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

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.

• The paper introduces a dynamic sensitivity framework using integral formulas to predict cumulative dose response monotonicity in incoherent feedforward motifs.
• Methodologically, it derives sensitivity formulas and applies them to canonical IFFM variants to distinguish monotonic and nonmonotonic responses.
• The findings offer critical insights for reverse-engineering network motifs and tailoring experimental designs in systems and synthetic biology.

Monotonicity of Cumulative Dose Response in Incoherent Feedforward Motifs

Background and Motivation

Adaptive biological systems exhibit complex transient dynamics in response to constant inputs, with the ability to exhibit homeostatic output levels regardless of sustained stimuli due to underlying network architecture. These phenomena, found in signaling and gene regulatory networks, are often encoded by so-called incoherent feedforward motifs (IFFMs), in which an input exerts both activating and antagonistic influences on an output via a direct and an indirect (delayed) pathway. IFFMs are canonical modules in systems and synthetic biology, extensively studied for their ability to enable transient responses, adaptation, and fold-change detection [Ankit & Sontag 2025; Sontag 2025; Shoval et al. 2011].

A classical focus has been on the instantaneous or steady-state dose response (DR), mapping constant input magnitudes to outputs at fixed (or steady) times. However, many physiological experiments and theoretical contexts are more concerned with the cumulative dose response (cDR), defined as the time-integral of the output over a finite window, i.e., the area under the transient response curve. This is especially relevant in immune signaling (e.g., T-cell cytokine accumulation), pharmacology, and transient phenotype discrimination. Strikingly, the cDR can exhibit monotonicity with respect to input magnitude even when the DR is nonmonotone—making it nontrivial to deduce input-output structure from standard observations.

This paper develops a rigorous dynamic framework for analyzing monotonicity in cDR across a class of IFFMs with linear intermediate filtering and nonlinear output nodes, identifies structural conditions governing monotonicity and non-monotonicity, and applies these results to multiple canonical IFFM variants.

Formal Problem Statement and Theoretical Framework

Let $u \in \mathbb{R}_+$ be a constant input, $x \in \mathbb{R}^n_+$ the intermediate (filtering) state with stable linear dynamics, and $y \in \mathbb{R}_+$ the output. The general IFFM framework is:

$\dot x = A x + b u$

$\dot y = F(x, y, u)$

with $A$ Hurwitz and Metzler, $b \geq 0$, and $F$ nonlinear, subject to sign-antagonistic input structure ($\partial_u F \cdot \partial_x F < 0$), representing incoherence. The DR and cDR at time $T$ are $x \in \mathbb{R}^n_+$0 and $x \in \mathbb{R}^n_+$1.

Crucially, monotonicity of the map $x \in \mathbb{R}^n_+$2 is not equated with monotonicity in $x \in \mathbb{R}^n_+$3 or with properties of trajectories in time; the former reflects an aggregated, input-to-area-under-curve property that may not be inferred from instantaneous behaviors.

The authors derive a sensitivity formula for $x \in \mathbb{R}^n_+$4 based on system trajectories and kernel integrals:

$x \in \mathbb{R}^n_+$5

where $x \in \mathbb{R}^n_+$6 is a combination of sensitivities of $x \in \mathbb{R}^n_+$7 and $x \in \mathbb{R}^n_+$8 to $x \in \mathbb{R}^n_+$9, constructed via the chain rule and system equations. The structure of $y \in \mathbb{R}_+$0 and $y \in \mathbb{R}_+$1 (sign definiteness, monotonicity) controls the behavior of cDR.

Main Theorems:

• If both $y \in \mathbb{R}_+$2 and $y \in \mathbb{R}_+$3 (or, alternately, $y \in \mathbb{R}_+$4 and the cumulative $y \in \mathbb{R}_+$5) are sign-definite for all $y \in \mathbb{R}_+$6 and $y \in \mathbb{R}_+$7, then $y \in \mathbb{R}_+$8 is sign-definite—hence $y \in \mathbb{R}_+$9 is monotone.
• If, for two inputs $\dot x = A x + b u$0, the sign of $\dot x = A x + b u$1 reverses, then $\dot x = A x + b u$2 is not monotone over $\dot x = A x + b u$3.

This dimensionally agnostic formalism generalizes previous episodic analyses [Ankit & Sontag 2025] to vector-valued intermediate states and more complex nonlinearities, encompassing numerous motif classes.

Application to Canonical IFFM Variants

The analytical framework is instantiated on four principal IFFM architectures, identified as IFFM1–IFFM4:

• IFFM1: Output $\dot x = A x + b u$4 receives direct activation from $\dot x = A x + b u$5, and antagonistic inhibition via $\dot x = A x + b u$6 (which integrates $\dot x = A x + b u$7). Monotonicity of cDR is demonstrated even in settings where DR is nonmonotone due to transient overshoots.
• IFFM2: $\dot x = A x + b u$8 receives direct activation from $\dot x = A x + b u$9 with nonlinear Michaelis-Menten-type saturation. Both DR and cDR are monotonic in $\dot y = F(x, y, u)$0, with monotonicity driven by persistent sign structure in sensitivity.
• IFFM3: Output is directly inhibited by $\dot y = F(x, y, u)$1 while also receiving activation via $\dot y = F(x, y, u)$2. Despite possible nonmonotonic DR, cDR is monotonic due to the sign pattern in $\dot y = F(x, y, u)$3 and $\dot y = F(x, y, u)$4.
• IFFM4: Combines normalization and inhibition in a structure where the kernel and sensitivity integrals can change sign. Here, both DR and cDR can display nonmonotonicity in $\dot y = F(x, y, u)$5, analytically shown via asymptotic expansions and confirmed via simulation.

Key Numerical Result: For IFFM1 and IFFM3, cDR is strictly monotonic for all simulated initial conditions and parameters despite complex, nonmonotonic transient DR. IFFM2 is monotonic at both levels, while IFFM4 clearly loses monotonicity in cDR—confirming the sufficiency and essential sharpness of the structural criteria.

Implications and Future Directions

This work introduces a powerful, dynamical sensitivity-based method for dissecting the cDR monotonicity question in a wide class of positive linear/nonlinear IFFMs. Its implications are significant for both theoretical systems biology and experimental design. In particular:

• Reverse Engineering: Theoretical sharpness means that persistent cDR monotonicity (or lack thereof) can be used as a constraint for falsifying whole classes of candidate mechanisms given experimental area-under-curve data.
• Motif Inference: Nonmonotonic cDR cannot be generated by certain IFFM motifs, clarifying earlier experimental paradoxes regarding immune signaling (cf. [Ankit & Sontag 2025]).
• Generic Methodology: The integral sensitivity approach is not limited to elementary motifs and could be extended to more complex networks, including those with integral feedback, nonpositive state variables, or time-varying inputs.

Future research avenues include: identifying minimal necessary conditions for cDR monotonicity in more general (e.g., non-positive, nonlinear, stochastic) networks; extending to networks with feedback or multi-layered cascade architectures; and connecting with model discrimination methodologies for high-dimensional biological experiments.

Conclusion

The paper establishes a rigorous mathematical foundation for analyzing the monotonicity of cumulative dose response in dynamic biological networks exhibiting incoherent feedforward structure. Through general sensitivity integral characterizations, it delivers both comprehensive constructive conditions for monotonicity and demonstrates their essential necessity in typical motifs. The approach is broadly applicable for discriminating between network hypotheses in data-rich biological and synthetic systems, and opens avenues for further exploration in nonlinear and high-dimensional regimes.

Reference:

"When is cumulative dose response monotonic? Analysis of incoherent feedforward motifs" (2604.01573)