The integrated WFHaldane (WFH) model of genetic drift resolving the many paradoxes of molecular evolution
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eLife assessment
This study presents a useful modification of a standard model of genetic drift by incorporating variance in offspring numbers, claiming to address several paradoxes in molecular evolution. It is unfortunate that the study fails to engage prior literature that has extensively examined the impact of variance in offspring number, implying that some of the paradoxes presented might be resolved within existing frameworks. In addition, while the modified model yields intriguing theoretical predictions, the simulations and empirical analyses are incomplete to support the authors' claims.
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Abstract
Genetic drift, the random changes in frequencies of neutral variants, is the fundamental force of molecular evolution. Underestimation of genetic drift is a major cause of misconclusions on evolution. However, the standard WrightFisher (WF) model of random sampling in population of size N only partially defines genetic drift with 1/ N or 1/ N e ( N e being a function of varying N ’s). In parallel, JBS Haldane proposed the branching process for genetic drift (Haldane 1927), whereby each gene copy is transmitted to K descendants with the mean and variance of E ( K ) and V ( K ). Genetic drift is simply V ( K ), i.e., the variance in transmission success, or V ( K )/ N when averaged over the population. Under the WF model, many paradoxes have emerged: i) Most curiously, genetic drift may often become stronger as N becomes larger at the ecological time scale, opposite to the model; ii) Sex chromosomes experience drift differently even with the same normalized N ; iii) Genetic drift operates on advantageous mutations depends on V ( K ) but not on N . iv) Irresolution and paradoxes emerge in multicopy gene systems, including diploidy, viruses and others, whereby evolution progresses both within and between individuals. We now show that the integration of the WF and Haldane (i.e., WFH) model can resolve these paradoxes. Most important, the WFH model can fully define genetic drift in molecular evolution.
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eLife assessment
This study presents a useful modification of a standard model of genetic drift by incorporating variance in offspring numbers, claiming to address several paradoxes in molecular evolution. It is unfortunate that the study fails to engage prior literature that has extensively examined the impact of variance in offspring number, implying that some of the paradoxes presented might be resolved within existing frameworks. In addition, while the modified model yields intriguing theoretical predictions, the simulations and empirical analyses are incomplete to support the authors' claims.

Reviewer #1 (Public Review):
Summary:
The authors present a theoretical treatment of what they term the "WrightFisherHaldane" model, a claimed modification of the standard model of genetic drift that accounts for variability in offspring number, and argue that it resolves a number of paradoxes in molecular evolution. Ultimately, I found this manuscript quite strange. The notion of effective population size as inversely related to the variance in offspring number is well known in the literature, and not exclusive to Haldane's branching process treatment. However, I found the authors' point about variance in offspring changing over the course of, e.g. exponential growth fairly interesting, and I'm not sure I'd seen that pointed out before. Nonetheless, I don't think the authors' modeling, simulations, or empirical data analysis are …
Reviewer #1 (Public Review):
Summary:
The authors present a theoretical treatment of what they term the "WrightFisherHaldane" model, a claimed modification of the standard model of genetic drift that accounts for variability in offspring number, and argue that it resolves a number of paradoxes in molecular evolution. Ultimately, I found this manuscript quite strange. The notion of effective population size as inversely related to the variance in offspring number is well known in the literature, and not exclusive to Haldane's branching process treatment. However, I found the authors' point about variance in offspring changing over the course of, e.g. exponential growth fairly interesting, and I'm not sure I'd seen that pointed out before. Nonetheless, I don't think the authors' modeling, simulations, or empirical data analysis are sufficient to justify their claims.
Weaknesses:
I have several outstanding issues. First of all, the authors really do not engage with the literature regarding different notions of an effective population. Most strikingly, the authors don't talk about Cannings models at all, which are a broad class of models with nonPoisson offspring distributions that nonetheless converge to the standard WrightFisher diffusion under many circumstances, and to "jumpy" diffusions/coalescents otherwise (see e.g. Mohle 1998, Sagitov (2003), Der et al (2011), etc.). Moreover, there is extensive literature on effective population sizes in populations whose sizes vary with time, such as Sano et al (2004) and Sjodin et al (2005). Of course in many cases here the discussion is under neutrality, but it seems like the authors really need to engage with this literature more.
