Convergence of the nuclear-to-cell ratio distribution under repeated divisions

Establish whether, in the growth–division model where proteins evolve via the stochastic translation reactions P_r → 2P_r, P_r → P_r + P_n, and P_r → P_r + P_c and cells divide when the total protein number doubles with binomial partitioning of molecules between daughters, the probability distribution of the nuclear-to-cell ratio NC_cell = p_n/(p_r + p_n + p_c) converges to a stable (stationary) distribution as the number of divisions tends to infinity.

Background

To incorporate extrinsic noise from cell division, the authors simulate a protein-doubling division rule and binomial partitioning of molecules between daughter cells within their stochastic translation model. They observe sample paths for the nuclear-to-cell ratio (NC_cell) across thousands of divisions and present corresponding histograms.

Based on these simulations, they posit that NC_cell may approach a stable distribution as the number of divisions increases but do not provide a proof or definitive characterization, identifying this as a conjectural point requiring further paper.