Learning Genetic Circuit Modules with Neural Networks: Full Version

Domitilla Del Vecchio; Eduardo D. Sontag; Jichi Wang

arxiv: 2509.19601 · v2 · submitted 2025-09-23 · 💻 cs.LG · cs.SY· eess.SP· eess.SY

Learning Genetic Circuit Modules with Neural Networks: Full Version

Jichi Wang , Eduardo D. Sontag , Domitilla Del Vecchio This is my paper

Pith reviewed 2026-05-18 13:42 UTC · model grok-4.3

classification 💻 cs.LG cs.SYeess.SPeess.SY

keywords modular identifiabilitygenetic circuitsneural networkssynthetic biologysystem identificationcompositional learninginput-output mapping

0 comments

The pith

Knowing the connections between modules in a genetic circuit allows a neural network to learn each module's input-output function from reduced system data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that prior knowledge of how modules are composed in a system enables identification of the individual modules' input and output behaviors using less training data than would otherwise be needed. It introduces modular identifiability to provide theoretical guarantees for recovering these module functions from partial observations of the overall system. A reader would care because this reduces the experimental burden in synthetic biology where data collection is expensive. The work demonstrates through theory and computation that a structure-aware neural network succeeds in module identification and in predicting system behavior on unseen inputs, while an agnostic network does not generalize.

Core claim

The authors claim that modular identifiability allows recovery of modules' input/output functions from a subset of the system's input/output data for systems motivated by genetic circuits, and that a neural network incorporating the compositional structure can learn these functions and predict outputs outside the training distribution.

What carries the argument

Modular identifiability, the property that the modules' input-output functions can be recovered from system-level data when the composition architecture is known in advance.

Load-bearing premise

The way the modules are connected in the overall system must be known ahead of time to use that knowledge in constraining the learning.

What would settle it

Observe whether the structure-aware neural network correctly recovers the individual module functions and accurately predicts system outputs for new input values not seen during training on a genetic circuit example with known architecture.

Figures

Figures reproduced from arXiv: 2509.19601 by Domitilla Del Vecchio, Eduardo D. Sontag, Jichi Wang.

**Figure 1.** Figure 1: Illustration of the system Σ. The system consists of n modules, each defined by yi = fi(ui) for i ∈ {1, . . . , n}, whose outputs are fed into a composition map G. In this paper, we will consider the problem of identifying the functions fi , i ∈ {1, ..., n}, and parameter θ from measurements of u = (u1, . . . , un) ∈ R n and Y . Specifically, we are interested in solving this problem when the inputs ui ar… view at source ↗

**Figure 2.** Figure 2: Regulation of a single gene expression module. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Define EG := maxk |G(fˆ(uk)) − G(f(uk))|/ maxk G(f(uk)) and Ef := maxk |fˆ(uk) − f(uk)|/ maxk f(uk), in which uk are the elements of the training set. (a) The plot shows the error convergence over 1000 epochs. The training dataset comprises 100 input/output pairs (uk, Yk), where inputs uk is uniformly sampled from the interval [0, 1]. We choose fˆ(u) as a fully connected feedforward NNET with four hidden l… view at source ↗

**Figure 4.** Figure 4: Illustration showing two transcriptional regulation modules that [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: Error convergence over 84,000 epochs for the case of two modules. ˆˆˆ [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

read the original abstract

In several applications, including in synthetic biology, one often has input/output data on a system composed of many modules, and although the modules' input/output functions and signals may be unknown, knowledge of the composition architecture can significantly reduce the amount of training data required to learn the system's input/output mapping. Learning the modules' input/output functions is also necessary for designing new systems from different composition architectures. Here, we propose a modular learning framework, which incorporates prior knowledge of the system's compositional structure to (a) identify the composing modules' input/output functions from the system's input/output data and (b) achieve this by using a reduced amount of data compared to what would be required without knowledge of the compositional structure. To achieve this, we introduce the notion of modular identifiability, which allows recovery of modules' input/output functions from a subset of the system's input/output data, and provide theoretical guarantees on a class of systems motivated by genetic circuits. We demonstrate the theory on computational studies showing that a neural network (NNET) that accounts for the compositional structure can learn the composing modules' input/output functions and predict the system's output on inputs outside of the training set distribution. By contrast, a neural network that is agnostic of the structure is unable to predict on inputs that fall outside of the training set distribution. By reducing the need for experimental data and allowing module identification, this framework offers the potential to ease the design of synthetic biological circuits and of multi-module systems more generally.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Knowing the exact wiring diagram lets a structure-aware neural net recover per-module functions from system data with less training and better generalization than black-box learning, but the whole thing rests on that diagram being perfect.

