Problem 3.7: Design principles for toggles

Consider two components, A and B, which regulate each other. A may activate or repress B, and B may activate or repress A. There are three possible architectures in this scenario, two with positive feedback and one with negative feedback, shown below.

Toggles

a) A circuit can behave like a toggle if it has two stable steady states, one with A high and B low and another with B high and A low. Only one of the above architectures can function as a toggle. Which one? Explain in words and sketches why only the one you chose can be a toggle.

b) A and/or B may have ultrasensitive regulation, which we describe with a Hill coefficient, \(n\), greater than one. Show that without ultrasensitive regulation, even the architecture you chose cannot have toggle behavior.