Statistics for the Luria-Delbrück distribution

Agnès Hamon,Bernard Ycart

doi:10.1214/12-ejs711

Abstract

The Luria-Delbr\"uck distribution is a classical model of mutations in cell kinetics. It is obtained as a limit when the probability of mutation tends to zero and the number of divisions to infinity. It can be interpreted as a compound Poisson distribution (for the number of mutations) of exponential mixtures (for the developing time of mutant clones) of geometric distributions (for the number of cells produced by a mutant clone in a given time). The probabilistic interpretation, and a rigourous proof of convergence in the general case, are deduced from classical results on Bellman-Harris branching processes. The two parameters of the Luria-Delbr\"uck distribution are the expected number of mutations, which is the parameter of interest, and the relative fitness of normal cells compared to mutants, which is the heavy tail exponent. Both can be simultaneously estimated by the maximum likehood method. However, the computation becomes numerically unstable as soon as the maximal value of the sample is large, which occurs frequently due to the heavy tail property. Based on the empirical generating function, robust estimators are proposed and their asymptotic variance is given. They are comparable in precision to maximum likelihood estimators, with a much broader range of calculability, a better numerical stability, and a negligible computing time.

Highlights

Luria and Delbruck (1943) reported an experiment on virus resistant bacteria: cultures of the same strain of Escherichia Coli having grown up independently for several generations, the cells were plated onto selective medium and surviving bacteria counted
As an alternative robust estimation technique, we propose to transform the data through X → zX, with 0 < z < 1, i.e. consider the empirical probability generating function
We have developed in R (R Development Core Team (2008)) a set of functions that perform the usual operations on the LD distributions, output Maximum Likelihood (ML) and Generating Function (GF) estimates, confidence regions and p-values for hypothesis testing

Summary

Introduction

Luria and Delbruck (1943) reported an experiment on virus resistant bacteria: cultures of the same strain of Escherichia Coli having grown up independently for several generations, the cells were plated onto selective medium and surviving bacteria counted. The generation time of any mutant cell is exponentially distributed with parameter μ;. The Luria-Delbruck distribution with parameters α and ρ and we denote it by LD(α, ρ) Observe that it depends on G (the generation time distribution of normal cells) only through its growth rate ν, matching the conclusions of. The time between the occurrence of a typical mutation and the end of the observation, asymptotically follows the exponential distribution with parameter ν This is the time for which any given mutant clone (population stemming from a single cell) will develop; 3. The Luria-Delbruck distribution is a compound Poisson of an exponential mixture of geometric distributions It is a heavy tail distribution, with tail exponent ρ: the higher the fitness of mutants compared to normal cells, the heavier the tail.

Bellman-Harris processes

Maximum Likelihood estimators

Generating function estimators

Simulation study