Haldane's dilemma
Encyclopedia
Haldane's Dilemma refers to a limit on the speed of beneficial evolution, first calculated by J. B. S. Haldane
in 1957, and clarified further by later commentators. Creationists
, and proponents of intelligent design
in particular, claim it remains unresolved. Contrary to creationist claims, Haldane's dilemma is of no importance in the evolutionary genetics literature. Today, Haldane's Dilemma is raised mostly by creationists opposed to evolution, who claim it is evidence against large-scale evolution, and a supposed example of negligence on the part of the scientific community
.
Haldane stated at the time of publication "I am quite aware that my conclusions will probably need drastic revision", and subsequent corrected calculations found that the cost disappears. He had made an invalid simplifying assumption which negated his assumption of constant population size, and had also incorrectly assumed that two mutations would take twice as long to reach fixation as one, while sexual recombination means that two can be selected simultaneously so that both reach fixation more quickly. The creationist claim is based on further errors and invalid assumptions.
That is, the problem for the cattle breeder is that keeping only the 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.)
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 lichen
s 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 cc moths might be halved.
Another potential problem is that if "ten other independently inherited characters had been subject to selection of the same intensity as that for colour, only
, 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.
, where 
is the quotient of those with the optimal genotype (or genotype) that survive to reproduce, and 
is the quotient for the entire population. The quotient of deaths 
for the entire population would have been 
, if all genotypes had survived as well as the optimal, hence 
is the quotient of deaths due to selection. As Haldane mentions, if 
, then 
- since ln(1) = 0.
Comparing to the above, we have that
, if we say that 
is the quotient of deaths for the 
selected locus and 
is again the quotient of deaths for the entire population.
The problem statement is therefore that the genes (actually alleles) in question are not particularly 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
gene is the sum 
of all values of 
over all generations of selection; that is, until fixation of the gene. Haldane states (loc. cit.) that he will show that 
depends mainly on 
, 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).
and 
in the 
generation. Their relative fitness is given by (cf. op. cit. p. 516)
where 0 ≤
≤ 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
generation then is
and the total number of deaths is the population size multiplied by
substituting q=1-p, the cost (given by the total number of deaths, 'D', required to make a substitution) is given by
Assuming λ < 1, this gives
where the last approximation assumes
to be small.
If λ = 1, then we have
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 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
.
Haldane then continues (op. cit. p. 521):
The number 300 of generations is a conservative estimate for a slowly evolving species not at the brink of extinction by Haldane's calculation. For a difference of at least 1,000 genes, 300,000 generations might be needed - maybe more, if some gene runs through more than one optimization.
in his 1963 paper "Haldane's Dilemma, Evolutionary Rates, and Heterosis".
At p. 185 Van Valen writes:
That is, since a high number of deaths are required to fix one gene rapidly, and dead organisms do not reproduce, fixation of more than one gene simultaneously would conflict. Note that Haldane's model assumes independence 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 0.9N times the original capacity. Therefore, if it is necessary for the population to fix more than one gene, it may not have reproductive capacity to counter the deaths.
J. B. S. Haldane
John Burdon Sanderson Haldane FRS , known as Jack , was a British-born geneticist and evolutionary biologist. A staunch Marxist, he was critical of Britain's role in the Suez Crisis, and chose to leave Oxford and moved to India and became an Indian citizen...
in 1957, and clarified further by later commentators. Creationists
Creationism
Creationism is the religious beliefthat humanity, life, the Earth, and the universe are the creation of a supernatural being, most often referring to the Abrahamic god. As science developed from the 18th century onwards, various views developed which aimed to reconcile science with the Genesis...
, and proponents of intelligent design
Intelligent design
Intelligent design is the proposition that "certain features of the universe and of living things are best explained by an intelligent cause, not an undirected process such as natural selection." It is a form of creationism and a contemporary adaptation of the traditional teleological argument for...
in particular, claim it remains unresolved. Contrary to creationist claims, Haldane's dilemma is of no importance in the evolutionary genetics literature. Today, Haldane's Dilemma is raised mostly by creationists opposed to evolution, who claim it is evidence against large-scale evolution, and a supposed example of negligence on the part of the scientific community
Scientific community
The scientific community consists of the total body of scientists, its relationships and interactions. It is normally divided into "sub-communities" each working on a particular field within science. Objectivity is expected to be achieved by the scientific method...
.
Haldane stated at the time of publication "I am quite aware that my conclusions will probably need drastic revision", and subsequent corrected calculations found that the cost disappears. He had made an invalid simplifying assumption which negated his assumption of constant population size, and had also incorrectly assumed that two mutations would take twice as long to reach fixation as one, while sexual recombination means that two can be selected simultaneously so that both reach fixation more quickly. The creationist claim is based on further errors and 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 for the cattle breeder is that keeping only the 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
Peppered moth
The peppered moth is a temperate species of night-flying moth. Peppered moth evolution is often used by educators as an example of natural selection.- Distribution :...
