General formulation of Luria-Delbrück distribution of the number of mutants

The Luria-Delbrück experiment is a cornerstone of evolutionary theory, demonstrating the randomness of mutations before selection. The distribution of the number of mutants in this experiment has been the subject of intense investigation during the past 70 years. Despite this considerable effort, most of the results have been obtained under the assumption of constant growth rate, which is far from the experimental condition. We derive here the properties of this distribution for arbitrary growth function for both the deterministic and stochastic growth of the mutants. The derivation we propose uses the number of wild-type bacteria as the independent variable instead of time. The derivation is surprisingly simple and versatile, allowing many generalizations to be taken easily into account.

Figure 1

The number of mutants $X$ at WT population size $N$ is the sum of the contributions of mutants lineages appearing at WT population size $n_i$. The size of the lineage of mutant $i$ appearing at WT population size $n_i$ is $m_i\left(n\right)=n/n_i$ for $n\ge n_i$. The $i\mathrm{th}$ mutant, appearing at size 1 at WT population $n_i$, will be present at size $N/n_i$ in the final population.

Figure 2

Contribution of mutants with different growth rate. The size of the lineage of mutant $i$ appearing at WT population size $n_i$ is $m_i\left(n\right)={(n/n_i)}^c$ for $n\ge n_i$.

Figure 3

The stochastic propagator $K_n^N$ of the mutant appearing at WT population size $n$.

Figure 4

Deterministic growth, random relative growth rate. The size of the lineage of mutant $i$ appearing at WT population size $n_i$ is $m_i\left(n\right)={(n/n_i)}^c$ for $n\ge n_i$. This time, however, $c$ is a random variable.