Operating Principles of Circular Toggle Polygons

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 toggle switch. Typically, it can allow for three stable states. A toggle triad – three mutually repressing regulators A, B and C, i.e. three toggle switches arranged circularly 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 the dynamics of different sized toggle polygons. We observed a pattern in their steady state frequency depending on whether the polygon was even or odd numbered. 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 networks enable usually twice the number of components with the states that follow 'circular permutation' patterns. 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.