When a selectively favourable gene substitution occurs in a population, changes in gene frequencies will occur at closely linked loci. In the case of a neutral polymorphism, average heterozygosity will be reduced to an extent which varies with distance from the substituted locus. The aggregate effect of substitution on neutral polymorphism is estimated; in populations of total size 106 or more (and perhaps of 104 or more), this effect will be more important than that of random fixation. This may explain why the extent of polymorphism in natural populations does not vary as much as one would expect from a consideration of the equilibrium between mutation and random fixation in populations of different sizes. For a selectively maintained polymorphism at a linked locus, this process will only be important in the long run if it leads to complete fixation. If the selective coefficients at the linked locus are small compared to those at the substituted locus, it is shown that the probability of complete fixation at the linked locus is approximately exp (− Nc), where c is the recombinant fraction and N the population size. It follows that in a large population a selective substitution can occur in a cistron without eliminating a selectively maintained polymorphism in the same cistron.

A mechanism for gene conversion is proposed which overcomes many of the difficulties that any copy choice model encounters. It is suggested that along with general genetic pairing of homologous genomes at meiosis, effective pairing over short regions of the genetic material occurs at the molecular level by the separation of the strands of the DNA double helices, followed by the annealing of strands from two homologous chromatids. If the annealed region happens to span a heterozygous site, mispairing of bases will occur. Such a situation may be analogous to that in DNA which is damaged by mutagens; the same or similar repair mechanisms may operate, and these, by adjusting the base sequences in order to restore normal base pairing, would bring about gene conversion in the absence of any genetic replication. The model indicates how precise breakage and rejoining of chromatids could occur in the vicinity of the conversion, so that conversion would frequently be accompanied by the recombination of outside markers. The model also proposes that the distance between two mutant sites on a fine structure map depends not so much on the frequency of a recombinational event occurring between them, but rather on the degree of inhibition of the processes of genetic pairing by the mutants themselves.

The model will explain almost all the data in a formal way, and it has the advantage over copy choice mechanisms for gene conversion in (1) being compatible with semi-conservative replication of DNA, (2) not invoking DNA synthesis during or after genetic pairing, (3) providing a molecular mechanism for close specific pairing, (4) making it unnecessary to postulate sister strand exchange or a process akin to this, (5) suggesting why rates of gene conversion in opposite directions are sometimes unequal and (6) providing an explanation of the clustering of mutant sites, a basis for map expansion and for the apparently capricious departure of fine structure maps from additivity. Although the model proposed is a general rather than a specific one, it suggests that the process of conversion and intragenic recombination is more complex than is usually believed, since it depends on several interacting factors. Nevertheless, it is hoped that the introduction of a model with this complexity will help to stimulate specific experiments, and that these will provide definitive information which would never be obtained if simpler models of conversion and intragenic recombination were believed to explain the genetic data sufficiently well.

(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. If N is the effective population size, i the 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 and c is the crossover distance between them, it was shown that the selection process can be completely specified by Niα, Niβand Nc and 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 by Nc. The relationship between chance of fixation at the limit and linkage distance (expressed as 2Nc/(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.

The effective population size is required to predict the rate of inbreeding and loss of genetic variation in wildlife. Since only census population size is normally available, it is critical to know the ratio of effective to actual population size (Ne/N). Published estimates of Ne/N (192 from 102 species) were analysed to identify major variables affecting the ratio, and to obtain a comprehensive estimate of the ratio with all relevant variables included. The five most important variables explaining variation among estimates, in order of importance, were fluctuation in population size, variance in family size, form of N used (adults υ. breeders υ. total size), taxonomic group and unequal sex-ratio. There were no significant effects on the ratio of high υ. low fecundity, demographic υ. genetic methods of estimation, or of overlapping υ. non-overlapping generations when the same variables were included in estimates. Comprehensive estimates of Ne/N (that included the effects of fluctuation in population size, variance in family size and unequal sex-ratio) averaged only 0·10–0·11. Wildlife populations have much smaller effective population sizes than previously recognized.

A new model of mutational production of alleles was proposed which may be appropriate to estimate the number of electrophoretically detectable alleles maintained in a finite population. The model assumes that the entire allelic states are expressed by integers (…, A−1, A0, A1, …) and that if an allele changes state by mutation the change occurs in such a way that it moves either one step in the positive direction or one step in the negative direction (see also Fig. 1). It was shown that for this model the ‘effective’ number of selectively neutral alleles maintained in a population of the effective size Ne under mutation rate υ per generation is given by

When 4Neυ is small, this differs little from the conventional formula by Kimura & Crow, i.e. ne = 1 + 4Neυ, but it gives a much smaller estimate than this when 4Neυ is large.

