Recommendation for preprint "Negative frequency-dependent selection is frequently confounding".
by Ignacio Bravo, 2017-06-17 22:43
Decision & reviews
Three experts in the field have provided with extensive, high-quality and fair scientific judgement on the revised version of th text. All three reviewers agree on the pertinence and timeliness of the subject, and all of them also agree on the need of sound review highlighting the false friends that may be mistaken by negative frequency-dependent selection. Nevertheless, two of the reviewers consider that some of the key terms used in the text are ill-defined, and that precisely because the aim of the review is to clarify and draw clear borders, semantics (i.e. meanings) are in this case of the utmost importance. I largely agree with this perception. Further, I understand that the author aims at a target audience that does not expect a hard mathematical description. Nevertheless, the experts' opinion remains that a sounder mathematical foundation will undoubtedly contribute to a richer and deeper manuscript. Finally, I think that the suggestion to provide clear published examples of such misinterpretations of NFDS may also be illuminating, but I understand that expurgating the literature may not be in the interest nor in the aim of the author. Globally, I consider that the manuscript is potentially very interesting, and I would suggest to revise it considering the comments provided by the authors, before it can be recommended by PCI Evol Biol.
--Thank you and the reviewers for taking the time to review this manuscript. I have changed the manuscript in accordance with the reviewers’ suggestions by, most importantly, clarifying the key terms identified by the reviewers. A consideration of the target audience with regard to this manuscript was noted by multiple reviewers and the editor. In this regard, I agree with the editor and the reviewers who noted that including formal descriptions of the models has the potential to alienate the very audience that could be most impacted by these types of concept articles. It is important to note that this paper is neither building nor critiquing the mathematical models that have been previously formalized, an important topic that is beyond the goal of this paper. The primary issue being addressed is the heuristic application of the concept of negative frequency dependent selection to polymorphisms in biological systems in which the general concept does it apply, as noted by the reviewers (most explicitly reviewer 2). The types of errors noted here are not being committed by math-bio researchers who, despite being the primary reviewers of this manuscript, are thus not the target audience. Critical evaluations of the mathematical models underlying negative frequency dependent selection has been done elsewhere (although more work is certainly necessary). Given the target audience, it is my opinion that re-describing the math underlying these models, which can be found in the cited work, will be counter-productive as (1) the models themselves are not being critiqued here, (2) no new models are being proposed, and (3) the primary issue is that the general concepts in those models is being misunderstood by those that do not dive into the math-bio weeds. Lastly, while I agree that providing examples of errors in the literature can be illuminating, it is likely to be detrimental to scientific progress in general by turning off the very audience that could benefit from this paper (I also think it is mean).
Reviewed by anonymous reviewer, 2017-06-17 22:43
The author has made some marginal changes to its original manuscript, but my main source of disagreement remains. However, I do not consider the paper is wrong, as our disagreement mostly revolves around semantics, so I do not have further requests for changes. I will however provide a brief reply to the author's response to my comments, as well as some minor formal suggestions on the text.
The main issue I have with the paper is that negative frequency-dependence is defined very narrowly: according to the author, "rare variants must be advantageous because of their relatively rarity, regardless of the ecological interaction that mediates it". With this definition, negative frequency-dependence will clearly be a very special case, and it is no surprise that the author can find many exceptions. At the other end of the spectrum, frequency dependence has been defined more broadly, with reference to the dimension of the environmental feedback (Heino et al 1998, Metz & Geritz 2016). With this broader definition, the distinctions pointed out by the author are not so relevant. So, whether or not one finds the author's contribution useful depends on where one sets the cursor.
--I believe the reviewer and I agree: negative frequency dependent selection is a narrow selective force that has nevertheless been applied broadly in a heuristic manner and often incorrectly. The point of the paper is that the negative frequency dependent selection framework, in the narrow sense, is regularly described as the force maintaining a natural polymorphism when it clearly is not. I agree with the reviewer that many of the issues dealt with here are relevant in different ways (or not at all) when discussing the dependence of natural selection on frequency of alleles in a broad sense, but these are outside the scope of this paper.
