Abstract: A recent attempt was made to resolve the heretofore unaddressed issue of the estimated time for evolution, concluding that there was plenty of time. This would have been a very significant result had it been correct. It turns out, however, that the assumptions made in formulating the model of evolution were faulty and the conclusion of that attempt is therefore unsubstantiated.
he standard neo-Darwinian theory accounts for evolution as the result of long sequences of random mutations each filtered by natural selection. The random nature of this basic mechanism makes evolutionary events random. The theory must therefore be judged by estimating the probabilities of those events. This probability calculation has,
however, not yet been addressed to justify the theory.
Wilf & Ewens [W&E] recently attempted to address this issue, but their attempt was unsuccessful. Their model of the evolutionary process omitted important features of evolution invalidating their conclusions. They considered a genome consisting of L loci (genes), and an evolutionary process in which each allele at these loci would eventually mutate so that the final genome would be of a more "superior" or "advanced" type. They let K-1 be the fraction of potential alleles at each gene locus that would contribute to the "superior" genome. They modeled the evolutionary process as a random guessing of the letters of a word. The word has L letters in an alphabet of K letters. In each round of guessing, each letter can be changed and could be converted to a "superior" letter with probability K-1.
At the outset they stated the two goals of their study, neither of which they achieved. Their first goal was to "to indicate why an evolutionary model often used to 'discredit' Darwin, leading to the 'not enough time' claim, is inappropriate." Their second goal was "to find the mathematical properties of a more appropriate model." They described what they called the "inappropriate model" as follows:
"The paradigm used in the incorrect argument is often formalized as follows: Suppose that we are trying to find a specific unknown word of L letters, each of the letters having been chosen from an alphabet of K letters. We want to find the word by means of a sequence of rounds of guessing letters. A single round consists in guessing all of the letters of the word by choosing, for each letter, a randomly chosen letter from the alphabet. If the correct word is not found, a new sequence is guessed, and the procedure is continued until the correct sequence is found. Under this paradigm the mean number of rounds of guessing until the correct sequence is found is indeed KL."
They gave no reference for such a model and, to my knowledge, no responsible person has ever proposed such a model for the evolutionary process to "discredit" Darwin. Such a model had indeed been suggested by many, not for the evolutionary process, but for abiogenesis [e.g., ] where it is indeed appropriate. Their first goal was not achieved.
They then described their own model, which they called "a more appropriate model." On the basis of their model, they concluded that the mean time for evolution increases as K log L, in contrast to KL of the "inappropriate" model. They called the first model "serial" and said that their "more correct" model of evolution was "parallel". Their characterization of "serial" and "parallel" for the above two models is mistaken. Evolution is a serial process, not a parallel one, and their model of the first, or "inappropriate", process is better characterized as "simultaneous" than "serial" because the choosing of the sequence (either nucleotides or amino acids) is simultaneous. What they called their "more appropriate" model is the following:
"After guessing each of the letters, we are told which (if any) of the guessed letters are correct, and then those letters are retained. The second round of guessing is applied only for the incorrect letters that remain after this first round, and so forth. This procedure mimics the 'in parallel' evolutionary process."
W&E were mistaken in thinking the evolutionary process to be an in-parallel one — it is an in-series one. A rare adaptive mutation may occur in one locus of the genome of a gamete of some individual, will become manifest in the genome of a single individual of the next generation, and will be heritable to future generations. If this mutation grants the individual an advantage leading to it having more progeny than its nonmutated contemporaries, the new genome's representation in the population will tend to increase exponentially and eventually it may take over the population.
Let p be the probability that in a particular generation (1) an adaptive mutation will occur in some individual in the population and (2) the mutated genome will eventually take over the population. If both these should happen, then we could say that one evolutionary step has occurred. The mean waiting time for the appearance of such a mutation is 1/p . After the successful adaptive mutation has taken over the population, the appearance of another adaptive mutation can start another step. In L steps of this kind, L new alleles will be incorporated into the mean genome of the population. These steps occur in series and the mean waiting time for L such steps is just L times the waiting time for one of them, or L/p. Thus the number of generations needed to modify L alleles is linear in L and not logarithmic as concluded from the flawed analysis of W&E.
The flaws in the analysis of W&E lie in the faulty assumptions on which their model is based. The "word" that is the target of the guessing game is meant to play the role of the set of genes in the genome and the "letters" are meant to play the role of the genes. A round of guessing represents a generation. Guessing a correct letter represents the occurrence of a potentially adaptive mutation in a particular gene in some individual in the population. There are K letters in their alphabet, so that the probability of guessing the correct letter is K-1. They wrote that
1- (1 - 1/K)r
is the probability that the first letter of the word will be correctly guessed in no more than r rounds of guessing. It is also, of course, the probability that any other specific letter would be guessed. Then they wrote that
[1- (1 - 1/K)r]L
is the probability that all L letters will be guessed in no more than r rounds. The event whose probability is the first of the above two expressions is the occurrence in r rounds of at least one correct guess of a letter. This corresponds to the appearance of an adaptive mutation in some individual in the population. That of the second expression is the occurrence of L of them. From these probability expressions we see that according to W&E each round of guessing yields as many correct letters as are lucky enough to be guessed. The correct guesses in a round remain thereafter unchanged, and guessing proceeds in successive rounds only on the remaining letters.
Their model does not mimic natural selection at all. In one generation, according to the model, some number of potentially adaptive mutations may occur, each most likely in a different individual. W&E postulate that these mutations remain in the population and are not changed. Contrary to their intention, this event is not yet evolution, because the mutations have occurred only in single individuals and have not become characteristic of the population. Moreover, W&E have ignored the important fact that a single mutation, even if it has a large selection coefficient, has a high probability of disappearing through random effects. They allow further mutations only in those loci that have not mutated into the "superior" form. It is not clear if they intended that mutations be forbidden in those mutated loci only in those individuals that have the mutation or in other individuals as well. They have ignored the fact that evolution does not occur until an adaptive mutation has taken over the population and thereby becomes a characteristic of the population. Their letter-guessing game is more a parody of the evolutionary process than a model of it. They have not achieved their second goal either.
Thus their conclusion that "there's plenty of time for evolution" is unsubstantiated. The probability calculation to justify evolutionary theory remains unaddressed.
 Wilf, H. S. & Ewens, W. J. (2010) There's plenty of time for evolution. Proc Natl Acad Sci USA 107 (52): 22454-22456. [RETURN TO TEXT]
 Hoyle, F. and N. C. Wickramasinghe, (1981). Evolution from Space, London: Dent. [RETURN TO TEXT]
 I am ignoring the generations needed for a successful adaptive mutation to take over the population. These generations must be added to the waiting time for a successful adaptive mutation to occur. [RETURN TO TEXT]
 Fisher, R. A. (1958). The Genetical Theory of Natural Selection, Oxford. Second revised edition, New York: Dover. [First published in 1929] [RETURN TO TEXT]