GENETBC CQNSEQUENCES CF SELEC'FEON, LER’KAGér Mifi QGMINANQ [N gMALL SEMULA'FEE PQFULATEONS That‘s for fire Dogma of DL D. MICHIGAN STATE UNEVERSITY Curtis C. Miner 1968 This is to certifg that the thesis entitled Genetic Consequences of Selection, Linkage, and Dominance in Small Simulated Populations presented by Curtis . C . Miller has been accepted towards fulfillment of the requirements for Ph. D. degree in Animal B_reeding '7 ..l/ ‘ /? (:7 A“ V4» or. gag/7 ""/"/ Major professor 0-169 Univcrgit ; LIBRAR" i. Michigan. Stair ABSTRACT GENETIC CONSEQUENCES OF SELECTION, LINKAGE, AND DOMINANCE IN SMALL SIMULATED POPULATIONS by Curtis C. Miller Use of Monte Carlo techniques allows one to partially bridge the gap between mathematical theory, restricted by unrealistic assumptions, and the genetic complexity of real populations. The chief goal of this study was to evaluate the influence of interactions among selection, linkage, and dominance upon genetic mean and variance and parameters of inbreeding and relationship in simulated populations. Parent populations of eight individuals--four of each sex--were simulated, each with 50 loci on a single autosome. Each gene was assumed to have equal effect on a single trait. Linkage was simulated by varying the probabil- ity of crossover (0.02, 0.26, or 0.5) between adjacent loci in the process of forming gametes. Models for partial, com- plete, and overdominance were established to assign geno- typic values. In each of 20 generations, 48 offspring were produced and evaluated for degree of inbreeding and full sib and half sib covariances, as well as the population mean, variance, and gene frequency. Selection of parents was made Curtis C. Miller at random or based on highest one—sixth of the genotypic or phenotypic values in a given generation. Tight linkage associated with effective selection produced lower values for inbreeding, gene frequency, and genotypic mean, and higher values for genotypic variance and covariances between full and half sibs, in a given genera- tion than were encountered under other conditions. GENETIC CONSEQUENCES OF SELECTION, LINKAGE, AND DOMINANCE IN SMALL SIMULATED POPULATIONS BY Curtis C. Miller A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Dairy 1968 ACKNOWLEDGMENTS I wish to eXpress my gratitude to Dr. John L. Gill for his counsel and guidance, both in and out of the class- room, during the period of my graduate study, and for his patient guidance and encouragement during the preparation of this thesis. The valuable assistance provided by Dr. Clinton E. Meadows during the period of my graduate study is also greatly appreciated. I am also greatly appreciative of the encouragement and valuable suggestions offered throughout my graduate career by Dr. Lon D. McGilliard. ii TABLE OF CONTENTS INTRODUCTION . . . . . . . . . . . . . . . REVIEW OF LITERATURE . . . . . . . . . . . Parameters . . . . . . . . . . . . . . Parameters of Relationship . . . . . . A Special Note on Degree of Linkage . METHODS AND PROCEDURE . . . . . . . . . . Experimental Design and Structure of the Initial Populations . . . . . . Mechanism of Simulation and Evaluation Evaluation of Population Parameters . Evaluation of Sibship Parameters . . . Selection of Parents . . . . . . . . . RESULTS AND DISCUSSION . . . . . . . . . . Population Parameters . . . . . . . . Coefficient of inbreeding . . . . Gene frequency . . . . . . . . . . Genotypic mean . . . . . . . . . . Genotypic variance . . . . . . . . Parameters of Relationship . . . . . . Relationship term rx -—full sibs . Relationship term rxy-—half sibs . Relationship term rxy--nonsibs . . Relationship terms Sxy and Syx . . Relationship term txy . . . . , . Relationship term uxy . . . . r . Relationship term ny . . . . . . Genotypic Covariance . . . . . . . . . APPLICATIONS OF RESULTS AND SUGGESTIONS FOR FIIRTI-ER RESEARCH O C O O O O O C C C O . SUWRY O O O O 0 0 O 0 0 O O 0 O O O 0 O - LITERATURE CITED . . . . . . . . . . . . . iii Page 18 23 26 26 29 31 32 38 42 42 42 60 71 78 9O 90 95 96 100 100 106 112 119 139 143 148 LIST OF TABLES Table Page 1. Average recombination rates (E) for linkage parameters of some previous Monte Carlo studies and the present study . . . . . . . . . 24 2. Mean coefficients of inbreeding with probabil— ity of recombination between adjacent loci of 0.50 . . . . . . . . . . . . . . . . . . . . 43 3. Mean coefficients of inbreeding with probabil- ity of recombination between adjacent loci Of 0026 O O O O O O O O O O O O O O O O O O 0 O 43 4. Mean coefficients of inbreeding with probabil- ity of recombination between adjacent loci Of 0002 O O O C O O O O O O O O O O O O O I O O 44 5. Mean coefficients of inbreeding for 27 treat- ment combinations and various combinations of the two and three factors--18th generation . . . . . . . . . . . . . . . . . . 45 6. Analysis of variance--mean coefficient of inbreeding--18th generation . . . . . . . . . . 47 7. .Mean frequencies of genes favored within inbred lines where recombination rates between adjacent loci were 0.50 . . . . . . . . 62 8. Mean frequencies of genes favored within inbred lines where recombination rates between adjacent loci were 0.26 . . . . . . . . 63 9. Mean frequencies of genes favored within inbred lines where recombination rates between adjacent loci were 0.20 . . . . . . . . 63 10. Mean frequencies of favored genes within inbred lines for 27 treatment combinations and combinations of the two and three fac- tors—-18th generation . . . . . . . . . . . . . 64 iv Table 11. 12. 13. 14. 15. l6. 17. 18. 19. 20. 21. 22. 23. Mean genotypic values for inbred lines where the average recombination rates between adjacent loci were 0.50 . . . . . Mean genotypic values for inbred lines where the average recombination rates between adjacent loci were 0.26 . . . . . Mean genotypic values for inbred lines where the average recombination rates between adjacent loci were 0.02 . . . . . Mean genotypic values for inbred lines for 27 treatment combinations and combinations of the two and three factors--18th generation . . . . . . . . . . . . . . . . Ratios of expected to observed genotypic variances within lines for complete dominance O C C O O O O O O O O O O O O O Genotypic variance within inbred lines when recombination rates between adjacent lOCi were 0.50 O O O O O O O O O O O O O O Genotypic variance within inbred lines when recombination rates between adjacent lOCi were 0.26 O O O O O O O O C O C C O O Genotypic variance within inbred lines when recombination rates between adjacent lOCi were 0.02 O O O O O O O O O O O O 0 0 Ratio of expected to observed genotypic variances within lines when selection was random . . . . . . . . . . . . . . . . . . Means of rx when recombination rates between adjacent loci were 0.50 . . . . . Means of rx when recombination rates between adjacent loci were 0.26 . . . . . Means of rx when recombination rates between adjgcent loci were 0.02 . . . . . Means of sxy where recombination rates between adjacent loci were 0.50 . . . . . Page 73 74 74 75 80 81 81 82 88 91 92 93 103 Table 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. Means of 3x where recombination rates between adjacent loci were 0.26 . . . . . . Means of sx where recombination rates between adjacent loci were 0.02 . . . . . . Means of tX where recombination rates between adjacent loci were 0.50 . . . . . . Means of tX where recombination rates between adjzcent loci were 0.26 . . . . . . Means of txy where recombination rates between adjacent loci were 0.02 . . . . . . Means of uX where recombination rates between adjacent loci were 0.50 . . . . . . Means of 11x where recombination rates between adjacent loci were 0.26 . . . . . . Means of uxy where recombination rates between adjacent loci were 0.02 . . . . . . Means of vx where recombination rates between adjacent loci were 0.50 . . . . . . Means of vx where recombination rates between adjacent loci were 0.26 . . . . . . Means of vX where recombination rates between adjacent loci were 0.02 . . . . . . Genotypic covariances between relatives where recombination rates between adjacent loci were 0.50 . . . . . . . . . . . . . . . Genotypic variances between relatives where recombination rates between adjacen loci were 0.02 . . . . . . . . . . . . . . . . . Ratio of expected to observed genotypic covariances between full, half, and non- sibs where selection was random . . . . . . vi Page 104 105 109 110 111 116 m 118 123 124 125 128 129 136 Figure 1. LIST OF FIGURES Mean inbreeding for partial, complete, and overdominance where recombination rates were free and selection was on genotype . . . . . . Mean inbreeding for three levels of recom— bination between adjacent loci where gene action was partial dominance and selection Wascnxgenotype . . . . . . . . . . . . . . . Mean inbreeding for three levels of recom- bination between adjacent loci where gene action was overdominance and selection was on genotype . . . . . . . . . . . . . . . . . . Mean inbreeding for three modes of selection where gene action was complete dominance and recombination rates were free . . . . . . . . Mean gene frequencies for partial, complete, and overdominance where recombination rates were free and selected was on genotype . . . Mean gene frequencies for three levels of recombination between adjacent loci where gene action was partial dominance and selection was on genotype . . . . . . . . . . Mean gene frequencies for three levels of recombination between adjacent loci where gene action was overdominance and selection was on genotype . . . . . . . . . . . . . . . .Mean gene frequencies over all linkage intensities for three modes of selection when gene action was complete dominance . . . Total and within line genotypic variance for three levels of dominance where recombination rates were free and selection was on genotype vii Page 54 55 56 57 67 68 69 70 83 Figure 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. Total and within line genotypic variance for three levels of dominance where recom- bination rates were free and selection was random . . . . . . . . . . . . . . . . . . . Within line genotypic variance for three intensities of linkage for overdominance with genotypic selection . . . . . . . . . . Total and within line genotypic variance for three modes of selection where gene action was complete dominance and recom— bination rates were free . . . . . . . . . . Means of rX for full, half, and nonsibs for random and genotypic selection when dominance was partial and recombination was free . . . . . . . . . . . . . . . . . . Means of rX for full and nonsibs for partial, and overdominance when selection was genotypic and recombination was free . . Means of sX for full, half, and nonsibs for random and genotypic selection when dominance was partial and recombination was free . . . . . . . . . . . . . . . . . . Means of sX for full sibs for partial, complete, and overdominance at low and free combination rates when selection was genotypic . . . . . . . . . . . . . . . . . . Means of tX for full, half, and nonsibs for random and genotypic selection when dominance was partial and recombination was free . . . . . . . . . . . . . . . . . . Means of tX for full sibs for complete and overdoanance at low and free recom- bination rates when selection was genotypic . .Means of ux for full, half, and nonsibs for random and genotypic selection when dominance was partial and recombination was free . . . Means of 11X for full sibs for complete and overdominance at low and free recombination rates when selection was genotypic . . . . . viii Page 84 85 86 97 99 101 102 107 108 113 114 Figure Page 21. Means of 11x for nonsibs for complete and overdominange at low and free recombination rates when selection was genotypic . . . . . . 115 22. Means of vx for full, half, and nonsibs for random and genotypic selection when dominance was partial and recombination was free . . . . 120 23. Means of vx for full sibs for complete and overdominange at low and free recombination rates when selection was genotypic . . . . . . 121 24. Means of vx for full sibs, for complete and overdominanze at low and free recombination rates when selection was genotypic . . . . . . 122 25. Means of genotypic covariance between full, half, and nonsibs where gene action was partial dominance, recombination rates were ~ free and selection was random . . . . . . . . . 131 26. Means of genotypic covariance between full sibs for three modes of selection where gene action was partial dominance and linkage was tight . . . . . . . . . . . . . . . . . . . . . 132 27. Means of genotypic covariance between full sibs where gene action was partial and com- plete dominance, recombination rates were free and tight, and selection was on genotype . 133 ix INTRODUCTION Maximal genetic improvement of economic traits in livestock depends on a knowledge of means, variances, and covariances. Genetic variation is necessary for the genetic improvement of the trait by selection while genotypic covar- iances between individuals for the trait are required in the formulation of equations to predict genotypes from pheno— types for selection. And it is the effectiveness of the selection by the particular prediction equation that will determine genetic progress in the population. Each of the above parameters is related to more basic parameters descriptive of the Mendelian mechanism of inheritance. Thus, to understand fully the effects of var— ious factors and their interactions on measurable parameters such as mean, variance, and covariance, a knowledge of their effects on underlying parameters such as gene frequency, amounts of fixation, and inbreeding is necessary. Mathematical theory has described each observable parameter in terms of the more basic parameters. But only equations for relatively simple models have evolved and even these are restricted by unrealistic assumptions. Some of these restrictions include infinite population size, equal numbers of parents of each sex, no linkage, and no epistasis. Some equations have been developed to allow for the effects of single ones of the restrictions mentioned, but virtually no work has been done to account for the joint effects of some of these factors. Many of the controversies that have developed in population genetics seem to have their basis in that the various factors have been considered singly. An alternative to theory is laboratory organisms. However, to the extent that laboratory animals and the species of interest differ, so may responses to a particular treatment. During the last decade the introduction and rapid development of Monte Carlo methods has provided a tool for study of population phenomena in more detail than has been possible with either laboratory organisms or mathematical theory. Basicallythis technique allows creation of popula- tions with high speed computers and members of the popula- tion to perform according to the probability associated with the particular system. Thus, one can simulate particular genetic systems and observe their forces directly on Mendel- ian parameters. Intelligent use of Monte Carlo techniques allows one to bridge the gap between the simple theory and the more complex models and provides insight into the conse- quences of the various effects and their interactions on the observable parameters. That the simulation process is based on current theory and complex genetic situations must be evaluated in light of existing theory. ,Furthermore, most Monte Carlo studies have restrictions directly comparing results from simulated populations with those from particular species impractical. The simulations do, however, allow some in— sight into the results of more complex models than we could otherwise evaluate with either theory or laboratory animals. This study was designed to use the Monte Carlo technique to ascertain the existence and magnitude of the effects of linkage, dominance, and mode of selection on parameters such as gene frequency, genotypic mean, geno- typic variance, and genotypic covariance of relatives through the simulation of small populations. REVIEW OF LITERATURE Genetic parameters such as genotypic means, geno- typic variances, and inbreeding coefficients are of value in describing or comparing populations. Of equal importance, especially in prediction of changes in parameters, are rela- tionships between individuals that constitute populations. Because of the wide scope of information about these param- eters and their use in quantitative genetics, the literature is quite diverse in origin and nature and understandably com- prises a massive volume of research. The impracticality of undertaking a comprehensive review of the entire field of quantitative genetics suggests that a brief review of the literature pertinent to this study would be preferable to a lengthy and perhaps superfi— cial treatment of all available information on the topics. Therefore, this review concerns studies in which simulated populations were used to evaluate effects of various factors and their interactions on population parameters. It also touches on the development of the theory of relationship between individuals in a population. Parameters Simple quantitative genetic problems have been math- ematically described for many years. Various parameters may be estimated by well—known formulae, but the validity of these equations is unknown because of limiting assumptions made in deriving them. Recently new techniques have been developed which allow one to simulate processes of genetics by repetitive sequences in a high-speed computer. This process (Monte Carlo Method) allows one to study more com— plex genetic systems than would otherwise be convenient. Fraser (1957a) applied the Monte Carlo procedure to simulation of genetic systems. He discussed binary repre- sentation of genotypes and use of logical arithmetic to identify the genetic nature of an individual at each locus.