When is cumulative dose response monotonic? Analysis of incoherent feedforward motifs

TL;DR

This paper analyzes when the cumulative dose response in incoherent feedforward motifs remains monotonic, providing conditions and characterizations for different system types.

Contribution

It introduces a unified integral framework to determine monotonicity of cDR in IFFM systems, revealing structural influences on response behavior.

Findings

IFFM1 and IFFM3 have monotone cDR despite non-monotone DR.

IFFM2's monotone DR implies monotone cDR.

IFFM4 can lose monotonicity due to structural properties.

Abstract

We study the monotonicity of the cumulative dose response (cDR) for a class of incoherent feedforward motifs (IFFM) systems with linear intermediate dynamics and nonlinear output dynamics. While the instantaneous dose response (DR) may be nonmonotone with respect to the input, the cDR can still be monotone. To analyze this phenomenon, we derive an integral representation of the sensitivity of cDR with respect to the input and establish general sufficient conditions for both monotonicity and non-monotonicity. These results reduce the problem to verifying qualitative sign properties along system trajectories. We apply this framework to four canonical IFFM systems and obtain a complete characterization of their behavior. In particular, IFFM1 and IFFM3 exhibit monotone cDR despite potentially non-monotone DR, while IFFM2 is monotone already at the level of DR, which implies monotonicity of cDR. In contrast, IFFM4 violates these conditions, leading to a loss of monotonicity. Numerical simulations indicate that these properties persist beyond the structured initial conditions used in the analysis. Overall, our results provide a unified framework for understanding how network structure governs monotonicity in cumulative input-output responses.