1. Nude is a new recessive gene causing hairlessness in the mouse. It is linked to rex and trembler in linkage group VII. The order of the three loci and the recombination frequencies are as follows:

2. In addition to hairlessness the new gene causes reduced body growth rate, very low fertility and a liver disease causing death. Nude mice may be classified at birth by the absence of vibrissae.

3. The hairlessness is due to abnormal keratinization of hair in the follicles. The skin histology resembles that of naked mice. The hair follicles were found to be deficient in free sulphydryl groups.

4. The majority of nude mice die of general body weakness within 2 weeks of birth. The survivors grow slowly and may live for a considerable period. But all nude mice eventually die, usually between 3 and 14 weeks of age.

5. The livers of dead or moribund nude mice are covered with lesions and scars. The defect has been traced histologically to its initial stage, namely, necrosis of small areas of tissue.

6. Attempts to relate the deficiency of sulphydryl groups in the hair follicles to abnormal sulphur metabolism in the liver were unsuccessful.

7. Pseudo-cysts of a parasitic protozoan, Toxoplasma gondii, were identified in the liver and brain of nude mice. In one case the free form of the organism was found.

8. The possible relationship between the liver disease and the pathogenic organism is discussed.

Data on the effects of inbreeding on fitness components are reviewed in the light of population genetic models of the possible genetic causes of inbreeding depression. Deleterious mutations probably play a major role in causing inbreeding depression. Putting together the different kinds of quantitative genetic data, it is difficult to account for the very large effects of inbreeding on fitness in Drosophila and outcrossing plants without a significant contribution from variability maintained by selection. Overdominant effects of alleles on fitness components seem not to be important in most cases. Recessive or partially recessive deleterious effects of alleles, some maintained by mutation pressure and some by balancing selection, thus seem to be the most important source of inbreeding depression. Possible experimental approaches to resolving outstanding questions are discussed.

A method is proposed for estimating effective population size (N) from data on linkage disequilibrium among neutral genes at several polymorphic loci or restriction sites. The efficiency of the method increases with larger sample size and more tightly linked genes; but for very tightly linked genes estimates of N are more dependent on long-term than on recent population history. Two sets of data are analysed as examples.

Method-of-moments estimators (MMEs) for the two-gene coefficients of relationship and inbreeding, and for thxe four-gene Cotterman coefficients, are described. These estimators, which use co-dominant genetic markers, are most appropriate for estimating pairwise relatedness or individual inbreeding coefficients, as opposed to their mean values in a group. This is because, compared to the maximum likelihood estimate (MLE), they show reduced small-sample bias and lack distributional assumptions. The ‘efficient’ MME is an optimally weighted average of estimates given by each allele at each locus. Generally, weights must be computed numerically, but if true coefficients are assumed zero, simplifiedestimators are obtained whose relative efficiencies are quite high. Population gene frequency is assumed to be assayed ina larger, ‘reference population’ sample, and the biases introduced by small reference samples and/or genetic drift of the reference population are discussed. Individual-level estimates of relatedness or inbreeding, while displaying high variance, are useful in several applications as a covariate in population studies.

It is assumed that a character under stabilizing selection is determined genetically by n linked, mutable loci with additive effects and a range of many possible allelic effects at each locus. A general qualitative feature of such systems is that the genetic variance for the character is independent of the linkage map of the loci, provided linkage is not very tight. A particular detailed model shows that certain aspects of the genetic system are moulded by stabilizing selection while others are selectively neutral. With reference to experimental data on characters of Drosophila flies, maize, and mice, it is concluded that large amounts of genetic variation can be maintained by mutation in polygenic characters even when there is strong stabilizing selection. The properties of the model are compared with those of heterotic models with linked loci.

Levels of neutral genetic diversity in populations subdivided into two demes were studied by multi-locus stochastic simulations. The model includes deleterious mutations at loci throughout the genome, causing ‘background selection’, as well as a single locus at which a polymorphism is maintained, either by frequency-dependent selection or by local selective differences. These balanced polymorphisms induce long coalescence times at linked neutral loci, so that sequence diversity at these loci is enhanced at statistical equilibrium. We study how equilibrium neutral diversity levels are affected by the degree of population subdivision, the presence or absence of background selection, and the level of inbreeding of the population. The simulation results are compared with approximate analytical formulae, assuming the infinite sites neutral model. We discuss how balancing selection can be distinguished from local selection, by determining whether peaks of diversity in the region of the polymorphic locus are seen within or between demes. The width of such diversity peaks is shown to depend on the total species population size, rather than local deme sizes. We show that, with population subdivision, local selection enhances between-deme diversity even at neutral sites distant from the polymorphic locus, producing higher FST values than with no selection; very high values can be generated at sites close to a selected locus. Background selection also increases FST, mainly because of decreased diversity within populations, which implies that its effects may be distinguishable from those of local selection. Both effects are stronger in selfing than outcrossing populations. Linkage disequilibrium between neutral sites is generated by both balancing and local selection, especially in selfing populations, because of linkage disequilibrium between the neutral sites and the selectively maintained alleles. We discuss how these theoretical results can be related to data on genetic diversity within and between local populations of a species.

