IN his response to our article (Huai and Woodruff 1997), Cutler takes it for granted that mutation rate and mutational event rate are the same in multicellular species, but there is fundamental distinction between them when premeiotic mutations give rise to clusters of identical mutant alleles. Cutler fails in understanding the essence of our cluster mutation model (Woodruffet al. 1996; Huai and Woodruff 1997), because his definition of mutation rate is conceptually wrong and biologically unrealistic. His use of unique mutational event rate instead of mutation rate is inappropriate even for his own cluster mutation model. One must use the mutation rate, which is determined by counting all members of a cluster, and not the mutational event rate, which counts each cluster as one.
Given that the molecular evolution rate depends on mutation rate, mutational processes will affect substitution processes. Premeiotic cluster mutations can elevate variancetomean ratio [R(t)] of the molecular clock above one, and our previous results hold true with a variety of neutral and selection models whenever the substitution rate is proportional to mutation rate (Huai and Woodruff 1997). Although Cutler's mathematical derivation and discussion is not relevant to premeiotic cluster mutations, due to the confusion of mutation rate and mutational event rate, it is important to clarify the connection and difference between these two concepts. In response to a similar argument by Cherry, we also try to build a straightforward link between mutation rate and substitution rate in the molecular evolution processes.
Mutation rate vs. mutational event rate: The controversy of whether to consider all new mutant alleles from a premeiotic mutation event as one or many in measuring mutation rate has a long history (Spencer and Stern 1948; Auerbach 1962; Russell 1964; Woodruff and Thompson 1992; Drost and Lee 1995; Woodruffet al. 1996; Selby 1998; J. N. Thompson, Jr., R. C. Woodruff and H. Huai, unpublished results). Our understanding of the substitution process critically depends on treatment of the complexity of the mutational process.
From its earliest definition, the term “mutation rate” is the measure of the impact of mutations on the gene pool. It stands for the rate or the probability of genetic change per generation and is a fundamental parameter in genetics and evolution. It does not need to fit in either the infinite allele, infinite site, or any other particular mutation model, since all of these have additional assumptions superimposed upon the mutation process for special purposes. The mutation rate concept also does not imply that any premeiotic mutational events will be frozen at their occurrence stage, while other similar nonmutated germline cells keep dividing to complete their developmental growth. If each of the nonmutated cell products are counted independently, why not treat the multiple mutant copies that arise from a single premeiotic mutational event in the same way (Auerbach 1962; Muller et al. 1963; Figure 1 of Woodruffet al. 1996; Figure 1 of Huai and Woodruff 1997; Selby 1998). In any mutation experiment, the unbiased estimation of the mean mutation rate is the proportion of new mutants among all those sampled, no matter whether the recovered new mutants are identical by descent (in clusters) or simply identical by chance (Tables 1 and 2 of Huai and Woodruff 1997).
When the average family size is n, the potential late mutations during meiosis (Figure 1B of Huai and Woodruff 1997) are sampled n times on average. Hence, it is biased if early premeiotic mutations that occur before germsomatic cell differentiation are counted only once, since all gametes have to experience the shared early development process (Figure 1A of Huai and Woodruff 1997). A cluster of identical new mutants should be sampled on average n times as well. Cutler obviously ignores or misunderstands our repeated emphasis that the mutational event rate is distinctly different from the traditional mutation rate when cluster mutations are involved (page 157 of Woodruffet al. 1996; and 340–341 of Huai and Woodruff 1997). Cutler's use of mutation rate (u), along with the infinite site model and multiple copies of unique mutational events, clearly indicates he is modeling the mutational event rate whenever he refers to mutation rate.
The mutational event rate requires information on identity by descent from allelism tests or sequencing data for these new mutants within a family. If it is possible to trace the cluster of identical new mutants to a single mutational event, then one can count it as “one” in calculating mutational event rate. As the development and life cycle of multicellular organisms get more complex, the difficulty of measuring mutational event rate increases dramatically. The reasonable denominator for mutational event rate is hard to determine in multicellular organisms. Cutler uses 2N (N is the stable population size) in calculating mutational event rate, which is a mistake. We believe it is only appropriate to measure the mutational event rate at the cellular division/generation level. If one combines the mutational event rate with the detailed information of the germline developmental process to evaluate the impacts of mutations, one will finally get the mutation rate.
