Fluctuation relations and fitness landscapes of growing cell populations

Abstract

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...

Universal constraints on selection strength in lineage trees

“…In fact, ideas from Stochastic Thermodynamics can be applied directly at the level of individual cell trajectories [6]. By following this kind of approach, we have derived general constraints on dynamical quantities characterizing the cell cycle such as the average number of divisions or the mean generation time [7,8]. These constraints are universal because they hold independently of the specific cell dynamic model and they are indeed verified in experimental data.…”

“…The statistical bias is captured by a relation between the two probability distributions for the number of divisions, similar to fluctuation relations in Stochastic Thermodynamics [7,8] p back (K, t) = p for (K, t) e K ln m−tΛt ,…”

“…Note that eq. (4) provides an estimator for the population growth rate Λ t , which requires only mother machine data [8]. This is useful, although in practice the convergence of this estimator is a challenging issue [16], which is one motivation to look for inequalities as done below.…”

“…and size [8], or for the number of divisions, these extrema values are given by the extremal trait values. The two terms in the r.h.s.…”

“…We now test our results on real data, extracted from [21], for the following traits: the number of divisions K, the size X and the age A, which are easily accessible and for which we predicted theoretical fitness landscapes [8]. The first step is to obtain the fitness landscapes of these traits, which are shown for three particular experimental conditions in Suppl.…”