Dense marker genotypes allow the construction of the realized relationship matrix between individuals, with elements the realized proportion of the genome that is identical by descent (IBD) between pairs of individuals. In this paper, we demonstrate that by replacing the average relationship matrix derived from pedigree with the realized relationship matrix in best linear unbiased prediction (BLUP) of breeding values, the accuracy of the breeding values can be substantially increased, especially for individuals with no phenotype of their own. We further demonstrate that this method of predicting breeding values is exactly equivalent to the genomic selection methodology where the effects of quantitative trait loci (QTLs) contributing to variation in the trait are assumed to be normally distributed. The accuracy of breeding values predicted using the realized relationship matrix in the BLUP equations can be deterministically predicted for known family relationships, for example half sibs. The deterministic method uses the effective number of independently segregating loci controlling the phenotype that depends on the type of family relationship and the length of the genome. The accuracy of predicted breeding values depends on this number of effective loci, the family relationship and the number of phenotypic records. The deterministic prediction demonstrates that the accuracy of breeding values can approach unity if enough relatives are genotyped and phenotyped. For example, when 1000 full sibs per family were genotyped and phenotyped, and the heritability of the trait was 0·5, the reliability of predicted genomic breeding values (GEBVs) for individuals in the same full sib family without phenotypes was 0·82. These results were verified by simulation. A deterministic prediction was also derived for random mating populations, where the effective population size is the key parameter determining the effective number of independently segregating loci. If the effective population size is large, a very large number of individuals must be genotyped and phenotyped in order to accurately predict breeding values for unphenotyped individuals from the same population. If the heritability of the trait is 0·3, and Ne=1000, approximately 5750 individuals with genotypes and phenotypes are required in order to predict GEBVs of un-phenotyped individuals in the same population with an accuracy of 0·7.

This paper describes the relationship between probabilities of identity by descent and the distribution of coalescence times. By using the relationship between coalescence times and identity probabilities, it is possible to extend existing results for inbreeding coefficients in regular systems of mating to find the distribution of coalescence times and the mean coalescence times. It is also possible to express Sewall Wright's FST as the ratio of average coalescence times of different pairs of genes. That simplifies the analysis of models of subdivided populations because the average coalescence time can be found by computing separately the time it takes for two genes to enter a single subpopulation and time it takes for two genes in the same subpopulation to coalesce. The first time depends only on the migration matrix and the second time depends only on the total number of individuals in the population. This approach is used to find FST in the finite island model and in one- and two-dimensional stepping-stone models. It is also used to find the rate of approach of FST to its equilibrium value. These results are discussed in terms of different measures of genetic distance. It is proposed that, for the purposes of describing the amount of gene flow among local populations, the effective migration rate between pairs of local populations, M^, which is the migration rate that would be estimated for those two populations if they were actually in an island model, provides a simple and useful measure of genetic similarity that can be defined for either allozyme or DNA sequence data.

The fast multipoint maximum likelihood mapping algorithm for crosses between inbred lines, introduced by Jansen et al. (2001), is extended for mapping in a full-sib family of an outbreeding species. The method accommodates different segregation types of markers and differences in recombination between parents. The two separate parental multipoint maximum likelihood maps are joined into an integrated map by averaging lengths over anchored segments and by interpolating or extrapolating for markers segregating in one parent only. The method is illustrated with simulated data. The method will enable a more accurate estimation of maps in outbreeding species than current methods.

Gene expression microarrays are an innovative technology with enormous promise to help geneticists explore and understand the genome. Although the potential of this technology has been clearly demonstrated, many important and interesting statistical questions persist. We relate certain features of microarrays to other kinds of experimental data and argue that classical statistical techniques are appropriate and useful. We advocate greater attention to experimental design issues and a more prominent role for the ideas of statistical inference in microarray studies.

