The advent of genome-wide association studies (GWAS) has been a great promise for our understanding of the connection between genotype and phenotype. Today, the NHGRI-EBI GWAS catalog contains 251,401 associations from 4,961 studies (1). This wealth of studies has also generated interest to use the summary statistics beyond the few top hits in order to make predictions for individuals without known phenotype, e.g. to predict polygenic risk scores or to study polygenic selection by comparing different groups. For instance, polygenic selection acting on the most studied polygenic trait, height, has been subject to multiple studies during the past decade (e.g. 2–6). They detected north-south gradients in Europe which were consistent with expectations. However, their GWAS summary statistics were based on the GIANT consortium data set, a meta-analysis of GWAS conducted in different European cohorts (7,8). The availability of large data sets with less stratification such as the UK Biobank (9) has led to a re-evaluation of those results. The nature of the GIANT consortium data set was realized to represent a potential problem for studies of polygenic adaptation which led several of the authors of the original articles to caution against the interpretations of polygenic selection on height (10,11). This was a great example on how the scientific community assessed their own earlier results in a critical way as more data became available. At the same time it left the question whether there is detectable polygenic selection separating populations more open than ever.

Generally, recent years have seen several articles critically assessing the portability of GWAS results and risk score predictions to other populations (12–14). Refoyo-Martínez et al. (15) are now presenting a systematic assessment on the robustness of cross-population signatures of polygenic adaptation in humans. They compiled GWAS results for complex traits which have been studied in more than one cohort and then use allele frequencies from the 1000 Genomes Project data (16) set to detect signals of polygenic score overdispersion. As the source for the allele frequencies is kept the same across all tests, differences between the signals must be caused by the underlying GWAS. The results are concerning as the level of overdispersion largely depends on the choice of GWAS cohort. Cohorts with homogenous ancestries show little to no overdispersion compared to cohorts of mixed ancestries such as meta-analyses. It appears that the meta-analyses fail to fully account for stratification in their data sets.

The authors based most of their analyses on the heavily studied trait height. Additionally, they use educational attainment (measured as the number of school years of an individual) as an example. This choice was due to the potential over- or misinterpretation of results by the media, the general public and by far right hate groups. Such traits are potentially confounded by unaccounted cultural and socio-economic factors. Showing that previous results about polygenic selection on educational attainment are not robust is an important result that needs to be communicated well. This forms a great example for everyone working in human genomics. We need to be aware that our results can sometimes be misinterpreted. And we need to make an effort to write our papers and communicate our results in a way that is honest about the limitations of our research and that prevents the misuse of our results by hate groups.

This article represents an important contribution to the field. It is cruicial to be aware of potential methodological biases and technical artifacts. Future studies of polygenic adaptation need to be cautious with their interpretations of polygenic score overdispersion. A recommendation would be to use GWAS results obtained in homogenous cohorts. But even if different biobank-scale cohorts of homogeneous ancestry are employed, there will always be some remaining risk of unaccounted stratification. These conclusions may seem sobering but they are part of the scientific process. We need additional controls and new, different methods than polygenic score overdispersion for assessing polygenic selection. Last year also saw the presentation of a novel approach using sequence data and GWAS summary statistics to detect directional selection on a polygenic trait (17). This new method appears to be robust to bias stemming from stratification in the GWAS cohort as well as other confounding factors. Such new developments show light at the end of the tunnel for the use of GWAS summary statistics in the study of polygenic adaptation.

