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| 12 | <DIV ALIGN=RIGHT> |
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| 13 | version 3.6 |
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| 14 | </DIV> |
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| 15 | <P> |
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| 16 | <DIV ALIGN=CENTER> |
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| 17 | <H1>GENDIST - Compute genetic distances from gene frequencies</H1> |
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| 18 | </DIV> |
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| 19 | <P> |
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| 20 | © Copyright 1986-2002 by the University of |
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| 21 | Washington. Written by Joseph Felsenstein. Permission is granted to copy |
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| 22 | this document provided that no fee is charged for it and that this copyright |
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| 23 | notice is not removed. |
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| 24 | <P> |
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| 25 | This program computes any one of three measures of genetic distance from a set |
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| 26 | of gene frequencies in different populations (or species). The three are |
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| 27 | Nei's genetic distance (Nei, 1972), Cavalli-Sforza's chord measure (Cavalli- |
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| 28 | Sforza and Edwards, 1967) and Reynolds, Weir, and Cockerham's (1983) genetic |
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| 29 | distance. These are written to an output file in a format that can be read by |
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| 30 | the distance matrix phylogeny programs FITCH and KITSCH. |
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| 31 | <P> |
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| 32 | The three measures have somewhat different assumptions. All assume that all |
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| 33 | differences between populations arise from genetic drift. Nei's distance is |
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| 34 | formulated for an infinite isoalleles model of mutation, in which there is a |
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| 35 | rate of neutral mutation and each mutant is to a completely new alleles. It |
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| 36 | is assumed that all loci have the same rate of neutral mutation, and that the |
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| 37 | genetic variability initially in the population is at equilibrium between |
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| 38 | mutation and genetic drift, with the effective population size of each |
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| 39 | population remaining constant. |
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| 40 | <P> |
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| 41 | Nei's distance is: |
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| 42 | <P> |
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| 43 | <PRE> |
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| 44 | __ __ |
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| 45 | \ \ |
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| 46 | /_ /_ p<SUB>1mi</SUB> p<SUB>2mi</SUB> |
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| 47 | m i |
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| 48 | D = - ln <FONT SIZE=+3>(</FONT> ------------------------------------- <FONT SIZE=+3>)</FONT>. |
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| 49 | __ __ __ __ |
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| 50 | \ \ \ \ |
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| 51 | [ /_ /_ p<SUB>1mi</SUB><SUP>2</SUP>]<SUP><SUP>1</SUP>/<SUB>2</SUB></SUP> [ /_ /_ p<SUB>2mi</SUB><SUP>2</SUP>]<SUP><SUP>1</SUP>/<SUB>2</SUB></SUP> |
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| 52 | m i m i |
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| 53 | </PRE> |
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| 54 | <P> |
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| 55 | where <EM>m</EM> is summed over loci, <EM>i</EM> over alleles at the <EM>m</EM>-th locus, and where |
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| 56 | <P> |
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| 57 | p<SUB>1mi</SUB> |
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| 58 | <P> |
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| 59 | is the frequency of the <EM>i</EM>-th allele at the <EM>m</EM>-th locus in population 1. |
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| 60 | Subject to the above assumptions, Nei's genetic distance is expected, for a |
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| 61 | sample of sufficiently many equivalent loci, to rise linearly with time. |
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| 62 | <P> |
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| 63 | The other two genetic distances assume that there is no mutation, and that all |
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| 64 | gene frequency changes are by genetic drift alone. However they do not |
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| 65 | assume that population sizes have remained constant and equal in all |
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| 66 | populations. They cope with changing population size by having expectations |
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| 67 | that rise linearly not with time, but with the sum over time of 1/N, where N |
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| 68 | is the effective population size. Thus if population size doubles, genetic |
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| 69 | drift will be taking place more slowly, and the genetic distance will be |
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| 70 | expected to be rising only half as fast with respect to time. Both genetic |
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| 71 | distances are different estimators of the same quantity under the same model. |
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| 72 | <P> |
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| 73 | Cavalli-Sforza's chord distance is given by |
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| 74 | <P> |
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| 75 | <PRE> |
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| 76 | __ __ __ |
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| 77 | \ \ \ |
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| 78 | D<SUP>2</SUP> = 4 /_ [ 1 - /_ p<SUB>1mi</SUB><SUP><SUP>1</SUP>/<SUB>2</SUB></SUP> p <SUB>2mi</SUB><SUP><SUP>1</SUP>/<SUB>2</SUB></SUP>] / /_ (a<SUB>m</SUB> - 1) |
