The paper has been modified and improved compared with the last version. The authors applied both theoretical and numerical methods to discuss the maintenance and the break-down of the POD (pseudo-overdominant) region. The factors such as spatial distribution of individual mutants, individual selection strength, recombination, dominance, and differentiated fitness between homozygotes, have been taken into consideration. The authors also learned the interactions between the POD and the background mutations occurring elsewhere. I think the manuscript is of great interest and could trigger further studies on the relevant topics.
Some mistakes and concerns are as follows:
1.     Line 99, “s_x = mins_1,s_2” should be changed to “s_x = min{s_1,s_2}”;
2.     Line 144, “see Eq. A2from Supp. File 1” should be modified as “see Eq. A2 from Supp. File 1”.
3.     Line 162, I think what the authors really meant was the “Ordinary Difference Equations” rather than the “Ordinary Differential Equations”, as the equations (10) do not contain the derivatives.
4.     Line 192, “The probability that a recombination event occurs between two trans-mutations is then l.” should be modified as “The expected number of recombination events occurring between two trans-mutations is then l.”
5.     Figure 3 a and b, I think this might be something wrong, Since the proportion of fixation and loss at some positions are negative (especially for the blue and green solid lines), I cannot see a reason for this.

Dear editor,

I have reviewed the manuscript entitled "The role of pseudo-overdominance in maintaining inbreeding depression" by Abu Awad and Waller for PCI Evol Biol. I was not involved in the first round of review, and have read the updated version of the MS as well as the comments and replies from the first round.

I really enjoyed reading the MS. Pseudo-overdominance is an important issue in evolutionary biology, and such theoretical models on how pseudo-overdominance zones erode over generations is, I think, a timely contribution. The MS reads well and the findings are interesting. It seems to me that the authors have well addressed the previous comments, such as a better justification for the origin of the POD zone and a better clarity of the results section and figures. I have thus only a few very minor comments.

MINOR COMMENTS

Title: To me, the main message of the MS is about how POD zones erode over time, which is not captured by the title.

line 70: The authors may want to indicate the estimated divergence time between Capsella and Arabidopsis.

lines 126-156: Having never worked on inbreeding depression, it is unclear to me what 'delta s' and 'delta od' capture. It may need to be explained.

line 222: The verb is missing in "The general map length R = 1 and 10 Morgans ...".

lines 258-261: This result may be briefly explained. It is not intuitive to me why, when no mutations have been cleaved off, the recombinant haplotype is selected against. It has the same number of deleterious mutations than the two other haplotype.

Figure 2: In the legend, is sH equal to 0.455, or to 0.45?

Figure S2: The fifth line of the legend seems to be encapsulated in the formula.

Review of: The role of pseudo-overdominance in maintaining inbreeding depression

In this manuscript, the authors explore how a “pseudo over dominant (POD) block of the genome” evolve, and how their evolution both impacts and is impacted by ongoing mutations at other regions of the genomes. Pseudo-over dominant genomic regions are portions of the genome for which heterozygotes have higher fitness than homozygotes, due to the action of numerous partially recessive mutations in repulsion phase disequilibrium, rather than the action of truly over variants.  The authors consider how the answers to these questions can change depending on mating system and population size.

The work has a few take-home messages – (1) POD regions are often unstable (2) POD regions are more stable when they are made of many tightly linked mutations, and/or other regions of the genome, and or heterozygosity is favored genome wide (3) POD regions are less stable when made of fewer more loosely linked mutations in populations are small (or effectively so because linked selection decreases Ne) and/or more selfing. (4)When POD regions are maintained high inbreeding depression can be maintained – even in largely selfing populations.

On the whole, the work seems correct, and like a nice contribution to the ongoing interest / appreciation of pseudo overdominance. However, I have numerous questions / concerns about decisions with regards to model conception and parameters choice etc. I also recommend the authors consult our recent preprint [Sianta et al. first posted may 2021, reposted Dec 2021 https://www.biorxiv.org/content/10.1101/2021.05.20.445016v2 ] which modelled the origin of POD, whose results complement this work (In hope to integrate the findings forom this paper into ours).

Concerns

*Beginning with a POD region*

The authors start their study with a stylized POD region already existing. Their underlying model is that two isolated populations have themselves accumulated a “local drift load” composed of n equally spaced deleterious mutations per haplotype in some specific genomic region, and with no other deleterious mutations elsewhere in the genome (although they can arise after this initial starting point). The biological basis for such a model is unclear. Harkness, Brandvain, and Goldberg (2019, JEB) began with a similar premise – assuming that isolated populations each fixed their local drift load but did not assume such variants to be highly localized to a single genomic region, and it is unclear why this would occur biologically. My sense is that it would not, and that rather than being strongly biologically motivated, this model is a hack to get at the interesting questions the authors hope to explore. If this is the case, the authors should make that clear, if not, they should provide more evidence that this is a reasonable biological scenario (e.g. perhaps such regions have exceptionally low recombination rates or high mutation rates, such that the load can be purged from other genomic regions, but not these?).  

*Composition of POD regions*

In addition to being surprised by the pre-existence of POD regions, I was confused by the properties of mutations in these regions. (as well as the ongoing mutations experienced by the population). The authors assume a dominance coefficient of 0.2 and a selective coefficient of 0.01. These choices were justified by pointing to the mean values from Agrawal and Whitlock’s efforts to estimating the distribution of these values from yeast knock outs. It is not obvious to me why this value is relevant for mutations that could make up a POD region – which should not be random draws from the 2D distribution of fitness and dominance effects – lets alone equal one value in the center of the distribution. A dominance value of h=0.2 is higher than what I imagine would make up PODs generated and maintained by a process other than the one considered in this model – so I worry about the applicability of these parameters.

*Explanation of results*

Often the explanation of the results seems pretty shallow. For example the authors state that “Counter-intuitively, POD heterozygosity persists much longer (relative to neutral heterozygosity) in smaller populations than in larger populations (compare the red and blue lines in Fig. S3 where N = 100 to Fig. 2 and Fig. S2 where N = 1000 and 5000).” I could logic why the authors found this to be counterintuitive, and I can logic an explanation for this result, but neither were provided.

*Limit discussion of the work to what was found*

The abstract and discussion suggest that the work is related to the formation and persistence of POD regions, however the methods and results focus only on the persistence. My suggestion is to not discuss the formation of POD regions as it is pretty far afield from what was modelled. It is also somewhat strange to have statements like “it seems unlikely that POD’s would arise in isolated populations”  which depend on modelling assumptions of h = 0.2, etc (see above)..

*Enhance clarity of figures, parameter names, etc etc*.

I found all the figures and parameters to be difficult to follow. Most increased my cognitive load substantially. I am afraid this could turn readers off and decrease the impact of this work. For example, denoting the population size as N, and the number of loci as n is a bit confusing, and made my life hard. Similarly, the sometimes the “control” (i.e. no POD region) was labelled as “no POD” and other times as s = 0. Direct labelling of figures would make results easier to process, and little changes like have the x-axis as “selfing rate (\sigma)” rather than “\sigma” would make it all easier to follow. Finally, I may be being dense, but I am having trouble seeing if there is any different information in the top (a & b) and bottom (c & d) parts of figure 5