The most interesting part of the manuscript, I think, is the discussion of the Density Dependent Haldane model (DDH). However, I feel like I did not fully understand some of the derivation presented in this section, which might be my own fault. For instance, I can't tell if Equation 5 is a result or an assumption  when I attempted a naive derivation of Equation 5, I obtained E(K_t) = 1 + r/c*(cn)*dt. It's unclear where the parameter z comes from, for example. Similarly, is equation 6 a derivation or an assumption? Finally, I'm not 100% sure how to interpret equation 7. I that a variance effective size at time t? Is it possible to obtain something like a coalescent Ne or an expected number of segregating sites or something from this?
Similarly, I don't understand their simulations. I expected that the authors would do individualbased simulations under a stochastic model of logistic growth, and show that you naturally get variance in offspring number that changes over time. But it seems that they simply used their equations 5 and 6 to fix those values. Moreover, I don't understand how they enforce population regulation in their simulationsis N_t random and determined by the (independent) draws from K_t for each individual? In that case, there's no "interaction" between individuals (except abstractly, since logistic growth arises from a model that assumes interactions between individuals). This seems problematic for their model, which is essentially motivated by the fact that early during logistic growth, there are basically no interactions, and later there are, which increases variance in reproduction. But their simulations assume no interactions throughout!
The authors also attempt to show that changing variance in reproductive success occurs naturally during exponential growth using a yeast experiment. However, the authors are not counting the offspring of individual yeast during growth (which I'm sure is quite hard). Instead, they use an equation that estimates the variance in offspring number based on the observed population size, as shown in the section "Estimation of V(K) and E(K) in yeast cells". This is fairly clever, however, I am not sure it is right, because the authors neglect covariance in offspring between individuals. My attempt at this derivation assumes that I_t  I_{t1} = \sum_{I=1}^{I_{t1}} K_{i,t1} where K_{i,t1} is the number of offspring of individual i at time t1. Then, for example, E(V(I_t  I_{t1})) = E(V(\sum_{i=1}^{I_{t1}} K_{i,t1})) = E(I_{t1})V(K_{t1}) + E(I_{k1}(I_{k1}1))*Cov(K_{i,t1},K_{j,t1}). The authors have the first term, but not the second, and I'm not sure the second can be neglected (in fact, I believe it's the second term that's actually important, as early on during growth there is very little covariance because resources aren't constrained, but at carrying capacity, an individual having offspring means that another individuals has to have fewer offspring  this is the whole notion of exchangeability, also neglected in this manuscript). As such, I don't believe that their analysis of the empirical data supports their claim.
Thus, while I think there are some interesting ideas in this manuscript, I believe it has some fundamental issues: first, it fails to engage thoroughly with the literature on a very important topic that has been studied extensively. Second, I do not believe their simulations are appropriate to show what they want to show. And finally, I don't think their empirical analysis shows what they want to show.
References:
Möhle M. Robustness results for the coalescent. Journal of Applied Probability. 1998;35(2):438447. doi:10.1239/jap/1032192859
Sagitov S. Convergence to the coalescent with simultaneous multiple mergers. Journal of Applied Probability. 2003;40(4):839854. doi:10.1239/jap/1067436085
Der, Ricky, Charles L. Epstein, and Joshua B. Plotkin. "Generalized population models and the nature of genetic drift." Theoretical population biology 80.2 (2011): 8099
Sano, Akinori, Akinobu Shimizu, and Masaru Iizuka. "Coalescent process with fluctuating population size and its effective size." Theoretical population biology 65.1 (2004): 3948
Sjodin, P., et al. "On the meaning and existence of an effective population size." Genetics 169.2 (2005): 10611070

Reviewer #2 (Public Review):
Summary:
This theoretical paper examines genetic drift in scenarios deviating from the standard WrightFisher model. The authors discuss Haldane's branching process model, highlighting that the variance in reproductive success equates to genetic drift. By integrating the WrightFisher model with the Haldane model, the authors derive theoretical results that resolve paradoxes related to effective population size.
Strengths:
The most significant and compelling result from this paper is perhaps that the probability of fixing a new beneficial mutation is 2s/V(K). This is an intriguing and potentially generalizable discovery that could be applied to many different study systems.
The authors also made a lot of effort to connect theory with various realworld examples, such as genetic diversity in sex chromosomes …
Reviewer #2 (Public Review):
Summary:
This theoretical paper examines genetic drift in scenarios deviating from the standard WrightFisher model. The authors discuss Haldane's branching process model, highlighting that the variance in reproductive success equates to genetic drift. By integrating the WrightFisher model with the Haldane model, the authors derive theoretical results that resolve paradoxes related to effective population size.