For all singlecell organisms, cells in culture, or similar cellular populations, the mutation rate per cell generation is equivalent to the mutational event rate. Only at the percelldivision or cellgeneration level do mutational event rates have perceptible meaning. In all species with somaticgermline cell differentiation, mutational event rate is an underestimation of the true mutation rate, and it has unavoidable logic problems within its definition. Even if there were mutational event rates per organism generation in multicellular species, one needs allelism information within a family to determine these rates and one also needs to track when these mutational events occurred to evaluate their impact on the gene pool. Thus, it is neither necessary nor adequate to use mutational event rate in population genetics and evolutionary theory. One should use the mutation rate instead, which is determined by counting all members of a cluster mutation.
Cutler uses a neutral model with infinite sites, constant population size N, and constant mutation rate u, so he actually uses the mutational event rate. He further states: “ … the number of unique mutations that occur in t generations is Poisson distributed with mean 2Nut. The probability that any one of these mutations will eventually fix in the population is simply the initial frequency of the mutant. For the neutral model without clusters, all mutations start at frequency 1/2N. For the model with clusters, we will assume mutations start at frequency P, where O < P < 1 is a random variable.” Again it is clear that he really means mutational event rate instead of mutation rate. Though he does not provide the biological model for deciding P, it is certain that 1/2N ≤ P < 1. Letting E(p)_{i} stand for the mean P of generation i, we know that E(p)_{i} > 1/2N with clusters, and E(p) > 1/2N over t generations. It can then be shown that the mean population size of any generation i is N + Nu[E(p)_{i} − 1/2N], which is slightly more than N, and over t generations is Nt + Ntu[E(p) − 1/2N], which is also more than Nt. In Cutler's model, the population size is neither stable nor constant at size N, either for each generation or for a number of generations as he has assumed. Worse yet, the substitution rate in the neutral model with constant “mutation rate” u is approximately u × 2NE(p)[2NE(p) is the mean cluster size including single mutants], which can be much larger than u. Cutler does not provide any biological basis to determine cluster size, so if several species have distinct 2NE(p) values, the molecular clock may not exist at all, even if these species had the same mutational event rates.
Clearly, Cutler's wrong definition of mutation rate, combined with the infinite sites model, is the root of all the above discrepancies. If Cutler adopted the correct definition of mutation rate and incorporated it in our premeiotic cluster mutation model, he would have observed that the number of unique mutations (M_{t}) is not independent from, but instead is negatively correlated with, the mean cluster size or E(p). Hence, his Equations 4–6 are incorrect, and all his conclusions based on these equations do not hold.
Overdispersed substitution process with premeiotic cluster mutations: What we have observed is that the variance to mean ratio of the number of mutants produced in any single generation can be much more than one with clusters (Huai and Woodruff 1997). Even if early premeiotic cluster mutations are grouped into the previous generation, as suggested by Cherry, late premeiotic clusters still inflate the variance to mean ratio of new mutations (see Figure 1B in Woodruffet al. 1996 and the Appendix in Huai and Woodruff 1997). This means the observed number of mutants can deviate more from the Poisson expectation than previously assumed, and the realized mutation rate is accordingly more variable. Although a constant mutation rate is assumed in molecular evolution, this theoretical rate may never be achieved within a lineage because of cluster mutations. In the presence of premeiotic cluster mutations, genetic drift and population dynamics cannot be separated from the mutational process as suggested by Cherry. Therefore, even if the mean mutation rates are the same for different evolutionary lineages, the observed or realized mutation rates within each population are less predictable due to clusters, and will have a larger dispersion index than the Poisson distribution.
The above conclusion can be applied over many generations because it is reasonable to assume that in each generation the genetic material mutates independently. Hence, both the mean and variance of the number of mutants produced during any number of generations are additive, as will be the observed mutation rates among species as shown in Equations 8 and 9 of Huai and Woodruff (1997). It is the mean realized mutation rate within each lineage over evolutionary history that is compared in molecular clock studies. Premeiotic cluster mutations can elevate the variancetomean ratio of the actual mutation rate within each lineage among evolutionary lineages. As far as molecular evolution rate is related to mutation rate, as predicted by the majority of neutral and selection models, premeiotic cluster mutations will have major effects on both mean and variance of substitution rate of each species even if the mean mutation rate remains constant. Clustered mutations do not change the mean mutation rate if one averages over many species and evolutionary time, but the resulting picture of molecular evolution is more varied and unpredictable.
Footnotes

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