PaperlayerView exact revision on arXiv

Source-linked AI summary

HGPRGDel: GPR-Aware Learning and Ranking for Growth-Coupled Gene Deletion Design

Ziwei Yang, Takeyuki Tamura

arXiv:2610.37478v1q-bio.QMcs.AI

TL;DR

Growth-coupled deletion design requires searching many gene combinations to find interventions that preserve growth while linking it to target production. This protocol learns and ranks candidate deletions from curated strategies, then evaluates constructed candidates with constraint-based analysis; HGPRGDel achieves the highest macro-F1 and AUC among evaluated learning methods across three GEMs.

  • Problem

    Growth-coupled deletion design must preserve growth while making target production necessary, but the possible gene-deletion combinations grow exponentially with gene count.

    Source: Introduction · p. 1

  • Method

    The protocol learns gene–target deletion priorities from curated strategies, constructs candidate deletion sets at adjustable search-space budgets, and evaluates them with constraint-based metabolic analysis.

    Source: Introduction · p. 2

  • Results

    HGPRGDel achieves the highest macro-F1 and AUC among evaluated learning methods on each of three GEMs.

    Source: Introduction · p. 3

  • Takeaways & Limitations

    Separating learned prioritization from constraint-based evaluation lets curated computational designs guide adjustable candidate searches while retaining mechanistic assessment.

    Source: Introduction · p. 2

    The evaluation reuses constraint-based reference strategies and assessment, so it evaluates computational design-knowledge reuse rather than independent biological ground truth.

    Source: Limitations and Future Work · p. 13

Abstract

from arXiv · show

Designing microbial strains for growth-coupled production, in which cellular growth is linked to the synthesis of a target compound through genetic interventions, is a fundamental goal in metabolic engineering. Constraint-based methods can identify growth-coupled deletion strategies, but design for each target typically requires a separate search across a large number of possible gene-deletion combinations. This motivates a reusable approach that learns from previously evaluated strain designs to prioritize candidate interventions before downstream constraint-based evaluation. We present a reproducible protocol for growth-coupled gene deletion design that learns and ranks gene-deletion priorities for each target metabolite. Using computationally curated deletion strategies as learning resources, the protocol generates continuous deletion-priority rankings, constructs candidate strategies at adjustable search-space budgets, and evaluates these candidates through constraint-based metabolic analysis. Its learning component, HGPRGDel, explicitly represents gene-protein-reaction (GPR) logic, capturing how multiple genes can jointly support a reaction or alternative genes can maintain its activity, while integrating molecular and reaction-level information. Across e_coli_core, iMM904, and iML1515, the protocol achieves the highest macro-F1 and AUC among the evaluated learning configurations. HGPRGDel's rankings recovered an average of 90% of the reference deletion genes from MetNetComp by examining fewer genes than the evaluated baseline methods. By separating learned intervention prioritization from downstream constraint-based assessment, the protocol provides a reproducible framework for reusing strain-design knowledge while controlling the size of the candidate space and retaining mechanistic evaluation of growth-coupled production.

Introduction

Growth-coupled deletion design must preserve growth while making target production necessary, but the number of possible gene combinations grows exponentially. This section motivates a reusable learning protocol that uses curated designs to prioritize candidates, while retaining constraint-based assessment and explicitly representing GPR logic.

Introduction

  • Growth-coupled production requires sufficient activity for growth while linking growth to target synthesis, and the number of possible deletion combinations grows exponentially with gene count.This combinatorial challenge motivates prioritizing candidate interventions before downstream evaluation.Source: Introduction · p. 1
  • The protocol learns gene–target deletion priorities from curated strategies, constructs candidate sets at adjustable search-space budgets, and evaluates each set independently with constraint-based analysis.MetNetComp strategies serve as learning targets and reference sets for assessing recovery of known designs.Source: Introduction · p. 2
  • HGPRGDel combines differentiable Boolean GPR aggregation, reaction-centric hypergraph context, and pretrained molecular representations to rank gene deletions without replacing downstream phenotype assessment.GPR-M encodes AND/OR rules, Graph-M propagates network context, and Gene-M/Meta-M provide frozen sequence and metabolite representations.Source: Introduction · pp. 2–3
  • Across three GEMs, HGPRGDel achieves the highest macro-F1 and AUC among the evaluated learning methods on each GEM.The contribution statement also reports recovery of computationally curated MetNetComp reference gene sets.Source: Introduction · p. 3