Strengths:
The most significant and compelling result from this paper is perhaps that the probability of fixing a new beneficial mutation is 2s/V(K). This is an intriguing and potentially generalizable discovery that could be applied to many different study systems.
The authors also made a lot of effort to connect theory with various realworld examples, such as genetic diversity in sex chromosomes and reproductive variance across different species.
Weaknesses:
One way to define effective population size is by the inverse of the coalescent rate. This is where the geometric mean of Ne comes from. If Ne is defined this way, many of the paradoxes mentioned seem to resolve naturally. If we take this approach, one could easily show that a large N population can still have a low coalescent rate depending on the reproduction model. However, the authors did not discuss Ne in light of the coalescent theory. This is surprising given that Eldon and Wakeley's 2006 paper is cited in the introduction, and the multiple mergers coalescent was introduced to explain the discrepancy between census size and effective population size, superspreaders, and reproduction variance  that said, there is no explicit discussion or introduction of the multiple mergers coalescent.
The WrightFisher model is often treated as a special case of the Cannings 1974 model, which incorporates the variance in reproductive success. This model should be discussed. It is unclear to me whether the results here have to be explained by the newly introduced WFH model, or could have been explained by the existing Cannings model.
The abstract makes it difficult to discern the main focus of the paper. It spends most of the space introducing "paradoxes".
The standard WrightFisher model makes several assumptions, including hermaphroditism, nonoverlapping generations, random mating, and no selection. It will be more helpful to clarify which assumptions are being violated in each tested scenario, as V(K) is often not the only assumption being violated. For example, the logistic growth model assumes no cell death at the exponential growth phase, so it also violates the assumption about nonoverlapping generations.
The theory and data regarding sex chromosomes do not align. The fact that \hat{alpha'} can be negative does not make sense. The authors claim that a negative \hat{alpha'} is equivalent to infinity, but why is that? It is also unclear how theta is defined. It seems to me that one should take the first principle approach e.g., define theta as pairwise genetic diversity, and start with deriving the expected pairwise coalescence time under the MMC model, rather than starting with assuming theta = 4Neu. Overall, the theory in this section is not well supported by the data, and the explanation is insufficient.

Reviewer #3 (Public Review):
Summary:
Ruan and colleagues consider a branching process model (in their terminology the "Haldane model") and the most basic WrightFisher model. They convincingly show that offspring distributions are usually nonPoissonian (as opposed to what's assumed in the WrightFisher model), and can depend on shortterm ecological dynamics (e.g., variance in offspring number may be smaller during exponential growth). The authors discuss branching processes and the WrightFisher model in the context of 3 "paradoxes": (1) how Ne depends on N might depend on population dynamics; (2) how Ne is different on the X chromosome, the Y chromosome, and the autosomes, and these differences do match the expectations base on simple counts of the number of chromosomes in the populations; (3) how genetic drift interacts with …
Reviewer #3 (Public Review):
Summary:
Ruan and colleagues consider a branching process model (in their terminology the "Haldane model") and the most basic WrightFisher model. They convincingly show that offspring distributions are usually nonPoissonian (as opposed to what's assumed in the WrightFisher model), and can depend on shortterm ecological dynamics (e.g., variance in offspring number may be smaller during exponential growth). The authors discuss branching processes and the WrightFisher model in the context of 3 "paradoxes": (1) how Ne depends on N might depend on population dynamics; (2) how Ne is different on the X chromosome, the Y chromosome, and the autosomes, and these differences do match the expectations base on simple counts of the number of chromosomes in the populations; (3) how genetic drift interacts with selection. The authors provide some theoretical explanations for the role of variance in the offspring distribution in each of these three paradoxes. They also perform some experiments to directly measure the variance in offspring number, as well as perform some analyses of published data.
Strengths:
(1) The theoretical results are welldescribed and easy to follow.
(2) The analyses of different variances in offspring number (both experimentally and analyzing public data) are convincing that nonPoissonian offspring distributions are the norm.
(3) The point that this variance can change as the population size (or population dynamics) change is also very interesting and important to keep in mind.
(4) I enjoyed the DensityDependent Haldane model. It was a nice example of the decoupling of census size and effective size.
Weaknesses:
(1) I am not convinced that these types of effects cannot just be absorbed into some timevarying Ne and still be wellmodeled by the WrightFisher process.
