version 3.69

Restdist -- Program to compute distance matrix
from restriction sites or fragments

© Copyright 2000-2008 by the University of Washington. Written by Joseph Felsenstein. Permission is granted to copy this document provided that no fee is charged for it and that this copyright notice is not removed.

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 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 or 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 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 theirs 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 (option N).

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 are given in my book Inferring Phylogenies (Felsenstein, 2004).) 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 distancen (available in this program using menu option N), 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 becoming 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 relevant 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.


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, 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.

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

Restriction site or fragment distances, version 3.69

Settings for this run:
  R           Restriction sites or fragments?  Sites
  N        Original or modified Nei/Li model?  Modified
  G  Gamma distribution of rates among sites?  No
  T            Transition/transversion ratio?  2.000000
  S                              Site length?  6.0
  L                  Form of distance matrix?  Square
  M               Analyze multiple data sets?  No
  I              Input sequences interleaved?  Yes
  0       Terminal type (IBM PC, ANSI, none)?  ANSI
  1       Print out the data at start of run?  No
  2     Print indications of progress of run?  Yes

  Y to accept these or type the letter for one to change

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

The R option toggles between a restriction sites distance, which is the default setting, and a restriction fragments distance. In both cases, another option appears, the N (Nei/Li) option. This allows the user to choose the original Nei and Li (1979) restriction sites or fragments distances rather than my modified versions, which are the defaults.

If the G (Gamma distribution) option is selected, the user will be asked to supply the coefficient of variation of the rate of substitution among sites. This is different from the parameters used by Nei and Jin, who introduced Gamma distribution of rates in DNA distances, but related to their parameters: their parameter a is also known as "alpha", the shape parameter of the Gamma distribution. It is related to the coefficient of variation by

     CV = 1 / a1/2


     a = 1 / (CV)2

(their parameter b is absorbed here by the requirement that time is scaled so that the mean rate of evolution is 1 per unit time, which means that a = b). As we consider cases in which the rates are less variable we should set a larger and larger, as CV gets smaller and smaller.

The Gamma distribution option is not available when using the original Nei/Li restriction fragments distance.

The T option 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 the resulting expected ratio of transitions to transversions. The default value of the T parameter if you do not use the T option is 2.0. The T option is not available when you choose the original Nei/Li restriction fragment distance, which assumes a Jukes-Cantor (1969) model of DNA change, for which the transition/transversion ratio is in effect fixed at 0.5.

The S option selects the site length. This is set to a default value of 6. It can be set to any positive integer. While in the Restml program there is an upper limit on the restriction site length (set by memory limitations), in Restdist there is no effective limit on the size of the restriction sites. A value of 20, which might be appropriate in many cases for RAPD or AFLP data, is typically not practical in Restml, but it is useable in Restdist.

Option L specifies that the output file will have a square matrix of distances. It can be used to change to lower-triangular data matrices. This will usually not be necessary, but if the distance matrices are going to be very large, this alternative can reduce their size by half. The programs which are to use them should then of course be informed that they can expect lower-triangular distance matrices.

The M, I, and 0 options are the usual Multiple data set, Interleaved input, and screen terminal type options. These are described in the main documentation file.

Option 1 specifies that the input data will be written out on the output file before the distances. This is off by default. If it is done, it will make the output file unusable as input to our distance matrix programs.

Option 2 turns off or on the indications of the progress of the run. The program prints out a row of dots (".") indicating the calculation of individual distances. Since the distance matrix is symmetrical, the program only computes the distances for the upper triangle of the distance matrix, and then duplicates the distance to the other corner of the matrix. Thus the rows of dots start out at full length, and then get shorter and shorter.


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

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

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.


The constants available to be changed are "initialv" and "iterationsr". The constant "initialv" is the starting value of the distance in the iterations. This will typically not need to be changed. The constant "iterationsr" is the number of times that the Newton-Raphson method which is used to solve the equations for the distances is iterated. The program can be speeded up by reducing the number of iterations from the default value of 20, but at the possible risk of computing the distance less accurately.


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

CONTENTS OF OUTPUT FILE (with all numerical options on)

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

    5 Species,   13 Sites

Name            Sites
----            -----

Alpha        ++-+-++--+ ++-
Beta         ++++--+--+ ++-
Gamma        -+--+-++-+ -++
Delta        ++-+----++ ---
Epsilon      ++++----++ ---

Alpha       0.000000  0.022368  0.107681  0.082634  0.095581
Beta        0.022368  0.000000  0.107681  0.082634  0.056895
Gamma       0.107681  0.107681  0.000000  0.192466  0.207319
Delta       0.082634  0.082634  0.192466  0.000000  0.015949
Epsilon     0.095581  0.056895  0.207319  0.015949  0.000000