‘ In a concurrent paper, Fraser (1957b) described the investigation of the effects of linkage and selection on the rate of genetic progress. He used six loci on a single chromosome, with varying degrees of recombination, two selec— tion intensities, and two population sizes, and found that linkage had no effect on genetic progress unless recombina- tion levels were smaller than 0.025 between bordering loci. Tight linkage had an exaggerated effect in the smaller popu— lation under intense selection and tended to restrain rate of progress in any population in which selection was intense. Barker (1958a, 1958b) continued the series of papers initiated by Fraser, but he attempted to simulate selection in a particular type of population. Simulation could be used to mimic actual populations, but validity of the simu- lated data depended on accurate estimates of parameters. His second paper included investigations with sex-linked loci in the model. Fraser (1960a) continued discussion of Monte Carlo methods, reemphasizing procedures and including results of selection where dominance and epistasis as well as linkage were included in the genetic system. Two later papers in the series (Fraser 1960b, 1960c) described the effects of epistatic combinations and differential reproductive rates. Martin and Cockerham (1960) applied Monte Carlo techniques in a study designed primarily to investigate effects of linkage on genetic progress in a small population. For the additive model with small populations, adjacent link— age of six loci had little or no effect on the genotypic mean regardless of the intensity of selection. Effects of linkage were absent for the dominance model, and selection had little effect on the mean after the early generations. When they expanded their model to include twenty loci, they found, for the additive model, that less intensely selected populations increased in mean value more rapidly at first, but were inferior to less intensely selected popula- tions by the 28th generation. Tight linkage slowed progress in all populations. Their dominance model with twenty loci showed the same effects, but less intensely selected popula- tions changed faster than in the additive model for both tight linkage and free recombination. Environmental vari- ances tended to slow progress and reduce some effects of selection, but effects of linkage were the same, qualitively, as in populations without environmental variance. Baker and Comstock (1961) used a model of complete dominance with 35 loci and recombination rates of 0.50 or 0.01 between adjacent loci. Contrary to previous authors, they did not find the genotypic mean depressed by tight link- age, in the free recombination model at the same levels of environmental variance and selection. A possible explana- tion for the differences in results may be that different values of parameters were used, particularly size of popula- tion. Qureshi (1963) considered effects of linkage, size of population, intensity of selection, and environmental variance on changes in genotypic mean for an additive model. All combinations of three levels of each of the four factors were applied to simulated populations continued for thirty generations. He also reported changes in genotypic variance and number of fixed loci. Initial responses to selection agreed closely with predicted values when free recombination was allowed. (All four factors were important in changing the genotypic mean and most interactions between two and three factors were important, particularly in latter genera— tions. Moderate linkage appeared to limit genetic advance, especially in smaller populations. In a later paper, Qureshi (1964) considered two other models, complete dominance of the desired gene and a specific case of overdominance where the two homozygotes were assigned equal merit. For the dominance model, tight linkage caused a negative response to selection in the smaller populations, primarily due to fixation of the undesirable genes. Intense selection produced a positive response in small populations with moderate linkage. Selection increased the genotypic mean of large populations regardless of the degree of linkage. Fixation of undesir- able genes was attributed almost entirely to random drift of gene frequency caused by small size of population and to the hindrance to selection caused by linkage. For the overdominance model, the mean response was dependent primarily upon degree of linkage and size of popu- lation although the rate of response was affected by inten- sity of selection. Gene frequency reached equilibrium in 30 generations only when populations were large, intensity of selection low, and recombination free or only moderately restricted. -Again, strong interaction between size of population and degree of linkage affected mean response to selection and the proportion of fixed loci. Gill (1965a, 1965b, 1965c) dealt with the effects of size of population, degree of selection, environmental vari- ation, and linkage on the response to selection. Additive, completely dominant, and overdominant gene action, plus six epistatic models were included. A one—sixteenth fractional replication of a 44 factorial plan was used to select combinations of the various factors to be simulated. Because of the confounding of interactions among themselves and with some main effects, caution is necessary in inter- preting the results. For example, size of population and amount of environmental variation were confounded with in- teractions of intensity of selection with size of population and with level of environmental variation, respectively, and these two interactions were suggested to be potentially important. The first paper (Gill, 1965a) evaluated changes in genotypic mean produced by limiting the size of population.- In general, size of population was an important factor only for models in which a substantial part of the total geno— typic variance was due to dominance. For the levels of selection simulated, Gill concluded that the critical size of a population, with respect to the random extinction of desired alleles, was between 16 and 32 individuals for a complete dominance model, while a population of thirty or more was required when overdominance existed and selection was intense. Although he used a finite population, Gill (1965b) applied a prediction equation formulated by Griffing (1960) to estimate genotypic progress. Since this equation was based on an assumption of an infinitely large population, Gill's results did not conform well with the hypothetical values. But he concluded that the magnitude of the 10 discrepancies possibly was larger than would be expected in a practical situation because of restrictions in simulation. The third paper by Gill (1965c) dealt with intensity of selection, degree of linkage, level of environmental variation, and mode of gene action on the response to selec- tion in populations of restricted size. In populations with complete dominance or complementary factors, little or no fixation of recessive alleles occurred at any level of selection, suggesting that the effect of random fixation on total response to selection should be small with intense selection. The amount and kind of fixation of genes in popula- tions having optimum epistatic or overdominant gene action appeared relatively unrelated to intensity of selection, being almost entirely dependent on size of population. Selection was most effective in populations where the geno- type of highest merit was homozygous, even in small popula- tions where random drift could be eXpected to cause fixation of some undesirable recessive genes. (Selection was most in- effective in small populations when heterozygous individuals had optimum genotypic value. Different levels of linkage had little effect, even in small populations, on genetic merit, gene frequency, or fixation, except for the first few generations of selection. Bias in the estimation of components of genetic variance in populations selected for intermediates seemed to result from ll linkage disequilibrium as well as inbreeding. Selection for intermediates probably maintained this disequilibrium. Under conditions of complete dominance the bias was caused by inbreeding rather than linkage disequilibrium. Gill and Clemmer (1966) considered the effects of linkage and selection on the coefficient of inbreeding. All combinations of three intensities of selection and three degrees of linkage were simulated in parent populations of four or eight individuals. Each of the eighteen populations was replicated twenty times, and each replicate was allowed to proceed for twelve generations. Theoretical coefficients of inbreeding were calculated for unselected populations and for those in which one—fourth of the individuals were select- ed each generation. For each generation the main effects on the average coefficient of inbreeding of linkage and selec- tion were highly significant, with each of 360 populations exhibiting more rapid increase in the mean level of inbreed- ing where either selection, linkage, or both were included in the model. Interactions between selection and linkage were not significant. When population size was adjusted for selection to an "effective number," the authors found their data in close agreement with theoretical inbreeding coeffi- cients for selected populations without linkage. When link- age was included in the model and the data averaged over the three levels of selection, the empirical and theoretical results diverged markedly during the first three generations. 12 Even when selection was excluded, the coefficient of in- breeding in populations with tight or moderate linkage diverged rapidly from the theoretical values. They sug- gested that this divergence may have resulted from an excess of heterozygous loci linked in coupling (rather than repul- sion) phase in the initial individuals of the population, since linkage equilibrium was not a condition of the model. After this disequilibrium dissipated, as it should in un- selected populations, the coefficient of inbreeding in- creased at a slower rate than during the first three gener- ations. Young (1966) considered large populations (1000 indi- viduals) to evaluate the effects of intensity of selection, degree of linkage, and level of environmental variance on genetic advance. His study included selecting the best 80, 50, or 10 percent of individuals in the population,(102, 0.2, or 0.5 recombination between adjacent loci, and heritability of 0.9, 0.4, or 0.1. Populations with additivity or complete dominance were simulated and continued for 30 generations. The trait under selection was controlled by ten loci with two alleles at each locus. The initial gene frequency was 0.5 at each locus, and the initial population was in linkage equilibrium. Under the additive model the realized advances were close to the expected advances for most cases, although the estimated advances were less accurate for populations selected intensely with low heritability. Linkage had no l3 effect on the predictions. Under the dominance model, pre— dictions of genetic advance were less accurate. Predictions tended to overestimate genetic advance for strong selection. For less intense selection, genetic advances tended to be underestimated, although agreement between realized and expected gains was fairly close. The decline of additive genetic variance under both models was rapid when selection was intense, particularly when heritability also was high. The effect of linkage on variance was small, although in the additive model, tight linkage tended to accelerate the rate of decline in variance in the early stages of selectiOn but" had an opposite effect in later generations. This trend was not observed for the dominance model. The initial rate of decline of additive variance was more rapid for the domi- nance model but later, when the variance was reduced to a low level, the rate of decline under either model was slower than in earlier generations. In a later study Young (1967) evaluated epistatic and mixed models using the same procedures as for the first paper. His epistatic model was the additive x additive one, and the mixed models included mixtures of additive plus dominant loci, additive plus dominant loci plus epistasis, and dominance plus epistasis. Agreement between expected and realized gains were close even under low selection intensity and low heritability for the model of additive plus dominance. He did not find any appreciable effect of 14 linkage on genetic progress nor did he find any evidence of loss of favorable alleles. The predictive ability of heritability using the A + D model was almost as accurate as using the additive model alone, but the inclusion of epistasis tended to cause erratic predictions. Additive genetic variance decreased faster for those models where epistasis was not present and linkage was not a factor in the decay of genetic variances. A number of recent investigations of multilocus models (Kimura, 1956; Kojima, 1959a, 1959b; Lewontin and Kojima, 1960; Bodmer and Parsons, 1962; Lewontin, 1964a, 1964b, 1965; Jain and Allard, 1966; and others) have shown that interactions between linkage and epistasis can, in fact, have significant effects on the structure of large, random- mating populations; and some of the features of such popula- tions can be understood only if the interactions between linkage and epistasis are known. Lush (1948) indicated that selection tends to create linkage disequilibrium when selec- tion is of favored genetic extremes, but the effect is magni- fied if selection is for a genetic intermediate. He explains this phenomenon with an epistatic model. Lewontin and Kojima (1960), working with a model of two loci showed that linkage does not affect the final equi— librium of a population mating at random. In a later paper, Lewontin (1964a) expanded his model and included five loci. He reiterated his comments concerning the necessity of 15 epistasis for linkage to be an important factor in natural selection. He also indicated a cumulative effect of linkage. Even if two genes are loosely linked, they may be held out of linkage equilibrium with each other if loci between them exhibit linkage disequilibrium. Latter (1965a) started from a base population in linkage equilibrium for two loci to ascertain the effect of linkage and the various interactions on genetic response as measured by change in gene frequency. In his model the genes acted additively at each locus, and no epistasis was involved. He also included different sizes of population, various proportionate effects of genes, and different inten- sities of selection. He showed that response (measured as change in gene frequency) to selection was progressively reduced as the degree of linkage was intensified but only when recombination was less than 0.10. He concluded that if the effective size of a population is restricted so that it reduces the response by 40 percent or more, linkage may be relatively unimportant for genes separated by as little as 5 map units. He concluded that the reduction in response was less pronounced for genes of smaller proportionate effect (a/o) or for populations of greater effective breed- ing size than he used (largest was N =40). Where size of population was smallest, linkage was most intense, and pro- portional effect was largest, he found the greatest reduc- tion in response. Using the notation of Lewontin and Kojima 16 (1960), viz D = (gllg22 - ngng)’ where gij denotes the frequency of the AiBj gamete, Latter computed the coeffi- cients of linkage disequilibrium. He found that the proba— bility of fixation of the least favored gamete, A2B2, was virtually unaffected by linkage, but the probability of fixation of the gametes Ale and A2Bl was increased. Fixa- tion, of course, refers to homozygosity and not necessarily to homogeneity. In a later paper, Latter (1965b) evaluated other parameters such as (l) the initial rate of response; (2) the half life and 0.95 life of the selection process; (3) the number of generations to 50 percent and 95 percent fixation, and (4) the mean coefficient of linkage disequi— librium in finite populations. The influence of linkage on the 0.95 life span of the selection process is rather complex since changes in the rate of approach to fixation and in the selection limit itself are involved. But when N was 10 or 20, the half life of the selection process was increased con- siderably as linkage increased in intensity. The effect of linkage on fixation (homozygosity) is less complex and, in almost all cases examined, Latter found a general increase in the number of generations required for fixation of the desired allele. He discussed three phases of the pattern of response to selection in finite populations. The first phase is characterized by the chance loss of the favored alleles due 17 to low initial frequency. The second phase is one possibly of appreciable duration, in which gene frequencies are intermediate in value, and the rate of genetic fixation is much reduced consequently. The third phase has to do with the fixation of favored alleles where high gene frequencies have been realized under selection. Linkage had its most obvious effects in the third phase of response; low recombi- nation rates were responsible for a reduced rate of fixation during this phase and for preservation of replicates in a state of joint segregation at the two loci. The reduction in total response due to linkage was due to fixation of repulsion gametes Ale and A2Bl more frequently than would be expected with free recombination without change in the probability of fixation of A2B2. Latter showed this phenom— enon was due to chance loss of the most favored gamete AlBl in many replicates followed by a period of continued segre- gation of gamete frequencies of gll = 0, g12 = l/2 + p, ng = 1/2 - p, and g22 = 0 where p is one-half the differ- ence between the frequencies of Al and B1. In such popula- tions the degree of linkage disequilibrium is maximal, and chance fixation of one of the repulsion gametes may readily occur before AlBl can be recovered by recombination. Felsenstein (1965) considered the effects of linkage on directional selection with and without epistasis. Where fitness was considered a function of a phenotype which it— self was determined additively by two nonoverdominant loci, 18 he showed that linkage disequilibrium generated by selection had the same sign as the second derivative of the function which related phenotype to fitness in the continuous-genera- tion model. If linkage disequilibrium generated by selec— tion was positive, tight linkage increased the rate of change of gene frequencies; but when the linkage disequilib- rium was negative, tight linkage decreased the rate of change of gene frequencies. He considered the additive model and showed negative linkage disequilibrium. Thus, tight linkage should reduce the rate of change in gene fre- quency by artificial selection. Parameters of Relationship The concept of genotypic covariance or correlations between relatives and the relation of these parameters to the more basic parameters describing the Mendelian mechanisms have been of special interest to geneticists for many years. Sturtevant (1965) credits Galton with developing the notion of correlation between relatives as a result of Galton's tabulation of heights of parents and their offspring. But Berge (1961) in a history of the development of animal breeding indicates that the ancient Greeks and Romans had, by comparison with the later periods, a relatively highly devel— oped standard of breeding of domestic animals. Approved methods of breeding were based on recognition that parents leave their mark on the offspring. According to Engeler (1936), lineal relationship was quantified as early as 1815 19 in the work of Krunitz, and the term Individual Potenzlehre gained importance in Germany about 1860 following the work of Weckherlin (1851). Other early investigators of the mathematical basis of covariation of relatives were Snow (1910), Brownlee (1910), Weinberg (1908a, 1908b, 1909), Jennings (1916, 1917), and Rollins (1917, 1918a, 1918b). It was not until 1918, however, that the matter was clarified by Fisher's classical work concerning "the corre- lation between relatives on the supposition of Mendelian inheritance." Using the theory of average effects of genes' and deviations from additivity caused by dominance, Fisher considered autosomal loci in random mating populations and developed general formulae for the various types of rela— tives. Working independently from Fisher, Wright (1921) used the method of path coefficients to develop expressions for the correlations between relatives. The resulting coef- ficient included only additive effects of genes and was termed "coefficient of relationship." . Malécot (1948) developed probabilistic arguments for the formulae of Fisher and defined "identity by descent" and "coefficient de parente." He also gave a probabilistic defi— nition of the "coefficient of inbreeding," which had pre— viously been defined by Wright (1922) in terms of correla- tion between uniting gametes. The definition of "coefficient 20 de parente" (rxy) follows: consider two individuals X and Y with genotypes "ab" and "Cd," where a, b, c, and d merely label the positions of genes at one locus. The rxy is the probability that a random gene from X is identical by descent with a random gene from the same locus of Y. "Coefficient de consanquinite," as defined by Malécot, is the probability that two genes carried by X at one locus are identical by descent. This concept can be shown to be identical to Wright's definition of "coefficient of inbreeding." Malécot showed that the covariance between any two individuals under random mating, X and Y, may be expressed as: Cov (X, Y) = Lg-f-gj 02A + (¢¢') 02D where 02A is the genic (additive genetic) variance, 02D is the dominance variance, and o and o' are, respectively, the probabilities that genes from the sires of X and Y are identical by descent and that genes from the dams of X and Y are identical by descent. Malécot's formulae do not account for inbreeding. Cockerham (1954) and.Kempthorne (1954, 1955, 1957) extended the formulae of previous workers to allow for epistatic gene action in a random mating population in which parents may or may not be inbred. The covariance between any two individuals (including epistasis among n loci) may be expressed as: 2 r s 269—513.) r (<1) ¢')5 0 A D Cov (X, Y) = r = 0, s = 0 l g_(r + s) g_n 21 where ozArDs is a component of genotypic variance involving additive effects of r loci and dominance effects of s loci. From the effects of linkage in the expectation of the covariances between certain relatives in random mating populations, Cockerham (1956) concluded in random mating diploid populations where genotypic frequencies are in link- age equilibrium, the covariances between lineal relatives are not affected by linkages if position effects are not present. In the treatments of linkage by Jones (1960, 1965), _Schne11 (1961, 1963), and Van Aarde (1963), the restrictions' were relaxed to allow for an arbitrary number of loci, each with an arbitrary number of alleles, arbitrary dominance and epistasis, and for any linkage relationships which are inde- pendent of gene effects. Both Schnell (1963) and Van Aarde (1963) have shown that Cockerham's (1956) generalization concerning linkage effects on the covariance of lineal rela— tives is true only for the special case of parent-offspring relationship. ,Harris (1964) defined nine mutually exclusive and exhaustive events, which could be used to derive eight different parameters of relationship, two of these being Malécot's (1948) rxy and uxy as defined by Kempthorne (1954). Harris used Kempthorne's and Malécot's parameters along with Fisher's work on effects of genes to describe the genotypif 22 covariances between individuals for quantitative genetic traits regardless of the system of mating. Rowe (1966) used genetic simulation to study degree of linkage, size of population, and intensity of selection on some methods frequently used to predict genetic improve- ment. Five methods of estimating relationships, components of variance among full sibs or paternal half sibs and three regression models, were evaluated. He analyzed results from the 23 factorial plan (of the main factors) for each of two levels of environmental variation. In general, the regres- sions proved more accurate than the components of variance in predicting genetic improvement. The results were more variable when environmental variation was included. Al- though he could not find a satisfactory explanation, Rowe reported a significant interaction between levels of selec— tion and linkage for the regression methods only when envi— ronmental variance was included in the model. At the low level of selection, tight linkage caused genetic progress to be less than predicted, but with free recombination at the same intensity of selection genetic progress tended to exceed the amount predicted. This trend was reversed for the higher level of selection and results were as predicted when environmental variance was excluded from the model. 