Initial Linkage Disequilibrium Research Articles

PSXII-4 Genomic Improvement of Disease Resistance Using Two Breeding Strategies in a Population of Penaeus Vannamei (Pacific White shrimp): a Simulation Study

Abstract This study aimed to investigate the impact of SNP density chip and genotyping strategies on the accuracy of genomic estimated breeding values (EBV) in a simulated population mimicking a commercial Pacific white shrimp (Penaeus vannamei) breeding program. The chromosome number and length mimicked the P. vannamei genome. To create an initial linkage disequilibrium (LD), a historical population of 5,000 animals was randomly bred for 1,000 generations and included a bottleneck on generation 750. In the last generation of the historical population, animals were randomly selected to create a breeding population with an effective population size (Ne) of 40. From the breeding population, a total of 40,000 animals were simulated over 20 generations. Phenotypes for a trait with a heritability of 0.1 was simulated, and genetic variance was 100% explained by 1,000 QTLs. Phenotypes and genotypes were saved for all animals in the last 10 generations, while pedigree from all 20 generations was used. To mimic the selection of young animals, a forward-in-time prediction was implemented by masking the phenotypes from the 20th generation in the data analysis. The accuracy of prediction was the correlation between the simulated true breeding value and the EBV in the last generation. Genomic information consisted of SNP chips with 6K, 11K, 22K, 45K, and 90K markers evenly spaced across the genome. The two breeding strategies considered in this study were 1) 25 families with 40 animals each, and 2) 50 families with 20 animals each. Additionally, the number of generations with phenotypes varied from nine to one. This was done to identify the minimum number of generations to collect data, as phenotyping individuals may be challenging in aquaculture. In all scenarios, genomic selection outperformed family selection. Results indicated that genotyping more individuals within families, instead of more families with fewer individuals, improved the accuracy of predictions. Furthermore, the maximum accuracy was achieved with 45K and 90K SNPs in both scenarios, showing that SNP density will reach a plateau. Finally, maximum accuracy plateaued at five generations. Finally, having one generation with phenotypes and genotypes yielded greater accuracies than nine generations with only phenotypes. Overall, the results from the present study suggest that genomic selection using the single-step genomic best linear unbiased prediction (ssGBLUP) methodology is a feasible tool to improve P. vannamei commercial populations, even for traits with low heritability. Moreover, it is recommended to collect data on more individuals per family than increasing the number of families, and data collection should be continued for at least four generations. Tests with real data will provide further insights into the ideal number of animals and SNP density necessary for maximum accuracy.

Four Parent Maize (FPM) Population: Effects of Mating Designs on Linkage Disequilibrium and Mapping Quantitative Traits.

Multiparent advanced generation inter-cross (MAGIC) populations can provide improved genetic mapping resolution by increasing allelic diversity and effective recombination. The Four Parent Maize (FPM; L.) population implemented five different mating designs used in MAGIC and bi-parental populations to compare empirical effects on genetic resolution and power of quantitative trait locus (QTL) detection; the combined population here comprised of 1149 individuals with 118,509 genetic markers. Measurements were recorded for plant height (PH), ear height (EH), days to anthesis (DTA) and silking (DTS) in seven environments, spanning three years. Linkage disequilibrium (LD) analysis of subpopulations indicated MAGIC population designs should incorporate generations of intermating to overcome initial LD increase caused by population admixture in a non-intermated four parent population (4way0sib). A 3- to 4-fold increase in genetic resolution (<0.8) and a 2.5-fold decrease in the extent of LD decay (<0.2) compared to the biparental populations was found for the four parent cross at the third generation of intermating (4way3sib). Power of QTL detection was affected to a greater extent by sample size rather than by mating designs. The FPM power simulations indicated that MAGIC populations have the ability to meet or exceed the mapping power of nested association panels with fewer individuals and diversity inputs. Using association mapping software we identified 2, 5, 7, and 6 QTL for PH, EH, DTA, and DTS, respectively. The FPM population is a valuable resource for quantifying empirical improvements of parent number, intermating, and the number of progeny for QTL linkage mapping.

