IN a recent paper, Huai and Woodruff (1997) make the mistake of equating substitution with mutation, and arrive, quite erroneously, at the conclusion that clustered mutations can have a significant effect on the pattern of molecular evolution. I will show that clustered mutations have no effect at all on the index of dispersion. A large index of dispersion remains a central unexplained observation in molecular evolution.
The neutral theory predicts that the number of mutations that arise in a population in t generations, which ultimately become fixed in the population, will be Poisson distributed with mean ut, where u is the per sequence, per generation mutation rate. Therefore, the variance in the number of substitutions, S_{t}, will equal the mean at any neutral locus. The ratio of the variance in the number of substitutions to the mean number is often called the index of dispersion, R(t),
Huai and Woodruff make two important observations. First, in a series of experiments, Woodruff et al. (1996) show that premeiotic mutations lead to clusters of identical mutations in many offspring of a single individual. Next, they note that the theory that predicts R(t) = 1 assumes that all mutations arrive in the population at frequency 1/2N. These two observations cause them to rightly conclude that our predictions for R(t) should be reanalyzed assuming that the starting frequency for new mutations can be larger than 1/2N and that the frequency can be variable.
The simplest neutral model that allows easy calculation of R(t) assumes infinite sites and constant population size N. By the neutral assumption, the probability that any mutation will eventually fix in the population is simply the initial frequency of that mutant. Mutations come in two types, meiotic, which enter the population at frequency 1/2N, and clusters, which enter the population at frequency P, where 0 < P < 1 is a random variable. Let u be the mutation rate to unique mutants. By the infinite sites assumption, all meiotic mutants are unique, as are all clusters, but each cluster counts as a single mutational event. We will consider two different levels of recombination. Sites will either be assumed to recombine freely (Kimura 1969), or no recombination will be permitted (Watterson 1975).
Let M_{t} be the number of unique mutations in a period of t generations. M_{t} is Poisson distributed with mean and variance 2Nut. Label each of these M_{t} mutations with a unique number between 1 and M_{t}. Associate with mutation j, 1 ≤ j ≤ M_{t}, a random variable X_{j}. Let X_{j} equal 1 if mutation j ultimately fixes in the population, and 0 otherwise. Thus, X_{j} is the indicator that mutation j ultimately fixes in the population. Even though X_{j} depends on the random variable P, it is, nonetheless, a simple Bernoulli random variable with moments
Huai and Woodruff's overestimation of R(t) stems from two errors. First, they model clustered mutations as if they cause distinct mutations to be copied into several offspring of a single individual, rather than a single unique mutation to be copied into those offspring. As a result, they allow u to be a random variable, rather that P. By failing to make the subtle distinction between u and P, Huai and Woodruff allow the same premeiotic mutation to fix more than once. This error is not substantial, however, and only leads one to conclude that R(t) ≈ 1 + O(1/2N). As population size increases, the distinction becomes unimportant.
The other error is far more serious. Huai and Woodruff derive the variance to mean ratio of the number of mutations, not the variance to mean ratio of the number of substitutions. As we can see from Equation 5, increases in the mutational variance will propagate into increases in the substitutional variance by a factor of (E[X_{j}])^{2} ≈ (1/2N)^{2}, whereas all other terms are of order 1/2N. Thus, by considering the variance to mean ratio of mutations, rather than substitutions, Huai and Woodruff overestimate the increase in R(t) by a factor 2N. Taking their two errors together they found R(t) ≈ 1 + 2N × O(1/2N) ≈ 1 + O(1), which they concluded was a significant effect. The analysis here shows that R(t) is exactly one. This analysis uses the infinite sites, freerecombination or norecombination model of the gene. The equivalent analysis for an arbitrary level of recombination (Hudson 1983) is more difficult, but should lead to the same conclusion that R(t) = 1.
Acknowledgments
I would like to thank John Gillespie and Hiroshi Akashi for many suggestions. This work was supported by fellowships from The Center for Population Biology, and The Institute for Theoretical Dynamics at UC Davis.
Footnotes

Communicating editor: N. Takahata
 Copyright © 1998 by the Genetics Society of America