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

Ziwei Yang; Takeyuki Tamura

q-bio.QMOct 27, 2026

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

Authors: Ziwei Yang, Takeyuki Tamura

Organizations: Bioinformatics Center, Institute for Chemical Research, Kyoto University, Kyoto, Japan

Abstract

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.

Figures & tables

Graphical Abstract. HGPRGDel is the growth-coupled gene deletion protocol’s learning model; MetNetComp provides its reference-strategy layer. Molecular representations, differentiable GPR logic, and reaction-centric hypergraph learning produce continuous gene-retention scores and deletion-priority rankings, which define an ordered search space for candidate deletion strategies and downstream constraint-based evaluation.

Figure 1: Toy model illustrating GPR-dependent deletion patterns. Circles represent metabolites, and rectangles represent reactions (

r_{7}

for growth and

r_{8}

for target production). The GPR rule for

r_{4}

requires both

g_{4}

and

g_{5}

(enzyme complex), whereas the rule for

r_{5}

requires either

g_{3}

g_{4}

(alternative isozymes). Deleting

\{g_{3},g_{4}\}

disables both reactions, removing the alternative route and the enzyme-complex-associated reaction. Among all

2^{5}=32

deletion combinations, only two achieve growth-coupled production of

m_{5}

, illustrating how GPR logic shapes feasible deletion strategies.

Figure 2: Overview of the growth-coupled gene deletion protocol. The protocol takes a GEM, a target metabolite, and molecular sequence information as input and uses computationally curated MetNetComp reference strategies for supervised learning. HGPRGDel serves as the learning component: GPR-M converts Boolean AND/OR GPR structure into reaction embeddings; Graph-M propagates metabolic context through a reaction-centric hypergraph; Gene-M and Meta-M provide frozen molecular representations; and Pred-M converts the resulting gene, metabolite, and reaction representations into a continuous retention score

\hat{y}_{g,m}

. The resulting gene–target scores define a deletion-priority ranking, which provides an ordered search space for constructing nested candidate deletion sets and downstream constraint-based assessment.

	e_coli_core	iMM904	iML1515
#Genes	137	905	1516
#Pre-determined genes	71	850	1483
#Target-dependent genes	66	55	33
#Reactions	95	1577	2712
#GCP target metabolites	48	782	1085
#Deletion strategies	41	105	443

Table 1: Metabolic model and dataset statistics. “#GCP target metabolites”: growth-coupled production (GCP) annotated targets; “#Deletion strategies”: targets with a usable curated maximal deletion set. “#Target-dependent genes” and “#Target-dependent pairs” exclude pre-determined genes (constant labels for

\geq 90\%

of targets; defined above), and, on iML1515 , gene–target slots without an explicit supervised label in the curated reference (Section S1.2).

Figure 3: Benchmark and component ablation across three GEMs. (A) Macro-F1 across evaluated methods. (B) Macro-F1 for HGPRGDel and its ablated variants. Values are mean

\pm

s.d. over five held-out metabolite folds.

Figure 4: Mechanistic analysis of GPR-aware reaction representations and their association with deletion prioritization. ( A ) SoftBoolean AND fidelity: soft-min versus hard-min leaf $L_{2}$ scalars across GEMs (linear axes; dashed $y{=}x$ ; black line, pooled LOWESS trend). Inset shows the distribution of soft-min $-$ hard-min residuals; mean $\pm$ SD is annotated. Spearman $r_{s}$ and $n$ are reported for each GEM. ( B ) SoftBoolean OR fidelity: log-sum-exp versus hard-max leaf $L_{2}$ scalars (log–log axes; dashed $y{=}x$ ). Grey hexbin density summarizes the pooled distribution, with GEM-specific points overlaid. Spearman $r_{s}$ and $n$ are reported for each GEM. ( C ) GPR-rule representation separation: mean silhouette score (cosine distance) for AND, OR, and single-gene reactions, computed jointly with empty reactions in the reference representation space. The number of reactions ( $n$ ) is annotated for each GEM. ( D ) Embedding diagnostics: nearest-neighbor subsystem-enrichment $z$ -scores across three GEMs, shown as bars; linear-probe accuracy for AND/OR rule classification (%) is annotated for each GEM. ( E ) Deletion F1 on target-dependent gene–metabolite pairs, stratified by genes whose associated reactions are exclusively AND, OR, or single-gene (MIXED-rule genes excluded). The number of genes ( $n$ ) is annotated for each GEM and GPR role. ( F ) Representation–output association: AUC of the continuous deletion-priority score ( $1-\hat{y}$ ) for distinguishing reference-set membership, stratified by exclusive GPR role. The number of target-dependent gene–metabolite pairs ( $n$ ) is annotated for each GEM and GPR role. UMAP embeddings and OR-branch attention entropy are shown in Supplementary Fig. S3.

