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1. Dr Phy ll m k 1 Reference 10 8 2 2 Differentiation of subdivided gene pools Let a collection of subpopulations have the subpopulation differentiation q at locus I l 1 L Then the unweighted subpopulation differentiation of the gene pool was proven to be the arithmetic mean of the subpopulation differentiation at each locus that is L DIN 1 de Reference 10 37 9 Acknowledgements and disclaimer I am extremely grateful to the many colleagues who have been of enormous help in com pleting this project Matthias K hle did some of the programming of the interactive sequence the configuration file being his idea His ability to find his way around in some body else s program which at that time was already complex is admirable I would like to thank numerous colleagues among them Fritz Bergmann Bernd Degen Reiner Finkeldey Hans Rolf Gregorius Hans H Hattemer Sven Herzog Bernhard Hosius Ger hard Miiller Starck Aristotelis Papageorgiou Rommy Starke Jozef Turok Martin Ziehe and too many master s students to list here for their eagerness to test the various stages of the program on their data and for their many suggestions for improvement Martin Ziehe in particular is to be credited for taking the trouble to recalculate many of the computed results thereby discovering a number of bugs before they could do any damage Spe cial thanks are due to Hans Rolf Gregorius for his val
2. ulations Theor Appl Genet 76 947 951 Gregorius H R 1989 Characterization and Analysis of Mating Systems Ekopan Verlag Witzenhausen Gregorius H R 1990 A diversity independent measure of evenness Amer Natur 136 701 711 Gregorius H R 1996 Differentiation between populations and its measurement Acta Biotheoretica 44 23 36 39 16 17 18 19 20 21 22 23 Gregorius H R Krauhausen J and M ller Starck G 1986 Spatial and temporal genetic differentiation among the seed in a stand of Fagus sylvatica L Heredity 57 255 262 Hattemer H H Bergmann F Ziehe M 1993 Einf hrung in die Genetik f r Studierende der Forstwissenschaft 2 Aufl J D Sauerl nder s Verlag Frankfurt am Main Ledwina T and Gnot S 1980 Testing for Hardy Weinberg equilibrium Biomet rics 36 161 165 Louis E J and Dempster E R 1987 An exact test for Hardy Weinberg and multiple alleles Biometrics 43 805 811 Miiller Starck G 1977 Untersuchungen ber die nat rliche Selbstbefruchtung in Best nden der Fichte Picea abies L Karst und Kiefer Pinus sylvestris L Silvae Genetica 26 207 217 Miiller Starck G 1977 Cross fertilization in a conifer stand inferred from enzyme gene markers in seeds Silvae Genetica 26 223 226 Pamilo P and Varvio Aho S 1984 Testing genotype frequencies and heterozygosi ties Marine Biology 79 99 100 Robertson A and
3. 10 for 10 samples 115 chars line as for DIN A4 paper crosswise 11 for 11 samples 125 chars line as in condensed mode Option 0 The width of the output medium e g paper can vary and with it the number of samples that fit onto one line If the available number of characters per line is known the formula on the second line above yields the maximal number of samples rounding down to the nearest integer if necessary If not all of the samples fit onto one line the tables of results are cut off after the specified number of samples and continued on the next lines of output The minimum number of characters per line is set to 75 One reason is that this is the length of the commentaries in the output Another is that the maximal number of columns of the contingency tables in the tests of genotypic structure that can be printed onto one line is also set to the chosen number of samples per line A reply of 0 zero causes the results of all samples to be printed onto one line Additional calculations using the same input file and locus configuration Option When this line appears the chosen calculations have been completed and either typed on the screen or stored in the output file Its purpose becomes apparent in the description of the options Option Y Type Y if additional frequency distributions measures or tests for the same or a different set of samples are desired for the same input file and locus
4. homozygosity in that only one allele or haplotype can be sampled per individual see 7 for proof alpha HWP In the case of alleles and haplotypes with arbitrary frequencies this rela tive frequency characterizes an analogous alpha for the best case situation for sampling haplotypes when only genotypes can be sampled 7 gives proof that this situation oc curs when the genotypes arose by random fertilization between alleles haplotypes which are then independently associated in the genotypes The resulting Hardy Weinberg Proportions HWP thus represent the optimal relationships between homozygosity and heterozygosity for sampling different alleles or haplotypes in genotypes In 7 it is shown that alpha HWP is equal to the value of alpha for a sample twice the size of the given sample Thus sampling haplotypes in a Hardy Weinberg population of genotypes is equivalent to drawing a sample of haplotypes singly as opposed to pairwise that is twice the size of the given sample of genotypes The output for the various measures and tests is described in the relevant Sec 6 7 8 The legend printed at the beginning of the output explains a few additional conventions 0 observed relative frequency expected absolute frequency in a test multilocus haplotype or genotype 9 999 undefined The designation of the different alleles haplotypes and genotypes in the output is demon strated in Tab 9 Th
5. 11 11 13 22 2 5 33 12 33 23 9999 3 0 10 20 30 40 3 32 33 12 11 23 3 19 13 12 13 23 3 4 13 11 33 23 3 7 11 11 13 22 3 25 33 12 33 23 3 4 13 11 13 23 3 3 33 12 33 23 3 3 13 11 13 33 3 3 13 12 13 23 9999 46 Example 2 This example demonstrates that it is still possible to input single locus genotypes at more than one locus even if they refer to the same population but not necessarily the same individuals The disadvantage of this type of input is that all indi vidual multilocus information is lost The 1 in the key lines indicates that the second field of each genotype line contains the frequency with which the respective genotype was found in the sample Note that several key lines per sample are possible Adapted from Kim Z S 1985 Viability selection at an allozyme locus during development in European beech Fagus sylvatica L Silvae Genetica 34 181 186 2 2 SAP A LAP A DEUTSCHLAND ECKERN DEUTSCHLAND KEIMLINGE GEWAECHSHAUS 1 214 1X 1 212 1X 1 0 20 1 151 11 1111 22 1107 33 1 6 44 1 51 12 1 83 13 1 5 14 1 68 23 1 7 24 1 3 34 9999 2 0 10 2 23 11 2 71 22 2 1 33 2 62 12 2 2 13 2 3 14 2 6 23 2 624 2 0 20 2 39 11 2 53 22 2 32 33 2 2 44 2 6 12 2 9 13 2 1 14 2 16 23 2 224 2 2 34 9999 47 Example 3 Here the number of loci is large necessitating continuation of the genotypes on additional lines Note that since all alleles are known i e 4 1 the format specification has bee
6. 2 Hmmm Examples of input files il 31 31 31 31 31 32 32 33 35 35 35 35 36 37 37 37 38 41 41 41 41 46 List of Tables Ee a N e 10 11 12 Frequency distributions calculated by GSED 2 222 22 Characterization of genetic structures 1 eee ee 0 Examples of format specification lines 2 22 Examples of key lines using the format specification LO TAS TRATO LID e da Se e ado as e e don El Example of a skeleton input file constructed by the auxiliary program GSEDINPT Using any ASCII text editor the lines beginning with lt must be replaced by the genotypic data for the respective population sam ple right justified within the fields defined by the key line Note that gametic sex is specified for all loci in Population 1 two loci in Population 2 and no loci in Population 6 3 254 23 Ae PR AA A A Interactive sequence for first run of GSED for an input file EXAMPLE DAT continued in next table ws E See Me sine A Geos ad Sh ae aes he Continuation of the interactive sequence begun in the previous table Start of the interactive sequence for a subsequent run using the input file EXAMPLE DAT The choices made during the first run were stored in the configuration file EXAMPLE CFG shown below This configuration can be adopted by replying with a Y in which case the interactive sequence will be skipped A reply of N allows a new choice of frequency
7. case each genotype line specifies the following integers sampleno indivno locus allele locus alleley locusnallelez locusn alleles where Table 4 Examples of key lines using the format specification 10 214 1X 10 212 Key line for sample 2 specifying that multilocus genotypes comprise loci 1 10 that each genotype is that of a single individual and that gametic sex is not specified for any locus 2 01020 30 40 50 60 70 80 90100 Key line for sample 5 specifying that multilocus genotypes comprise loci 1 10 that each genotype is accompanied by its frequency in the sample and that gametic sex is specified for all loci 5 O 11 21 31 41 51 61 71 81 91101 sample number referring to list of samples in header number designating the individual whose genotype is listed locus allele designation of first allele at locus 7 as an integer gt 1 if gametic sex is specified for this locus then locus allele is the allele contributed by the maternal parent locus allele i 2 n designation of second allele at locus i as an integer gt 1 if gametic sex is specified for this locus then locus alleleg is the allele contributed by the paternal parent 2 3 3 Genotype frequencies in sample If the sign of locdefj in the key line of a sample equals see Sec 2 3 1 then each genotype is interpreted as having been found in a number of individuals The frequency of a genotype in the sample is giv
