Haldane's dilemma

Substitution cost

In the introduction to The Cost of Natural Selection Haldane writes that it is difficult for breeders to simultaneously select all the desired qualities, partly because the required genes may not be found together in the stock; but, he writes,^[5]

especially in slowly breeding animals such as cattle, one cannot cull even half the females, even though only one in a hundred of them combines the various qualities desired.^[5]

That is, the problem for the cattle breeder is that breeding only from the specimens with the desired qualities will yield too few individuals in the next generation. Haldane states that this same problem arises with respect to natural selection. The problem is greatest for 'K-selected' species with low fecundity, and is less of an issue for 'r-selected' species with high fecundity. Haldane attempts ^[5]

to make quantitative the fairly obvious statement that natural selection cannot occur with great intensity for a number of characters at once unless they happen to be controlled by the same genes.^[5]

In the case of selection on peppered moths, if "ten other independently inherited characters had been subject to selection of the same intensity as that for colour, only ${\displaystyle (1/2)^{10}}$, or one in 1024, of the original genotype would have survived." The species would most likely have become extinct; but it might well survive ten other selective periods of comparable selectivity, if they happened in different centuries.^[5]

Selection intensity

Haldane proceeds to define the intensity of selection regarding "juvenile survival" (that is, survival to reproductive age) as ${\displaystyle I=\ln(s_{0}/S)}$, where ${\displaystyle s_{0}}$ is the proportion of those with the optimal genotype (or genotypes) that survive to reproduce, and ${\displaystyle S}$ is the proportion of the entire population that similarly so survive. The proportion for the entire population that die without reproducing is thus ${\displaystyle 1-S}$, and this would have been ${\displaystyle 1-s_{0}}$ if all genotypes had survived as well as the optimal. Hence ${\displaystyle s_{0}-S}$ is the proportion of "genetic" deaths due to selection. As Haldane mentions, if ${\displaystyle s_{0}\approx S}$, then ${\displaystyle I\approx s_{0}-S}$.^[6]

Haldane's cost

Haldane writes

I shall investigate the following case mathematically. A population is in equilibrium under selection and mutation. One or more genes are rare because their appearance by mutation is balanced by natural selection. A sudden change occurs in the environment, for example, pollution by smoke, a change of climate, the introduction of a new food source, predator, or pathogen, and above all migration to a new habitat. It will be shown later that the general conclusions are not affected if the change is slow. The species is less adapted to the new environment, and its reproductive capacity is lowered. It is gradually improved as a result of natural selection. But meanwhile, a number of deaths, or their equivalents in lowered fertility, have occurred. If selection at the ${\displaystyle i^{th}}$ selected locus is responsible for ${\displaystyle d_{i}}$ of these deaths in any generation the reproductive capacity of the species will be ${\displaystyle \prod \left(1-d_{i}\right)}$ of that of the optimal genotype, or ${\displaystyle \exp \left(-\sum d_{i}\right)}$ nearly, if every ${\displaystyle d_{i}}$ is small. Thus the intensity of selection approximates to ${\displaystyle \sum d_{i}}$.^[5]

Comparing to the above, we have that ${\displaystyle d_{i}=s_{0i}-S}$, if we say that ${\displaystyle s_{0i}}$ is the quotient of deaths for the ${\displaystyle i^{th}}$ selected locus and ${\displaystyle S}$ is again the quotient of deaths for the entire population.

Note that Haldane's model as stated here allows for more than one gene to move towards fixation at a time; but each of them adds to the cost of substitution.

Haldane states that he will show that ${\displaystyle D_{i}}$ depends mainly on ${\displaystyle p_{0}}$, the low frequency at which the allele is at the time of environmental change.^[5]

A mathematical model of the cost of diploids

Let A and a be two alleles with frequencies ${\displaystyle p_{n}}$ and ${\displaystyle q_{n}}$ in the ${\displaystyle n^{\mbox{th}}}$ generation. Their relative fitness is given by^[5]

Genotype	AA	Aa	aa
Frequency	${\displaystyle p_{n}^{2}}$	${\displaystyle 2p_{n}q_{n}}$	${\displaystyle q_{n}^{2}}$
Fitness	1	${\displaystyle 1-\lambda K}$	${\displaystyle 1-K}$

where 0 ≤ ${\displaystyle K}$ ≤ 1, and 0 ≤ λ ≤ 1.