Personally, I don't think a narrow definition of frequency-dependence is useful. Unless we look at very simple models, I don't see how fitness can depend on alele frequency only, and not on population densities or other environmental variables. This is the main reason why I was (and remain) rather unconvinced by the author's essay. However, I may be wrong, and since there is scope for debate, I think the author's contribution should play a role in this debate.
--It is important to note that I am not attempting to define frequency dependence narrowly, but to describe a specific slice of frequency dependence that, in some areas of evolutionary biology, is important and is often invoked. An additional detail - negative frequency dependent selection does not require that relative fitness is a function of allele frequency only, but that allele frequency must be part of the function (that is, there must be a strict dependence on frequency). I have tried to clarify this in the text. I agree with the reviewer that this manuscript, which discusses the areas to which this concept should apply, needs to be openly discussed and debated.
A few additional remarks:
• I don't agree with the interpretation of the paper by Kisdi 1999. In fact, by definition, when the population sits at a fitness minimum, any rare variant can invade, resulting in evolutionary branching. For me this qualifies as negative frequency-dependence, although it does not fit, if I understand correctly, the author's definition.
--I agree with the reviewer about the definition of a fitness minimum. But the conditions of the model presented in the paper prevent invasion due to free recombination (according to the interpretation of the authors). This is a topic that could be discussed at length, although beer will likely be necessary for any real progress.
• About short-term and long-term evolution: my remark was that, for long-term evolution, you also need to account for the genetic variation due to new mutations, not only for the standing genetic variation.
--I see. I agree with the reviewer about the importance of this topic but think this is beyond the general scope for target audience.
• The author uses the concepts of hard and soft selection to discuss density- and frequency-dependent selection. At the risk of passing for an iconoclast, I do not think these concepts, which have their origin in very specific models that are best viewed as caricatures, are really empirically or theoretically very useful. But this is best left for another discussion!
--I would very much like to not discuss hard or soft selection in this manuscript.
Some minor comments on the text:
• top of p. 4: "the overwhelming majority OF this broad field"?
• bottom of p. 6: it could be useful to explain in detail how this is different from the self-incompatibility allele example.
--I am not sure how to make this more clear, but welcome suggestions. The main concept I was trying to make clear is that this example does not rely on relative rarity.
• I think there are too many "luminaries" in this text. Some rewording could be welcome lest the reader be dazzled!
Reviewed by David Baltrus, 2017-06-17 22:43
I have read and reviewed the revised version of this manuscript, and have also read and taken into account the well reasoned critiques of other reviewers.
To me, the revised manuscript does read as more precise in its language and more limited in scope (almost to a fault, but not quite). I can definitely see the anonymous reviewer's points and agree that these are quite relevant critiques. This is certainly a situation that can be semantically challenging, and I think it's also going to be the case where specific viewpoints, backgrounds, and research foci are going to influence how the article is viewed.
To me, this article is well placed and hits the right notes for an audience that has heard the words "frequency dependent selection" but which are not necessarily truly experts in that topic. It's an article that will do well, especially given the examples, to clear up some confusion across research topics that might seek to invoke the phrase "frequency dependent selection". I get the feeling that this is the intent, and I think the article does deliver in this context and it will make people think. Although I thought the first version was pretty easy to digest, this version does set up the contrasts better and lays out the necessary details more clearly than the first.
In the context of my own research background, and with the acknowledgement that others might not see it this way, I think this article will be well received by the audiences at the intended levels. I would encourage others to read it at a broad level, but with the point of view that some of the specific details could be debatable. I don't view this as a problem, but as an admonition that nature and evolutionary scenarios can be complicated. This is actually an improvement from many of the discussions/invocations of frequency dependent selection I've seen lately.