References

1. Buniello A, MacArthur JAL, Cerezo M, Harris LW, Hayhurst J, Malangone C, et al. The NHGRI-EBI GWAS Catalog of published genome-wide association studies, targeted arrays and summary statistics 2019. Nucleic Acids Research. 2019 Jan 8;47(D1):D1005–12. doi: https://doi.org/10.1093/nar/gky1120

2. Turchin MC, Chiang CW, Palmer CD, Sankararaman S, Reich D, Hirschhorn JN. Evidence of widespread selection on standing variation in Europe at height-associated SNPs. Nature Genetics. 2012 Sep;44(9):1015–9. doi: https://doi.org/10.1038/ng.2368

3. Berg JJ, Coop G. A Population Genetic Signal of Polygenic Adaptation. PLOS Genetics. 2014 Aug 7;10(8):e1004412. doi: https://doi.org/10.1371/journal.pgen.1004412

4. Robinson MR, Hemani G, Medina-Gomez C, Mezzavilla M, Esko T, Shakhbazov K, et al. Population genetic differentiation of height and body mass index across Europe. Nature Genetics. 2015 Nov;47(11):1357–62. doi: https://doi.org/10.1038/ng.3401

5. Mathieson I, Lazaridis I, Rohland N, Mallick S, Patterson N, Roodenberg SA, et al. Genome-wide patterns of selection in 230 ancient Eurasians. Nature. 2015 Dec;528(7583):499–503. doi: https://doi.org/10.1038/nature16152

6. Racimo F, Berg JJ, Pickrell JK. Detecting polygenic adaptation in admixture graphs. Genetics. 2018. Arp;208(4):1565–1584. doi: https://doi.org/10.1534/genetics.117.300489

7. Lango Allen H, Estrada K, Lettre G, Berndt SI, Weedon MN, Rivadeneira F, et al. Hundreds of variants clustered in genomic loci and biological pathways affect human height. Nature. 2010 Oct;467(7317):832–8. doi: https://doi.org/10.1038/nature09410

8. Wood AR, Esko T, Yang J, Vedantam S, Pers TH, Gustafsson S, et al. Defining the role of common variation in the genomic and biological architecture of adult human height. Nat Genet. 2014 Nov;46(11):1173–86. doi: https://doi.org/10.1038/ng.3097

9. Bycroft C, Freeman C, Petkova D, Band G, Elliott LT, Sharp K, et al. The UK Biobank resource with deep phenotyping and genomic data. Nature. 2018 Oct;562(7726):203–9. doi: https://doi.org/10.1038/s41586-018-0579-z

10. Berg JJ, Harpak A, Sinnott-Armstrong N, Joergensen AM, Mostafavi H, Field Y, et al. Reduced signal for polygenic adaptation of height in UK Biobank. eLife. 2019 Mar 21;8:e39725. doi: https://doi.org/10.7554/eLife.39725

11. Sohail M, Maier RM, Ganna A, Bloemendal A, Martin AR, Turchin MC, et al. Polygenic adaptation on height is overestimated due to uncorrected stratification in genome-wide association studies. eLife. 2019 Mar 21;8:e39702. doi: https://doi.org/10.7554/eLife.39702

12. Martin AR, Kanai M, Kamatani Y, Okada Y, Neale BM, Daly MJ. Clinical use of current polygenic risk scores may exacerbate health disparities. Nature Genetics. 2019 Apr;51(4):584–91. doi: https://doi.org/10.1038/s41588-019-0379-x

13. Bitarello BD, Mathieson I. Polygenic Scores for Height in Admixed Populations. G3: Genes, Genomes, Genetics. 2020 Nov 1;10(11):4027–36. doi: https://doi.org/10.1534/g3.120.401658

14. Uricchio LH, Kitano HC, Gusev A, Zaitlen NA. An evolutionary compass for detecting signals of polygenic selection and mutational bias. Evolution Letters. 2019;3(1):69–79. doi: https://doi.org/10.1002/evl3.97

15. Refoyo-Martínez A, Liu S, Jørgensen AM, Jin X, Albrechtsen A, Martin AR, Racimo F. How robust are cross-population signatures of polygenic adaptation in humans? bioRxiv, 2021, 2020.07.13.200030, version 5 peer-reviewed and recommended by Peer community in Evolutionary Biology. doi: https://doi.org/10.1101/2020.07.13.200030

16. Auton A, Abecasis GR, Altshuler DM, Durbin RM, Abecasis GR, Bentley DR, et al. A global reference for human genetic variation. Nature. 2015 Sep 30;526(7571):68–74. doi: https://doi.org/10.1038/nature15393

