## TAYLOR Jesse ("Jay")

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*16 May 2023*

### A new and almost perfectly accurate approximation of the eigenvalue effective population size of a dioecious population: comparisons with other estimates and detailed proofs

#### All you ever wanted to know about Ne in one handy place

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**Charles Baer**based on reviews by Jesse ("Jay") Taylor and 1 anonymous reviewerOf the four evolutionary forces, three can be straightforwardly summarized both conceptually and mathematically in the context of an allele at a genomic locus. Mutation (the mutation rate, μ) is simply captured by the per-site, per-generation probability that an allele mutates into a different allele. Recombination (the recombination rate, r) is captured as the probability of recombination between two sites, wherein alleles that are in different genomes in one generation come together in the same genome in the next generation. Natural selection (the selection coefficient, s) is captured by the probability that an allele is present in the next generation, relative to some reference.

Random genetic drift – the random fluctuation in allele frequency due to sampling in a finite population - is not so straightforwardly summarized. The first, and most common way of characterizing evolutionary dynamics in a finite population is the Wright-Fisher model, in which the only deviation from the assumptions of Hardy-Weinberg conditions is finite population size. Importantly, in a W-F population, mating between diploid individuals is random, which implies self-fertile monoecy, and generations are non-overlapping. In an ideal W-F population, the probability that a gene copy leaves i descendants in the next generation is the result of binomial sampling of uniting gametes (if the locus is biallelic). The – and the next word is meaningful – magnitude/strength/rate/power/amount of genetic drift is proportional to 1/2N, where N is the size of the population. All of the following are affected by genetic drift: (1) the probability that a neutral allele ultimately reaches fixation, (2) the rate of loss of genetic variation within a population, (3) the rate of increase of genetic variance among populations, (4) the amount of genetic variation segregating in a population, (5) the probability of fixation/loss of a weakly selected variant.

Presumably no real population adheres to ideal W-F conditions, which leads to the notion of "effective population size", Ne (Wright 1931), loosely defined as "the size of an ideal W-F population that experiences an equivalent strength of genetic drift". Almost always, Ne<N, and any violation of W-F assumptions can affect Ne. Importantly, Ne can be defined in different ways, and the specific formulation of Ne can have different implications for evolution. Ne was initially defined in terms of the rate of decrease of heterozygosity (inbreeding effective size) and increase in variance among populations (variance effective size). Ewens (1979) defined the Eigenvalue effective size (equivalent to the "random extinction" effective size) and elaborated on the conditions under which the various formulations of Ne differ (Ewens 1982). Nordborg and Krone (2002) defined the effective size in terms of the coalescent, and they identified conditions in which genetic drift cannot be described in terms of a W-F model (Sjodin et al. 2005); also see Karasov et al. (2010); Neher and Shraiman (2011).

Distinct from the issue of defining Ne is the issue of calculating Ne from data, which is the focus of this paper by De Meeus and Noûs (2023). Pudovkin et al. (1996) showed that the Eigenvalue effective size in a dioecious population can be formulated in terms of excess heterozygosity, which the current authors note is equivalent to formulating Ne in terms of Wright's FIS statistic. As emphasized by the title, the marquee contribution of this paper is to provide a better approximation of the Eigenvalue effective size in a dioecious population. Science marches onward, although the empirical utility of this advance is obviously limited, given the tremendous inherent sources of uncertainty in real-world estimates of Ne. Perhaps more valuable, however, is the extensive set of appendixes, in which detailed derivations are provided for the various formulations of effective size. By way of analogy, the material presented here can be thought of as an extension of the material presented in section 7.6 of Crow and Kimura (1970), in which the Inbreeding and Variance effective population sizes are derived and compared. The appendixes should serve as a handy go-to source of detailed theoretical information with respect to the different formulations of effective population size.

**REFERENCES**

Crow, J. F. and M. Kimura. 1970. An Introduction to Population Genetics Theory. The Blackburn Press, Caldwell, NJ.

De Meeûs, T. and Noûs, C. 2023. A new and almost perfectly accurate approximation of the eigenvalue effective population size of a dioecious population: comparisons with other estimates and detailed proofs. Zenodo, ver. 6 peer-reviewed and recommended by Peer Community in Evolutionary Biology. https://doi.org/10.5281/zenodo.7927968

Ewens, W. J. 1979. Mathematical Population Genetics. Springer-Verlag, Berlin.

Ewens, W. J. 1982. On the concept of the effective population size. Theoretical Population Biology 21:373-378. https://doi.org/10.1016/0040-5809(82)90024-7

Karasov, T., P. W. Messer, and D. A. Petrov. 2010. Evidence that adaptation in Drosophila Is not limited by mutation at single sites. Plos Genetics 6. https://doi.org/10.1371/journal.pgen.1000924

Neher, R. A. and B. I. Shraiman. 2011. Genetic Draft and Quasi-Neutrality in Large Facultatively Sexual Populations. Genetics 188:975-U370. https://doi.org/10.1534/genetics.111.128876

Nordborg, M. and S. M. Krone. 2002. Separation of time scales and convergence to the coalescent in structured populations. Pp. 194–232 in M. Slatkin, and M. Veuille, eds. Modern Developments in Theoretical Population Genetics: The Legacy of Gustave Malécot. Oxford University Press, Oxford. https://www.webpages.uidaho.edu/~krone/malecot.pdf

Pudovkin, A. I., D. V. Zaykin, and D. Hedgecock. 1996. On the potential for estimating the effective number of breeders from heterozygote-excess in progeny. Genetics 144:383-387. https://doi.org/10.1093/genetics/144.1.383

Sjodin, P., I. Kaj, S. Krone, M. Lascoux, and M. Nordborg. 2005. On the meaning and existence of an effective population size. Genetics 169:1061-1070. https://doi.org/10.1534/genetics.104.026799

Wright, S. 1931. Evolution in Mendelian populations. Genetics 16:0097-0159. https://doi.org/10.1093/genetics/16.2.97