In this work, we study the behavior of a time-delayed mutually repressive auto-activating three-gene system. Delays are introduced to account for the location difference between DNA transcription that leads to production of messenger RNA and its translation that result in protein synthesis. We study the dynamics of the system using numerical simulations, computational bifurcation analysis and mathematical analysis. We find Hopf bifurcations leading to stable and unstable rotation in the system, and we study the rotational behavior as a function of cyclic mutual repression parameter asymmetry between each gene pair in the network. We focus on how rotation co-exists with a stable heteroclinic flow linking the three saddles in the system. We find that this coexistence allows for a transition between two markedly different types of rotation leading to strikingly different phenotypes. One type of rotation belongs to Hopf-induced rotation while the other type, belongs to heteroclinic cycling between three saddle nodes in the system. We discuss the evolutionary and biological implications of our findings.

在这项工作中，我们研究了延时相互抑制的自激活三基因系统的行为。引入延迟是为了解释导致产生信使 RNA 的 DNA 转录与其导致蛋白质合成的翻译之间的位置差异。我们使用数值模拟、计算分岔分析和数学分析来研究系统的动力学。我们发现了导致系统稳定和不稳定旋转的 Hopf 分岔，我们研究了旋转行为作为网络中每个基因对之间循环相互抑制参数不对称的函数。我们关注旋转如何与连接系统中三个鞍座的稳定异斜流共存。我们发现这种共存允许两种明显不同类型的旋转之间的过渡，从而导致显着不同的表型。一种类型的旋转属于 Hopf 引起的旋转，而另一种类型属于系统中三个鞍节点之间的异宿循环。我们讨论了我们发现的进化和生物学意义。