Constraint-Based Methods for Growth-Coupled Strain Design

  • Constraint-based optimization provides a general route to growth-coupled strain design, but each new GEM–target combination still requires a separate optimization or search procedure.Methods include bilevel programming and later search-efficiency improvements such as local search, mixed-integer formulations, and minimum-cut-set enumeration.Source: Constraint-Based Methods for… · p. 3

Machine Learning for Metabolic Gene Perturbation

  • DeepGDel and GraphGDel established learning-based deletion prioritization, while the cited approaches do not explicitly formulate it around differentiable GPR logic.The section motivates a reaction-centric formulation that treats GPR structure as a biological prior and adds metabolic topology context.Source: Introduction · p. 2, Machine Learning for Metabolic Gene… · p. 3

Foundation Models in Systems Biology

The protocol combines pretrained molecular representations with GPR-aware reaction learning and metabolic context to prioritize gene deletions. It turns those priorities into nested candidate sets for separate constraint-based assessment, with the learning stage designed around gene–reaction logic.

Foundation Models in Systems Biology

  • Frozen ESM-2 protein and ChemBERTa metabolite representations contribute molecular information, while the framework combines curated design knowledge, GPR logic, and metabolic topology in a reusable protocol.The pretrained backbones are used without fine-tuning on limited per-GEM labels.Source: Foundation Models in Systems Biology · pp. 3–4

Protocol Overview

  • The five-stage protocol runs from target-specific problem definition through HGPRGDel scoring and ranking to candidate construction and downstream constraint-based assessment.The full protocol is distinct from HGPRGDel, its GPR-aware learning component.Source: Protocol Overview · p. 4

Problem Definition and Reference Strategies

  • A candidate is growth-coupled under the stated weak criterion when biomass is maximized and fixed, target production is then maximized, and both reach at least 0.001.MetNetComp deletion gene sets provide supervised reference labels; ranked genes yield nested candidate sets of increasing size, while exhaustive feasible-set enumeration is outside the learning task's scope.Source: Problem Definition and Reference… · p. 4

Encoding Molecular Information

  • Gene-M and Meta-M encode protein sequences and metabolite SMILES with frozen ESM-2 and ChemBERTa backbones, training only linear projection layers for their representations.The projections limit trainable parameters under the available per-GEM supervision.Source: Encoding Molecular Information · p. 4

Biological Rationale

  • GPR AND rules represent enzyme complexes requiring all subunits, whereas OR rules represent isozymes that can sustain activity through alternatives.Deleting any required AND subunit blocks its reaction; blocking an OR-supported reaction requires deleting all alternatives.Source: Biological Rationale · p. 4

GPR-Specific Differentiable Aggregator

  • The parser converts Boolean rules into recursive AND/OR trees, and the differentiable aggregator represents them using a soft bottleneck for conjunctions and attention with log-sum-exp for disjunctions.The OR aggregation encodes the any-sufficient principle for isozyme alternatives.Source: GPR-Specific Differentiable Aggregator · pp. 4–5

Reaction-Centric Hypergraph Learning

HGPRGDel represents reactions as hyperedges over participating metabolites and conditions message passing on GPR-derived reaction attributes. It pools reaction-level context with gene and target representations to predict gene-retention scores and rank deletion priorities.

Hypergraph Formulation

  • The hypergraph keeps each reaction as a hyperedge over its participating metabolites, preserving reaction identity and multi-metabolite structure during message passing.Currency metabolites are filtered to limit topological hubs' influence.Source: Hypergraph Formulation · p. 5

GPR-Conditioned Message Passing (Graph-M)

