I have defended a specific explication of one dimension of the notion of “drift” in
evolutionary theory. Specifically, I have sought to explicate the notion of drift when that notion is used in a causal sense. When we talk about the causal influence of drift as an influence that is quantified by Nev, we are talking about non-interactive, non-pervasive, indiscriminate causal influences over the reproduction of variant types in a population. That’s not the only conception of drift at work in evolutionary theory, but that is all the more reason to break up the notion of drift and offer different, rather than rival, specifications of its various meanings.
Appendix
Consider a population of a stable size, N, whose members (i = 1, 2, 3, … N) contribute each k gametes apiece to the next generation. For a diploid population of stable size, the total
number of gametes contributed by the organisms is 2N, and the mean value for the ki variables must be 2. The variance in progeny number per parent for the ki can be expressed using the standard formula for variance:
(1)
Now consider the proportion of cases in which two random gametes come from the same parent. Imagine we choose some gamete in the descendant generation at random. The chance that another subsequent gamete chosen at random comes from the same parent can be expressed this way:
(2)
The probability of initially drawing at random one of the gametes of the ith parent is ki/2N, the number of gametes the ith parent produces divided by the number of gametes in total. Having drawn that gamete, the pool of available gametes for the second draw is one gamete smaller, consisting now of 2N – 1 gametes; the number of gametes in the pool descended from the ith parent is one less too. So the probability of drawing another gamete with the same parent as the first is (ki –1)/(2N – 1). The probability of both events in sequence is just the product of each event separately, giving the expression from the above equation.
The trick at this point is to modify the above expression such that the expression for the variance in offspring number per parent can be substituted into the expression (2). Here’s one
way to accomplish such a substitution. First, expand the expression for the variance in offspring number in this way:
Next, manipulate the expression (2) for the probability of drawing two gametes from the same parent in two random draws in the following manner:
Since the sum of all the gametes produced by the parents (the sum in the second term in the
numerator) must equal 2N and given that is just 4N, we end up with the following
At this point, we have an abstract expression for the probability of randomly drawing two gametes from the same parent in our population, which can have any degree of variance in offspring number per parent because we have set things up so that quantity is expressed by a variable. We further know, from the statistical theory of independent sampling processes, that the probability of drawing two gametes from the same parent when the gametes are sampled independently is 1/N. This makes sense, because the independent sampling process with
replacement is such that there are two descendants per parent on average, so that you have a 2 in 2N chance of getting a descendant of the same parent on the second draw, having gotten an arbitrary gamete on the first draw.
It is now possible to relate the sampling process in which the variance in offspring number per parent is expressed by a variable to an independent sampling process. The first gives the following value for the probability of drawing at random two gametes of the same parent:
The second gives a value of 1/N. Setting these values equal to one another gives the following formula for the effective size of a population with an arbitrary variance in offspring number:
A population with an arbitrary variance in progeny number will exhibit a next-generation variance in allele frequency as if it were a population in which independent sampling was going on with a size of Nev, when Nev is fixed according to the above formula.
Acknowledgements
Thanks to several anonymous referees for their helpful suggestions for improvements to this paper.
References
Beatty, J. [1984]: 'Chance and Natural Selection', Philosophy of Science, 51, pp. 183-211. Beatty, J. [1992]: 'Random Drift', in E. K. Keller and E. Lloyd (eds), Keywords in Evolutionary Biology, Cambridge, MA: Harvard University Press, pp. 273-281.
Brandon, R. [1990]: Adaptation and Environment, Princeton: Princeton University Press.
Brandon, R. [2005]: 'The Difference between Drift and Selection: A Reply to Millstein', Biology and Philosophy, 20(1), pp. 153-170.
Brandon, R. and Nijhout, F. [2006]: 'The Empirical Nonequivalence of Genic and Genotypic Models of Selection: A (Decisive) Refutation of Genic Selectionism and Pluralistic Genic Selectionism', Philosophy of Science, 73, pp. 277-297.
Brandon, R. N. and Carson, S. [1996]: 'The Indeterministic Character of Evolutionary Theory: No "No Hidden Variables Proof" but No Room for Determinism Either', Philosophy of Science, 63(3), pp. 315-337.
Buri, A. [1956]: 'Gene Frequency in Small Populations of Mutant Drosophila', Evolution, 10, pp. 367-402.
Christiansen, F. B. and Prout, T. [2000]: 'Aspects of Fitness', in R. S. Singh and C. B. Krimbas (eds), Evolutionary Genetics: From Molecules to Morphology, Cambridge: Cambridge
University Press, pp.
Ewens, W. J. [2004]: Mathematical Population Genetics: 1. Theoretical introduction, New York: Springer-Verlag.
Frankham, R. [1995]: 'Effective population size/adult population size ratios in wildlife: a review', Genetical Research, 66, pp. 95-107.
