GENDIST - Compute genetic distances from gene frequencies

CONTENTS:

• DESCRIPTION
• INPUT FORMAT
• OUTPUT FORMAT
• CONSTANTS
• RUN TIMES
• FUTURE OF THIS PROGRAM
• TEST DATA SET
• TEST SET OUTPUT

The three measures have somewhat different assumptions. All assume that all differences between populations arise from genetic drift. Nei's distance is formulated for an infinite isoalleles model of mutation, in which there is a rate of neutral mutation and each mutant is to a completely new alleles. It is assumed that all loci have the same rate of neutral mutation, and that the genetic variability initially in the population is at equilibrium between mutation and genetic drift, with the effective population size of each population remaining constant.

Nei's distance is:

                                      __  __
                                      \   \
                                      /_  /_  p   p
                                       m   i   1mi 2mi
           D  =  - ln  ( ---------------------------------------- ).
                           __  __               __  __
                           \   \      2  1/2    \   \      2  1/2
                         [ /_  /_  p    ]     [ /_  /_  p    ]
                            m   i   1mi          m   i   2mi

             p
              1mi

The other two genetic distances assume that there is no mutation, and that all gene frequency changes are by genetic drift alone. However they do not assume that population sizes have remained constant and equal in all populations. They cope with changing population size by having expectations that rise linearly not with time, but with the sum over time of 1/N, where N is the effective population size. Thus if population size doubles, genetic drift will be taking place more slowly, and the genetic distance will be expected to be rising only half as fast with respect to time. Both genetic distances are different estimators of the same quantity under the same model.

Cavalli-Sforza's chord distance is given by

                   __              __                       __
      2            \               \      1/2     1/2       \
     D     =    4  /_  [  1   -    /_   p      p       ]  / /_  (a  - 1)
                    m               i    1mi    2mi          m    m

Reynolds, Weir, and Cockerham's (1983) genetic distance is

                       __   __
                       \    \                    2
                       /_   /_  [ p     -  p    ]
        2               m    i     1mi      2mi
       D     =       ------------------------------------
                         __              __
                         \               \
                      2  /_   [  1   -   /_  p    p     ]
                          m               i   1mi  2mi

Having computed one of these genetic distances, one which you feel is appropriate to the biology of the situation, you can use it as the input to the programs FITCH, KITSCH or NEIGHBOR. Keep in mind that the statistical model in those programs implicitly assumes that the distances in the input table have independent errors. For any measure of genetic distance this will not be true, as bursts of random genetic drift, or sampling events in drawing the sample of individuals from each population, cause fluctuations of gene frequency that affect many distances simultaneously. While this is not expected to bias the estimate of the phylogeny, it does mean that the weighing of evidence from all the different distances in the table will not be done with maximal efficiency. One issue is which value of the P (Power) parameter should be used. This depends on how the variance of a distance rises with its expectation. For Cavalli-Sforza's chord distance, and for the Reynolds et. al. distance it can be shown that the variance of the distance will be proportional to the square of its expectation; this suggests a value of 2 for P, which the default value for FITCH and KITSCH (there is no P option in NEIGHBOR).

If you think that the pure genetic drift model is appropriate, and are thus tempted to use the Cavalli-Sforza or Reynolds et. al. distances, you might consider using the maximum likelihood program CONTML instead. It will correctly weigh the evidence in that case. Like those genetic distances, it uses approximations that break down as loci start to drift all the way to fixation. Although Nei's distance will not break down in that case, it makes other assumptions about equality of substitution rates at all loci and constancy of population sizes.

The most important thing to remember is that genetic distance is not an abstract, idealized measure of "differentness". It is an estimate of a parameter (time or cumulated inverse effective population size) of the model which is thought to have generated the differences we see. As an estimate, it has statistical properties that can be assessed, and we should never have to choose between genetic distances based on their aesthetic properties, or on the personal prestige of their originators. Considering them as estimates focuses us on the questions which genetic distances are intended to answer, for if there are none there is no reason to compute them. For further perspective on genetic distances, I recommend my own paper evaluating properties of some genetic distances (Felsenstein, 1985c), the paper by Reynolds, Weir, and Cockerham (1983), and the material in Nei's book (Nei, 1987).