, Biston betularia, whose color is determined by two allele
Allele
An allele is one of two or more forms of a gene or a genetic locus . "Allel" is an abbreviation of allelomorph. Sometimes, different alleles can result in different observable phenotypic traits, such as different pigmentation...
genes C and c. The CC and Cc moths are dark, while the cc moths are light. Against the originally pale lichen
Lichen
Lichens are composite organisms consisting of a symbiotic organism composed of a fungus with a photosynthetic partner , usually either a green alga or cyanobacterium...
s 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 cc moths might be halved.
Another potential problem is that if "ten other independently inherited characters had been subject to selection of the same intensity as that for colour, only
, 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, where is the quotient of those with the optimal genotype (or genotype) that survive to reproduce, and is the quotient for the entire population. The quotient of deaths for the entire population would have been , if all genotypes had survived as well as the optimal, hence is the quotient of deaths due to selection. As Haldane mentions, if , then - since ln(1) = 0.The cost
At p. 514 Haldane writesI 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 theselected locus is responsible for
of these deaths in any generation the reproductive capacity of the species will be
of that of the optimal genotype, or
nearly, if every
is small. Thus the intensity of selection approximates to
.
Comparing to the above, we have that
, if we say that is the quotient of deaths for the selected locus and is again the quotient of deaths for the entire population.The problem statement is therefore that the genes (actually alleles) in question are not particularly 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
gene is the sum of all values of over all generations of selection; that is, until fixation of the gene. Haldane states (loc. cit.) that he will show that depends mainly on , 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).A mathematical model of the cost of diploids
Let A and a be two alleles with frequencies and in the generation. Their relative fitness is given by (cf. op. cit. p. 516)
Genotype AA Aa aa
Frequency
Fitness 1
where 0 ≤
≤ 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
generation then is
and the total number of deaths is the population size multiplied by
The important number 300
Haldane (op. cit. p. 517) approximates the above equation by taking the continuum limit of the above equation. This is done by multiplying and dividing it by dq so that it is in integral formsubstituting q=1-p, the cost (given by the total number of deaths, 'D', required to make a substitution) is given by
Assuming λ < 1, this gives
where the last approximation assumes
to be small.If λ = 1, then we have
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 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
.Haldane then continues (op. cit. p. 521):
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.
The number 300 of generations is a conservative estimate for a slowly evolving species not at the brink of extinction by Haldane's calculation. For a difference of at least 1,000 genes, 300,000 generations might be needed - maybe more, if some gene runs through more than one optimization.
Origin of the term "Haldane's Dilemma"
Apparently the first use of the term "Haldane's Dilemma" was by paleontologist Leigh Van ValenLeigh Van Valen
Leigh Maiorana Van Valen was an American evolutionary biologist. He was professor emeritus in the Department of Ecology and Evolution at the University of Chicago....
in his 1963 paper "Haldane's Dilemma, Evolutionary Rates, and Heterosis".
At p. 185 Van Valen writes:
Haldane (1957 [= The Cost of Natural Selection]) drew attention to the fact that in the process of the evolutionary substitution of one allele for another, at any intensity of selection and no matter how slight the importance of the locus, a substantial number of individuals would usually be lost because they did not already possess the new allele. Kimura (1960, 1961) has referred to this loss as the substitutional (or evolutional) load, but because it necessarily involves either a completely new mutation or (more usually) previous change in the environment or the genome, I like to think of it as a dilemma for the population: for most organisms, rapid turnover in a few genes precludes rapid turnover in the others. A corollary of this is that, if an environmental change occurs that necessitates the rather rapid replacement of several genes if a population is to survive, the population becomes extinct.
That is, since a high number of deaths are required to fix one gene rapidly, and dead organisms do not reproduce, fixation of more than one gene simultaneously would conflict. Note that Haldane's model assumes independence 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 0.9N times the original capacity. Therefore, if it is necessary for the population to fix more than one gene, it may not have reproductive capacity to counter the deaths.
See also
- J.B.S. Haldane
- Genetic driftGenetic driftGenetic drift or allelic drift is the change in the frequency of a gene variant in a population due to random sampling.The alleles in the offspring are a sample of those in the parents, and chance has a role in determining whether a given individual survives and reproduces...
- Journal of GeneticsJournal of GeneticsThe Journal of Genetics is a peer-reviewed scientific journal in the field of genetics and evolution. It was founded in 1910 by the British geneticists William Bateson and Reginald Punnett and is one of the oldest genetics journals. It was later edited by J.B.S...
- Error catastropheError catastropheError catastrophe is a term used to describe the extinction of an organism as a result of excessive RNA mutations. The term specifically refers to the predictions of mathematical models similar to that described below, and not to an observed phenomenon.Like every organism, viruses 'make mistakes'...
- Genetic LoadGenetic loadIn population genetics, genetic load or genetic burden is a measure of the cost of lost alleles due to selection or mutation...
- Swamping argumentSwamping argumentSwamping argument is an objection against Darwinism made by Fleeming Jenkin. He asserts that accidentally appeared profitable variety can't be preserved by natural selection in population, but should be 'swamped' with ordinary traits...