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Drift as constitutive: conclusions from a formal reconstruction of population genetics

Abstract

This article elaborates on McShea and Brandon’s idea that drift is unlike the rest of the evolutionary factors because it is constitutive rather than imposed on the evolutionary process. I show that the way they spelled out this idea renders it inadequate and is the reason why it received some (good) objections. I propose a different way in which their point could be understood, that rests on two general distinctions. The first is a distinction between the underlying mathematical apparatus used to formulate a theory and a concept proposed by that theory. With the aid of a formal reconstruction of a population genetic model, I show that drift belongs to the first category. That is, that drift is constitutive of population genetics in the same sense that multiplication is constitutive in classical mechanics, or that circle is constitutive in Ptolemaic astronomy. The second distinction is between eliminating a concept from a theory and setting its value to zero. I will show that even though drift can be set to zero just like the rest of the evolutionary factors (as others have noted in their criticism of McShea and Brandon), eliminating drift is much harder than eliminating those other factors, since it would require changing the entire mathematical apparatus of standard population genetic theory. I conclude by drawing some other implications from the proposed formal reconstruction.

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Notes

  1. There is quite a large literature discussing the adequacy of the analogy in various different aspects (see Sect. 5 for some examples). In this article I focus on only one of those aspects, which is presented in what follows.

  2. In their 2010 book, and in Brandon and McShea (2012, p. 739) they also treat mutation as a constitutive constraint. I will have more to say about mutation and its status in Sect. 4.

  3. For example, as I show below, because considering it allows us to get a better understanding of how certain models within PG are put together, and how they explain.

  4. For more on their definition of constitutive and imposed constraints, see Brandon and McShea (2012).

  5. Again, this is for clarity. The axioms could be established in an entirely formal manner.

  6. Alternative presentations, for example, calculate the probability of every possible type of mating in the population, together with the sampling probabilities for the descendants of each of those types. Generational transitions are then modeled with recourse to more than two sampling processes. There are other formal reconstructions that focus on these kind of presentation; see for example Lloyd (1994) and Lorenzano (2014).

  7. Following usual talk, I call the type of a gene an allele-type (for example, when one says that “gene g1 is of allele-type A, while gene g2 is of allele-type a”), and the type of a pair of genes (of an individual) a genotype, even though "genotype" would have been a better fitting name to the type of a gene. The term "genotype" is also used ambiguously in the literature to refer to a particular pair of genes (not to its type); here, I use the term "genotype" to refer exclusively to the former not the latter.

  8. I also assume, for simplicity, that fitness coefficients remain constant over the generations, though this could be easily modified in a more complex version of the reconstruction.

  9. In a sampling process without replacement, already chosen individuals from a population to form part of the sample cannot be chosen again. In the typical examples, where one samples marbles from an urn, marbles that have already been sampled are not put back inside the urn. In contrast, in a sampling process with replacement, marbles are put back into the urn after being sampled and can be chosen again. Obviously, when one speaks of biological sampling processes there is no one choosing the individuals from the samples; for instance, in parental sampling, the sample consists simply of the individuals who survived.

    Probability assignments for sampling processes with and without replacement are different; however, it can be mathematically proven that, when the samples are large, the probabilities of the second approximate (and are equal, at the limit) to the first (see Feller 1971, Chapters II, VI).

  10. Notice that, for this second process, it is not the case that Ii+1 ⊆ Ii* (the "sample" is not a subset of the original population), because that would violate Axiom 1. In fact (because of Axiom 1) Ii+1 and Ii* do not share any elements. However, the genotype distribution of Ii+1 depends on that found in Ii* exactly in the same way as if it was a sample (a subset) properly speaking—see the constraints listed below.

  11. Fitness coefficients do not play a role here because, as said before, I am only considering selection by viability. In a more complete version, fitnesses should appear.

  12. I assume that migration plays a role especially in the first sampling process, and mutation in the second.

  13. Logicians (and to a lesser extent, mathematicians) usually do specify the language in which their theories are built, while empirical scientists typically do not do this. However, formal reconstructions of empirical theories, done usually by philosophers, do make the language explicit. For example, for a formal reconstruction of CM that makes explicit all the terms used in its language, see Balzer et al. (1987).

  14. Of course, there is also the issue of whether the theory itself remains the same if one of its concepts is eliminated. I will not go into this problem here.

  15. Moreover, the concept of drift was originally introduced into PG to account for certain empirical phenomena that could not be explained purely by selection (e.g. Gulick 1872, discusses the puzzling geographical distribution of the genera within a family of land snails in the Hawaiian islands, which are phenotypically very distinct, but live in environments that are very similar to one another; see also Hagedoorn and Hagedoorn 1921; Brooks 1899, for other antecedents). None of this would make much sense if drift was just a purely mathematical phenomenon.

