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Large population solution of the stochastic Luria-Delbruck evolution model

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TitleLarge population solution of the stochastic Luria-Delbruck evolution model
Publication TypeJournal Article
Year of Publication2013
AuthorsKessler DA, Levine H
JournalProceedings of the National Academy of Sciences of the United States of America Proceedings of the National Academy of Sciences of the United States of America
Volume110
Pagination11682-7
Date Published2013-Jul-16
ISBN Number1091-6490
Accession NumberMEDLINE:23818583
Abstract

Luria and Delbruck introduced a very useful and subsequently widely adopted framework for quantitatively understanding the emergence of new cellular lineages. Here, we provide an analytical treatment of the fully stochastic version of the model, enabled by the fact that population sizes at the time of measurement are invariably very large and mutation rates are low. We show that the Lea-Coulson generating function describes the "inner solution," where the number of mutants is much smaller than the total population. We find that the corresponding distribution function interpolates between a monotonic decrease at relatively small popula

tions, (compared with the inverse of the mutation probability), whereas it goes over to a Levy alpha-stable distribution in the very large population limit. The moments are completely determined by the outer solution, and so are devoid of practical significance. The key to our solution is focusing on the fixed population size ensemble, which we show is very different from the fixed time ensemble due to the extreme variability in the evolutionary process.

Short TitleProc. Natl. Acad. Sci. U. S. A.Proc. Natl. Acad. Sci. U. S. A.
Alternate JournalProceedings of the National Academy of Sciences of the United States of America