On the Distribution of State Values of Reproducing Cells: the General   Evolution Equation and its Applications
  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. 
Katsuhiko Sato and Kunihiko Kaneko