On natural selection, genetic fitness and the role of math

I really am not quite sure what to make of this one.

Last week at the blog Dynamic Ecology it was argued that natural selection behaves like a “risk-averse” money investor. That is, assuming that fitness varies over time (due to e.g. changing environmental variables or other selective factors), natural selection favors situations in which the mean fitness is maximized while the variance is minimized. The idea is explained in this short paper by Orr (2007), whose goal was to explain previous findings (Gillespie, 1973) intuitively. This presumes that knowledge of investor behavior is commonplace, but for my money, an examination of the math details and assumptions is what’s really needed.

This conclusion seems entirely problematic to me.


Some background. Biological fitness, most generally, is the directional change in a population variable over time, due to the effect of natural selection. In population genetics it is applied to heritable genetic variants, i.e. genotypes or alleles, at one or more genetic loci. There are both absolute and relative fitness definitions. Absolute fitness, call if F, is just like an interest rate: a fractional increase or decrease of a starting quantity, per biological generation (g), above 1.0 if increasing and below if decreasing. If F is constant, it’s then an exponential function with base F and exponent g, F^g. Allele proportions are always fractions defined on [0,1] and must collectively sum to 1.0. Relative fitness standardizes absolute fitness, to either the mean or the maximum, of all the absolute fitnesses. Orr (2007) uses the simplest possible situation for explanation: one genetic locus having just two alleles in the population, equal starting frequencies, changes over time due to selection only, just one allele per individual (i.e. haploids, to avoid the complications of mating), non-overlapping generations, and constant population size.

What seems to me a major problem is the conclusion that natural selection is “risk-averse” based strictly on the existing mathematics. This is what Orr and others have concluded. The claim is that if the fitness of alleles varies over time, natural selection will (attempt to?, should?, must?, does?) favor that allele which both maximizes the fitness mean and minimizes the variance. Depending on just what you read, it’s not always clear exactly what this claim refers to: relative or absolute fitness (and how each are defined), and arithmetic or geometric mean. And unfortunately, I don’t have access to Gillespie’s original paper to consult to see what he said.

If you instead just use absolute fitness and the geometric mean*, it’s easy to demonstrate that this claim is wrong, which in turn means that claiming “risk-aversion” to be an inherent property of natural selection is not correct. It’s instead just an artifact of the mathematics used. This is because the value of a multiplicative series depends only on the geometric mean of that series (= the nth root of the product, where n = series length), which can be produced in infinitely many ways. How much the series varies in getting there is entirely irrelevant. Obviously.

If you assume their conclusion you will have to eventually explain, at lower complexity levels (ultimately, the molecular), just how this could work. It’s been a long while since I looked at the literature on that topic, but I think it would be hard to explain just how an allele could maintain a high-mean, low-variance fitness over a wide range of truly selective conditions. It would arguably be just as likely, possibly moreso, that such a finding would result from either (1) not actually realizing the latter conditions, or (2) testing a gene (or heritable phenotype) that is not in fact responsive to the selective factor(s) you’ve identified. Both issues are tied to the very much more important question of whether or not one can reliably identify selection operating in wild populations, where major violations of these super simple theoretical conditions, not to mention other sources of noise, always exist.

The following results demonstrate the point. I simulate Orr’s conditions (see above), tested with 20 trials of g = 50 generations randomly sampled from each of two fitness distributions having very different variances but identical means. The starting allele numbers are 50 for each allele, thus frequencies equal at a = b = 0.50. If their claim is right, trials with lower fitness variance should show a greater relative increase, for allele a, than those with a higher variance (since the mean is unchanged). Allele a is the focus since it has higher mean fitness, but I track both.

Added Note. You cannot just set the variance around a geometric mean by specifying a symmetric distribution around the mean, such as the normal: that will alter (lower) the mean. You must instead do so with the logarithms thereof. Also, the mean geometric fitness values of the two alleles can have any possible relative magnitudes to each other. Code and results updated for clarity.

