Inherited or produced? Inferring protein production kinetics when protein counts are shaped by a cell's division history

Inferring protein production kinetics for dividing cells is complicated due to protein inheritance from the mother cell. For instance, fluorescence measurements -- commonly used to assess gene activation -- may reflect not only newly produced proteins but also those inherited through successive cell divisions. In such cases, observed protein levels in any given cell are shaped by its division history. As a case study, we examine activation of the glc3 gene in yeast involved in glycogen synthesis and expressed under nutrient-limiting conditions. We monitor this activity using snapshot fluorescence measurements via flow cytometry, where GFP expression reflects glc3 promoter activity. A na"ive analysis of flow cytometry data ignoring cell division suggests many cells are active with low expression. Explicitly accounting for the (non-Markovian) effects of cell division and protein inheritance makes it impossible to write down a tractable likelihood -- a key ingredient in physics-inspired inference, defining the probability of observing data given a model. The dependence on a cell's division history breaks the assumptions of standard (Markovian) master equations, rendering traditional likelihood-based approaches inapplicable. Instead, we adapt conditional normalizing flows (a class of neural network models designed to learn probability distributions) to approximate otherwise intractable likelihoods from simulated data. In doing so, we find that glc3 is mostly inactive under stress, showing that while cells occasionally activate the gene, expression is brief and transient.