UW Aquatic & Fishery Sciences Quantitative Seminar

# Robin Waples

Northwest Fisheries Science Center

**Effective population size: two new twists on an old idea**

## Abstract

The concept of effective population size (*Ne*) is based on an elegantly simple idea which, however, rapidly becomes very complex in most real world situations. Here I discuss two recent projects that reveal new twists in our understanding of effective size.

I. *Inbreeding effective size and parentage analysis without parents*

In a typical parentage analysis, multilocus genotypes are scored in both parents and progeny, and these data are used to ‘assign’ progeny to parents. This type of study has produced many novel insights, including the ability to directly calculate the number of genes contributed to the next generation by each parent (*ki*). From this information, and knowing the total number of parents (*N*), one can use standard formulas to calculate *Ne*. But what if your sample of parents is incomplete or missing entirely? In theory, it should be possible to reconstruct parental genotypes based entirely on a sample of progeny. This is not quite feasible yet but might be in the near future if large enough numbers of SNPs become available. If the two parents for each individual in a sample could be identified, it would be possible to construct a vector of parental *ki* values, but this would provide information only about parents that actually contributed offspring to your sample. What about the ‘null’ parents that produced no offspring in your sample, and how would they affect an estimate of *Ne*? The surprising answer is that null parents have no effect at all. I show the following: 1) It is possible to rewrite the standard formula for inbreeding effective size so that it is only a function of ∑(*ki*) and ∑(*ki*2). That is, it is not necessary to know *N* to calculate inbreeding *Ne*. This same relationship does not hold for variance *Ne*. 2) It is easy to prove #1 assuming a complete sample of all progeny. However, even if only a sample of progeny is taken, simulated data show that the estimate of *Ne* using the simple formula that ignores null parents is unbiased. This means that parentage analysis without parents can be used to provide an unbiased, single-sample estimate of inbreeding *Ne*, provided parental contributions to the offspring sample can be resolved. 3) It is not necessary to actually reconstruct parental genotypes; from a matrix of pairwise relationships (as can be estimated by some current software programs) it is possible to construct the vector of *ki* values and estimate *Ne*. 4) Accuracy and precision of the new method based on parentage analysis without parents compares favorably with single-sample estimators of *Ne* currently in use.

II. *A hybrid Felsenstein-Hill method for calculating Ne in species with overlapping generations*

Like most population genetics theory, the concept of *Ne* was developed under a discrete-generation model. During the Nixon administration, two papers (Felsenstein 1971; Hill 1972) showed how to calculate *Ne* for age-structured species. The two approaches produce the same answer under certain conditions and have contrasting advantages and disadvantages. I describe a hybrid approach that combines the best features of both. Like Felsenstein (1971), the new method is based on age-specific survival and fertility rates and therefore can be directly applied to any species for which life-table data are available. However, like Hill (1972), the hybrid method is more general as it relaxes Felsenstein’s assumption of Poisson variance in reproductive success each time period. The basic principle underlying the new method is that age structure stratifies a population into winners and losers in the game of life: individuals that live longer have more opportunities to reproduce and therefore have a higher mean lifetime reproductive success than do individuals that die at a younger age. This creates different classes of individuals within the population based on age at death. The new method has the following features:

- It uses demographic information of the type found in a life table;
- It can accommodate two sexes with unequal sex ratio and/or different vital rates;
- It can accommodate overdispersed variance in reproductive success and different ratios of variance to mean reproductive success in the two sexes;
- It can calculate effective size in species that change sex during their lifetime;
- It can calculate
*Ne*and*Ne*/*N*based on various ways of defining*N*; - It can allow one to explore the relationship between the effective number of breeders per year (
*Nb*) and effective size per generation (*Ne*) in age-structured species; - It (will soon be) implemented in freely available software with a flexible user interface.