Association Between KCNJ6 (GIRK2) Gene Polymorphism rs2835859 and Post-operative Analgesia, Pain Sensitivity, and Nicotine Dependence

G-protein-activated inwardly rectifying potassium (GIRK) channels are expressed in many tissues and activated by several Gi/o protein-coupled receptors, such as opioid and dopamine receptors, and thus are known to be involved in the modulation of opioid-induced analgesia, pain, and reward. We focused on a GIRK-channel subunit that plays a pivotal role in the brain, GIRK2, and investigated the contribution of genetic variations of the GIRK2 (KCNJ6) gene to individual differences in the sensitivity to opioid analgesia. In our initial linkage disequilibrium analysis, a total of 27 single-nucleotide polymorphisms (SNPs) were selected within and around the regions of the KCNJ6 gene. Among them, the rs2835859 SNP, for which associations with analgesia and pain have not been previously reported, was selected in the exploratory study as a potent candidate SNP associated with opioid analgesic sensitivity. The results were corroborated in further confirmatory study. Interestingly, this SNP was also found to be associated with sensitivity to both cold and mechanical pain, susceptibility to nicotine dependence, and successful smoking cessation. The results indicate that this SNP could serve as a marker that predicts sensitivity to analgesic and pain and susceptibility to nicotine dependence.

Effect of epistasis and linkage on fixation probability in three-locus models: An ancestral recombination–selection graph approach

We study the probability of ultimate fixation of a single new mutant arising in an individual chosen at random at a locus linked to two other loci carrying previously arisen mutations. This is done using the Ancestral Recombination–Selection Graph (ARSG) in a finite population in the limit of a large population size, which is also known as the Ancestral Influence Graph (AIG). An analytical expansion of the fixation probability with respect to population-scaled recombination rates and selection intensities is obtained. The coefficients of the expansion are expressed in terms of the initial state of the population and the epistatic interactions among the selected loci. Under the assumption of weak selection at tightly linked loci, the sign of the leading term, which depends on the signs of epistasis and initial linkage disequilibrium, determines whether an increase in recombination rates increases the chance of ultimate fixation of the new mutant. If mutants are advantageous, this is the case when epistasis is positive or null and the initial linkage disequilibrium is negative, which is an expected state in a finite population under directional selection. Moreover, this is also the case for a neutral mutant modifier coding for higher recombination rates if the same conditions hold at the selected loci. Under the same conditions, deleterious mutants are disfavored for ultimate fixation and neutral modifiers for higher recombination rates still favored. The recombination rates between the modifier locus and the selected loci do not come into play in the leading terms of the approximation for the fixation probability, but they do in higher-order terms.

An Alternative Way of Constructing Ancestral Graphs Using Marker Allele Ages from Population Linkage Disequilibrium Information

Abstract An alternative way of constructing ancestral graphs, which is different from the coalescent-based approach, is proposed using population linkage disequilibrium (LD) data. The main difference from the existing method is the construction of the ancestral graphs based on var-iants instead of individual sequences. Therefore, the key of the proposed method is to use the order of allele ages in the graphs. Distinct from the previous age-esti-mation methods, allele ages are estimated from full hap-lotype information by examining the number of gen-erations from the initial complete LD to the current de-cayed state for each two variants depending on the di-rection of LD decay between variants. Using a simple algorithmic procedure, an ancestral graph can be de-rived from the expected allele ages and current LD de-cay status. This method is different in many ways from previous methods, and, with further improvement, it might be a good replacement for the current appro-aches.Keywords: allele age, ancestral graph, haplotypes, link-age disequilibrium, recombination

