Fluctuation Analysis and the Luria-Delbrück Distribution

Research project P 14682 Fluctuation analysis and the Luria-Delbrück distribution Reinhard BÜRGER 09.10.2000 The Luria-Delbrück distribution arises in connection with populations of self-reproducing entities, such as bacteria or chromosomes, whose progeny are not necessarily completely accurate copies of their parents. It is most intimately connected with Luria and Delbrück`s ground-breaking experiments on bacterial mutation and the concomitant introduction of fluctuation analysis by these authors. In this classical context, the Luria-Delbrück distribution refers to the distribution of the number of mutant colonies that emerge on solid medium from a large number of bacteria if a certain hypothesis concerning the origin of mutant bacteria is satisfied. Since then, fluctuation analysis has become a valuable tool for the measurement of mutation rates in single-cell populations. A satisfactory theory of fluctuation analysis must provide a model for the calculation and the analysis of the Luria- Delbrück distribution under a variety of different assumptions. In particular, it should be sufficiently realistic such that it admits inferences about parameters of biological interest by comparison with the empirical distribution obtained from a fluctuation experiment. The purpose of this project is to establish such a theory on the basis of the approach recently initiated by Angerer, who was able to formulate the first model for the study of the Luria- Delbrück distribution that is explicitly solvable. However, for both theoretical as well as practical reasons, it will be necessary to study generalizations of Angerer`s model for growth and mutation processes that are more general than those already discussed in that paper. For instance, what is the influence of multi-stage forward and of backward mutation on the Luria-Delbrück distribution? If there is selection on mutant or non-mutant cells, what does this mean for the distribution of mutant colonies observed in the experiment? Do different models of the growth of mutant or non-mutant cells necessarily lead to different Luria-Delbrück distributions? Questions like these are also important in the study of the inverse problem, namely, does the distribution of mutant colonies observed in the experiment allow to draw inferences about the mutation process, and what are these?

The fluctuation analysis of bacterial mutations is an experimental protocol to determine certain parameters characterizing the process of mutation in bacteria. The parameter of prime importance in this context is the mutation rate, which is the probability that the division of a non-mutant cell ends up with the formation of a mutant daughter cell. A cell is referred to as `mutant` if its physiological properties differ from those of the bulk of the population. The procedure is as follows: A culture of bacteria is split into a certain number of `sister cultures`. These are then supplied with fresh medium and grown into larger cultures which, at a suitable time, are put on solid selection medium that only supports the proliferation of mutant cells. After one or two days, the mutant bacteria will have grown into visible colonies. The exact number of colonies per sister culture constitutes the outcome of the experiment. The main difficulty is the mathematical analysis of the distribution of mutant colonies that would theoretically be expected, and the evaluation of the data. Making use of the analytic theory of branching processes, we have been able to answer these questions in a satisfactory manner. In particular, it is now possible to take into account realistic models of cell proliferation. In comparison with previous theories of fluctuation analysis, this is the main advantage of the models established in the course of this project.