Reviewed by anonymous reviewer, 2017-06-17 22:43
(First, I have to declare that I did not have access to the first version of this manuscript, therefore I am not commenting on the previous revision round.)
The understanding of a non trivial phenomenon such as biological evolution requires identifying the processes that underlie it. A sharp, unambiguous circumscription of these processes is however needed so that research remains able to reveal yet undetected causes of observed effects, thus allowing theory to be improved, not to mention the accuracy of its prediction.
In the present work, the author discusses the alleged key role of negative frequency-dependent selection (NFDS) in persistent genetic variation and how a maintained polymorphism could be explained by other processes – namely directional selection with environmental change, density-dependent selection, multiple niche selection and community diversity – that could be misinterpreted as NFDS.
Both the topic and the purpose of this paper are of significant interest in the field of evolutionary biology and I can only support its ambition to disentangle NFDS from concurrent selective contexts that can as well lead to stable genetic diversity – one of the main questions in evolutionary biology. I find the text interesting, richly documented and I really think the reflection it brings can be helpful to a large community of theoretical and field or experimental evolutionary biologists, mainly because it tries to link conceptual thinking to concrete examples, and as such, I fully agree with the conclusion paragraphs.
Nonetheless, in my opinion, the present version of the manuscript fails to reach its praiseworthy goal. The reason why I cannot suggest its recommendation as it stands is mainly because the central principle of NFDS is still loosely defined and consequently prevents further rigorous discussion. This is in particular due to the lack of formalism that could have greatly contributed to the needed clarification and lead to compelling proof. As exposed by , mathematical models act as “proof-of-concept” tests of verbal explanations, the scope of which is otherwise restricted to trivial cases.
I do not claim that this paper should turn into a heavy mathematical demonstration (it may even be self-defeating as part of that format could repel part of the aimed audience). I nevertheless suggest the author to be very neat on the definitions, to state the assumptions and to try to translate them into simple mathematical conditions in a way similar to e.g. reference 36 of the paper (Heino et al. 1998), while clearly indicating the conceptual novelty (with respect to NFDS) the manuscript brings compared to this same reference (which already contains, technically, key arguments that should be helpful for the present discussion -- see Table 1 of Heino et al. 1998).
--I agree with the reviewer that Heino et al did a very good job describing frequency dependence to the target audience of mathematically-oriented evolutionary biologists and the mathematical conditions were well presented. I also agree with the reviewer that the addition of a more mathematical demonstration will repel the very audience that this manuscript should reach. In the revised manuscript, I have pointed to the underlying models where possible for those that would like to discuss the mathematical details of the models. I would also point out that this paper does not describe any conceptual novelty with respect to negative frequency dependent selection. The descriptions in this article draw directly from the assumptions and underlying foundations of published models.
It is also important to remember that the primary objective of this paper is not to describe the key role of negative frequency-dependent selection in persistent genetic variation per se but to point out that negative frequency dependent selection is often used as a heuristic to describe genetic variation in systems where it does not apply as they violate the key assumption/condition of the process. The issue at hand is not a critique of prior modeling nor is it that mathematically-oriented research has made errors that need to be rectified. Further, a critical evaluation of the mathematical models underlying negative frequency dependent selection has been done well and, while more rigorous discussion of model details is always necessary, this manuscript is not intended spark those discussions.
Precisely, when the author introduces NFDS in its paper, it is not clear if he does acknowledge this definition as the sentence starts with “In models…” (and further “In these models…”). But more importantly, it is not clear if the fact that “rare variants have a selective advantage specifically because etc.” is part of the definition of NFDS, one of its necessary conditions, or a sufficient condition. The same ambiguity holds for the next sentence “Thus, NFDS can maintain genetic polymorphisms…”, while I wonder if “can” implies potentiality or systematic ability.
--I have made these definitions more clear throughout.