Soft selective sweep near a gene that increases plant height in wheat

Strong selection within a given population locally reduces genetic variability not only in the selected gene itself but also in neighbouring loci. This so-called hitch-hiking effect is related to the initial linkage disequilibrium between markers and the selected gene, and depends mainly on the number of copies of the beneficial allele at the start of the selection phase. Contrary to the classical case, in which selection acts on a single, newly arisen beneficial mutation, we considered selection from standing variation (soft selective sweeps) on a gene (Rht-B1) with a major effect on plant height, a selected trait in an experimental wheat population grown for 17 generations, and we documented the evolution of gene diversity and linkage disequilibrium near this gene. As expected, Rht-B1 was found to be under strong selection (s = 0.15) and its variation in frequency accounted for 15% of the total trait evolution. This led to a smaller genetic effective population size at Rht-B1 (N(eg) = 18) compared to the whole genome estimation (N(eg) = 167). When compared with expectations under genetic drift only, no significant decrease in gene diversity was found at the closest loci. We computed expected di-locus frequencies for any linked marker-Rht-B1 pair due to hitch-hiking effects. We found that hitch-hiking was expected to affect the two most closely linked loci, but expected reduction in gene diversity was not greater than that due to genetic drift, which was consistent with the observations. Such limited effect was attributed to the low level of linkage disequilibrium (0.16) estimated after parental intercrosses, together with a relatively high initial frequency of the gene. This situation is favourable to candidate gene approaches where small linkage disequilibrium around selected genes is expected.

Linkage disequilibrium in two European F2 flint maize populations under modified recurrent full-sib selection

According to quantitative genetic theory, linkage disequilibrium (LD) can hamper the short- and long-term selection response in recurrent selection (RS) programs. We analyzed LD in two European flint maize populations, KW1265 x D146 (A x B) and D145 x KW1292 (C x D), under modified recurrent full-sib selection. Our objectives were to investigate (1) the decay of initial parental LD present in F(2) populations by three generations of intermating, (2) the generation of new LD in four (A x B) and seven (C x D) selection cycles, and (3) the relationship between LD changes and estimates of the additive genetic variance. We analyzed the F(2) and the intermated populations as well as all selection cycles with 104 (A x B) and 101 (C x D) simple sequence repeat (SSR) markers with a uniform coverage of the entire maize genome. The LD coefficient D and the composite LD measure Delta were estimated and significance tests for LD were performed. LD was reduced by intermating as expected from theory. A directional generation of negative LD between favorable alleles could not be observed during the selection cycles. However, considerable undirectional changes in D were observed, which we attributed to genetic sampling due to the finite population size used for recombination. Consequently, a long-term reduction of the additive genetic variance due to negative LD was not observed. Our experimental results support the hypothesis that in practical RS programs with maize, LD generated by selection is not a limiting factor for obtaining a high selection response.

On the evolution of epistasis II: A generalized Wright–Kimura framework

The evolution of fitness interactions between genes at two major loci is studied where the alleles at a third locus modify the epistatic interaction between the two major loci. The epistasis is defined by a parameter ε and a matrix structure that specifies the nature of the interactions. When ε = 0 the two major loci have additive fitnesses, and when these are symmetric the interaction matrices studied here produce symmetric viabilities of the Wright [1952. The genetics of quantitative variability. In: Reeve, E.C.R., Waddington, C.H. (Eds.), Quantitative Inheritance. Her Majesty's Stationary Office, London]–Kimura [1956. A model of a genetic system which leads to closer linkage by natural selection. Evolution 10, 278–281] form. Two such interaction matrices are studied, for one of which epistasis as measured by | ε | always increases, and for the other it increases when the linkage between the major loci is tight enough and there is initial linkage disequilibrium. Increase of epistasis does not necessarily coincide with increase in equilibrium mean fitness.

Fine-scale mapping using Hardy-Weinberg disequilibrium.