8. configuration Since the frequency data is already stored the input file is not reread and results are obtained quickly This option provides a means of ordering the output differently from that reflected by the interactive sequence It also allows calculation of measures of variation between samples for different sets of samples 19 Option N An answer of N terminates the program 3 2 Configuration file In subsequent runs of GSED for an input file a configuration file may exist This will be the case if the question Should these choices be stored in a file for later use was answered with Y in an earlier run for the same input file The configuration file contains the previous answers to the questions listed under the headings of Choice of frequency distributions and Choice of calculations see Tab 6 Its name is composed of the filename of the input file and the extension CFG If a configuration file exists then a configuration table such as that presented in Tab 8 is typed on the screen after the Locus configuration has been specified see Tab 6 If the answer to the subsequent question Do you want to adopt this configuration is answered by Y then Choice of frequency distributions and Choice of calculations are skipped The question Should gametic sex specification if given be retained is still posed since in the case of gametic sex specification one
9. e Maternal allele haplotype frequencies Unknown maternal alleles are assumed to be arandom sample of all maternal alleles and are thus left out of the calculation In the same manner incomplete maternal multilocus haplotypes containing an unknown allele at one or more loci are also treated as a random sample of haplotypes and are ignored e Paternal allele haplotype frequencies Unknown paternal alleles and haplotypes are treated in the same way as maternal ones e Allele haplotype frequencies Only those alleles are taken into account that are part of a completely known genotype Thus if one allele is known and the other is unknown e g because the primary endosperm of a conifer seed was analyzed but the embryo lost the known allele will not be counted in the allele frequency distribution e Genotype frequencies Unknown genotypes are assumed to be a random sample and are not counted 27 6 Measures of variation The following measures of variation can be calculated for any of the types of frequency distributions listed in Sec 5 6 1 Measures of variation within samples 6 1 1 Diversity v Let a collection be characterized by a frequency vector p p1 P2 Pn of its genetic types where n N and for k 1 n py gt 0 and Xg pk 1 The diversity v p of the collection is defined as v p En v p measures the differentiation effective number of types it is less than or equal to the actual number of types an
10. form genotypes is provided by tests of two models describing mating systems The input to GSED consists of the genotypes found in a sample of individuals taken from a collection e g population deme cohort An individual s genotype refers to the alleles present at a single gene locus single locus genotype or more commonly at each of a number of gene loci multilocus genotype If known designation of the gametic sex of each allele at a locus that is the sex of the contributing parent can be included in the genotype Many of the measures of variation calculated by the program can be applied not only to genetic types but also to any system of classification by which each individual of a population can be assigned one of a finite set of discrete types e g phenotypes ecotypes Although the assumption that data input to GSED concerns genetic types is reflected in its commentaries one or higher dimensional non genetic classifications can be disguised as maternal alleles or haplotypes at loci for which gametic sex is specified and paternal type unknown An input file would be analogous to that of Example 4 in B Output headings would have to be reinterpreted accordingly When using GSED to analyze genetic types however it is essential that the alleles at each locus be known In other words the phenotype produced by the genes at each locus must be a gene marker in that the phenotype enables identification of all invo
11. format specification even if a line is terminated after the first field they seek the missing data in the next lines which probably already define the next sample and print an error message as soon as the mix up leads to ambiguity One solution is to fill out all of the remaining fields defined by the format specification with blanks including those in continuation lines As this is often forgotten GSED uses the following strategy If 9999 is encountered in field 1 the current sample is ended The program then backspaces in the input file to this same end of sample line and begins to search line for line for the next key line recognizable by the 0 in field 2 In order for this to work the sample number and the 0 in the first two fields of the next key line must be separated either by one or more blank characters or by a comma By this means reading errors should be avoided 2 4 End of input Reading of data is terminated when the end of the input file is encountered that is after the end of sample line for the final sample Operating systems apparently differ in their handling of end of files Whereas TOS rec ognizes the end of an input file if no further characters appear after the last 9999 not even lt Return gt is necessary DOS expects to find the cursor at the beginning of the Table 5 Example of a skeleton input file constructed by the auxiliary program GSED INPT Using any ASCII text edito
12. information consult any FORTRAN language reference manual The I field descriptor indicates that n integers are to be read in consecutive fields of width w 7 e number of character positions The X field descriptor indicates that n character positions are to be skipped The repeat count r indicates that the contents of the parentheses are to be repeated r times ME Separates field descriptors Separates field descriptors and causes reading to continue on a new line Parentheses enclose the entire format specification 2 3 Input format for each sample The main part of an input file consists of the genotypes found in each sample The data for each sample has the following form e A key line describing how the following genotypes are to be interpreted e The list of genotypes found in the sample e End of sample line consisting of the integer 9999 in the first I field The end of sample line for one sample is followed immediately no empty lines by the key line of the next sample The samples can appear in any order The input file terminates with the end of sample line of the last sample All data lines including key line and end of sample line are read according to the format specification line described above see Sec 2 2 6 2 3 1 Key line The key line of a sample fulfills four functions e designation of the sample number e assignment of fields 7 e blocks of character positions in a data line to
13. large enough for most applications Except for very small input files it is not recommended to use a floppy disk as the default drive since calculation is extremely slow A 2 Limitations on data Tab 11 gives a list of variables the maximal allowed values of which were fixed at compile time and can be changed only by altering the source code Error messages are printed if any of these maxima is exceeded and execution is stopped The test of homogeneity allows a maximum of Maz No Alleles genetic types All tests ac comodate up to 100 degrees of freedom Violation does not cause termination of execution in these cases All integers are of type INTEGER 4 and range between 2147483647 and 2147483647 Reals are of single precision type REAL 4 with approximately 7 digit accuracy and range from 1073 to 10 The output formats accomodate 5 digit integers up to 99999 and floating point numbers with up to 5 digits in front of the decimal point Floating point calculations are printed with 3 decimal places the one exception are the expected absolute frequencies in tests which have two decimal places A 3 Temporary direct access storage GSED stores intermediate results in eight direct access files Depending on the compiler they receive names such as FORnnn DAT F77 nnmn or a seemingly arbitrary sequence of letters and numbers They are stored on the default directory or drive see 41 Table 11 Interpretation and maxi
14. loci showing dominance cannot be used for the analysis of genetic types unless additional inheritance analysis has revealed the true genotype of each individual at the locus GSED is written in FORTRAN 77 and has been compiled for the operating systems MS DOS PC DOS Version 2 1 or higher VMS Version 5 0 or higher and TOS Atari The program reads data from a previously constructed ASCII input file In each run the user interactively requests calculations of frequency distributions measures and tests and samples The output is printed either directly onto the screen or stored in a file for later processing The next three sections 2 3 4 of this manual deal with practical matters namely constructing an input file running the program and understanding the output The following four sections 5 6 7 8 are concerned with the concepts behind the program The first of these sections 5 reviews the different types of frequency distribution that can be calculated from samples of multilocus genotypes Three sections 6 7 8 outline the measures and tests that are performed for the various frequency distributions including references to mostly original articles containing detailed descriptions of the underlying concepts A list of references follows denoted in the text by numbers in square brackets Two appendices A B are devoted to technical considerations hardware storage space and examples of input files 2 Constructing an i