  • GPR-derived reaction attributes condition metabolite message passing, making gene-level Boolean structure available as reaction context.Two stacked hypergraph convolution layers refine compound representations while preserving reactions as explicit multi-metabolite units.Source: GPR-Conditioned Message Passing… · p. 6
  • For each gene–target pair, Pred-M pools reactions associated with either the gene or target, retaining context even when they share no reaction.The reduced reaction context is combined with gene and target representations before an MLP predicts the gene-retention score.Source: GPR-Conditioned Message Passing… · p. 6
  • The sigmoid retention score ranges from 0 to 1, with lower scores indicating higher deletion priority; ranking is retained independently of threshold selection.Training uses entropy-weighted binary cross-entropy, reducing the influence of genes with nearly constant labels across targets; molecular encoders remain frozen while other modules are jointly trained.Source: GPR-Conditioned Message Passing… · p. 6

Results

Across three GEMs, HGPRGDel achieves higher macro-F1 than all four baselines and beats GraphGDel on every fold. Evaluation uses metabolite-based five-fold cross-validation, with performance comparisons focused on relative gains over matched baselines.

Experimental Setup

  • Dataset statistics cover three GEMs and counts of annotated GCP targets, usable deletion strategies, and target-dependent genes and pairs.Target-dependent counts exclude near-constant-label genes and, on iML1515, gene–target slots without explicit supervised labels.Source: Experimental Setup · p. 6, Table 1 · p. 7
  • Evaluation uses metabolite-based five-fold cross-validation, with target-dependent genes and pairs defining the primary reported metrics.Threshold selection uses an inner validation split; the held-out test metabolites do not contribute to training or threshold selection.Source: Experimental Setup · pp. 6–7

The Core Learning Component Performs Consistently Across Three GEMs

  • HGPRGDel achieves higher macro-F1 than all four baselines across three GEMs and outperforms GraphGDel on every fold.Table 2 and Fig. 3A report the comparison.Source: The Core Learning Component Performs… · p. 7
  • HGPRGDel's macro-F1 gains over GraphGDel are 14.9, 7.1, and 18.4 percentage points on e coli core, iMM904, and iML1515, respectively.The largest gain is on iML1515, which has the largest reaction count and absolute number of multi-gene rules.Source: The Core Learning Component Performs… · p. 7
  • HGPRGDel's AUC–macro-F1 gap is 7.3, 7.6, and 6.7 percentage points across the three GEMs, versus 0.3–3.7 for baselines.Because fold-level macro-F1 reflects threshold choice as well as ranking, the authors use continuous-score and ranking metrics to assess prioritization independently.Source: The Core Learning Component Performs… · p. 7
  • HGPRGDel was applied across GEMs spanning a 30-fold reaction-count range without GEM-specific hyperparameter tuning; absolute cross-GEM scores require caution because pair counts and label distributions differ.Macro-F1 was 88.61% on iMM904 and 90.38% on iML1515.Source: The Core Learning Component Performs… · p. 7

Molecular, GPR, and Topology Priors Provide Complementary Signal

The ablations quantify contributions from pretrained molecular, GPR, and topology priors; subsequent analyses examine learned reaction embeddings and target-level deletion rankings.

Molecular, GPR, and Topology Priors Provide Complementary Signal

  • Removing pretrained weights costs 6.6–12.6 pp macro-F1; GPR-M removal costs 7.3–13.8 pp, the largest single-component loss, while Graph-M removal costs 5–12 pp.On e_coli_core, AND/OR distinctions contribute approximately 10.3 pp and gene–reaction connectivity approximately 3.5 pp of the total 13.8 pp GPR-M gain.Source: Molecular, GPR, and Topology Priors… · pp. 7–8, Table 3 · p. 8

Learned Reaction Embeddings Recover GPR Logic and Metabolic Organization

  • AND/OR reactions separate more than single-gene reactions (silhouette 0.47–0.83 vs. 0.05–0.25), while deletion-priority AUC spans 0.84–0.99 across roles and GEMs.On iML1515, OR-exclusive genes have the highest deletion F1 at 88.8%, versus 81.1% for AND-exclusive and 79.6% for single-gene genes.Source: Learned Reaction Embeddings Recover… · p. 9
  • SoftBoolean gates track hard min/max (Spearman rs ≥ 0.86); AND/OR types are near-ceiling decodable, with subsystem enrichment z=7.7–104.8.The enrichment strengthens with GEM scale, and these properties are reported without explicit pathway supervision.Source: Learned Reaction Embeddings Recover… · p. 8, Figure 4 · p. 10