Figure 5: Deletion-priority ranking recovers curated reference gene sets. All methods are evaluated on the same held-out metabolites, gene roster, and curated labels; scores are continuous retention probabilities ranked by

1-\hat{y}_{g,m}

(higher

=

earlier deletion). ( A ) Mean average precision. ( B ) Cumulative recall as the top fraction of the ranked gene list increases, for HGPRGDel, GraphGDel, NMA, DeepGDel, DNN, and a random ranking. Circles mark each method’s

k_{90}

. ( C )

k_{90}

, the fraction of genes required to recover 90% of reference genes; lower is better. Reference gene sets are computationally curated MetNetComp annotations and are not experimentally validated.

GEM	#Held-out	#Scored	Avg. $\|GT\|$	Avg. $\|Pred\|$	Rec. (%)	Prec. (%)	F1 (%)	AP (%)
e_coli_core	41	41	67.1	66.4	92.03	92.53	91.98	94.71
iMM904	105	105	670.4	672.5	98.91	98.62	98.76	99.97
iML1515	443	408	1205.3	1201.6	99.02	99.34	99.18	99.53

Table 4: Recovery of curated MetNetComp reference gene sets. Gene-level recovery metrics are computed for targets with explicit supervised labels; definitions are provided in Methods and Supplementary Section S1.2.

Figure 6: Target-level case study of GPR-aware deletion prioritization for succinate ( succ_c ) on e_coli_core (target-dependent genes; Supplementary Section S1). ( A ) Central-carbon schematic highlighting reactions hit by the ten lowest-retention predicted deletions. Colors denote GPR type: blue, AND complex; gold, OR isozymes; purple, SINGLE. Labels show reaction, lowest-retention gene(s), and retention score (lower $=$ higher deletion priority); bold labels mark genes in the curated reference. ( B ) Ten highest- and ten lowest-retention scores; filled/open markers denote GT deleted/retained. ( C ) Retention heatmap for top-ranked genes across succ_c , akg_c , and fum_c ; $*$ denotes GT deleted. akg_c ( $\alpha$ -ketoglutarate) and fum_c (fumarate) are TCA-neighbor targets detailed in Supplementary Section S5.

GEM	$\|M\|$	$\|R\|$	$\|G\|$	GCP targets	Deletion strategies	Target-dep. pairs	Del. rate (%)
e_coli_core	72	95	137	48	41	2,706	43.4
iMM904	1226	1577	905	782	105	5,775	64.6
iML1515	1877	2712	1516	1085	443	13,464	52.7

Table S1: Genome-scale metabolic model overview (extends main-text Table 1). “GCP targets”: GCP-annotated targets in the GEM. “Deletion strategies”: targets with a usable curated maximal deletion set used for supervision (the supervised-target count). “Target-dep. pairs” excludes pre-determined genes (Section S1.2).

GEM	Roster pairs	Target-dep. pairs	Del. pairs (target-dep.)	Pre-det. genes	Del. rate (%)
e_coli_core	5,617	2,706	1,175	71	43.4
iMM904	95,025	5,775	3,730	850	64.6
iML1515	671,588	13,464	7,097	1,483	52.7

Table S2: Label composition across the three GEMs (curated supervision). “Roster pairs” is

|\mathit{G}|\times|\mathcal{M}_{\mathrm{target}}|

(on iML1515 ,

53{,}060

roster slots lack explicit curated labels and are excluded from supervised pair counts). “Target-dep. pairs” and “Del. pairs (target-dep.)” exclude pre-determined genes (majority-label consistency

