  Fluctuations of cell state, e.g., abundances of some proteins, have attracted
much attention both theoretically and experimentally. The distribution of such
state over cells, however, is not only a result of intracellular stochastic
process, but is also influenced by the growth in cell numbers that depends on
the state. By incorporating the growth-death process into the standard
Fokker--Planck equation for the probability distribution, a nonlinear temporal
evolution equation of distribution is obtained that includes a self-consistent
growth term. The derived equation is generally solved analytically by means of
eigenfunction expansions. By focusing on the case with linear relaxation, two
examples are considered as applications of the proposed general formalism.
First, by assuming that the growth rate of a cell increases linearly with the
state value $x$, the shift of the average state value $x$ due to the growth
effect is shown to be proportional to the variance of the state $x$ and the
relaxation time, similarly with the biological fluctuation- response
relationship. Second, when there is a gap in the growth rate at some threshold
value for the state $x$, existence of a critical gap value is demonstrated,
beyond which the average growth rate starts to increase. This critical value is
again obtained in terms of the relaxation time and the variance of $x$, all of
which are experimentally measurable quantities. The relevance of the results to
the analysis of biological data on the distribution of cell states, as obtained
for example by flow cytometry, is discussed.
