Breitsch Nathan, Moses Gregory, Boczko Erik, Young Todd
Department of Mathematics, Ohio University, Athens, OH, USA,
J Math Biol. 2015 Apr;70(5):1151-75. doi: 10.1007/s00285-014-0786-7. Epub 2014 May 10.
We study a model of cell cycle ensemble dynamics with cell-cell feedback in which cells in one fixed phase of the cycle S (Signaling) produce chemical agents that affect the growth and development rate of cells that are in another phase R (Responsive). For this type of system there are special periodic solutions that we call k-cyclic or clustered. Biologically, a k-cyclic solution represents k cohorts of synchronized cells spaced nearly evenly around the cell cycle. We show, under very general nonlinear feedback, that for a fixed k the stability of the k-cyclic solutions can be characterized completely in parameter space, a 2 dimensional triangle T. We show that T is naturally partitioned into k(2) sub-triangles on each of which the k-cyclic solutions all have the same stability type. For negative feedback we observe that while the synchronous solution (k = 1) is unstable, regions of stability of k ≥ 2 clustered solutions seem to occupy all of T. We also observe bi-stability or multi-stability for many parameter values in negative feedback systems. Thus in systems with negative feedback we should expect to observe cyclic solutions for some k. This is in contrast to the case of positive feedback, where we observe that the only asymptotically stable periodic orbit is the synchronous solution.
我们研究了一种具有细胞间反馈的细胞周期总体动力学模型，其中处于周期S（信号传导）的一个固定阶段的细胞会产生化学物质，这些化学物质会影响处于另一阶段R（响应）的细胞的生长和发育速率。对于这种类型的系统，存在特殊的周期解，我们称之为k循环或聚集解。从生物学角度来看，k循环解代表了在细胞周期中几乎均匀分布的k组同步细胞。我们表明，在非常一般化的非线性反馈下，对于固定的k值，可以在二维三角形参数空间T中完全表征k循环解 的稳定性特征。我们证明，T自然地被划分为k²个子三角形，在每个子三角形上，k循环解都具有相同的稳定性类型。对于负反馈情况我们观察到，虽然同步解（k = 1）是不稳定的，但k≥2聚集解 的稳定区域似乎占据了整个T。我们还在负反馈系统中观察到许多参数值存在双稳态或多稳态现象。因此，在具有负反馈的系统中，我们应该预期会观察到某些k值的循环解。这与正反馈的情况形成对比，在正反馈情况下我们观察到唯一渐近稳定的周期轨道是同步解。
