A general pattern observed across species with sex chromosomes (XX/XY or ZW/ZZ sex determination) is that the sex-specific chromosome (Y or W) and the paired chromosome (X or Z) evolve to suppress recombination along their length, often via structural rearrangements. This is a key transition in the evolution of sex chromosomes, as this shutdown of recombination allows for (1) divergence between the sex chromosome pair and (2) the eventual accumulation of deleterious mutations and loss of gene content on the sex-specific chromosome (‘degradation’). Understanding what causes the initial shutdown of recombination across the sex chromosomes, and what maintains this recombination shutdown, is therefore a key question in the field of sex chromosome evolution.

This paper provides a survey of multiple theories that explain the shutdown of recombination, but focuses on determining whether the recently proposed sheltering theory put forward in Jay et al. 2022 is a viable explanation of recombination. The initial step in the Jay et al. 2022 model involves the fixation of a ‘less-loaded’ or ‘lucky’ inversion: an inversion that happens to capture a genetic region with fewer deleterious mutations than the average haplotype. The dynamics of such inversions on autosomes are well-investigated (Nei et al. 1967). If the inverted haplotype is sufficiently more fit than the corresponding haplotypes, it may fix in the population, which restores recombination in the region captured by the inversion (as homozygotes for the inversion may recombine freely). However, if such an inverted haplotype captures a sex-determining locus (e.g. a dominant male-determining factor, such as SRY in mammals), it remains permanently heterozygous even if it reaches fixation because the male-determining factor is never homozygous (i.e., YY individuals are never produced). The fixation of the inverted haplotype therefore induces a shutdown of recombination in inverted region between the X and Y chromosomes.

Jay et al. 2022 argue that if the 'less-loaded' inversion haplotype also carries recessive deleterious mutations, its presence in a permanently heterozygote form will be beneficial because these deleterious recessive mutations are sheltered from selection in a heterozygote state. As a result, more and more overlapping inversions of this type will accumulate along the Y chromosome, leading to a chromosome-wide shutdown of recombination. However, Jay et al. 2022 assume that the possibility of a reversion of the initial inverted region, which restores recombination between the X and Y, is highly unlikely. This is contrast to a recent model presented in Lenormand and Roze 2022 (the authors of the present paper), which suggested the fixation of such an initial ‘lucky’ inversion is not sufficient to maintain recombination shutdown between the X and Y, because eventually the non-recombining region on the Y will accumulate enough deleterious mutations that a reversion of the inversion will appear and fix.

The focus of the current paper is to determine the rate at which reversions of the initial inverted haplotype must appear to support the Lenormand & Roze model vs. the Jay et al. model. In summary, the authors find that this rate of reversion must be very low relative to the rate of inversions in the population for the initial inversion not to be reverted (approximately 4 orders of magnitude lower than the rate of inversions). The authors argue that additional factors (such as the regulatory divergence on the X and Y proposed in their 2022 paper) are therefore required to maintain the presence of an inversion.

Overall, I found this to be a straightforward paper that clarifies why these two recent and superficially similar models reached such different conclusions about the stability of inversions on the sex chromosomes. While the data on the rate of reversions of inversions is quite limited, this paper will presumably direct more attention to empirical work attempting to measure this parameter. I have some suggestions, given below, for how to strengthen the paper.

--- I feel that the paper would be improved by a more detailed discussion of the molecular biology of inversions (and their potential subsequent reversions). Some of this is provided in lines 353-363, but more background on how & why inversions (and reversions) occur would be very helpful. This is particularly the case in terms of the data on reversions from Drosophila, where there is a proposed mechanism in lines 357-358; this mechanism would increase the probability that a reversion occurs with the same (or similar) breakpoints as the initial inversion, which is the point of contention between the two focal models.

--- A key component of the Jay et al. 2022 is the assumption that deleterious mutations are partially recessive, which allows these mutations to be ‘sheltered’ in their permanently heterozygous form. Here, the dominance of all deleterious mutations is also assumed to be partially recessive (fixed at 0.3). But it would be nice to see a bit more discussion of the implications of this assumption in the paper. For example, if mutations were (on average) even more recessive, would that slow the fitness decline on the inverted region of the Y and allow more time for another inversion to occur (thereby potentially making reversion less likely or slowing the male fitness decline leading to extinction)? Would this also be the case if there was an inverse relationship between h and s as predicted by theories of dominance (e.g. Kacser & Burns 1981, Manna et al. 2011)?

--- Lines 310-313: The authors suggest that the possibility of gene conversions and double-crossovers within the inversions will provide occasional recombination events. Would this then slow the accumulation of deleterious mutations on the Y chromosome? It seems that these occasional recombination events would perhaps stabilize the presence of the inversion on the Y, but simultaneously decrease the rate of degradation of the Y as well (e.g. as in lines 230-240). Further elaboration of the potential impacts of occasional recombination within the inversion on these processes would be helpful.

---Figure S1 – I would suggest moving this to the main text, if there is space; it summarizes the authors data about reversion rate well.

--- Lines 360-363: It would be useful to have citations for this point, as the fact recombination may be suppressed outside the inverted region can certainly make the accumulation of secondary inversions more likely and the possibility of reversion therefore more remote, in line with the Jay et al. 2022 model. E.g. from Drosophila melanogaster, this area of reduced recombination outside the inversion can range between 3-12 mb (Koury 2023)

Kacser, H., & Burns, J. A. (1981). The molecular basis of dominance. Genetics, 97(3-4), 639-666.

Koury, S. A. (2023). Predicting recombination suppression outside chromosomal inversions in Drosophila melanogaster using crossover interference theory. Heredity, 130(4), 196-208.

Manna, F., Martin, G., & Lenormand, T. (2011). Fitness landscapes: an alternative theory for the dominance of mutation. Genetics, 189(3), 923-937.

Nei, M., Kojima, K. I., & Schaffer, H. E. (1967). Frequency changes of new inversions in populations under mutation-selection equilibria. Genetics, 57(4), 741.