On the distribution of state values of reproducing cells

Characterizing a cell state by measuring the degree of gene expression as well as its noise has gathered much attention. The distribution of such state values (e.g., abundances of some proteins) over cells has been measured, and is not only a result of intracellular process, but is also influenced by the growth in cell number that depends on the state. By incorporating the growth-death process into the standard Fokker-Planck equation, a nonlinear temporal evolution equation of distribution is derived and then solved by means of eigenfunction expansions. This general formalism is applied to the linear relaxation case. First, when the growth rate of a cell increases linearly with the state value x, the shift of the average x due to the growth effect is shown to be proportional to the variance of x and the relaxation time, similar to the biological fluctuation-response relationship. Second, when there is a threshold value of x for growth, the existence of a critical growth rate, represented again by the variance and the relaxation time, is demonstrated. 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.