Haldane's dilemma

Haldane's Dilemma refers to a paradox introduced by J. B. S. Haldane concerning the efficacy of natural selection. The dilemma is based upon the substitution cost introduced by Haldane in his 1957 paper The Cost of Natural Selection. It has been used by some creationists to show that "evolution could not have happened", though it is not a problem for evolutionary biology since it consists of a poor model of natural selection that makes invalid assumptions.

The 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, writes Haldane (p. 511),

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.

That is, the problem is for the cattle breeder is that keeping only those specimens with the desired qualities will lower the reproductive capability too much to keep a useful breeding stock.

Haldane states that this same problem arises with respect to natural selection. Characters that are positively correlated at one time may be negatively correlated at a later time, so simultaneous optimization of more than one character is a problem also in nature. And, as Haldane writes (loc. cit.)

[i]n this paper I shall try 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.

In faster breeding species there is less of a problem. Haldane mentions (loc. cit.) the peppered moth, Biston betularia, whose color is determined by two allele genes C and c. The CC and Cc moths are dark, while the cc moths are light. Against the originally pale lichens the darker moths were easier for birds to pick out, but in areas, where pollution has darkened the lichens, the cc moths had become rare. Haldane mentions that in a single day the frequency of of cc moths might be halved.

But even here there is a potential problem, if "ten other independently inherited characters had been subject to selection of the same intensity as that for colour, only $(1/2)^{10}$, 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.

Selection Intensity

Haldane proceeds to define (op. cit. p. 512) the intensity of selection regarding "juvenile survival" (that is, survival to reproductive age) as $I=$ ln $(s_{0}/S)$, where $s_{0}$ is the quotient of those with the optimal genotype (or genotype) that survive to reproduce, and $S$ is the quotient for the entire population. The quotient of deaths $1-S$ for the entire population would have been $1-s_{0}$, if all genotypes had survived as well as the optimal, hence $s_{0}-S$ is the quotient of deaths due to selection. As Haldane mentions, if $s_{0}$ ≈ $S$, then $I$ ≈ $s_{0}-S$ - since ln(1) = 0.

The Cost

At p. 514 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 $i^{th}$ selected locus is responsible for $d_{i}$ of these deaths in any generation the reproductive capacity of the species will be Π( 1 - $d_{i}$ ) of that of the optimal genotype, or exp( -Σ$d_{i}$ ) nearly, if every $d_{i}$ is small. Thus the intensity of selection approximates to Σ$d_{i}$.

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

The problem statement is therefore that the genes (actually alleles) in question are not particular beneficial under the previous circumstances; but a change in environment favors these genes by natural selection. The individuals without the genes are therefore disfavored, and the favorable genes spread in the population by the death (or lowered fertility) of the individuals without the genes. Note that Haldane's model as stated here allows for more than one gene to move towards fixation at a time; but each such will add to the cost of substitution.

The total cost of substitution of the $i^{th}$ gene is the sum $D_{i}$ of all values of $d_{i}$ over all generations of selection; that is, until fixation of the gene. Haldane states (loc. cit.) that he will show that $D_{i}$ depends mainly on $p_{0}$, the small frequency of the gene in question, as selection begins - that is, at the time that the environmental change occurs (or begins to occur).

Let A and a be two alleles with frequencies $p_{n}$ and $q_{n}$ in the $n^{th}$ generation. Their relative fitness is given by (cf. op. cit. p. 516)

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

where 0 ≤ $K$ ≤ 1, and 0 ≤ λ ≤ 1.

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

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

and the to tal number of deaths is the population size multiplied by

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

The Magic Number 300

Haldane (op. cit. p 517) approximates the above equation by

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

Assuming λ < 1, this gives

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

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

If λ = 1, then we have

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

In his discussion Haldane writes (op. cit. p. 520) 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.

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 hight intensities - such as in the case of the peppered moth - have occured frequently and estimates that a value of n = 300 is a probable number of generations. This gives a selection intensity of $I=30/300=0.1$.

Haldane then continues:

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 existing environment and the residual environment.

So the number 300 of generations is a conservative estimate for a slowly evolving species and not at the brink of extinction. For a difference of at least 1,000 genes, then 300,000 generations are needed - maybe more, if some gene runs through more than one optimization.

The Dilemma

According to Robert William (see Haldane's dilemma by Robert Williams), the term "Haldane's Dilemma" was first mentioned by paleontologist Leigh Van Valen in his 1963 paper "Haldane's Dilemma, Evolutionary Rates, and Heterosis". Van Valen saw the dilemma as the observation that "for most organisms, rapid turnover in a few genes precludes rapid turnover in the others." Haldane's dilemma has come to mean this limit upon the rate of evolution.

That is, since a high number of deaths are required to fix one gene rapidly, and dead organisms do not reproduce, fixation of two genes simultaneously would conflict. Note that Haldanes's model assumes independency of genes at different loci; if the selection intensity is 0.1 for each gene moving towards fixation, and there are N such genes, then the reproductive capacity of the species will be lowered to 0.9^N.

Walter ReMine and Haldane's Dilemma

Baraminologist Walter ReMine in his book The Biotic Message from 1993 introduces a somewhat different version of the dilemma.

Since the lines leading to hominids and the other apes are supposed to have split 10 million years ago, we find a maximum of 10⁷/(300×20) ≈ 1,667 genes fixed, if we assume 20 years between generations, much fewer than the 4.8×10⁷ genetic differences between humans and chimpanzees.

Note here that ReMine calculates with base pairs (point substitutions), not with whole genes. However, in Haldane's original formulation (p. 521 quoted above), he states that "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." So ReMine's formulation appears to not be compatible with Haldane's.

See also

• J.B.S. Haldane
• Genetic drift
• Journal of Genetics
• Error catastrophe

References

• Haldane, J.B.S., "The Cost of Natural Selection", J. Genet. 55:511-524, 1957.
• Van Valen, L., "Haldane’s Dilemma, evolutionary rates, and heterosis", Amer. Nat. 47:185-190, 1963.
• ReMine, W.J., The Biotic Message, St. Paul Science, Saint Paul, MN, pp. 208-236; 499-507, 1993.

External links

• Haldane's Dilemma by Walter ReMine.
• Haldane's Dilemma by Robert Williams.
• Rebuttal to Williams' arguments.
• talk.origins index to creationist claims, claim CB121.