Universal constraints on selection strength in lineage trees

We obtain general inequalities constraining the difference between the average of an arbitrary function of a phenotypic trait, which includes the fitness landscape of the trait itself, in the presence or in the absence of natural selection. These inequalities imply bounds on the strength of selection, which can be measured from the statistics of traits or divisions along lineages. The upper bound is related to recent generalizations of linear response relations in Stochastic Thermodynamics, and is reminiscent of the fundamental theorem of Natural selection of R. Fisher and of its generalization by Price. The lower bound follows from recent improvements on Jensen inequality and is typically less tight than the upper bound. We illustrate our results using numerical simulations of growing cell colonies and with experimental data of time-lapse microscopy experiments of bacteria cell colonies.