23 A Special Note on Degree of Linkage In all studies reviewed, simulation of intensity of linkage was defined as the probability of crossing—over between adjacent loci on the same chromosome. Intensity also may be defined in terms of the average relationship n-l 2 are shown in Table l for some previously mentioned studies. between n( ) pairs of loci. The average relationships The probability of recombination for all possible pairs of loci is ra,b = l/2 [l — (l-2r)b_a], where §_and b are lin- early ordered loci numbered consecutively on a chromosome, and 3 is the probability of recombination between adjacent loci. All relationships between loci on different chromo- somes are assumed to be 0.50. Rowe (1966) defined the aver— age probability of recombination for loci on the same chromosome 2 ni-l r. = 1/2 1 - $233 + ”'er 1 - (1—2r) l n.r 2 1 ni(ni-l)r where ni is the number of loci on the chromosome. For n loci distributed over several chromosomes, the average probability of recombination, Er is: ni(ni—l) ’r'=1/2+ 2. a. n(n"-_1) " r1 ‘ ”2 Recombination rates between adjacent loci are all that can be controlled in simulated populations. However, the above treatment points out that there is a high proba— bility of recombination between a large number of loci in a 24 Table 1. Average recombination rates (E) for linkage param— eters of some previous Monte Carlo studies and the present study Recombination Between __ Number of Loci per Adjacent Loci r Autosomes Autosome Reference 0.500 0.500 2 3 Fraser (1957) 0.250 0.417 0.050 0.326 0.250 0.313 0.005 0.303 0.500 0.500 1 5 Martin and 0.010 0.020 Cockerham (1960) 0.500 0.500 1 35 Baker and 0.010 0.102 Comstock (1961) 0.500 0.500 8 5 Gill (1963) 0.200 0.479 0.050 0.458 0.005 0.450 0.500 0.500 4 10 Qureshi (1963) 0.050 0.420 0.005 0.389 0.300 0.500 5 5 Gill and 0.200 0.466 Clemmer (1966) 0.005 0.418 0.500 0.500 1 40 Rowe (1966) 0.010 0.114 0.500 0.500 1 10 Young (1966, 1967) 0.200 0.350 0.050 0.150 0.500 0.500 1 50 Present study 0.260 0.481 0.020 0.185 25 polygenic system. Low recombination rates such as 0.05 and 0.005 between adjacent loci on a chromosome contribute little to reducing the average recombination values. And because all linkages between loci must be considered in evaluating the effect of various recombination rates on different genetic parameters, the average recombination rate seems to be more descriptive of the linkage intensity than the rate between adjacent loci only. METHODS AND PROCEDURE Experimental Design and Structure of the Initial Populations The intent of this study was to ascertain the exis- tence and magnitude of the effects of linkage, dominance, and mode of selection on parameters such as gene frequency, genotypic mean, genotypic variance, and genotypic covariance of relatives through the simulation of small populations consisting of full, half, and nonsibs. A study could include all combinations of a large number of factors, including many models of gene action, various levels of environmental variation, different frac- tions of recombination between adjacent and nonadjacent loci, and several methods and intensities of selection. The mat- ing scheme and number of individuals maintained in the popu- lation, as well as initial gene frequency per locus, the number of loci affecting the trait and the number of alleles per locus could be varied beyond the scope of the present study. However, time and cost limit the size of any study to an arbitrary number of factors and levels deemed most important by the investigator. This study was limited to a 33 factorial plan with additional base populations. .Such a plan compared the effects and interactions among the three 26 27 factors at three levels. The variable factors and levels were: 1. Intensity of linkage. One autosome was simu— lated with recombination fractions of 0.50, 0.26, or 0.02 between adjacent loci. The 0.50 level is functionally the same as if each locus were on a different autosome. 2. Degree of dominance. Three models of gene action were included, partial, complete, and overdominance, or more precisely, [h - (d + r)/2] = 2, 4, and 6, respec- tively, where d, h, and r represent genotypic values of homozygous dominant, heterozygous, and homozygous recessive‘ genotypes. 3. Mode of selection. Eight individuals from each population of 48 were selected by one of three methods to become parents of the succeeding generation. They were selected at random, for highest genotypic values, or for highest phenotypic values. The environmental variance in- cluded in the latter case was three times as large as the expected genotypic variance in the original population. In addition to the 27 treatment combinations described above, a model of additive gene action with free recombination was included for each mode of selection to facilitate comparisons of empirical results with well—known theoretical expressions. Three replicates of populations per treatment com- bination were simulated. Since the effects of all factors 28 were fixed and three equally spaced levels of linkage and dominance were included, orthogonal polynomials could be used to evaluate further main effects and interactions of these two factors. Each initial parental population included four indi- viduals of each of two sexes, and each individual contained a single pair of autosomes with 50 loci, two alleles per locus. Simulation forced each locus to be in Hardy—Weinberg equilibrium at gene frequency of one—half. Each male was paired with four females and produced three offspring per mating. The resulting population of offspring consisted of' 48 individuals. Relationships included 48 full sib pairs, 432 possible half sib pairs, and 648 possible nonsib pairs. On this and succeeding generations of offspring measurements of parameters of relationship and population were made. Parental genes were transferred to the offspring one locus at a time with probability of recombination between adjacent loci specified by the level of linkage. From the 48 off- spring, eight individuals were selected by the appropriate method to be parents for the next generation. The first four selected individuals were considered as one sex and the last four the opposite sex. Each population was continued for twenty generations. Additional restrictions to the study were: (1) no interlocus interactions (epistasis) were included, (2) link- age was applied to adjacent loci only and no interference 29 was included, (3) no multiple alleles, and (4) individuals were unisexual diploid units. Mechanism of Simulation and Evaluation The computer system was the CONTROL DATA 3600, a general purpose digital computing system. Each word in the storage module has a 51 bit structure made up of 48 bits of data and three parity bits. Magnetic core storage of 65,536 of these 48 bit words is available. Each of the 48 bit words can be "packed" or partially "packed" with the contents of several "incomplete" words. This feature allows a single word of storage to identify completely each gene by its origin as well as its allelic structure (0 or 1) with FORTRAN language. Each of the original 800 genes was labeled according to allelic structure and given a unique identification number. The first gene assigned to an indi— vidual at each locus was labeled with an odd identification number and the homologue appropriately identified. At the time of pairing, the identification number of each gene as well as the allelic value was transferred to the offspring. A parent number (i.e., one through four for males, five through eight for females) as well as a sib number (1 through 48) was incorporated into the "packed" word at the time of transfer of the gene from parent to offspring. First a random gene from the first locus of the male was transferred to the offspring, then a random gene from the first locus of the female. Next a gene from the 30 second locus of the male was transferred, the choice of the two available genes depending on the outcome of a probabil- ity operation (r-x), where r is a random number between zero and one and x is a predetermined value selected as the desired recombination rate between adjacent loci. For exam- ple, when the recombination rate is one-half, the outcome of the operation (r-x) is equally likely to be positive or nega- tive. Either the plus or minus can indicate a crossover between adjacent loci on the chromosome. But when the recom- bination rate is 0.02, the negative result from the opera— tion (r-x) represents a crossover between adjacent loci, and the equal or plus values represent no crossover, linkage of alleles passing to the offspring. Linkage requires that the gene originating from the parent in question be located on the same homologue as the preceding one from the same parent. A crossover means that the second gene originates from the opposite homologue of the chromosome pair. This probability process was continued for all adjacent loci. .At this point a gene was identified by: (1) allele value (0 or 1), (2) gene number (1 through 800), (3) parent number (1—4 for males, 5—8 for femalesL'(4) sib number (1-48), and (5) homologue identification (0 or 1). All Monte Carlo investigations use pseudo—random numbers. These are random only conceptually, but they have fulfilled many criteria of randomness. A library program was available at Michigan State University for the generation 31 of uniformly distributed pseudo—random numbers that range from 0 to l. The random numbers are produced by the standard multipicative congruential method. This system was first proposed by Lehmer in 1951 and modified by Rotenberg (1960) and Greenberger (1961). The equation takes on the form: Xi+l = (2p+l) Xi +.C with p 2.2 and C being odd. The ini- tial value, Xi, must be supplied. A detailed description of this method is in Gill's thesis (1963) and to a lesser extent in Parker's thesis (1967). Evaluation of ngulation Parameters Among the parameters evaluated for each generation of offspring were frequency of the favored allele and geno- typic mean and variance. .Since each replicate population was an inbred line, the parameters computed were averages within lines. Genotypes of offspring were evaluated by operations of masking. These masking operations compared two operands (chromosomes), bit by bit (locus by locus), and assigned either a zero or one at that location to two logi- cal variables, depending on the particular logical operation and bit structure. Following is a list of the masking oper— ations and the resulting binary bit values. 32 Operand Logical Operation EL____XL EEAND-V' FLOR. v 1 l l 1 Binary representation 1 O O l 0 l 0 l 0 0 0 o By this process each locus was identified as homozy- gous dominant (1,1), homozygous recessive (0,0), or hetero- zygous. Genotypic values were then assigned to each locus according to the genotype of the locus and the level of dominance. Then the genotypic mean and variance of the inbred line were computed. The genotypic values per locus were: Level of Dominance Genotype None Partial Complete Over 11 (AA) 12 10 8 6 10 (Aa) 6 7 8 9 00 (aa) 0 0 0 0 Expected initial mean (per locus) 6 6 6 6 Expected initial variance (per locus) 18.0 13.5 12.0 13.5 Evaluation of Sibship Parameters Each generation parameters of relationship were measured for full sibs, half sibs, and nonsibs. Full sibs are offspring of the same parents, half sibs have one parent the same, and nonsibs have different parents. 33 Since each mating produced three offspring, three pairs of full sibs per mating were possible or 48 pairs in all. Likewise, since there were 54 possible pairs of half sibs per parent [C(l2,2) - 4C(3,2)], 432 pairs of half sibs existed. Of 1128 possible pairs in the population [C(48,2)], 48 were full sibs, 432 were half sibs,and the remainder (648) were nonsibs. All of the pairs of full sibs were evaluated, whereas only 48 half and 48 nonsib pairs were chosen for evaluation at random, by sampling without replacement. Malécot (1948) gave a precise probabilistic defini— tion to the concept of "alikeness by descent" or "identity by descent" and, thus, clarified the concept. Two genes, "a" and "b", say, are alike by descent if they are both copies in the reproductive process of a single gene that occurred at some previous time or one is a copy of the other. The probability of "a" and "b" being alike by descent may be conveniently symbolized as P(a==b). This probability is con- sidered relative to some base population in which all genes are considered as not being alike by descent. An individual X has a genotype at a certain locus which may be represented as A where the subscripts dixs indicate the origin of the gene; Axs was transmitted to X by its sire, and AX was obtained from the dam of X. The d coefficient of inbreeding, FX, of individual X is defined as the probability that the two genes possessed by X at a certain locus are alike by descent to each other; i.e., 34 F = P(AXS =led) where Axs = A is inter reted as " is x p Axs xd alike by descent to Axd‘" Malécot also defined the "coeffi- cient de parente" between individuals X and Y (symbolized as rxy) as being the probability that a random gene of X at a given locus is alike by descent to a random gene from Y at the same locus. Representing the genotype of Y as Ays’ Axs’ the term rxy may be represented as rxy = 1/4 [P(AXS = AYS) + P(AXS = Ayd) + P(Axd = AYS) + P (Axd = Ayd)]. The uxy quantity which occurs in the parametrization of Kempthorne (1954, 1957) for covariances between relatives in random mating populations may be represented symbolically as uxy = P(Axs = Ays # Axd = Ayd + P(Axs = Ayd # Axd = Ays) Harris (1964) then defined in a like manner four additional probabilities of alikeness by descent (other than rxy’ uxy’ Fx' Fy). These are sXy = 1/2 [P(AXS = Axd = AYS) + P(AXS = Axd = AYd)], Syx = 1/2 [P(AYS = Ayd = AXS) + P(AYS = Ayd = Axd)], tXy = P (Axd = Axs = Ayd = AYS), and vxy = P (Axs = Axd # Ays = Ayd)° 35 Examination shows that Sxy is the probability that both genes of X are alike by descent to a random gene of Y at a certain locus; Syx is the probability that both genes of Y are alike by descent to a random gene of X; tXy is the prob- ability that all four genes are alike by descent to each other; and VXY is the probability that the two genes of X are alike by descent to each other, the two genes of Y are alike by descent to each other, but the genes of X are not alike by descent to the genes of Y. Note that txy = t and yx v = v x but Sxy is not necessarily equal to s . XY Y yx Considering the two random individuals, X and Y, Harris (1964) described a set of nine mutually exclusive and exhaustive events pertaining to the alikeness and nonalike- ness of the four genes that are relevant for a given locus. These events and symbolic representations for their proba- bilities were described by Harris as follows: Event Probability Number Event of Event 1 Axs = Axd = Ays = Ayd Pl 2 Axs : Axd # Ays = Ayd P2 3 Axs = Axd = Ays # Ayd P or AXS=Axd=AYd7SAYS 3 4 Axs # Axd = Ays = Ayd P or A # A = A _ 4 xd xs ys - Ayd 5 Axs = Ays # Axd = Ayd P or AXS = Ayd # Axd = Ays 5 36 Event Probability Number Event of Event 6 Ays # Axs = Axd # Ayd P and AYS # Ayd 6 7 ~A # A = A = A xs ys yd xd P7 and AXS C'Axd 8 Axd # AXS = Ays # Ayd andAxd # Ayd or Axd # AXS = AYd # Ays and Axd # Ays P or AXS # Axd = Ays # Ayd and AXS # Ayd 8 or AXS # Axd = Ayd # Ays andAXS # Ays 9 Axs # Axd # Ays # Ayd P and AYS # AXS # Ayd ¢ Axd 9 9 It follows that 2 Pi = 1.0. i=1 Note then that FX = P1 + P2 + P3 + P6, F = P + P + P + P y l 2 4 7’ rXy = 1/4[4Pl + 2P3 + 2P4 + 2P5 + P8], sxy = P1 + 1/2 P3, syx = P1 + 1/2 P4, tXY = P1, uXy = P5, and vxy = P2. 37 Combining this notation with Fisher's (1941) defi- nitions of effects of genes, Harris described the genotypic variance for a partially inbred (PI) population (total, i.e., not within lines) as _ 2 V(G)PI — (l-FX) VR (GX) + FX VI (GX) + FX (l-FX) Dl' Similarly, Cov (G G ) = 2r 2 + (s + s ) + t 2 x’ y PI xy OAI xy yx GADI xy ODI 2 2 + uXY GDR +(tXY + ny - FX Fy) D1. VR(Gx) was defined to be the sum of additive and dominance variance as computed for a random mating population, . 2 2 2 i.e., V (G ) = o = o + o - R x GR AR DR‘ VI (GX) was defined as the variance of additive effects of genes in a completely in- A2 was defined as that portion of the I genotypic variance due to the portions of the effects of the bred population. 0 homozygous genotypes that were additive effects of genes in the original random mating population. In terms of gene frequencies, additive effects, and dominance deviations, Kempthorne (1957) defined the follow- ing terms: 2 2 2 o = 4 2 p- a- = 20 . AI 1 l 1 AR 2 2 o = = ADI 2 E Pi 91 511' D1 § Pl 511 ' and o 2 = z p. 5.. - (ESP. 5. ) where D . 1 11 1 1i . 38 the pi represent the frequency of the ith allele, oi the average effect of the ith allele, and oij the deviation from additivity due to dominance for the genotype AIAJ in a ran- dom mating population with genotypic array E § pi pj Ai Aj. Computation of probabilities of the nine—mutually exclusive events required evaluation locus by locus for each pair of sibs. Genes alike by descent were determined by comparing identification numbers of genes (1 through 800). This procedure required "unpacking" one word of computer memory into several identifiable parts. First, the number of each gene was isolated, and a probability was assigned appropriately to each comparison by a series of "compare statements" involving the four genes representing a single locus for a pair of sibs. The allelic structure also was evaluated at that locus for the pair of sibs by the masking operation previously described. This procedure was required to compute the actual covariance between pairs of sibs. Each of the 50 loci per pair of sibs was evaluated in this manner before another pair of sibs was evaluated. These procedures were continued for the 48 pairs of sibs of each type and the averages per sibship were calculated. Selection of Parents After the three sibships were evaluated, eight indi— viduals were selected to become parents to produce the 48 offspring for the next generation. Individuals were 39 selected randomly, for highest genotypic value, or for highest phenotypic value. For the random selections, random numbers between zero and one were multiplied by 48, incremented by one, and truncated to whole integers. This provided a uniformly distributed series of numbers ranging from one through 48. Selection was without replacement to insure that eight dif- ferent individuals would be chosen. The first four selected individuals were considered to be of one sex and the last four the opposite sex. Genotypic selection involved identifying eight indi-' viduals having highest genotypic values by the Bucharest sort. This process, one of a group of procedures known as merge sorts, consisted of a sequence of successive mergings of two sets of numbers. A set of n numbers (genotypic values) to be sorted in ascending order was divided into two sets for which the number of elements differed by no more than one. These sets were merged to form two new sets of partially ordered subsets. These two groups of partially ordered subsets were then merged to form two sets of larger partially ordered subsets. The length of the partial order- ings increased by a factor of at least 2 with each merge. The merging process was stopped when the merge produced a single ordered sequence. A detailed description of this process is given by Evans and Perry (1961). 