Abstract
The probability that the fitter of two alleles will increase in frequency in a population goes up as the product of N (the effective population size) and s (the selection coefficient) increases. Discovering the distribution of values for this product across different alleles in different populations is a very important biological task. However, biologists often use the product Ns to define a different concept; they say that drift “dominates” selection or that drift is “stronger than” selection when Ns is much smaller than some threshold quantity (e.g., ½) and that the reverse is true when Ns is much larger than that threshold. We argue that the question of whether drift dominates selection for a single allele in a single population makes no sense. Selection and drift are causes of evolution, but there is no fact of the matter as to which cause is stronger in the evolution of any given allele.
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Notes
There now is disagreement among philosophers about this claim. Some hold that neither causes evolution (Walsh 2000; Walsh et al. 2002), others hold that only selection is a cause (Brandon 2008), while still others hold that both are causes (Sober 1984; Millstein 2002; Stephens 2004; Reisman and Forber 2005; Shapiro and Sober 2007, McShea and Brandon 2010). We count ourselves in this last category, though the main point of the present paper is not to criticize alternative positions.
The effective population size (usually represented by the symbol “Ne”) is the same as the census size when various assumptions hold (for example, random mating and equal numbers of males and females). The technical details of how the two quantities are related (see Charlesworth 2009 for details) won't matter in what follows.
It is a further question whether “causal difference making” should be calibrated by arithmetic differences or by ratios; see Fitelson and Hitchcock (2011) for discussion.
Or (as pointed out by an anonymous referee), the distribution might be bimodal where the expected value is in the valley between two peaks and the actual value is identical with the expected value.
Here, we are envisioning a maximum likelihood estimate of the true probability of cancer from frequencies in the population, though actual epidemiological studies may introduce assumptions based on theoretical considerations about the underlying probability distribution.
It might be objected here that the problem can be solved as follows: if x cigarettes and y grams of asbestos both contain the same weight of carcinogenic materials, then we should compare how much each change affects the probability of lung cancer. The problem with this suggestion is that asbestos and cigarette smoke may contain different carcinogenic ingredients and these may differ in their potencies. An ounce of one poison need not have the same impact as an ounce of another. The fact that it is possible to measure two causes by using a “common currency” does not mean that it is appropriate to do so. For example, suppose that performance on a math test is influenced by a person’s years of education and by the minutes spent studying for the test. While these causes are both measured in units of time, there is no reason to think that the strengths of these two causes should be compared by considering the effect of an additional 30 minutes of prior education with an additional 30 minutes of time spent studying for the test.
There is a complication: the impact of a unit change in cigarettes may depend on what asbestos dosage is considered, and vice versa. If so, one can compute the average effect of each.
This does not mean that if the product were smaller, the less fit allele would be more likely to evolve than the fitter allele.
Matthen and Ariew (2002) make a similar claim: “Suppose that over a period of time a population stays exactly the same, or changes in some determinate way. The proposition that drift was involved to degree p in this history generally has no determinate truth value” (65). We agree with their conclusion but differ in our reasons. We take their argument to be that it does not make sense to talk about drift with respect to the births and deaths of token organisms or resulting changes to token populations; doing so would be a category mistake because drift only emerges when we consider types of populations. Our reasons for thinking that there is no fact of the matter regarding which cause was stronger persist even if we properly treat drift and selection as population-level causes.
Most theoretical and philosophical discussions of drift assume the Wright-Fisher model. However, there are alternative models of population dynamics under drift and selection, and some of these models make different predictions about how drift and selection will interact (see Der et al. 2011 for a discussion). Our discussion in what follows will focus on the Wright-Fisher model.
As an anonymous reviewer pointed out, this probability distribution is derived theoretically from a model. The probability distributions of lung cancer conditional on smoking and asbestos that we considered above were generated by inferring conditional probabilities from frequency data.
We have not included mutation in our discussion until now. Doing so increases the realism of the example, but does not affect what we want to say about the comparison of selection and drift. Without mutation, the population frequencies of 0 and 1 in Fig. 1 would be absorbing states and there would be more weight at those points in the distribution.
A similar pattern arises when we consider genetic set-ups different from heterozygote superiority. When there is selection for a dominant gene, or for a recessive gene, different values of Ns will correspond to different density curves. Curves with the same value of s but different values of N will have the same mean, but will differ in variance (Roughgarden 1979, pp. 77–78).
