Abstract
Several recent arguments by philosophers of biology have challenged the traditional view that evolutionary factors, such as drift and selection, are genuine causes of evolutionary outcomes. In the case of drift, advocates of the statistical theory argue that drift is merely the sampling error inherent in the other stochastic processes of evolution and thus denotes a mathematical, rather than causal, feature of populations. This debate has largely centered around one particular model of drift, the Wright–Fisher model, and this has contributed to the plausibility of the statisticalists’ arguments. However, an examination of alternative, predictively inequivalent models shows that drift is a genuine cause that can be manipulated to change population outcomes. This case study illustrates the influence of methodological assumptions on ontological judgments, particularly the pernicious effect of focusing on a particular model at the expense of others and confusing its assumptions and idealizations for true claims about the phenomena being modeled.
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Notes
This example is an instance of a standard Wright–Fisher model interpretation of drift. I do not mean to imply that this is the only possible type of drift process, and I will consider others in Sect. 4.
In a pure drift model, each individual in the population has an identical expected offspring distribution. In mixed models, drift can be combined with selection, mutation, and other factors.
X denotes the number of individuals of the X trait. X/N gives the frequency of the trait.
The terms in the right-hand side of the equation below denote, from left to right, the number of individuals with trait X, the number of individuals of the alternative trait, and a term accounting for variance in the offspring distribution of an individual with the X trait. The right-most term is used to calculate the variance expected population size, which I will discuss in more detail Section 4. See Der et al. (2011) for a more thorough explanation.
For example, Millstein (2002) and Hodge (1987) argue that drift is indiscriminate sampling (that is, sampling without respect to intrinsic physical differences among individuals). According to Gildenhuys (2009), drift refers to “causal influences over a population” that are “non-interactive, non-pervasive, and indiscriminate causes” (p. 522). Other prominent defenses of the causal theory, such as Shapiro and Sober (2007), Stephens (2004), and Reisman and Forber (2005) do not take an explicit stand on this issue.
This is stronger than the nomological necessity constitutive of causal explanations.
For example, Sober (1984, p. 117) argues that the sampling size of a stochastic process is a cause of that process’s outcomes. The probability distribution over outcomes of a series of coin-flips is affected by both the bias of the coin and the number of times you flipped it. See also Sober (2011) for a defense of the claim that there are a priori causal claims in evolutionary biology.
According to some views, drift just is the actual deviation of trait frequencies from expectation (Brandon 2005). On most causal views, \(N \)is an indication of drift’s power to cause such deviations.
Here, we can model this by supposing that the ball drawing is no longer indiscriminate, and a token yellow ball is twice as likely to be drawn than a token green ball. See Brandon (2005, pp. 157–158) for a similar example.
From Wright, the probability of fixation (\(\pi )\) of a beneficial allele introduced at frequency 1/2\(N =\) 2s/(1\(-\)e\(^{-4Ns})\). Notice that as Ns increases, the value of the denominator goes to 1, so 2\(s\) will be a good approximation of the probability of fixation for most values of \(s\). For a more detailed derivation and discussion, see Kimura (1962, pp. 715–716). I am grateful to a reviewer for helpful comments on this issue.
I am not claiming that philosophers have wholly ignored the distinction between population size and effective population size; for instance, Stephens (2004) uses the effective population size as an indicator of the strength of drift. However, few in this debate have paid close attention to the significant causal factors which determine effective population size. Gildenhuys (2009) is a welcome exception.
As Der et al. (2011) show, the Wright–Fisher model, as well as the other models I consider, also obey the formal characterizations of drift described by (Mean) and (Variance). The question of whether there are drift models that exhibit the informal properties of drift while not obeying these formal properties is interesting but beyond the scope of this paper. For my purposes, it suffices to show that there are models which meet the formal requirements which population geneticists use to define drift processes but differ interestingly in their resulting population dynamics.
The probability that green will go to fixation is the probability the population will go through a bottleneck event in that generation (1/N \(=\) 0.01) times the probability that it will be a green ball that is chosen to populate the next generation (its starting frequency, 0.6).
The Eldon–Wakeley model is an extension of the Moran framework. For a more detailed discussion, see Der et al. (2012, p. 1332).
One of their primary arguments is that the statisticalist view cannot adequately capture actual scientific practice since the development of alternative models of drift was motivated by the desire for models that made more realistic biological assumptions than the Wright–Fisher model, which Millstein et al. claim “makes no sense if drift were only a statistical outcome” (p. 6). I agree with their contention that biologists have often conceived of drift as a causal process, yet I do not think this argument is a particularly compelling refutation for statistical theorists who want to substantially revise the way that biologists (as well as philosophers) think about drift.
I will return to this point in Sect. 6.
I will ignore meiotic drive or other segregation distorters.
Filler (2009) argues that drift’s tendency to reduce heterozygosity shows that drift is an evolutionary force.
This does not mean that every gamete will contribute a single descendent to the daughter population. Sampling with replacement allows for the possibility that a gamete could be sampled more than once. However, this condition does ensure that the expected offspring distribution of a gamete is binomial.
The quote from Der et al. (2012) suggests ways of manipulating the probability distribution over offspring number so as to increase skew; when mortality rates are high, individuals may increase the number of gametes they produce in a given breeding episode, so we could intervene on offspring distributions by changing the environment in a way that increases mortality rates.
The proof of this is simple. In \((1- 1/2N)\) of the generations, the probability of a jump from 0.5 \(A\) to \(>\) 0.8 \(A\) or 0.5 \(a\) to \(>\)0.8 \(a\) is given by the binomial equation, but in 1/2\(N\) of the generations, the probability of such a jump is 1, which by necessity makes the probability of a jump greater than it would be in a binomial process. The exact probability of such a jump will depend on the value of \(\lambda \), \(N\), and the frequency of \(A \)and \(a\) when a replacement event occurs. The probability of fixation of a selectively favored allele will also increase.
This is another way of saying that drift is “indiscriminate”. See Millstein (2002) for a more thorough explication of the concept.
The question of whether we should be realists or instrumentalists about models is somewhat orthogonal to this debate. Antirealists will argue that it is a mistake to go beyond the models to make any judgments whatsoever about ontology. I will not respond to this argument since none of the parties to the debate in question are antirealists. Instead, here I defend the more modest claim that if we are engaged in the practice of making ontological judgments in science, then we should not read them directly off of mathematical models.
Though see Lange and Rosenberg (2011) for a defense of the latter claim’s plausibility.
Another case of a methodological device in population genetics being given an erroneous ontological interpretation is discussed in Clatterbuck et al. (2013).
While an exploration of the empirical ramifications of alternative drift models is beyond the scope of this paper, I suspect that the work of Der et al. will be fruitful in making more accurate predictions about drift. For instance, debates over the plausibility of Wright’s Shifting Balance Theory have appealed to features of Wright–Fisherian drift—such as the probability of fixation of a novel mutant in a small population, time until fixation of neutral alleles, and the interaction of selection and drift—which may be different if the populations are undergoing other types of drift (for an overview of the debate, see Coyne et al. (1997)).
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Acknowledgments
I am grateful to the editors of this special issue, three anonymous reviewers, Elliott Sober, Trevor Pearce, Naftali Weinberger, Brian McLoone, Martin Barrett, and an audience at the Ontology and Methodology conference held at Virginia Tech in May, 2013 for helpful comments on earlier versions of this paper.
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Clatterbuck, H. Drift beyond Wright–Fisher. Synthese 192, 3487–3507 (2015). https://doi.org/10.1007/s11229-014-0598-8
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DOI: https://doi.org/10.1007/s11229-014-0598-8