Haldane's Dilemma

Preamble

The paper by Haldane (1957) discusses and attempts to quantify the rate at which otherwise harmful genetic mutations become fixed within an animal population due to natural selection by way of environmental change which favours individuals possessing the mutation. Based on his calculations, Haldane determined that an animal population can only fix this mutation in the population at a rate of 300 generations, at best, and avoid the collapse of the species. The implication being, by my opponent, that there would not be enough time for certain species (ie. humans from the other great apes) to have successfully diverged in the time frame mainstream science says that they have.

The Haldane (1957) paper, according to my H-Index Calculator, has been cited 5613 times and has an h-index >10 and a g-index >10.

Methodology

Haldene first begins by looking at Haploid, Clonal or Self-Fertilizing organisms, or Maternally passed cytoplasmic traits. Stating that if A and a represent allalomorphic genes in these populations then for the representative nth population, prior to selection, we have the frequencies:

p_nA, q_na where p_n + q_n = 1.

The cost incurred to fix a gene in the population of such organisms is calculated to be:

The deaths over all generations for the species in question which Haldane calculated to be between 5-15 times the number of species members of a population each generation, 10 being typical.

Then Haldane looks at autosomal locus with pairs A and a in diploids giving their fitness as:

AA frequency: p_n² fitness: 1

Aa frequency: 2p_nq_n fitness: 1-k

aa frequency: q_n² fitness: 1-K

For the members of these species the total deaths over all generations was calculated to be between 10-100 times the number of members each generation with 30 being typical.

Haldane then examines autosomal locus in inbreeding diploids with the following frequency table:

AA frequency: p_n²+fp_nq_n

Aa frequency: 2(1-k)(1-f)p_nq_n

aa frequency: (1-K)(fp_nq_n+q_n²)

Here the deaths of heterozygotes can be ignored if the organisms employ full inbreeding and the cost is similar to that of in the first group of organisms. If there is partial inbreeding then the cost is slightly less than the first group and the breeding coefficient f is introduced where f>0. The cost is not significantly reduced unless A is recessive.

After this Haldane looks sex-linked characteristics in diploids. The frequency and fitness table produced is:

AA frequency: p_n² fitness: 1

Aa frequency: p_nq_n fitness: 1-k

aa frequency: q_n² fitness: 1-K

A frequency: p_n fitness: 1

a frequency: q_n fitness: 1-l

For this group Haldane calculates the death rate over all generations to carry a cost of 10-40 times the population of a generation where 20 is typical.

Lastly Haldane examines the cost to heterozygotes.According to Haldane's calculations the cost to heterozygotes is small and therefore little fixation can occur.

Discussion

Haldane then concludes that the rate of change, for adapting a previously harmful gene to survive, over 300 generations is reasonable because if change occurred over too few generations this would render the gene unstable and the population would languish and collapse.

Interestingly, Haldane makes the assertion that multiple mutations would take the same amount to fix simultaneously as if they were fixed in sequence. Literally, Haldane claims that three mutations being selected for will take three times as long.

Can this slowness be avoided by selecting several genes at a time? I doubt it, for the following reason. Consider clonally reproducing bacteria, in which a number of disadvantageous genes are present, kept in being by mutation, each with frequencies of the order of 10^-4. They become slightly advantageous through a change of environment or residual genotype. Among 10¹² bacteria there might be one which possessed three such mutants. But since the cost of selection is proportional to the negative logarithm of the initial frequency the mean cost of selecting its descendants would be the same as that of selection for the three mutants in series, though the process might be quicker. The same argument applies to mutants linked by an inversion. Once several favourable mutants are so linked the inversion may be quickly selected. But the rarity of inversions containing several rare and favourable mutants will leave the cost unaltered. (Haldane, 1957, p. 522)

However, aside from mentioning the negative logarithm Haldane offers no good reason why multiple mutations could not be fixed any faster than one mutation. In his example, Haldane cites bacteria that reproduce clonally. Organisms that utilize sexual reproduction multiple mutations can be selected simultaneously, via sexual recombination, and likely have those traits fixed in the population sooner.