\geq 0.9

; Section S1.2 ). “Del. rate” is the fraction of target-dependent pairs with

y_{g,m}=0

GEM	Fold 1	Fold 2	Fold 3	Fold 4	Fold 5
e_coli_core	1,096	1,096	1,096	1,096	1,233
iMM904	19,005	19,005	19,005	19,005	19,005
iML1515	133,408	133,408	133,408	133,408	137,956

Table S3: Metabolite-based five-fold cross-validation partitions (seed 42). Entries are the numbers of roster gene–metabolite pairs in the held-out metabolite set for each fold. Counts use the full gene roster, including pre-determined genes. Target-dependent pair counts are reported in Table S2 . On iML1515 , counts include unlabeled roster slots.

GEM	$\|R\|$	Med. depth	Max depth	Single gene (%)	Flat AND (%)	Flat OR (%)	Nested (%)	Empty (%)
e_coli_core	95	1	4	28.4	10.5	28.4	5.3	27.4
iMM904	1577	1	3	43.9	4.5	13.6	4.2	33.9
iML1515	2712	1	4	48.0	7.6	24.0	3.9	16.5

Table S4: GPR rule complexity across the three GEMs. Depth is the root-to-leaf length of the parsed Boolean tree. Percentages are over all reactions; the largest absolute number of multi-gene rules occurs on iML1515 (2,712 reactions), whereas multi-gene proportions are highest on e_coli_core .

Parameter	Value
Gene representation dim ( $d_{g}$ )	320
Metabolite representation dim ( $d_{m}$ )	256
Hypergraph conv dim	128
Hypergraph conv layers	2
GPR-M reaction-embedding dim	$d_{g}$ ( $=320$ ; GPR-M output)
Reaction-context dim ( $D_{\text{rxn}}$ )	128 (association-MLP output, $d_{g}\to D_{\text{rxn}}$ )

Table S5: Full hyperparameter configuration for HGPRGDel. All reported experiments use these settings without per-GEM retuning.

Method	Input representation	Pretrained backbone	Optimization	Training
DNN	Sequence encoders (from scratch), no graph	None	Adam, 0.001	50 epochs, patience 5
DeepGDel	LSTM sequence encoders (from scratch)	None	Adam, 0.001	50 epochs, patience 5
NMA	DeepGDel encoders + fixed one-hop neighborhood-mean graph feature	None	Adam, 0.001	50 epochs, patience 5
GraphGDel	LSTM sequence encoders + GCN (from scratch) on pairwise metabolite graph	None	Adam, 0.001	50 epochs, patience 5
HGPRGDel	ESM-2 (gene) + ChemBERTa (metabolite)	ESM-2 / ChemBERTa (frozen)	Adam, 0.001	50 epochs, patience 5

Table S6: Training configurations of the evaluated methods.

M-Prec. $=$ macro-precision; M-Recall $=$ macro-recall; AUC $=$ area under the ROC curve.
GEM	Method	Accuracy (%)	M-Prec. (%)	M-Recall (%)	Macro-F1 (%)	AUC (%)
e_coli_core	DNN	$62.77\pm 3.84$	$64.31\pm 3.66$	$63.58\pm 3.38$	$58.94\pm 3.53$	$62.18\pm 3.96$
	DeepGDel	$70.83\pm 3.57$	$71.92\pm 3.62$	$72.41\pm 3.60$	$70.16\pm 3.73$	$70.47\pm 3.90$
	NMA	$71.54\pm 3.63$	$72.65\pm 3.67$	$73.03\pm 3.64$	$70.91\pm 3.78$	$71.25\pm 3.94$
	GraphGDel	$77.16\pm 3.39$	$76.12\pm 3.31$	$75.91\pm 3.42$	$74.22\pm 3.27$	$76.33\pm 3.34$
	HGPRGDel	$\mathbf{89.34\pm 3.76}$	$\mathbf{89.33\pm 3.75}$	$\mathbf{89.27\pm 3.82}$	$\mathbf{89.11\pm 3.81}$	$\mathbf{96.42\pm 1.57}$

Table S7: Full baseline comparison: all five metrics across three GEMs (mean

\pm

s.d. over five held-out metabolite test folds on target-dependent pairs). Bold: best per GEM block. Every row reports per-fold metrics with each fold’s own frozen

\theta

, under the convention of Section S2.6.1 . Tabulated metrics are computed on the target-dependent pairs (evaluable pairs: 2706, 5775 and 13464 for e_coli_core , iMM904 and iML1515 ). The target-dependent gene sets are defined with the label-consistency threshold

\rho_{\mathrm{label}}{=}0.9

(Section S1.2 ) in all three GEMs.

