications of progress of run | Boolean value Yes/No | Yes
Advanced (Unprompted) qualifiers | Allowed values | Default
(none)

## Input file format

fdnainvar reads any normal sequence USAs.

### Input files for usage example

### File: dnainvar.dat

       4   13
    Alpha     AACGTGGCCAAAT
    Beta      AAGGTCGCCAAAC
    Gamma     CATTTCGTCACAA
    Delta     GGTATTTCGGCCT

## Output file format

fdnainvar output consists first (if option 1 is selected) of a reprinting of the input data, then (if option 2 is on) tables of observed pattern frequencies and pattern type frequencies. A table will be printed out, in alphabetic order AAAA through TTTT of all the patterns that appear among the sites and the number of times each appears. This table will be invaluable for computation of any other invariants. There follows another table, of pattern types, using the 1234 notation, in numerical order 1111 through 1234, of the number of times each type of pattern appears. In this computation all sites at which there are any ambiguities or deletions are omitted. Cavender's invariants could actually be computed from sites that have only Y or R ambiguities; this will be done in the next release of this program.

If option 3 is on the invariants are then printed out, together with their statistical tests. For Lake's invariants the two sums which are expected to be equal are printed out, and then the result of an one-tailed exact binomial test which tests whether the difference is expected to be this positive or more. The P level is given (but remember the multiple-tests problem!).

For Cavender's L invariants the contingency tables are given. Each is tested with a one-tailed chi-square test. It is possible that the expected numbers in some categories could be too small for valid use of this test; the program does not check for this. It is also possible that the chi-square could be significant but in the wrong direction; this is not tested in the current version of the program. To check it beware of a chi-square greater than 3.841 but with a positive invariant. The invariants themselves are computed, as the difference of cross-products. Their absolute magnitudes are not important, but which one is closest to zero may be indicative. Significantly nonzero invariants should be negative if the model is valid. The K invariants, which are simply differences among the L invariants, are also printed out without any test on them being conducted. Note that it is possible to use the bootstrap utility SEQBOOT to create multiple data sets, and from the output from sunning all of these get the empirical variability of these quadratic invariants.

### Output files for usage example

### File: dnainvar.fdnainvar

    Nucleic acid sequence Invariants method, version 3.6b

     4 species,  13  sites

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

    Alpha        AACGTGGCCA AAT
    Beta         ..G..C.... ..C
    Gamma        C.TT.C.T.. C.A
    Delta        GGTA.TT.GG CC.

       Pattern   Number of times

         AAAC         1
         AAAG         2
         AACC         1
         AACG         1
         CCCG         1
         CCTC         1
         CGTT         1
         GCCT         1
         GGGT         1
         GGTA         1
         TCAT         1
         TTTT         1

    Symmetrized patterns (1, 2 = the two purines  and  3, 4 = the two pyrimidines
                      or  1, 2 = the two pyrimidines  and  3, 4 = the two purines)

         1111         1
         1112         2
         1113         3
         1121         1
         1132         2
         1133         1
         1231         1
         1322         1
         1334         1

    Tree topologies (unrooted):

        I:  ((Alpha,Beta),(Gamma,Delta))
       II:  ((Alpha,Gamma),(Beta,Delta))
      III:  ((Alpha,Delta),(Beta,Gamma))

      [Part of this file has been deleted for brevity]

    different purine:pyrimidine ratios from 1:1.

      Tree I:

       Contingency Table

          2     8
          1     2

       Quadratic invariant =             4.0

       Chi-square =    0.23111 (not significant)

      Tree II:

       Contingency Table

          1     5
          1     6

       Quadratic invariant =            -1.0

       Chi-square =    0.01407 (not significant)

      Tree III:

       Contingency Table

          1     2
          6     4

       Quadratic invariant =             8.0

       Chi-square =    0.66032 (not significant)

    Cavender's quadratic invariants (type K) using purines vs. pyrimidines
     (these are expected to be zero for the correct tree topology)
    They will be misled if there are substantially
    different evolutionary rate between sites, or
    different purine:pyrimidine ratios from 1:1.
    No statistical test is done on them here.

      Tree I:              -9.0
      Tree II:              4.0
      Tree III:             5.0

## 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
--- | ---
ednacomp | DNA compatibility algorithm
ednadist | Nucleic acid sequence Distance Matrix program
ednainvar | Nucleic acid sequence Invariants method
ednaml | Estimates phylogenies from nucleic acid sequence Maximum Likelihood
ednamlk | Estimates phylogenies from nucleic acid sequence Maximum Likelihood with molecular 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
fdnaml | Estimates 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
fpromlk | Protein phylogeny by maximum likelihood
fprotdist | Protein distance algorithm
fprotpars | Protein pasimony algorithm
frestboot | Bootstrapped restriction sites algorithm
frestdist | Computes 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.

Although we take every care to ensure that the results of the EMBOSS version are identical to those from the original package, we recommend that you check your inputs give the same results in both versions before publication.

Please report all bugs in the EMBOSS version to the EMBOSS bug team, 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.