(2) Along these lines, there is wellestablished literature showing that a broad class of processes (a large subset of Cannings' Exchangeable Models) converge to the WrightFisher diffusion, even those with nonPoissonian offspring distributions (e.g., Mohle and Sagitov 2001). E.g., equation (4) in Mohle and Sagitov 2001 shows that in such cases the "coalescent Ne" should be (N1) / Var(K), essentially matching equation (3) in the present paper.
(3) Beyond this, I would imagine that branching processes with heavytailed offspring distributions could result in deviations that are not well captured by the authors' WFH model. In this case, the processes are known to converge (backwardintime) to Lambda or Xi coalescents (e.g., Eldon and Wakely 2006 or again in Mohle and Sagitov 2001 and subsequent papers), which have welldefined forwardintime processes.
(4) These results that Ne in the WrightFisher process might not be related to N in any straightforward (or even onetoone) way are wellknown (e.g., Neher and Hallatschek 2012; Spence, Kamm, and Song 2016; Matuszewski, Hildebrandt, Achaz, and Jensen 2018; Rice, Novembre, and Desai 2018; the work of Lounès Chikhi on how Ne can be affected by population structure; etc...)
(5) I was also missing some discussion of the relationship between the branching process and the WrightFisher model (or more generally Cannings' Exchangeable Models) when conditioning on the total population size. In particular, if the offspring distribution is Poisson, then conditioned on the total population size, the branching process is identical to the WrightFisher model.
(6) In the discussion, it is claimed that the last glacial maximum could have caused the bottleneck observed in human populations currently residing outside of Africa. Compelling evidence has been amassed that this bottleneck is due to serial founder events associated with the outofAfrica migration (see e.g., Henn, CavalliSforza, and Feldman 2012 for an older review  subsequent work has only strengthened this view). For me, a more compelling example of changes in carrying capacity would be the advent of agriculture ~11kya and other more recent technological advances.

Author response:
We thank you for the opportunity to provide a concise response. The criticisms are accurately summarized in the eLife assessment:
the study fails to engage prior literature that has extensively examined the impact of variance in offspring number, implying that some of the paradoxes presented might be resolved within existing frameworks.
The essence of our study is to propose the adoption of the Haldane model of genetic drift, based on the branching process, in lieu of the WrightFisher (WF) model, based on sampling, usually binomial. In addition to some extensions of the Haldane model, we present 4 paradoxes that cannot be resolved by the WF model. The reviews suggest that some of the paradoxes could be resolved by the WF model, if we engage prior literature sufficiently.
We certainly could not review all the …
Author response:
We thank you for the opportunity to provide a concise response. The criticisms are accurately summarized in the eLife assessment:
the study fails to engage prior literature that has extensively examined the impact of variance in offspring number, implying that some of the paradoxes presented might be resolved within existing frameworks.
The essence of our study is to propose the adoption of the Haldane model of genetic drift, based on the branching process, in lieu of the WrightFisher (WF) model, based on sampling, usually binomial. In addition to some extensions of the Haldane model, we present 4 paradoxes that cannot be resolved by the WF model. The reviews suggest that some of the paradoxes could be resolved by the WF model, if we engage prior literature sufficiently.
We certainly could not review all the literature on genetic drift as there must be thousands of them. Nevertheless, the literature we do not cover is based on the WF model, which has the general properties that all modifications of the WF model share. (We should note that all such modifications share the sampling aspect of the WF model. To model such sampling, N is imposed from outside of the model, rather than selfgenerating within the model. Most important, these modifications are mathematically valid but biologically untenable, as will be elaborated below. Thus, in concept, the WF and Haldane models are fundamentally different.)
In short, our proposal is general with the key point that the WF model cannot resolve these (and many other) paradoxes. The reviewers disagree (apparently only partially) and we shall be specific in our response below.
We shall first present the 4th paradox, which is about multicopy gene systems (such as rRNA genes and viruses, see the companion paper). Viruses evolve both within and between hosts. In both stages, there are severe bottlenecks. How does one address the genetic drift in viral evolution? How can we model the effective population sizes both within and between hosts? The inability of the WF model in dealing with such multicopy gene systems may explain the difficulties in accounting for the SARSCoV2 evolution. Given the small number of virions transmitted between hosts, drift is strong which we have shown by using the Haldane model (Ruan, Luo, et al. 2021; Ruan, Wen, et al. 2021; Hou, et al. 2023).