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| 79 | m i m |
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| 80 | </PRE> |
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| 81 | <P> |
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| 82 | <P> |
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| 83 | where m indexes the loci, where i is summed over the alleles at the m-th |
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| 84 | locus, and where a is the number of alleles at the m-th locus. It can be |
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| 85 | shown that this distance always satisfies the triangle |
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| 86 | inequality. Note that as given here it is divided by the number of degrees |
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| 87 | of freedom, the sum of the numbers of alleles minus one. The quantity which |
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| 88 | is expected to rise linearly with amount of |
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| 89 | genetic drift (sum of <EM>1/N</EM> over time) is <EM>D</EM> squared, the quantity computed |
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| 90 | above, and that is what is written out into the distance matrix. |
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| 91 | <P> |
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| 92 | Reynolds, Weir, and Cockerham's (1983) genetic distance is |
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| 93 | <P> |
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| 94 | <PRE> |
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| 95 | |
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| 96 | __ __ |
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| 97 | \ \ |
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| 98 | /_ /_ [ p<SUB>1mi</SUB> - p<SUB>2mi</SUB>]<SUP>2</SUP> |
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| 99 | m i |
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| 100 | D<SUP>2</SUP> = -------------------------------------- |
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| 101 | __ __ |
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| 102 | \ \ |
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| 103 | 2 /_ [ 1 - /_ p<SUB>1mi</SUB> p<SUB>2mi</SUB> ] |
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| 104 | m i |
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| 105 | </PRE> |
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| 106 | <P> |
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| 107 | <P> |
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| 108 | where the notation is as before and D<SUP>2</SUP> is the quantity that is |
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| 109 | expected to rise linearly with cumulated genetic drift. |
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| 110 | <P> |
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| 111 | Having computed one of these genetic distances, one which you feel is |
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| 112 | appropriate to the biology of the situation, you can use it as the input to |
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| 113 | the programs FITCH, KITSCH or NEIGHBOR. Keep in mind that the statistical |
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| 114 | model in |
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| 115 | those programs implicitly assumes that the distances in the input table have |
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| 116 | independent errors. For any measure of genetic distance this will not be true, |
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| 117 | as bursts of random genetic drift, or sampling events in drawing the sample of |
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| 118 | individuals from each population, cause fluctuations of gene frequency that |
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| 119 | affect many distances simultaneously. While this is not expected to bias the |
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| 120 | estimate of the phylogeny, it does mean that the weighing of evidence from all |
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| 121 | the different distances in the table will not be done with maximal |
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| 122 | efficiency. One issue is which value of the P |
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| 123 | (Power) parameter should be used. This depends on how the variance of a |
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| 124 | distance rises with its expectation. For Cavalli-Sforza's chord distance, and |
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| 125 | for the Reynolds et. al. distance it can be shown that the variance of the |
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| 126 | distance will be proportional to the square of its expectation; this suggests |
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| 127 | a value of 2 for <EM>P</EM>, which the default value for FITCH and KITSCH |
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| 128 | (there is no P option in NEIGHBOR). |
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| 129 | <P> |
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| 130 | If you think that the pure genetic drift model is appropriate, and are thus |
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| 131 | tempted to use the Cavalli-Sforza or Reynolds et. al. distances, you might |
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| 132 | consider using the maximum likelihood program CONTML instead. It will |
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| 133 | correctly weigh the evidence in that case. Like those genetic distances, it |
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| 134 | uses approximations that break down as loci start to drift all the way to |
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| 135 | fixation. Although Nei's distance will not break down in that case, it |
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| 136 | makes other assumptions about equality of substitution rates at all loci and |
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| 137 | constancy of population sizes. |
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| 138 | <P> |
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| 139 | The most important thing to remember is that genetic distance is not an |
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| 140 | abstract, idealized measure of "differentness". It is an estimate of a |
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| 141 | parameter (time or cumulated inverse effective population size) of the |
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| 142 | model which is thought to have generated the differences we see. As an |