First, I am unsatisfied by the lack of definition of “Darwinian fitness” (p.3) (how is it computed, is it absolute, relative?) and their related notions of “per capita fitness” (p.8), “per capita growth rate” (p.11), “per capita rate of increase” (p.11) “per capita reproductive advantage” (p.5), “per capita fitness advantage” (p.6), “per capita selective advantage” (p.6), the author inconsistently uses throughout the paper.
--I agree with the reviewer that the multitude of terms can be confusing. I have limited the number of terms and defined them throughout the manuscript.
Second, if I assume that what the author means by fitness is equivalent to the instantaneous growth rate (as done in classical frameworks that link evolution to population dynamics), that is to say that one can write
dn/dt = w.n,
where n is the density of the focal genotype and w if the so-called fitness, therefore it is not sufficient that w is a decreasing function of the frequency of the focal genotype p = n/N (N being the total population density) for the selection to be negative frequency-dependent. Indeed, if we take e.g. w1 = 2 – p and w2 = 1 – q, with q = 1 – p, then both fitnesses are decreasing function of the respective frequency of each genotype but the genotype 1 will always increase in frequency even where frequent, as the following short R script can show:
Y=lsoda(c(1,10),Times,ODE1) ; plot(Y[,1],Y[,2]/(Y[,2]+Y[,3]),type='l',lwd=2,ylim=c(0,1))
Y=lsoda(c(10,1),Times,ODE1) ; plot(Y[,1],Y[,2]/(Y[,2]+Y[,3]),type='l',lwd=2,ylim=c(0,1))
As a result, the fitness decrease with frequency is not a sufficient condition of NFDS (I acknowledge that an ecological mechanism that would generate the fitness functions as assumed here seems unrealistic but this then highlights the fact that fitness cannot be any function of frequency and therefore it must be stated in the definition).
--I agree with the reviewer that it is not sufficient that that w is a decreasing function of the frequency and I apologize for the confusion. Negative frequency dependent selection occurs when relative fitness (not absolute fitness) is a function of relative abundance. While this is a necessary condition for negative frequency dependent selection, it is not sufficient for a stable polymorphism. However, it is beyond the scope of this paper to review the conditions necessary to achieve a stable polymorphism through negative frequency dependent selection as (1) the issue at hand is that natural polymorphisms are identified and then explained (incorrectly) using negative frequency dependent selection in the literature and (2) this has been done elsewhere.
Third, the fitness decrease with frequency is neither a necessary condition of NFDS. Indeed, one can imagine w1 to be a constant but w2 to decrease with frequency, e.g. w2 = 1/(2q). The following short R script then shows that despite the fact that the fitness of the focal genotype (1) is frequency-independent, its frequency increases if rare because of it rarity and decreases when frequent because of its commonness and that polymorphism is maintained, although the causality here is mechanistically indirect:
Y=lsoda(c(1,10),Times,ODE1) ; plot(Y[,1],Y[,2]/(Y[,2]+Y[,3]),type='l',lwd=2,ylim=c(0,1))
Y=lsoda(c(10,1),Times,ODE1) ; plot(Y[,1],Y[,2]/(Y[,2]+Y[,3]),type='l',lwd=2,ylim=c(0,1))
--Sorry again for the confusion, the explanation to this is similar that given above. Relative fitness decreases with increasing frequency is a necessary condition of negative frequency dependent selection, absolute fitness is not. I have made this more clear in the manuscript.
As a conclusion, the definition of NFDS provided in this paper is logically flawed. Rather, one should investigate the following mathematical condition derived from the replicator’s equation and little calculus (making plain the “advantageous”/”disadvantageous” dichotomy by the way):
NFDS <=> (w > w’ if p~0) and (w < w’ if p~1)
where w is the fitness of the focal genotype the frequency of which is p and w’ is the fitness of the other genotype(s) (if several, an appropriate weighted mean must be taken). If one does agree with this formulation, therefore attesting or contesting NFDS in a given eco-evolutionary context should be reduced to the estimation of four fitness limits.