Hardy-Weinberg disequilibrium (HWD) among affected individuals has recently been proposed for fine-scale mapping of disease susceptibility genes. We investigate the statistical properties of several available HWD measures and develop a new HWD measure J for fine-scale mapping. It is shown both theoretically and through simulations that the available HWD measures depend not only on the genetic distance between the marker locus of interest and the disease susceptibility locus, but also on the allele frequencies at the marker locus. On the contrary, the new measure is not affected by the allele frequencies at the marker locus under the following assumptions: (a) there is initial complete linkage disequilibrium between the marker and the disease loci, (b) there are no new mutations at the marker and the disease loci, and (c) the population under study is large. We develop a novel method to estimate the location of the disease susceptibility gene based on the HWD measure J. The estimator is robust to low mutation rates at the marker and the disease loci. We compare the standard error of the estimated disease gene loci using P excess for case-control studies with the standard error using J for case-only studies under various disease models. The newly developed method is successfully applied to a data set on hereditary haemochromatosis (HH).

Haplotypes vs single marker linkage disequilibrium tests: what do we gain?

The genetic dissection of complex diseases represents a formidable challenge for modern human genetics. Recently, it has been suggested that linkage disequilibrium (LD) based methods will be a powerful approach for delineating complex disease genes. Most proposed LD test statistics search for association between a single marker and a putative trait locus. However, the power of a single marker association test may suffer because LD information contained in flanking markers is ignored. Intuitively, haplotypes (which can be regarded as a collection of ordered markers) may be more powerful than individual, unorganised markers. In this study, we derive the analytical tools based on standard chi-square statistics to directly investigate and compare the power between multilocus haplotypes and single marker LD tests. More specifically, novel formulas are obtained in order to calculate expected haplotype frequencies of unlimited size. This study demonstrates that the use of haplotypes can significantly improve the power and robustness of mapping disease genes. Additionally, we detail how the power of haplotype based association tests are affected by important population genetic parameters such as the genetic distance between markers and disease locus, mode of disease inheritance, age of trait causing mutation, frequency of associated marker allele, and level of initial LD. Finally, published data from the Hereditary Hemochromatosis disease region is used to illustrate the utility of haplotypes.

Linkage Detection Adaptive to Linkage Disequilibrium: The Disequilibrium Maximum-Likelihood–Binomial Test for Affected-Sibship Data

It has been demonstrated in the literature that the transmission/disequilibrium test (TDT) has higher power than the affected-sib-pair (ASP) mean test when linkage disequilibrium (LD) is strong but that the mean test has higher power when LD is weak. Thus, for ASP data, it seems clear that the TDT should be used when LD is strong but that the mean test or other linkage tests should be used when LD is weak or absent. However, in practice, it may be difficult to follow such a guideline, because the extent of LD is often unknown. Even with a highly dense genetic-marker map, in which some markers should be located near the disease-predisposing mutation, strong LD is not inevitable. Besides the genetic distance, LD is also affected by many factors, such as the allelic heterogeneity at the disease locus, the initial LD, the allelic frequencies at both disease locus and marker locus, and the age of the mutation. Therefore, it is of interest to develop methods that are adaptive to the extent of LD. In this report, we propose a disequilibrium maximum-binomial-likelihood (DMLB) test that incorporates LD in the maximum-binomial-likelihood (MLB) test. Examination of the corresponding score statistics shows that this method adaptively combines two sources of information: (a) the identity-by-descent (IBD) sharing score, which is informative for linkage regardless of the existence of LD, and (b) the contrast between allele-specific IBD sharing score, which is informative for linkage only in the presence of LD. For ASP data, the proposed test has higher power than either the TDT or the mean test when the extent of LD ranges from moderate to strong. Only when LD is very weak or absent is the DMLB slightly less powerful than the mean test; in such cases, the TDT has essentially no power to detect linkage. Therefore, the DMLB test is an interesting approach to linkage detection when the extent of LD is unknown.