15. may want to perform the same calculations with and without regard of gametic sex see Sec 5 If the answer to Do you want to adopt this configuration is N then Choice of frequency distributions and Choice of calculations must be made anew as in Tab 6 3 3 Sorting of haplotypes and genotypes An answer of Y to any of the following questions causes the lists of encountered haplo types and genotypes to be printed in lexicographic order Frequency distributions Test of homogeneity of the sample distributions Test of Hardy Weinberg structure and heterozygosity Test of product structure only if gametic sex is specified Since sorting of multilocus types can take an extreme amount of computing time it is advisable not to choose these calculations for multilocus combinations only the first two questions apply unless they themselves are of interest A test of homogeneity for a large number of multilocus types may well exceed the capacity of the program anyway see Sec A 2 Often heterozygosity is the only calculation desired for multilocus genotypes it is performed quickly if it alone is selected 20 4 Understanding GSED output GSED output is divided into the output for each locus combination t e for each single locus or multilocus combination see Sec 6 7 If calculations are requested only for single loci results for the gene pool and hypothetical gametic output defined by t
16. statistic G 2 Y N In N In E N types For one degree of freedom x goodness of fit test with continuity correction c 3 with statistic N E N 4 Mees DE ERES T3 a E N ypes N and E N represent observed and expected sample frequencies respectively of the different types These statistics are asymptotically x distributed the number of degrees of freedom de pending on the model Thus it must be kept in mind that these tests are accurate only for large sample sizes A warning is printed in the output if a type is found to have expected frequency less than 5 Exact tests have recently been devised in some cases but these seemed too time consuming in terms of computing time to allow their inclusion in the larger framework of GSED In borderline cases i e statistic near critical value of x of small sample size it may be advisable to retest structures using special statistics programs that perform exact hypothesis testing References 19 25 pp 71ff 7 2 1 Test of Hardy Weinberg structure and heterozygosity To each unordered genotypic structure with relative frequencies P of genotypes A A Pi Pi Nic Pij 1 there corresponds a Hardy Weinberg structure with genotypic frequencies P defined by P p and Pi 2pip fort Aj andi j 1 k In this definition p is the relative frequency of allele A from the original genotypic structure i e pi Pi 5 oy jf Pi Hardy Weinberg structu
17. 1971 Probability Models and Statistical Methods in Genetics John Wiley amp Sons Inc New York London Sydney Toronto Emigh T H 1980 A comparison of tests for Hardy Weinberg equilibrium Biomet rics 36 627 642 Gregorius H R 1974 Genetischer Abstand zwischen Populationen I Zur Konzep tion der genetischen Abstandsmessung Silvae Genetica 23 22 27 Gregorius H R 1974 On the concept of genetic distance between populations based on gene frequencies Proc Joint IUFRO Meeting S02 04 1 3 Stockholm Session I 17 26 Gregorius H R 1978 The concept of genetic diversity and its formal relationship to heterozygosity and genetic distance Math Biosciences 41 253 271 Gregorius H R 1980 The probability of losing an allele when diploid genotypes are sampled Biometrics 36 643 652 Gregorius H R 1984 A unique genetic distance Biometrical J 26 13 18 Gregorius H R 1984 Measurement of genetic differentiation in plant populations Pp 276 285 in Gregorius H R ed Population Genetics in Forestry Springer Verlag Berlin Heidelberg New York Tokyo Gregorius H R and Roberds J H 1986 Measurement of genetical differentiation among subpopulations Theor Appl Genet 71 826 834 Gregorius H R 1987 The relationship between the concepts of genetic diversity and differentiation Theor Appl Genet 74 397 401 Gregorius H R 1988 The meaning of genetic variation within and between subpop
18. GSED Genetic Structures from Electrophoresis Data User s Manual Elizabeth M Gillet Institut f r Forstgenetik und Forstpflanzenz chtung Universitat Gottingen Biisgenweg 2 37077 Gottingen Germany April 1998 GSED User s Manual April 1998 EM Gillet Institut fiir Forstgenetik Univ Gottingen 1994 1997 1998 http www uni forst gwdg de forst fg software htm Contents 1 Introduction 1 2 Constructing an input file 4 IL Header wits xray gether a Gide i A E BS Se oh 4 2 2 Format specification line te ae deuce ae Mo ee era BB Ob BS ek 6 2 3 Input format for each sample e aia ida 22 Jamie 6 23 1 ey Ind A E O E ae BMA 7 2 3 2 Genotypes of single individuals o0 a nen 7 2 3 3 Genotype frequencies in sample 8 2 3 4 End of sample line sra a ara 9 2 4 End okinpit tagia oo Gee sen oe Ge gente Se ele ER Sete ge wt eth ety 9 25 Auxili ry programs AE ra GOA Rigel BO WN da EE A da hE 10 33 1 GSEDINE D oc 4 225 2 2 a at a A DA Gee Be 10 2027 MS GTO Bie gt La eae ar are are e peed 11 3 Running GSED 12 9L First runs 2 ns a ake a o AS ee Ow Some Bed 12 3 2 Configuration AAA Ae ah Sh Gee tee eon Ge Se ee a nse Gb tar 20 3 3 Sorting of haplotypes and genotypes 00 20 4 Understanding GSED output 21 5 Frequency distributions 25 6 Measures of variation 28 6 1 Measures of variation within samples o 28 Gabel A A ra 28 6 1 2 Total popula
19. Hill W G 1984 Deviations from Hardy Weinberg proportions Sampling variances and use in estimation of inbreeding coefficients Genetics 107 703 718 Weber E 1978 Mathematische Grundlagen der Genetik VEB Gustav Fischer Ver lag Jena Weir B S 1990 Genetic Data Analysis Sinauer Associates Inc Publ Sunderland Mass 40 A Technical considerations A 1 Hardware requirements GSED is written in the programming language FORTRAN 77 and can be implemented under any operating system for which a FORTRAN 77 compiler is available To date it has been compiled for the operating systems MS DOS PC DOS Version 2 1 or higher VMS Version 5 0 or higher and TOS Atari A math co processor is not used GSED spends most of its time writing intermediate results to direct access files and retrieving them from these see Sec A 3 Since these files are stored in the default directory VMS TOS or in the root directory of the default drive DOS this process can be speeded up considerably if the executable program is started from a RAM Disk RAM Random Access Memory as the default drive The executable program itself may be read from a different device e g floppy disk as long as the prompt points to the RAM Disk as the default drive Locating the input and output files in the RAM Disk additionally speeds up calculation Moreover operating in a RAM Disk prevents lots of wear and tear on the hard disk drive A 1 MB RAM Disk should be
20. The hypothetical gametic output is defined by the set of gametes that results from stochas tically independent association between loci free recombination and equal gametic pro duction for all members vgam therefore measures the potential of a population for pro ducing genetically diverse gametes Reference 6 8 1 3 Total population differentiation r of the gene pool Let a collection of subpopulations have the total population differentiation rq at locus l 1 L Then the total population differentiation r of the gene pool was proven to equal the arithmetic mean of the total population differentiation at each locus that is 1 bp L ore l 1 E Reference 11 36 8 2 Measures of variation between samples 8 2 1 Distance dy between gene pools Let one collection be characterized by the frequency vectors of the different genes alleles at L gene loci that is by the L frequency vectors p pu Par lt lt Pnu 1 L where n IN is the number of alleles at locus l and py gt 0 and NL pk 1 holds for all k 1 n Let a second collection be characterized by the L frequency vectors p Pii Pa al Pr l 1 L at the same L loci and for the same numbering of alleles at each locus The gene pool genetic distance dy between the two collections was proven to be the arithmetic mean of the single locus distances t e E do gt do pi pi El ll m n Ms tape Esla Ni
21. Y Measures of variation within samples Diversity v_2 Y Total population differentiation delta_T Y Evenness Y finite population size Y infinite population size Y Measures of variation between samples Genetic distance d_0 Y Subpopulation differentiation D_j delta Y subpopulations weighted proportional to sample size Y subpopulations equally weighted Y Test of homogeneity of the sample distributions Y Analysis of genotypic structure Heterozygosity Y Tests of single locus structure Test of Hardy Weinberg structure and heterozygosity Y Test of product structure only if gametic sex is specified 13 su Y Table 7 Continuation of the interactive sequence begun in the previous table Should these choices be stored in a file for later use Y Configuration stored in file EXAMPLE CFG Sample 1 Population 1 Sample 2 Population 2 Sample 3 Population 3 Samples for output 0 all samples 1 some samples Option 0 Output unit S screen F file Option F Output file EXAMPLE OUT Width of output min of 75 characters line as number of samples per line No samples line 1 10 No characters line 15 For example 0 for ALL 3 samples 75 char line 6 for 6 samples 75 chars line as for DIN A4 paper upright 10 for 10 samples 115 chars line as for DIN A4 paper crosswise 11 for 11 samples 125 chars line as in condensed mode Op
22. able 10 Example of output for a test of genotypic structure TEST OF HARDY WEINBERG STRUCTURE GAMETIC SEX DISREGARDED Allele 1 3 Freq 1 10 80 16 00 3 60 30 120 48 00 36 00 200 Level of C V of CHI 2 Test statistics significance DF 1 G 6 438 0 050 3 841 X 2 6 250 0 010 6 635 X 2 c 5 5 486 0 001 10 828 23 the given degrees of freedom DF and Level of significance The symbol n s found to the right of a statistic in other tables means not significant Self explanatory messages are printed on the screen if difficulties of the following types are encountered Files cannot be opened read or closed an erroneous answer is given during the interactive sequence limitations on data are exceeded see Sec A 2 Messages are printed in the output in the following cases A requested frequency distribution measure or test cannot be calculated differences in definition of genetic types between samples prohibit comparison of the samples special situations arise during a test Some messages are followed by cause and a number the latter referring to a compiler specific list of I O status values 24 5 Frequency distributions The input to GSED usually consists of the genotypes observed at one or more gene loci in a sample of diploid individuals It is also possible to input haplotypes observed in a sample of gametophytes of one sex by listing th
