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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:
\ \ /_ /_ p1mi p2mi m i D = - ln ( ------------------------------------- ). \ \ \ \ [ /_ /_ p1mi2]1/2 [ /_ /_ p2mi2]1/2 m i m i
where m is summed over loci, i over alleles at the m-th locus, and where
is the frequency of the i-th allele at the m-th locus in population 1. Subject to the above assumptions, Nei's genetic distance is expected, for a sample of sufficiently many equivalent loci, to rise linearly with time.
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
\ \ \ D2 = 4 /_ [ 1 - /_ p1mi1/2 p 2mi1/2] / /_ (am - 1) m i m
where m indexes the loci, where i is summed over the alleles at the m-th locus, and where a is the number of alleles at the m-th locus. It can be shown that this distance always satisfies the triangle inequality. Note that as given here it is divided by the number of degrees of freedom, the sum of the numbers of alleles minus one. The quantity which is expected to rise linearly with amount of genetic drift (sum of 1/N over time) is D squared, the quantity computed above, and that is what is written out into the distance matrix.
Reynolds, Weir, and Cockerham's (1983) genetic distance is
\ \ /_ /_ [ p1mi - p2mi]2 m i D2 = -------------------------------------- \ \ 2 /_ [ 1 - /_ p1mi p2mi ] m i
where the notation is as before and D2 is the quantity that is expected to rise linearly with cumulated genetic drift.
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.
qThe 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 Reynolds, Weir, and Cockerham (1983), and the material in Nei's book (Nei, 1987).
% fgendist Compute genetic distances from gene frequencies Phylip gendist program input file: gendist.dat Phylip gendist program output file [gendist.fgendist]: Distances calculated for species European . African .. Chinese ... American .... Australian ..... Distances written to file "gendist.fgendist" Done.
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Compute genetic distances from gene frequencies Version: EMBOSS:220.127.116.11 Standard (Mandatory) qualifiers: [-infile] frequencies File containing one or more sets of data [-outfile] outfile [*.fgendist] Phylip gendist program output file Additional (Optional) qualifiers: -method menu [n] Which method to use (Values: n (Nei genetic distance); c (Cavalli-Sforza chord measure); r (Reynolds genetic distance)) -[no]progress boolean [Y] Print indications of progress of run -lower boolean [N] Lower triangular distance matrix Advanced (Unprompted) qualifiers: (none) Associated qualifiers: "-outfile" associated qualifiers -odirectory2 string Output directory General qualifiers: -auto boolean Turn off prompts -stdout boolean Write first file to standard output -filter boolean Read first file from standard input, write first file to 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 and exit. 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 dying program messages -version boolean Report version number and exit
|Standard (Mandatory) qualifiers|
|frequencies||File containing one or more sets of data||Frequency value(s)|
|outfile||Phylip gendist program output file||Output file||<*>.fgendist|
|Additional (Optional) qualifiers|
|-method||list||Which method to use||
|-[no]progress||boolean||Print indications of progress of run||Boolean value Yes/No||Yes|
|-lower||boolean||Lower triangular distance matrix||Boolean value Yes/No||No|
|Advanced (Unprompted) qualifiers|
|"-outfile" associated outfile qualifiers|
|string||Output directory||Any string|
|-auto||boolean||Turn off prompts||Boolean value Yes/No||N|
|-stdout||boolean||Write first file to standard output||Boolean value Yes/No||N|
|-filter||boolean||Read first file from standard input, write first file to standard output||Boolean value Yes/No||N|
|-options||boolean||Prompt for standard and additional values||Boolean value Yes/No||N|
|-debug||boolean||Write debug output to program.dbg||Boolean value Yes/No||N|
|-verbose||boolean||Report some/full command line options||Boolean value Yes/No||Y|
|-help||boolean||Report command line options and exit. More information on associated and general qualifiers can be found with -help -verbose||Boolean value Yes/No||N|
|-warning||boolean||Report warnings||Boolean value Yes/No||Y|
|-error||boolean||Report errors||Boolean value Yes/No||Y|
|-fatal||boolean||Report fatal errors||Boolean value Yes/No||Y|
|-die||boolean||Report dying program messages||Boolean value Yes/No||Y|
|-version||boolean||Report version number and exit||Boolean value Yes/No||N|
When the gene frequencies data are used in CONTML or GENDIST, this involves the following assumptions:
How these assumptions affect the methods will be seen in my papers on inference of phylogenies from gene frequency and continuous character data (Felsenstein, 1973b, 1981c, 1985c).