A common problem for genome-wide association analysis (GWAS) is lack of power for detection of quantitative trait loci (QTLs) and precision for fine mapping. Here, we present a statistical method, termed single-step GBLUP (ssGBLUP), which increases both power and precision without increasing genotyping costs by taking advantage of phenotypes from other related and unrelated subjects. The procedure achieves these goals by blending traditional pedigree relationships with those derived from genetic markers, and by conversion of estimated breeding values (EBVs) to marker effects and weights. Additionally, the application of mixed model approaches allow for both simple and complex analyses that involve multiple traits and confounding factors, such as environmental, epigenetic or maternal environmental effects. Efficiency of the method was examined using simulations with 15 800 subjects, of which 1500 were genotyped. Thirty QTLs were simulated across genome and assumed heritability was 0·5. Comparisons included ssGBLUP applied directly to phenotypes, BayesB and classical GWAS (CGWAS) with deregressed proofs. An average accuracy of prediction 0·89 was obtained by ssGBLUP after one iteration, which was 0·01 higher than by BayesB. Power and precision for GWAS applications were evaluated by the correlation between true QTL effects and the sum of m adjacent single nucleotide polymorphism (SNP) effects. The highest correlations were 0·82 and 0·74 for ssGBLUP and CGWAS with m=8, and 0·83 for BayesB with m=16. Standard deviations of the correlations across replicates were several times higher in BayesB than in ssGBLUP. The ssGBLUP method with marker weights is faster, more accurate and easier to implement for GWAS applications without computing pseudo-data.

A general representation of multilocus selection is extended to allow recombination to depend on genotype. The equations simplify if modifier alleles have small effects on recombination. The evolution of such modifiers only depends on how they alter recombination between the selected loci, and does not involve dominance in modifier effects. The net selection on modifiers can be found explicitly if epistasis is weak relative to recombination. This analysis shows that recombination can be favoured in two ways: because it impedes the response to epistasis which fluctuates in sign, or because it facilitates the response to directional selection. The first mechanism is implausible, because epistasis must change sign over periods of a few generations: faster or slower fluctuations favour reduced recombination. The second mechanism requires weak negative epistasis between favourable alleles, which may either be increasing, or held in check by mutation. The selection (si) on recombination modifiers depends on the reduction in additive variance of log (fitness) due to linkage disequilibria (υ1 < 0), and on non-additive variance in log (fitness) (V′2, V′3,.. epistasis between 2, 3.. loci). For unlinked loci and pairwise epistasis, si = − (υ1 + 4V2/3)δr, where δr is the average increase in recombination caused by the modifier. The approximations are checked against exact calculations for three loci, and against Charlesworth's analyses of mutation/selection balance (1990), and directional selection (1993). The analysis demonstrates a general relation between selection on recombination and observable components of fitness variation, which is open to experimental test.

An estimator is proposed for the parameter C = 4Nc. where N is the population size and c is the recombination rate. The estimator is appropriate for use with sequence or restriction site data from random samples from within populations. Properties of the estimator are investigated for an infinite-sites neutral model using Monte Carlo simulations. The median and mode of the distribution of the estimator are close to the true value for all parameter values examined, but large data sets are required to obtain reliable estimates.

Knowledge of the genetic relatedness among individuals is essential in diverse research areas such as behavioural ecology, conservation biology, quantitative genetics and forensics. How to estimate relatedness accurately from genetic marker information has been explored recently by many methodological studies. In this investigation I propose a new likelihood method that uses the genotypes of a triad of individuals in estimating pairwise relatedness (r). The idea is to use a third individual as a control (reference) in estimating the r between two other individuals, thus reducing the chance of genes identical in state being mistakenly inferred as identical by descent. The new method allows for inbreeding and accounts for genotype errors in data. Analyses of both simulated and human microsatellite and SNP datasets show that the quality of r estimates (measured by the root mean squared error, RMSE) is generally improved substantially by the new triadic likelihood method (TL) over the dyadic likelihood method and five moment estimators. Simulations also show that genotyping errors/mutations, when ignored, result in underestimates of r for related dyads, and that incorporating a model of typing errors in the TL method improves r estimates for highly related dyads but impairs those for loosely related or unrelated dyads. The effects of inbreeding were also investigated through simulations. It is concluded that, because most dyads in a natural population are unrelated or only loosely related, the overall performance of the new triadic likelihood method is the best, offering r estimates with a RMSE that is substantially smaller than the five commonly used moment estimators and the dyadic likelihood method.

Genetic parameters widely used to monitor genetic variation in conservation programmes, such as effective number of founders, founder genome equivalents and effective population size, are interrelated in terms of coancestries and variances of contributions from ancestors to descendants. A new parameter, the effective number of non-founders, is introduced to describe the relation between effective number of founders and founder genome equivalents. Practical recommendations for the maintenance of genetic variation in small captive populations are discussed. To maintain genetic diversity, minimum coancestry among individuals should be sought. This minimizes the variances of contributions from ancestors to descendants in all previous generations. The method of choice of parents and the system of mating should be independent of each other because a clear-cut recommendation cannot be given on the latter.