# Lab Wiki

haldane_1937

### Haldane 1937

Haldane, J. B. S. (1937). The effect of variation of fitness. The American Naturalist, 71(735), 337-349.

Abstract: In a species in equilibrium variation is mainly due to two causes. Some deleterious genes are being weeded out by selection at the same rate as they are produced by mutation. Others are preserved because the heterozygous form is fitter than either homozygote. In the former case the loss of fitness in the species is roughly equal to the sum of all mutation rates and is probably of the order of 5 per cent. It is suggested that this loss of fitness is the price paid by a species for its capacity for further evolution.

Takeaway: Haldane considers the effects of deleterious mutations. Estimates are given for mutation-selection equilibrium allele frequencies and average reduction of fitness in the population. Fitness reduction is only a function of, and proportional to, mutation rates. The average individual in a population has a fitness that is a small fraction of the maximum theoretically possible if the genome were free of deleterious mutations.

#### Notes

##### Symbols
• $\mu$ is the mutation rate per locus per generation.
• $N$ is the population size.
• $x$ is the frequency of individuals carrying a copy of the mutant allele.
• $f$ is the relative average fitness of individuals carrying the mutant allele.
• $y$ is half the number of heterozygotes, which are $2y$.
• $p$ is the mutant allele frequency.
• $F$ is the total fitness of an individual over all loci subject to purifying selection in the genome.
• Here, $s$ is the reduction in fitness of the mutation. $s = 1-f$.
##### Dominant Fitness Effect

Pages 341–342 deal with a dominant fitness effect and the assumptions that the species is diploid, deleterious allele is rare (so most individuals with a copy only have one copy), and the unmutated fitness is one.

A fraction of the unmutated loci are expected to mutate each generation.

$$N\mu (2-x)$$

Mutated copies are removed by selection each generation ($f$ survive).

$$N x (1-f)$$

At equilibrium the rate of removal is equal to the rate of input.

$$N x (1-f) = N\mu (2-x)$$

$$Nx - Nxf = 2N\mu - N\mu x$$ $$Nx - Nxf + N\mu x = 2N\mu$$ $$x (N - Nf + N\mu) = 2N\mu$$ $$x = \frac{2N\mu}{N - Nf + N\mu} = \frac{2\mu}{1 - f + \mu}$$

Assuming $\mu$ is much smaller than one and that the mutant alleles are rare gives

$$x = 2 p (1-p) \approx 2 p \approx \frac{2\mu}{s}\mbox{.}$$

$$p \approx \frac{\mu}{s}\mbox{.}$$

The loss of fitness (from one) to the species is

$$x (1-f) = x - fx = \frac{2\mu}{1 - f + \mu} - \frac{2\mu f}{1 - f + \mu} = 2\mu \left(\frac{1-f}{1-f+\mu}\right) \approx 2\mu\mbox{.}$$

This is approximately $2\mu$ if $\mu$ is small relative to $1-f$. The interesting thing about this is that, at equilibrium, the loss of fitness is only a function of the mutation rate and is independent of the average fitness of individuals carrying the mutant alleles. The reason for this is the inverse relationship between the loss of fitness and the equilibrium frequency of the allele. Mutant alleles with a higher fitness attain a higher frequency in the population. So even though the loss of fitness is small more individuals are affected. And vice versa, Mutants with a large loss of fitness are maintained at a low frequency and fewer individuals are affected by the large loss. On average over an entire population these factors cancel out and the average loss of fitness is only twice the mutation rate (for dominant effects).

##### Recessive Fitness Effect

Page 344 treats the case of recessive loss of fitness.

A fraction of the unmutated loci are expected to mutate each generation. (Only half of the heterozygous loci are unmutated.)

$$2N\mu (1-x-y)$$

Mutated copies are removed by selection each generation. (Each loss removes two copies of the mutant alleles so $2x$.)

$$N 2 x (1-f)$$

At equilibrium

$$2N\mu (1-x-y) = N 2 x (1-f)$$ $$2N\mu -2N\mu x-2N\mu y = N 2 x - N 2 x f$$ $$2N\mu -2N\mu y = N 2 x - N 2 x f + 2N\mu x$$ $$2N\mu -2N\mu y = x (N 2 - N 2 f + 2N\mu )$$ $$x =\frac{2N\mu -2N\mu y}{ N 2 - N 2 f + 2N\mu }$$ $$x =\frac{\mu -\mu y}{ 1 - f + \mu } = \frac{\mu (1 - y)}{ 1 - f + \mu }$$

If $y$ and $\mu$ are small

$$x \approx \frac{\mu}{ 1 - f }$$

$$p^2 = x \approx \frac{\mu}{ 1 - f } = \frac{\mu}{s}$$ $$p \approx \sqrt{\frac{\mu}{s}}$$

The average loss of fitness in the population is

$$x (1-f) = x - fx \approx \frac{\mu}{1 - f} - \frac{\mu f}{1 - f} = \mu \left(\frac{1-f}{1-f}\right) = \mu\mbox{.}$$

Again, this is independent of the fitness effect and only a function of the mutation rate.

##### Multiple Loci

Pages 345–346 describe the predictions over all loci in the genome. Assuming mutations are independent both in occurrence and in fitness effects the average individual in the population has an expected fitness of

$$F = \prod_i (1-m_i)$$

where $m$ is the mutation rate $\mu$ at the $i$th locus for recessive mutations and $m=2\mu$ for dominant effects.

If the per nucleotide per generation mutation rate is $10^{-8}$, a tenth of our 3.3 billion base pair genome is under purifying selection, and the majority of mutations are recessive in fitness effects then

$$F \approx (1-10^{-8})^{3.3 \times 10^8}\approx 0.037\mbox{.}$$

Therefore, our fitness is predicted to be a small fraction, approximately 3–4%, of its theoretical maximum without mutations (both new mutations that have occurred in our own genomes and mutations that we have inherited from our ancestors).

##### Other things

Haldane discusses the effects of inbreeding and sex-linkage on these prediction and goes through quite a bit of additional logical details in the introduction as well as some examples from insects later in the paper.