40 Phenotypic selection used the same sorting routine for phenotypic values. An environmental contribution to each phenotypic value required generation of standardized random normal deviates. Each deviate multiplied by the desired environmental standard deviation provided the random environmental contribution to a phenotypic value. Use of the desired environmental standard deviation was an attempt to standardize the initial degree of heritability (in the broad sense) of the trait. For example, if the genotypic variance were 18 per locus, then for initial heritability of 0.25, the environmental variance needs to be 54 per locus 30‘ that the total phenotypic variance per locus is 72. Thus, the environmental contribution to a phenotypic value at one locus needs to be J54 x, where x is a standardized random normal deviate. The procedure used to obtain standardized random normal deviates involved summing uniformly distributed ran— dom numbers. The validity of this procedure is supported by the Central Limit Theorem, and the deviates have been shown to be sufficiently well—behaved unless one is vitally inter— ested in extreme values. Gill (1963) described the procedure in detail. A specified number of uniformly distributed ran— dom numbers 0 < ri < l are generated, added together, and the variance is coded so that the sum is normally distributed with mean equal to zero and standard deviation equal to one. 41 In this study twelve random numbers were generated by standard subroutine, added together and 6.0 subtracted 12 from the total (ei = Z ri — 6). The range after subtrac- i=1 tion was -6 < ei < 6, and the expected value of ei was zero. The variance of a uniformly distributed variable is equal to the square of the range divided by twelvei2 Therefore, V(ri) was (1)2/12 or 1/12. Then V(ei) was V (“Z ri) which was 12(1/12) or 1. Thus, ei were distributedbwith mean zero and standard deviation one as required for standardized normal deviates. Samples of random deviates conformed closely to expectations. Expected environmental variation was fixed for the first generation and remained the same for all generations. EXpected heritability was fixed at 0.25 in the first genera- tion but decreased thereafter due to loss of genotypic vari- ance within inbred lines. After the appropriate individuals were selected for parents, a new generation of offspring was produced as described earlier. At each generation the identification and the allelic value of the gene were transferred to the offspring. RESULTS AND DISCUSS ION Population Parameters Coefficient of inbreeding.——To evaluate differences in the effects of various factors on inbreeding, variance was analyzed in each of four different generations, 3, 8, l3, and 18. Since levels of the three factors (i.e., selec— tion, linkage, and dominance) were fixed and, for two (link— age and dominance) factors were quantitative and equally spaced, partition of the sums of squares of the effects and their interactions into orthogonal polynomial fractions was possible. The coefficient of inbreeding, as described earlier, for each of the 27 treatment combinations in generations 3, 8, 13, and 18 is given in Tables 2, 3, 4, and 5. Values listed for a given generation were averages for that genera- tion and the two preceding and two succeeding generations for all three replications. For example, entries under Generation 3 represent averages of 15 means of 48 individuals from generations one through five of each of three replicate populations. Preliminary checks of the data validated the assump- tions underlying the analysis of variance of differences in inbreeding due to varying levels of selection, linkage, and 42 43 Table 2. Mean coefficients of inbreeding with probability of recombination between adjacent loci of 0.50 Generation Selection Dominance 3 8 13 18 Partial .12 .32 .51 .64 Random Complete .10 .32 .49 .61 Over .09 .29 .46 .59 Partial .15 .42 .65 .80 Genotypic Complete .16 .44 .65 .76 Over .12 .34 .50 .60 Partial .ll .35 .53 .68 Phenotypic Complete .12 .35 .51 .64 Over .11 .34 .51 .64_ Table 3. Mean coefficients of inbreeding with probability of recombination between adjacent loci of 0.26 Generation Selection Dominance 3 8 13 18 Partial .12 .33 .51 .62 Random Complete .11 .31 .50 .63 Over .12 .37 .53 .67 Partial .14 .41 .64 .78 Genotypic Complete .15 .46 .64 .78 Over .13 .37 .52 .61 Partial .10 .29 .47 .61 Phenotypic Complete .13 .40 .55 .67 Over .10 .31 .47 .60 44 Table 4. Mean coefficients of inbreeding with probability of recombination between adjacent loci of 0.02 Generation Selection Dominance 3 8 13 18 Partial .10 .33 .47 .64 Random Complete .08 .29 .49 .63 Over .11 .29 .49 .63 Partial .18 .51 .70 .82 Genotypic Complete .14 .38 .48 .54 Over .12 .35 .42 .48 Partial .11 .37 .60 .73 Phenotypic Complete .09 .32 .50 .67 Over .10 .28 .42 .51 45 Table 5. Mean coefficients of inbreeding for 27 treatment combinations and various combinations of the two and three factors——l8th generation Dominance Selection Recombination Partial Complete Over Average 0.50 .64 .62 .57 E61 Random 0.26 .63 .61 .67 .64 0.02 .65 .65 .54 .61 Average .64 .63 .59 .62 0.50 .79 .75 .59 .71 Genotypic 0.26 .78 .79 .60 .72 0.02 .82 .55 .48 .61 Average .80 .70 .56 .69 0.50 .67 .64 .64 .65 Phenotypic 0.26 .62 .67 .60 .63 0.02 .74 .66 .59 .64 Average .68 .66 .59 .64 0.50 .70 .67 .60 .66 Average 0.26 .67 .69 .63 .67 0.02 .74 .62 .51 .62 Average .71 .66 .58 .65 46 dominance. Although several treatment combinations exhib- ited large error variances with respect to the pooled error term, these were not sufficiently large to require transfor- mation of the data. Since the primary purpose of this study was to eval- uate the effects of various factors on several parameters in inbred populations, results about the 18th generation (aver- age of the last five) will be discussed first. Data from the last five generations did not provide sufficient evidence to show a significant effect of the interaction of all three factors. However, each of the interactions of two factors were statistically significant. Variance analysis for the mean coefficients of inbreeding is shown in Table 6. Therefore differences among means of treatment combinations, rather than main effects are perti— nent. The interaction of selection and dominance appeared to be the most prominent. Interaction of selection with linear effect of dominance was the largest source of varia- tion among the orthogonal polynomial fractions. Populations affected by partial dominance achieved the highest inbreed— ing regardless of the mode of selection. The difference was largest when selection was on genotypic value, intermediate when selection was on phenotypic value, and hardly measure- able when individuals were selected at random. 47 Table 6. .Analysis of variance-—mean coefficient of inbreed- ing—-18th generation Source D.F. S.S. M.S. F Selection 2 .0602 .0301 7.8** Dominance 2 .2221 .1110 28.8** Linkage 2 .0303 .0151 3.9** S x D 4 .0957 .0240 6.2** -S x D linear 2 .0950 .0475 12.2** -S x D quadratic 2 .0007 .0004 o cam .ouoagfioo .HMHDHMQ Mom meflooouncfl emu: .H ousmflm meowumuocmw ON ma 0H ea NH OH m o e N o . _ . _ _ . _ j . .800 o Ill 1 .EOQ U ll \\ .soo m . \ 1 \\ i \\O\\. \\\ \ \\D I \\ \ \o\ \ \ \ \\ \- \ I \\\ \\\hT\\ J J OH. ON. om. o¢. om. 00. Oh. ow. om. OO.H burpeexqux go querorjgeoo ueew 55 .oguocom co mmB soauooaom pew oocmcwEoc Hmauumm mmB cowuom meow muons» Hooa “swoonpm smokuon cowumcflnaooou mo mao>oa mourn Mom mcflcoouncfl Com: .m ousmflm meowumuocow ON ma 0H ¢H NH OH m 0 ¢ N o _ _ _ . _ _ u n . 0 L3. \. \\\ \\\ .. on. w \ e \ \x u \ . . 0 \\ I on O \ \ m... \ . I. \ I. O“. T... .\O m. \.\ .\\ w \ .\o\ n om. 4 \\ .\ m. \ \ \\O t om. I \ \\ W \ \. 1 \\o\ .\o a \ \\ l 05. p \\ \\ NO. H .H Ioll. m. \ \\ 5 \ \\.\ 0N0 H... H llllll l om. \ I ut\ o I\\ om H «H @9323 .. om. OO.H 56 .ommuocom co mmBGOADUoHom cam oocmaHEOUHQ/o mm3 coauom ocom ouosB flooa ucoomflom coo3uon coaumcflnaooou mo mHo>oH women now msflooouecw cmoz ON .m ousmflh meowumumcow ma 3 3 NH 3 m o a m o _ _ _ _ _ _ _ _ . o .\.. 1 S. .\ \u \\ 1 ON. \ . 1 On. \Q\va\ .\ul-l- \\\ 1 3. \\ \\ \.\o\ \\ -\.|¢I.I.lo\. \\ . \\\ t I oo. 1 on. No. n u -l-l mm. H H III 1 om. om. n u ommxcflq 1 om. oo.H burpeexquI go querorgjeoa ueew 57 .ooum ouo3.moumu coaumcflnaooou cam ooemcHEOp ouoam IEoo mm3 coauom meow ono£3 coauooaom mo moooe money How mcflooouncfl emu: .6 ousmflm mCOaumHmcow 0N ma 0H ea NH OH m m m N o 4 l1 dl A 4 q 4 1 J O m I OH. ounmmuoamnm lulu \\ uaguocmol l l . \ I om. m e soocmm \\X\\\ .\\i u I om. m. e n IL . . ow m T- m l.om. 1. O J l 00. I m a Iron. a D. To .w l om. l_om. oo.H 58 size of the breeding population is eight. By Malécot's approximation, the expected values of'g are 0.118, 0.354, 0.528, and 0.654 for generations 3, 8, 13, and 18 where the number of generations of inbreeding is one less than the number of a particular generation in this case. By Wright's recurrence relation, the corresponding expected values are 0.119, 0.341, 0.513, and 0.640. While the observed inbreed- ing values were consistently lower than the expected values (by Malécot's approximation), the differences were not sta4 tistically significant. In the second generation the coef- ficient of inbreeding was slightly lower than expected (0.0625). Since inbreeding is cumulative the early discrep- ancy may have been responsible for the lower levels in later generations. However, there is no reason other than sam— pling error why results from the nine relevant treatment combinations should not average to the expected value. The effect of selection on inbreeding values did not agree well with projected figures from Robertson's (1961) theory. Robertson showed that selection for a character with high heritability can cause an important reduction in the effective population size and thus result in increased rates of inbreeding. However, Robertson's paper had the assumption that genes act additively, whereas all models in this study included dominance. .Also, two-thirds of the popu— lations involved linkage which was not considered in Robert- son's theory. The "new" effective number "Ne" for perfect 59 heritability was 3.5. For example, Malécot's approximation gave an expected level of inbreeding of 0.911 for the 18th generation. The corresponding figure from Wright's equation was 0.895. The group of populations with partial dominance had a level of inbreeding closer than others to the theoretical value in the 18th generation. Since gene action was not completely additive, one would not expect complete agreement with theoretical figures. Another possibility for lack of agreement was parents were selected on genotype, and the first four selected were one sex. This deliberate forcing relatives to be the same sex violated the assumptions in the derivation of the prediction formula. Where environmental variation constituted three- fourths of the total phenotypic variation in the first generation (heritability = 0.25), Ne was 4.3 and the expected inbreeding in the 18th generation was 0.86. The group selected on phenotype did not approach the theoretical g in the 18th generation as closely as the group selected on genotype. Part of the divergence of the group selected on phenotype can be explained by the decrease in heritability after several generations of selection had decreased the genotypic variance within the line. Essentially, the selec- tion process became less and less effective in this group; and as it did, the selected individuals were not as closely 60 related as they would have been if heritability had been constant. Data from this study do not agree completely with results published by Gill and Clemmer (1966). They found a definite increase in inbreeding when there was tight linkage between adjacent loci, but this study does not show the same trend. They also reported that linkage and selection jointly caused an increase in inbreeding in early genera- tions, an effect not evident in this study although the limited number of replications did not allow as precise an evaluation in early generations as Gill and Clemmer's work where 20 replications were used. If, as they suggested, excess coupling existed in the initial population, this could account for the increase in inbreeding. This study did not expose any linkage disequilibrium in initial gener- ations. Gene frequenpy.-—The frequency of a gene in a popu- lation is a descriptive term to characterize the proportion of a "favored" or other allele in the genetic system and can assume numerical values from zero to one. If the frequency is one,the population consists entirely of members homozy- gous for the allele; if zero, the allele is not present in the population. Lush (1948) described four major forces, migration, mutation, selection, and chance which are capable of affect— ing gene frequency. This study concerns the influence of 61 selection, chance, and the interaction of the two on gene frequency in inbred populations with various degrees of link- age and dominance.‘ Lush (1948),.Kempthorne (1957), and others have described the genotypic array of a population as follows: Frequengy Genotype Genotypic Value Non—Inbred Inbred to Degree F AA Y + 2x p2 p2+Fp(l-p) Aa Y + hX 2p(l-p) 2p(l-p) (l-F) aa Y (1-p)2 (1-p)2 + Fp(1-p) The "p" term represents the frequency of the "A" allele, (l-p) the frequency of the "a" allele, and "F" the degree of inbreeding at that locus. Without selection as inbreeding approaches unity the genotypic frequencies of the population approach their respective gene frequencies. However, for a single locus the gene frequency of "A" becomes either zero or one for a given line. The number of lines where the gene frequency is unity for this allele is a function of original gene fre- quency. For many loci within an inbred line the analogy holds for loci as well as lines. For example, the frequency of the allele "A" may be "p" for the line over all loci but at each locus will assume a frequency of either one or zero with the number of loci having the "A" allele fixed depen— dent upon the original gene frequency in the line. The latter case is discussed for this study. The original 62 population was forced to be in Hardy-Weinberg equilibrium with a gene frequency of exactly one-half at each of 50 loci. Mean frequencies of the favored genes for the 27 treatment combinations for generations 3, 8, l3, and 18 are listed in Tables 7, 8, 9, and 10. Table 7. Mean frequencies of genes favored within inbred lines where recombination rates between adjacent loci were 0.50 Generation Selection Dominance 3 8 13 18. Partial .48 .49 .49 .48 Random Complete .49 .49 .48 .50 Over .49 .54 .54 .52 Partial .63 .80 .94 .98 Genotypic Complete .57 .72 .84 .89 Over .55 .65 .70 .75 Partial .55 .57 .60 .61 Phenotypic Complete .52 .56 .61 .63 Over .51 .53 .54 .57 In the 18th generation the three main effects in the 33 factorial plan and the interactions of selection with linkage and with dominance were statistically significant (P < .05). Mean gene frequencies from populations with dif- ferent levels of dominance over all levels of linkage were fairly constant when individuals were selected at random. 63 Table 8. Mean frequencies of genes favored within inbred lines where recombination rates between adjacent loci were 0.26 Generation Selection Dominance 3 8 13 18 Partial .51 .49 .48 .50 Random Complete .50 .53 .53 .54 Over .51 .55 .54 .53 Partial .58 .78 .92 .99 Genotypic Complete .59 .76 .87 .93 Over .56 .67 .72 .77 Partial .52 .57 .60 .63 Phenotypic Complete .52 .55 .56 .59‘ Over .52 .53 .54 .54 Table 9. Mean frequencies of genes favored within inbred lines where recombination rates between adjacent loci were 0.02 Generation Selection Dominance 3 8 13 18 Partial .50 .49 .51 .51 Random Complete .50 .54 .52 .50 Over .50 .52 .54 .54 Partial .63 .74 .81 .86 Genotypic Complete .58 .65 .70 .69 Over .55 .61 .65 .64 Partial .51 .51 .54 .54 Phenotypic Complete .52 .53 .58 .58 Over .53 .57 .58 .59 64 Table 10. Mean frequencies of favored genes within inbred lines for 27 treatment combinations and combinations of the two and three factors--l8th generation Dominance Selection Recombination Partial Complete Over Average 0.50 .48 .49 .52 .50 Random 0.26 .50 .53 .55 .53 0.02 .51 .50 .47 .49 Average .49 .51 .52 .51 0.50 .98 .89 .75 .87 Genotypic 0.26 .99 .94 .77 .90 0.02 .86 .69 .65 .73 Average .94 .84 .72 .84 0.50 .61 .63 .57 .60 Phenotypic 0.26 .63 .59 .54 .59 0.02 .54 .58 .59 .57 Average .59 .60 .57 .59 0.50 .69 .67 .62 .66 Average 0.26 .71 .69 .62 .67 0.02 .64 .59 .57 .60 Average .68 .65 .60 .64 65 This was expected since there was no force other than chance. acting to change the frequency, and that effect should have been random. As expected, the mean gene frequency by the 18th generation was highest for populations with partial domi- nance when evaluation was over all linkage levels but restricted to selection on genotype. In the group of popu- lations with selection on phenotype, the mean gene frequency was the same for partial or complete dominance and both frequencies were slightly higher than in populations with overdominance. ,With partial dominance and selection on genotype, those populations with moderate or no linkage reached fixation. Gene frequency reached its ultimate level of 0.98 in the 17th generation for populations with free recombination, with the average of one locus being fixed in the recessive state for the three replications. This fixa— tion of the undesirable gene probably occurred by chance before selection could change the gene frequency at that locus. At the time of complete fixation, the inbreeding for the particular replication was 0.78. Thus, while fixation was achieved in terms of homozygosity, complete homogeneity was not reached where homogeneity is defined as likeness by descent. One needs to be cautious in using the two terms. If homozygosity is to measure homogeneity, error is only in one direction; i.e., to report higher levels of inbreeding than actually exist. Populations with moderate linkage also 66 reached fixation, this time in the 18th generation, and again only an average of one locus per 3 replications was fixed in the undesirable condition. The level of inbreeding for this group was 0.67. Populations affected by other treatment combinations failed to reach the homozygous condi- tion within 20 generations. Generation trends for gene frequency are in Figures 5, 6, 7, and 8. Over all levels of dominance, no trends in gene fre- quency due to linkage were evident when parents were selected at random. However, when selection was on the basis of geno- type or phenotype, populations with tight linkage had a lower mean gene frequency than occurred in populations with other levels of linkage. Tight linkage appears to be acting as a restraint in the process of changing gene frequency. By the 8th generation a noticeable effect of tight linkage was evident, and this remained in later generations for all three levels of dominance. Tight linkage restraining increase in the frequency of the favored allele with partial dominance may be caused by fixation of unfavorable alleles at some loci before selection can remove them from the population. This seems consistent with the higher level of inbreeding than in popu— lations with higher rates of recombination but with the same levels of selection and dominance. 67 .omhuocom so mm3 cofluooaom our menu oHoB moumu coaumswneooou ouo£3 mocmcweoouo>o cam .ouonEoo .HMHuHmm Mom moflososvoum osom emu: .m mcoflumuocow 0N ma 0H ea NH 0H m N o q _ . . . q a .800 o Il-l 1 .809 U nl.|l. .800 m l l L J ousmflm OH. ON. Om. cm. 00. Oh. om. om. OO.H AouanbeJJ sues 68 . .ommu locum so mmB coauooaom pew ooamcfiEoo HMfluumm mmB coauom ocom muonz HUGH unmounom cooBDon coaumCHQEoooH mo mHo>oH mouse new moaoeosvoum meow com: .0 onsmflm meowumuoeow om ma 0H «H NH» 0H m _ . d ._ _ 1.. 4 n 0 Ln Aouenbexa sues OO.H 69 ON .ommuoeom so mm3 QOHuooHom pom oocmcHEOUHo>o mmB COHuom ocom ouonB HooH ucoomflom cooBDoQ :OHumcHQEooou mo mHo>mH mouse How mmHocosqoum memo coo: mGOHumumcoo OH OH. vH NH OH O O m N _ _ _ _ L n d- _J mo. u o -|-I mm. H A ll... om. H .H .5 musmHm OH. ON. on. 0%. om. OO. Kouanbexa sues Oh. Ow. 70 .moemcHEOO oumHmEoo mm3 COHuUm memo con3 eOHuoonm mo moUoE mounu How mOHuHmcoucH ommMCHH HHm Ho>o mOHocosvmum memo smog .m musmHm mQOHumnocoo 0N OH OH ¢H NH OH O O o o _ _ A H . . 1 _ o I.OH. onmuocorm Il-.|. I ON. UHQNDOQmO IIIII. EOUGmm I om. I 0%. fl} l.omo (\\\.‘I IlllCllltlllTa'l.