Toward a Universal Model for Spatially Structured Populations
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Evaluation Summary:
This very clear paper, which will be of interest to scientists in the field of evolutionary biology, investigates the effect of population structure on the efficacy of selection on a single locus. The results are based on analytical computations and numerical simulations, conducted in a scientifically rigorous manner. Although the conclusions are currently limited, the paper could serve as a solid basis for a more ambitious investigation.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #1 and Reviewer #2 agreed to share their names with the authors.)
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Evaluation Summary:
This very clear paper, which will be of interest to scientists in the field of evolutionary biology, investigates the effect of population structure on the efficacy of selection on a single locus. The results are based on analytical computations and numerical simulations, conducted in a scientifically rigorous manner. Although the conclusions are currently limited, the paper could serve as a solid basis for a more ambitious investigation.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #1 and Reviewer #2 agreed to share their names with the authors.)
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Reviewer #1 (Public Review):
The authors were trying to understand published claims about the effect of population structure on fixation probabilities by generalizing the models, effectively introducing more parameters to separate possible causes. The paper is for the most part extremely well written, and the thinking crystal clear. From mathematical point of view, everything appears to hold up.
The results certainly support the limited conclusion that whether a particular model of population structure helps or hinders selection can depend crucially on whether migration in symmetric not, but does not provide much general insight into why that is so.
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Reviewer #2 (Public Review):
This paper focusses on how the probability of fixation of a favourable mutation can become more sensitive to its fitness in certain population structures, thus "amplifying" selection. This intriguing phenomenon was identified in [3], and has received some attention, but within a restricted niche in the literature. There has been extensive study of such questions in population genetics, and this paper does cite much of this, thus making a connection between literatures that had been quite separate. In particular, their key method of separating the process into fixation within and across demes goes back to Slatkin [8].
"Amplification" of selection suggests that this phenomenon could be harnessed in practice, and this is suggested in the last sentence of the paper. Indeed, quantitative genetics theory rests …
Reviewer #2 (Public Review):
This paper focusses on how the probability of fixation of a favourable mutation can become more sensitive to its fitness in certain population structures, thus "amplifying" selection. This intriguing phenomenon was identified in [3], and has received some attention, but within a restricted niche in the literature. There has been extensive study of such questions in population genetics, and this paper does cite much of this, thus making a connection between literatures that had been quite separate. In particular, their key method of separating the process into fixation within and across demes goes back to Slatkin [8].
"Amplification" of selection suggests that this phenomenon could be harnessed in practice, and this is suggested in the last sentence of the paper. Indeed, quantitative genetics theory rests ultimately on the fixation of favourable alleles at the selection limit (as pointed out by Robertson, 1960). However, the overall efficiency depends on the rate of improvement, given some constraints (say, on total number of individuals), and depends on rates of fixation, not just probabilities. The results here do not directly show that population structure can increase the net efficiency of selection, and the consensus in the theory for evolution of sex is that a well-mixed population is most effective.
There is a strong connection with Wright's (1931) "shifting balance" theory, which relied on a two-stage process of selection, within and between demes (Rouhani & Barton, 1993, Genet. Res; Coyne et al., 1997, Evolution). This involves the fixation of a favourable combination of alleles in one deme, and then the spread of that combination between demes. This is a more complex process than is studied here, but it shares a reliance on asymmetric migration, and a sensitivity to population structure.
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Reviewer #3 (Public Review):
This work addresses the question that how likely a mutant takes over in a spatially structured population. It is known that some spatial structures, compared to a well-mixed population, may increase or decrease the fixation probability of a mutant and in this way amplify or suppress the effect of natural selection. This article provides new insights in the understanding of the situations where such amplification or suppression of the effect of natural selection occurs.
The paper studies a model of structured populations on graphs, where each node of the graph contains a well-mixed deme. The model considers birth, death and migration events which arise independently of each other. The birth rate includes a local density dependent competition term at each deme. The authors consider an initial state where all …
Reviewer #3 (Public Review):
This work addresses the question that how likely a mutant takes over in a spatially structured population. It is known that some spatial structures, compared to a well-mixed population, may increase or decrease the fixation probability of a mutant and in this way amplify or suppress the effect of natural selection. This article provides new insights in the understanding of the situations where such amplification or suppression of the effect of natural selection occurs.
The paper studies a model of structured populations on graphs, where each node of the graph contains a well-mixed deme. The model considers birth, death and migration events which arise independently of each other. The birth rate includes a local density dependent competition term at each deme. The authors consider an initial state where all the demes are at their demographic equilibrium size and that all the individuals are of wild type. The objective is to compute the fixation probability of a mutant introduced in this population. Considering different forms of graphs, they investigate which spatial structures amplify or suppress the effect of natural selection.
The authors consider a particular framework where the migration rate is small so that migration is a rare event compared to birth and death events. They also consider large carrying capacities in the demes, so that in the time period of the study no deme extinction occurs and the deme sizes fluctuate around their deterministic steady values.
In this regime they propose a coarse-grained model that approximates the original model. In this model, as a consequence of scarcity of migration events, each deme is of mutant or wild type and the evolution of the population can then be described as a Markov process where elementary steps are migration events, which change the state of the system if fixation occurs. The advantage of such coarse-grained model is that one can compute analytically the fixation probability of a mutant, as a function of the structure of the graph. Note however that in this analytical computation, the fixation probability of the mutant in a single deme is estimated by the fixation probability given by the Moran process, where the size of the population in a deme is considered constant. Numerical simulations, considering small migration rates, confirm that the fixation probabilities estimated by this method approach well the fixation probabilities in the original model. It is not investigated though how small the migration rate should be for the analytical results to remain valid.
The fixation probabilities for different graph structures are computed via this method, identifying situations where amplification or suppression of the effect of natural selection occurs depending on the spatial structure and the asymmetry of migration rates.
The results are compared to previous related works, using population models on graphs, but where each node corresponds whether to an individual or to a deme with constant population size. In these previous works usually the evolutionary outcome depends on the order of birth and death events, while in the present work these events are considered independent. Note however that in those previous works the migration, birth and death events occur at the same time scale, while here the migration events are assumed to be rare compared to the birth/death events. For this reason, I would not consider this work as a universal analysis of the problem but as a complementary one to the previous literature.
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