Generalized Euler-Lotka equation for correlated cell divisions

Cell division times in microbial populations display significant fluctuations. These fluctuations impact the population growth rate in a non-trivial way. If fluctuations are uncorrelated among different cells, the population growth rate is predicted by the Euler-Lotka equation, which is a classic result in mathematical biology. However, cell division times can present significant correlations, due to physical properties of cells that are passed from mothers to daughters. In this paper, we derive an equation remarkably similar to the Euler-Lotka equation which is valid in the presence of correlations. Our exact result is based on large deviation theory and does not require particularly strong assumptions on the underlying dynamics. We apply our theory to a phenomenological model of bacterial cell division. We find that the discrepancy between the growth rate predicted by the Euler-Lotka equation and our generalized version is relatively small, but large enough to be measurable in experiments.