\ \\ c - .uIIlTII..I|9|IIIIIfi|| \\b\\\ I.OO \\\R\\\. .1 On. Q\\\ \\\ \\\A\\ I om. \\\bII IIIIOIlIl IlIIIIIII fll I om. oo.H Kouenbela sues 71 The restraint on gene frequency in populations with overdominance or complete dominance and tight linkage may be a function of both inefficient selection and low recombina- tion rates. Selection would favor the heterozygotes at least equally with the best homozygotes. With tight linkage, gene frequency would not have as much opportunity to increase as when recombination rates were high because of the time required to eliminate double heterozygotes in repulsion phase. Many of the controversies in population genetics appear to have their basis in that various factors have been. considered singly. Results in this study give numerical substance to the often expressed idea that multigenic systems cannot be understood with analysis of the isolated effect of any single variable but only through characterization of the interplay of various factors acting simultaneously. That is especially true in the investigation of effects of linkage on population parameters. Furthermore, descriptive param- eters such as mean and variance, while useful, often give misleading indications of the effects of various factors, particularly when there are interactions between the factors. Genotypic mean.--Because the genotypic mean of a population is related to gene frequency, gene action, and average inbreeding, the effects of the different factors on genotypic mean will not be discussed in detail. Rather the effects of inbreeding and gene frequency on the genotypic 72 mean will be considered. The effects of dominance, selec- tion, and linkage on inbreeding and gene frequency param- eters were discussed earlier. Kempthorne (1957) showed the relationship between the mean of a random mating population and the expected mean of an unselected inbred population without epistasis or link— age is of the form: UF = UR + qu(d-2h+r) where d, h, and r represent the genotypic values of AA, Aa, and aa and UF and UR represent genotypic means of inbred and random mating populations, respectively. The terms p and q refer to the frequencies of the two alleles at a given locus. Or this F R 11 The p. are allelic frequencies, and 5.. are deviations of 1 11 can be represented as U = U + FDl where D1 = 5 pi 6... 1 homozygotes from additivity. Both forms show clearly the involvement of dominance in changes in the genotypic mean upon inbreeding. If d + r = 2h, no dominance, the mean of an unselected inbred population is not expected to change. The mean of one inbred population is expected to be less than the mean obtained under random mating when dominance is included in the model, except for negative overdominance where both homozygotes are favored over heterozygotes. In this case, the mean of the inbred population should be above the mean of a randomly mated population. Thus, the magni- tude of the depression of the mean of an inbred population depends upon the degree of dominance as well as the level 73 of inbreeding. Selection can be considered a force opposing the effects of inbreeding and dominance upon the mean. The mean of an inbred line over all loci should be identical to the genotypic mean of the entire population. Thus, while the actual values obtained are values within lines, the expected values are based on the estimates of population parameters. The explanation given for gene fre- quency relative to this situation is similar for the geno— typic mean. Differences between the three levels each of the three main factors and the various two-way interactions were statistically highly significant (P< .01) by the 18th gener-3 ation. Tables 11, 12, 13, and 14 list the genotypic means for the five generations averaged for each of the 27 treat- ment combinations. Table 11. Mean genotypic values for inbred lines where the average recombination rates between adjacent loci were 0.50 Generation Selection Dominance 3 8 13 18 Partial 5.78 5.68 5.40 5.12 Random Complete 5.77 5.30 4.87 4.79 Over 5.81 5.43 5.01 4.43 Partial 7.12 8.48 9.50 9.80* Genotypic Complete 6.27 6.92 7.38 7.56 Over 6.09 6.23 6.20 6.20 Partial 6.41 6.40 6.49 6.46 Phenotypic Complete 6.02 5.85 5.94 5.77 Over 6.02 5.46 5.04 4.84 *Complete homozygosity. 74 Table 12. Mean genotypic values for inbred lines where the average recombination rates between adjacent loci were 0.26 Generation Selection Dominance 3 8 13 18 Partial 5.98 5.65 5.31 5.43 Random Complete 5.92 5.72 5.34 5.08 Over 5.95 5.35 4.96 4.47 Partial 6.74 8.30 9.45 9.93* Genotypic Complete 6.52 7.27 7.62 7.83 Over 6.26 6.29 6.33 6.25 Partial 6.09 6.42 6.52 6.68 Phenotypic Complete 6.03 5.71 5.57 5.49_ Over 6.05 5.55 5.10 4.78 *Complete homozygosity. Table 13. Mean genotypic values for inbred lines where the average recombination rates between adjacent loci were 0.02 Generation Selection Dominance 3 8 13 18 Partial 5.93 5.61 5.68 5.48 Random Complete 5.91 5.83 5.23 4.77 Over 5.84 5.47 4.96 4.45 Partial 7.06 7.87 8.39 8.75 Genotypic Complete 6.40 6.55 6.73 6.62 Over 6.09 6.09 6.23 5.93 Partial 6.05 5.81 5.84 5.73 Phenotypic Complete 6.12 5.61 5.66 5.36 Over 6.03 5.70 5.29 5.16 75 Table 14. Mean genotypic values for inbred lines for 27 treatment combinations and combinations of the two and three factors--18th generation Dominance Selection Recombination Partial Complete Over Average 0.50 5.12 4.79 4.43 4.78 Random 0.26 5.43 5.05 4.19 4.89 0.02 5.78 4.77 4.44 5.00 Average 5.44 4.87 4.35 4.89 0.50 9.80 7.56 6.20 7.86 Genotypic 0.26 9.93 7.83 6.25 8.00 0.02 8.75 6.62 5.93 7.10 Average 9.49 7.34 6.12 7.65 0.50 6.46 5.77 4.84 5.69 Phenotypic 0.26 6.68 5.49 4.78 5.65 0.02 5.73 5.36 5.16 5.41 Average 6.29 5.54 4.92 5.58 0.50 7.13 6.04 5.16 6.11 Average 0.26 7.35 6.12 5.07 6.18 0.02 6.75 5.58 5.18 5.84 Average 7.08 5.91 76 Linkage did not alter the genotypic means in the 18th generation for a particular dominance level when selec- tion was random. Tight linkage in the presence of genotypic selection tended to reduce the genotypic mean by the 18th generation for both partial and complete dominance, when compared between linkage levels, and to a lesser extent for the overdominant model. This decrease was related to gene frequency and mean inbreeding. With partial dominance, mean inbreeding increased, and the combination of decreased gene frequency and in- creased inbreeding in the presence of dominance resulted in the marked decrease of genotypic mean. While mean inbreed— ing actually decreased as linkage intensity increased for the complete dominance model, the decrease in gene frequency created the net loss measured by the genotypic mean of the population. Genotypic means for the three linkage by geno- typic selection levels for overdominant models were fairly constant. This was due to a slight decrease in mean in- breeding with only a very slight decrease in gene frequency as linkage intensities increased; thus, the two forces tended to cancel the action of each other. Results of phenotypic selection were similar but dif- fered less than genotypic selection at the different linkage intensities over levels of dominance and within each level. Orthogonal partition of the interaction of dominance by linkage showed the linear by quadratic effect to be the 77 significant component. This implied that dominance effects were linear while the linkage effects at the different levels of dominance were quadratic. Tables 11, 12, 13, and 14 indicate that over all modes of selection the intense linkage (r==0.02) at complete and partial dominance levels showed the lowest genotypic means while either moderate link— age (r==0.26) or free recombination (r==0.50) at the same dominance levels showed higher values. The two linkage intensities were not different for the considered dominance levels. This pattern also showed for overdominance, but only when selection was on genotype. As for gene frequency and mean inbreeding, the 18th generation showed the most differences between main and interaction effects. Selection and dominance were the earliest to express their effects on genotypic means with linkage and the two way interactions following. Because the predicted genotypic means agreed so well with those observed for populations where individuals were selected at random,those data have not been included. The predicted mean values, however, underestimated the genotypic means when individuals were selected for superior genotypic or phenotypic values. This should be expected because selection changes gene frequency and tends to eliminate those individuals that have loci fixed in the "unfavorable" state. Selection tends to reduce the influence of chance as a force in fixing unfavorable alleles. The underestimation 78 of the genotypic mean was not as drastic when selection was based on phenotype as when based on genotype. Martin and Cockerham (1960) reported findings simi- lar to those in this study with regard to the genotypic mean. They tried tight linkage and free recombination, two inten- sities of selection, and two types of gene action. Environ- mental variance was included in one of the runs with 20 loci acting in an additive manner. They reported a marked effect of linkage producing a genotypic plateau when selection was intense and no environmental variance was present. They found the less intensely selected group progressed slowly at' first but overtook the more intensely selected group by the 20th generation, especially when linkage was tight. Envi- ronmental variance generally slowed the progress from selec- tion, but qualitatively the effects of linkage were the same with or without environmental variance. Their populations with dominance showed the same effect of linkage except progress was not as rapid as for the additive model. Genotypic variance.-—Kempthorne (1954) and Cockerham (1954) provided formulae which partition the total genotypic variance into its additive, dominance, and epistatic compo- nents. However, these formulae are valid only under random mating and no linkage. Robertson (1952) discussed the prob- lem of variation within lines when the breeding population remains constant in size and there is no selection. Within 79 a line in which gene frequency was q, the genetic variance was q2(l-q2) so that the average genetic variance within lines was U - U4, where the U's are the moments of the q 2 distribution about zero. By a similar argument he showed the genetic variance between lines was U4 - U3, and the total genetic variance in these terms was U2 - U2. The variation within lines was represented as vw = a(l—F) + b(l-F)3 + C(l-F)6 where a = 0.8 q(l—q) b = —q(l-q)(l-2q) and 2 2 c = 0.2q(l-q) - q (l-q) . Note that when F = 0, Vw = a+b+c = q2(l-q2) and when F = 1, VW = 0. As F increases, the last two terms in VW decrease until VW becomes proportional to (l-F). These formulae were applied to this study to compute expected genotypic variances within lines for generations 3, 8, 13, and 18 for complete dominance. For random selection ratios of expected to observed variances were near one (Table 15). But as Robertson (1952) stated, the expectations were biased upward when selection was on phenotype or espe- cially genotype. The average ratio for all generations com- puted when selection was at random was 1.06. This figure agrees remarkably well with Kemp's (1967), where his data resulted from simulating two inbred lines, selecting on two traits, and measuring the response to selection in these Table 15. within lines for complete dominance Ratios of expected to observed genotypic variances Generation Recombination Selection Level 3 8 13 18 Avg. .50 0.94 1.10 1.08 1.05 1.04 Random .26 1.05 1.11 1.07 1.07 1.07 .02 0.98 1.23 0.99 0.89 1.05 .50 1.28 1.37 1.42 1.15 1.31 Genotypic .26 1.39 1.53 1.07 0.73 1.18 .02 1.43 1.31 1.30 1.02 1.26 .50 1.12 1.20 1.29 1.14 1.19 Phenotypic .26 1.15 1.03 1.04 0.82 1.01 .02 1.19 1.11 1.20 1.05 1.14, traits along with two other correlated traits. He used 20 loci with complete dominance and a constant selection pres- sure on the number of individuals kept for parents. ratios, 1.06. both traits was 0.30 averaged 1.09. Ratios for selected traits where heritability for Similar from his data for two unselected traits averaged This compares to 1.11 averaged over linkage and generations in this study where heritability was slightly lower (0.25). on genotype, linkage on the ratios do not show directly but they are the ratio increased to 1.25. When selection was The effects of evident indirectly by their influence on inbreeding and gene frequencies which were used to estimate variances. Variances within lines decreased with time for all treatment combinations (Tables 16, 17, and 18). This was 81 Table 16. Genotypic variance within inbred lines when recombination rates between adjacent loci were 0.50 Generation Selection Dominance 3 8 13 18 Partial 579 441 296 232 Random Complete 549 391 296 234 Over 629 489 405 293 Partial 433 200 49 0 Genotypic Complete 446 238 91 58 Over 567 384 328 263 Partial 533 412 274 205 Phenotypic Complete 501 348 251 198. Over 636 490 428 309 Table 17. Genotypic variance within inbred lines when recombination rates between adjacent loci were 0.26 Generation Selection Dominance 3 8 13 18 Partial 553 417 331 264 Random Complete 522 399 301 219 Over 585 467 433 316 Partial 498 254 76 0 Genotypic Complete 432 201 109 52 Over 536 379 324 265 Partial 566 433 327 224 Phenotypic Complete 485 375 287 209 Over 603 497 424 366 82 Table 18. Genotypic variance within inbred lines when recombination rates between adjacent loci were 0.02 Generation Selection Dominance 3 8 13 18 Partial 596 454 320 213 Random Complete 512 400 315 255 Over 651 540 436 311 Partial 424 225 145 94 Genotypic Complete 417 317 237 262 Over 571 487 410 447 Partial 548 456 343 220 Phenotypic Complete 503 402 271 203 Over 590 491 423 331 expected since inbreeding was increasing monotonically per locus within each line. When recombination was free (r==0.50) and selection was random, variances in populations with com- plete dominance or overdominance decreased less, proportion- ately, than in populations with partial dominance. Popula- tions showing more dominance should maintain more genotypic variance than those with additive or nearly additive gene action. Total genotypic variance (computed from average gene frequency and inbreeding) and within line genotypic vari- ance are in Figures 9, 10, 11, and 12. Genotypic selection reduced variation within lines to zero for populations with partial dominance and also decreased variation considerably for populations with complete dominance. Variance decreased 83 .mmmpocmm so mmz coauomamm 0cm «mum mum3 mmumu coaumcflneoumu mnmn3 mocmcflEOU mo mam>ma mmnnu Mom mocmwum> owmmuocmm mafia cflnufl3 paw Hmuoe .m musmflm mcoaumumcmw ..~ "A 3 3 S 3 m o v m o o n ooa 1 com .. com 1 03 A D. I 1 com m. u D 3 .08 cflnufis I O . Hmuoe .. O 1 ooh .309 O '3' .son 0 III oom éoo m L oom oooa 84 .EOCCMH mmS coauomamm cam mmum mum? mwumn coflumcHQEoomH mHmSB wocmcflfioc mo maw>ma mmufip How oucmaum> Uflmmuocmm mafia GflSDHB cam Hmuoe .OH musmflm mQOHumumcmw ON ma 0H fla NH OH w 0 ¢ N O u . . . . . . _ _ o HmDOB I O I OOH :3qu u 0 VI / .EOQ O .l-l IOON .EOQ U .EOQ m 00m 00% A 9 com I T.- e w 000 a 005 com com OOOH OOHH 85 .coauomamm Udmmuocmm nufls mucmcHEOG Inm>o uom mmmxcfla mo mmwuflmcmucfl moms» How mommaum> Uflmmuocmm mafia cflnuflz .HH mnsmflm mGOHumHmeU om ma ma ¢H NH OH m o g N o 4 _ . . . _ _ . . o .nooa loom I ' ’I‘I I JOOM I I, [00¢ A f-"+‘|II.II|II-| OI+C'U+O /’ WM Ic‘lqul‘ / . I’ll loom m IJ’ w .4: e I»?! loom /./ loom me. u unlil loom mm. H H.lilll om. u H _Ioom mmmxcflq oooa 86 .mmum mHmB mmumu coaumcflneoomu paw mocmcfleon mumHmEou mmz cofluum msmm mnm£3 :oHuomHmm mo mmpoE woman you mocmflum> Uflmmuocmm mafia cflnuw3 cam Hmuoe .NH musmflm macaumumcmw 0N ma ma ga NH 0H m o g N o I d . . . . _ 1 . . o lo/ .2309 u 0 III Uflmmuocmnm II-|I. . .10]. [0,: /7/ 383056 I II n 08 . nlLfll. [0.]! EOGCmm nlAYll . .IIATII. IIJI/ . lol. 1.03 / IO/ulo// /o/ , / 1 cos A /9/ / m lid. .Lflllfllv |.oom m. / w IIAYII. A. a . e -IIIOIlIt c o 000 . I o I D ‘I¥'I+O 0 \\Ill 0 . . I.oon s c o . . I.oom I.oo¢ oooa 87 slightly in populations with overdominance; but since the heterozygote was favored, effects of selection and inbreed- ing on fixation of genes were somewhat counterbalanced, and genotypic variance was preserved. Phenotypic selection tended to produce results between those for the randomly selected groups and those selected on genotype. Table 19 includes ratios of expected to observed genotypic variances within lines. The approximation used to calculate expected variances was (1—F)o: where 0: represents total genotypic variance in the first generation. The approx- imation is a comparatively good indicator of total genotypic. variance within inbred lines. But usefulness of the approx- imation is restricted because only the additive portion of the total genotypic variance is of primary concern in the development of prediction equations for selection. An additive model was used for a limited number of trials as a convenient check on the computer program. The ratio of predicted to observed variances was no closer to unity for the additive model than for populations with partial or complete dominance. This adds nothing to the study other than to check on the program. Robertson (1952) indicated that 0.75a(1-F) + b(1-F)3 + 2c(1-F)6 approximates well additive genetic variance with— in 1ines. Unfortunately, the remainder of variance may not be only dominance in an inbred population. Where mating is random, the genotypic variance is the sum of additive and 88 Table 19. Ratio of expected to observed genotypic variances within lines when selection was random Generation Recombination Rate Dominance 3 8 13 18 Partial 0.95 0.95 1.00 0.99 0.50 Complete 0.95 0.96 1.01 0.98 Over 0.98 0.93 0.87 0.89 Partial 0.97 1.00 0.93 0.92 0.26 Complete 0.96 0.94 0.91 0.95 Over 0.95 0.84 1.02 0.93 Partial 0.97 0.94 1.01 1.03 0.02 Complete 0.99 0.98 0.91 0.84 Over 0.92 0.88 0.80 0.76. dominance components of variance if epistasis is excluded. But when mating is not random, genotypic variance is com- posed of variance from additive and dominance effects of genes, and a new portion, covariance of additive and domi- nance effects. Covariance could be derived (Kempthorne, 1957), but this part was not calculated. However, additive variance at any given generation conformed with values from Robertson's approximation with no selection regardless of inbreeding. Mather (1949) showed that linkages between loci do not affect the genotypic mean, but linkage disequilibrium causes the genotypic variance to increase with prevailing coupling phases or to decrease with repulsion phases. 89 When selection is random, there is no reason to expect pro- portions of repulsion and coupling phases to differ other than by chance. Large size of sample should minimize varia- tion from chance. In this study trend in variance with level of linkage was indefinite when selection was random. When selection was on genotype, linkage preserved genotypic variance, especially tight linkage. This is an interaction of selection and linkage in variance. Changes in gene frequency and fixation may help to explain the interaction of selection and linkage. For when selection was intense and linkage tight, the gene frequen- cies did not increase as rapidly as when linkage was absent. Within a given line and at a particular locus, a delaying effect on the increase of gene frequency tended to maintain genotypic variance for a longer time. Truncation selection, in additive gene action, increases linkage disequilibrium by and AP an amount--APl APZ in one cycle where AP are changes 1 2 by selection in gene frequencies of two linked genes (Nei, 1963). Selection, therefore, causes disequilibrium in favor of repulsion if APl and AP are positive. Also, when genes 2 act additively, or nearly so as with partial dominance, the rates of fixation of the minus as well as of the plus genes are increased, and this also contributes to decreasing the variance within a line. Both of these conditions were shown by populations with partial dominance. That is, when selec- tion on genotype was intense, the gene frequency increased at 90 a slower rate than when linkage was tight. Furthermore, more fixation took place, but probably more of the fixed genes were unfavorable than when recombination was unrestricted. Populations with complete or overdominance showed a similar trend in genotypic variances, possibly for different reasons. When selection was on phenotype, the variances with- in lines were only slightly higher, if at all, than when selection was based on genotype. Parameters of Relationship Relationship term ruy—-full sibs.