Gale, J. S. [1990]: Theoretical Population Genetics, London: Unwin Hyman Ltd.
Gillespie, J. [1998]: Population Genetics: A Concise Guide, Baltimore, Maryland: The Johns Hopkins University Press.
Gould, S. J. and Lewontin, R. C. [1979]: 'The Spandrels of San Marco and the Panglossian Paradigm', Proceedings of the Royal Society of London, 205(1161), pp. 581-598.
Hartl, D. and Clark, A. [2007]: Principles of Population Genetics, Sunderland, MA: Sinauer Associates.
Hedgecock, D. [1994]: 'Does variance in reproductive success limit effective population sizes of marine organisms?' in A. R. Beaumont (ed), Genetics and Evolution of Aquatic Organisms, London: Chapman & Hall, pp. 122-134.
Hedrick, P. W. [2005a]: Genetics of Populations, Boston: Jones and Bartlett.
Hedrick, P. W. [2005b]: 'Large Variance in Reproductive Success and the Ne/N Ratio', Evolution, 59(7), pp. 1596-1599.
Hill, W. G. [1972]: 'Effective Size of Populations with Overlapping Generations', Theoretical Population Biology, 3, pp. 278-289.
Hodge, M. J. S. [1987]: 'Natural Selection as a Causal, Empirical, and Probabilistic Theory', in L. Krüger (ed), The Probabilistic Revolution, Cambridge: MIT Press, pp. 43-62.
Imaizumi, Y., Nei, M. and Furusho, T. [1970]: 'Variability and Heritability of Human Fertility', Annals of Human Genetics, 33, pp. 251-259.
Jacquard, A. [1974]: The Genetic Structure of Populations, New York: Springer-Verlag.
Lewontin, R. C. [1970]: 'The Units of Selection', Annual Review of Ecology and Systematics(1), pp. 1-18.
Maynard Smith, J. [1987]: 'How to Model Evolution', in J. Dupre (ed), The Latest on the Best, Cambridge, Mass,: MIT Press, pp.
Millstein, R. [1996]: 'Random Drift and the Omniscient Viewpoint', Philosophy of Science, 63, pp. S10-S18.
Millstein, R. [2002]: 'Are Random Drift and Natural Selection Conceptually Distinct?' Biology and Philosophy, 17(1), pp. 33-53.
Millstein, R. [2003]: 'Interpretations of Probability in Evolutionary Theory', Philosophy of Science, 70, pp. 1317-1328.
Millstein, R. [2005]: 'Selection vs. Drift: a response to Brandon's reply', Biology and Philosophy, 20(1), pp. 171-175.
Millstein, R. [2006]: 'Natural Selection as a Population-Level Causal Process', British Journal for the Philosophy of Science, 57, pp. 627-653.
Okasha, S. [2006]: Evolution and the Levels of Selection, New York: Oxford University Press. Plutynski, A. [forthcoming]: 'Drift: A Historical and Conceptual Overview', Biological Theory. Reisman, K. and Forber, P. [2005]: 'Manipulation and the Causes of Evolution', Philosophy of Science, 72, pp. 1113-1123.
Rice, S. [2004]: Evolutionary Theory: Mathematical and Conceptual Foundations, Sunderland, MA: Sinauer and Associates.
Richardson, R. C. [2006]: 'Chance and Patterns of Drift: A Natural Experiment', Philosophy of Science, 73, pp. 642-654.
Robertson, A. [1962]: 'Selection for Heterozygotes in Small Populations', Genetics, 47, pp. 1291- 1300.
Rosenberg, A. and Bouchard, F. [2004]: 'Fitness, Probability and the Principles of Natural Selection', British Journal for the Philosophy of Science, 55, pp. 693-712.
Rosenberg, A. and Bouchard, F. [2005]: 'Matthen and Ariew's Obituary for Fitness: Reports of its Death Have Been Greatly Exaggerated', Biology and Philosophy, 20, pp. 343-353.
Roughgarden, J. [1979]: Theory of Population Genetics and Evolutionary Ecology, New York: MacMillan.
Santiago, E. and Caballero, A. [1995]: 'Effective Size of Populations under Selection', Genetics, 139, pp. 1013-1030.
Santiago, E. and Caballero, A. [1998]: 'Effective Size and Polymorphism of Linked Neutral Loci in Populations under Directional Selection', Genetics, 149, pp. 2105-2117.
Sterelny, K. I. M. [2003]: 'Last Will and Testament: Stephen Jay Gould's The Structure of Evolutionary Theory', Philosophy of Science, 70, pp. 255-263.
Walsh, D. M., Lewens, T. and Ariew, A. [2002]: 'The Trials of Life: Natural Selection and Random Drift', Philosophy of Science, 69(3), pp. 452-473.