read the letter

This paper's main point is that if you know the wiring between genetic circuit modules, a neural network can recover the input-output behavior of each module from overall system measurements, using less data than a black-box approach would need. They provide theoretical guarantees for modular identifiability on systems inspired by genetic circuits. What stands out is the introduction of modular identifiability as a way to factor the learning problem using the known architecture. The computational studies are straightforward and show the structure-aware network succeeding at out-of-distribution prediction where the agnostic one fails. That comparison is useful and gives credit to the idea. The soft spot is the reliance on exact prior knowledge of how the modules connect. In practice, synthetic biology often involves unknown crosstalk or incomplete diagrams, and the paper's simulations assume the architecture is given perfectly. Without tests on misspecified graphs, it's hard to know how well this holds up when the assumption is violated even mildly. The central claim would weaken if that prior is off. No signs of circular reasoning or data issues in what is described. The work builds on prior modular identification but applies it to NNs in this context. This is for synthetic biologists and people working on learning in modular systems. A reader who cares about scaling circuit design with fewer experiments would find the framework and results relevant. It has enough new theory and clear experiments to merit a serious referee, who could probe the proofs and the practical assumptions. I recommend sending it to peer review rather than desk rejecting it.

Referee Report

2 major / 2 minor

Summary. The paper presents a modular learning framework that incorporates prior knowledge of a system's compositional architecture to identify the input/output functions of composing modules from overall system I/O data, while reducing data requirements. It introduces the notion of modular identifiability and provides theoretical guarantees for a class of systems motivated by genetic circuits. Computational studies with neural networks show that structure-aware models can recover module functions and generalize to out-of-distribution inputs, whereas structure-agnostic networks cannot.

Significance. If the results hold, the framework could meaningfully reduce experimental data needs in synthetic biology for characterizing and redesigning multi-module genetic circuits via module identification and reuse. The combination of a new identifiability notion with theoretical guarantees and empirical evidence of improved generalization is a strength; the explicit use of known architecture as prior knowledge to factor the I/O map is a clear technical contribution.

major comments (2)

[Abstract and theoretical section] Abstract and theoretical development (modular identifiability definition): The central guarantees require exact knowledge of the composition graph to factor the overall I/O map into per-module functions. The manuscript should add a robustness analysis (e.g., bounds or experiments) showing how small errors in the assumed wiring—such as missing/extra edges from crosstalk—affect identifiability and data efficiency, as this is a load-bearing assumption for the genetic-circuit motivation.
[Computational studies] Computational studies section: All reported experiments use perfectly known architectures. To substantiate the claim that the approach eases design of synthetic biological circuits, the paper should include results under modest architecture misspecification or added unmodeled interactions; without them the generalization advantage over agnostic NNs is not shown to survive realistic conditions.

minor comments (2)

[Theoretical results] Clarify the precise class of systems (e.g., linearity, monotonicity, or other properties) for which the modular identifiability theorem holds; a concise statement would help readers assess scope.
[Experimental setup] Provide full details on neural-network architectures, training procedures, and data-exclusion rules to support reproducibility of the computational studies.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of the significance of our work and for the constructive major comments. We respond to each point below and indicate the revisions we will make to the manuscript.

read point-by-point responses

Referee: [Abstract and theoretical section] Abstract and theoretical development (modular identifiability definition): The central guarantees require exact knowledge of the composition graph to factor the overall I/O map into per-module functions. The manuscript should add a robustness analysis (e.g., bounds or experiments) showing how small errors in the assumed wiring—such as missing/extra edges from crosstalk—affect identifiability and data efficiency, as this is a load-bearing assumption for the genetic-circuit motivation.