  16. It might be thought that my conclusions regarding drift as part of the background mathematical vocabulary lend some credibility to the statisticalist position, defended chiefly by Walsh, Ariew, Lewens and Matthen (I thank an anonymous reviewer for this suggestion). I have some reservations about this, but for reasons of space I cannot explore this issue here, since going into it would require introducing that debate more fully. Instead, leave it open as a suggestion.

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Acknowledgements

This work has been funded by the research projects SAI 827-223/19 and PUNQ 1401/15 (National University of Quilmes, Argentina), UNTREF 32/15 255 (Universidad Tres de Febrero, Argentina) and UBACyT 20020170200106BA (Universidad de Buenos Aires, Argentina).

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Appendix: Proofs of theorems

Appendix: Proofs of theorems

Theorem A1

Let R be the relation \(\left\{ {\left\langle {x,\left\{ {x,y} \right\}} \right\rangle /x \in G_{i} \,\& \,\left\{ {x,y} \right\} \in I_{i} } \right\}\). Note that R is a function, since Axiom 3 specifies that R satisfies the uniqueness and existence requisites for functions (each gene has one and only one corresponding individual to which it belongs). Thusly, R can be seen as a GiIi function. Additionally, Axiom 2 implies that each gene is present only once inside each individual (in the gjgk part). All of these tells us that, for each gene, there exists one and only one individual to which it belongs, and which also contains a different gene. In that way, every individual from generation i will be the value assigned by function R to two different genes (arguments) from that generation. Therefore, if |Gi| is the number of genes in generation i, function R establishes |Gi|/2 partitions in its domain Gi. Notice also that R is suryective (every element of the codomain, in this case Ii, is the value of some argument), since, by Axiom 3, no gene can be “loose”. Therefore, the number of partitions also coincides with the number of individuals. Therefore, |Ii| = |Gi|/2, which immediately gives us the desired result.

Theorem A2

For simplicity’s sake, I only prove this for finite populations, and for allele type A (the proof for a is almost identical). By applying the definitions of FreqAT and FreqGT, what needs to be proved is that:

$$\frac{{\left| {\left\{ {g_{k} \in G_{i} /f_{1} \left( {g_{k} } \right) \, = A} \right\}} \right|}}{{\left| {G_{i} } \right|}} \, = \, \frac{{\left( {\left| {\left\{ {i_{k} \in I_{j} /f_{2} \left( {i_{k} } \right) \, = \{ A,A\} } \right\}} \right| \, + \, 1/2\left| {\left\{ {i_{k} \in I_{j} /f_{2} \left( {i_{k} } \right) \, = \{ A,a\} } \right\}} \right|} \right)}}{{\left| {I_{i} } \right|}}.$$

It shall be convenient to call Gi(x) the set \(\left\{ {g_{k} \in G_{i} /f_{1} \left( {g_{k} } \right) \, = x} \right\}\) (i.e. the set of genes of type x present in generation i), and Ii({x, y}) the set \(\left\{ {i_{k} \in I_{j} /f_{2} \left( {i_{k} } \right) \, = \{ x,y\} } \right\}\) (the set of individuals of genotype {x, y} in generation i). With this terminology, what needs to be proved is that:

$$\frac{{\left| {G_{i} \left( A \right)} \right|}}{{\left| {G_{i} } \right|}} \, = \, \frac{{\left| {I_{i} \left( {\{ A,A\} } \right)} \right| + \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \left| {I_{i} \left( {\{ A,a\} } \right)} \right|}}{{\left| {I_{i} } \right|}}$$

Theorem A1 establishes that |Gi| = 2 × |Ii|. Thus, for the previous equality to hold, what needs to happen is that:

$$\begin{aligned} \left| {G_{i} \left( A \right)} \right| & = 2\left( {\left| {I_{i} \left( {\{ A,A\} } \right)} \right| + \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \left| {I_{i} \left( {\{ A,a\} } \right)} \right|} \right) \\ & = 2\left| {I_{i} \left( {\{ A,A\} } \right)} \right| + \left| {I_{i} \left( {\{ A,a\} } \right)} \right| \\ \end{aligned}$$

To prove this, I define two new sets, called Gi(A, {A, A}) and Gi(A, {A, a}). These sets will represent the set of A genes present in an {A, A} kind of individual, and in an {A, a} kind of individual. Formally,

$$G_{i} \left( {x, \, \{ x,y\} } \right) = \, \{ g_{k} \in G_{i} /f_{1} (g_{k} ) = x{\text{ and }}\exists g_{j} \in G_{i} {\text{ such that }}f_{1} (g_{j} ) = y\quad \& \quad \{ g_{k} ,g_{j} \} \in I_{i} \}$$

Axioms 3 and 4 imply that \(G_{i} (A,\{ A,A\} ) \cap G_{i} (A,\{ A,a\} ) = \varnothing\) and that \(G_{i} (A,\{ A,A\} ) \cup G_{i} (A,\{ A,a\} ) = G_{i} (A)\) (the first follows from the fact that no gene is inside more than one individual, while the second from the fact that every gene is inside at least one individual, along with another gene—which must be either of the same or different type). Both of these facts imply that \(\left| {G_{i} (A,\{ A,A\} )} \right| + \left| {G_{i} (A,\{ A,a\} )} \right| = \left| {G_{i} (A)} \right|\).

