Temporal Dose-Response Inversion in a Biological Circuit

Dose-response curves are a standard approach to quantify the behavior of a biological system in response to a range of input concentrations. Similarly, one can examine the temporal dose-response of a biological system by measuring how long the system's output remains sustained in response to inputs of different duration. Experience suggests that a direct correlation should emerge between the duration of a stimulus and the corresponding duration of the output, but we have recently observed that some adaptive biological circuits have the ability for temporal dose-response inversion. We focus in particular on the capacity of these circuits to generate short-lived outputs in response to sustained inputs, and vice-versa generate sustained outputs in response to short-lived inputs. Building on these observations, here we provide a formal notion of inverse dose duration, or IDD, for positive systems, and we show that linear positive systems cannot exhibit IDD. Then, we consider a second order nonlinear model for the incoherent feedforward loop (IFFL) motif in biology, a well-known adaptive system, and we derive exact IDD results. In particular we identify and characterize parametric conditions for IDD. We also show that IDD can only emerge over a finite range of input durations, hence it is not a global property.