  The mechanism of bacterial cell size control has been a mystery for decades,
which involves the well-coordinated growth and division in the cell cycle. The
revolutionary modern techniques of microfluidics and the advanced live imaging
analysis techniques allow long term observations and high-throughput analysis
of bacterial growth on single cell level, promoting a new wave of quantitative
investigations on this puzzle. Taking the opportunity, this theoretical study
aims to clarify the stochastic nature of bacterial cell size control under the
assumption of the accumulation mechanism, which is favoured by recent
experiments on species of bacteria. Via the master equation approach with
properly chosen boundary conditions, the distributions concerned in cell size
control are estimated and are confirmed by experiments. In this analysis, the
inter-generation Green's function is analytically evaluated as the key to
bridge two kinds of statistics used in batch-culture and mother machine
experiments. This framework allows us to quantify the noise level in growth and
accumulation according to experimental data. As a consequence of non-Gaussian
noises of the added sizes, the non-equilibrium nature of bacterial cell size
homeostasis is predicted, of which the biological meaning requires further
investigation.