As the reviewers suggest, it is possible to modify the WF model to account for some of these paradoxes. However, the modifications are often mathematically convenient but biologically dubious. Much of the debate is about the progeny number, K. (We shall use haploid model for this purpose but diploidy does not pose a problem as stated in the main text.) The modifications relax the constraint of V(k) = E(k) inherent in the WF sampling. One would then ask how V(k) can be different from E(k) in the WF sampling even though it is mathematically feasible (but biologically dubious)? Kimura and Crow (1963) may be the first to offer a biological explanation. If one reads it carefully, Kimura's modification is to make the WF model like the Haldane model. Then, why don't we use the Haldane model in the first place by having two parameters, E(k) and V(k), instead of the oneparameter WF model?
The Haldane model is conceptually simpler. It allows the variation in population size, N, to be generated from within the model, rather than artificially imposed from outside of the model. This brings us to the first paradox, the densitydependent Haldane model. When N is increasing exponentially as in bacterial or yeast cultures, there is almost no drift when N is very low and drift becomes intense as N grows to near the carrying capacity. We do not see how the WF model can resolve this paradox, which can otherwise be resolved by the Haldane model.
The second and third paradoxes are about how much mathematical models of population genetic can be detached from biological mechanisms. The second paradox about sex chromosomes is rooted in the realization of V(k) ≠ E(k). Since E(k) is the same between sexes but V(k) is different, how does the WF sampling give rise to V(k) ≠ E(k)? We are asking a biological question that troubled Kimura and Crow (1963) alluded above. The third paradox is acknowledged by two reviewers. Genetic drift manifested in the fixation probability of an advantageous mutation is 2s/V(k). It is thus strange that the fundamental parameter of drift in the WF model, N (or Ne), is missing. In the Haldane model, drift is determined by V(k) with N being a scaling factor; hence 2s/V(k) makes perfect biological sense,
We now answer the obvious question: If the model is fundamentally about the Haldane model, why do we call it the WFHaldane model? The reason is that most results obtained by the WF model are pretty good approximations and the branching process may not need to constantly rederive the results. At least, one can use the WF results to see how well they fit into the Haldane model. In our earlier study (Chen, et al. (2017); Fig. 3), we show that the approximations can be very good in many (or most) settings.
We would like to use the modern analogy of gasengine cars vs. electricmotor ones. The Haldane model and the WF model are as fundamentally different concepts as the driving mechanisms of gaspowered vs electric cars. The old model is now facing many problems and the fixes are often not possible. Some fixes are so complicated that one starts thinking about simpler solutions. The reservations are that we have invested so much in the old models which might be wasted by the switch. However, we are suggesting the integration of the WF and Haldane models. In this sense, the WF model has had many contributions which the new model gratefully inherits. This is true with the legacy of gasengine cars inherited by EVs.
The editors also issue the instruction: while the modified model yields intriguing theoretical predictions, the simulations and empirical analyses are incomplete to support the authors' claims.
We are thankful to the editors and reviewers for the thoughtful comments and constructive criticisms. We also appreciate the publishing philosophy of eLife that allows exchanges, debates and improvements, which are the true spirits of science publishing.
References for the provisional author responses
Chen Y, Tong D, Wu CI. 2017. A New Formulation of Random Genetic Drift and Its Application to the Evolution of Cell Populations. Mol. Biol. Evol. 34:20572064.
Hou M, Shi J, Gong Z, Wen H, Lan Y, Deng X, Fan Q, Li J, Jiang M, Tang X, et al. 2023. Intra vs. Interhost Evolution of SARSCoV2 Driven by Uncorrelated SelectionThe Evolution Thwarted. Mol. Biol. Evol. 40.
Kimura M, Crow JF. 1963. The measurement of effective population number. Evolution:279288.
Ruan Y, Luo Z, Tang X, Li G, Wen H, He X, Lu X, Lu J, Wu CI. 2021. On the founder effect in COVID19 outbreaks: how many infected travelers may have started them all? Natl. Sci. Rev. 8:nwaa246.
Ruan Y, Wen H, He X, Wu CI. 2021. A theoretical exploration of the origin and early evolution of a pandemic. Sci Bull (Beijing) 66:10221029.
Review comments
eLife assessment
This study presents a useful modification of a standard model of genetic drift by incorporating variance in offspring numbers, claiming to address several paradoxes in molecular evolution.
It is unfortunate that the study fails to engage prior literature that has extensively examined the impact of variance in offspring number, implying that some of the paradoxes presented might be resolved within existing frameworks.
We do not believe that the paradoxes can be resolved.
In addition, while the modified model yields intriguing theoretical predictions, the simulations and empirical analyses are incomplete to support the authors' claims.
