再生细胞状态值分布的一般演化方程及其应用
On the Distribution of State Values of Reproducing Cells: the General Evolution Equation and its Applications
摘要：
细胞状态的波动，例如，一些蛋白质的丰度，引起了理论和实验上的广泛关注。然而，这种状态在细胞上的分布不仅是细胞内随机过程的结果，而且还受到依赖于状态的细胞数量增长的影响。通过将增长-死亡过程纳入概率分布的标准Fokker-Planck方程，得到了一个包含自洽增长项的非线性时间分布演化方程.导出的方程一般通过本征函数展开解析求解。通过关注线性松弛的情况下，两个例子被认为是所提出的一般形式主义的应用。首先，通过假设细胞的生长速率随着状态值$x$线性地增加，由于生长效应导致的平均状态值$x$的偏移被示出为与状态$x$和弛豫时间的方差成比例，类似于生物波动-响应关系。第二，当在状态$x$的某个阈值处的增长率存在差距时，证明存在临界差距值，超过该临界差距值，平均增长率开始增加。该临界值再次根据弛豫时间和$x$的方差获得，所有这些都是实验上可测量的量。的相关性的结果，以分析的生物数据的细胞状态的分布，例如通过流式细胞术获得的，进行了讨论。
英文摘要：
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.
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