Distance matrix from restriction sites or fragments


Distances calculated from restriction sites data or restriction fragments data. The restriction sites option is the one to use to also make distances for RAPDs or AFLPs.


Restdist reads the same restriction sites format as RESTML and computes a restriction sites distance. It can also compute a restriction fragments distance. The original restriction fragments and restriction sites distance methods were introduced by Nei and Li (1979). Their original method for restriction fragments is also available in this program, although its default methods are my modifications of the original Nei and Li methods.

These two distances assume that the restriction sites are accidental byproducts of random change of nucleotide sequences. For my restriction sites distance the DNA sequences are assumed to be changing according to the Kimura 2-parameter model of DNA change (Kimura, 1980). The user can set the transition/transversion rate for the model. For my restriction fragments distance there is there is an implicit assumption of a Jukes-Cantor (1969) model of change, The user can also set the parameter of a correction for unequal rates of evolution between sites in the DNA sequences, using a Gamma distribution of rates among sites. The Jukes-Cantor model is also implicit in the restriction fragments distance of Nei and Li(1979). It does not allow us to correct for a Gamma distribution of rates among sites.

Restriction Sites Distance

The restriction sites distances use data coded for the presence of absence of individual restriction sites (usually as + and - or 0 and 1). My distance is based on the proportion, out of all sites observed in one species or the other, which are present in both species. This is done to correct for the ascertainment of sites, for the fact that we are not aware of many sites because they do not appear in any species.

My distance starts by computing from the particular pair of species the fraction

   f =  ---------------------
         n++ + 1/2 (n+- + n-+)

where n++ is the number of sites contained in both species, n+- is the number of sites contained in the first of the two species but not in the second, and n-+ is the number of sites contained in the second of the two species but not in the first. This is the fraction of sites that are present in one species which are present in both. Since the number of sites present in the two species will often differ, the denominator is the average of the number of sites found in the two species.

If each restriction site is s nucleotides long, the probability that a restriction site is present in the other species, given that it is present in a species, is


`where Q is the probability that a nucleotide has no net change as one goes from the one species to the other. It may have changed in between; we are interested in the probability that that nucleotide site is in the same base in both species, irrespective of what has happened in between. The distance is then computed by finding the branch length of a two-species tree (connecting these two species with a single branch) such that Q equals the s-th root of f. For this the program computes Q for various values of branch length, iterating them by a Newton-Raphson algorithm until the two quantities are equal.

The resulting distance should be numerically close to the original restriction sites distance of Nei and Li (1979) when divergence is small. Theirs computes the probability of retention of a site in a way that assumes that the site is present in the common ancestor of the two species. Ours does not make this assumption. It is inspired by theirs, but differs in this detail. Their distance also assumes a Jukes-Cantor (1969) model of base change, and does not allow for transitions being more frequent than transversions. In this sense mine generalizes theres somewhat. Their distance does include, as mine does as well, a correction for Gamma distribution of rate of change among nucleotide sites.

I have made their original distance available here

Restriction Fragments Distance

For restriction fragments data we use a different distance. If we average over all restriction fragment lengths, each at its own expected frequency, the probability that the fragment will still be in existence after a certain amount of branch length, we must take into account the probability that the two restriction sites at the ends of the fragment do not mutate, and the probability that no new restriction site occurs within the fragment in that amount of branch length. The result for a restriction site length of s is:

          f = --------
               2 - Qs

(The details of the derivation will be given in my forthcoming book Inferring Phylogenies (to be published by Sinauer Associates in 2001). Given the observed fraction of restriction sites retained, f, we can solve a quadratic equation from the above expression for Qs. That makes it easy to obtain a value of Q, and the branch length can then be estimated by adjusting it so the probability of a base not changing is equal to that value. Alternatively, if we use the Nei and Li (1979) restriction fragments distance, this involves solving for g in the nonlinear equation

       g  =  [ f (3 - 2g) ]1/4

and then the distance is given by

       d  =  - (2/r) loge(g)

where r is the length of the restriction site.

Comparing these two restriction fragments distances in a case where their underlying DNA model is the same (which is when the transition/transversion ratio of the modified model is set to 0.5), you will find that they are very close to each other, differing very little at small distances, with the modified distance become smaller than the Nei/Li distance at larger distances. It will therefore matter very little which one you use.

A Comment About RAPDs and AFLPs

Although these distances are designed for restriction sites and restriction fragments data, they can be applied to RAPD and AFLP data as well. RAPD (Randomly Amplified Polymorphic DNA) and AFLP (Amplified Fragment Length Polymorphism) data consist of presence or absence of individual bands on a gel. The bands are segments of DNA with PCR primers at each end. These primers are defined sequences of known length (often about 10 nucleotides each). For AFLPs the reolevant length is the primer length, plus three nucleotides. Mutation in these sequences makes them no longer be primers, just as in the case of restriction sites. Thus a pair of 10-nucleotide primers will behave much the same as a 20-nucleotide restriction site, for RAPDs (26 for AFLPs). You can use the restriction sites distance as the distance between RAPD or AFLP patterns if you set the proper value for the total length of the site to the total length of the primers (plus 6 in the case of AFLPs). Of course there are many possible sources of noise in these data, including confusing fragments of similar length for each other and having primers near each other in the genome, and these are not taken into account in the statistical model used here.


Here is a sample session with frestdist

% frestdist 
Distance matrix from restriction sites or fragments
Input file: restdist.dat
Output file [restdist.frestdist]: 

Restriction site or fragment distances, version 3.6b

Distances calculated for species
    Alpha        ....
    Beta         ...
    Gamma        ..
    Delta        .