Authors: We concur that robustness to inaccuracies in the assumed composition graph is crucial for the applicability to genetic circuits, where phenomena like crosstalk could introduce wiring errors. Our theoretical results on modular identifiability are derived under the assumption of precise knowledge of the composition architecture, which enables the factorization of the overall I/O map. In the revised manuscript, we will incorporate a new subsection in the theoretical development that provides a preliminary robustness analysis. This will include analytical bounds on the perturbation of module function estimates for small graph errors in a simplified setting, as well as a brief discussion of implications for data efficiency. We believe this addresses the core concern while noting that a exhaustive study of all possible misspecifications lies outside the current scope. revision: partial
Referee: [Computational studies] Computational studies section: All reported experiments use perfectly known architectures. To substantiate the claim that the approach eases design of synthetic biological circuits, the paper should include results under modest architecture misspecification or added unmodeled interactions; without them the generalization advantage over agnostic NNs is not shown to survive realistic conditions.

Authors: We appreciate this suggestion to enhance the relevance of our computational studies to real-world synthetic biology scenarios. While the primary experiments demonstrate the benefits under exact architecture knowledge, we will add new results in the revised version. Specifically, we will include simulations with modest misspecifications, such as randomly adding or removing a small percentage of edges in the composition graph or introducing unmodeled nonlinear interactions. These experiments will compare the performance of the structure-aware neural network against the agnostic baseline in terms of module recovery and out-of-distribution generalization. We expect to show that the modular approach retains some advantages even under mild misspecification, thereby strengthening the claims regarding reduced data needs in circuit design. revision: yes

Circularity Check

0 steps flagged

Modular identifiability uses externally supplied architecture as prior; derivation remains self-contained

full rationale

The paper defines modular identifiability to enable recovery of module I/O functions from system-level data when the composition architecture is known in advance. This architecture functions as an independent input constraint rather than a quantity derived or fitted within the learning process. Theoretical guarantees are stated for a specific class of systems motivated by genetic circuits, and the computational studies contrast the structured NNET against an agnostic baseline to show improved extrapolation. No quoted equations or steps reduce a claimed prediction to a fitted parameter by construction, nor does any load-bearing result collapse to a self-citation chain or self-referential definition. The framework is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the composition architecture is known and that the class of systems motivated by genetic circuits satisfies conditions for modular identifiability; no free parameters or invented entities are mentioned.

axioms (2)

domain assumption The composition architecture of the system is known a priori and can be used to constrain module identification.
Stated in the abstract as the key prior knowledge that reduces data needs.
domain assumption The systems belong to a class motivated by genetic circuits for which modular identifiability holds.
The theoretical guarantees are provided specifically for this class.

pith-pipeline@v0.9.0 · 5807 in / 1288 out tokens · 46407 ms · 2026-05-18T13:42:02.932214+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1: modular identifiability on uni-modular input set U ... G(ˆf1(u1),…,ˆfn(un),ˆθ)=G(f1(u1),…,fn(un),θ) for all u∈U implies ˆfi=fi and ˆθ=θ
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Y = θ f(u)/(1+f(u)) and multi-module resource-competition form Gi=θi fi/(1+sum fj)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Retroactivity controls the temporal dynamics of gene transcription,

S. Jayanthi, K. S. Nilgiriwala, and D. D. Vecchio, “Retroactivity controls the temporal dynamics of gene transcription,”ACS Synthetic Biology, vol. 2, no. 8, pp. 431–441, 2013

work page 2013

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Modular analysis and design of biological circuits,

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work page 2020

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A load driver device for engineering modularity in biological networks,

D. Mishra, P. M. Rivera, A. Lin, D. D. Vecchio, and R. Weiss, “A load driver device for engineering modularity in biological networks,” Nature Biotechnology, vol. 32, no. 12, pp. 1268–1275, 2014

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Modular cell biology: retroactivity and insulation,

D. D. Vecchio, A. J. Ninfa, and E. D. Sontag, “Modular cell biology: retroactivity and insulation,”Molecular Systems Biology, vol. 4, no. 1, p. 161, 2008

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Resource competition shapes the response of genetic circuits,

Y . Qian, H.-H. Huang, J. I. Jim ´enez, and D. D. Vecchio, “Resource competition shapes the response of genetic circuits,”ACS Synthetic Biology, vol. 6, no. 7, pp. 1263–1272, 2017

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Understanding resource competition to achieve predictable synthetic gene expression in eukaryotes,

R. D. Blasi, J. Gabrielli, K. Shabestary, I. Ziarti,et al., “Understanding resource competition to achieve predictable synthetic gene expression in eukaryotes,”Nature Reviews Bioengineering, vol. 2, no. 9, pp. 721– 732, 2024