17. Stern AJ, Speidel L, Zaitlen NA, Nielsen R. Disentangling selection on genetically correlated polygenic traits using whole-genome genealogies. bioRxiv. 2020 May 8;2020.05.07.083402. doi: https://doi.org/10.1101/2020.05.07.083402

In most animal species, males and females display distinct survival prospect, a phenomenon known as sex gap in longevity (SGL, Marais et al. 2018). The study of SGLs is crucial not only for having a full picture of the causes underlying organisms’ health, aging and death but also to initiate the development of sex-specific anti-aging interventions in humans (Austad and Bartke 2015). Three non-mutually evolutionary causes have been proposed to underlie SGLs (Marais et al. 2018). First, SGLs could be the consequences of sex-differences in life history strategies. For example, evolving dimorphic traits (e.g. body size, ornaments or armaments) may imply unequal physiological costs (e.g. developmental, maintenance) between the sexes and this may result in differences in longevity and aging. Second, mitochondria are usually transmitted by the mother and thus selection is blind to mitochondrial deleterious mutations affecting only males. Such mutations can freely accumulate in the mitochondrial genome and may reduce male longevity, a phenomenon called the mother’s curse (Frank and Hurst 1996). Third, in species with sex chromosomes, all recessive deleterious mutations will be expressed on the single X chromosome in XY males and may reduce their longevity (the unguarded X effect). In addition, the numerous transposable elements (TEs) on the Y chromosome may affect aging. TE activity is normally repressed by epigenetic regulation (DNA methylation, histone modifications and small RNAs). However, it is known that this regulation is disrupted with increasing age. Because of the TE-rich Y chromosome, more TEs may become active in old males than in old females, generating more somatic mutations, accelerating aging and reducing longevity in males (the toxic Y effect, Marais et al. 2018).
The relative contributions of these different effects to SGLs remain unknown. Sex-differences in life history strategies have been considered as the most important cause of SGLs for long (Tidière et al. 2015) but this effect remain equivocal (Lemaître et al. 2020) and cannot explain alone the diversity of patterns observed across species (Marais et al. 2018). Similarly, while studies in Drosophila and humans have shown that the mother’s curse contributes to SGLs in those organisms (e.g. Milot et al. 2007), its contribution may not be strong. Recently, two large-scale comparative analyses have shown that in species with XY chromosomes males show a shorter lifespan compared to females, while in species with ZW chromosomes (a system in which the female are the heterogametic sex and are ZW, and the males ZZ) the opposite pattern is observed (Pipoly et al. 2015; Xirocostas et al. 2020). Apart from these correlational studies, very little experimental tests of the effect of sex chromosomes on longevity have been conducted. In Drosophila, the evidence suggests that the unguarded X effect does not contribute to SGLs (Brengdahl et al. 2018). Whether a toxic Y effect exists in this species was unknown.