INPUT FORMAT

The input to this program is standard and is as described in the Gene Frequencies and Continuous Characters Programs documentation file above. It consists of the number of populations (or species), the number of loci, and after that a line containing the numbers of alleles at each of the loci. Then the gene frequencies follow in standard format.

The options are selected using a menu:

Genetic Distance Matrix program, version 3.5c

Settings for this run:
  A   Input file contains all alleles at each locus?  One omitted at each locus
  N                        Use Nei genetic distance?  Yes
  C                Use Cavalli-Sforza chord measure?  No
  R                   Use Reynolds genetic distance?  No
  L                         Form of distance matrix?  Square
  M                      Analyze multiple data sets?  No
  0              Terminal type (IBM PC, VT52, ANSI)?  ANSI
  1            Print indications of progress of run?  Yes

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

OUTPUT FORMAT

The output file simply contains on its first line the number of species (or populations). Each species (or population) starts a new line, with its name printed out first, and then and there are up to nine genetic distances printed on each line, in the standard format used as input by the distance matrix programs. The output, in its default form, is ready to be used in the distance matrix programs.

CONSTANTS

The constants available to be changed by the user if the program is recompiled are "namelength" the length of a species name, set to 10 in the distribution and "epsilon" which defines a small quantity that is used when checking whether allele frequencies at a locus sum to more than one: if all alleles are input (option A) and the sum differs from 1 by more than epsilon, or if not all alleles are input and the sum is greater than 1 by more then epsilon, the program will see this as an error and stop. You may find this causes difficulties if you gene frequencies have been rounded. I have tried to keep epsilon from being too small to prevent such problems.

RUN TIMES

The program is quite fast and the user should effectively never be limited by the amount of time it takes. All that the program has to do is read in the gene frequency data and then, for each pair of species, compute a genetic distance formula for each pair of species. This should require an amount of effort proportional to the total number of alleles over loci, and to the square of the number of populations.

FUTURE OF THIS PROGRAM

The main change that will be made to this program in the future is to add provisions for taking into account the sample size for each population. The genetic distance formulas have been modified by their inventors to correct for the inaccuracy of the estimate of the genetic distances, which on the whole should artificially increase the distance between populations by a small amount dependent on the sample sizes. The main difficulty with doing this is that I have not yet settled on a format for putting the sample size in the input data along with the gene frequency data for a species or population.

I may also include other distance measures, but only if I think their use is justified. There are many very arbitrary genetic distances, and I am reluctant to include most of them.

TEST DATA SET

    5    10
2 2 2 2 2 2 2 2 2 2
European   0.2868 0.5684 0.4422 0.4286 0.3828 0.7285 0.6386 0.0205
0.8055 0.5043
African    0.1356 0.4840 0.0602 0.0397 0.5977 0.9675 0.9511 0.0600
0.7582 0.6207
Chinese    0.1628 0.5958 0.7298 1.0000 0.3811 0.7986 0.7782 0.0726
0.7482 0.7334
American   0.0144 0.6990 0.3280 0.7421 0.6606 0.8603 0.7924 0.0000
0.8086 0.8636
Australian 0.1211 0.2274 0.5821 1.0000 0.2018 0.9000 0.9837 0.0396
0.9097 0.2976

    5
European    0.0000  0.0780  0.0807  0.0668  0.1030
African     0.0780  0.0000  0.2347  0.1050  0.2273
Chinese     0.0807  0.2347  0.0000  0.0539  0.0633
American    0.0668  0.1050  0.0539  0.0000  0.1348
Australian  0.1030  0.2273  0.0633  0.1348  0.0000