### Reviewer's report 1

Sergei Maslov, Brookhaven National Laboratory, USA

This reviewer provided no comments for publication.

### Reviewer's report 2

Alexander Gordon, University of North Carolina at Charlotte, USA

**Review of the paper "On the relationship between the generic load and the variance of relative fitness" by Emmanuil E. Shnol, Elena E. Ermakova, and Alexey S. Kondrashov**

The authors show that among all probability distributions of fitness w supported on an interval [0,*w*_{max}] with a given variance V of the relative fitness *w*/, the smallest possible value of the gene load *L* = 1 - *w*/_{max} equals *V*/(1 + *V*) and is attained if and only if the fitness takes only the values 0 and w_{max} with probabilities *V*/(1 + *V*) and 1/(1 + *V*), respectively. The authors do this for discrete probability distributions with finitely many values *w*_{j} but mention that this restriction is not important.

The statement and the proof are correct (although quite a few changes are necessary, see "Corrections and suggestions" below). However, this result is purely mathematical in nature. Its significance for population genetics should be assessed by an expert in that field.

...

**Review of the revised paper "On the relationship between the load and the variance of relative fitness" by Emmanuil E. Shnol, Elena E. Ermakova, and Alexey S. Kondrashov**.

**A generalization**.

Here we will show how, using a different idea, the theorem can be established in its full generality - without assuming that the fitness *w* has a discrete distribution. The proof is straightforward and relies on three simple lemmas that follow the proof. The case *V* = 0 of the theorem is trivial, so we will assume that *V* > 0.

Let *w* have a probability distribution supported on [0,*w*_{max}] and not concentrated at its endpoints (that is, Pr{0 < *w* < *w*_{max}} > 0). We can replace this distribution by a new one that is supported on the two-point set {0,*w*_{max}} and has the same mean . (By Lemma 1, such a distribution exists and is unique.) This new distribution has a strictly greater variance (Lemma 2) and consequently a strictly greater value of *V* than the original one: *V* > *V*_{0}. Let us further change the new distribution by continuously increasing the mass of the atom at *w*_{max} and decreasing the mass of the atom at 0 so that the total mass remains equal to 1. Then will be strictly increasing, while both and Var *w*, and hence *V* = Var *w*/()^{2}, will be changing continuously. At the end, *V* = 0. Therefore, at some intermediate point we will have *V* = *V*_{0} and a strictly increased .

This shows that, in order to maximize (or equivalently, minimize *L*) for a given *V* > 0, it suffices to consider distributions supported on the two-point set {0,*w*_{max}} and having the prescribed value of *V*. But there is exactly one such distribution (Lemma 3).This completes the proof of the theorem.

*Lemma 1*. Given *w*_{0} Є (0,*w*_{max}), there exists exactly one distribution consisting of two atoms, at 0 and *w*_{max}, and whose mean equals *w*_{0}.

*Proof*. Let Pr{*w* = *w*_{max}} = *p*; then *w*_{0} = *pw*_{max}, and we should have *p* = *w*_{0}/*w*_{max}.

*Lemma 2*. Suppose a random variable *w* is supported on the interval [0,*w*_{max}] and has a given mean = *w*_{0} Є (0,*w*_{max}). The variance Var *w* attains its maximum over all such random variables if and only if *w* takes only two values 0 and *w*_{max} (see Lemma 1).

*Proof*. Let *u* := *w* - *w*_{max}/2, so that -*w*_{max}/2 ≤ u ≤ *w*_{max}/2 and we want to maximize Var *w* ≡ Var *u* = **E** *u*^{2} - (**E** *u*)^{2} = **E** *u*^{2} - (*w*_{0} - *w*_{max}/2) ^{2}. Since |*u*| ≤ *w*_{max}/2, the maximum is attained if and only if |*u*| ≡ *w*_{max}/2, or equivalently *w* Є {0,*w*_{max}}, with probability 1.

*Lemma 3*. There exists exactly one distribution consisting of two atoms, at 0 and *w*_{max}, and having a given value of *V* (*V* > 0).

*Proof*. Let Pr{*w* = *w*_{max}} = *p*, so that Pr{*w* = 0} = 1 - *p*. Then *V* = Var *w*/()^{2} = (*p*(1 - *p*) *(w*_{max})^{2})/(*pw*_{max}) ^{2} = (1 - *p*)/*p* = 1/*p* - 1, and we should have *p* = 1/(1 + *V*).

### Reviewer's report 3

Eugene V. Koonin, National Center for Biotechnology Information, NIH, USA

This very brief manuscript defines and solves an important problem in the theory of evolution. Shnol *et al*. introduce the concept of the minimal genetic load and prove that genetic load assumes the minimal value when the fitness landscape is defined in the simplest possible way, namely with only two fitness values ("fit" and "unfit" individuals). In the extremely short but constructive Discussion, the authors explain why minimal genetic load can be useful to describe population dynamics. I have a variety of questions and minor suggestions none of which invalidates the conclusions but that could be useful to address. To begin with, I think the Discussion is a bit too brief. I am not talking about any extensive elaboration but I believe it would be useful to try and give the reader some intuitive feeling WHY the condition of the minimal genetic load is what they show it is. I think this is doable and would improve the manuscript.

The more important point comes here, at the start of the Discussion: "The same results can be proven if *w* is a continuous random variable." First, it is not 100% clear to me what "the same" means in this context. That, although W is now a continuous variable, the minimal value of L is reached when and only when all individuals accumulate in just two points on the landscape? It would be best to explain. Second, "can be proven" leaves the reader with some uneasiness. If it is simple, why not give the proof for this, more general case? If it is hard, perhaps, briefly explain why and why confidence it can be proven. If simple but tedious, perhaps, make it an Appendix?

Then, I am somewhat uncertain about the exact meaning of this key statement: "Indeed, if the minimal genetic load, consistent, for example, with a particular genomic rate of deleterious mutations or a particular rate of changes of the environment, is high, this means that the population under such conditions cannot survive unless it consists of very fecund individuals". I this paper, the minimal genetic load is defined in relation to the variance of relative fitness; does the quoted statement imply that V is deterministically depends on the other variables mentioned in that sentence? I think clarification would be helpful. Finally, a couple of minor issues: I think it would be best to say in the Background section of the abstract that L and V are AMONG the most fundamental characteristics of selection.

The proof of the theorem is written in a somewhat unusual form, interspersed with comments which I think mostly distract from the logic of the proof. It would be best, I believe, to give the proof the way it is normally done, then comment.