Public Reviews:
Reviewer #1 (Public Review):
Summary:
The authors present a theoretical treatment of what they term the "WrightFisherHaldane" model, a claimed modification of the standard model of genetic drift that accounts for variability in offspring number, and argue that it resolves a number of paradoxes in molecular evolution. Ultimately, I found this manuscript quite strange.
The notion of effective population size as inversely related to the variance in offspring number is well known in the literature, and not exclusive to Haldane's branching process treatment. However, I found the authors' point about variance in offspring changing over the course of, e.g. exponential growth fairly interesting, and I'm not sure I'd seen that pointed out before.
Nonetheless, I don't think the authors' modeling, simulations, or empirical data analysis are sufficient to justify their claims.
Weaknesses:
I have several outstanding issues. First of all, the authors really do not engage with the literature regarding different notions of an effective population. Most strikingly, the authors don't talk about Cannings models at all, which are a broad class of models with nonPoisson offspring distributions that nonetheless converge to the standard WrightFisher diffusion under many circumstances, and to "jumpy" diffusions/coalescents otherwise (see e.g. Mohle 1998, Sagitov (2003), Der et al (2011), etc.). Moreover, there is extensive literature on effective population sizes in populations whose sizes vary with time, such as Sano et al (2004) and Sjodin et al (2005).
Of course in many cases here the discussion is under neutrality, but it seems like the authors really need to engage with this literature more.
The most interesting part of the manuscript, I think, is the discussion of the Density Dependent Haldane model (DDH). However, I feel like I did not fully understand some of the derivation presented in this section, which might be my own fault. For instance, I can't tell if Equation 5 is a result or an assumption  when I attempted a naive derivation of Equation 5, I obtained E(K_t) = 1 + r/c*(cn)*dt. It's unclear where the parameter z comes from, for example. Similarly, is equation 6 a derivation or an assumption? Finally, I'm not 100% sure how to interpret equation 7. I that a variance effective size at time t? Is it possible to obtain something like a coalescent Ne or an expected number of segregating sites or something from this?
Similarly, I don't understand their simulations. I expected that the authors would do individualbased simulations under a stochastic model of logistic growth, and show that you naturally get variance in offspring number that changes over time. But it seems that they simply used their equations 5 and 6 to fix those values. Moreover, I don't understand how they enforce population regulation in their simulationsis N_t random and determined by the (independent) draws from K_t for each individual? In that case, there's no "interaction" between individuals (except abstractly, since logistic growth arises from a model that assumes interactions between individuals). This seems problematic for their model, which is essentially motivated by the fact that early during logistic growth, there are basically no interactions, and later there are, which increases variance in reproduction. But their simulations assume no interactions throughout!
The authors also attempt to show that changing variance in reproductive success occurs naturally during exponential growth using a yeast experiment. However, the authors are not counting the offspring of individual yeast during growth (which I'm sure is quite hard). Instead, they use an equation that estimates the variance in offspring number based on the observed population size, as shown in the section "Estimation of V(K) and E(K) in yeast cells". This is fairly clever, however, I am not sure it is right, because the authors neglect covariance in offspring between individuals. My attempt at this derivation assumes that I_t  I_{t1} = \sum_{I=1}^{I_{t1}} K_{i,t1} where K_{i,t1} is the number of offspring of individual i at time t1. Then, for example, E(V(I_t  I_{t1})) = E(V(\sum_{i=1}^{I_{t1}} K_{i,t1})) = E(I_{t1})V(K_{t1}) + E(I_{k1}(I_{k1}1))*Cov(K_{i,t1},K_{j,t1}). The authors have the first term, but not the second, and I'm not sure the second can be neglected (in fact, I believe it's the second term that's actually important, as early on during growth there is very little covariance because resources aren't constrained, but at carrying capacity, an individual having offspring means that another individuals has to have fewer offspring  this is the whole notion of exchangeability, also neglected in this manuscript). As such, I don't believe that their analysis of the empirical data supports their claim.
Thus, while I think there are some interesting ideas in this manuscript, I believe it has some fundamental issues:
first, it fails to engage thoroughly with the literature on a very important topic that has been studied extensively. Second, I do not believe their simulations are appropriate to show what they want to show. And finally, I don't think their empirical analysis shows what they want to show.