23. amples 3 2 1 x No Locus Combinations x Maz No Loci Per Combination Max No Genotypes 1 x No Locus Combinations x No Samples Maz No Genotypes 1 x No Locus Combinations x 2 j Max No Haplotypes 1 No Locus Combinations Maz No Genotypes 1 x No Locus Combinations Maz No Identified x No Chosen Samples x 2 Sec A 1 and are deleted automatically upon successful completion of the program If the program is interrupted in mid run the files may remain and can be deleted by hand Since they are not in ASCII code they cannot be read by a text editor Sufficient storage space must therefore be available not only for the executable program the input file and the output file but also for the direct access files The maximum size of these files can be calculated by the formulae given in Tab 12 which use the variables defined in Tab 11 The total storage space for the eight files as number of 4 byte integers is less than or equal to Maz No Haplotypes 1 No Locus Combinations x x 3 x No Samples Maz No Loci Per Combination 1 Maz No Genotypes 1 x No Locus Combinations x No Samples 3 Maz No Loci Per Combination 2 No Locus Combinations x No Samples 2x Maz No Identified x No Chosen Samples 8 x Size of Enlargement where Maz No Haplotypes is set to Maz No Alleles if all locus combinations refer to single loci Example
24. cy distributions and calculations Input file EXAMPLE DAT Input file EXAMPLE DAT Locus 1 LAP A Locus 2 LAP B Sample 1 Population 1 Locus 3 IDH Sample 2 Population 2 Locus 4 PGI Sample 3 Population 3 Locus configuration 0 all single loci 2 multilocus some loci 1 some single loci 3 multilocus all loci Frequency distributions Y Maternal contributions Y MEASURES OF VARIATION WITHIN SAMPLES Paternal contributions y Diversity Y Allele haplotype frequencies Y Total population differentiation deltaT Y Genotype frequencies YI Evenness ii A finite population size Y infinite population size Y l MEASURES OF VARIATION BETWEEN SAMPLES Genetic distance Y Subpopulation differentiation D_j delta Y weights proportional to sample size Y equal weights Y Test of homogeneity Y da EAE ANALYSIS OF GENOTYPIC STRUCTURE Heterozygosity Y l Test of Hardy Weinberg structure heterozygosity single locus Y Test of product structure single locus Y Do you want to adopt this configuration Answer Y yes or N no Y 17 Option 1 The output contains the results for only those samples given in reply to the following question This option allows measures of variation between samples see Sec 6 2 to be calculated for differing sets of samples As an example samples 1 and 3 are chosen in reply to the fol
25. d equals this number only for a uniform distribution Reference 6 11 6 1 2 Total population differentiation r Let a collection of size N be characterized by a frequency vector p p p2 Pn of its genetic types where n IN and for k 1 n px gt 0 and Pz pr 1 The total population differentiation r of the collection is defined as r 1 E TW 2 it or letting N N pz be the absolute frequency of the kth type N N Nk ee ee EN Nel It holds that 0 lt r lt 1 with r 0 for monomorphy and r 1 if no two sample members are of the same genetic type References 11 12 6 1 3 Evenness e Given a distribution of types of individuals in a collection the evenness of the distri bution is considered to measure the degree to which these types are equally represented 14 28 The evenness e is defined to equal one minus the minimal distance of the frequency dis tribution to all plateaus each consisting of equally frequent types in effectively infinite collections In small collections the plateaus are defined by the respective distributions closest to uniformity If dmin equals this minimal distance the absolute evenness is given by L Amin for the definition of d see genetic distance below e 1 holds only for uniform distributions As e approaches a lower bound of 0 5 the unevenness increases As a transformation of evenness which varies betw
26. distributions OH CaIGUIATIOUSS ei A oh a ha ie ge oe AR Be do Gg h Examples demonstrating designation of alleles haplotypes and genotypes di VVE OUL Us sae ed oo St EN As te Sk oe wh a Os oe Example of output for a test of genotypic structure Interpretation and maximal values of variables used in calculating storage space needed for direct access files 2 2 aa a Maximum size of the direct access files as number of 4 byte integers Mul tiplication of each result by 4 yields total number of bytes Notation 1 If additional storage space is added blockwise add Size of Enlargement 2 Maz No Haplotypes Maz No Alleles if only single loci are considered ill 43 iv 1 Introduction The computer program GSED Genetic Structures from Electrophoresis Data is based on a conceptually and mathematically unified system of data analysis for the characteriza tion of genetic structures in population genetic investigations which has been developed at this institute in recent years Tabs 1 and 2 contain an overview All measures of variation within and between samples and gene pools are based on the single measure of genetic distance do which quantifies the proportion of genetic types by which two collections of individuals or populations differ 4 5 6 8 Heterozygosity goes even deeper by measuring variation at the level of the individual Inference on the manner of association of alleles to
27. e of each endosperm and thus the diploid genotype of each tree 1 Inference of the genotype of a diploid embryo and subtraction of the haplotype of the corresponding endosperm then reveals the haplotype of the paternal gamete for codominant alleles of enzyme loci 20 21 If the gametic sex of the alleles i e the sex of the parent contributing each allele at all involved loci is specified a number of additional frequency distributions can be calculated For a single locus e Allele frequencies among maternal contributions The set of alleles contributed by the maternal parents of the sampled individuals represents a sample of the alleles in the population of successful maternal gametes e Allele frequencies among paternal contributions In like manner the set of alleles contributed by the paternal parents of the sampled individuals represents a sample of the alleles in the population of successful paternal gametes 25 e Ordered genotype frequencies maternal x paternal alleles The set of ordered geno types represents a sample out of the population of successful fusions between fe male and male gametes Ordered genotypes take into account the gametic sex speci fication of the alleles at the locus distinguishing for example between the genotypes 1 x 3 and 3 x 1 see Tab 9 Over a set of loci e Haplotype frequencies among maternal contributions The set of maternal haplo types repr
28. e output of the statistical tests is similar to the example in Tab 10 The upper table in 10 contains the observed frequencies of the genotypes 4 4 A143 and A3A3 and beneath each in square brackets the frequencies expected under the null hypothesis of Hardy Weinberg structure The observed allele frequencies are given to the right of the table The lower table entitled Test statistics contains the re sults of the likelihood ratio test G Pearson s x test X and in tables such as this with one degree of freedom DF 1 the x test with continuity correction of 0 5 X 2 c 5 The symbol here directly to the right of each statistic indicates its level of significance which can be inferred from the two rightmost columns of the table The abbreviation C V of CHI 2 stands for critical value of the x distribution for 22 Table 9 Examples demonstrating designation of alleles haplotypes and genotypes in the output ST Allele allele 1 Single locus genotype alleles 1 and 3 Ordered single locus genotype maternal allele 3 x paternal al lele 1 Haplotype 241 allele 2 at first locus allele 4 at second locus allele 1 at third locus Multilocus genotype 14 23 13 single locus genotypes 1 x 4 at first locus 2 x 3 at second lo cus 1 x 3 at third locus Ordered multilocus genotype 12 40 3 2 maternal haplo type 1 4 3 x paternal hap lotype 2 0 2 T
29. e second allele at each locus as unknown i e 1 and gametic sex as specified see Sec 2 3 From genotype data it is possible to construct the following frequency distributions For a single locus e Allele frequencies Each sampled diploid individual contributes two alleles to the overall sample so that heterozygotes reveal more allelic types than homozygotes The association between alleles in genotypes genotypic structure therefore deter mines the degree to which a sample detects the allelic types in a population see Sec 4 alpha alpha HWP e Genotype frequencies The genotype of each sampled individual is counted without regard to gametic sex specification Over a set of loci e Multilocus genotype frequencies The multilocus genotype of each sampled individual is counted without regard to gametic sex specification Gametic sex specification In some organisms it is possible to determine which allele at a nuclear gene locus was contributed by the maternal parent For example the seed of most coniferous species contains not only the diploid embryo but also nutritive tissue genetically identical to the maternal gametophyte the primary endosperm or megagametophyte If the endosperm of a seed is subjected to isoenzyme electrophoresis the maternal phenotype is revealed Inheritance analysis of the phenotypes of the endosperm produced by single trees allows inference of the haploid genotype haplotyp
