Fluctuation relations and fitness landscapes of growing cell populations

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. Known models of cell size control are studied with a formalism based on path integrals or on operators. We investigate some consequences of this fluctuation relation, which constrains the distributions of the number of cell divisions and leads to inequalities between the mean number of divisions and the doubling time of the population. We finally study the concept of fitness landscape, 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.