GEM	Mean $\Delta$ (pp)	$95\%$ CI	Wilcoxon (two-sided)	Wilcoxon (one-sided)	NB $t$ -test
e_coli_core	$14.89$	$[10.47,19.32]$	$0.0625$	$0.0312$	$3.4\times 10^{-3}$
iMM904	$7.07$	$[5.21,8.94]$	$0.0625$	$0.0312$	$2.2\times 10^{-3}$
iML1515	$18.38$	$[16.95,19.82]$	$0.0625$	$0.0312$	$1.9\times 10^{-5}$

Table S8: Paired tests of HGPRGDel vs. GraphGDel on held-out test macro-F1 (five folds per GEM). “Mean

\Delta

” is the mean macro-F1 difference (pp); the

95\%

CI is the paired

t

interval on the mean difference. “NB

t

-test” is the Nadeau–Bengio corrected resampled

t

-test, in which the paired standard error

s/\sqrt{n}

is replaced by

s\sqrt{1/n+n_{\mathrm{test}}/n_{\mathrm{train}}}

; with

n{=}5

folds and

n_{\mathrm{test}}/n_{\mathrm{train}}{=}1/4

this inflates the standard error by exactly

1.5\times

, to account for the dependence between the training sets of different folds.

Ranking source	Successes	Genes examined	Mean GT recall	Mean Jaccard
HGPRGDel deletion priority	26	41.7	0.617	0.616
Random order (mean of $20$ )	4.3 [2–7]	9.2	0.077	0.069
Curated reference (reference-informed)	41	38.7	0.591	0.591

Table S9: Localization of weakly growth-coupled operating points from the continuous deletion ranking on e_coli_core (

n{=}41

targets,

137

roster genes). Genes are added one at a time in decreasing deletion priority

1-\hat{y}_{g,m}

; the search stops at the first prefix that is weakly growth coupled, and “Genes examined” is that stopping rank (equal to

|D^{*}|

by construction). Recall and Jaccard are averaged over each row’s solved targets. Random order is repeated

20

times per target (mean over orders; min–max in brackets); reference-informed ordering places the curated reference genes first.

Figure S1: Localization of weakly growth-coupled operating points from continuous deletion rankings on e_coli_core (

n{=}41

targets): cumulative fraction reaching weak growth coupling versus genes examined. The

x

-axis counts every gene examined, including those added before a lethal break, so such targets advance the

x

-axis without ever contributing to the cumulative count and a plateau reflects targets that ordering cannot solve. HGPRGDel priority (solid); random order (mean over

20

shuffles, band

=

min–max); curated reference genes placed first (reference-informed ordering, dashed).

Ranking source	Successes	Genes examined	Mean GT recall	Mean Jaccard
HGPRGDel deletion priority	44	455.0	0.681	0.680
Random order (mean of $20$ )	9.9 [8–12]	3.8	0.005	0.005
Curated reference (reference-informed)	105	377.4	0.563	0.563

Table S10: Localization of weakly growth-coupled operating points from the continuous deletion ranking on iMM904 (

n{=}105

targets,

905

roster genes), computed identically to Table S9 .

Figure S2: Localization of weakly growth-coupled operating points from continuous deletion rankings on iMM904 (

n{=}105

targets): cumulative fraction reaching weak growth coupling versus genes examined. The

x

-axis counts every gene examined, including those added before a lethal break, so such targets advance the

x

-axis without ever contributing to the cumulative count and a plateau reflects targets that ordering cannot solve. HGPRGDel priority (solid); random order (mean over

20

shuffles, band

=

min–max); curated reference genes placed first (reference-informed ordering, dashed).