30. een 0 and 1 the relative evenness of the population is defined as e 1 2 Amin Reference 14 6 2 Measures of variation between samples 6 2 1 Genetic distance do Let two collections be characterized by frequency vectors p p1 P2 Pn and p p po Pp of their genetic types where n IN and for k 1 n Pk pi gt 0 and Xg Pk 1 X Py The genetic distance d p p is defined as dolp p pr Phl k 1 NI The genetic distance between two collections is specified as the proportion of genetic elements alleles genes at multiple loci gametes genotypes which the two collections do not share Thus d 1 if and only if the two collections have no types in common References 4 5 6 8 29 6 2 2 Subpopulation differentiation D and 6 Let a population be divided into demes subpopulations collections The amount of genetic differentiation of one subpopulation to the remainder of the population is specified as the proportion of genetic elements alleles genes at multiple loci gametes genotypes by which a deme differs from the remainder of the population in type 9 This proportion is defined as D do p Pj where p and p are the frequency distributions of the types in deme j and in the re mainder of the population respectively and do is the genetic distance defined above The subpopulation differentiation is then defined by b dig D where the weights c expr
31. en in the second field In this case each genotype line specifies the following integers sampleno frequency locus allele locus alleley locusnallelez locusn alleles where sample number referring to list of samples in header number of individuals possessing the genotype locus allele designation of first allele at locus as an integer gt 1 if gametic sex is specified for this locus then locus allele stems from the maternal parent locus alleleg i 2 n designation of second allele at locus 7 as an integer gt 1 if gametic sex is specified for this locus then locus alleley stems from the paternal parent A null allele at locus 7 is designated by locus allele 0 zero j 1 or 2 An unknown allele is specified by locus allele 1 j Lor 2 Note that unknown alleles in a random sample of genotypes can present a problem for the calculation of frequency distributions see Sec 5 2 3 4 End of sample line The end of sample line contains only the following integer in the first field sampend 9999 Data for the remaining samples are appended to the previous end of sample line according to the same pattern Since GSED does not know when to expect an end of sample line when encountered these lines are read according to the same format specification see Sec 2 2 as the key and genotype lines Some compilers specify that data be read in all of the fields defined in the
32. esents a sample of the haplotypes in the population of successful ma ternal gametes e Haplotype frequencies among paternal contributions The set of paternal haplotypes represents a sample of the haplotypes in the population of successful paternal gametes e Haplotype frequencies A sample of the haplotypes of successful gametes is con structed by counting both the maternal and the paternal haplotypes of the sampled individuals Since each sampled individual contributes two haplotypes the associ ation between haplotypes in genotypes genotypic structure determines the degree to which a sample detects the haplotypes present in a population see Allele fre quencies above and see Sec 4 alpha alpha HWP e Ordered multilocus genotype frequencies maternal x paternal haplotypes The set of ordered genotypes represents a sample of the genotypes in the population of successful fusions between female and male gametes Ordered multilocus geno type frequencies distinguish between maternal and paternal haplotypes For exam ple whereas the ordered genotype 1 2 1 2 results from fusion of the mater nal haplotype 1 1 and the paternal haplotype 2 2 the ordered genotype 21 1 2 isthe product of maternal haplotype 2 1 and paternal haplotype 1 2 see Tab 9 Obtaining unordered genotypes when gametic sex is specified Note that if gametic sex is specified and the response to the question Sh
33. ess the proportion of genetic elements present in the jth deme References 9 10 12 15 6 2 3 Test of homogeneity Let m collections of individuals each be characterized by a frequency distribution defined by the number of individuals of each of n types in the collection A test of homogeneity of the m frequency distributions tests the hypothesis that these m collections all originated from a single large collection of individuals conditioned on the marginal distributions given by the m sample sizes as proportions of the sum of sample sizes and the mean relative frequencies of the n types over the samples Goodness of fit tests see Sec 7 2 are performed for m 1 n 1 degrees of freedom References 2 pp 365ff 24 pp 96ff 30 7 Analysis of genotypic structure The following measures and tests aid in the characterization of genotypic structures In contrast to other measures quantifying variation within and between samples see Sec 6 8 heterozygosity measures genetic variation within individuals Tests of single locus structure investigate the association of gametes in observed zygotic genotypic structures by comparing the observed structures to the corresponding expected structures under certain models of association 7 1 Heterozygosity 7 1 1 Proportion of heterozygosity of single locus genotypes Given the genotypes of all individuals in a collection at a single gene locus the proportion of heterozygosity e
34. f and only if the Locus configuration comprises only single loci t e if either option 0 or 1 is chosen Since the gene pool measures are formulated as means of the respective single locus measures at all loci contributing to the gene pool the single locus measures must already be available Option 3 Calculations will be performed for multilocus genotypes defined by the genotypes at all of the single loci in EXAMPLE DAT for all four loci Choice of frequency distributions Answer Y yes or N no Choices can be made among the four types of frequency distribution offered by the sub sequent questions and described below see Sec 5 Should gametic sex specification if given be retained Y 15 If no gametic sex is specified at any locus then the answer to this question is meaningless If gametic sex is specified at some or all loci then an answer of N will cause this specification to be ignored at all of them For example in such a case both of the genotypes AA and A24 where the first allele is that contributed by the maternal parent would be counted as the genotype A As Choice of calculations Answer Y yes or N no Frequency distributions Y An answer of Y causes the calculated frequency distributions to be included in the output If the answer is N they will be omitted Measures of variation within samples Measures of variation between samples Analy
35. gene loci e specification of whether each of the genotypes is to be interpreted as that of a single individual see Sec 2 3 2 or of a number of individuals see Sec 2 3 3 e indication of whether gametic sex is specified for the alleles at each locus see Sec 5 The key line specifies the following integers for n gene loci sampleno keyline 0 locdef gamsex locdef3 gamsery locdefn gamsexn where sample number referring to list of samples in header keyline 0 indicator of key line locdefj locus number of first locus in multilocus genotype referring to list A of gene loci in header usually 1 locdef or locdef indicates that each multilocus genotype is that of a single individ Bee ual see Sec 2 3 2 locdef indicates that each multilocus genotype is accompanied by its fre ARA quency in the sample see Sec 2 3 3 gamsez indicator of gametic sex specification of first locus gamsex 1 AR if gametic sex is specified 0 otherwise locus number of th locus in multilocus genotype gamsex i 2 n indicator of gametic sex specification of ith locus gamsex 1 if gametic sex is specified 0 otherwise Examples of key lines are given in Tab 4 2 3 2 Genotypes of single individuals If the sign of locdef in the key line of a sample equals or is blank see Sec 2 3 1 then each genotype is interpreted to be that of a single individual In this
36. he relative frequency of the ordered genotype A A i e A is the maternal contribution and A the paternal so that gt Pi 1 p is the relative frequency of allele A among maternal gametic contributions and p is the relative frequency of allele A among paternal gametic contributions Given a random sample of N individuals from a large population the test of a product structure is performed as a test of independence of association between maternal and pa ternal allelic contributions conditioned on marginal distributions given by the frequencies of these contributions in the sample For absolute frequencies N i j 1 k of the ordered genotypes in the sample the absolute frequency N of the allele A in the sample of N maternal alleles equals N gt N and the frequency N of the allele A in the sample of N paternal alleles equals N Ss Nji Conditioning on the allele frequen cies in the sample i e assuming that the true frequency p and p of allele A among the maternal and paternal gametes produced in the population equals p N N and po N N respectively the genotypic frequencies expected under the null hypothesis of a product structure equal E Nij NFN8 N i j 1 k The number of degrees of freedom equals k 1 kd 1 where k and k are the numbers of alleles with non zero frequency among maternal and paternal contributions respectively Reference 2 pp 360ff 34 8 A
37. hese loci are included at the close of the output see Sec 8 Each locus combination single or multilocus is in turn divided into the output for each of the chosen frequency distributions measures of variation within samples between samples followed by the analysis of genotypic structure heterozygosity tests of single locus structure Results for measures of variation and heterozygosity appear in tables each column con taining the results for one of the chosen samples If the chosen width of output see Sec 3 1 Width of output is not sufficient to allow inclusion of all samples onto one line each table is truncated vertically and continued on the next lines If the current locus combination consists only of a single locus the output for this combination closes with the results of the chosen tests of single locus structure for each sample The output for each frequency distribution begins with a heading which provides the following information about each sample Sample No Number of the sample in accordance with the list of samples in the input file Gam sex spec Abbreviation of Gametic sex specification yes if the sex of the parent contributing each allele is known in the entire sample no otherwise Sample size Total number of individuals whose genotypes are included in the input file regardless of whether they contain unknown alleles or not No identified N