Furthermore, there is a problem in Haldane's calculations. In his paper, Haldane uses 1-k to represent the fitness of gene a. If k = 0.01 then the fitness j of gene a works out to be, by Haldane's calculations, j = 1 + 0.01 = 1.01. Using Haldan's equation with this fitness value, if population N = 100 000, the frequency that A is passed to next generations is p = 1/N = 1e-5 and the frequency that a is passed to the next generations q = 1 - p = 0.99999 we get the following value for deaths.

(q^2/j)*N = ((0.99999)^2)/1.01)*100 000 = 9.90e4

So a bit over 99 000 individuals need to die to pass a to the next generations and eventually cause fixation, 99 009 with better precision. To put that into perspective; 99 009/100 000 * 100 =~ 99%! Haldane perpetuates this error throughout his paper. The point is, there is no reasonable expectation that individuals carrying A have to die, particularly at such rates, but Haldane kept doing so because he normalized his fitness factor incorrectly by counting the previously killed off carriers of A in the calculation for the next generation.

Fixing a gene in a population is not restricted to adopting a previously harmful mutation either. Here, again, the fixation could possibly occur at a higher rate without destabilizing the population as noted by Valen (1963).

Dodson (1962) seized on this estimate of 300 generations, applied it to evolution within the genus Homo, and needless to say for this case, found a poor fit with observed and inferred facts. The most probable interpretation of this difference is that much of the evolution of this genus has been such to present no dilemma to the populations. The most commonly noted difference between successive species, and one that is probably responsible for some of the other differences, is the increase in brain size and presumably intelligence. There is no reason to believe that the part of this increase that occurred in the transition from H. erectus to H. sapiens was a direct adaptation to middle Pliestocene environmental changes (that is, would not have been valuable to the environments of Homo erectus), although it is conceivable that environmental changes (such as possible changes in predators and food) increased the selective advantage for greater intelligence or lowered a threshold of disadvantage for the cocomitant structural disturbance of the skull. The other functional complex known to have changed importantly is posture (and presumably locomotion); it is very possible that the only difficulties in the change of this complex also, were the reorganizations of the anatomy and the genome and not a greater than previous loss of individuals with the ancestral genotype. (Valen, 1963, p. 186-187)

Conclusions

It must first be said that Haldane was not presenting any fault with evolution, nor was he in any way defending Creationism or its poorly disguised brother Intelligent Design. Haldane, in his paper, was trying to come up with a means to calculate how quickly a population evolves if survival meant adopting a previously harmful gene. Nor is there any discussion or other indication about genomes decaying.

Secondly, Haldane's estimate of 300 generations was falsified by Dodson when he compared changes in genus Homo. There is no empirical grounds to accept 300 generations as sacrosanct and suggests that there is no dilemma.

Thirdly, Haldane's calculations are erroneous because he did not properly normalize his fitness factor correctly. Why should individual carrying gene A need to die in such large numbers to fix gene a? Clearly, the empirical evidence suggests that this does not happen on the norm and yet evolution proceeds unabated.

Lastly, although much more can probably yet be said, Haldane was a scientist and admitted that his work probably was not without error. His efforts, nonetheless, helped scientists better understand the evolutionary process. This is how science works!

To conclude, I am quite aware that my conclusions will probably need drastic revision. But I am convinced that quantitative arguments of the kind here put forward should play a part in all future discussions of evolution. (Haldane, 1957, p. 523)

Questions to my Opponent

How did you reach the conclusion that this paper supported a decaying genome?

What is it about the 300 generations estimate that you find significant seeing that it is evidently incorrect?

Did you actually spend time reading the paper?

Bibliography

• The Cost of Natural Selection Haldane, 1957

Haldane's Dilemma, Evolutionary Rates, and Heterosis Valen, 1963