Effective sizes and time to migration-drift equilibrium in geographically subdivided populations.

Many versions of the effective population size (Ne) exist, and they are important in population genetics in order to quantify rates of change of various characteristics, such as inbreeding, heterozygosity, or allele frequencies. Traditionally, Ne was defined for single, isolated populations, but we have recently presented a mathematical framework for subdivided populations. In this paper we focus on diploid populations with geographic subdivision, and present new theoretical results. We compare the haploid and diploid versions of the inbreeding effective size (NeI) with novel expression for the variance effective size (NeV), and conclude that for local populations NeV is often much smaller than both versions of NeI, whenever they exist. Global NeV of the metapopulation, on the other hand, is close to the haploid NeI and much larger than the diploid NeI. We introduce a new effective size, the additive genetic variance effective size NeAV, which is of particular interest for long term protection of species. It quantifies the rate at which additive genetic variance is lost and we show that this effective size is closely related to the haploid version of NeI. Finally, we introduce a new measure of a population's deviation from migration-drift equilibrium, and apply it to quantify the time it takes to reach this equilibrium. Our findings are of importance for understanding the concept of effective population size in substructured populations and many of the results have applications in conservation biology.

Metapopulation effective size and conservation genetic goals for the Fennoscandian wolf (Canis lupus) population.

The Scandinavian wolf population descends from only five individuals, is isolated, highly inbred and exhibits inbreeding depression. To meet international conservation goals, suggestions include managing subdivided wolf populations over Fennoscandia as a metapopulation; a genetically effective population size of Ne⩾500, in line with the widely accepted long-term genetic viability target, might be attainable with gene flow among subpopulations of Scandinavia, Finland and Russian parts of Fennoscandia. Analytical means for modeling Ne of subdivided populations under such non-idealized situations have been missing, but we recently developed new mathematical methods for exploring inbreeding dynamics and effective population size of complex metapopulations. We apply this theory to the Fennoscandian wolves using empirical estimates of demographic parameters. We suggest that the long-term conservation genetic target for metapopulations should imply that inbreeding rates in the total system and in the separate subpopulations should not exceed Δf=0.001. This implies a meta-Ne of NeMeta⩾500 and a realized effective size of each subpopulation of NeRx⩾500. With current local effective population sizes and one migrant per generation, as recommended by management guidelines, the meta-Ne that can be reached is ~250. Unidirectional gene flow from Finland to Scandinavia reduces meta-Ne to ~130. Our results indicate that both local subpopulation effective sizes and migration among subpopulations must increase substantially from current levels to meet the conservation target. Alternatively, immigration from a large (Ne⩾500) population in northwestern Russia could support the Fennoscandian metapopulation, but immigration must be substantial (5-10 effective immigrants per generation) and migration among Fennoscandian subpopulations must nevertheless increase.

Relative risks and effective number of meioses: a unified approach for general genetic models and phenotypes.

Many common diseases are known to have genetic components, but since they are non-Mendelian, i.e. a large number of genetic factors affect the phenotype, these components are difficult to localize. These traits are often called complex and analysis of siblings is a valuable tool for mapping them. It has been shown that the power of the affected relative pairs method to detect linkage of a disease susceptibility locus depends on the locus contribution to increased risk of relatives compared with population prevalence (Risch, 1990a,b). In this paper we generalize calculation of relative risk to arbitrary phenotypes and genetic models, but also show that the relative risk can be split into the relative risk at the main locus and the relative risk due to interaction between the main locus and loci at other chromosomes. We demonstrate how the main locus contribution to the relative risk is related to probabilities of allele sharing identical by descent at the main locus, as well as power to detect linkage. To this end we use the effective number of meioses, introduced by Hössjer (2005a) as a convenient tool. Relative risks and effective number of meioses are computed for several genetic models with binary or quantitative phenotypes, with or without polygenic effects.

Kriging with nonparametric variance function estimation.

A method for fitting regression models to data that exhibit spatial correlation and heteroskedasticity is proposed. It is well known that ignoring a nonconstant variance does not bias least-squares estimates of regression parameters; thus, data analysts are easily lead to the false belief that moderate heteroskedasticity can generally be ignored. Unfortunately, ignoring nonconstant variance when fitting variograms can seriously bias estimated correlation functions. By modeling heteroskedasticity and standardizing by estimated standard deviations, our approach eliminates this bias in the correlations. A combination of parametric and nonparametric regression techniques is used to iteratively estimate the various components of the model. The approach is demonstrated on a large data set of predicted nitrogen runoff from agricultural lands in the Midwest and Northern Plains regions of the U.S.A. For this data set, the model comprises three main components: (1) the mean function, which includes farming practice variables, local soil and climate characteristics, and the nitrogen application treatment, is assumed to be linear in the parameters and is fitted by generalized least squares; (2) the variance function, which contains a local and a spatial component whose shapes are left unspecified, is estimated by local linear regression; and (3) the spatial correlation function is estimated by fitting a parametric variogram model to the standardized residuals, with the standardization adjusting the variogram for the presence of heteroskedasticity. The fitting of these three components is iterated until convergence. The model provides an improved fit to the data compared with a previous model that ignored the heteroskedasticity and the spatial correlation.