The input formats are fairly similar to the discrete-character programs, but with one difference. When CONTML is used in the gene-frequency mode (its usual, default mode), or when GENDIST is used, the first line contains the number of species (or populations) and the number of loci and the options information. There then follows a line which gives the numbers of alleles at each locus, in order. This must be the full number of alleles, not the number of alleles which will be input: i. e. for a two-allele locus the number should be 2, not 1. There then follow the species (population) data, each species beginning on a new line. The first 10 characters are taken as the name, and thereafter the values of the individual characters are read free-format, preceded and separated by blanks. They can go to a new line if desired, though of course not in the middle of a number. Missing data is not allowed - an important limitation. In the default configuration, for each locus, the numbers should be the frequencies of all but one allele. The menu option A (All) signals that the frequencies of all alleles are provided in the input data -- the program will then automatically ignore the last of them. So without the A option, for a three-allele locus there should be two numbers, the frequencies of two of the alleles (and of course it must always be the same two!). Here is a typical data set without the A option:
5 3 2 3 2 Alpha 0.90 0.80 0.10 0.56 Beta 0.72 0.54 0.30 0.20 Gamma 0.38 0.10 0.05 0.98 Delta 0.42 0.40 0.43 0.97 Epsilon 0.10 0.30 0.70 0.62
whereas here is what it would have to look like if the A option were invoked:
5 3 2 3 2 Alpha 0.90 0.10 0.80 0.10 0.10 0.56 0.44 Beta 0.72 0.28 0.54 0.30 0.16 0.20 0.80 Gamma 0.38 0.62 0.10 0.05 0.85 0.98 0.02 Delta 0.42 0.58 0.40 0.43 0.17 0.97 0.03 Epsilon 0.10 0.90 0.30 0.70 0.00 0.62 0.38
The first line has the number of species (or populations) and the number of loci. The second line has the number of alleles for each of the 3 loci. The species lines have names (filled out to 10 characters with blanks) followed by the gene frequencies of the 2 alleles for the first locus, the 3 alleles for the second locus, and the 2 alleles for the third locus. You can start a new line after any of these allele frequencies, and continue to give the frequencies on that line (without repeating the species name).
If all alleles of a locus are given, it is important to have them add up to 1. Roundoff of the frequencies may cause the program to conclude that the numbers do not sum to 1, and stop with an error message.
While many compilers may be more tolerant, it is probably wise to make sure that each number, including the first, is preceded by a blank, and that there are digits both preceding and following any decimal points.
CONTML and CONTRAST also treat quantitative characters (the continuous-characters mode in CONTML, which is option C). It is assumed that each character is evolving according to a Brownian motion model, at the same rate, and independently. In reality it is almost always impossible to guarantee this. The issue is discussed at length in my review article in Annual Review of Ecology and Systematics (Felsenstein, 1988a), where I point out the difficulty of transforming the characters so that they are not only genetically independent but have independent selection acting on them. If you are going to use CONTML to model evolution of continuous characters, then you should at least make some attempt to remove genetic correlations between the characters (usually all one can do is remove phenotypic correlations by transforming the characters so that there is no within-population covariance and so that the within-population variances of the characters are equal -- this is equivalent to using Canonical Variates). However, this will only guarantee that one has removed phenotypic covariances between characters. Genetic covariances could only be removed by knowing the coheritabilities of the characters, which would require genetic experiments, and selective covariances (covariances due to covariation of selection pressures) would require knowledge of the sources and extent of selection pressure in all variables.
CONTRAST is a program designed to infer, for a given phylogeny that is provided to the program, the covariation between characters in a data set. Thus we have a program in this set that allow us to take information about the covariation and rates of evolution of characters and make an estimate of the phylogeny (CONTML), and a program that takes an estimate of the phylogeny and infers the variances and covariances of the character changes. But we have no program that infers both the phylogenies and the character covariation from the same data set.
In the quantitative characters mode, a typical small data set would be:
5 6 Alpha 0.345 0.467 1.213 2.2 -1.2 1.0 Beta 0.457 0.444 1.1 1.987 -0.2 2.678 Gamma 0.6 0.12 0.97 2.3 -0.11 1.54 Delta 0.68 0.203 0.888 2.0 1.67 Epsilon 0.297 0.22 0.90 1.9 1.74
Note that in the latter case, there is no line giving the numbers of alleles at each locus. In this latter case no square-root transformation of the coordinates is done: each is assumed to give directly the position on the Brownian motion scale.
For further discussion of options and modifiable constants in CONTML, GENDIST, and CONTRAST see the documentation files for those programs.
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 0.000000 0.078002 0.080749 0.066805 0.103014 African 0.078002 0.000000 0.234698 0.104975 0.227281 Chinese 0.080749 0.234698 0.000000 0.053879 0.063275 American 0.066805 0.104975 0.053879 0.000000 0.134756 Australian 0.103014 0.227281 0.063275 0.134756 0.000000
|egendist||Genetic Distance Matrix program|
|fcontml||Gene frequency and continuous character Maximum Likelihood|
Please report all bugs to the EMBOSS bug team (emboss-bug © emboss.open-bio.org) not to the original author.
Converted (August 2004) to an EMBASSY program by the EMBOSS team.