Identifying the mechanism of intercellular feedback regulation is critical for the basic understanding of tissue growth control in organisms. In this paper, we analyze a tissue growth model consisting of a single lineage of two cell types regulated by negative feedback signaling molecules that undergo spatial diffusion. By deriving the fixed points for the uniform steady states and carrying out linear stability analysis, phase diagrams are obtained analytically for arbitrary parameters of the model. Two different generic growth modes are found: blow-up growth and final-state controlled growth which are governed by the nontrivial fixed point and the trivial fixed point, respectively, and can be sensitively switched by varying the negative feedback regulation on the proliferation of the stem cells. Analytic expressions for the characteristic timescales for these two growth modes are also derived. Remarkably, the trivial and nontrivial uniform steady states can coexist and a sharp transition occurs in the bistable regime as the relevant parameters are varied. Furthermore, the bistable growth properties allows for the external control to switch between these two growth modes. In addition, the condition for an early accelerated growth followed by a retarded growth can be derived. These analytical results are further verified by numerical simulations and provide insights on the growth behavior of the tissue. Our results are also discussed in the light of possible realistic biological experiments and tissue growth control strategy. Furthermore, by external feedback control of the concentration of regulatory molecules, it is possible to achieve a desired growth mode, as demonstrated with an analysis of boosted growth, catch-up growth and the design for the target of a linear growth dynamic.

中文翻译：

---

两阶段细胞谱系模型中组织生长动力学的调节反馈效应

确定细胞间反馈调节的机制对于生物体组织生长控制的基本理解至关重要。在本文中，我们分析了一个组织生长模型，该模型由两个细胞类型的单一谱系组成，这些细胞类型由经历空间扩散的负反馈信号分子调节。通过导出均匀稳态的不动点并进行线性稳定性分析，可以解析地获得模型任意参数的相图。发现了两种不同的通用增长模式：爆发式增长和终态受控增长，它们分别由非平凡不动点和平凡不动点控制，并且可以通过改变对增殖的负反馈调节来灵敏地切换干细胞。还导出了这两种增长模式的特征时间尺度的解析表达式。值得注意的是，平凡和非平凡的均匀稳态可以共存，并且随着相关参数的变化，双稳态区域会发生急剧转变。此外，双稳态生长特性允许外部控制在这两种生长模式之间切换。此外，还可以推导出先加速生长后生长迟缓的条件。这些分析结果通过数值模拟进一步验证，并提供对组织生长行为的见解。我们的结果还根据可能的现实生物实验和组织生长控制策略进行了讨论。此外，通过调节分子浓度的外部反馈控制，