Improved inference of mutation rates: II. Generalization of the Luria-Delbrück distribution for realistic cell-cycle time distributions

In the first paper of this series (Kepler and Oprea, Theor. Popul. Biol. 2001) we found a continuum approximation of the Luria-Delbrück distribution in terms of a scaled variable related to the proportion of mutants in the culture. Here we show that the Luria-Delbrück distribution is inaccurate when realistic division processes are being considered due to the non-Markovian character of the cell cycle. We derive the expectation of the proportion of mutants in the culture for arbitrary cell-cycle time distributions. We then introduce a two-parameter generalization of the continuum Luria-Delbrück distribution for two of the more commonly used cell-cycle time distributions: gamma and shifted exponential. We obtain the generalized distribution by defining a map from the actual parameters to "effective" parameters. The effective mutation rate is obtained analytically, while the effective population size is obtained by fitting simulation data. Our simulations show that the second parameter depend mostly on the coefficient of variation of the cell-cycle time distribution.