References:
Möhle M. Robustness results for the coalescent. Journal of Applied Probability. 1998;35(2):438447. doi:10.1239/jap/1032192859
Sagitov S. Convergence to the coalescent with simultaneous multiple mergers. Journal of Applied Probability. 2003;40(4):839854. doi:10.1239/jap/1067436085
Der, Ricky, Charles L. Epstein, and Joshua B. Plotkin. "Generalized population models and the nature of genetic drift." Theoretical population biology 80.2 (2011): 8099
Sano, Akinori, Akinobu Shimizu, and Masaru Iizuka. "Coalescent process with fluctuating population size and its effective size." Theoretical population biology 65.1 (2004): 3948
Sjodin, P., et al. "On the meaning and existence of an effective population size." Genetics 169.2 (2005): 10611070
Reviewer #2 (Public Review):
Summary:
This theoretical paper examines genetic drift in scenarios deviating from the standard WrightFisher model. The authors discuss Haldane's branching process model, highlighting that the variance in reproductive success equates to genetic drift. By integrating the WrightFisher model with the Haldane model, the authors derive theoretical results that resolve paradoxes related to effective population size.
Strengths:
The most significant and compelling result from this paper is perhaps that the probability of fixing a new beneficial mutation is 2s/V(K). This is an intriguing and potentially generalizable discovery that could be applied to many different study systems.
The authors also made a lot of effort to connect theory with various realworld examples, such as genetic diversity in sex chromosomes and reproductive variance across different species.
Weaknesses:
One way to define effective population size is by the inverse of the coalescent rate. This is where the geometric mean of Ne comes from. If Ne is defined this way, many of the paradoxes mentioned seem to resolve naturally. If we take this approach, one could easily show that a large N population can still have a low coalescent rate depending on the reproduction model. However, the authors did not discuss Ne in light of the coalescent theory. This is surprising given that Eldon and Wakeley's 2006 paper is cited in the introduction, and the multiple mergers coalescent was introduced to explain the discrepancy between census size and effective population size, superspreaders, and reproduction variance  that said, there is no explicit discussion or introduction of the multiple mergers coalescent.
The WrightFisher model is often treated as a special case of the Cannings 1974 model, which incorporates the variance in reproductive success. This model should be discussed. It is unclear to me whether the results here have to be explained by the newly introduced WFH model, or could have been explained by the existing Cannings model.
The abstract makes it difficult to discern the main focus of the paper. It spends most of the space introducing "paradoxes".
The standard WrightFisher model makes several assumptions, including hermaphroditism, nonoverlapping generations, random mating, and no selection. It will be more helpful to clarify which assumptions are being violated in each tested scenario, as V(K) is often not the only assumption being violated. For example, the logistic growth model assumes no cell death at the exponential growth phase, so it also violates the assumption about nonoverlapping generations.
The theory and data regarding sex chromosomes do not align. The fact that \hat{alpha'} can be negative does not make sense. The authors claim that a negative \hat{alpha'} is equivalent to infinity, but why is that? It is also unclear how theta is defined. It seems to me that one should take the first principle approach e.g., define theta as pairwise genetic diversity, and start with deriving the expected pairwise coalescence time under the MMC model, rather than starting with assuming theta = 4Neu. Overall, the theory in this section is not well supported by the data, and the explanation is insufficient.
{Alpha and alpha' can both be negative. X^2 = 0.47 would yield x = 0.7}
Reviewer #3 (Public Review):
Summary:
Ruan and colleagues consider a branching process model (in their terminology the "Haldane model") and the most basic WrightFisher model. They convincingly show that offspring distributions are usually nonPoissonian (as opposed to what's assumed in the WrightFisher model), and can depend on shortterm ecological dynamics (e.g., variance in offspring number may be smaller during exponential growth). The authors discuss branching processes and the WrightFisher model in the context of 3 "paradoxes": (1) how Ne depends on N might depend on population dynamics; (2) how Ne is different on the X chromosome, the Y chromosome, and the autosomes, and these differences do match the expectations base on simple counts of the number of chromosomes in the populations; (3) how genetic drift interacts with selection. The authors provide some theoretical explanations for the role of variance in the offspring distribution in each of these three paradoxes. They also perform some experiments to directly measure the variance in offspring number, as well as perform some analyses of published data.
Strengths:
(1) The theoretical results are welldescribed and easy to follow.
(2) The analyses of different variances in offspring number (both experimentally and analyzing public data) are convincing that nonPoissonian offspring distributions are the norm.
(3) The point that this variance can change as the population size (or population dynamics) change is also very interesting and important to keep in mind.