It is unclear which populations should be included in this data set. Should we focus on a specific allele and look at every population in which the same allele is present? Only those in which the allele has a non-zero selection coefficient? A positive coefficient? What about those populations from which the allele has been eliminated? Should we consider only conspecific populations or populations from a larger taxonomic unit?
For a derivation and a more detailed discussion of the probability of fixation, see Kimura (1962, pp. 715–716).
There are two additional features of this equation that are worth noting. Consider the following three possible solutions to the equation: (A) Ns = 1, N = 1000, S = 0.001, π ≈ 0.002. (B) Ns = 1, N = 100, S = 0.01, π ≈ 0.02, (C) Ns = 0.1, N = 100, S = 0.001, π ≈ 0.006. Comparing A and B shows that the value of Ns does not fix the probability of fixation. Comparing A and C shows that reducing the value of Ns can increase the probability that the favorable allele will reach fixation. This last comparison may seem to contradict our earlier statement that increasing Ns increases the probability that the favored allele will increase in frequency; in fact, it does not. Increase in frequency is one thing; reaching fixation is another.
Here and elsewhere, our discussion of what happens in “infinite” populations should be understood as shorthand for what happens as N approaches infinity. Discussing the frequency of a gene as N→∞ makes sense even if it makes no sense to ask what the frequency is of a gene in a population that is literally infinite.
We have assumed that drift plays a causal role in a population if and only if the population is finite. Here we are using “drift” to name a causal property of a process, not to indicate a possible outcome of a process (e.g., deviation from an expected value). For helpful discussion of the distinction, see Stephens (2004).
A similar analysis can be carried out for the question of how selection and drift differentially affect the probability that a disadvantageous allele will increase in frequency.
Okasha (2009, p. 28) endorses a zeroing-out strategy for determining the relative causal contributions of drift and selection. Such causal decomposition is possible, he argues, when we can identify the relevant causes and say, for each, what the effect would have been had the factor not been operative.
This is not to say that drift processes never have a direction. For instance, we expect that a population with two alternate alleles, A and a, evolving under pure drift (with no mutational input) will eventually go to fixation for A or a. Which allele is expected to go to fixation is determined entirely by the starting frequencies of the two alleles. If the population starts at 80 % A, the probability of A’s going to fixation is 0.8, so we do expect that the population will move in that direction in the long term (Filler 2009). However, in a population evolving under both drift and selection, we still cannot say that if the population evolved in the direction predicted by drift alone that drift was therefore the stronger cause.
A coin-flipping analogy is helpful here. If we flip a fair coin 5 times, we would not be too surprised to observe an outcome of 5 heads. If we flip a fair coin 1,000 times, it is much less probable that we will get 1,000 heads. However, it is still possible, and given enough coin-flipping experiments, the probability that we would eventually observe this outcome approaches unity.
Beatty (1984), Hodge (1987), and Millstein (2002) embrace this “separate process view” when they define drift as an “indiscriminate sampling process” and selection as a “discriminate sampling process”. This position entails that no single process can be both drift and selection, since a sampling process that is discriminating can’t be indiscriminate, and vice versa. Sober and Shapiro (2007, p. 256) make the mistake of saying that selection and drift are distinct processes, a claim that was extraneous to their larger argument. For a defense of the view that drift and selection are different aspects of a single process, see Abrams (2007).
Sober (2011) distinguishes three types of conventionalism about causes. Qualitative conventionalism about X and Y says that there is no fact of the matter as to whether X and Y both cause Z. Comparative conventionalism says that when X and Y both cause Z, there is no fact of the matter as to which is stronger. And quantitative conventionalism says that when X and Y both cause Z there is no fact of the matter as to how strong each cause is. Sober was discussing group and individual selection in that paper, and was describing alternatives to the three-fold realism of Okasha (2009), but the three-fold distinction and the associated triplet of realist positions pertain to the present topic of selection versus drift.
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Acknowledgments
We are grateful to Martin Barrett, David Baum, Michael Goldsby, Daniel Hausman, Trevor Pearce, Reuben Stern, Elena Spitzer, Mike Steel, Naftali Weinberger, and to the anonymous referees of this journal for useful comments on an earlier draft.
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Clatterbuck, H., Sober, E. & Lewontin, R. Selection never dominates drift (nor vice versa). Biol Philos 28, 577–592 (2013). https://doi.org/10.1007/s10539-013-9374-2
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DOI: https://doi.org/10.1007/s10539-013-9374-2