Using a population dynamics inspired by an ensemble of growing cells, a set of fluctuation theorems linking observables measured at the lineage and population levels are derived. One of these relations implies inequalities comparing the population doubling time with the mean generation time at the lineage or population levels. We argue that testing these inequalities provides useful insights into the underlying mechanism controlling the division rate in such branching processes.

We construct a pathwise formulation of a growing population of cells, based on two different samplings of lineages within the population, namely the forward and backward samplings. We show that a general symmetry relation, called fluctuation relation relates these two samplings, independently of the model used to generate divisions and growth in the cell population. these relations lead to estimators of the population growth rate, which can be very efficient as we demonstrate by an analysis of a set of mother machine data. These fluctuation relations lead to general and important inequalities between the mean number of divisions and the doubling time of the population. We also study the fitness landscape, a concept based on the two samplings mentioned above, which quantifies the correlations between a phenotypic trait of interest and the number of divisions. We obtain explicit results when the trait is the age or the size, for age and size-controlled models. While the growth of cell populations appears deterministic, many processes occurring at the single cell level are stochastic. Among many possibilities, stochasticity at the single cell level can arise from stochasticity in the generation times 1 , from stochasticity in the partition at division 2,3 , or from the stochasticity of single cell growth rates, which are usually linked to stochastic gene expression 4. Ideally one would like to be able to disentangle the various sources of stochasticity present in experimental data 5. This would allow to understand and predict how the various sources of stochasticity affect macroscopic parameters of the cell population, such as the Malthusian population growth rate 6,7. Beyond this specific question, research in this field attempts to elucidate the fundamental physical constraints which control growth and divisions in cell populations. With the advances in single cell experiments, where the growth and divisions of thousand of individual cells can be tracked, robust statistics can be acquired. New theoretical methods are needed to exploit this kind of data and to relate experiments carried out at the population level with experiments carried out at the single cell level. For instance, one would like to relate single-cell time-lapse video microscopy experiments of growing cell populations 8 , which provide information on all the lineages in the branched tree, with experiments carried out with the mother machine configuration, which provide information on single lineages 9,10. Let us now review quickly how the issue was addressed theoretically. In 2015, a pathwise thermodynamic framework was built for population dynamics using large deviation theory. One important result was a variational principle for the population growth rate 11 , which was formulated in terms of two key path distributions, namely the chronological and the retrospective probability distributions. Then, in order to explain their experimental observation that populations of Escherichia coli double faster than the mean doubling time of their constituent...