Budget (%)	Genes allowed	Successes	Mean search iterations	Mean GT recall	Mean Jaccard
5	7	2/41	1.00	0.074	0.074
10	14	2/41	1.00	0.181	0.181
20	28	4/41	1.00	0.404	0.403
30	42	10/41	1.10	0.607	0.601
50	69	32/41	1.59	0.885	0.822
100	137	41/41	1.59	0.874	0.653

Table S11: Search-space restriction and downstream gDel-minRN outcomes on held-out e_coli_core targets (

n{=}41

). Mean recall and Jaccard are computed against the MetNetComp reference strategy and are reported for completeness rather than as the primary endpoint.

Figure S3: Restricting gDel-minRN’s candidate gene pool to the top-

B\%

of a gene ranking, on held-out e_coli_core targets (

n{=}41

), under three ranking sources: HGPRGDel’s deletion-priority ranking, GraphGDel’s retention scores, and random gene order (mean over

5

independent shuffles per target). ( A ) Fraction of targets for which a feasible growth-coupled solution exists within the allowed gene pool, as a function of

B

. ( B ) Mean number of search iterations required by gDel-minRN among the runs that found a feasible solution, as a function of

B

GEM	Target-dependent Macro-F1	Target-dependent AUC	Full-gene Macro-F1	Full-gene AUC
e_coli_core	$89.04\%$	$96.48\%$	$92.20\%$	$97.93\%$
iMM904	$88.81\%$	$96.25\%$	$97.60\%$	$99.76\%$
iML1515	$90.40\%$	$97.10\%$	$97.98\%$	$99.67\%$

Table S12: Full-gene supplementary evaluation. “Target-dependent” reproduces the primary evaluation subset (pooled over the fold-averaged predictions); “Full gene roster” additionally includes pre-determined genes.

GEM	Targets evaluated	Mean top-10 hit rate	Targets with $10/10$ hits
e_coli_core	$41$	$98.0\%$	$36/41$ ( $87.8\%$ )
iMM904	$105$	$100.0\%$	$105/105$ ( $100.0\%$ )
iML1515	$408$	$99.5\%$	$400/408$ ( $98.0\%$ )

Table S13: All-target top-10 recovery, computed identically to the main-text case study for every evaluated target.

GEM	$\rho$	Target-dependent genes	Macro-F1	AUC
e_coli_core	$0.80$	$41$	$87.98\%$	$96.19\%$
	$0.90$	$66$	$89.04\%$	$96.48\%$
	$0.95$	$80$	$89.54\%$	$96.83\%$
iMM904	$0.80$	$21$	$77.39\%$	$84.95\%$
	$0.90$	$55$	$88.81\%$	$96.25\%$
	$0.95$	$96$	$91.73\%$	$97.86\%$

Table S14: Sensitivity to the target-dependent gene definition.

\rho{=}0.90

reproduces the primary evaluation.

GEM	Targets	Macro-F1 ( $95\%$ CI)	AUC ( $95\%$ CI)
e_coli_core	$41$	$89.04\%$ $[85.55,\,92.35]$	$96.48\%$ $[94.61,\,98.00]$
iMM904	$105$	$88.81\%$ $[87.44,\,90.05]$	$96.25\%$ $[95.44,\,96.94]$
iML1515	$443$	$90.40\%$ $[89.60,\,91.20]$	$97.10\%$ $[96.70,\,97.50]$

Table S15: Target-level bootstrap

95\%

confidence intervals for HGPRGDel (target-dependent genes,

2{,}000

resamples over target metabolites, seed

42

GEM	Method	Fold 1	Fold 2	Fold 3	Fold 4	Fold 5
e_coli_core	DNN	54.67	62.23	57.09	62.94	57.77
	DeepGDel	65.27	73.43	68.22	74.28	69.60
	NMA	67.03	74.19	68.71	75.67	68.95
	GraphGDel	70.05	76.10	75.07	71.77	78.11
	HGPRGDel	88.30	92.90	92.37	83.45	88.55
iMM904	DNN	63.48	67.01	64.54	67.23	64.89

Table S16: Per-fold held-out test macro-F1 (%) on target-dependent gene–metabolite pairs for all methods. Each fold evaluates target metabolites excluded from training in that round (Section S1.4).