On the evolution of recombination in haploids and diploids: I. Deterministic models

AbstractIn population genetic theory, most analytical and numerical studies of the evolution of recombination have focused on diploid genetics. In studies of the foundations and applications of genetic algorithms (GA's), however, the bit‐strings are usually treated as haploid genotypes. In this paper and its companion paper (Bergman et al., 1995), we compare results for the evolutionary dynamics of modifiers of recombination in haploids with results derived for diploids. In this paper, we study the evolution of an allele that controls the rate of recombination between two loci subject to directional selection. It is shown analytically that the fate of a recombination modifier in both haploids and diploids is determined in a complicated way by the sign of the epistasis (interaction in fitness) between the loci, the sign of the initial linkage disequilibrium, and the amount of recombination between the modifier and the genes under selection. This theory is deterministic in that the population is regarded as infinite and no sampling occurs to produce offspring from parents. In the companion paper (Bergman et al., Complexity, 1(2) 1995), we expand upon this work by addressing epistatic interactions among several loci in finite populations.

Decay of linkage disequilibrium in a finite island model.

Time-dependent behaviour of linkage disequilibrium when there was initial linkage disequilibrium is studied in a finite island model assuming neutrality. Explicit expressions for linkage disequilibrium parameters are obtained. From these expressions, the initial and the ultimate decay rates of linkage disequilibrium parameters are found to be increased and decreased, respectively, by finiteness of the population when recombination rate, migration rate and inverse of subpopulation size are of comparable order. Thus, linkage disequilibrium created in the past may persist longer in smaller subdivided populations. Also, differentiation of the gametic parameter of linkage disequilibrium among subpopulations is found to diminish quickly compared to the linkage disequilibrium in the whole population. Implications of these results for the interpretation of linkage disequilibria in natural populations are discussed.

Computer simulation of marker-assisted selection utilizing linkage disequilibrium

The value of marker-assisted selection (MAS) using linkage disequilibrium between genetic markers and quantitative trait loci (QTL) was examined. To simulate the disequilibrium, four base populations were created, F2, F5, F10 and F20, by random mating from a cross between two inbred lines. Selections were on breeding values estimated from: (1) marker QTL (MQTL) associations (MAS); (2) conventional best linear unbiased prediction (BLUP) methods; and (3) a combination of 1 and 2 (COMB). Alternative cases were studied by varying the parameters (heritability, initial linkage disequilibrium, and distribution of QTL effects). A genome with 100 QTL and 100 markers randomly (but equally) spread over 20 chromosomes, each 100 centiMorgans (cM) in length, was generated. Linkage disequilibrium (over 30 replicates) of QTLs with their nearest marker averaged 0.153, 0.104, 0.068, and 0.047 for the four base populations, and fell to 0.035, 0.025, 0.021, and 0.018, respectively, after ten generations of MAS selection (heritability 0.25). The initial linkage disequilibrium had the greatest effect on the genetic gain by MAS with the responses for the base populations F2>F5>F10>F20. Genetic gains by conventional BLUP selection were usually greater than by MAS. However, MAS contributed to the combined selection (COMB) to give appreciably higher genetic responses. Hybridization of selected lines after several generations of selection contributed little to generating further linkage disequilibrium. Detection of markers closer to the QTL will increase the linkage disequilibrium available for selection. Eventually with very close linkage each QTL allele can be uniquely identified in selection, and selection will then be equivalent to selection on the QTLs themselves.

Effects of two-locus linkage disequilibrium on progress from reciprocal recurrent selection in maize

Because each population in a reciprocal recurrent selection program can be dually described as a tester for its reciprocal and as a population subject to selection, selection progress can be partitioned into algebraic terms that are (a) unaffected by initial linkage disequilibrium, (b) modified exclusively by initial linkage disequilibrium in the selection populations, (c) influenced solely by initial linkage disequilibrium in the tester populations, and (d) modulated jointly by initial linkage disequilibrium in both populations. If only additive genetic effects are present in the intercross population, linkage disequilibrium affects selection progress via only the selection populations. Initial linkage disequilibrium affects selection progress via the tester populations, and/or the selection and tester populations jointly, only in the presence of epistatic effects for which, out of all alleles involved in the interaction, two alleles emanate from the tester population. If only additive x additive epistasis is present, initial linkage disequilibrium affects progress via both the selection and tester populations, but the effect is permanent via only the selection populations. The effect of initial linkage disequilibrium via the tester populations, and/or the selection and tester populations jointly, is permanent only in the presence of additive x dominance and/or dominance x dominance epistasis in the intercross population.

