Effective population size (N e ) is a key parameter in population genetics. It has important applications in evolutionary biology, conservation genetics and plant and animal breeding, because it measures the rates of genetic drift and inbreeding and affects the efficacy of systematic evolutionary forces, such as mutation, selection and migration. We review the developments in predictive equations and estimation methodologies of effective size. In the prediction part, we focus on the equations for populations with different modes of reproduction, for populations under selection for unlinked or linked loci and for the specific applications to conservation genetics. In the estimation part, we focus on methods developed for estimating the current or recent effective size from molecular marker or sequence data. We discuss some underdeveloped areas in predicting and estimating N e for future research.Heredity ( INTRODUCTIONThe concept of effective population size, introduced by Sewall Wright (1931, 1933), is central to plant and animal breeding (Falconer and Mackay, 1996), conservation genetics (Frankham et al., 2010;Allendorf et al., 2013) and molecular variation and evolution (Charlesworth and Charlesworth, 2010), as it quantifies the magnitude of genetic drift and inbreeding in real-world populations. A substantial number of extensions to the basic theory and predictions were made since the seminal work of Wright, with main early developments by James Crow and Motoo Kimura (Kimura and Crow, 1963a;Crow and Kimura, 1970) and later by a list of contributors. Several review papers (Crow and Denniston, 1988;Caballero, 1994;Wang and Caballero, 1999;Nomura, 2005a) and population genetic books (Fisher, 1965;Wright, 1969;Ewens, 1979;Nagylaki, 1992) have summarised the existing theory in predicting the effective size of a population at different spatial and timescales under various inheritance modes and demographies. Comparatively, methodological developments (reviewed by Schwartz et al., 1999;Beaumont, 2003a;Wang, 2005;Palstra and Ruzzante, 2008;Luikart et al., 2010;Gilbert and Whitlock, 2015) in estimating the effective size of natural populations from genetic data lag behind but are accelerating in the past decade, thanks to the rapid developments of molecular biology.The classical developments of effective population size theory are based on the rate of change in gene frequency variance (genetic drift) or the rate of inbreeding. The effective population size is defined in reference to the Wright-Fisher idealised population, that is, a hypothetical population with very simplifying characteristics where genetic drift is the only factor in operation, and the dynamics of allelic and genotypic frequencies across generations merely depend on the population census (N) size. The effective size of a real population is then defined as the size of an idealised population, which would give rise to the rate of inbreeding and the rate of change in variance of gene frequencies actually observed in the population under consideration,

We use regression models to investigate the effects of inbreeding in 119 zoo populations, encompassing 88 species of mammals, birds, reptiles and amphibians. Meta-analyses show that inbreeding depression for neonatal survival was significant across the 119 populations although the severity of inbreeding depression appears to vary among taxa. However, few predictors of a population's response to inbreeding are found reliable. The models are most likely to detect inbreeding depression in large populations, that is, in populations in which their statistical power is maximised. Purging was found to be significant in 14 populations and a significant trend of purging was found across populations. The change in inbreeding depression due to purging averaged across the 119 populations is o1%, however, suggesting that the fitness benefits of purging are rarely appreciable. The study re-emphasises the necessity to avoid inbreeding in captive breeding programmes and shows that purging cannot be relied upon to remove deleterious alleles from zoo populations.

Using computer simulations, we evaluate the effects of genetic purging of inbreeding load in small populations, assuming genetic models of deleterious mutations which account for the typical amount of load empirically observed. Our results show that genetic purging efficiently removes the inbreeding load of both lethal and non-lethal mutations, reducing the amount of inbreeding depression relative to that expected without selection. We find that the minimum effective population size to avoid severe inbreeding depression in the short term is of the order of N≈70 for a wide range of species' reproductive rates. We also carried out simulations of captive breeding populations where two contrasting management methods are performed, one avoiding inbreeding (equalisation of parental contributions (EC)) and the other forcing it (circular sib mating (CM)). We show that, for the inbreeding loads considered, CM leads to unacceptably high extinction risks and, as a result, to lower genetic diversity than EC. Thus we conclude that methods aimed at enhancing purging by intentional inbreeding should not be generally advised in captive breeding conservation programmes.

The genetic relatedness between individuals because of their recent common ancestry is now routinely estimated from marker genotype data in molecular ecology, evolutionary biology and conservation studies. The estimators developed for this purpose assume that marker allele freque218 in a population are known without errors. Unfortunately, however, these frequencies, upon which both the definition and the estimation of relatedness are based, are rarely known in reality. Frequently, the only data available in a relatedness analysis are a sample of multilocus genotypes from which both allele frequencies and relatedness must be deduced. Furthermore, because of various constraints, sample sizes of individuals can be quite small (say <50 individuals) in practice. This study shows, for the first time, that the widely used relatedness estimators become severely biased when they use allele frequencies calculated from an extremely small sample (say <10 individuals). The extent of bias depends on the sample size, the (unknown) population allele frequencies, the actual relatedness and the estimators. It also shows that relatedness estimators become even more biased when they use allele frequencies calculated from a sample by omitting a focal pair of individuals whose relatedness is being estimated. This study modifies two estimators to suit small samples and shows, both analytically and by analysing simulated and empirical data, that the two modified estimators are much less biased, more precise and more accurate than the original estimators. These performance advantages of the modified estimators are shown to increase with a decreasing sample size of individuals and with an increasing value of actual relatedness.

Parentage exclusion probabilities are now routinely calculated in genetic marker-assisted parentage analyses to indicate the statistical power of the analyses achievable for a given set of markers, and to measure the informativeness of a set of markers for parentage inference. Previous formulas invariably assume that parentage is to be sought for a single offspring, while in practice multiple full siblings might be sampled (for example, seeds, eggs or young from a pair of monogamous parents) and their father, mother or both are to be assigned among a number of candidates. In this study, I derive formulas for parentage exclusion probabilities for an arbitrary number (n) of fullsibs, which reduce to previous equations for the special case of n ¼ 1. I also derive sibship exclusion probabilities, and investigate the power of differentiating half-sib, avuncular and grandparent-grandoffspring relationships using unlinked autosomal markers among different numbers of tested individuals. Applications of the formulas are demonstrated using both theoretical and empirical data sets of allele frequencies. The results from the study highlight the conclusion that the power of genealogical relationship inferences can be enhanced enormously by analysing multiple individuals for a given set of markers. The equations derived in this study allow more accurate determination of marker information and of the power of a parentage/sibship analysis. In addition, they can be used to guide experimental designs of parentage analyses in selecting markers and determining the number of offspring to be sampled and genotyped.