The Statistical Theory of Bacterial Populations Subject to Mutation

Summary This paper is a study of the growth of a mixed bacterial population composed of two types of organisms, and subject to mutation from one type to the other, in one or both directions. The bacteria divide by binary fission, but the rates of division for the two types may differ. The paper is intended as a unified and critical presentation of statistical results previously published by other authors, together with a number of new results. The first part presents a mathematical theory, from both the deterministic and the stochastic viewpoints. The definition of a mutation rate is discussed in some detail. Particular attention is directed to the situation in which the proportion of organisms of one type remains negligible throughout the period of growth (so-called short-term experiments); the effects of delayed phenotypic expression and of the sampling of replicate cultures are considered. For the more general situations, approximate expressions are obtained for the first few cumulants of the joint distribution of the numbers of organisms of each type present at any time. The second part is concerned with methods of estimation of the parameters of a mutation process. In particular I examine the suitability of various methods of estimating the mutation rate in short-term experiments, when phenotypic delay is present. Methods based on the upper portion of the distribution of the number of mutants in replicate cultures, such as the upper quartile, are preferable. The verification of the theory by certain data published by other authors is considered in the third part. An attempt is made to fit some observed distributions of the number of mutants in replicate cultures, using a particular model for phenotypic delay; the fit is unsatisfactory, the fitted distributions having too high a frequency of cultures with two or three mutants.