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.

中文翻译：

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圆形肘节多边形的工作原理

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常见网络主题是两个自激活“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，它可以允许三个稳定状态-（高A，低B），（低A，高B）和（中A，中B）。触发三合一开关-三个相互抑制的调节器A，B和C，即三个圆形排列的触发开关（在A和B之间，在B和C之间以及在A和C之间）-可以允许六个稳定状态：三个“单正”和三个“双重肯定”的。但是，较大的拨动多边形的操作原理，即圆形布置以形成多边形的拨动开关，仍然不清楚。这里，我们使用离散和连续方法来模拟不同大小的切换多边形的动力学。我们根据多边形是偶数还是奇数观察到了处于稳态频率的模式。偶数拨动多边形导致两个主导状态，网络的连续分量表示交替的高电平和低电平。另一方面，奇数切换多边形启用更多数量的状态，通常是组件数量的两倍，且其成分遵循“圆形置换”模式。结合自我激活可以保留这些趋势，同时增加相应网络中多稳定性的频率。我们的研究结果可洞悉参与细胞命运决策的监管部门循环安排的设计原理，