—-Mean values of rXY for full, half, and nonsibs for the 27 treatment combina- tions are in Tables 20, 21, and 22. By the 18th generation the interaction of all three factors was highly significant (P < .01) for full, half, and nonsibs. Intensity of linkage had little or no effect on rxy regardless of dominance when selection was random. But when linkage was tight and genes either completely or overdominant, populations selected on genotype showed markedly lower rxy compared to populations with less restricted recombination. Since rxy represents the probability that randomly selected genes from the same locus of each of two full sibs are alike by descent, the effect of linkage can be likened to that which occurred with E, the difference being that §_measures the probability of alikeness by descent of genes at one locus within a given individual. 91 Table 20. Means of rx when recombination rates between adjacent loci were 0.50 Generations Selection Dominance Sibship 3 8 13 18 F5 .32 .48 .62 .72 Partial HS .22 .40 .57 .68 NS .11 .32 .51 .63 PS .32 .48 .61 .70 Random Complete HS .20 .40 .55 .66 NS .10 .32 .49 .61 PS .31 .46 .59 .68 Over HS .20 .37 .52 .64. NS .10 .29 .46 .59 PS .35 .55 .73 .84 Partial HS .25 .49 .69 .83 NS .15 .42 .65 .81 F8 .35 .56 .73 .81 Genotypic Complete HS .26 .50 .69 .78 NS .17 .44 .66 .76 PS .32 .48 .60 .68 Over HS .22 .41 .55 .64 NS .11 .34 .50 .60 F8 .32 .50 .64 .75 Partial HS .22 .42 .58 .72 NS .11 .34 .53 .68 F8 .33 .51 °62 .72 Phenotypic Complete HS .23 .43 .57 .68 NS .13 .36 .51 .64 F8 .31 .49 .63 .73 Over HS .21 .41 .57 .68 NS .10 .33 .51 .64 92 Table 21. Means of rX when recombination rates between adjacent loci were 0.26 Generations Selection Dominance Sibship 3 8 13 18 F5 .33 .49 .62 .71 Partial HS .22 .42 .57 .67 NS .12 .34 .51 .63 PS .32 .47 .61 .72 Random Complete HS .21 .39 .55 .67 NS .11 .31 .49 .63 F8 .32 .46 .61 .72 Over HS .20 .37 .55 .67. NS .09 .29 .49 .63 F8 .34 .55 .72 .83 Partial HS .24 .48 .68 .81 NS .14 .41 .64 .79 PS .34 .57 .72 .83 Genotypic Complete HS .25 .52 .68 .80 NS .15 .46 .64 .78 F8 .33 .50 .62 .68 Over HS .23 .44 .57 .65 NS .13 .37 .52 .61 PS .32 .46 .60 .70 Partial HS .21 .38 .53 .66 NS .10 .30 .47 .61 PS .33 .54 .66 .74 Phenotypic Complete HS .23 .47 .61 .70 NS .13 .40 .55 .66 PS .32 .47 .59 .69 Over HS .21 .39 .53 .64 NS .10 .31 .47 .60 93 Table 22. Means of rx when recombination rates between adjacent loci were 0.02 Generations Selection Dominance Sibship 3 8 13 18 PS .32 .49 .60 .73 Partial HS .20 .41 .54 .68 NS .10 .33 .47 .64 F5 .30 .46 .61 .72 Random Complete HS .20 .38 .55 .67 NS .10 .29 .49 .64 F8 .32 .49 .60 .70 Over HS .21 .40 .54 .66- NS .10 .32 .48 .61 PS .35 .59 .74 .85 Partial HS .27 .54 .71 .84 NS .19 .50 .69 .82 F8 .35 .49 .58 .62 Genotypic Complete HS .24 .43 .53 .58 NS .14 .38 .50 .56 ES .32 .46 .51 .54 Over HS .22 .40 .47 .52 NS .12 .34 .42 .48 F8 .33 .51 .69 .80 Partial HS .22 .44 .64 .76 NS .10 .37 .59 .73 F8 .31 .47 .61 .74 Phenotypic Complete HS .21 .40 .55 .71 NS .10 .33 .49 .67 F8 .31 .45 .55 .61 Over HS .21 .35 .48 .56 NS .10 .28 .40 .51 94 When linkage is tight in a closed population, and selection is on genotype, probabilities of alikeness by descent should increase if selection is effective in locat- ing genetically superior animals, regardless of whether within an individual or between related individuals. When genes exhibit either complete or overdominance, errors will be made in selecting genetically superior individuals. Thus, more unfavorable alleles for the trait will be maintained in the line, and the probability of fixation in this state is larger than for populations where the genes act either additively or nearly so. For populations with partial dominance, rXy did not increase significantly (P<:.05) when linkage was intensified, but this may be, at least in part, due to the mode of sexing selected individuals as was explained earlier for coeffi- cients of inbreeding. The effect of linkage on rxy for populations selected on phenotype was similar to the effect for those selected on genotype, but the response to in— creased linkage was smaller, except for populations with partial dominance, where there was a substantial increase in the rxy values when linkage was intensified. The three-way interaction was significant in the 13th generation but not in the 8th or 3rd. The same pattern of rxy was evident in earlier generations as in the 18th, but in the 3rd generation differences were small enough that selection, dominance, and their interaction were the only 95 factors significantly affecting this parameter. Selection, in all generations, tended to increase mean rxy as it did with F, and higher degrees of dominance tended to produce lower means of rxy' Another way to describe the trend for this parameter was to fit a polynomial equation to time. For all degrees of dominance and all recombination rates, where selection was random, the second degree polynomial was of the form rxy = 0.2126 + 0.0364 t — 0.0006 t2 where t represents gener- ation. When selection was on genotype, genes were partially dominant and freely recombined, the equation took the form rxy = 0.2003 + 0.0520 t — 0.0009 t2; and for the same gene action and selection but tight linkage, the best fit was rxy = 0.1941 + 0.0585 t - 0.0012 t2. When selection was on genotype and genes were overdominant, equations for popula- tions with free recombination or tight linkage were rxy = 0.2036 + 0.0422 t — 0.0009 t2 and rXy = 0.2211 + 0.0373 t - 0.0011 t2. The last equation provides the poorest fit by accounting for the least variance due to the regression (R2). R2 was 0.97 for the last model but was 0.99 or more for all others. Relationship term rX.--half sibs.-—The relation 1 between rXy and F for half sibs is similar to that for full sibs. Likewise the analyses of variance in rxy for the various generations are similar. The main difference be- tween the sib groups is at starting level of rxy' All full 96 sib groups averaged 0.25 for rxy in the first generation, and half sibs averaged 0.125; rxy for nonsibs was zero as expected. At this point perhaps one should be cautioned about the terms full, half, and nonsibs. This terminology is unambiguous only if the parents were unrelated animals in a randomly mating population. Since, after the first genera- tion of mating, selected individuals were related, the "full sibs" were more closely related than if the parents were not related. This increase in inbreeding of individuals, which was a result of mating related animals, expresses itself in' the parameter rxy' rXy for half sibs tended to approach that of full sibs by the 18th generation, when selection was on genotype regardless of rate of recombination. Relationship term rxy-—nonsibs.--As mentioned above, rxy reflects the average inbreeding within the line at the time it is measured. Because rxy and F are both zero for true nonsibs, rX must reflect directly average inbreeding. From Tables 20, 21, and 22 and Tables 2, 3, and 4, when F is subtracted from rxy of nonsibs, the panmictic value of rxy (i.e., zero) results. Similar results are obtained for full sibs if one subtracts 0.75 F and likewise for half sibs if 0.875 F is removed. In most cases a second degree polynomial can be used to describe the time trend of rxy for the nonsibs. Figure 13 shows the trend of rxy for full, half, and nonsibs for random 97 .mmum mm3 GOHDMCHQEoumH paw Hmauumm 003 mucosafiop c033 coauomamm vamauocmm 6cm EOUGMH How mnamco: cam .mam: Hasw wow mxu. Mo mamas mcoflpmumcmo ON m._.. 0H fiH NH m 0 1v p — L F p — s — - - . — _ b m I D “\\\K\\\ .UIQ \\ \\\ mzlnl \o\\\ \\ mmlul| \\\\o\\\ \\\\\ \ . \ R \O \ \ . \ \\ .\ \o \\\\.\ \ s |\|-.\ ‘.\ \ u‘.“ -\\\ \\ \\ \ O m . \ NV] _ INH.O IwN.o I Om.o l m¢.O [00.0 I.NN..O l. Om.o l om.o .MH musmflm Xx 98 and genotypic selection when dominance was partial and recombination was free. Trends for rxy between full and nonsibs when selected was on genotype, gene action was either partial or overdominant, and recombination rates between adjacent loci were 0.50 are shown in Figure 14. Kempthorne (1957) defined Fx-y as the expected inbreeding of a random offspring (Z) produced by individuals X and Y. From previous notation F r , and rzz (the x-y xy coefficient of parentage of the offspring with itself) is l/2(l+rxy)' In this study the average rxy values for the entire population at a given generation are combinations of those for full, half, and nonsibs. Since there are 1128 possible pairs; 48 full sibs, 432 half sibs, and 648 nonsibs, the average rxy is weighted accordingly. For example, F(2) may be expressed as rxy(l)' where rxy(l) is the weighted term for generation one. Under random selection rxy(l) was 0.0585, and the average F in the first generation was 0.0587. This same procedure could not be applied when individuals were selected nonrandomly because the average rxy(l) would not be indicative of those selected to be parents. However, where one knows F may be computed because in— , r . (l) xy(1-l) breeding in generation (i+1) is representative of rxy for the selected parents in the preceeding generation. Values of F for various combinations of selection, dominance, and linkage gives some idea of the extent that selected parents are related. 99 .mmum mm3 cowumcwneoomu 0cm oamhuocmm mM3 coauumamm x :053 mocmcflaocum>o 0cm .HMAuHmm How mnwmcoc cam HHDM How H MD mama: .wa musmflm mGOHumnmcmo ON _mH NH 1v...” NW. OH O o .v N O p p p p p — - L s — p — p b filo \ Q». 0 I O \\ .NH.O m u o \\ \ 62 I ll \\ \o\ 6.626 mm \\D \\\\ \\\ \ \O\ \O\ . \ \ \ .IQMoO \\\b \. . \. \\LK\ \ \‘ \O\\ o levuo \ \ \ O \\ \O\ \ \o\ \0\ . .068 \ \o\\ . \ lNh.O \ \O. . \\ Y\\ O 1¢moo Tom.o Ax 100 Relationship_terms sxy and syx.—-sxy and Syx are probabilities that both genes of one locus for one individ- ual are alike by descent to a random gene of the other indi- vidual. Although sxy and syx are not necessarily equal (Harris, 1964), the two parameters were not significantly different. In the first generation, that sxy and syx were zero for all sib groups was expected because when either F or rxy are zero, sxy and Syx must be zero also. Both response over time and pattern of relative values for dif- ferent treatment combinations were similar to those de- scribed earlier for rxy' although the magnitude was differ- ent. Graphic representation of trends of Sxy are in Fig- ures 15 and 16. Relationship term txy'__txy takes nonzero values only where all four genes at a given locus of the two indi— viduals are identical by descent. Both individuals must be inbred and related. Thus, txy must always be less than or equal to F, and txy can never be larger than either sXy or syx' The difference between tXy and either Sxy or Syx pro- vides the probability of exactly 3 of the 4 genes being alike by descent at a locus. This event is P3 or P4 in Harris' (1964) description. Again, the general pattern for this parameter follows that of rxy’ but txy by the 18th generation for populations with complete or overdominance, tight linkage, and selection on genotype was much lower than for other groups. Thus, not only inbreeding was reduced in 101 .mmum mmB Gowumcwnfioomu cam Hmfluumm 003 mocmcfleoc 20:3 cofluumamm oammuosmm can Soccmn How mnflmcoc cam .mamn .HHDM How mxm mo mamas mnOHumumamw em 6.3.we.we.mm.em.w wrw. \ \\\ .o\ \\ .\ \ \‘ p \O ‘ \\ \ .7. .\.\\\M«\\\\. U l 0 \.w a \ mzl-..| \ \\. . \XK . \\ mm \ \ o \ o ..mH onsmflm W310 F¢N.o I om.o sz 102 .oflmmuocmm mm3 coauomamm G033 mwumu coflpmcflnfioomu mmum 0am 30H um mucmcaaowuo>o 0cm .mpmHmEoo .HMHuHmm How mnflm HHSM How xm mo mcmwz .oH musmflm mcoflumuwcmo 3.8.ma.fl.fl.£.m.m.w.fl.o INH.0 I¢N.0 10m.0 rmv.0 sz 100.0 uNh.0 I¢0.0 .00.0 103 Table 23. Means of sX where recombination rates between adjacent loci were 0.50 Generations Selection Dominance Sibship 3 8 13 18 F8 .07 .22 .41 .55 Partial HS .05 .19 .37 .52 NS .02 .13 .32 .48 F8 .06 .22 .39 .52 Random Complete HS .04 .18 .35 .49 NS .01 .14 .29 .44 F5 .05 .19 .36 .60 Over HS .05 .15 .31 .46 NS .02 .11 .26 .42 FS .09 .31 .56 .75 Partial HS .07 .27 .53 .73 NS .04 .23 .50 .71 F8 .10 .33 .57 .69 Genotypic Complete HS .07 .29 .54 .66 NS .05 .24 .51 .64 F8 .07 .23 .39 .49 Over HS .05 .19 .35 .46 NS .02 .14 .30 .43 PS .07 .25 .43 .60 Partial HS .05 .20 .38 .57 NS .02 .16 .34 .54 F8 .07 .25 .41 .55 Phenotypic Complete HS .05 .21 .37 .51 NS .03 .17 .32 .48 FS .05 .23 .41 .56 Over HS .04 .19 .37 .52 NS .02 .14 .32 .48 104 Table 24. Means of sX where recombination rates between adjacent lozi were 0.26 Generations Selection Dominance Sibship 3 8 13 18 ES .07 .24 .41 .54 Partial HS .05 .20 .36 .50 NS .03 .16 .32 .47 PS .06 .21 .40 .55 Random Complete HS .04 .17 .35 .51 NS .02 .13 .31 .48 F8 .06 .20 .40 .55 Over HS .04 .15 .36 .51 NS .02 .12 .32 .48 F8 .09 .31 .55 .73 Partial HS .06 .27 .51 .71 NS .04 .22 .48 .69 F8 .09 .35 .55 .72 Genotypic Complete HS .06 .31 .52 .70 NS .04 .27 .48 .68 F8 .07 .26 .40 .50 Over HS .05 .21 .36 .47 NS .03 .17 .32 .43 F5 .06 .20 .37 .52 Partial HS .04 .16 .32 .48 NS .02 .ll .27 .45 ES .07 .29 .46 .58 Phenotypic Complete HS .06 .25 .42 .55 NS .03 .20 .38 .51 F8 .06 .21 .36 .50 Over HS .04 .17 .31 .46 NS .02 .13 .27 .42 105 Table 25. Means of 5x where recombination rates between adjacent loci were 0.02 Generations Selection Dominance Sibship 3 8 13 18 F8 .06 .24 .36 .56 Partial HS .04 .20 .32 .53 NS .02 .14 .28 .47 F8 .05 .20 .39 .55 Random Complete HS .03 .16 .35 .51 NS .01 .10 .30 .48 F8 .06 .23 .38 .52 Over HS .04 .19 .33 .48 NS .02 .14 .29 .44 PS .09 .39 .60 .77 Partial HS .08 .34 .57 .75 NS .06 .32 .56 .73 F8 .08 .24 .34 .40 Genotypic Complete HS .06 .21 .29 .36 NS .04 .17 .27 .34 F8 .06 .21 .27 .30 Over HS .04 .17 .24 .28 NS .02 .14 .21 .27 F3 .07 .27 .51 .67 Partial HS .04 .22 .46 .64 NS .02 .18 .43 .61 F8 .05 .21 .39 .58 Phenotypic Complete HS .03 .18 .34 .56 NS .01 .14 .29 .52 F8 .06 .17 .30 .40 Over HS .03 .13 .26 .35 NS .02 .10 .21 .31 106 these groups but also relationship between two individuals. As a result, values of tXy were less than half as large as when recombination was less restricted. The second degree polynomial for txy with time fit closely for all 27 treat- ment combinations. Figures 17 and 18 show responses of txy over the 20 generations. is the probability that Relationship term u" .--u AY XY each of the two genes of x is alike by descent to a differ- ent one of the two genes of y with the restriction that the two members of the pair of genes possessed by either x or y cannot be alike by descent to each other. This event is described by P5 in Harris' (1964) notation. The value of uxy was 0.25 for all full sib groups in the first generation and zero for the half and nonsib groups as expected. This was only one of the parameters of rela- tionship that did not closely follow a second degree polyno- mial response with time. This difference could be expected since inbreeding in both related individuals would cause P5 to decrease and some other probabilities such as P1, P3, and P to increase. The resulting effect was an increase in s 4 s , r , and F. YX XY xy' Selection on genotype or phenotype tended to produce smaller values of uxy by the 18th generation than did selec- tion at random, and linkage accelerated this effect when genes exhibited partial dominance. For populations with complete or overdominance, tight linkage in genotypic 107 .mmnm mm3 GOHumcHQEOUmH 0am HMHuHmm mm3 mucmcHEov :mn3 GOHuomHmm UHmmuocmm 0cm EGCGMH How mQHmcoc 0cm .MHmn .HHDM How xx“ 00 mama: mcoHumumcmo 0N . fim 0H ¢H NH JH. m 0 ¢ N p r - p P b L b - mz IIIIIII \. \ mull-I \ \o.\ \o\ . \ \ \o\ mm s \x \ . \\ \ox \ \ .\.\ \ .\Q\ \.\ \ . x \\ \ \\fi\\.\ \\ .\.\b .\ . \. \x m \o. \\ C \. .hH musmHm INH.0 uwN.0 10m.0 um¢.0 [00.0 INh.0 uvm.0 I00.0 Xxq 108 .onmuocmm mm3 COHuUmHmm G033 mmumu COHumcHnfioomH 0000 0am 30H um mocmaHEOCH0>o 0cm mumHgEoo How mnwm HHsm MOM xx» 00 mammx MCOHHMHmcmw ON 0H 0H ¢H NH 0H 1m 0 w n — P P p — L by b p h b I? p .i \.A“ \ |.\.IQ‘ . I‘ *\ \ \Ol. I |.|l.\\\ \O\ \ \ \ O \ \\ OI ll lulu I |o ox \ \ o \\O\ \ o \O‘ \O\ NO. " .H I . om. " .H I. o canon 0 III .EOQ 0 .0H musmHm 0 0 I..NH.0 l ¢N.0 r 3.0 Axq 109 Table 26. Means of tX where recombination rates between adjacent loci were 0.50 Generations Selection Dominance Sibship 3 8 13 18 F8 .04 .16 .34 .49 Partial HS .02 .11 .29 .44 NS .01 .07 .24 .39 PS .03 .16 .31 .45 Random Complete HS .01 .11 .25 .40 NS .01 .08 .21 .36 F8 .03 .13 .29 .43 Over HS .01 .08 .22 .37- NS .01 .05 .18 .34 F8 .05 .24 .50 .71 Partial HS .03 .18 .45 .68 NS .01 .14 .41 .66 F8 .06 .25 .50 .64 Genotypic Complete HS .03 .19 .45 .60 NS .02 .16 .43 .57 PS .04 .16 .31 .42 Over HS .01 .11 .25 .37 NS .01 .07 .21 .34 F8 .04 .18 .35 .54 Partial HS .01 .12 .29 .49 NS .01 .09 .25 .45 ES .04 .19 .34 .48 Phenotypic Complete HS .02 .13 .27 .43 NS .01 .09 .23 .39 F3 .03 .17 .34 .49 Over HS .01 .11 .28 .44 NS .01 .08 .23 .40 llO Table 27. Means of tx where recombination rates between adjacent loci were 0.26 Generations Selection Dominance Sibship 3 8 13 18 PS .04 .17 .34 .48 Partial . HS .02 .12 .27 .42 NS .01 .09 .23 .38 F8 .03 .15 .33 .49 Random Complete HS .01 .09 .27 .43 NS .01 .06 .23 .39 PS .04 .14 .33 .49 Over HS .01 .09 .27 .43 ‘ NS .01 .06 .23 .40 PS .05 .24 .49 .68 Partial HS .02 .18 .43 .65 NS .01 .14 .40 .63 F3 .05 .27 .48 .67 Genotypic Complete HS .03 .22 .44 .64 NS .02 .18 .40 .62 F8 .04 .18 .32 .42 Over HS .02 .12 .26 .37 NS .01 .10 .22 .33 PS .03 .13 .29 .46 Partial HS .01 .08 .22 .40 NS .01 .05 .18 .36 F8 .04 .22 .39 .52 Phenotypic Complete HS .02 .16 .33 .47 NS .01 .12 .29 .44 F8 .03 .14 .29 .43 Over HS .01 .09 .22 .37 NS .01 .06 .18 .33 111 Table 28. Means of tx where recombination rates between adjacent loci were 0.02 Generations Selection Dominance Sibship 3 8 13 18 F8 .03 .17 .29 .49 Partial HS .01 .ll .22 .45 NS .01 .07 .18 .38 F8 .02 .14 .31 .49 Random Complete HS .01 .08 .25 .43 NS .01 .04 .20 .40 F8 .03 .17 .30 .45 Over HS .01 .ll .24 .40- NS .01 .07 .20 .35 F8 .05 .31 .53 .73 Partial HS .04 .25 .50 .70 NS .02 .22 .48 .68 F8 .04 .16 .24 .30 Genotypic Complete HS .02 .11 .19 .26 NS .01 .09 .17 .24 F8 .03 .13 .17 .20 Over HS .01 .09 .15 .18 NS .01 .06 .ll .16 PS .04 .20 .44 .62 Partial HS .01 .14 .38 .57 NS .01 .10 .34 .54 F8 .03 .14 .31 .51 Phenotypic Complete HS .01 .10 .24 .48 NS .01 .06 .19 .44 F8 .03 .ll .23 .32 Over HS .01 .06 .16 .26 NS .01 .04 .12 .22 112 selection caused an increase in uxy above that observed with less restricted recombination. Figures 19, 20, and 21 show the fluctuations of uxy over the 20 generations for various groups. uxy for full sibs decreased to the lowest level of any sib groups by the 20th generation when selection was on genotype, recombination was free, and gene action was partial dominance. Figure 20 shows the effect <15 tight linkage on uxy between full sibs when selection was on genotype, and gene action was either complete or overdominance. After the 8th generation the group with tight linkage fluctuated more than did the groups- with free recombination, but for both complete and overdomi— nance tight linkage tended to maintain the mean uxy between full sibs. Figure 21 shows the same treatment combinations but for nonsibs. Relationship term_yxy.——VXY represents the probabil- ity that both individuals are inbred at a particular locus but none of the genes of x are identical by descent to any of the genes of y at that locus. As expected, the nonsibs had larger mean vX than did either the half or full sibs. Small vXy for full sibs are attributable to close relation- ship which results in identity by descent of all four genes at more loci than for nonsibs. The increase over generations in ny for full sibs when selection was random expresses the gradual degeneration of heterozygosity in the population, but since there was no directional force, loci were equally 113 .mmum mm3 GoHumcHneoomn 0cm HMHuHmm mm3 woxmcHEoc cmnz GOHuomHmm owmmuocmm 0cm Eocewn Mom mQHmcoc 0cm .mamn .HHSM mom x5 00 mamas .mH 0Hdem chHumumqmw ON mH 0H ¢H NH 0% m P . b . _ L _ . — . r _ b "\O 0 INI-.IQI. " - .1 III?" . IIoI \IHM“\.\ :36 II I. a I. Wmummm.IJw“”Iw “JH\I\.