Transient amplifiers of selection and reducers of fixation for death-Birth updating on graphs

The spatial structure of an evolving population affects the balance of natural selection versus genetic drift. Some structures amplify selection, increasing the role that fitness differences play in determining which mutations become fixed. Other structures suppress selection, reducing the effect of fitness differences and increasing the role of random chance. This phenomenon can be modeled by representing spatial structure as a graph, with individuals occupying vertices. Births and deaths occur stochastically, according to a specified update rule. We study death-Birth updating: An individual is chosen to die and then its neighbors compete to reproduce into the vacant spot. Previous numerical experiments suggested that amplifiers of selection for this process are either rare or nonexistent. We introduce a perturbative method for this problem for weak selection regime, meaning that mutations have small fitness effects. We show that fixation probability under weak selection can be calculated in terms of the coalescence times of random walks. This result leads naturally to a new definition of effective population size. Using this and other methods, we uncover the first known examples of transient amplifiers of selection (graphs that amplify selection for a particular range of fitness values) for the death-Birth process. We also exhibit new families of “reducers of fixation”, which decrease the fixation probability of all mutations, whether beneficial or deleterious.

First, the analysis investigates a limit of the fixation probability as epsilon approaches 0. But in that limit, the graphs are no longer well-behaved in the sense that their fixation time goes to infinity. In other words, the graph cannot achieve fixation in finite time in the limit that the blades are not connected to the hub. So this limit does not make sense to me, because the graph reduces to independent Moran processes, one on each blade.
We understand the confusion here, as it is true that the fixation probability is undefined at $\epsilon=0$. However, the limit is well-defined in the usual mathematical sense that the fixation probability can be made arbitrarily close to its limiting value by choosing $\epsilon$ sufficiently small. In other words, the limiting value is an accurate approximation for small but positive $\epsilon$. This point is corroborated by our new Figure S1, in which we show excellent agreement between the $\epsilon \to 0$ limit and Monte Carlo simulations for $\epsilon = 0.001$.
Since this confusion is likely to be shared with other readers, we have added some explanation about the $\epsilon \to 0$ limit on lines 222-225.
Second, there is an error in computing that limit for the fan from Eq. 13. Lim eps -> 0 of Neff (Eq. 13) is mn. Either the equation is misreported or there is a deeper issue to be addressed.
We thank you very much for identifying this mistake. Eq. (13) was misreported--we had simply pasted in the wrong result. We have replaced this equation with its correct version (now Eq. (14)), which resolves the issues you mentioned. The new Eq. (14) result has the correct limits, agrees with the corresponding result for Separated Hubs in the SI, and also agrees with the results for nonweak selection. We thank you for the suggestion to run Monte Carlo simulations. We have run simulations on all three graph structures, in two cases: (i) for $\epsilon=0.1$, to verify our weak-selection results, and (ii) for $\epsilon = 0.001$, to verify our $\epsilon \to 0$ results. Both sets of simulations show excellent agreement with our analytical results. As expected, the simulated fixation probability coincides with the weak-selection tangent line around r=1, but deviates as r moves away from 1.
We present the Monte Carlo results in Supplementary Figures 1 and 2.
There are a number of additional minor revisions that I include in the attached, including suggested references, interpretations, a request for an additional clarifying figure, etc.
We thank you for all of the corrections and suggestions in the PDF comments. We have accepted all of them, with the following comments: • With regard to the initial placement of the mutant, we clarify on lines 83-86 that we are considering the initial mutant location to be uniformly distributed over all vertices. This is appropriate for death-Birth updating, since each vertex is replaced with uniform probability at every time-step. • While there is probably some connection between Eq. (7) and Wald's identity (and we thank you for suggesting this), the more direct connection is to Kac's formula for mean return times. We have added a reference to Kac's formula after Eq. (7). • A fan with just one blade, or with one vertex per blade, has very different properties that are not characteristic of the fan structure overall. (Specifically, a fan with one vertex per blade is simply a star, which behaves very differently from a fan.) Some of our formulas do not apply in these cases. Rather than introducing special cases for n=1 and m=1, we found it simpler and clearer to restrict to n>=2 and m>=2.
• For these figures (now 3CD, 4BCD, and 5BCD), we did not use a numerical value of $\epsilon$, but rather we used the analytically-calculated limits as $\epsilon \to 0$. This is specified in the figure captions. These figures are now complemented by the Monte Carlo simulations in Supplementary Figure 2, which show excellent agreement between $\epsilon \to 0$ limits and simulated fixation probabilities for $\epsilon = 0.0001$.
There are unlabelled equations and references in the SI as well.
Thank you for pointing this out. We have gone through the SI carefully to make sure all equations and references are correctly labeled.

I'd like to mention that the situation of amplifiers under dB is similar to the situation of suppressors under Bd. In general, there seems to be a rather low number of suppressors under Bd, and most of them seem to be transient and/or weak, at least for unweighted and undirected graphs. The fan also seems very interesting for Bd. A quick estimate left me with the conclusion that it is probably an unusually strong suppressor under Bd, in essence only fixating if the mutant lands in the central node as long as r << 1= and n << 1= if I didn't make a mistake.
We thank you for this suggestion. We are currently working on a follow-up manuscript that will apply this same methodology to Birth-death updating. Interestingly, the Fan with Bd updating displays several different regimes of behavior, depending on the value of $\epsilon$.

Could it be possible to use the effective population size to prove that there is no undirected, unweighted graph which is an amplifier under dB (at least in weak selection) by showing that Neff > N requires weights or directedness? If not, what big classes of graphs can be proven to have Neff < N?
We have taken up your suggestion to search for unweighted amplifiers of weak selection for death-Birth. We have found that such graphs do exist--the smallest we found is size 12. Our search was informed by a perturbation argument, which suggests that amplifiers of weak selection can be found by starting with an isothermal graph and then reducing the edge weight to vertices with large remeeting time.
We present our newly-discovered unweighted amplifier in Figure 2. The perturbation argument referenced above is now included under main-text results, lines 194-208, with derivation in Section 3 of the S1 Appendix.

Likewise, what is the relationship between the coalescence/remeeting time and fixation time? It seems intuitive that there should be a connection, and the paper shows how the remeeting time relates to fixation probability, at least in weak selection. Since there seems to be a broad correlation between in fixation probability and time (positive in Bd, negative in dB), this could be a good angle to explain this correlation.
We agree with you that there is likely a connection between effective population size and fixation time. Our preliminary investigations suggest that this is a deep and interesting question, which probably merits its own paper.

Comments on the Supplementary Information
As already mentioned, there are a few oversights in the supplementary that I'd like to be remedied. 1. The numbering of the equations is wrong on almost every reference. It seems like one or two equations were removed without changing the references, since it is usually just off by one or two.
We thank you for pointing out these oversights. We have fixed the equation numbering. You are correct in your derivation. Fortunately, we had used the correct equations in our derivations--the mistakes only occurred when transcribing these equations into the SI.
We fixed the typos in Eqs. (33), (34) and (36)  We thank you for pointing out these oversights. We read through the SI again and fixed mistakes, including: • Changed the instances of wh to wH • Fixed the ordering of the subscripts on SH in Theorem 3 to n , m , h • Changed b = nm+n from b = nm+m as noted by reviewer • In the first paragraph of subcase 3.2 of Theorem 3 on page 13, changed subscript from ρ K nm+n-1 to ρ K nm+h • Fixed exponent in numerator of Eq. (86) (now labeled Eq. (93)) from -( m -1) to -( m + h )