GEM	Configuration	Accuracy (%)	M-Prec. (%)	M-Recall (%)	Macro-F1 (%)	AUC (%)
e_coli_core	Full	$\mathbf{89.34\pm 3.76}$	$\mathbf{89.33\pm 3.75}$	$\mathbf{89.27\pm 3.82}$	$\mathbf{89.11\pm 3.81}$	$\mathbf{96.42\pm 1.57}$
	Mean Pooling	$79.24\pm 3.29$	$78.96\pm 3.18$	$79.51\pm 3.01$	$78.80\pm 3.21$	$88.10\pm 3.24$
	No Graph-M	$78.31\pm 3.26$	$78.28\pm 3.09$	$77.28\pm 3.03$	$77.58\pm 3.29$	$86.55\pm 3.32$
	No pretrained	$76.80\pm 3.30$	$76.50\pm 3.20$	$76.90\pm 3.10$	$76.50\pm 3.30$	$87.00\pm 3.20$
	No GPR-M	$76.68\pm 3.28$	$76.28\pm 3.12$	$76.43\pm 3.05$	$75.30\pm 3.29$	$84.60\pm 3.34$
iMM904	Full	$\mathbf{89.65\pm 1.21}$	$\mathbf{88.64\pm 1.27}$	$\mathbf{88.61\pm 1.59}$	$\mathbf{88.61\pm 1.42}$	$\mathbf{96.22\pm 1.12}$

Table S17: Full ablation study: all five metrics across three GEMs (mean

\pm

s.d. over five held-out metabolite test folds on target-dependent pairs). Bold: best per GEM block.

AND-node configuration	Macro-F1 (%)	AUC (%)
Combined (default)	$\mathbf{89.11\pm 3.81}$	$\mathbf{96.42\pm 1.57}$
Mean-only	$68.64\pm 4.08$	$78.08\pm 4.23$
Soft-min-only	$69.38\pm 2.22$	$81.43\pm 1.87$

Table S18: AND-branch ablation of GPR-M on e_coli_core (mean

\pm

s.d. over five held-out folds). “Mean-only” uses

\mathbf{z}_{\text{and}}{=}\mathbf{m}

; “Soft-min-only” uses

\mathbf{z}_{\text{and}}{=}\mathbf{s}

; “Combined” is the default. Bold: best.

Graph encoder	Macro-F1 (%)
Hypergraph (default)	$\mathbf{89.11\pm 3.81}$
Bipartite MPNN	$79.44\pm 2.77$
Pairwise GCN	$75.99\pm 2.91$

Table S19: Graph-encoding comparison on e_coli_core (macro-F1, mean

\pm

s.d. over five folds). GPR-M and Pred-M are fixed. Hypergraph row matches main-text Table 3 Full.

Target	AP	Top-10	Precision	Recall	F1	$\|$ Pred $\|$	Jaccard
succ_c	0.91	$10/10$	0.84	0.61	0.71	32	0.72
akg_c	0.96	$10/10$	0.96	0.87	0.92	28	0.92
fum_c	0.94	$10/10$	0.94	0.80	0.86	17	0.86

Table S20: Ranking and threshold summary for three TCA-neighbor targets on e_coli_core (target-dependent genes,

n{=}66

). Top-10 hit: fraction of the ten lowest-retention calls labeled Deleted in the curated reference. Precision / Recall / F1: target-dependent-subset overlap after the global macro-F1 threshold. Jaccard: full-roster gene-set overlap versus the curated reference. All three targets have top-10 hit

10/10

(no false positives in the lowest-retention ten).

Gene	Retention score	Curated	Example reaction	GPR type
b1773	0.007	Deleted	FBA	OR
b1276	0.009	Deleted	ACONTa	OR
b0733	0.046	Deleted	CYTBD	AND
b0734	0.050	Deleted	CYTBD	AND
b2587	0.058	Deleted	AKGt2r	SINGLE
b3739	0.065	Deleted	ATPS4r	AND

Table S21: Top-10 lowest-retention HGPRGDel predicted deletions for succ_c on e_coli_core (target-dependent genes). Retention score: model output

\hat{y}_{g,m}

(lower

=

stronger deletion preference). The curated reference label is

0

for deleted genes.