\I ‘ A I D . \’ . [£5 nomé Im¢.0 Kxn l00.0 mlO 0|. rNh.o mZ Illunlll mm I II .86 mm r0m.0 114 .UHmauoamm mw3 COHuumHom 20:3 mmumu COHuMGHnEOUmH mwum 0cm 30H um mUGMCHEOGHw>o 0cm mumHmaoo How mnHm HHSM How mxd Ho mama! .0N mudem mQOHUMHmGQG m - mH F 0%... p ”H - N—H P QM - my _ w — W P m n o o INHoo II to, I IIIIm II II Ior’dtl-J’I I JIIIII . IIIIIP I II ,/ I o r0M.O 10¢.O n X TA [00.0 N0. H H I O INN. O om. u u I O .aon 0 III Ivmo .EOQ U .86 115 .onmuocmm 003 COH000H00 c053 00u0H 00H00GHAEOU0H 0000 0:0 30H 00 00:0:H80000>o 0:0 000H0800 How 0QH0G0: Mom xxs mo 0G00z .HN 0H50Hm 0G0Hu0n0c0o om 3 3 3 3.3 w 0 LFIP._.?r_p_pt._ T NH.O I I. \O\ IIIIIIII HVIO\ \ I, II0 \ AI .II \ a O n s X rA luemé N0. H H I my cm. H H I O. .500 0 III .260 0 I020 116 Table 29. Means of ux where recombination rates between adjacent loci were 0.50 Generations Selection Dominance Sibship 3 8 13 18 F8 .25 .27 .23 .19 Partial HS .06 .13 .14 .13 NS .03 .09 .12 .11 F8 .25 .26 .24 .20 Random Complete HS .05 .13 .16 .13 NS .02 .10 .13 .12 F8 .25 .28 .25 .21 Over HS .05 .13 .15 .15 NS .02 .09 .12 .12 F8 .25 .26 .18 .11 Partial HS .07 .16 .13 .09 NS .04 .13 .11 .07 F8 .25 .24 .17 .13 Genotypic Complete HS .08 .15 .12 .10 NS .05 .12 .11 .09 F8 .24 .25 .23 .20 Over HS .06 .14 .16 .15 NS .03 .ll .14 .14 F8 .25 .26 .24 .17 Partial HS .05 .13 .16 .13 NS .02 .09 .13 .11 F8 .25 .27 .23 .19 Phenotypic Complete HS .06 .15 .15 .14 NS .03 .11 .13 .13 F8 .25 .27 .24 .19 Over HS .05 .14 .15 .14 NS .02 .10 .13 .12 117 Table 30. Means of uX where recombination rates between adjacent logi were 0.26 Generations Selection Dominance Sibship 3 8 13 18 F8 .25 .27 .24 .20 Partial HS .06 .14 .16 .14 NS .03 .10 .13 .13 F8 .26 .27 .24 .19 Random Complete HS .05 .13 .15 .13 NS .02 .09 .11 .11 F8 .26 .27 .22 .18 Over HS .04 .12 .14 .14 . NS .02 .07 .09 .10 F5 .25 .25 .18 .12 Partial HS .07 .15 .13 .10 NS .04 .11 .ll .09 FS .25 .23 .18 .12 Genotypic Complete HS .07 .15 .14 .09 NS .04 .ll .11 .08 F8 .25 .25 .23 .20 Over HS .06 .16 .16 .16 NS .04 .ll .15 .14 F8 .25 .28 .25 .20 Partial HS .05 .14 .16 .14 NS .02 .09 .12 .12 F8 .25 .26 .22 .17 Phenotypic Complete HS .06 .15 .15 .13 NS .03 .12 .12 .12 F8 .25 .27 .25 .21 Over HS .05 .14 .16 .15 NS .02 .09 .13 .13 118 Table 31. Means of uX where recombination rates between adjacent loci were 0.02 Generation Selection Dominance Sibship 3 8 13 18 PS .26 .27 .26 .19 Partial HS .05 .14 .17 .12 NS .02 .10 .13 .12 F5 .25 .28 .25 .20 Random Complete HS .06 .13 .16 .14 NS .02 .09 .12 .12 F8 .26 .27 .24 .21 Over HS .05 .13 .15 .14 NS .02 .09 .12 .12‘ F8 .24 .20 .14 .09 Partial HS .09 .14 .ll .07 NS .07 .11 .10 .07 F8 .27 .24 .24 .22 Genotypic Complete HS .07 .15 .21 .21 NS .03 .12 .19 .19 PS .25 .24 .24 .23 Over HS .07 .16 .18 .20 NS .04 .12 .15 .18 PS .26 .25 .20 .14 Partial HS .07 .14 .14 .10 NS .03 .10 .ll .08 PS .26 .27 .24 .17 Phenotypic Complete HS .05 .15 .16 .13 NS .03 .10 .14 .12 F8 .26 .28 .27 .23 Over HS .06 .13 .16 .17 NS .03 .09 .12 .14 119 likely to become fixed for either allele. Chance determines in which state the locus becomes fixed, and the process of achieving fixation is slow. If one studied additional gen- erations, this term probably would decrease at the expense of other terms. When selection was on genotype, vxy was lower in populations with partial or complete dominance and free recombination than when genes were linked. Populations with overdominance showed larger ny than those with less dominance because selection favored heterozygotes. When selection was not random, tight linkage caused larger vX than when recombination was less restricted. The time trends of vXy were similar to but smaller than those of uXY with respect to linkage effects. Geno- typic selection tended to decrease this parameter within a given sib—group by the 18th generation. Figures 22, 23, and 24 show the trends in time for ny° Genotypic Covariance In general, the genotypic covariances between rela— tives increased over generations, but there were some notable exceptions to this trend. When selection was random, covar- iances between full, half, and nonsibs reached almost the same value by the 18th generation for a given degree of dominance. This result is similar to trends in other param- eters of relationship when selection was random even though the values were quite different in the first generation. 120 .0000 003 :0H00:HQEOU0H 0:0 H0Huu0m 003 00:0:HEO0 :003 :0H000H00 0Hm>00:0m 0:0 Eo0:0u How 0QH0:0: 0:0 .MHms .Hasm How hkb 00 0:00: 0:0Hu0u0:00 R . WH b @—H . .Vfi . “H . QH . & P W . W p l H 21 I - w. ‘ 1.. .1... .. ,-%v..\..\.1 “Hulk-I II \\‘II’IOIIIII‘Q‘ “ldlII-IIOI .\|| 0 \ .IIOI. IQ... \OI..I.O\. |II|.IIIOI \ \\IO\ C(‘I‘I‘ . (.‘IL‘III .NN 0H50Hm 0 0 INH.0 A X .A l0N.0 I0m.O 121 .UHQMuo:0m 003 :OHUU0H00 :033 0000H :0H00:HABOU0H 0000 0:0 30H 00 00:0:HE00H0>0 0:0 0u0HmEOU How 0QH0 HHDM Mom mx> mo 0:002 .mN 0H50Hm 0:0Hu0u0d0w a we a 3. 0.0.0.011 N . \IIfIlO/ \ ‘IQI . Till-OIII /b\ FNH A x K I¢N N0. "HI 0 om. nu: O .EOQOI-I .Eonolll .EonmI I0m 122 .oflmmuo:0m 003 :0H000H00 :003 00u0H :0H00:HAEOU0H 0000 0:0 30H 00 00:0:HEO0H0>0 0:0 0u0HmEOU How .0QH0 HHDm Mom >x> mo 0:002 .vN 0usmHm 0:0H00H0:0w £m JH 0H 0H NH OH O . . _ . _ . _ . _ . O I' l O U ' II .NH.O A X rA I¢N.O 8. u 0 no om. u u u. oEOQ 0 III .68 0 I 13.6 123 Table 32. Means of vx where recombination rates between adjacent logi were 0.50 Generations Selection Dominance Sibship 3 8 13 18 PS .01 .01 .02 .02 Partial HS .01 .02 .03 .03 NS .01 .05 .05 .06 F8 .01 .01 .02 .02 Random Complete HS .01 .02 .03 .03 NS .01 .04 .06 .06 F8 .01 .01 .02 .02 Over HS .01 .02 .03 .03 NS .01 .05 .06 .06 PS .01 .02 .02 .01 Partial HS .01 .03 .03 .02 NS .02 .06 .05 .03 F8 .01 .02 .02 .01 Genotypic Complete HS .01 .04 .03 .03 NS .02 .06 .05 .04 F3 .01 .02 .02 .03 Over HS .01 .03 .04 .04 NS .01 .06 .07 .06 F8 .01 .01 .02 .02 Partial HS .01 .03 .03 .03 NS .01 .05 .07 .06 F8 .01 .01 .02 .02 Phenotypic Complete HS .01 .02 .03 .03 NS .02 .05 .06 .06 FS .01 .02 .02 .02 Over HS .01 .03 .04 .03 NS .01 .05 .06 .06 124 Table 33. Means of vX where recombination rates between adjacent loci were 0.26 Generations Selection Dominance Sibship 3 8 13 18 F8 .01 .01 .02 .02 Partial HS .01 .03 .03 .03 NS .02 .05 .06 .06 F3 .01 .01 .02 .01 Random Complete HS .01 .02 .03 .03 NS .01 .05 .06 .06 PS .01 .01 .01 .02 Over HS .01 .02 .03 .03 NS .01 .04 .05 .05 F8 .01 .02 .02 .01 Partial HS .01 .03 .03 .02 NS .02 .05 .05 .04 PS .01 .02 .02 .01 Genotypic Complete HS .01 .03 .03 .02 NS .02 .06 .06 .04 F8 .01 .02 .03 .03 Over HS .01 .03 .04 .04 NS .02 .05 .07 .07 F8 .01 .01 .02 .02 Partial HS .01 .02 .03 .03 NS .01 .04 .06 .06 PS .01 .02 .02 .02 Phenotypic Complete HS .01 .03 .03 .03 NS .02 .06 .06 .06 F5 .01 .01 .02 .02 Over HS .01 .02 .03 .04 NS .01 .05 .06 .06 125 Table 34. Means of vx where recombination rates between adjacent lozi were 0.02 Generations Selection Dominance Sibship 3 8 13 18 F5 .01 .01 .02 .02 Partial HS .01 .03 .04 .03 NS .01 .04 .06 .06 PS .01 .01 .02 .02 Random Complete HS .01 .03 .04 .03 NS .01 .05 .06 .06 ES .01 .01 .02 .02 Over HS .01 .03 .03 .03- NS .01 .05 .06 .05 F8 .02 .03 .03 .02 Partial HS .02 .04 .04 .02 NS .03 .06 .06 .04 ES .01 .03 .04 .04 Genotypic Complete HS .01 .05 .06 .06 NS .02 .06 .08 .08 PS .01 .03 .04 .06 Over HS .01 .05 .06 .07 NS .01 .06 .07 .09 F8 .01 .01 .02 .01 Partial HS .01 .03 .03 .02 NS .01 .05 .06 .04 F5 .01 .02 .02 .02 Phenotypic Complete HS .01 .03 .04 .03 NS .01 .06 .08 .06 FS .01 .01 .02 .03 Over HS .01 .02 .04 .04 NS .01 .05 .07 .07 126 For example, the nonsibs had zero genotypic covariance in the first generation, and the full sibs had twice as much or more than half sibs. The degree of covariance in the first generation depends on level of dominance. In an additive model, run only as a check, the genotypic covariance was 1/2 the genotypic variance for generation one for full sibs, 1/4 for half sibs, and 0 for nonsibs. However, by the 20th generation, the genotypic covariance equaled 0.87 of the total genotypic variance in the population for all sib groups. This was identical with a result calculated from the theoret— ical genetic correlation given by Kempthorne (1957): 2r 0% 2r [(1+F£) ° (1+FY)] cg (1+Fx) For the additive case Harris' (1964) theory is identical with Kempthorne's. ’With random selection as the level of dominance increased, the genotypic covariance for any genera— tion decreased. Cockerham (1956) reported that the presence of link- age in randomly mating populations would affect the covari- ances of relatives but only where one relative was not an ancestor of the other. He contended that only epistatic components of the covariance were subject to bias by linkage. Schnell (1963) also pointed out that linkage would affect the epistatic component of covariance, and Van Aarde (1963) showed that linkage could influence the covariances between 127 lineal descendants more distant than parent—offspring. Indications from this study (Tables 35 and 36) are that linkage in the absence of selection does not significantly affect covariances of inbred relatives although the covar- iances were consistently slightly smaller with tight linkage. and only collateral relatives were evaluated. When selection was on genotype, covariance decreased much as total genotypic variance did. This is a necessity since the upper limit of genotypic covariance is the geno- typic variance. Parker (1966) showed some of the relation- ships between covariance, variance, and correlation between two traits. In complete dominance and selection on genotype, the covariance did not show as marked increase with tight link- age as in populations with partial dominance. This is con— sistent with previous results of this study since the mean inbreeding was less for tight linkage with complete or over- dominance and genotypic selection. ,When selection was intense Parker (1966) found genetic covariance quickly declined but variance also tended to decrease at a proportionate rate to maintain a constant genetic correlation where heritability was moderate. While his study concerned genetic correlation between two traits and inbreeding was not considered, there are parallels between this study and his work particularly with respect to effects of selection on variances and covariances. 128 Table 35. Genotypic covariances between relatives where recombination rates between adjacent loci were 0.50 Generations Selection Dominance Sibship 3 8 13 18 F8 7.4 1.7 16.0 18.7 Partial HS 4.5 9.5 14.4 17.7 NS 1.5 7.1 13.0 16.3 FS 5.6 9.2 11.3 12.3 Random Complete HS 2.7 7.2 9.9 11.2 NS 1.0 5.6 8.9 10.5 FS 4.5 6.2 7.5 8.6 Over HS 1.8 3.8 5.4 7.1 NS 0.7 2.8 4.5 6.4 FS 6.0 5.2 2.6 2.0 Partial HS 3.7 4.2 2.4 2.0 NS 1.7 3.4 2.2 2.0 FS 4.6 3.8 2.9 2.4 Genotypic Complete HS 2.6 3.0 2.5 2.2 NS 1.3 2.3 2.2 2.1 FS 3.1 2.7 2.2 1.7 Over HS 1.0 1.5 1.2 1.1 NS 0.3 1.0 1.0 0.9 FS 6.7 0.8 14.3 17.1 Partial HS 4.0 8.5 12.8 16.1 NS 1.0 6.5 11.3 15.1 FS 5.0 7.9 9.1 10.3 Phenotypic Complete HS 2.7 6.2 7.8 9.3 NS 1.0 5.0 6.9 8.7 FS 4.1 6.1 7.0 7.4 Over HS 1.4 4.0 5.1 6.1 NS 0.3 2.9 4.3 5.4 129 Table 36. Genotypic variances between relatives where recombination rates between adjacent loci were 0.02 Generations Selection Dominance Sibship 3 8 13 18 F8 7.2 11.2 14.2 17.9 Partial HS 3.9 9.0 12.5 17.0 NS 0.9 6.7 10.8 15.6 FS 5.2 7.6 10.2 12.0 Random Complete HS 2.5 5.6 8.6 10.7 NS 0.7 4.1 7.4 9.9 FS 4.5 6.1 7.0 8.3. Over HS 1.6 3.9 5.1 6.7 NS 0.6 2.8 4.0 5.8 FS 6.3 8.7 8.6 7.9 Partial HS 4.5 7.7 8.1 7.5 NS 2.6 7.0 7.8 7.3 FS 4.3 4.4 4.4 4.4 Genotypic Complete HS 2.3 3.5 3.7 3.8 NS 0.9 2.8 3.5 3.4 FS 2.8 2.2 1.9 1.6 Over HS 1.0 1.0 0.9 1.2 NS 0.1 0.4 0.5 1.0 FS 7.3 11.0 15.7 18.9 Partial HS 4.2 8.7 13.9 17.7 NS 1.2 6.7 12.4 16.8 FS 5.0 7.9 9.7 11.1 Phenotypic Complete HS 2.6 6.0 8.3 10.5 NS 0.8 4.8 7.3 9.7 FS 4.1 4.8 5.8 6.4 Over HS 1.3 2.6 3.9 5.2 NS 0.4 1.7 2.9 4.4 130 When selection was intense and heritability high, Parker (1966) found a rather distinct reduction in the correlation coefficient; and this reduction came only after 15 generations of selection. He concluded that genetic correlations may not provide reliable estimates of genetic covariances. The interaction of dominance, selection and linkage was marked; but analysis of variance was not used to evaluate the covariances in this study because of lack of information concerning their distributions. Genotypic selection in free recombination and partial dominance seemed to cause a sharp decrease in the covariance between full sibs. This covariance remained fairly stable until the 8th or 9th generation of selection then decreased markedly. The covariance of half sibs increased until the 8th or 9th gener- ation, then decreased rather rapidly. Figures 25, 26, and 27 show tinmztrends for this parameter. The nonsibs followed a similar pattern, but increases in early generations were larger. Fixation caused leveling of values of this param- eter by the 17th generation. Because populations with_free recombination or moderate linkage reacted similarly, only results for tight linkage and free recombination will be discussed. Parameters evaluated previously indicated that r'is a more descriptive parameter than simple recombination rates between adjacent loci to describe effects of linkage in a population. 131 .EO0:00 003 :0H000H00 0:0 000% 0003 00000 :0H00:HQE0000 .00:0:H800 H0Hu00m 003 :00000 0:00 000:3 0200:o: 0:0 .MHmn .HHSM :003u0n 00:0H00>oo UHQNuo:0m mo 0:002 0:0Hu000g00 ON mH “H ¢H NH OH L P p n — P.I— - — — P0) .mN 005000 [.NH O o A e 1 To 2 u o e .I 0N 132 .000Hu 003 000x:HH 0:0 00:0:HEO0.H0H0000 003 :OHuu0 0:00 000:3 :0H000H00 NO mmUOE GQHSH HOM.mQHm HHS“ Gmw3umfl GUSMHHM>OU UHQkfiocmm MO mcmmz .ON mhdmflh mGOHHMHmCmO ON 0 H H NH 0H 0 0 N 0 L w L WP W L — . _ L $ _ _ p b . _ O TtlI-l u\.\I\ IIATII .IIQIII .IIICIIIIIILTIII \\ .I \ IL I NH m \\\ A \\Q\ m \\\ T. E H II _I O a I 0N OIII mIII 0 mm .I 0m .0mmuo:00 :0 003 :0H000H00 0:0 .un0Hu 0:0 000m 0003 00000 :0H00:HQEOU00 .00:0:HEO0 000Hmaou 0:0 H0Hu000 003 :0H000 0:00 000:3 00H0 HHSM :003002 00:0H00>00 0H0>00000 HO 0:002 .nN 00:0Hm 133 mGOHHMHmflmw 0N 0H 0H NH NH 0H 0 0 .0 N 0 _ . _ p F L b — b '— p b . — b L L _ . O I I0: H lolldIIuIlull .r HmIIHWIanIIt. [III-Ill O\Q l.NH 3 O A E I T. . 2 u D 8 I.¢N me. nu- .68 onlllu om. u 0 .. .ao: olIIlI NO.H.HI.EOQm°.I.O l_0m 134 When genes exhibited complete dominance and selec- tion was on genotype, the degeneration of genotypic covar- iance started near the 5th generation for full sibs but did not decrease proportionately as much as when dominance was only partial. Results for populations with overdominance and the same linkage and selection tended to mimic those for populations with complete dominance except for size. When linkage was tight, the trend in values over time changed considerably. By the third generation the mean covariance for full, half, and nonsibs with partial dominance was larger than when recombination was free. Covariance in- creased or stayed at the same level until about the 18th generation when it started to decrease for all sib groups. It should be recalled that populations with tight linkage did not reach fixation due to selection on genotype as did those with free or moderate linkage and partial dominance. The mechanism involved is unknown, but the following would be consistent with the observations. Two related individ- uals are more likely to have the same genotypic configura- tion at a given locus when linkage is intense than when it is nonexistant. Tight linkage tends to increase mean in- breeding, even though more unfavorable genes are fixed than when recombination is free. Closely related individuals then should have increased covariance. Tight linkage also makes selection less effective. Thus, the decrease in covariance due to selection comes about much later with 135 tight linkage than when recombination is free. Variation within lines showed a similar trend. Phenotypic selection in the absence of linkage gave results similar to those for lines with random selection. The final levels of covariance were only slightly less for phenotypically selected individuals, and most of this dis- crepancy resulted in the early generations when selection was most effective. Tight linkage tended to increase covar- iances through the 18th generation for partial dominance. This was possible because the low selective force did not cause degeneration of genotypic variance. The increased inbreeding in the presence of tight linkage probably was responsible for the increased covariances. Linkage in the absence of epistasis but presence of inbreeding and selection caused covariances between rela- tives to be larger than when recombination was unrestricted, if genes exhibited partial dominance. Selection decreased covariances as it did genotypic variances, and the level of dominance affected the covariances. The interactions between selection, linkage, and dominance appeared to play a major role in determining the covariance. Table 37 shows the ratios of expected to observed genotypic covariances between relatives when selection was at random. Expected covariances were obtained with Harris' (1964) theory. Linkage appeared to have little effect on these ratios, and the expectations seemed more valid for 136 Table 37. Ratio of expected to observed genotypic covar— iances between full, half, and nonsibs where selection was random Generations Recombination Rate Dominance Sibship 3 8 13 18 F8 1.15 1.07 1.00 0.99 Partial HS 1.24 1.08 1.01 0.99 NS 1.88 1.14 1.01 0.99 FS 1.15 1.03 1.04 1.08 0.50 Complete HS 1.33 1.04 1.04 1.08 NS 1.77 1.06 1.02 1.06 FS 1.30 1.32 1.32 1.25 Over HS 1.47 1.52 1.44 1.32 NS 1.88 1.44 1.47 1.33- FS 1.16 1.12 1.08 1.04 Partial HS 1.33 1.18 1.10 1.03 NS 2.71 1.27 1.14 1.04 FS 1.19 1.21 1.16 1.14 0.02 Complete HS 1.42 1.25 1.21 1.17 NS 2.43 1.29 1.24 1.18 FS 1.31 1.38 1.42 1.31 Over HS 1.66 1.52 1.56 1.42 NS 2.17 1.67 1.70 1.47 populations with partial dominance than for other levels of dominance. In most cases, predictions for nonsibs were less accurate than those for full or half sibs. expected rather than In applying Harris' equation, observed variances were used. Thus, if random fluctuations caused deviations of observed variances from expectations, larger errors of estimation could occur than if observed 137 values were used. This would be most noticeable in the early generations when covariance terms were rather small. Covariances between relatives are normally used to estimate heritability. In the narrow sense heritability is described by Lush (1948) as 03/06 where 0% is additive genetic variance and 0% is phenotypic variance for the trait in the population. Anything that biases either numerator and/or denominator of this ratio will also tend to bias genetic progress predicted by heritability. When a model includes epistasis, for example, genotypic covariance between full sibs contains components other than additive variance. Cockerham (1956) has indicated that only the epistatic components of the covariances between relatives which are subject to bias from linkage are affected. Like- wise the model of Schnell (1961) included epistasis, and the epistatic component was affected by linkage. Rowe (1966) indicated that the family component of variance was the portion biased by linkage in the model, and to this extent his variance component method failed to esti- mate heritability accurately. Upper truncation selection favored repulsion gametes and tended to generate negative linkage disequilibrium or a negative correlation between the effects of different loci. He also expressed the potential genic variance in the form Epigiai. He then compared the 138 actual genic variance with the potential genic variance and found the ratio of potential to actual variances to be high- est (>2.0) when selection was intense and linkage tight. Data in this study do not offer a direct comparison to Rowe's (1966) data because he evaluated the individuals as a unit rather than per locus except for the comparison of potential to actual genic variances. If individuals were evaluated as single units in this study, some parameters, particularly variances and covariances, likely would differ from observed results. For example, if negative disequilib- rium existed, as expected under selection, and linkage were tight so that loss of the disequilibrium was minimal, then the genotypic variances would not simply be the sum of the variances at all loci, but the negative correlation between loci would also need to be considered. APPLICATIONS OF RESULTS AND SUGGESTIONS FOR FURTHER RESEARCH Any simulated population has limitations, and direct application of observed results to real genetic populations, particularly large animals, may be misleading. Yet, several points seem worthy of mention. Linkage did affect all genetic parameters measured when there was inbreeding and selection--even in the absence of epistasis. The extent to which linkage affects param- eters appears to be related more closely with the average recombination rate between all loci (f) affecting a trait rather than to the recombination rate between adjacent loci (r). Large numbers of loci per chromosome and increased number of chromosomes both decrease E: For given r and num- ber of loci, fewer chromosomes lead to decreased f. Latter (1965), working with only two loci, also showed tight link- age (r==0.05) had little effect on the response to selection when the effective size of population (N) was 40 or greater. The best 10 percent of the population was selected each generation. He also varied the proportionate effects of genes, a/o, where "a" represents the difference in genotypic value between the two homozygotes, and 0 represents the standard deviation of the trait under consideration. He 139 140 found a significant interaction between N, a/o, and the level of recombination when he measured response to selec- tion. For any given N and proportionate