A Target-Level Case Study of GPR-Aware Deletion Prioritization

  • For succinate in e_coli_core, ranking AP is 0.91 across 66 target-dependent genes, and all ten lowest-retention calls match the curated reference.Those calls include OR-type isozyme genes, AND-type complex subunits, and a single-gene transport call.Source: A Target-Level Case Study of GPR-Aware… · p. 10

Learned Reaction Embeddings Recover GPR Logic and Metabolic Organization

Deletion-Priority Rankings Provide an Ordered Search Space for Candidate Strategy Construction

Rankings turn each target’s deletion priorities into candidate sets that can be evaluated at adjustable search budgets. In held-out targets, feasibility depended on downstream constraint-based assessment, while reuse of curated designs and GPR-aware representations structured the search.

Deletion-Priority Rankings Provide an Ordered Search Space for Candidate Strategy Construction

  • For each target, progressively larger candidate sets take the first q genes in the priority ranking, making strategy size a budget choice rather than a model hyperparameter.Figure 6 illustrates succinate rankings, including GPR types and retention scores; lower retention means higher deletion priority.Source: Deletion-Priority Rankings Provide an… · p. 11, Figure 6 · p. 12
  • Ranked candidates contained a weakly growth-coupled set for 26 of 41 held-out e coli core targets (63.4%) and 44 of 105 iMM904 targets (41.9%).These are coverage results, not prediction rates: every unsolved target reached a lethal prefix, while reference genes ordered first succeeded for all evaluated targets.Source: Deletion-Priority Rankings Provide an… · p. 11
  • Feasible-solution coverage on e coli core rose from 4.9% at 5% and 10% of genes to 9.8% at 20%, 24.4% at 30%, 78.0% at 50%, and 100% with all genes.Increasing the candidate-space budget trades search scope against coverage; targets missed at smaller budgets were recovered by unrestricted search.Source: Deletion-Priority Rankings Provide an… · p. 12

Reusing Computational Design Knowledge

  • Previously obtained design strategies provide gene–target priorities, while constraint-based analysis evaluates the resulting interventions; the ranking can be reused across targets with candidate budget explicit.The learned ranking structures the downstream intervention space rather than replacing mechanistic optimization.Source: Reusing Computational Design Knowledge · p. 12

Structured Biological Knowledge in Learning

  • GPR structure captures dependencies from enzyme complexes and alternative isozymes, shaping learning without requiring it to reproduce the full constraint-based model.Molecular and network information provide complementary context.Source: Structured Biological Knowledge in… · p. 12

Search-Space Control and Scope

  • The framework is intended for settings with prior computational design knowledge, and its scope depends on the reference strategies, target-dependent gene definition, and validation design.A feasible strategy may fall outside the selected candidate space at a restricted budget.Source: Search-Space Control and Scope · p. 13

Limitations and Future Work

The framework reuses computationally curated designs to prioritize interventions while retaining downstream mechanistic assessment. Its evidence is bounded by the reference strategies and evaluation design, and future work points toward experimental outcomes and multiple feasible strategies.

Limitations and Future Work

  • Reference strategies do not exhaustively represent feasible or experimentally observed deletions, and evaluation is limited by target-dependent gene definitions and cross-validation folds.Both the reference strategies and downstream assessment rely on constraint-based metabolic modeling, so the study does not evaluate against independent biological ground truth.Source: Limitations and Future Work · p. 13

Conclusion

  • The framework converts computational design knowledge into gene–target priorities that structure downstream constraint-based search while retaining mechanistic assessment.This makes previously obtained design knowledge reusable across related design problems.Source: Conclusion · p. 13

Key Points

  • HGPRGDel achieved the highest macro-F1 and AUC among evaluated methods across three GEMs, while rankings recovered 90% mean cumulative recall with fewer genes examined than baseline methods.Ablation and representation analyses support contributions from the integrated biological information sources.Source: Key Points · p. 13

Supplementary Information

  • Supplementary material provides implementation details and extended analyses that complement or extend main-text results.Cross-references identify where those details appear.Source: Supplementary Information · p. 16