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| 143 | estimate, it has statistical properties that can be assessed, and we should |
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| 144 | never have to choose between genetic distances based on their aesthetic |
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| 145 | properties, or on the personal prestige of their originators. Considering them |
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| 146 | as estimates |
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| 147 | focuses us on the questions which genetic distances are intended to answer, |
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| 148 | for if there are none there is no reason to compute them. For further |
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| 149 | perspective on genetic distances, I recommend my own paper evaluating |
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| 150 | Reynolds, Weir, and Cockerham (1983), and the material in Nei's book (Nei, |
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| 151 | 1987). |
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| 152 | <P> |
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| 153 | <H2>INPUT FORMAT</H2> |
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| 154 | <P> |
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| 155 | The input to this program is standard and is as described in the Gene |
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| 156 | Frequencies and Continuous |
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| 157 | Characters Programs documentation file above. It consists of the number of |
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| 158 | populations (or species), the number of loci, |
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| 159 | and after that a line containing the numbers of alleles at each of the |
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| 160 | loci. Then the gene frequencies follow in standard format. |
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| 161 | <P> |
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| 162 | The options are selected using a menu: |
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| 163 | <P> |
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| 164 | <TABLE><TR><TD BGCOLOR=white> |
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| 165 | <PRE> |
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| 166 | |
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| 167 | Genetic Distance Matrix program, version 3.6a3 |
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| 168 | |
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| 169 | Settings for this run: |
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| 170 | A Input file contains all alleles at each locus? One omitted at each locus |
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| 171 | N Use Nei genetic distance? Yes |
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| 172 | C Use Cavalli-Sforza chord measure? No |
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| 173 | R Use Reynolds genetic distance? No |
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| 174 | L Form of distance matrix? Square |
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| 175 | M Analyze multiple data sets? No |
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| 176 | 0 Terminal type (IBM PC, ANSI, none)? (none) |
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| 177 | 1 Print indications of progress of run? Yes |
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| 178 | |
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| 179 | Y to accept these or type the letter for one to change |
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| 180 | |
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| 181 | </PRE> |
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| 182 | </TD></TR></TABLE> |
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| 183 | <P> |
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| 184 | The A (All alleles) option is described in the Gene Frequencies and |
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| 185 | Continuous Characters Programs documentation file. As with CONTML, it is |
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| 186 | the signal that all alleles are represented in the gene frequency input, |
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| 187 | without one being left out per locus. C, N, and R are the signals to |
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| 188 | use the Cavalli-Sforza, Nei, or Reynolds et. al. genetic distances |
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| 189 | respectively. The Nei distance is the default, and it will be computed |
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| 190 | if none of these options is explicitly invoked. The L option is the signal |
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| 191 | that the distance matrix is to be written out in Lower triangular form. |
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| 192 | The M option is the usual Multiple Data Sets option, useful for |
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| 193 | doing bootstrap analyses with the distance matrix programs. It allows |
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| 194 | multiple data sets, but does not allow multiple sets of weights (since |
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| 195 | there is no provision for weighting in this program). |
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| 196 | <P> |
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| 197 | <H2>OUTPUT FORMAT</H2> |
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| 198 | <P> |
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| 199 | The output file simply contains on its first line the number of species (or |
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| 200 | populations). Each |
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| 201 | species (or population) starts a new line, with its name printed out |
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| 202 | first, and then and there are up to nine |
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| 203 | genetic distances printed on each line, in the standard format used as input |
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| 204 | by the distance matrix programs. The output, in its default form, is |
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| 205 | ready to be used in the distance matrix programs. |
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| 206 | <P> |
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| 207 | <H2>CONSTANTS</H2> |
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| 208 | <P> |
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| 209 | The constants |
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| 210 | available to be changed by the user if the program is recompiled are |
