Dosage effects in morphogenetic gradients of transcription factors: insights from a simple mathematical model

The classical Hill equation is the simplest way to model sharp transcriptional responses (TR) to changing concentrations of a regulator. Such steep sigmoidal transitions of the TR are often involved in the creation of boundaries or domains during development. Here, I use the Hill function to explore the response of a promoter to a gradient of a transcription factor (TF). Specifically, I examine how various sources of nonlinearity (such as that imposed by the number of TF-binding sites) and the nonlinearity of the gradient itself influence TR and modulate its robustness to variations in the dosage of morphogen. This qualitative analysis underscores how nonlinearities can modulate the robustness of TR and is a convenient way to convey concepts in a teaching setting.