(4) I enjoyed the DensityDependent Haldane model. It was a nice example of the decoupling of census size and effective size.
Weaknesses:
(1) I am not convinced that these types of effects cannot just be absorbed into some timevarying Ne and still be wellmodeled by the WrightFisher process.
(2) Along these lines, there is wellestablished literature showing that a broad class of processes (a large subset of Cannings' Exchangeable Models) converge to the WrightFisher diffusion, even those with nonPoissonian offspring distributions (e.g., Mohle and Sagitov 2001). E.g., equation (4) in Mohle and Sagitov 2001 shows that in such cases the "coalescent Ne" should be (N1) / Var(K), essentially matching equation (3) in the present paper.
(3) Beyond this, I would imagine that branching processes with heavytailed offspring distributions could result in deviations that are not well captured by the authors' WFH model. In this case, the processes are known to converge (backwardintime) to Lambda or Xi coalescents (e.g., Eldon and Wakely 2006 or again in Mohle and Sagitov 2001 and subsequent papers), which have welldefined forwardintime processes.
(4) These results that Ne in the WrightFisher process might not be related to N in any straightforward (or even onetoone) way are wellknown (e.g., Neher and Hallatschek 2012; Spence, Kamm, and Song 2016; Matuszewski, Hildebrandt, Achaz, and Jensen 2018; Rice, Novembre, and Desai 2018; the work of Lounès Chikhi on how Ne can be affected by population structure; etc...)
(5) I was also missing some discussion of the relationship between the branching process and the WrightFisher model (or more generally Cannings' Exchangeable Models) when conditioning on the total population size. In particular, if the offspring distribution is Poisson, then conditioned on the total population size, the branching process is identical to the WrightFisher model.
(6) In the discussion, it is claimed that the last glacial maximum could have caused the bottleneck observed in human populations currently residing outside of Africa. Compelling evidence has been amassed that this bottleneck is due to serial founder events associated with the outofAfrica migration (see e.g., Henn, CavalliSforza, and Feldman 2012 for an older review  subsequent work has only strengthened this view). For me, a more compelling example of changes in carrying capacity would be the advent of agriculture ~11kya and other more recent technological advances.
Recommendations for the authors:
Reviewing Editor Comments:
The reviewers recognize the value of this model and some of the findings, particularly results from the densitydependent Haldane model. However, they expressed considerable concerns with the model and overall framing of this manuscript.
First, all reviewers pointed out that the manuscript does not sufficiently engage with the extensive literature on various models of effective population size and genetic drift, notably lacking discussion on Cannings models and related works.
Second, there is a disproportionate discussion on the paradoxes, yet some of the paradoxes might already be resolved within current theoretical frameworks. All three reviewers found the modeling and simulation of the yeast growth experiment hard to follow or lacking justification for certain choices. The analysis approach of sex chromosomes is also questioned.
The reviewers recommend a more thorough review of relevant prior literature to better contextualize their findings. The authors need to clarify and/or modify their derivations and simulations of the yeast growth experiment to address the identified caveats and ensure robustness. Additionally, the empirical analysis of the sex chromosome should be revisited, considering alternative scenarios rather than relying solely on the MSE, which only provides a superficial solution. Furthermore, the manuscript's overall framing should be adjusted to emphasize the conclusions drawn from the WFH model, rather than focusing on the "unresolved paradoxes", as some of these may be more readily explained by existing frameworks. Please see the reviewers' overall assessment and specific comments.
Reviewer #2 (Recommendations For The Authors):
In the introduction  "Genetic drift is simply V(K)"  this is a very strong statement. You can say it is inversely proportional to V(K), but drift is often defined based on changes in allele frequency.
Page 3 line 86. "sexes is a sufficient explanation."> "sex could be a sufficient explanation"
The strongest line of new results is about 2s/V(K). Perhaps, the paper could put more emphasis on this part and demonstrate the generality of this result with a different example.
The math notations in the supplement are not intuitive. e.g., using i_k and j_k as probabilities. I also recommend using E[X] and V[X]for expectation and variance rather than \italic{E(X)} to improve the readability of many equations.
Eq A6, A7, While I manage to follow, P_{10}(t) and P_{10} are not defined anywhere in the text.
Supplement page 7, the term "probability of fixation" is confusing in a branching model.
E.q. A 28. It is unclear eq. A.1 could be used here directly. Some justification would be nice.
Supplement page 17. "the biological meaning of negative..". There is no clear justification for this claim. As a reader, I don't have any intuition as to why that is the case.