Sexual recombination under the joint effects of mutation, selection, and random sampling drift

The advantage or disadvantage of sexual reproduction or recombination for the accumulation of mutant genes in a population is studied under the joint effects of recurrent mutations, selection, and random sampling drift. To obtain the rate at which mutant genes are incorporated three different methods are used; numerical integration of Kolmogorov backward equations, simulation of stochastic difference equations, and Monte Carlo experiments. The first two methods are used in a two-locus system to obtain the fixation probability of double mutants and other related quantities under five different selection models. The third one is conducted for a multiple-locus system and the rate of accumulation of mutant genes per locus is studied. Comparison of the results between sexual and asexual populations shows that the effect of recombination depends on initial linkage disequilibrium, mutation rate v, selection intensity s, and population size N e. The mode of selection is also an important factor and the large effect of recombination is observed when mutant genes are individually deleterious but collectively favorable. Under a given model of selection, the great advantage or disadvantage of recombination is achieved when a large extent of genetic polymorphism is produced not by mutation but by recombination. Extreme values of N e s and N e v make the effect insignificant. The results of Monte Carlo experiments also reveal the presence of interaction between selection and sampling drift even when the loci segregate independently and selection is multiplicative. Although this interaction is usually small, there are cases in which one locus theory cannot be used freely. In those cases, the effect of recombination is prominent and one locus theory gives an overestimate of the rate.

Two-locus theory in recurrent selection for general combining ability in maize

Two-locus theory for recurrent selection for general combining ability in maize was developed. The theory featured: (a) recombination of the selfed progeny of selected parents; and (b) linkage disequilibrium in the initial gametic array. The theory indicated: (a) that initial linkage disequilibrium exerts a permanent influence upon selection progress; (b) that interposition of one or more generations of random mating before each cycle reduces the permanent effect in ensuing cycles; and (c) that random mating done before initiation of selection is more efficient in removing the influence of linkage disequilibrium on selection progress than random mating done between subsequent cycles.

The effect of positive assortative mating at one locus on a second linked locus

Considerations proceed from a model of positive assortative mating based on genotype at one locus, with an arbitrary number of alleles, assuming no selection, mutation, or migration, hypothetically infinite population size, and discrete non-overlapping generations. From these conditions, inferences are made about the genotypic structure at a linked locus, as well as about the corresponding 2-locus gametic structure.The following main results are presented: in the course of the generations, the genotypic structure at the second locus and the 2-locus gametic structure always tend to a limit responsive to the initial conditions concerning the joint genotypic structure at the two loci and the degree of assortativity and linkage. A complete, analytical representation of the limits is given. In particular, if assortative mating is only partial and at the same time linkage is not complete, a population is not able to maintain a permanent deviation of the gametic structure from linkage equilibrium, and thus the genotypic structure at the second locus tends to Hardy-Weinberg proportions. On the other hand, if initial linkage disequilibrium is combined with partial assortative mating and complete linkage (or with complete assortative mating and unlinked loci) the population maintains this disequilibrium and thus the genotypic structure at the second locus need not tend to Hardy-Weinberg proportions. It turns out that the conditions not only of complete linkage, but also of unlinked loci together with complete assortativity, imply no change in gametic structure from the initial structure.In order to demonstrate the influence of several parameters on the speed of convergence to and the magnitude of the respective limits, several graphs are included.