Distances written to file "restdist.frestdist"


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

Command line arguments

   Standard (Mandatory) qualifiers:
  [-data]              discretestates File containing one or more sets of
                                  restriction data
  [-outfile]           outfile    Output file name

   Additional (Optional) qualifiers (* if not always prompted):
   -[no]restsites      boolean    Restriction sites (put N if you want
                                  restriction fragments)
   -neili              boolean    Use original Nei/Li model (default uses
                                  modified Nei/Li model)
*  -gamma              boolean    Gama distributed rates among sites
*  -gammacoefficient   float      Coefficient of variation of substitution
                                  rate among sites
   -ttratio            float      Transition/transversion ratio
   -sitelength         integer    Site length
   -lower              boolean    Lower triangular distance matrix
   -printdata          boolean    Print data at start of run
   -[no]progress       boolean    Print indications of progress of run

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

   "-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
(Parameter 1)
File containing one or more sets of restriction data Discrete states file  
(Parameter 2)
Output file name Output file <sequence>.frestdist
Additional (Optional) qualifiers Allowed values Default
-[no]restsites Restriction sites (put N if you want restriction fragments) Boolean value Yes/No Yes
-neili Use original Nei/Li model (default uses modified Nei/Li model) Boolean value Yes/No No
-gamma Gama distributed rates among sites Boolean value Yes/No No
-gammacoefficient Coefficient of variation of substitution rate among sites Number 0.001 or more 1
-ttratio Transition/transversion ratio Number 0.001 or more 2.0
-sitelength Site length Integer 1 or more 6
-lower Lower triangular distance matrix Boolean value Yes/No No
-printdata Print data at start of run Boolean value Yes/No No
-[no]progress Print indications of progress of run Boolean value Yes/No Yes
Advanced (Unprompted) qualifiers Allowed values Default

Input file format

frestdist 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, but there is also a third number, which is the number of different restriction enzymes that were used to detect the restriction sites. Thus a data set with 10 species and 35 different sites, representing digestion with 4 different enzymes, would have the first line of the data file look like this:

   10   35    4

The site data are in standard form. Each species starts with a species name whose maximum length is given by the constant "nmlngth" (whose value in the program as distributed is 10 characters). The name should, as usual, be padded out to that length with blanks if necessary. The sites data then follows, one character per site (any blanks will be skipped and ignored). Like the DNA and protein sequence data, the restriction sites data may be either in the "interleaved" form or the "sequential" form. Note that if you are analyzing restriction sites data with the programs DOLLOP or MIX or other discrete character programs, at the moment those programs do not use the "aligned" or "interleaved" data format. Therefore you may want to avoid that format when you have restriction sites data that you will want to feed into those programs.

The presence of a site is indicated by a "+" and the absence by a "-". I have also allowed the use of "1" and "0" as synonyms for "+" and "-", for compatibility with MIX and DOLLOP which do not allow "+" and "-". If the presence of the site is unknown (for example, if the DNA containing it has been deleted so that one does not know whether it would have contained the site) then the state "?" can be used to indicate that the state of this site is unknown.

Input files for usage example

File: restdist.dat

   5   13   2
Alpha     ++-+-++--+++-
Beta      ++++--+--+++-
Gamma     -+--+-++-+-++
Delta     ++-+----++---
Epsilon   ++++----++---

Output file format

frestdist output 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 or 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.

The distances printed out are scaled in terms of expected numbers of substitutions per DNA site, 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. Thus when the G option is used, the rate of change at one site may be higher than at another, but their mean is expected to be 1.

Output files for usage example

File: restdist.frestdist

Alpha       0.000000  0.022368  0.107681  0.068767  0.082639
Beta        0.022368  0.000000  0.107681  0.068767  0.044216
Gamma       0.107681  0.107681  0.000000  0.176484  0.192466
Delta       0.068767  0.068767  0.176484  0.000000  0.019724
Epsilon     0.082639  0.044216  0.192466  0.019724  0.000000

Data files








Diagnostic Error Messages


Exit status

It always exits with status 0.

Known bugs


See also

Program nameDescription
ednacompDNA compatibility algorithm
ednadistNucleic acid sequence Distance Matrix program
ednainvarNucleic acid sequence Invariants method
ednamlPhylogenies from nucleic acid Maximum Likelihood
ednamlkPhylogenies from nucleic acid Maximum Likelihood with clock
ednaparsDNA parsimony algorithm
ednapennyPenny algorithm for DNA
eprotdistProtein distance algorithm
eprotparsProtein parsimony algorithm
erestmlRestriction site Maximum Likelihood method
eseqbootBootstrapped sequences algorithm
fdiscbootBootstrapped discrete sites algorithm
fdnacompDNA compatibility algorithm
fdnadistNucleic acid sequence Distance Matrix program
fdnainvarNucleic acid sequence Invariants method
fdnamlEstimates nucleotide phylogeny by maximum likelihood
fdnamlkEstimates nucleotide phylogeny by maximum likelihood
fdnamoveInteractive DNA parsimony
fdnaparsDNA parsimony algorithm
fdnapennyPenny algorithm for DNA
fdolmoveInteractive Dollo or Polymorphism Parsimony
ffreqbootBootstrapped genetic frequencies algorithm
fpromlProtein phylogeny by maximum likelihood
fpromlkProtein phylogeny by maximum likelihood
fprotdistProtein distance algorithm
fprotparsProtein pasimony algorithm
frestbootBootstrapped restriction sites algorithm
frestmlRestriction site maximum Likelihood method
fseqbootBootstrapped sequences algorithm
fseqbootallBootstrapped sequences algorithm


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.


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.