S1.1 Genome-Scale Metabolic Model Overview

  • Evaluation uses e coli core, iMM904, and iML1515, spanning a curated E. coli core model, a yeast genome-scale reconstruction, and a comprehensive E. coli reconstruction.These models are accessed from COBRA-style JSON files, with GPR Boolean strings parsed by HGPRGDel’s GPRRuleParser.Source: S1.1 Genome-Scale Metabolic Model… · p. 16

S1.2 MetNetComp Label Construction

  • MetNetComp maximal strategies supply deleted-versus-retained supervision labels and recovery references; reported metrics use target-dependent genes whose labels vary across supervised targets.On iML1515, 53,060 roster slots lack explicit strategy-table rows and are omitted from supervised-pair counts.Source: S1.2 MetNetComp Label Construction · p. 16

S1.3 Label Composition and Class Imbalance

S1.4 Cross-Validation Partitions

  • All reported results use metabolite-based five-fold cross-validation with seed 42, keeping each target metabolite’s gene labels together in one fold.Across the five rounds, every supervised target metabolite is evaluated once.Source: S1.4 Cross-Validation Partitions · p. 16

S1.5 GPR Rule Complexity

The supplement documents the models, preprocessing, evaluation, and training protocol used for HGPRGDel. It specifies frozen molecular encoders, common settings across GEMs, and nested validation procedures.

S2.1 Hardware and Software Environment

  • Experiments ran on an NVIDIA A100-PCIE-40GB GPU with a Xeon Gold 6258R CPU, using PyTorch and PyG hypergraph convolution.Software dependencies and versions are provided with the code release.Source: S2.1 Hardware and Software Environment · p. 17

S2.2 Downstream FBA Evaluation Protocol

  • COBRApy re-simulates FBA, testing growth first and then target production at optimum biomass; weak coupling requires both growth and production fluxes of at least 0.001.MetNetComp curated sets are used as reference labels and are not re-verified.Source: S2.2 Downstream FBA Evaluation Protocol · p. 17

S2.3 Pre-trained Models and Sequence Processing

  • Frozen ESM-2 and ChemBERTa encoders represent gene sequences and metabolite SMILES, with sequences truncated at 256 tokens; compounds without SMILES receive a shared fixed embedding.Compound-node structure coverage is 100.0% on e coli core, 86.3% on iMM904, and 89.9% on iML1515; the no-pretrained ablation instead trains randomly initialized backbones.Source: S2.3 Pre-trained Models and Sequence… · p. 17

S1.5 GPR Rule Complexity

  • GPR-rule complexity is summarized across the three GEMs; the largest absolute number of multi-gene rules occurs on iML1515, while their proportions are highest on e coli core.Rule depth is measured as root-to-leaf length in the parsed Boolean tree.Source: S1.5 GPR Rule Complexity · p. 17, Table S4 · p. 18

S2.4 Full Hyperparameter Settings

  • The same HGPRGDel hyperparameter configuration is used across all three GEMs without per-GEM retuning.The settings include batch size 16, up to 50 epochs, and early-stopping patience of 5.Source: S2.4 Full Hyperparameter Settings · p. 18

S2.5 Training Procedure

  • Training uses inner-validation AUC for early stopping and selects each fold’s macro-F1 threshold on the inner validation split, then applies it unchanged to held-out metabolites.Five-fold splits are by target metabolite; the per-gene entropy weighting down-weights near-constant genes, and reported metrics are unweighted.Source: S2.5 Training Procedure · pp. 18–19

S2.6 Baseline Comparison Protocol

HGPRGDel’s macro-F1 exceeded GraphGDel on every fold across all three GEMs, with positive mean differences. The supplement describes the shared comparison protocol and qualifies the significance evidence by fold count and multiple-testing correction.