In a very elegant study, Brown et al. (2020) provided strong evidence for such a toxic Y effect in Drosophila melanogaster. First, they checked that in the D. melanogaster strain that they were studying (Canton-S), males were indeed dying younger than females. They also confirmed that in this strain, as in others, the male genomes include more repeats and heterochromatin than the female ones using cytometry. A careful analysis of the heterochromatin (using H3K9me2, a repressive histone modification typical of heterochromatin, as a proxy) in old flies revealed that heterochromatin loss was much more important in males than in females, in particular on the Y chromosome (but also to a lesser extent at the pericentromric regions of the autosomes). This change in heterochromatin had two outputs, they found. First, the expression of the genes in those regions was affected. They highlighted that many of such genes are involved in immunity and regulation with a potential impact on longevity. Second, they found a striking TE reactivation. These two effects were stronger in males. While females showed clear reactivation of 6 TEs, with the total fraction of repeats in the transcriptome going from 2% (young females) to 4.6% (old females), males experienced the reactivation of 32 TEs, with the total fraction of repeats in the transcriptome going from 1.6% (young males) to 5.8% (old males). It appeared that most of these TEs are Y-linked. And when focusing on Y-linked repeats, they found that 32 Y-linked TEs became upregulated during male aging and the fraction of Y-linked TEs in the transcriptome increased ninefold.
All these observations clearly suggested that male longevity was decreased because of a toxic Y effect. To really uncover a causal relationship between having a Y chromosome and shorter longevity, Brown et al. (2020) artificially produced flies with atypical karyotypes: X0 males, XXY females and XYY males. This is very interesting as they could uncouple the effect of the phenotypical sex (being male or female) and having a Y chromosome or not, as in fruit flies sex is determined not by the Y chromosome but by the X/autosome ratio. Their results are striking. They found that longevity of the X0 males was the highest (higher than XX females in fact), and that of the XYY males the lowest. Females XXY had intermediate longevities. Importantly, this was found to be robust to genomic background as results were the same using crosses from different strains. When analysing TEs of these flies, they found a particularly strong expression of the Y-linked TEs in old XXY and XYY flies. Interestingly, in young XXY and XYY flies Y-linked TEs expression was also strong, suggesting the chromatin regulation of the Y chromosome is disrupted in these flies.
This work points to the idea that SGLs in D. melanogaster are mainly explained by the toxic Y effect. The molecular details however remain to be elucidated. The effect of the Y chromosome on aging might be more complex than envisioned in the toxic Y model presented above. Brown et al. (2020) indeed found that heterochromatin loss was globally faster in males, both at the Y chromosome and the autosomes. The organisation of the nucleus, in particular of the nucleolus, which is involved in heterochromatin maintenance, involves the sex chromosomes in D. melanogaster as discussed in the paper, and may explain this observation. The epigenetic status of the Y chromosome is known to affect that of all the autosomes in Drosophila (Lemos et al. 2008). Also, in Brown et al. (2020) most of the work (in particular the genomic part) has been done on Canton-S. Only D. melanogaster was studied but limited data suggest different Drosophila species may have different SGLs. The TE analysis is known to be tricky, different tools to analyse TE expression exist (e.g. Lerat et al. 2017; Lanciano and Cristofari 2020). Future work should focus on testing the toxic Y effect on other D. melanogaster strains and other Drosophila species, using different tools to study TE expression, and on dissecting the molecular details of the toxic Y effect.

