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Continuous surrogates versus threshold Boolean networks for modeling Arabidopsis ISR gene regulation

This study compares continuous machine learning models (Random Forest and MLP) with a discrete threshold Boolean network for modeling *Arabidopsis* ISR gene regulation, revealing that while continuous surrogates excel in local numerical accuracy, threshold Boolean networks offer superior global qualitative dynamical fidelity, suggesting the two approaches are complementary tools for biological modeling.

Original authors: Gonzalo A. Ruz

Published 2026-07-28

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CC BY 4.0

Original authors: Gonzalo A. Ruz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine a bustling city where every building is a cell, and inside each building, thousands of tiny workers (genes) are constantly talking to one another. They don't speak in full sentences; instead, they send simple signals: "Turn on the lights!" or "Shut the door!" This complex conversation is called a Gene Regulatory Network. It's the brain of the cell, deciding how to react when things change, like when a plant gets attacked by a bug or needs to grow. Scientists have been trying to build computer models to predict how these workers will talk next. But they face a tricky choice: should they build a model that tries to guess the exact volume of every voice (a continuous model), or should they build a model that only cares if the voice is "loud" or "quiet" (a discrete model)? One approach is great for math, but the other is easier for humans to understand. The big question is: which one actually tells the true story of how the cell behaves over time?

This paper dives into that exact dilemma using a specific plant called Arabidopsis thaliana (a tiny weed that scientists love to study) and its defense system against bacteria. The researchers wanted to see if a fancy, flexible computer program could predict the plant's future better than a simpler, rule-based logic machine. They tested two "continuous" models—Random Forest (think of it as a committee of many decision trees voting on the answer) and a Multi-Layer Perceptron (a simple type of artificial brain)—against a Threshold Boolean Network (a strict logic machine that only sees "on" or "off" switches).

Here is what they found, and it's a bit of a plot twist. When looking at just the next single moment in time, the Random Forest was the best at guessing the exact numbers. It had an average error (MAE) of 1.910 and a root mean squared error (RMSE) of 2.836, beating the artificial brain (MLP) which had errors of 2.089 and 3.106. However, when the scientists asked these models to keep predicting step-by-step into the future—like rolling a ball down a hill to see where it ends up—the story changed completely.

The Threshold Boolean Network was the undisputed champion of the long game. When it tried to recreate the plant's defense story over time, it got it 100% correct (a trajectory binary accuracy of 1.000). It matched the observed pattern perfectly, with zero mistakes. The artificial brain (MLP) was also a hero, getting almost everything right with a 0.986 accuracy and only one tiny mistake. But the Random Forest, despite being the best at the single-step math, stumbled badly when rolling forward. Its path drifted away from reality, ending up with a much lower accuracy of 0.708 and accumulating 21 total errors.

The paper suggests that being good at guessing the exact number right now doesn't mean you'll understand the whole story later. The simple, logic-based model (the Boolean Network) was better at capturing the "shape" of the plant's reaction, even though it ignored the exact numbers. The complex models were great at the math but lost the plot over time. The authors conclude that these aren't rivals to be beaten, but rather complementary tools. If you need to know the precise volume of a gene's voice, use the continuous models. But if you want to understand the logical story of how the plant defends itself, the simple, interpretable Boolean network is the way to go. It turns out, sometimes a simple "on/off" switch tells a truer story than a complex calculator.

Technical Summary: Continuous Surrogates versus Threshold Boolean Networks for Modeling Arabidopsis ISR Gene Regulation

Problem Statement
Gene regulatory network (GRN) modeling faces a persistent tradeoff between predictive accuracy and mechanistic interpretability. While flexible machine learning models often excel at local numerical prediction, they may lack the explicit dynamical representation required to understand regulatory logic. Conversely, discrete models like Boolean networks offer interpretability but may struggle with complex, nonlinear dynamics. This study addresses the question of how continuous surrogate models compare to discrete mechanistic models when applied to the same biological system, specifically examining whether high local numerical accuracy in the continuous domain translates to faithful qualitative dynamical behavior in the binary domain.

