Size-structured populations with growth fluctuations: Feynman-Kac formula and decoupling

We study a size-structured population model in which individual cells accumulate size at a rate determined by a fluctuating internal variable (e.g., gene expression levels). Together, the internal variable and the size variable constitute a cell's phenotypic state. Many previous models of phenotypically heterogeneous populations can be viewed as special cases of this model. In some of these models, the internal variable is known to become asymptotically independent of cell size. Here, we generalize this decoupling phenomenon and connect it to the Feynman-Kac formula, which relates the lineage dynamics to the distribution of phenotypes in a growing population where selection generates a bias towards faster-growing cells. To this end, we derive conditions for decoupling between the internal variable and size, both in the lineage and population ensembles. When decoupling occurs in both ensembles, the size dynamics can be transformed, via a random time change, into a growth-homogeneous process, and population expectations can be evaluated by an exponential tilting procedure that follows from the Feynman–Kac formula. We further characterize weaker, ensemble-specific forms of decoupling that hold in either the lineage or the population ensemble, but not both. Finally, we provide a more general interpretation of tilted expectations in terms of the mass-weighted phenotype distribution.