Maximum likelihood estimation of spontaneous mutation rates from large initial populations

When estimating a spontaneous mutation rate from either a single culture (C=1) or from the C parallel cultures (C>1) of a fluctuation experiment, the use of a large initial population size N0 to seed each culture will permit a gaussian approximation for the probability distribution of the number M of mutants at the time when the culture(s) has (have) grown to size N=N02g, i.e., experienced g doublings. Using this gaussian approximation we find that the maximum likelihood estimate mu of the expected number mu of mutants present in a culture in generation g is (exactly) (equation: see text) where r = 2g / g and M 2 is the average of the squares of the C mutant counts. The maximum likelihood estimate p of the unknown mutation rate p is p = 2 mu / gN assuming an 'ideal' experiment and that there were no mutants in the initial population. A well-behaved maximum likelihood estimate is known to be efficient in large samples and we illustrate by Monte Carlo simulation that indeed p is better (has smaller mean squared error) than our previous (Rossman et al., 1995) estimator (equation: see text) (M is the average mutant count) provided N0 is of the order 1/p or larger. This advantage exists even without a fluctuation experiment, i.e., for C = 1.