Methodology
The study utilizes a time-series dataset of eight defense-related genes (PR1, PDF1.2, WRKY70, WRKY54, WRKY33, MYC2, ERF1, LOX2) from Arabidopsis thaliana responding to induced systemic resistance (ISR) triggered by Paraburkholderia phytofirmans PsJN. The dataset comprises nine temporal observations (0 to 24 hours).

The authors formulated a unified comparison using two representations of the same data:

Continuous Representation: Raw gene expression values used to train and evaluate continuous surrogate models.
Discrete Representation: A sign-binarized version of the data (1 if expression > 0, 0 otherwise) used to train and evaluate a discrete model.

Three models were compared:

Random Forest (RF) Regression: A non-parametric ensemble method trained to predict the next continuous state vector. It included time increments ( $\Delta t$ ) as an input feature.
Multi-Layer Perceptron (MLP): A neural network with a single hidden layer, trained to predict the next continuous state vector.
Threshold Boolean Network (TBN): A discrete mechanistic model where gene states update based on weighted sums and thresholds. It was trained node-wise using a perceptron rule on the binarized transitions.

Evaluation Protocol
Given the small dataset size (eight one-step transitions), the study employed a rolling-origin evaluation scheme to preserve temporal causality. Models were trained on an expanding set of initial transitions and tested on the immediate next unseen transition. Performance was assessed via:

One-step Prediction: Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) for continuous models; Binary Accuracy and Hamming Distance for all models (after binarizing continuous predictions).
Recursive Multi-step Rollout: Starting from the first observed state, models were iteratively applied to generate future trajectories. This assessed global dynamical fidelity and error accumulation.
Interpretability: RF feature importance scores were analyzed, while the TBN's learned weight matrix was visualized to identify putative activating (positive) and inhibitory (negative) interactions.

Key Results

One-Step Numerical Performance: In the continuous domain, Random Forest achieved the best average performance (MAE: 1.910, RMSE: 2.836), outperforming the MLP (MAE: 2.089, RMSE: 3.106).
One-Step Qualitative Performance: In the binary domain, the TBN achieved the highest average binary accuracy (0.550) and lowest Hamming distance (3.600), outperforming both RF (0.500) and MLP (0.495).
Recursive Rollout Fidelity:
- The TBN exactly reproduced the observed binarized trajectory (Trajectory Accuracy: 1.000, Hamming Distance: 0).
- The MLP demonstrated near-perfect fidelity in the binary domain (Trajectory Accuracy: 0.986, Hamming Distance: 1), despite not having the best one-step numerical scores.
- The RF accumulated substantial deviation over time, resulting in a trajectory binary accuracy of 0.708 and a total Hamming distance of 21.
Interpretability: The RF feature importance was diffuse, lacking a clear regulatory structure. In contrast, the TBN provided an explicit signed interaction map, revealing a nontrivial regulatory structure with specific genes (e.g., ERF1, WRKY70, WRKY33, MYC2) playing dominant roles.

Significance and Claims
The paper claims that local numerical accuracy and global qualitative dynamical fidelity are not necessarily aligned. The results demonstrate that a model performing best in one-step continuous prediction (RF) may fail to preserve the qualitative dynamics of the system during recursive rollout, whereas a discrete mechanistic model (TBN) can perfectly recover the qualitative trajectory despite being trained only on binarized data.

The study concludes that continuous surrogates and threshold Boolean networks should be viewed as complementary tools rather than competing approaches. Continuous models are well-suited for capturing expression magnitudes, while TBNs offer a more interpretable representation of qualitative regulatory logic. The authors position this work as a proof-of-concept case study highlighting the value of evaluating both predictive accuracy and dynamical fidelity when modeling gene regulatory systems, rather than establishing a universal ranking of modeling approaches. The findings suggest that for systems where qualitative regulatory progression is the primary interest, discrete mechanistic models may be preferable, even when continuous data is available.

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