--I agree with the reviewer that additional work can be done on the modeling of negative frequency dependent selection, which is well beyond the scope of this paper. I encourage the reviewer to create a formal rubric that one could apply to data or models to determine the applicability of negative frequency dependent selection in a given system. This would be an important addition to the literature and could help clear up these issues in a very practical way. Here, I have made the definition more clear to avoid the misunderstanding between relative and absolute fitness as it is described in these models.
Apart from the NFDS definition, I think the author should insist on the literature that has mistaken the concurrent processes chosen by the author for NFDS instead of focusing on few great names of evolutionary biology. More precisely, it is well established in the vast biomathematical literature that Lotka-Volterra systems are not examples of NFDS. In addition, from the quote provided, it is quite unclear that Lewontin (not “Lewinton”, p.8) was thinking about NFDS rather than density-dependence.
--I agree with the reviewer that pointing out mistakes in the literature can be illuminating, but it is often counter-productive because it can alienate the very audience one hopes to reach (I also consider it to be mean). I agree that Lewontin was thinking about density-dependence. However, this section of the book is explicitly about negative frequency dependent selection (the preceding paragraphs include “Rare-genotype advantage is an attractive hypothesis” and “a rare allele will increase in frequency but will not become fixed in the population, because as it gets commoner the fitness of its carriers decreases, and the other alleles are now favored.” That he was thinking of density dependence while discussion negative frequency dependent selection is indeed the point of this section.
My next comments are minor and hereafter given linearly:
p.6 – “A prominent … lines of thought”. This historical digression sounds a bit off topic.
--I have found this a useful literary device and would prefer to keep it.
p.7 - “The presence or frequency …”. It is worth to mention that this does not hold for all parasites and requires cross-reactivity, within-host competition and epidemiological feedback to be negligible.
--this example is specifically about flu and is used only to illustrate of the ideas presented. Adding the complications of other systems could add confusion and obscure the primary point.
p.8 – “the logic suggests”. The implied logic is not straightforward for me.
--I agree, the logic is incorrect, but is what was stated in the literature. Describing the incomplete logic is the point of this section.
p.8 – “is always at 100%”. Better “equal to 1”?
--For the target audience, I believe 100% is more clear
p.9 – A system brace and a centered label would be more aesthetic for Eq.1.
p.9 – Low frequency and low density are more explicit than “relatively rarity” and “numerical rarity”.
--For the target audience and to agree with the other language to avoid confusion, I think these terms are more clear.
Figure 1. Perhaps mention, for the sake of completeness, that the self-incompatibility investigated here is gametophytic. “Pollinated” instead of “pollenated”? The legend seems written for two alleles while three alleles are shown in the graphs.
--I have fixed the legend such that the third allele does not feel left out
Figure 2. The graphs should be labeled. The legends of panels A and C do not indicate what the quantity corresponding to the y-axis is. “at 100% frequencies” is unclear.
--I have added more descriptions to the figures to clarify
Figure 3. The graphs should be labeled and indicated in the legend. The first sentence of the legend should be rephrased. The parameter values can be given in the supplementary material only.
Supplementary material. The outline of this appendix is inconsistent. I have checked the R scripts and they work fine, producing the Figures (1 and 3) provided in the main text. A word is missing at the end of “by percent each is found in the”. Use “seedling” instead of “baby plant”/”babies”? Only the first plot command is relevant for Figure 1. As for Figure 3, the script lacks some comments within the for loops, and the final plot.
Summary: I request a substantial revision of this manuscript, essentially with respect to the definition and formalization of the NFDS and an emphasis on the novelty the discussion here provided compared to previous literature.
 Servedio MR, Brandvain Y, Dhole S, Fitzpatrick CL, Goldberg EE, et al. (2014) Not Just a Theory—The Utility of Mathematical Models in Evolutionary Biology. PLOS Biology 12(12): e1002017. https://doi.org/10.1371/journal.pbio.1002017