Mutations and infinity: improved statistical methods for estimating spontaneous rates

Certain mathematical artifacts which had been appended by others to Luria and Delbrück's [Genetics 28: 491-511, 1943] model of spontaneous mutagenesis in bacterial populations have added confusion to the modeling and measurement of spontaneous mutation rates. Additional confusion arises when models which had been tuned for experiments with bacterial cultures grown from a small inoculum are adapted for use with mammalian cell cultures grown from a large initial population. As one consequence, biologists still tend to grow the large number of parallel cultures required by the fluctuation test in order to avoid large errors due to the high variability in the number of mutants in a growing culture. By avoiding models with infinite mean values and certain mathematical approximations that lead to conceptual and practical difficulties, the large variance of the number of mutants can be avoided (and the precision of the estimated mutation rate controlled) through the use of sufficiently large initial cell populations. A direct consequence is that simpler experiments with fewer cultures may suffice. In this paper, after a discussion of the confusions, we extend our previous approach [Rossman et al.: Mutat Res 328:21-30, 1995] by giving improved formulas for the standard error of the estimated mutation rate. The improvement results from using a more inclusive model based on consideration of the variability due to both the biological phenomenon of the growing culture (growth and mutation) and the protocols used for selection (sampling and plating efficiency). Also included is the situation where the initial cell population is not assumed to be free of mutants but the initial mutant fraction is measured instead. These standard error formulas are useful in planning experiments that yield mutation rate estimates with planned precision and for comparing and testing hypotheses about mutation rates in two or more populations which are grown under different conditions.