      mean.a mean.b  sd.a  sd.b final.a final.b rel.a rel.b
 [1,] 1.0094 1.0037 0.855 1.003      80      60 0.571 0.429
 [2,] 1.0094 1.0037 0.769 0.650      80      60 0.571 0.429
 [3,] 1.0094 1.0037 0.900 0.937      80      60 0.571 0.429
 [4,] 1.0094 1.0037 0.946 0.675      80      60 0.571 0.429
 [5,] 1.0094 1.0037 0.940 0.918      80      60 0.571 0.429
 [6,] 1.0094 1.0037 1.065 0.834      80      60 0.571 0.429
 [7,] 1.0094 1.0037 0.720 0.937      80      60 0.571 0.429
 [8,] 1.0094 1.0037 0.775 0.853      80      60 0.571 0.429
 [9,] 1.0094 1.0037 0.717 0.931      80      60 0.571 0.429
[10,] 1.0094 1.0037 0.604 0.829      80      60 0.571 0.429
[11,] 1.0094 1.0037 0.885 0.735      80      60 0.571 0.429
[12,] 1.0094 1.0037 0.627 0.796      80      60 0.571 0.429
[13,] 1.0094 1.0037 0.727 0.742      80      60 0.571 0.429
[14,] 1.0094 1.0037 0.880 0.764      80      60 0.571 0.429
[15,] 1.0094 1.0037 0.971 0.780      80      60 0.571 0.429
[16,] 1.0094 1.0037 0.864 0.940      80      60 0.571 0.429
[17,] 1.0094 1.0037 0.786 0.551      80      60 0.571 0.429
[18,] 1.0094 1.0037 1.026 0.872      80      60 0.571 0.429
[19,] 1.0094 1.0037 0.775 1.002      80      60 0.571 0.429
[20,] 1.0094 1.0037 0.853 0.734      80      60 0.571 0.429
[21,] 1.0094 1.0037 0.180 0.199      80      60 0.571 0.429
[22,] 1.0094 1.0037 0.137 0.151      80      60 0.571 0.429
[23,] 1.0094 1.0037 0.166 0.187      80      60 0.571 0.429
[24,] 1.0094 1.0037 0.178 0.171      80      60 0.571 0.429
[25,] 1.0094 1.0037 0.174 0.169      80      60 0.571 0.429
[26,] 1.0094 1.0037 0.183 0.186      80      60 0.571 0.429
[27,] 1.0094 1.0037 0.144 0.204      80      60 0.571 0.429
[28,] 1.0094 1.0037 0.170 0.189      80      60 0.571 0.429
[29,] 1.0094 1.0037 0.164 0.149      80      60 0.571 0.429
[30,] 1.0094 1.0037 0.152 0.121      80      60 0.571 0.429
[31,] 1.0094 1.0037 0.160 0.213      80      60 0.571 0.429
[32,] 1.0094 1.0037 0.159 0.174      80      60 0.571 0.429
[33,] 1.0094 1.0037 0.157 0.183      80      60 0.571 0.429
[34,] 1.0094 1.0037 0.149 0.159      80      60 0.571 0.429
[35,] 1.0094 1.0037 0.159 0.171      80      60 0.571 0.429
[36,] 1.0094 1.0037 0.172 0.172      80      60 0.571 0.429
[37,] 1.0094 1.0037 0.153 0.198      80      60 0.571 0.429
[38,] 1.0094 1.0037 0.175 0.194      80      60 0.571 0.429
[39,] 1.0094 1.0037 0.177 0.156      80      60 0.571 0.429
[40,] 1.0094 1.0037 0.136 0.194      80      60 0.571 0.429

First 20 rows are high variance trials and 2nd 20 are low. First pair of columns are the geometric mean fitnesses, second pair the std deviations, third pair the (rounded) final allele numbers, and fourth pair the relative frequencies thereof.

Note that all results are exactly identical, even though the standard deviations differ between trials by a factor of 1.12/0.12 = 9.3. So if the problem is defined and analyzed in this way, instead of defining absolute fitnesses based on changes in allele frequencies (Orr, 2007), and/or using arithmetic means to define relative fitnesses derived from them, then variance becomes irrelevant. Note however that, if the mean fitnesses are close to each other and the variances large, as they are here, there can be long sequences of years in which the asymptotically less fit allele appears to be more fit. This point leads straight into the very tough issues involving actually detecting and measuring real fitness differences in the real messy real world.

*R code:

## Script tests the claims of Orr 2007 regarding natural selection being risk averse
# Orr (2007) doi:10.1111/j.1558-5646.2007.00237.x; Evolution 61-12: 2997–3000

# 1. Parameters and templates
options(digits=5)
a.1 = 50; b.1 = 50						# initial allele sizes
a.mult = 1.6; b.mult = 1.2					# factors by which a and b change, over g generations; values > 1 = increase
a.g = a.1*a.mult; b.g=b.1*b.mult				# ending allele sizes
F.a=end.a^(1/g); F.b=end.b^(1/g)				# geometric mean, per generation, absolute fitnesses, (F.a^g = end.a)
sd.scaler = seq(1.0, 0.25, -0.75)				# scales the fitness std deviations
probs = seq(0.005, 0.995, 0.005)				# for getting the cdf quantiles
trials = 20								# trials per experiment

n = trials*length(sd.scaler)
results = matrix(NA, n, ncol=8)
colnames(results) =c("mean.a","mean.b","sd.a","sd.b","final.a","final.b","rel.a","rel.b")

# 2. Test
c=0
for (i in 1:(n/trials)){
# fitness rate distributions; must preserve geometric mean in each random sample:
# random sample g/2 times, take differences from mean, join the two 
 sd.a=sd.scaler[i]*F.a/3; sd.b=sd.scaler[i]*F.b/3					# sets the sd around the geometric mean
 quants.a = qnorm(probs, mean=log(F.a), sd=sd.a)
 quants.b = qnorm(probs, mean=log(F.b), sd=sd.b)
 for (j in 1:trials){
  c=c+1
  samp.a1 = sample(quants.a, size=g/2, replace=T)
  samp.a2 = log(F.a) - samp.a1; samp.a = c(samp.a1, samp.a2)
  samp.b1 = sample(quants.b, size=g/2, replace=T)
  samp.b2 = log(F.b) - samp.b1; samp.b = c(samp.b1, samp.b2)
  fitnesses.a = exp(2*samp.a); fitnesses.b = exp(2*samp.b)
  F.a == (prod(fitnesses.a))^(1/g); F.b == (prod(fitnesses.b))^(1/g)		# tests; should be equal
  sd.a = sd(fitnesses.a); sd.b = sd(fitnesses.b)
  pa.g = a.g/(a.g+b.g); pb.g = b.g/(a.g+b.g)
  results[c,1:8] = c(F.a,F.b,sd.a,sd.b,a.g,b.g,pa.g,pb.g)
 }
}
results[,c(3,4,7,8)] = round(results[,c(3,4,7,8)], 3); results

Refs:
Gillespie, J.H. (1973). Natural selection with varying selection coefficients—a haploid model. Genet. Res. 21:115–120.

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