Effect of initial linkage disequilibrium and epistasis on fixation probability in a small population, with two segregating loci

The probability of ultimate fixation was studied for 2 loci small populations by the method of Monte Carlo simulation and also partly by analytical treatment. The analytical solutions for fixation probability known at present and the difficulty of their application to more complex situations were discussed. Monte Carlo experiments were carried out for large values ofN e s 1,N e s 2 andN e ε, for which the analytical solutions have not been obtained.One of the main purposes was to investigate the effect of initial linkage disequilibrium (D) on fixation probability. Whens 1,s 2 and ε are 0, the effect of disequilibrium is given by {1-2N e c/(2N e c+1)}D withKIMURA'S model and {1-2N e c/(2 N e c-c+1)}D with the model ofKARLIN andMCGREGOR. When |N e s 1| and |N e s 2| were small and ε=0, the effect of disequilibrium was shown to be almost the same as in the selectively neutral case (formulas 8). When those parameters are not small, the effect of disequilibrium is larger because of the rapid approach to fixation.Another purpose of this study was to investigate the effect of epistasis on fixation probability.KIMURA obtained the solution for joint fixation ofA andB when the selective advantage is given to genotypeAB as the joint effect ofA andB. The present study has verified his formula and showed that linkage does not affectu(AB) under initial linkage equilibrium. Also some cases of additive × additive epistasis were studied.

The effect of linkage on limits to artificial selection

(i) A computer simulation study has been made of selection on two linked loci in small populations, where both loci were assumed to have additive effects on the character under selection with no interaction between loci. IfNis the effective population size,ithe intensity of selection in standard units, α and β measure the effects of the two loci on the character under selection as a proportion of the pheno-typic standard deviation andcis the crossover distance between them, it was shown that the selection process can be completely specified byNiα,NiβandNcand the initial gene frequencies and linkage disequilibrium coefficient. It is then easily possible to generalize from computer runs at only one population size. All computer runs assumed an initial population at linkage equilibrium between the two loci. Analysis of the results was greatly simplified by considering the influence of segregation at the second locus on the chance of fixation at the first (defined as the proportion of replicate lines in which the favoured allele was eventually fixed).(ii) The effects of linkage are sufficiently described byNc.The relationship between chance of fixation at the limit and linkage distance (expressed as2Nc/(2Nc+ 1)) was linear in the majority of computer runs.(iii) When gene frequency changes under independent segregation were small, linkage had no effect on the advance under selection. In general, segregation at the second locus had no detectable influence on the chance of fixation at the first if the gene effects at the second were less than one-half those at the first. With larger gene effects at the second locus, the chance of fixation passed through a minimum and then rose again. For two loci to have a mutual influence on one another, their effects on the character under selection should not differ by a factor of more than two.(iv) Under conditions of suitable relative gene effects, the influence of segregation at the second locus was very dependent on the initial frequency of the desirable allele. The chance of fixation at the first, plotted against initial frequency of the desirable allele at the second, passed through a minimum when the chance of fixation at the second locus was about 0·8.(v) A transformation was found which made the influence of segregation at the second locus on the chance of fixation at the first almost independent of initial gene frequency at the first and of gene effects at the first locus when these are small.(vi) In the population of gametes at final fixation, linkage was not at equilibrium and there was an excess of repulsion gametes.(vii) The results were extended to a consideration of the effect of linkage on the limits under artificial selection. Linkage proved only to be of importance when the two loci had roughly equal effects on the character under selection. The maximum effect on the advance under selection occurred when the chance of fixation at both of the loci was between 0·7 and 0·8. When the advance under selection is most sensitive to changes in recombination value, a doubling of the latter in no case increased the advance under selection by more than about 6%. The proportion selected to give maximum advance under individual selection (0·5 under independent segregation) was increased, but only very slightly, when linkage is important.(viii) These phenomena could be satisfactorily accounted for in terms of the time scale of the selection process and the effective size of the population within which changes of gene frequency at the locus with smaller effect must take place.