References

Austad, S. N., and Bartke, A. (2015). Sex differences in longevity and in responses to anti-aging interventions: A Mini-Review. Gerontology, 62(1), 40–46. 10.1159/000381472
Brengdahl, M., Kimber, C. M., Maguire-Baxter, J., and Friberg, U. (2018). Sex differences in life span: Females homozygous for the X chromosome do not suffer the shorter life span predicted by the unguarded X hypothesis. Evolution; international journal of organic evolution, 72(3), 568–577. 10.1111/evo.13434
Brown, E. J., Nguyen, A. H., and Bachtrog, D. (2020). The Y chromosome may contribute to sex-specific ageing in Drosophila. Nature ecology and evolution, 4(6), 853–862. 10.1038/s41559-020-1179-5 or preprint link on bioRxiv
Frank, S. A., and Hurst, L. D. (1996). Mitochondria and male disease. Nature, 383(6597), 224. 10.1038/383224a0
Lanciano, S., and Cristofari, G. (2020). Measuring and interpreting transposable element expression. Nature reviews. Genetics, 10.1038/s41576-020-0251-y. Advance online publication. 10.1038/s41576-020-0251-y
Lemaître, J. F., Ronget, V., Tidière, M., Allainé, D., Berger, V., Cohas, A., Colchero, F., Conde, D. A., Garratt, M., Liker, A., Marais, G., Scheuerlein, A., Székely, T., and Gaillard, J. M. (2020). Sex differences in adult lifespan and aging rates of mortality across wild mammals. Proceedings of the National Academy of Sciences of the United States of America, 117(15), 8546–8553. 10.1073/pnas.1911999117
Lemos, B., Araripe, L. O., and Hartl, D. L. (2008). Polymorphic Y chromosomes harbor cryptic variation with manifold functional consequences. Science (New York, N.Y.), 319(5859), 91–93. 10.1126/science.1148861
Lerat, E., Fablet, M., Modolo, L., Lopez-Maestre, H., and Vieira, C. (2017). TEtools facilitates big data expression analysis of transposable elements and reveals an antagonism between their activity and that of piRNA genes. Nucleic acids research, 45(4), e17. 10.1093/nar/gkw953
Marais, G., Gaillard, J. M., Vieira, C., Plotton, I., Sanlaville, D., Gueyffier, F., and Lemaitre, J. F. (2018). Sex gap in aging and longevity: can sex chromosomes play a role?. Biology of sex differences, 9(1), 33. 10.1186/s13293-018-0181-y
Milot, E., Moreau, C., Gagnon, A., Cohen, A. A., Brais, B., and Labuda, D. (2017). Mother's curse neutralizes natural selection against a human genetic disease over three centuries. Nature ecology and evolution, 1(9), 1400–1406. 10.1038/s41559-017-0276-6
Pipoly, I., Bókony, V., Kirkpatrick, M., Donald, P. F., Székely, T., and Liker, A. (2015). The genetic sex-determination system predicts adult sex ratios in tetrapods. Nature, 527(7576), 91–94. 10.1038/nature15380
Tidière, M., Gaillard, J. M., Müller, D. W., Lackey, L. B., Gimenez, O., Clauss, M., and Lemaître, J. F. (2015). Does sexual selection shape sex differences in longevity and senescence patterns across vertebrates? A review and new insights from captive ruminants. Evolution; international journal of organic evolution, 69(12), 3123–3140. 10.1111/evo.12801
Xirocostas, Z. A., Everingham, S. E., and Moles, A. T. (2020). The sex with the reduced sex chromosome dies earlier: a comparison across the tree of life. Biology letters, 16(3), 20190867. 10.1098/rsbl.2019.0867

Organisms often show robustness to genetic or environmental perturbations. Whether these two components of robustness can evolve separately is the focus of the paper by Le Rouzic [1]. Using theoretical analysis and individual-based computer simulations of a gene regulatory network model, he shows that multiple aspects of robustness can be investigated as a set of pleiotropically linked quantitative traits. While genetically correlated, various robustness components (e.g., mutational, developmental, homeostasis) of gene expression in the regulatory network evolved more or less independently from each other under directional selection. The quantitative approach of Le Rouzic could explain both how unselected robustness components can respond to selection on other components and why various robustness-related features seem to have their own evolutionary history. Moreover, he shows that all components were evolvable, but not all to the same extent. Robustness to environmental disturbances and gene expression stability showed the largest responses while increased robustness to genetic disturbances was slower. Interestingly, all components were positively correlated and remained so after selection for increased or decreased robustness.

This study is an important contribution to the discussion of the evolution of robustness in biological systems. While it has long been recognized that organisms possess the ability to buffer genetic and environmental perturbations to maintain homeostasis (e.g., canalization [2]), the genetic basis and evolutionary routes to robustness and canalization are still not well understood. Models of regulatory gene networks have often been used to address aspects of robustness evolution (e.g., [3]). Le Rouzic [1] used a gene regulatory network model derived from Wagner’s model [4]. The model has as end product the expression level of a set of genes influenced by a set of regulatory elements (e.g., transcription factors). The level and stability of expression are a property of the regulatory interactions in the network.

Le Rouzic made an important contribution to the study of such gene regulation models by using a quantitative genetics approach to the evolution of robustness. He crafted a way to assess the mutational variability and selection response of the components of robustness he was interested in. Le Rouzic’s approach opens avenues to investigate further aspects of gene network evolutionary properties, for instance to understand the evolution of phenotypic plasticity.

Le Rouzic also discusses ways to measure his different robustness components in empirical studies. As the model is about gene expression levels at a set of protein-coding genes influenced by a set of regulatory elements, it naturally points to the possibility of using RNA sequencing to measure the variation of gene expression in know gene networks and assess their robustness. Robustness could then be studied as a multidimensional quantitative trait in an experimental setting.