S2.6 Baseline Comparison Protocol

S2.6.1 Graph Convention and the Declared Inductive Sensitivity

  • Folds differ in supervised gene–metabolite pairs, not the complete GEM graph presented to the model; trainable parameter counts were not reported consistently across methods.Per-fold metrics use each fold’s frozen threshold, while mechanism and recovery analyses average scores before thresholding.Source: S2.6.1 Graph Convention and the… · pp. 19–20

S3.1 Full Performance Comparison (All Metrics)

S3.2 Fold-Level Paired Significance Tests

  • HGPRGDel exceeded GraphGDel on every fold, with mean macro-F1 differences of 14.9, 7.1, and 18.4 pp across the three GEMs; corrected t-tests retained p ≤ 3.4 × 10^-3.The two-sided Wilcoxon test reached its minimum attainable p-value at five folds, while Bonferroni correction across GEMs raised the one-sided result to 0.094, above α=0.05.Source: S3.2 Fold-Level Paired Significance… · pp. 21–22, Table S8 · p. 21

S3.3 Operating-Point Analysis from Deletion-Priority Rankings

Continuous deletion-priority rankings can localize weakly growth-coupled operating points on both e coli core and iMM904, while restricting a constraint-based search has budget-dependent effects. The analyses distinguish ranking-based prefix evaluation from search over a restricted candidate pool.

S3.3.1 Operating-Point Analysis on e coli core: Weak Growth Coupling from Continuous Rankings

S3.3.2 Operating-Point Analysis on iMM904: Weak Growth Coupling at Genome Scale

  • On iMM904, HGPRGDel localized weak growth coupling for 44/105 targets (41.9%) after examining a mean of 455.0 genes, compared with 9.9/105 successes on average for random orders.Reference-informed ordering reached 105/105 targets; HGPRGDel solved 30 targets that no random order solved.Source: S3.3.2 Operating-Point Analysis on… · pp. 22, 24

S3.3.3 Search-Space Restriction under Alternative Gene Rankings

  • At 50% of the gene pool, HGPRGDel achieved a 78.0% success rate, versus 70.7% for GraphGDel and 35.1% for random ranking; at 5–20%, random ranking performed better.The search-space restriction experiment was a controlled e coli core case study; it does not establish computational savings on larger GEMs.Source: S3.3.3 Search-Space Restriction under… · p. 24
  • Among successful restricted-pool searches, mean iterations remained between 1.00 and 1.90 across rankings and budgets, with no consistent ranking-dependent pattern.The reported ranking effect was on whether a feasible solution was contained in the restricted pool, rather than iterations once one was reachable.Source: S3.3.3 Search-Space Restriction under… · p. 24
  • The search-space restriction results support a ranking- and budget-dependent coverage–feasibility trade-off, not a uniform benefit from learned rankings.At very small budgets, random ranking had higher success rates; a possible pathway-diversity explanation was not directly tested.Source: S3.3.3 Search-Space Restriction under… · p. 24

S3.4 Full-Gene Supplementary Evaluation

  • The primary evaluation excludes pre-determined genes from reported metrics, although they remain part of the supervised training signal.The supplementary full-gene evaluation reuses fold-averaged predictions and includes those genes.Source: S3.4 Full-Gene Supplementary Evaluation · p. 24

S3.5 All-Target Top-10 Recovery

The supplementary analyses extend top-10 recovery across targets, test sensitivity to how target-dependent genes are defined, and characterize uncertainty in HGPRGDel’s metrics. They also specify the controlled variants used in the component ablations.

S3.5 All-Target Top-10 Recovery

  • Table S13 extends the top-10 recovery diagnostic from the succinate case study to every target with at least ten evaluable target-dependent genes.It counts matches between the ten lowest-retention calls and curated reference deletions.Source: S3.5 All-Target Top-10 Recovery · p. 24, Table S13 · p. 26

S3.6 Sensitivity to the Target-Dependent Gene Definition

  • Full-gene macro-F1 and AUC were higher than target-dependent-subset scores on all three GEMs, consistent with pre-determined genes being easier to classify from near-constant labels.The pooled supplementary values differ slightly from the main-text per-fold means and standard deviations.Source: S3.6 Sensitivity to the… · p. 26, Table S12 · p. 26
  • Across the three GEMs, macro-F1 increased monotonically with the target-dependent label-consistency threshold ρ; the qualitative pattern was consistent across GEMs.On iMM904 and iML1515, the stricter ρ=0.80 threshold retained a smaller, more label-diverse subset and corresponded to lower macro-F1.Source: S3.6 Sensitivity to the… · pp. 24, 26, Table S14 · p. 26

