Genetic regulatory networks that count to 3.

Sensing a graded input and differentiating between its different levels is at the core of many developmental decisions. Here, we want to examine how this can be realized for a simple system. We model gene regulatory circuits that reach distinct states when setting the underlying gene copy number to 1, 2 and 3. This distinction can be considered as counting the copy number. We explore different circuits that allow for counting and keeping memory of the count after resetting the copy number to 1. For this purpose, we sample different architectures and parameters, only considering circuits that contain repressive links, which we model by Michaelis-Menten terms. Interestingly, we find that counting to 3 does not require a hierarchy in Hill coefficients, in contrast to counting to 2, which is known from lambda phage. Furthermore, we find two main circuit architectures: one design also found in the vertebrate neural tube in a development governed by the sonic hedgehog morphogen and the more robust design of a repressilator supplemented with a weak repressilator acting in the opposite direction.