38. i between 1 and 50 separated by a comma or blank e Gene locus names maximum of 12 characters each on consecutive lines e Sample names maximum of 40 characters each on consecutive lines Examples of format specification lines are given in Tab 3 Table 3 Examples of format specification lines The format specification line 10 214 1X 10 212 reads 22 integers including a 10 locus genotype from the following line of data namely 2 190 7222 2123 12 23 35 21 23 21 11 11 22 33 The format specification line 20 I4 14 10 212 9X 10 212 reads 42 integers including a 20 locus genotype from the following two lines of data 2 2345 2 1 2 3 1 1 22345 21 23 24 22 33 11 21 21 23 21 32 33 12 23 23 31 32 23 32 11 2 2 Format specification line e The format specification line contains the number of loci constituting the multilocus genotypes in character posi tions 1 2 and a FORTRAN format specification beginning in character position 5 defining which character positions of the subsequent data lines contain the information required by the program As explained in 2 3 the FORTRAN format specification must specify the character posi tions of 2 2 x n integers in each of the subsequent input records where n is the number of gene loci in the multilocus genotypes For present purposes a general description of the following elements of a FORTRAN format specification will suffice for further
39. iological meaning while in others it is a mere construct for defining storage space The reason is that all genotypes are regarded as the product of haplotypes which themselves are numbered consecutively in the order in which they are encountered in the input This manner of storage is the most space saving since storage space is used only for genetic types that actually have been found and thus need not be reserved for potential types that never appear Haplotypes in this sense are defined as follows 1 If all locus combinations specify single loci Maz No Haplotypes is set to Maz No Alleles as mentioned above Each genotype is stored as the product of its two alleles 2 In the case of locus combinations specifying ordered multilocus genotypes which are composed of two known haplotypes contributed by the parents Maz No Haplotypes is the maximum number of different haplotypes regardless of gametic sex encoun tered for any locus combination and in any sample Each genotype is stored as the product of its two haplotypes 3 In the case of multilocus combinations lacking gametic sex specification haplo types are formed artificially as follows Before a multilocus genotype is stored the alleles at each of the constituent single loci are put in nondecreasing order e g genotype 2 1 becomes 1 2 One haplotype is then formed as the list of the alleles appearing in the first position at each constituent locus the
40. le must be inserted between the respective key line and the end of sample line with the help of any ASCII 10 text editor Positioning of the genotype data over the character positions of a line obeys the example given by the key line 2 e the integers are placed right justified within the fields defined by the key line Examples of completed input files are given in B 2 5 2 GSEDTEST GSEDTEST is a self explanatory program to quickly test new or altered input files for their readability by GSED GSEDTEST reads the lines in an input file as would GSED and types them on the screen or into a file An error message is printed if an unreadable data line is encountered 11 3 Running GSED Execution of GSED begins interactively with a sequence of questions to be answered by the user After specifying a preconstructed input file see Sec 2 the user may choose any or all of the frequency distributions listed in Tab 1 and request calculation of any of the measures and tests listed in Tab 2 Additional questions concern the format of the output After all questions have been answered GSED performs the desired calculations Results can either be written into an output file or typed on the screen A sample interactive sequence is given in Tabs 6 and 7 3 1 First run The first time an input file is read into GSED the sequence of questions listed in Tabs 6 and 7 appears on the screen The meaning of these questions will be described in mo
41. lowing questions How many samples 2 Which samples separated by commas and using as many lines as necessary 1 3 Output unit S screen F file Option Output can be directed to one of two units as follows Option S All results are typed on the screen They are not saved elsewhere and thus are lost as soon as they disappear off the screen Option F Output file EXAMPLE OUT Results are output as ASCII text to the designated file A maximum of 60 characters are allowed for the file name and any necessary specification of path Since the output is in ASCII code it is possible to alter its format later using any text editor The finished file can then be printed on any printer If the output file EXAMPLE OUT already exists the following message appears File already exists A append new output O overwrite old output Option Option A The new output will be added to the end of the existing file thus preserving the previous contents 18 Option O The new output will replace the previous contents of the file which are thus lost Note that this option is indicated by the letter 0 and not the numeric character 0 zero Width of output min of 75 characters line as number of samples per line No samples line 1 10 No characters line 15 For example 0 for ALL 3 samples 75 char line 6 for 6 samples 75 chars line as for DIN A4 paper upright
42. lved Table 1 Frequency distributions calculated by GSED P Alllele frequencies among paternal contributions Allele frequendies OSS Genotype frequencies SSCS 7 aplotype frequencies among paternal contributions Haploiype Frequencies LT Genotype frequencies If gametic sex of the alleles at each locus is specified Table 2 Characterization of genetic structures e ANALYSIS OF ALLELIC HAPLOTYPIC AND GENOTYPIC STRUCTURES Measures of variation within samples Diversity v x Total population differentiation r Evenness e Measures of variation between samples x Genetic distance do Subpopulation differentiation D and 6 x Test of homogeneity e ANALYSIS OF GENOTYPIC STRUCTURE Heterozygosity single locus and multilocus Test of Hardy Weinberg structure and heterozygosity Test of product structure e ANALYSIS OF THE GENE POOL Measures of variation within samples Diversity v of the gene pool Diversity Ugam of the hypothetical gametic output x Total population differentiation r of the gene pool Measures of variation between populations Distance dy between gene pools Differentiation 6 of subdivided gene pools alleles Isoenzyme phenotypes resulting from gene loci possessing a codominant mode of inheritance are gene markers Dominance such as is caused by a recessive null allele at a locus in isoenzyme investigations does not give rise to a gene marker
43. mal values of variables used in calculating storage space needed for direct access files No Chosen Samples Maz No Identified No Locus Combinations Maz No Loci Per Combination Maz No Alleles Maz No Haplotypes Maz No Genotypes Size of Enlargement Interpretation Number of samples in input file Number of chosen samples see Sec 3 1 How many samples Maximum number of individuals identi fied per sample see Sec 4 Number of locus combinations see Sec 3 1 Locus configuration Maximum number of loci per locus com bination 1 if single locus only see Sec 3 1 Locus configuration 40 Maximum number of different alleles at any locus and in any sample Maximum number of different haplotypes that occur for any multilocus combina tion and any sample see Sec A 3 Maximum number of different genotypes that occur for any locus combination and sample Size of storage space added when file is enlarged expressed as the equivalent number of 4 byte integers 42 Table 12 Maximum size of the direct access files as number of 4 byte integers Multi plication of each result by 4 yields total number of bytes Notation 1 If additional storage space is added blockwise add Size of Enlargement 2 Max No Haplotypes Maz No Alleles if only single loci are considered Unit File size as number of 4 byte integers Maz No Haplotypes 1 xNo Locus Combinations x No S
44. n altered to leave out the blank between the alleles at a single locus Lack of gametic sex specification is indicated by the 0 following each locus number in the key line 1 30 Locus 1 Locus 2 Locus 30 Beech forest 30 214 20 12 11 8x 10 i2 1i1 1 0 10 20 30 40 50 60 70 80 90100110120130140150160170180180200 210220230240250260270280290300 1 1 11 22 31 11 21 12 44 34 35 43 31 32 33 22 13 13 22 11 11 22 43 23 22 33 11 22 31 12 11 00 1 2 22 33 33 22 44 11 22 11 31 11 22 31 12 14 34 42 13 23 21 22 35 43 21 12 11 23 21 11 22 12 9999 Example 4 This example shows the input for a sample of successful maternal haplo types such as could be found by sampling the bulk seed of a stand of a conifer species and subjecting only the primary endosperm of each seed to isoenzyme electrophoresis see Sec 5 The specification of gametic sex is indicated by the 1 following each locus number in the key line The unknown paternal contribution at each locus is designated by 1 1 4 Locus 1 Locus 2 Locus 3 Locus 4 Scots pine forest 4 2i4 4 13 12 O 11 21 Prrrr 1 2 1 2 2 1 3 1 1 4 1 1 9999 48