S3.7 Target-Level Bootstrap Uncertainty

  • The target-level bootstrap estimates uncertainty in HGPRGDel’s metrics from 2,000 resamples, but is not a paired comparison with other methods.The analysis resamples target metabolites from fold-averaged predictions because matched per-target baseline predictions are unavailable.Source: S3.7 Target-Level Bootstrap Uncertainty · p. 27, Table S15 · p. 27

S4.1 Ablation Variant Definitions

  • The ablation variants change one architectural component at a time while retaining the same encoders, prediction MLP, classifier input dimension, folds, seeds, and threshold protocol.The variants remove or replace GPR aggregation, hypergraph propagation, GPR-derived information, or pretrained encoder weights.Source: S4.1 Ablation Variant Definitions · p. 28

S4.2 Per-Fold Held-Out Results

This block reports the scope of held-out and ablation evaluations alongside target-level case studies. The supplied results describe ranking patterns and feasibility checks, but do not provide the numerical cells of the cited performance tables.

S4.2 Per-Fold Held-Out Results

S4.3 Full Ablation Study (All Metrics)

S4.4 AND-Branch Ablation of GPR-M

  • The AND-branch comparison evaluates mean-only, soft-min-only, and combined aggregation variants on e coli core, with the combined variant as the default.The table caption identifies the best variant but the supplied text gives no outcome values.Source: S4.4 AND-Branch Ablation of GPR-M · p. 28, Table S18 · p. 30

S4.5 Graph-Encoding Comparison

S5.1 Succinate Production

  • For succinate, the highest-confidence deletion calls span OR-type isozyme reactions, AND-type complexes, and a single transport call; the top ten contain no false positives.The analysis uses target-dependent genes, and its overlap metrics are not directly comparable to full-roster metrics.Source: S5 Additional Case Studies · p. 28, iML1515 · p. 29, Table S20 · p. 30

S5.2 α-Ketoglutarate Production

  • For cytosolic α-ketoglutarate, the lowest-retention calls prioritize PFL and fermentation OR calls ahead of several TCA consumers, unlike the succinate ordering.The default-threshold predicted deletion set is not weakly growth-coupled, although the top ten contain no false positives.Source: S5.2 α-Ketoglutarate Production · p. 31
  • Cytosolic and extracellular α-ketoglutarate have the same top-ten target-dependent genes in the same order, but neither thresholded predicted set is weakly growth-coupled.Both curated reference sets satisfy the weak-growth-coupling check.Source: S5.2 α-Ketoglutarate Production · p. 31

S5.3 Fumarate Production

The fumarate case study shows target-specific deletion priorities, while the supplementary methods detail parser scope, embeddings, hypergraph construction, and differentiable GPR aggregation. Together, these passages describe how the model represents complex and alternative gene support for reactions.

S5.3 Fumarate Production

  • Fumarate’s earliest deletion ranks emphasize respiratory and ATP-synthase AND complexes alongside aconitase OR and downstream TCA calls, unlike the succinate and α-ketoglutarate rankings.No false positives appear among the top ten target-dependent genes.Source: S5.3 Fumarate Production · p. 31, Table S23 · p. 32

S6.1 GPR Parser: Scope and Algorithmic Complexity

  • The GPR parser accepts COBRA-style infix rules with and/or, parentheses, and whitespace-separated identifiers, but not negation or nonstandard dialects.Parsing takes O(L) time and O(D) stack space for rule length L and tree depth D.Source: S6.1 GPR Parser: Scope and Algorithmic… · p. 32

S6.2 ESM-2 Embedding Coverage

  • Every gene referenced in the evaluated GEMs’ GPR rules has an ESM-2 embedding, so zero-vector substitution was not used in these experiments.Every GPR-tree leaf therefore received a non-zero gene embedding during training and evaluation.Source: S6.2 ESM-2 Embedding Coverage · p. 32, Table S24 · p. 32

S6.3 Currency Metabolite Filtering

  • Hypergraph construction excludes high-flux currency cofactors from compound–reaction incidence unless a metabolite is a supervised production target.The listed filtered base metabolites include ATP, NAD, water, phosphate, and oxygen.Source: S6.3 Currency Metabolite Filtering · p. 32, Table S25 · p. 33