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D. D. Vecchio, “Modularity, context-dependence, and insulation in engineered biological circuits,”Trends in Biotechnology, vol. 33, no. 2, pp. 111–119, Feb 2015

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A. Gy ¨orgy, J. I. Jim´enez, J. Yazbek, H.-H. Huang, H. Chung, R. Weiss, and D. D. Vecchio, “Isocost lines describe the cellular economy of genetic circuits,”Biophysical Journal, vol. 109, no. 3, pp. 639–646, 2015

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An endoribonuclease-based feedforward controller for decoupling resource-limited genetic modules in mammalian cells,

R. D. Jones, Y . Qian, V . Siciliano, B. DiAndreth, J. Huh, R. Weiss, and D. D. Vecchio, “An endoribonuclease-based feedforward controller for decoupling resource-limited genetic modules in mammalian cells,” Nature Communications, vol. 11, p. 5690, 2020

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Characterization and mitigation of gene expression burden in mammalian cells,

T. Frei, F. Cella, F. Tedeschi, J. Guti ´errez, G.-B. Stan, M. Khammash, and V . Siciliano, “Characterization and mitigation of gene expression burden in mammalian cells,”Nature Communications, vol. 11, p. 4641, 2020

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A. Pandey, W. Poole, A. Swaminathan, V . Hsiao, and R. M. Murray, “Fast and flexible simulation and parameter estimation for synthetic biology using bioscrape,”Journal of Open Source Software, vol. 8, no. 83, p. 5057, 2023

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W. Poole, A. Pandey, A. Shur, Z. A. Tuza, and R. M. Murray, “Biocrn- pyler: Compiling chemical reaction networks from biomolecular parts in diverse contexts,”PLoS Computational Biology, vol. 18, no. 4, p. e1009987, 2022

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Combining model-based and data-driven models: an application to synthetic biology resource competition,

A. Darabi, Z. An, M. A. Al-Radhawi, W. Cho, M. Siami, and E. D. Sontag, “Combining model-based and data-driven models: an application to synthetic biology resource competition,”bioRxiv, 2025, preprint, posted March 2025

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Machine learning for synthetic gene circuit engineering,

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G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics-informed machine learning,”Nature Reviews Physics, vol. 3, no. 6, pp. 422–440, 2021

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M. Eslami, A. E. Borujeni, H. Eramian, M. Weston, G. Zheng, J. Ur- rutia, C. Corbet, D. Becker, K. Maschhoff, A. Clowers, A. Cristofaro, H. D. Hosseini, D. B. Gordon, Y . Dorfan, J. Singer, M. Vaughn, N. Gaffney, J. Fonner, C. A. V . Stubbs, and E. Yeung, “Prediction of whole-cell transcriptional response with machine learning,”Bioin- formatics, vol. 38, ...

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M. A. Alcantar, M. A. English, J. A. Valeri, and J. J. Collins, “A high-throughput synthetic biology approach for studying combinatorial chromatin-based transcriptional regulation,”Molecular Cell, vol. 84, no. 12, pp. 2382–2396.e9, 2024

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Identification of dynamic models in complex networks with prediction error methods: Basic methods for consistent module estimates,

P. M. J. van den Hof, A. G. Dankers, P. S. C. Heuberger, and X. J. A. Bombois, “Identification of dynamic models in complex networks with prediction error methods: Basic methods for consistent module estimates,”Automatica, vol. 49, no. 10, pp. 2994–3006, 2013

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Identification of dynamic models in complex networks with predic- tion error methods: Predictor input selection,

A. Dankers, P. M. J. V . den Hof, X. Bombois, and P. S. C. Heuberger, “Identification of dynamic models in complex networks with predic- tion error methods: Predictor input selection,”IEEE Transactions on Automatic Control, vol. 61, no. 4, pp. 937–952, 2016

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Nonlinear network identifiability: The static case,

R. Vizuete and J. M. Hendrickx, “Nonlinear network identifiability: The static case,” in2023 62nd IEEE Conference on Decision and Control (CDC). IEEE, 2023, pp. 443–448

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Nonlinear network identifiability with full excitations,

——, “Nonlinear network identifiability with full excitations,”arXiv preprint arXiv:2405.07636, 2024

work page arXiv 2024

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