Reviewer #1 (Public Review):
Summary:
The authors present a theoretical treatment of what they term the "Wright-Fisher-Haldane" 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 non-Poisson offspring distributions that nonetheless converge to the standard Wright-Fisher 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*(c-n)*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 individual-based 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 simulations---is 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_{t-1} = \sum_{I=1}^{I_{t-1}} K_{i,t-1} where K_{i,t-1} is the number of offspring of individual i at time t-1. Then, for example, E(V(I_t | I_{t-1})) = E(V(\sum_{i=1}^{I_{t-1}} K_{i,t-1})) = E(I_{t-1})V(K_{t-1}) + E(I_{k-1}(I_{k-1}-1))*Cov(K_{i,t-1},K_{j,t-1}). 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):438-447. doi:10.1239/jap/1032192859
Sagitov S. Convergence to the coalescent with simultaneous multiple mergers. Journal of Applied Probability. 2003;40(4):839-854. 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): 80-99
Sano, Akinori, Akinobu Shimizu, and Masaru Iizuka. "Coalescent process with fluctuating population size and its effective size." Theoretical population biology 65.1 (2004): 39-48
Sjodin, P., et al. "On the meaning and existence of an effective population size." Genetics 169.2 (2005): 1061-1070