Heritability allows a comparison of the relative importance of genes and environment to the variation of traits within and across populations. The concept of heritability and its definition as an estimable, dimensionless population parameter was introduced by Sewall Wright and Ronald Fisher nearly a century ago. Despite continuous misunderstandings and controversies over its use and application, heritability remains key to the response to selection in evolutionary biology and agriculture, and to the prediction of disease risk in medicine. Recent reports of substantial heritability for gene expression and new estimation methods using marker data highlight the relevance of heritability in the genomics era.

A theoretical investigation has been made of the influence of population size (N) and recombination fraction (c) on linkage disequilibrium (D) between a pair of loci. Two situations were studied: (i) where both loci had no effect on fitness and (ii) where they showed heterozygote superiority, but no epistacy.If the populations are initially in linkage equilibrium, then the mean value ofD remains zero with inbreeding, but the mean ofD (2) increases to a maximum value and decreases until fixation is reached at both loci. The tighter the linkage and the greater the selection, then the later is the maximum in the mean ofD (2) reached, and the larger its value. The correlation of gene frequencies,r, in the population of gametes within segregating lines was also studied. It was found that, for a range of selection intensities and initial gene frequencies, the mean value ofr (2) was determined almost entirely byN c and time, measured proportional toN.The implication of these results on observations of linkage disequilibrium in natural populations is discussed.

(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.

Recently, we reported a method to estimate the proportion of phenotypic variance explained by all SNPs from genome-wide association studies, and estimated that half of the heritability for human height was captured by common SNPs. Here we partition genetic variation for height, body mass index (BMI), von Willebrand factor (vWF) and QT interval (QTi) onto chromosomes and chromosome segments, using 586,898 SNPs genotyped on 11,586 unrelated individuals. We estimate that ~45%, ~17%, ~25% and ~21% of variance in height, BMI, vWF and QTi, respectively, can be explained by considering all autosomal SNPs simultaneously, and a further ~0.5–1% by X-chromosome SNPs for height, BMI and vWF. We show that variance explained by each chromosome for height and QTi is proportional to the total gene length on that chromosome. In genome-wide analyses, common SNPs in or near genes explain more variation than SNPs between genes. We propose a novel approach to estimate variation due to cryptic relatedness and population stratification. Our results provide further evidence that a substantial proportion of heritability is accounted for by causal variants in linkage disequilibrium with common SNPs; that height, BMI and QTi are highly polygenic traits; and that the additive variation explained by a part of the genome is approximately proportional to the total length of DNA contained within genes therein.

SUMMARYA 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.

A moment estimator of, the coancestry coefficient for alleles within a population, was described by Weir & Cockerham in 1984 (100) and is still widely cited. The estimate is used by population geneticists to characterize population structure, by ecologists to estimate migration rates, by animal breeders to describe genetic variation, and by forensic scientists to quantify the strength of matching DNA profiles. This review extends the work of Weir & Cockerham by allowing different levels of coancestry for different populations, and by allowing non-zero coancestries between pairs of populations. All estimates are relative to the average value of theta between pairs of populations. Moment estimates for within- and between-population theta values are likely to have large sampling variances, although these may be reduced by combining information over loci. Variances also decrease with the numbers of alleles at a locus, and with the numbers of populations sampled. This review also extends the work of Weir & Cockerham by employing maximum likelihood methods under the assumption that allele frequencies follow the normal distribution over populations. For the case of equal theta values within populations and zero theta values between populations, the maximum likelihood estimate is the same as that given by Robertson & Hill in 1984 (70). The review concludes by relating functions of theta values to times of population divergence under a pure drift model.

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