Scaling Solution in the Large Population Limit of the General Asymmetric Stochastic Luria–Delbrück Evolution Process

David A. Kessler, Herbert Levine

Research output: Contribution to journalArticlepeer-review

27 Scopus citations

Abstract

One of the most popular models for quantitatively understanding the emergence of drug resistance both in bacterial colonies and in malignant tumors was introduced long ago by Luria and Delbrück. Here, individual resistant mutants emerge randomly during the birth events of an exponentially growing sensitive population. A most interesting limit of this process occurs when the population size N is large and mutation rates are low, but not necessarily small compared to 1/N. Here we provide a scaling solution valid in this limit, making contact with the theory of Levy α-stable distributions, in particular one discussed long ago by Landau. One consequence of this association is that moments of the distribution are highly misleading as far as characterizing typical behavior. A key insight that enables our solution is that working in the fixed population size ensemble is not the same as working in a fixed time ensemble. Some of our results have been presented previously in abbreviated form [12].

Original languageEnglish
Pages (from-to)783-805
Number of pages23
JournalJournal of Statistical Physics
Volume158
Issue number4
DOIs
StatePublished - Feb 2015

Bibliographical note

Publisher Copyright:
© 2014, Springer Science+Business Media New York.

Keywords

  • Alpha-stable distribution
  • Growth
  • Luria–Delbrück
  • Mutants

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