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| 211 | "namelength" the length of a species name, set to 10 in the distribution |
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| 212 | and "epsilon" which defines a small quantity that is used when checking |
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| 213 | whether allele frequencies at a locus sum to more than one: if all |
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| 214 | alleles are input (option A) and the sum differs from 1 by more than epsilon, |
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| 215 | or if not all alleles are input and the sum is greater than 1 by more |
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| 216 | then epsilon, the program will see this as an error and stop. You may |
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| 217 | find this causes difficulties if you gene frequencies have been rounded. |
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| 218 | I have tried to keep epsilon from being too small to prevent such problems. |
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| 219 | <P> |
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| 220 | <H2>RUN TIMES</H2> |
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| 221 | <P> |
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| 222 | The program is quite fast and the user should effectively never be limited by |
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| 223 | the amount of time it takes. All that the program has to do is read in the |
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| 224 | gene frequency data and then, for each pair of species, compute a genetic |
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| 225 | distance formula for each pair of species. This should require an amount of |
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| 226 | effort proportional to the total number of alleles over loci, and to the |
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| 227 | square of the number of populations. |
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| 228 | <P> |
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| 229 | <H2>FUTURE OF THIS PROGRAM</H2> |
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| 230 | <P> |
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| 231 | The main change that will be made to this program in the future is to add |
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| 232 | provisions for taking into account the sample size for each population. The |
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| 233 | genetic distance formulas have been modified by their inventors to correct for |
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| 234 | the inaccuracy of the estimate of the genetic distances, which on the whole |
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| 235 | should artificially increase the distance between populations by a small |
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| 236 | amount dependent on the sample sizes. The main difficulty with doing this is |
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| 237 | that I have not yet settled on a format for putting the sample size in the |
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| 238 | input data along with the gene frequency data for a species or population. |
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| 239 | <P> |
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| 240 | I may also include other distance measures, but only if I think their use is |
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| 241 | justified. There are many very arbitrary genetic distances, and I am |
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| 242 | reluctant to include most of them. |
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| 243 | <P> |
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| 244 | <HR> |
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| 245 | <P> |
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| 246 | <H3>TEST DATA SET</H3> |
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| 247 | <P> |
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| 248 | <TABLE><TR><TD BGCOLOR=white> |
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| 249 | <PRE> |
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| 250 | 5 10 |
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| 251 | 2 2 2 2 2 2 2 2 2 2 |
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| 252 | European 0.2868 0.5684 0.4422 0.4286 0.3828 0.7285 0.6386 0.0205 |
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| 253 | 0.8055 0.5043 |
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| 254 | African 0.1356 0.4840 0.0602 0.0397 0.5977 0.9675 0.9511 0.0600 |
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| 255 | 0.7582 0.6207 |
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| 256 | Chinese 0.1628 0.5958 0.7298 1.0000 0.3811 0.7986 0.7782 0.0726 |
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| 257 | 0.7482 0.7334 |
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| 258 | American 0.0144 0.6990 0.3280 0.7421 0.6606 0.8603 0.7924 0.0000 |
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| 259 | 0.8086 0.8636 |
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| 260 | Australian 0.1211 0.2274 0.5821 1.0000 0.2018 0.9000 0.9837 0.0396 |
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| 261 | 0.9097 0.2976 |
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| 262 | </PRE> |
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| 263 | </TD></TR></TABLE> |
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| 264 | <P> |
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| 265 | <HR> |
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| 266 | <P> |
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| 267 | <H3>TEST SET OUTPUT</H3> |
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| 268 | <P> |
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| 269 | <TABLE><TR><TD BGCOLOR=white> |
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| 270 | <PRE> |
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| 271 | 5 |
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| 272 | European 0.0000 0.0780 0.0807 0.0668 0.1030 |
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| 273 | African 0.0780 0.0000 0.2347 0.1050 0.2273 |
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| 274 | Chinese 0.0807 0.2347 0.0000 0.0539 0.0633 |
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| 275 | American 0.0668 0.1050 0.0539 0.0000 0.1348 |
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| 276 | Australian 0.1030 0.2273 0.0633 0.1348 0.0000 |
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| 277 | </PRE> |
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| 278 | </TD></TR></TABLE> |
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| 279 | <P> |
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