  Cell size varies between different cell types, and between different growth
and osmotic conditions. However, the nuclear-to-cell volume ratio (N/C ratio)
remains nearly constant. In this paper, we build on existing deterministic
models of N/C ratio homeostasis and develop a simplified gene translation model
to study the effect of stochasticity on the N/C ratio homeostasis. We solve the
corresponding chemical master equation and obtain the mean and variance of the
N/C ratio. We also use a Taylor expansion approximation to study the effects of
the system size on the fluctuations of the N/C ratio. We then combine the
translation model with a cell division model to study the effects of extrinsic
noises from cell division on the N/C ratio. Our model demonstrates that the N/C
ratio homeostasis is maintained when the stochasticity in cell growth is taken
into account, that the N/C ratio is largely determined by the gene fraction of
nuclear proteins, and that the fluctuations in the N/C ratio diminish as the
system size increases.
