TY - JOUR
T1 - Large population solution of the stochastic Luria–Delbrück evolution model
JF - Proceedings of the National Academy of Sciences
JO - Proc Natl Acad Sci USA
SP - 11682
LP - 11687
DO - 10.1073/pnas.1309667110
VL - 110
IS - 29
AU - Kessler, David A.
AU - Levine, Herbert
Y1 - 2013/07/16
UR - http://www.pnas.org/content/110/29/11682.abstract
N2 - Luria and Delbrück 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 populations, (compared with the inverse of the mutation probability), whereas it goes over to a Lévy α-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.
ER -