The following two facts, along with the equation just derived, directly give us the desired result.

  1. 1.

    \(2\left| {I_{i} \left( {\{ A,A\} } \right)} \right| = \left| {G_{i} (A,\{ A,A\} )} \right|\)

    This follows from the fact that the members of Ii({A, A}) are pairs of different members of Gi(A,{A, A}), since every member of Gi(A,{A, A}) belongs to an AA individual, along with another Gi(A,{A, A}) member.

  2. 2.

    \(\left| {I_{i} \left( {\{ A,a\} } \right)} \right| = \left| {G_{i} (A,\{ A,a\} )} \right|\)

This follows from the fact that every member of Gi(A,{A, a}) belongs to a (different) Aa individual (i.e. member of Ii({A, a})). This individual also contains a member not belonging to Gi(A,{A, a}, but to Gi(a,{A, a}). Thus, a bijection can be established between both sets, which means that they have the same number of elements.

Theorem A3

This theorem follows from the following two facts. First, that if:

$$\frac{n!}{{x!y!\left( {n - x - y} \right)!}}p^{x} \times q^{y} \times r^{n - x - y}$$

is a multinomial distribution (with p, q, r being the frequencies of three kinds of objects in the population), then the expected value of the frequencies in the sample is exactly p: q: r (i.e. that all kinds of objects maintain their original frequency). Note that the probability assignment for the first sampling process is multinomially distributed.

The second fact is the probabilistic “Law of large numbers”. This law states (with terminology modified to fit the one I am using) that:

$$\mathop {\lim}\limits_{n \to \infty} P\left({\hat{p} - E_{{\hat{p}}} (Sampl(x)) > \epsilon} \right) = 0$$

where n is the size of a sample in a probabilistic sampling process Sampl(x), \(\hat{p}\) is the frequency of one kind of object in the sample, \(E_{{\hat{p}}} \left( {Sampl\left( x \right)} \right)\) is the expected value of \(\hat{p}\) in Sampl(x), and ϵ is an arbitrary number (thus, an arbitrarily small one). In other words, at infinite sample sizes, the probability that deviations from the expected value occur (for the frequency of one kind of object) reduce to zero (for more on this law, see Feller 1971, chapter X). If this is applied to the first sampling process, the result is that actual outcomes should equal their expected outcomes. These two facts together directly imply the desired result.

Theorem A4

The proof of this result uses the same two facts as the proof from above. Since |Ii*| is infinite, by Theorem A3, we get that for every genotype {x, y}:

$$FreqGT\left( {I_{i} *, \, \{ x,y\} } \right) \, = \frac{{w_{x,y} \times FreqGT(I_{i} ,\{ x,y\} )}}{{\bar{w}(i)}}$$

Now, since we assume that all fitnesses are equal, then there exists a number m, such that m = wA,A = wA,a = wa,a. Thus (by definition of \(\bar{w}(i)\)):

$$\begin{aligned} FreqGT\left( {I_{i} *, \, \{ x,y\} } \right) \, & = \frac{{m \times FreqGT(I_{i} ,\{ x,y\} )}}{{m \times FreqGT(I_{i} ,\{ A,A\} ) + m \times FreqGT(I_{i} ,\{ A,a\} ) + m \times FreqGT(I_{i} ,\{ a,a\} )}} \\ & = \frac{{m \times FreqGT(I_{i} ,\{ x,y\} )}}{{m \times \left( {FreqGT(I_{i} ,\{ A,A\} ) + FreqGT(I_{i} ,\{ A,a\} ) + FreqGT(I_{i} ,\{ a,a\} )} \right)}} \\ \end{aligned}$$

Since the m’s cancel out in the numerator and denominator, and what is left in the denominator (the sum of all the genotype frequencies) equals 1, all of this equals FreqGT(Ii, {x, y}). Thus, after the first sampling process, the frequencies of the genotypes remain identical. Additionally, Theorem A2 implies that allele-type frequencies will also be identical to the originals (since they can be calculated from genotype frequencies, which are themselves identical to the originals). Thus, for every allele-type x, FreqAT(\(\cup I_{i} *\), x) = FreqAT(Gi, x) (the union set of a set of individuals equals the set of genes present in those individuals).

By using the same two facts as in the demonstration of Theorem A3, and the probability distribution of the second sampling process, we get the desired result.

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Roffé, A.J. Drift as constitutive: conclusions from a formal reconstruction of population genetics. HPLS 41, 55 (2019). https://doi.org/10.1007/s40656-019-0294-6

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Keywords

  • Population genetics
  • Drift
  • Formal reconstruction
  • Constitutivity
  • Classical mechanics
  • Analogy