Decoding the dynamics of cellular decision-making and cell differentiation is a central question in cell and developmental biology. A common network motif involved in many cell-fate decisions is a mutually inhibitory feedback loop between two self-activating ‘master regulators’ A and B, also called as toggle switch. Typically, it can allow for three stable states—(high A, low B), (low A, high B) and (medium A, medium B). A toggle triad—three mutually repressing regulators A, B and C, i.e. three toggle switches arranged circularly (between A and B, between B and C, and between A and C)—can allow for six stable states: three ‘single positive’ and three ‘double positive’ ones. However, the operating principles of larger toggle polygons, i.e. toggle switches arranged circularly to form a polygon, remain unclear. Here, we simulate using both discrete and continuous methods the dynamics of different sized toggle polygons. We observed a pattern in their steady state frequency depending on whether the polygon was an even or odd numbered one. The even-numbered toggle polygons result in two dominant states with consecutive components of the network expressing alternating high and low levels. The odd-numbered toggle polygons, on the other hand, enable more number of states, usually twice the number of components with the states that follow ‘circular permutation’ patterns in their composition. Incorporating self-activations preserved these trends while increasing the frequency of multistability in the corresponding network. Our results offer insights into design principles of circular arrangement of regulatory units involved in cell-fate decision making, and can offer design strategies for synthesizing genetic circuits.

解码细胞决策和细胞分化的动力学是细胞和发育生物学的核心问题。许多细胞命运决定中涉及的一个常见网络基序是两个自激活“主调节器”A 和 B 之间的相互抑制反馈回路，也称为拨动开关。通常，它可以允许三种稳定状态——（高 A、低 B）、（低 A、高 B）和（中 A、中 B）。拨动三元组——三个相互抑制的调节器 A、B 和 C，即三个循环排列的拨动开关（A 和 B 之间、B 和 C 之间、A 和 C 之间）——可以允许六个稳定状态：三个“单正”和三个“双阳性”。然而，较大的拨动多边形的工作原理，即圆形排列形成多边形的拨动开关，仍不清楚。这里，我们使用离散和连续方法模拟不同大小的切换多边形的动态。我们观察到它们稳态频率的模式取决于多边形是偶数还是奇数。偶数切换多边形导致两个主要状态，网络的连续组件表示交替的高低电平。另一方面，奇数切换多边形启用更多数量的状态，通常是其组成中遵循“圆形排列”模式的状态的组件数量的两倍。结合自激活保留了这些趋势，同时增加了相应网络中多重稳定性的频率。我们的结果提供了对参与细胞命运决策的调节单元循环排列的设计原则的见解，