References

[1] Le Rouzic, A (2022) Gene network robustness as a multivariate character. arXiv: 2101.01564, ver. 5 peer-reviewed and recommended by Peer Community in Evolutionary Biology. https://arxiv.org/abs/2101.01564

[2] Waddington CH (1942) Canalization of Development and the Inheritance of Acquired Characters. Nature, 150, 563–565. https://doi.org/10.1038/150563a0

[3] Draghi J, Whitlock M (2015) Robustness to noise in gene expression evolves despite epistatic constraints in a model of gene networks. Evolution, 69, 2345–2358. https://doi.org/10.1111/evo.12732

[4] Wagner A (1994) Evolution of gene networks by gene duplications: a mathematical model and its implications on genome organization. Proceedings of the National Academy of Sciences, 91, 4387–4391. https://doi.org/10.1073/pnas.91.10.4387

Sterols and hopanoids are sometimes used as biomarkers to infer the origin of certain groups of organisms. Traditionally, hopanoid-derived products in ancient rocks have been considered to indicate the presence of bacteria, whereas sterol derivatives have been considered to be exclusive to eukaryotes. However, a closer look at the topic reveals a rather complex distribution of either compound in both bacteria and eukaryotes. (1). The known biosynthetic pathways for sterols and hopanoids are similar but diverge at a critical step where two different enzymes are used: squalene-hopene cyclase (SHC) and oxidosqualene cyclase (OSC), the latter requiring oxygen. These two enzymes belong to the same gene family, whose complex evolutionary history is difficult to reconcile with the known species phylogeny.

In this study (2), Dr. Warren R. Francis revisits the evolution of this gene family using an extended dataset with a broader taxonomic representation. In contrast to the traditional representation of the tree rooted between SHC and OSC paralogs (i.e., based on function), the author proposes that rooting the tree within bacterial SHCs and assuming a secondary origin of OSC is more parsimonious. This postulates SHC to be the ancestral function –retained in many extant bacteria and some eukaryotes– and OSC to have emerged later within bacteria –currently being mostly present in eukaryotes–. The reconstructed evolutionary history is arguably complex and can only be reconciled with the species' phylogeny by invoking many secondary losses. These losses are considered likely because many extant species acquire sterols and hopanoids by diet and lack one or both enzymes. Some cases of recent horizontal gene transfer are also proposed.

In contrast to the dichotomy between bacterial SHCs and eukaryote OSCs, the new proposed scenario suggests that the eukaryote ancestor likely inherited both enzymes from bacteria and thus could be able to synthesize both sterols and hopanoids. Under this hypothesis, not only bacteria but also eukaryotes could be responsible for the hopane found in old rocks. This agrees with eukaryote fossils dating back to more than 1 billion years ago (3). Also, the observed increase of sterane levels in rocks ~600-700 million years old cannot be associated with the origin of eukaryotes, which is a much older event, but could rather reflect changes in atmospheric oxygen levels because oxygen is required for the synthesis of sterols by OSC.

References

1. Santana-Molina C, Rivas-Marin E, Rojas AM, Devos DP (2020) Origin and Evolution of Polycyclic Triterpene Synthesis. Molecular Biology and Evolution, 37, 1925–1941. https://doi.org/10.1093/molbev/msaa054

2. Francis WR (2022) The Eukaryotic Last Common Ancestor Was Bifunctional for Hopanoid and Sterol Production. Preprints, 2020040186, ver. 5 peer-reviewed and recommended by Peer Community in Evolutionary Biology.  https://doi.org/10.20944/preprints202004.0186.v5

3. Butterfield NJ (2000) Bangiomorpha pubescens n. gen., n. sp.: implications for the evolution of sex, multicellularity, and the Mesoproterozoic/Neoproterozoic radiation of eukaryotes. Paleobiology, 26, 386–404.  https://doi.org/10.1666/0094-8373(2000)026<0386:BPNGNS>2.0.CO;2

​Of 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