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Microbial populations adapt to their environment by acquiring advantageousmutations, but in the early twentieth century, questions about how theseorganisms acquire mutations arose. The experiment of Salvador Luria and MaxDelbr\"uck that won them a Nobel Prize in 1969 confirmed that mutations don'toccur out of necessity, but instead can occur many generations before there isa selective advantage, and thus organisms follow Darwinian evolution instead ofLamarckian. Since then, new areas of research involving microbial evolutionhave spawned as a result of their experiment. Determining the mutation rate ofa cell is one such area. Probability distributions that determine the number ofmutants in a large population have been derived by D. E. Lea, C. A. Coulson,and J. B. S. Haldane. However, not much work has been done when time of celldivision is dependent on the cell age, and even less so when cell division isasymmetric, which is the case in most microbial populations. Using probabilitygenerating function methods, we rigorously construct a probability distributionfor the cell population size given a life-span distribution for both mother anddaughter cells, and then determine its asymptotic growth rate. We use this toconstruct a probability distribution for the number of mutants in a large cellpopulation, which can be used with likelihood methods to estimate the cellmutation rate.

Age-dependent Branching Processes and Applications to the Luria-Delbr\"uck Experiment