Processing Oscillatory Signals by Incoherent Feedforward Loops

Authors: Anna Barredo Porta, Álvaro Borachok Fernandez, Pau Boncompte Carre, Gonzalo Plaza, Adriana Royuela Bermudez
Institution: Universitat Pompeu Fabra
Date: October 26th, 2023

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Overview

This repository contains the numerical simulations accompanying the study "Processing Oscillatory Signals by Incoherent Feedforward Loops". The work extends the analysis of Zhang et al. (2016) to characterise all four canonical types of Incoherent Feedforward Loop (IFFL) network motifs, examining their ability to process pulsatile/oscillatory biological signals.

IFFLs are three-node gene-regulatory motifs in which an input signal S reaches an output node R via two opposing paths, one direct and one indirect through an intermediate node X, always with an odd number of inhibitory arrows. Their key functional property is the capacity to either count input pulses (accumulating R proportionally to the number of pulses received) or adapt to sustained signals (resetting R to baseline), depending on pulse duration and kinetic parameters.

The central question explored here is: does the steady-state level of R encode the number of input pulses, their duration, or neither?

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Biological Context

Each IFFL type has documented biological realisations:

Type	Biological example
I1-FFL	Transcriptional Network of TNF-induced necroptosis: TNF activates NF-κB to counteract necrotic cell death
I2-FFL	Arabidopsis thaliana circadian clock: double-negative feedback loop between CCA1/LHY, PRR5/TOC1, and PRR9/7
I3-FFL	Synthetic stripe-forming circuit: two viral RNA polymerases (SP6-RNAP and T7-RNAP) and LacI for repression of GFP protein
I4-FFL	m-xylene degradation in TOL plasmid: P. putida mt-2 Pu gene induced upon IHF and XylR, which is repressed by IHF

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Repository Structure

.
├── IFFL_type1.ipynb   # I1-FFL: S→X→|R, S→R  (X represses R via Hill function)
├── IFFL_type2.ipynb   # I2-FFL: S→|X, AND(x<1, inp<1) → R
├── IFFL_type3.ipynb   # I3-FFL: S→X→R, S→|R  (input degrades R directly)
├── IFFL_type4.ipynb   # I4-FFL: S→|X, AND(x>1, inp>1) → R  (low cooperativity)
└── Processing_Oscillatory_Signals_by_Incoherent_Feedforward_Loops.pdf

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Model Equations

The input signal is a periodic square-wave φ(t) with k pulses, period T, and duration D:

φ(t) = 1  if  t₀ + iT ≤ t < t₀ + iT + D   (i = 0, …, k−1)
φ(t) = 0  otherwise

Type 1 (I1-FFL)

dx/dt = β·φ(t) − x
dr/dt = β·φ(t) − [γᵣ · xⁿ/(xⁿ+1) + γ₀] · r

X follows the input directly; once it crosses a threshold it represses R via a high-cooperativity Hill function. Short pulses allow X to return below threshold between pulses, enabling R to count; long pulses keep X above threshold, resetting R each cycle (adaptation).

Type 2 (I2-FFL)

dx/dt = αₓ / (1 + (β·φ(t))ⁿ) − x
dr/dt = αᵣ · AND(x < 1, β·φ(t) < 1) − γ₀ · r

The signal represses X; R is produced only when both X and the input are below threshold (AND gate). The production window opens after each pulse ends (when the input is off but X is still below threshold), making R accumulation sensitive to pulse duration. For β < 1, X never crosses its threshold and counting is not possible.

Type 3 (I3-FFL)

dx/dt = β·φ(t) − γₓ·x
dr/dt = xⁿ/(xⁿ+1) − r·(γᵣ·β·φ(t) + γ₀)

X is activated by the input; R is activated by X but repressed directly by the input signal through an additive degradation term γᵣ·S. This creates a distinctive inverse (decreasing) counting behaviour: R decreases with each additional pulse, the opposite of I1-, I2-, and I4-FFLs. The I3-FFL is the least common IFFL motif in known transcription networks (S. cerevisiae, E. coli), likely due to its limited biological utility in this dual-response mode.

Type 4 (I4-FFL)

dx/dt = αₓ / (1 + (β·φ(t))ⁿ⁼²) − x
dr/dt = αᵣ · AND(x > 1, β·φ(t) > 1) − γ₀ · r

Structurally similar to I2-FFL, but with low cooperativity (n=2) for X and an AND gate that opens at pulse onset (when X is still above threshold and the input has just turned on). For long pulses, X is driven below threshold before the input turns off, closing the gate and enabling adaptation; for short pulses, the gate stays briefly open per cycle, enabling counting.

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Counting Quality Metric

Following Zhang et al. (2016), counting quality is assessed by fitting a linear regression to the final value of R (at a fixed endpoint) as a function of the number of pulses k:

• High-quality counting: R² ≥ 0.99
• Low-quality counting: 0 < R² < 0.99
• Adaptation: R² low or negative slope (for I3-FFL)

The slope of the regression is used as a proxy for the response magnitude per pulse.

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Analyses Performed (per notebook)

Each notebook covers:

1. Time-series visualisation: S, X, and R trajectories for short (D=1) and long (D=3–3.5) pulses, illustrating the counting vs. adaptation duality.
2. Counting calibration curve: Final R vs. number of pulses, with linear regression overlay, R², and slope reported.
3. Effect of basal degradation γ₀: How reporter stability affects counting quality; high γ₀ degrades linearity.
4. Parameter heatmaps: R² and slope swept over 2D grids of:
   • β (input amplitude) × D (pulse duration)
   • γᵣ or αₓ × D (depending on topology)
5. Analytical boundary (I1-FFL only): The theoretical counting/adaptation boundary (1 − e^−D)/(1 − e^−T) = 1/β plotted in log-log scale for several β values.

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Key Findings

Type	Counting direction	Key parameter controlling counting boundary
I1-FFL	Upward	β and γᵣ; analytical boundary exists
I2-FFL	Upward	Pulse duration D; β < 1 always prevents counting
I3-FFL	Downward	γᵣ; requires high initial R concentration
I4-FFL	Upward	αₓ; low αₓ prevents sufficient X repression

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Requirements

Package	Purpose
numpy	Numerical arrays
scipy	ODE integration (odeint), linear regression
matplotlib	Plotting

Install with:

pip install numpy scipy matplotlib

Python 3.8+ recommended.

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Usage

Each notebook is self-contained. Open any of them in Jupyter Lab, Jupyter Notebook, or VS Code and run all cells in order.

jupyter lab IFFL_type1.ipynb

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Key Parameters

Symbol	Description
β	Input signal amplitude (must be > 1 for I1/I3; < 1 for I2/I4 in counting regime)
D	Pulse duration
T	Inter-pulse period (default T = 4)
n	Hill-function cooperativity (n = 110 for near-step; n = 2 for I4-FFL)
γ₀	Basal degradation rate of R (high values reduce counting quality)
γᵣ	Input-dependent or X-dependent degradation rate of R
γₓ	Basal degradation rate of X (I3-FFL)
αₓ, αᵣ	Maximum production rates of X and R

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References

• Zhang, C., Tsoi, R., Wu, F., and Li, Y. (2016). Processing oscillatory signals by incoherent feedforward loops. PLOS Computational Biology, 12(9): e1005101. (primary reference)
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