effect, the popu- 1ations with tight linkage responded less than those with less restricted recombination when genes acted additively. For the same N and tight linkage but "high" proportionate effects, populations responded less. When N was increased to 40 the tight linkage (r==0.01) had some effect, but it was not nearly so marked as when N was small. Even with small N, populations with the larger proportionate effects responded considerably more for all linkage intensities than' did those with small a/c. All of this suggests that one needs to be cautious in generalizing results of "small" simulated studies to the general case. For if economic traits in dairy cattle and other species of livestock are controlled by many pairs of genes, "a/o" likely will be smaller than values used in this study. Also there probably will not be many instances where populations will approach the sizes used in most simulated studies. This, combined with the more realistic r values to describe recombination rates, will probably insure that all results from simulated studies are magnified compared to large animal populations. The most significant effects of linkage and domi- nance and their interactions were in populations selected on genotype. For large animals, where heritabilities of 141 most economic traits tend to be low to moderate and a high percentage of any population must be saved for parents, the selection process would not maintain nearly so much linkage disequilibrium as when selection is more intense. Many controversies that have developed in population genetics appear to have their basis in that various factors have been considered singly. This study, along with others, has demonstrated the fallacy of assuming that important factors do not interact. Although levels of some factors such as gene frequency, degree of linkage, and others may never be known for economic traits in large animals, their effects on estimates of genetic parameters should not be discounted, and the directions of biases are important, even if the magnitude of bias cannot be measured or applied. Knowledge of potential biases seems especially important for small populations. This study, like most others, has raised many new questions. For this particular type of study to be most usefully applied to animal breeding, a population should be simulated in as near agreement with known results for a particular species as possible. Because the pertinent num— ber of loci is unknown for most economic traits, one might wish to consider upper and lower bounds. This could involve simulation of as many as one thousand pairs of genes, each with small proportionate effects. Likewise various gene frequencies and recombination rates between various loci 142 could be simulated. And finally "realistically large" numbers should be used to avoid chance phenomena as much as possible. Other questions such as the effects of epistasis, disproportionate gene effects, varying recombination rates between loci on one chromosome, and other complications have not been considered. All of these could affect the measur- able parameters, mean and variance. Thus, their effects and interactions should be evaluated. SUMMARY Changes in mean and variance of genotypes in gene frequencies, and in genotypic covariances between relatives from differences in selection, linkage, and dominance were measured in original populations of 8 individuals, each with 50 loci on a single chromosome. Two alleles per locus were simulated by computer in the initial population in Hardy- Weinberg equilibrium where gene frequencies per locus were exactly 0.5. Four individuals were males and four females. Mating each male to each female produced three offspring per mating, 48 offspring per generation. The population of off—4 spring was characterized each generation by gene frequency, genotypic mean, and genotypic variance per locus for the 48 individuals averaged over the 50 loci. Each generation 48 full sib pairs, 48 half sib pairs, and 48 nonsib pairs were evaluated for the actual genotypic covariance and to measure exactly relationship parameters, including coefficient of inbreeding, between the related individuals. Eight off- spring were selected to become parents for the next gener- ation. Twenty generations of offspring were evaluated. Genes were linked by controlling recombination between adjacent loci on the chromosome at three rates, unlinked or recombination rates of 0.50 between adjacent 143 144 loci, moderate linkage or recombination rates of 0-26 between loci, and tight linkage where recombination rates between adjacent loci were 0.02. Three levels of dominance were partial dominance with genotypic values of 10, 7, and 0 for the three genotypic arrays per locus, complete dominance where the genotypes were 8, 8, 0, and overdominance where the genotypic values were 6, 9, 0 for the corresponding genotypic arrays per locus. Means of genotypes per locus were all 6.0 and genotypic variances for partial, complete, and overdominance models were 13.5, 12.0, and 13.5, respectively, for the initial population of individuals. Genotypic, phenotypic, and random selection were used to choose parents for the succeeding generation. The eight individuals ranking highest among the 48 offspring on either genotype or phenotype when selection was other than random were selected. Phenotypic variance in the original population was four times as large per locus as genotypic variance. The first four individuals selected or chosen at random were one sex and the remaining ones the opposite sex. Analysis of variance of the 33 factorial with three replicates showed interaction of selection and linkage to be particularly important in the mean level of inbreeding. Tight linkage in the presence of intense (genotypic) selec- tion depressed the mean level of inbreeding over all domi- nance levels. H0wever, the inbreeding level increased 145 slightly for the partial dominance model but decreased markedly for both the complete and overdominance models. Linkage in the presence of random selection had no effect on inbreeding levels. Theoretical coefficients of inbreed- ing conformed quite nicely with obsefved values when selec- tion of parents was at random. Gene frequency was not affected by any of the link- age or dominance levels when selection was at random. By the 18th generation gene frequency showed a rather sharp decrease within a particular dominance level when linkage became tight and selection was intense. This was most noticeable for complete dominance. A similar trend was exhibited when selection was based on phenotype, but the magnitude was not as great. Genotypic mean decreased more for the overdominance group than the other two dominance levels by the 18th gener- ation for random selection. This decrease was attributable to the dominance deviations and their relationship with mean inbreeding values in depressing the mean. Genotypic means for groups exhibiting partial dominance were depressed the least. Linkage did not affect the genotypic means when selection was random. By the 18th generation the corresponding genotypic means for phenotypic selection were higher than for randomly selected groups and less for groups selected on genotype. 146 Genotypic variances within lines decreased less when linkage was tight than when recombination rates were either free or moderate. The variance within lines also decreased more for the groups selected on genotype than for the other two modes of selection. Within the genotypic selection, partial dominance with free and moderate recombination rates decreased variances more than the other combinations. Geno- typic variances within lines for complete dominance agreed quite well with the theoretical values. Covariances between relatives tended to increase over the generations, becoming almost the same for full, half, and nonsibs by the 18th generation for each level of dominance. Tight linkage, particularly in the presence of intense selection, retained more genotypic covariance than free recombination. This result was most evident within genotypic selection when gene action was partial dominance. When gene action was overdominance, linkage had little effect on covariance for the three groups of sibs. A similar trend was evident in phenotypic selection. The relationship parameter rxy tended to increase over the generations for all treatment combinations. In- creases were more rapid when selection was intense and dominance levels least. Tight linkage in the presence of complete or overdominance and directional selection tended to slow the rate of increase of relationship. Three other relationship parameters (sxy, Syx' and txy) followed a 147 pattern similar to rxy’ and the different factors tended to affect these three in a manner not unlike rxy’ In all cases a second degree polynomial could be used to describe the trend over the 20 generations. uxy did not show a continual increase over time for most combinations. Instead this parameter tended to peak about the 10th or 12th generation, then decrease. The peak usually came later for half and nonsibs than for full sibs. Tight linkage, partial dominance, and genotypic selection caused this parameter to decrease for full sibs almost from the first generation while groups with complete and over- dominance tended to lag in the peak and rate of decline when linkage was tight. vxy followed a pattern somewhat similar to that of uxy' and vxy was the least prevalent of all expressions used to describe the relationship between full, half, and nonsibs. Tight linkage associated with effective selection produced lower values for inbreeding, gene frequency, and genotypic mean and higher values for genotypic variances and covariances between full and half sibs in a given popu— lation, than were encountered under other conditions. LITERATURE C ITED Baker, L. H., and R. E. Comstock. 1961. Linkage and heterozygosity in finite populations. Mimeographed. Johnston, Iowa, Hy-Line Poultry Farms. Baker, L. H., and R. E. Comstock. 1962. Monte Carlo studies of linkage effects in population genetics. Genetics 47:949 (Abstr.). Barker, J. S. F. 1958a. Simulation of genetic systems by automatic digital computers. III. Selection between alleles at an autosomal locus. Australian Journal of Biological Sciences 11:603. Barker, J. S. F. 1958b. Simulation of genetic systems by automatic digital computers. IV. Selection between alleles at a sex-linked locus. ‘Australian Journal of Biological Sciences 11:613. - Berge, S. 1961. The historical development of animal breeding. In Schilling, E., ed. Schriftenreihe des Max-Panck, Instits fur Tierzucht und Tierernahrung, Special Volume, 1961. 109 pp. ,Mariensee, German, Max-Planck-Gesellschaft-Dokumentationsstelle. Bodmer, W. F., and P. A. Parsons. 1962. Linkage and recombination in evolution. Advance Genetics 11:1. Brownlee, J. 1910. The significance of the correlation coefficient when applied to Mendelian distribution. Proceedings of the Royal Society, Edinburgh, 30:473. Cockerham, C. C. 1954. An extension of the concept of partitioning hereditary variance for analysis of covariances among relatives when epistasis is present. Genetics 39:859. Cockerham, C. C. 1956. Effects of linkage on the covar- iances between relatives. Genetics 41:138. Engeler, W. 1936. Die Entwicklung des Herdebuchwesens unter dem Einfluss der Lehren von der Vererbung and .Zfichtung bei den landwirtschaftlichen Haustieren. In Neue forschungen in tierzucht und abstammungslehre. 39 pp. Bern, Switzerland, Verbandsdruckerei A. G. 148 149 Evans, G. W., and C. L. Perry. 1961. Programming and coding for automatic digital computers. McGraw-Hill Book Company, Inc., New York. Felsenstein, J. 1965. selection. The effect of linkage on directional Genetics 52:349. Fisher, R. A. 1918. The correlation between relatives on the supposition of Mendelian inheritance. Transactions of the Royal Society, Edinburgh, 52, part 2:399. Fisher, R. A. 1941. Average excess and average effect of a gene substitution. Annals of Eugenics (London) 11:53. Fraser, A. S. 1957a. Simulation of genetic systems by automatic digital computers. I. Introduction. Austra- lian Journal of Biological Sciences 10:484. Fraser, A. S. 1957b. Simulation of genetic systems by automatic digital computers. II. Effects of linkage on, rates of advance under selection. .Australian Journal of Biological Sciences 10:492. Fraser, A. S. 1960a. Simulation of genetic systems by automatic digital computers. V. Linkage, dominance, and epistasis. In Kempthorne, 0., ed. Biometrical genetics. 70 pp. Pergamon Press, New Ybrk. Fraser, A. S. 1960b. Simulation of genetic systems by automatic digital computers. VI. Epistasis. Australian Journal of Biological Sciences 13:150. Fraser, A. S. 1960c. Simulation of genetic systems by automatic digital computers. VII. Effects of reproduc- tive rate and intensity of selection on genetic struc- ture. Australian Journal of Biological Sciences 13:344. Gill, J. L. 1963. Effect of population size, selection intensity, linkage, and nonadditive variability upon genetic change in simulated genetic populations. Ph.D. thesis, Iowa State Univ., Ames. Gill, J. L. 1965a. Effects of finite size on selection advance in simulated genetic populations. .Australian Journal of Biological Sciences 18:599. Gill, J. L. 1965b. A Monte Carlo evaluation of predicted selection response. Australian Journal of Biological Sciences 18:1007. 150 Gill, J. L. 1965c. Selection and linkage in simulated genetic populations. .Australian Journal of Biological Sciences 18:1171. Gill, J. L., and B. A. Clemmer. 1966. Effects of selection on linkage on degree of inbreeding. Australian Journal of Biological Sciences 19:307. Greenberger, M. 1961. Notes on a new pseudo—random number generator. Journal of the Association of Computing Machinery 8:163. Griffing, B. 1960. Theoretical consequences of truncation selection based on the individual phenotype. Australian Journal of Biological Sciences 13:307. Harris, D. L. 1964. Genotypic covariances between inbred relatives. Genetics 50:1319. Jain, J. K., and R. W. Allard. 1966. The effects of link—~ age, epistasis, and inbreeding on population changes under selection. Genetics 53:633. Jennings, H. S. 1916. The numerical results of diverse systems of breeding. Genetics 1:53. Jennings, H. S. 1917. The numerical results of diverse systems of breeding with respect to two pairs of char- acters linked or independent, with special relation to the effects of linkage. Genetics 2:97. Jones, R. M. 1960. Linkage distributions and epistacy in quantitative inheritance. Heredity 15:153. Jones, R. M. 1965. Components of variation under sib- mating with linkage. Genetica 36:147. Kempthorne, O. 1954. The correlation between relatives in a random mating population. Proceedings of the Royal Society, London, B, 143:103. _Kempthorne, O. 1955. The theoretical values of correlations between relatives in random mating populations. Genetics 40:153. Kempthorne, O. 1957. An introduction to genetic statistics. John Wiley and Sons, Inc., New Ybrk. Kemp, K. E. 1967. Evaluation by genetic simulation of changes in a cross bred population resulting from selection in a purebred population. Ph.D. thesis, Michigan State Univ., East Lansing. 151 Kimura, M. 1956. A model of a genetic system which leads to closer linkage by natural selection. Evolution 10: 278. Kojima, K. 1959a. Role of epistasis and overdominance in stability of equilibria with selection. Proceedings of the National Academy of Sciences, U.S. 45:984. Kojima, K. 1959b. Role of epistasis and overdominance in stability of equilibria with selection. Proceedings of the National Academy of Sciences, U.S. 45:984. Latter, B. D. H. 1965. The response to artificial selec- tion due to autosomal genes of large effect. II. The effects of linkage on limits to selection in finite ,populations. Australian Journal of.Biological Sciencegy. 18:1009. ‘ Latter, B. D. H. 1966. The response to artificial selec- tion due to autosomal genes of large effect. III. The . effects of linkage on the rate of advance and approach to fixation in the finite populations. Australian Journal of Biological Sciences 19:131. Lehmer, D. H. 1951. Mathematical methods in large scale computing units. Annals of the Computer Laboratory, Harvard University 26:141. Lewontin, R. C., and K..Kojima. 1960. The evolutionary dynamics of complex polymorphisms. Evolution 14:458. Lewontin, R. C. 1964a. The interaction of selection and linkage. I. General considerations; heterotic models. Genetics 49:49. Lewontin, R. C. 1964b. The interaction of selection and linkage. II. Optimum models. Genetics 50:757. Lewontin, R. C. 1965. The role of linkage in natural selection. Proceedings of the 11th International Congress of Genetics 3:517. Lush, J. L. 1948. The genetics of populations. Mimeo- graphed. Department of Animal Science, Iowa State Univ., Ames. Malécot, G. 1948. Les mathématiques de L'hérédité. Paris, Masson et Cie. 152 Martin, F. G., and Cockerham, C. C. 1960. High speed selection studies. In Kempthorne, 0., ed. Biometrical genetics. Pergamon Press, New YOrk. 35 pp. Mather, K. 1949. Biometrical Genetics. Methuen and Co., Ltd., London. Nei, M. 1963. Effect of selection on the components of genetic variance. Edited by W. D. Hansen and H. F. Robinson. National Academy of Sciences, National Resource Council Publication 982:501. Parker, R. J. 1966. Genetic correlation and response to selection in simulated populations. Ph.D. thesis, Michigan State Univ., East Lansing. Qureshi, A. W. 1963. A Monte Carlo evaluation of the role of finite population size and linkage in response to continuous mass selection. Technical report MC 6. Mimeographed. Statistical Laboratory, Iowa State Univ., Ames. Qureshi, A. W. 1964. -A Monte Carlo evaluation of the role of finite population size and linkage in response to continuous mass selection. II. Dominance and overdomi- nance. Technical report MC 9. Statistical Laboratory, Iowa State Univ., Ames. Robertson, Alan. 1952. The effect of inbreeding on the variation due to recessive genes. Genetics 37:189. Robertson, Alan. 1961. Inbreeding in artificial selection programmes. Genetical Research 2:189. Robbins, R. B. 1917. Applications of mathematics to breeding problems. Genetics 2:489. Robbins, R. B. 1918a. Applications of mathematics to breeding problems. II. Genetics 3:73. Robbins, R. B. 1918b. .Applications of mathematics to breeding problems. III. Genetics 3:375. Rotenberg, A. 1960. .A new pseudo-random number generator. Journal of the Association of Computing Machinery 7:75. Rowe, K. E. 1966. Prediction of genetic improvement in a finite population under selection. Ph.D. thesis, Iowa State Univ., Ames. Schnell, F. W. 1961. Some general formulations of linkage effects in inbreeding. Genetics 46:947. 153 Schnell, F. W. 1963. The covariance between relatives in the presence of linkage. National Academy of Sciences, National Research Council Publication 982. Snow, E. C. 1910. On the determination of the chief corre- lations between collaterals in the case of a simple Mendelian population mating at random. Proceedings of the Royal Society of London, A, 83:37. .Sturtevant, A. H. 1965. A history of genetics. Harper and Row, New York. VanAarde, I. M. R. 1963. Covariances of relatives in random mating populations with linkage. Ph.D. thesis, Iowa State Univ., Ames. Weckherlin, A von. 1851. Landwirtschaftliche Tierproduk- tionslehre. vol. e. Schafzucht. second ed. Stuttgart. Original not available; cited in Engeler, W. 1936. Die Entwicklung des Herdebuchwesens unter dem Einfluss- der Lehren von der Vererbung and Zuchtung bei den land- wirtschaftlichen haustieren. In Neue forschungen in tierzucht und abstammungslehre. 43 pp. Bern, Switzerland, Verbandsdruckerei AG. Weinberg, W. 1908a. fiber Verebungsgesetze Beim Menschen. Zeitschift fur induktive Abstammungs and verebunglehre 1:377. Weinbert, W. 1908b. fiber Verebungsgesetze Beim Menschen. Zeitschrift fur induktive Abstammungs und Verebungslehre 1:440. Weinberg, W. 1909. fiber Verebungsgesetze Beim Menschen. Zeitschrift fur induktive Abstammungs und Verebungslehre 2:276. Wright,.S. 1921a. .Systems of mating. I. The biometric relations between parent and offspring. Genetics 6:111. Wright, S. 1922. Coefficients of inbreeding and relation- ship. American Naturalist 56:330. Wright, S. 1931. Evolution in Mendelian populations. Genetics 16:97. Wright, S. 1951. The genetical structure of populations. Annals of Eugenics 15:323. 154 Ybung, S. S. Y. 1966. Computer simulation of directional selection in large populations. I. The programme, the additive and the dominance models. Genetics 53:189. YOung, S. S. Y. 1967. Computer simulation of directional selection in large populations. II. The additive x additive and mixed models. Genetics 56:73. M'TlTli‘l’InjflwiflujiIaHE/tMIEIWLIWIWES