S7 Mathematical Details of Differentiable GPR

AND Node (Enzyme Complexes)

  • For AND rules, the aggregator combines mean and soft-min branches to represent obligate enzyme subunits, with the temperature-scaled soft minimum approaching the limiting subunit as temperature approaches zero.The resulting vector encodes a bottleneck across embedding dimensions.Source: AND Node (Enzyme Complexes) · p. 33

OR Node (Isozyme Alternatives)

  • For OR rules, attention and a dimension-wise log-sum-exp branch focus the representation on the strongest available isozyme.The OR branch has no temperature parameter; its log-sum-exp is a smooth upper bound on the strongest isozyme coordinate.Source: OR Node (Isozyme Alternatives) · pp. 33–34

Single-Gene and Empty Rules

S7.2 Gradient Propagation

The gradient rules route training signals differently for AND and OR nodes, emphasizing the limiting enzyme-complex subunit or the strongest isozyme, respectively.

Soft-Min Gradient (AND Node)

  • At AND nodes, the smallest gene-embedding value receives the largest gradient weight, directing loss toward the bottleneck subunit.This routing encourages accurate representation of the subunit that determines deletion feasibility.Source: Soft-Min Gradient (AND Node) · p. 34

Log-Sum-Exp Gradient (OR Node)

  • At OR nodes, the largest gene-embedding value receives the largest gradient weight, emphasizing the strongest isozyme as the critical blocking target.The reaction is blocked only when all alternative isozymes are deleted.Source: Log-Sum-Exp Gradient (OR Node) · p. 34

S7.3 Numerical Stability

The supplementary analyses address numerical stability, fidelity of soft Boolean approximations, and organization of learned reaction embeddings. They report strong agreement with hard GPR aggregation and above-random neighborhood enrichment, alongside explicit limits on interpreting some diagnostics.

S7.3 Numerical Stability

  • Log-sum-exp shifting keeps exponent arguments non-positive, while a softplus-parameterized temperature guarantees τ > ϵ > 0 and prevents division by zero in gradients.The shifts address overflow or underflow from direct exponentiation at small τ or large |u|; the initial temperature is approximately 1.Source: S7.3 Numerical Stability · p. 34

S7.4 Empirical Fidelity of Soft Boolean Approximations

  • Soft-min/hard-min Spearman correlations were 1.000, 0.987, and 0.857 across e coli core, iMM904, and iML1515; log-sum-exp/hard-max correlations were 0.992, 0.998, and 0.924.The lower AND correlation on iML1515 is reported as an observation, not attributed to a mechanism.Source: S7.4 Empirical Fidelity of Soft… · p. 35
  • OR-branch attention entropy on iML1515 generally remained below ln K, indicating non-uniform isozyme weighting but not demonstrating that attention is functionally necessary.Entropy increased with isozyme group count K.Source: S7.4 Empirical Fidelity of Soft… · p. 35, Figure S4 · p. 36

S8.1 Reaction Representation Analysis

  • Deletion F1 (AND/OR/SINGLE) was 88.2/87.4/88.9% on e coli core, 96.2/86.8/92.5% on iMM904, and 81.1/88.8/79.6% on iML1515.The linear probe covers only AND- and OR-type reactions; single-gene and empty reactions are excluded because their GPR type is recoverable from reaction degree.Source: S8.1 Reaction Representation Analysis · p. 35
  • Nearest-neighbor subsystem enrichment exceeded random in all three GEMs, with z-scores of 7.7, 36.5, and 104.8 for e coli core, iMM904, and iML1515.Silhouette scores are reported for completeness but are not used as embedding-quality evidence because subsystem overlap can penalize biologically meaningful shared intermediates.Source: S8.1 Reaction Representation Analysis · p. 35
  • The iML1515 GPR-type UMAP shows partial co-localization, whereas coarse-category color mixing is expected and is not quantitative evidence.The quantitative organization evidence is reported through the main-text diagnostics and nearest-neighbor analysis.Source: S8.1 Reaction Representation Analysis · p. 35, Figure S4 · p. 36, Figure S5 · p. 36
Loading 2610.37478v1…