45. nalysis of the gene pool The gene pool of a population with respect to the number Z of non homologous gene loci located at a certain section of the genome is thought of as the set of all gene alleles at these loci realized in all individuals 6 The following types of gene pool can be constructed the first two only if gametic sex is specified at all contributing loci e Gene pool of maternal contributions e Gene pool of paternal contributions e Gene pool of single locus genotypes 8 1 Measures of variation within samples 8 1 1 Diversity v of the gene pool Let a collection be characterized at each of L loci by the frequency vector Pi Pu Pal Pnl for 1 L where n N and for i 1 n py gt 0 and X Pu 1 Denoting by m va gt 1 1 the allelic diversity at the l th locus the gene pool genic diversity v of the collection was proved to equal the harmonic mean of the single locus diversities t e 1 1 L v See z gt va ees Xi Pa Reference 11 8 1 2 Diversity vgam of the hypothetical gametic output Let a collection be characterized at locus l J 1 L by the frequency vector Pi Pu Pas Pri where n N and for i 1 n pa gt 0 and Ef py 1 Denoting by n 1 va gt ri 1 the allelic diversity at the th locus the hypothetical gametic diversity vgam of the collection is defined as 2 vgam vo l 1 35
46. nput file The input to GSED consists of the genotypes of the individuals included in samples taken from one or more populations demes provenances etc GSED reads the genotypes represented in each sample from an external file that was constructed beforehand using any ASCII text editor Each input file for GSED consists of three parts each of which will be explained in detail subsequently e a header defining the names and numbers of gene loci and samples see Sec 2 1 e a format specification line giving both the number of loci per genotype and a FOR TRAN format specification defining the character positions in which data appear in subsequent lines see Sec 2 2 e the single locus or multilocus genotypes observed in each sample Either the geno type is given for each single individual in the sample see Sec 2 3 2 or each genotype found in the sample is accompanied by its frequency i e the number of individuals possessing it see Sec 2 3 3 Two auxiliary programs aid input file construction see Sec 2 5 e GSEDINPT writes header and key lines to a file according to information interac tively supplied by the user see Sec 2 5 1 e GSEDTEST tests finished input files by reading them and typing them on the screen see Sec 2 5 2 Examples of input files are given in B 2 1 Header The header of an input file has the following form e One line containing number of samples maximum of 100 and number of gene loc
47. ould gametic sex specification if given be retained is Y see Sec 3 1 then the ordered genotype frequency distribution will be calculated and all measures will be based on this distribution In order to obtain the unordered distribution and measures calculated for it GSED must be restarted using the same input file but a reply of N must be given to the above question Gene pool If all of the locus combinations that were chosen for calculation were single loci then the gene pool made up of the genes at these loci is automatically constructed This will be the case if option 0 or 1 was given in answer to Locus configuration of the interactive sequence see Sec 3 1 Although the frequency distribution of the gene pool is not explicitly included in the output all of the chosen measures of variation within 26 and between samples are also calculated for the gene pool see Sec 8 and listed at the end of the output Unknown alleles and genotypes Sometimes it is not possible to determine the genotype or if gametic sex is specified one of the parental contributions to an individual at one or more of the investigated loci In this case it is up to the user to make sure that the unknown types represent random samples of the respective types in the population GSED treats unknown alleles denoted 1 in input and haplotypes and genotypes containing them as follows for each frequency distribution
48. quals the proportion of heterozygous individuals in the collection Reference 16 7 1 2 Conditional heterozygosity of single locus genotypes The conditional heterozygosity at a single gene locus takes into account that the pro portion of heterozygosity is conditional on the allele frequencies It results from division of the actual heterozygosity proportion of heterozygosity at a single locus by the corresponding maximum proportion of heterozygosity Hmax obtainable for the underlying allele frequencies where Hmax equals 1 if all allele frequencies are less than or equal to 0 5 and Hmax 2 1 p if the most frequent allele has frequency p greater than 0 5 References 6 16 7 1 3 Degree of heterozygosity of multilocus genotypes The degree of heterozygosity is defined for an individual with respect to a specified number of gene loci and is identical to the proportion of loci at which this individual is heterozygous The average degree of heterozygosity refers to the distribution of this degree in a collection of individuals Hence it can be proven that the average degree of heterozygosity equals the mean proportion of heterozygotes at the single loci Reference 6 31 7 2 Tests of single locus structure The following goodness of fit tests are performed for two models of single locus genotypic structure Pearson s y goodness of fit test with statistic Gay types N E N E N Likelihood ratio test with
49. r the lines beginning with lt must be replaced by the genotypic data for the respective population sample right justified within the fields defined by the key line Note that gametic sex is specified for all loci in Population 1 two loci in Population 2 and no loci in Population 3 PGI Population 1 Population 2 Population 3 4 214 4 1X 212 1 O11 21 31 41 lt Enter sample no individual no and genotype of all sampled individuals lt listing maternal allele at each locus first 9999 2 0 11 21 30 40 lt Enter sample no individual no and genotype of all sampled individuals lt listing maternal allele first if gametic sex is specified 9999 3 0 10 20 30 40 lt Enter sample no frequency and genotype for all sampled genotypes 9999 next empty line To compensate for these differences an arbitrary number of empty lines can appear after the final 9999 for DOS at least one such line is mandatory 2 5 Auxiliary programs 2 5 1 GSEDINPT GSEDINPT is a self explanatory interactive program with the help of which a skeleton input file can be constructed In accordance with the user s answers to questions about the data the following lines are written into the new file header format specification line and for each sample the key line and the end of sample line Tab 5 gives an example of a skeleton file EXAMPLE DAT constructed by GSEDINPT After completion of GSEDINPT the genotype data for each samp
50. re detail in the following In cases where choice of option is specified by a capital letter small letters are also accepted e g y instead of Y Input file Type the name of the input file see Sec 2 including path specification if necessary A maximum of 60 characters is allowed Locus configuration 0 all single loci 2 multilocus some loci 1 some single loci 3 multilocus all loci Option The four options are explained as follows Option O Calculations will be carried out for every single locus Option 1 Calculations will be carried out for some of the single loci As an example the loci 2 and 4 are specified in reply to the following question 12 Table 6 Interactive sequence for first run of GSED for an input file EXAMPLE DAT continued in next table Input file Locus 1 LAP A Locus 2 LAP B Locus 3 IDH Locus 4 PGI Choice of frequency Allele haplotype Allele haplotype Allele haplotype GSED logo EXAMPLE DAT Sample 1 Population 1 Sample 2 Population 2 Sample 3 Population 3 0 all single loci 2 multilocus some loci 1 some single loci 3 multilocus all loci distributions Answer Y yes or N no frequencies among maternal contributions Y frequencies among paternal contributions Y frequencies Y Genotype frequencies Y Choice of calculations Answer Y yes or N no Frequency distributions
51. res result from special mating systems such as are specified e g in 13 pp 20ff 68ff and 17 pp 175ff The purpose here is to detect deviations of 1 an actual genotypic structure P from its corresponding Hardy Weinberg structure P and 2 actual heterozygosity from the 32 corresponding Hardy Weinberg heterozygosity Actual heterozygosity is defined by Phet 1 Y Pu and its corresponding Hardy Weinberg heterozygosity by Pi 1 Pe Assume that a sample of N individuals was randomly drawn from a large population and consider their genotypes at a locus with k alleles Gametic sex if specified is disregarded i e genotypes A A and A A are not distinguished For unordered absolute genotype frequencies Ni i j 1 k Nig Nj Dic Ni N in the sample the absolute frequency N of allele A in the sample of 2N alleles equals N 2N 2 4 Nij Conditioning on the allele frequencies in the sample i e assuming that the true frequency pi of allele A in the population equals p N 2N the genotypic frequencies expected under the null hypothesis of Hardy Weinberg structure equal E Na N 4N and E N NiN 2N i j 1 k The N and E N for i lt j are the observed and expected sample frequencies respec tively entering the test statistics described above The number of degrees of freedom equals k k 1 2 The observed numbers of homozygotes and heterozygotes in a sample of N individual
52. s from a large population equal gt gt Ni and Nij respectively The numbers of homozy gotes and heterozygotes expected under the assumption of a Hardy Weinberg structure equal i i i lt j i respectively Again the expected frequencies are conditioned on the allele frequencies in the sample One degree of freedom remains By definition a genotypic structure shows an excess of homozygotes heterozygotes if its proportion of homozygotes heterozygotes exceeds the proportion of homozygotes het erozygotes in the corresponding Hardy Weinberg structure If the genotypic structure is a Hardy Weinberg structure then such an excess will not be statistically significant if the test for Hardy Weinberg structure is not significant an excess still may or may not be significant Tests for homozygote excess frequently form the first step in an analysis of so called inbreeding structures Detailed tests for realization of inbreeding structures require consideration of various cases as specified e g in 23 References 2 13 pp 20ff 68 f 17 pp 175ff 18 22 23 and references therein 25 pp 71ff 7 2 2 Test of product structure for ordered genotypes In a large population random fusion of gametes from the set of maternal and the set of paternal gametes gives rise to a zygotic genotypic structure at a locus with k alleles that fulfills the properties of a product structure Py p p ij 1 k 33 where P is t