If λ = 0, then Aa has the same fitness as AA, e.g. if Aa is phenotypically equivalent with AA (A dominant), and if λ = 1, then Aa has the same fitness as aa, e.g. if Aa is phenotypically equivalent with aa (A recessive). In general λ indicates how close in fitness Aa is to aa.

The fraction of selective deaths in the ${\displaystyle n^{\mbox{th}}}$ generation then is

${\displaystyle d_{n}=2\lambda Kp_{n}q_{n}+Kq_{n}^{2}=Kq_{n}[2\lambda +(1-2\lambda )q_{n}]}$

and the total number of deaths is the population size multiplied by

${\displaystyle D=K\sum _{0}^{\infty }q_{n}\;[2\lambda +(1-2\lambda )q_{n}].}$

Origin of the number 300

Haldane approximates the above equation by taking the continuum limit of the above equation.^[5] This is done by multiplying and dividing it by dq so that it is in integral form

${\displaystyle dq_{n}=-Kp_{n}q_{n}[\lambda +(1-2\lambda )q_{n}]}$

substituting q=1-p, the cost (given by the total number of deaths, 'D', required to make a substitution) is given by

${\displaystyle D=\int _{0}^{q_{_{0}}}{\frac {[2\lambda +(1-2\lambda )q]}{(1-q)[\lambda +(1-2\lambda )q]}}dq={\frac {1}{1-\lambda }}\int _{0}^{q_{_{0}}}\left[{\frac {1}{1-q}}+{\frac {\lambda (1-2\lambda )}{\lambda +(1-2\lambda )q}}\right]dq.}$

Assuming λ < 1, this gives

${\displaystyle D={\frac {1}{1-\lambda }}\left[-{\mbox{ln }}p_{0}+\lambda {\mbox{ ln }}\left({\frac {1-\lambda -(1-2\lambda )p_{0}}{\lambda }}\right)\right]\approx {\frac {1}{1-\lambda }}\left[-{\mbox{ln }}p_{0}+\lambda {\mbox{ ln }}\left({\frac {1-\lambda }{\lambda }}\right)\right]}$

where the last approximation assumes ${\displaystyle p_{0}}$ to be small.

If λ = 1, then we have

${\displaystyle D=\int _{0}^{q_{_{0}}}{\frac {2-q}{(1-q)^{2}}}dq=\int _{0}^{q_{_{0}}}\left[{\frac {1}{1-q}}+{\frac {1}{(1-q)^{2}}}\right]dq=p_{0}^{-1}-{\mbox{ ln }}p_{0}+O(\lambda K).}$

In his discussion Haldane writes that the substitution cost, if it is paid by juvenile deaths, "usually involves a number of deaths equal to about 10 or 20 times the number in a generation" – the minimum being the population size (= "the number in a generation") and rarely being 100 times that number. Haldane assumes 30 to be the mean value.^[5]

Assuming substitution of genes to take place slowly, one gene at a time over n generations, the fitness of the species will fall below the optimum (achieved when the substitution is complete) by a factor of about 30/n, so long as this is small – small enough to prevent extinction. Haldane doubts that high intensities – such as in the case of the peppered moth – have occurred frequently and estimates that a value of n = 300 is a probable number of generations. This gives a selection intensity of ${\displaystyle I=30/300=0.1}$.

Haldane then continues:^[5]

The number of loci in a vertebrate species has been estimated at about 40,000. 'Good' species, even when closely related, may differ at several thousand loci, even if the differences at most of them are very slight. But it takes as many deaths, or their equivalents, to replace a gene by one producing a barely distinguishable phenotype as by one producing a very different one. If two species differ at 1000 loci, and the mean rate of gene substitution, as has been suggested, is one per 300 generations, it will take 300,000 generations to generate an interspecific difference. It may take a good deal more, for if an allele a1 is replaced by a10, the population may pass through stages where the commonest genotype is a1a1, a2a2, a3a3, and so on, successively, the various alleles in turn giving maximal fitness in the existing environment and the residual environment.^[5]