Coarse-graining and stochastic oscillations in a phenomenological model of cell-size homeostasis
Within a continuous-time, stochastic model of single-cell size homeostasis, we study how the structure of feedback from size to growth rates and cell-cycle progression shapes overall size dynamics, both within and across cell cycles. We focus on a model in which the feedback from cell size to these other processes occurs only through the size deviations, defined as the difference between the absolute size and the progression through the cell cycle. In a linear regime of this model, the dynamics reduce to a stochastically forced simple harmonic oscillator, yielding closed-form expressions for mother-daughter size correlations. We compare these to the higher order regression coefficients that measure the size memory over many generations. Our analysis reveals how the interplay between cell-cycle timing and intrinsic fluctuations shapes the apparent coarse-grained size control strategy, and in-particular, that coarse-grained correlations may not reflect the mechanistic feedback structure. We compare this model to a more commonly used approach where the coarse-grained dynamics are hard-coded into the model; hence, the first order autoregressive model for sizes is a perfect description of the size dynamics and therefore more accurately reflects the feedback structure.