53. s follow Sample calculations of storage space requirements for direct access files single locus genotypes An input file consists of 10 locus unordered genotypes of 100 individuals in each of 10 samples A maximum of 4 different alleles appears at each locus so that at 43 most 10 genotypes can be constructed at each locus The values of the variables are shown in the following list The total storage space required for all eight direct access files equals 5330 integers For 4 byte integers the total storage space is 21328 bytes lt 22 kilobytes Maz No Haplotypes 4 Maz No Genotypes 10 No Locus Combinations 10 No Samples 10 Maz No Loci Per Combination 1 Maz No Identified 100 No Chosen Samples 10 Size of Enlargement 0 Sample calculations of storage space requirements for direct access files multilocus genotypes For the data described in the previous example the multilocus genotypes defined by all 10 loci are considered If all individuals in any given sample have different 10 locus genotypes then the values of the variables are as shown in the following list and 7574 integers 30296 bytes lt 31 kilobytes storage space are needed Maz No Haplotypes 100 Maz No Genotypes 100 No Locus Combinations 1 No Samples 10 Maz No Loci Per Combination 10 Maz No Identified 100 No Chosen Samples 10 Size of Enlargement 0 A note on Max No Haplotypes In some cases this variable has a b
54. second hap lotype is formed accordingly Maz No Haplotypes is thus the maximal number of 44 such haplotypes for any locus combination and in any sample Finally if a mix ture of single locus and ordered and unordered multilocus combinations is chosen Maz No Haplotypes is the maximum over all of these values 45 B Examples of input files Example 1 The file EXAMPLE DAT was already introduced see Sec 2 3 and espe cially Tab 5 The first sample Population 1 consists of the 4 locus genotypes of 5 individuals des ignated 1 3 4 5 and 6 gametic sex is specified at all loci Population 2 consists of the 4 locus genotypes of the 5 individuals 1 5 gametic sex is specified at loci 1 and 2 but not loci 3 and 4 In Population 3 nine different genotypes were found in a sample of 100 individuals the frequencies of the different genotypes in the sample equalling 32 19 gametic sex is not specified at any locus This constellation of gametic sex specification may not be very realistic but it demonstrates the form of data input and in particular the meaning of the key line see Sec 2 3 1 3 4 LAP A LAP B IDH PGI Population 1 Population 2 Population 3 4 2i4 4 1x 2i2 1 011 21 31 41 1 113 12 33 23 1 3 31 11 33 22 1 4 31 11 33 22 1 5 11 11 33 22 1 6 33 21 13 33 9999 2 0 11 21 30 40 2 1 33 12 11 23 2 2 13 21 13 23 2 3 31 11 33 23 2 4
55. sis of genotypic structure If selected the measures and tests offered in the subsequent questions and described in Sec 6 7 8 are calculated for each of the chosen frequency distributions Should these choices be stored in a file for later use Y If the answer is Y the above choices of frequency distributions measures and tests are saved in a configuration file The configuration file has the same filename as the input file and the extension CFG All subsequent runs using the same input file will first print the stored configuration table and then ask whether it should be adopted If the answer is N new choices can be made An example for the case in which a configuration file already exists is given in Tab 8 The sequence of questions continues as follows Samples for output 0 all samples 1 some samples Option The two options are explained as follows Option 0 The output contains the results for all of the samples in the input file Measures of variation between samples see 6 2 are calculated using ALL of the samples 16 Table 8 Start of the interactive sequence for a subsequent run using the input file EX AMPLE DAT The choices made during the first run were stored in the configuration file EXAMPLE CFG shown below This configuration can be adopted by replying with a Y in which case the interactive sequence will be skipped A reply of N allows a new choice of frequen
56. tion 0 Reading input file EXAMPLE DAT sample 1 Population 1 7 records found sample 2 Population 2 7 records found sample 3 Population 3 11 records found Reading of input file completed Sorting haplotypes Sorting genotypes Calculating and outputting results for locus No 1 LAP A for locus No 2 LAP B for locus No 3 IDH for locus No 4 PGI Additional calculations using the same input file and locus configuration Option N FORTRAN STOP 14 Number of different single loci 2 Which gene loci separated by commas and using as many lines as necessary 2 4 Option 2 Calculations will be carried out for multilocus genotypes defined by the genotypes at different sets of gene loci the so called multilocus combinations As an example one multilocus combination comprising the gene loci 1 and 2 and a second comprising only the single locus 1 are specified in reply to the following questions as in the second case a multi locus combination can refer to the genotypes at a single locus Number of different multilocus combinations 2 Combination 1 How many gene loci 2 Which loci separated by commas and using as many lines as necessary 1 2 Combination 2 How many gene loci 1 Which loci separated by commas and using as many lines as necessary It is important to note here that measures characterizing the gene pool see Sec 8 are calculated i
57. tion differentiation dr 2 004 28 O Lo Evenness dls e REA te DA a ee es 28 6 2 Measures of variation between samples 020 29 6 2 1 Genetic distance d atk Soh IA EEE I 29 6 2 2 Subpopulation differentiation D and Gn whee ae w dae aH ew 023 Test of homogeneity ej gh eae ui a Bee Be Analysis of genotypic structure 74 Heterozygosity ata TN okt Bone tio Se ee es ee ee 7 1 1 Proportion of heterozygosity of single locus genotypes 7 1 2 Conditional heterozygosity of single locus genotypes 7 1 3 Degree of heterozygosity of multilocus genotypes 7 2 Tests of single locus structure 255 452 ll alae A a ed 7 2 1 Test of Hardy Weinberg structure and heterozygosity 7 2 2 Test of product structure for ordered genotypes Analysis of the gene pool 8 1 Measures of variation within samples 8 1 1 Diversity v of the gene pool os pa ra ie ale 8 1 2 Diversity vgam Of the hypothetical gametic output 8 1 3 Total population differentiation r of the gene pool 8 2 Measures of variation between samples nn nn 8 2 1 Distance dy between gene pools nen 8 2 2 Differentiation of subdivided gene pools 2 222222 Acknowledgements and disclaimer Technical considerations A 1 Hardware requirements sn er ehe As Limitations on data sur ae er er ee bee OG a A 3 Temporary direct access storage 2
58. uable instruction over the years on the meaning of genetic variation and mating systems in general and on the implemented measures and tests in particular as well as for concrete suggestions for improvement of the output Finally I am grateful to Hans Rolf Gregorius Hans H Hattemer Bernhard Hosius and Martin Ziehe for helpful suggestions for improvement of this manual Generous financial support without which this project could not have been completed so soon was provided by two institutions the Bundesforschungsanstalt fir Forst und Holzwirtschaft BFH in Hamburg and the Hessische Forstliche Versuchsanstalt in Hann Miinden Florian Scholz and Bernd Degen of the Institut fiir Forstgenetik BFH Gro hansdorf and Alwin Jan en of the Hess Forstl Versuchsanstalt were instrumen tal in obtaining this support and I thank them for their very considerable efforts I have tried my best to find all programming errors Nevertheless the user is advised to check the correctness of the results as I can assume no liability for any errors I ve missed I would be very grateful for news of any real mistakes that remain in the program 38 References 1 Bergmann F 1971 Genetische Untersuchungen bei Picea abies mit Hilfe der 10 11 12 13 14 15 Isoenzym Identifizierung II M glichkeiten f r genetische Zertifizierung von Forst saatgut Allgemeine Forst und Jagdzeitung 142 278 280 Elandt Johnson R C
59. umber of individuals whose genetic types with respect to the cur rent frequency distribution are completely identified no unknown alleles see Sec 5 Relative frequencies refer to this number alpha HWP see below No unknown Number of individuals whose genetic types with respect to the cur rent frequency distribution are unknown and thus are not counted see Sec 5 Sample size equals the sum of No identified and No unknown p q alpha In loose terms alpha tells how frequent a type allele haplotype genotype must be in the base population in order for it to have a probability of 0 95 or greater of being represented in a sample of the given size No identified More precisely the 21 probability of having sampled and identified all types occurring with relative frequency greater than or equal to alpha is 0 95 or greater Obviously the larger the sample size the smaller alpha becomes see 7 for derivation of alpha Alleles and multilocus haplotypes occur in pairs in the form of genotypes If the only way to sample haplotypes is by sampling genotypes it must be remembered that the manner of association between the different haplotypes making up the genotypes homozygosity heterozygosity has a great influence on the probability of finding the rarer haplotypes alpha describes the worst case situation for finding the rarer haplotypes namely pure

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