  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.