Gene	Retention score	Curated	Example reaction	GPR type
b3952	0.000	Deleted	PFL	AND
b1478	0.000	Deleted	ALCD2x	OR
b0727	0.001	Deleted	AKGDH	AND
b0811	0.002	Deleted	GLNabc	AND
b3739	0.004	Deleted	ATPS4r	AND
b0724	0.011	Deleted	SUCDi	AND

Table S22: Top-10 lowest-retention HGPRGDel predicted deletions for akg_c on e_coli_core (target-dependent genes).

Gene	Retention score	Curated	Example reaction	GPR type
b0733	0.007	Deleted	CYTBD	AND
b1276	0.008	Deleted	ACONTa	OR
b3739	0.089	Deleted	ATPS4r	AND
b3731	0.093	Deleted	ATPS4r	AND
b0734	0.097	Deleted	CYTBD	AND
b0727	0.104	Deleted	AKGDH	AND

Table S23: Top-10 lowest-retention HGPRGDel predicted deletions for fum_c on e_coli_core (target-dependent genes).

GEM	GPR-referenced genes	Genes with ESM-2	Gene coverage	Reactions (complete GPR)
e_coli_core	137	137	100%	69 / 69 (100%)
iMM904	905	905	100%	1,043 / 1,043 (100%)
iML1515	1,516	1,516	100%	2,266 / 2,266 (100%)

Table S24: ESM-2 embedding coverage for genes referenced in GPR rules. “Reactions with complete GPR coverage” counts reactions whose every gene has an ESM-2 embedding.

GEM	Unique metabolites	Currency matched	Non-currency	Incidence kept / total
e_coli_core	72	20	52	175 / 360
iMM904	1226	88	1138	3,749 / 6,517
iML1515	1877	41	1836	6,333 / 10,575

Table S25: Currency metabolites excluded from hypergraph incidence construction (unless a supervised target). The 20 filtered base metabolites are atp , adp , amp , nad , nadh , nadp , nadph , coa , h , h2o , pi , ppi , co2 , nh4 , o2 , glutathione , na1 , k , mg2 , and ca2 . Effect on each GEM: incidence kept / total.

GEM	$\|\mathit{R}\|$	NN same-subsystem	Random NN	$z$	NMI (subsystem)	NMI (GPR type)
e_coli_core	95	0.225	$0.123\pm 0.013$	$7.7$	0.404	0.471
iMM904	1577	0.196	$0.042\pm 0.004$	$36.5$	0.357	0.411
iML1515	2712	0.411	$0.076\pm 0.003$	$104.8$	0.287	0.504

Table S26: Reaction embedding organization tests. NN same-subsystem fraction compares observed neighbor purity to label-permutation baselines (

k{=}10

1{,}000

shuffles of subsystem labels) in

\mathbf{Z}_{\text{rxn}}

space;

z

-scores indicate standard deviations above the shuffled mean. NMI uses KMeans with one cluster per annotated subsystem (or per GPR type); NMI is reported descriptively, with no random baseline computed, and is not used as standalone evidence of embedding quality.

Figure S4: Supporting mechanism evidence. ( A ) UMAP of iML1515 reaction embeddings ( $|\textit{R}|{=}2{,}712$ ) colored by top-level GPR type (AND, OR, Single-gene, Empty). Partial co-localization indicates that the learned representation encodes GPR type without explicit supervision; quantitative assessment is reported in main-text Fig. 4C,D. ( B ) OR-branch attention entropy $H_{\mathrm{att}}$ vs. isozyme group size $K$ ( iML1515 ); solid line $=\ln K$ (maximum entropy). Most OR reactions fall below $\ln K$ , indicating non-uniform isozyme weighting rather than degenerate uniform attention.

Figure S5: UMAP of GPR-M reaction embeddings on iML1515 , colored by four coarse metabolic categories (transport, amino-acid metabolism, energy metabolism, cofactor metabolism). Reactions outside these categories or without a coarse mapping are shown in grey ( $n$ shown in figure). Global color mixing is expected because pathways share intermediates and is not used as quantitative evidence.

Learning to Rank

Learning to rank develops models that order items by relevance or utility for search, recommendations, and other decisions.

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Jul 7, 2026

Canopy: A Heterograph Foundation Model for Metabolic Engineering

Designing microbial strains that produce high-value chemicals at commercially viable titers remains a central challenge in metabolic engineering. Existing computational approaches either rely on stoichiometric constraint-based models that cannot learn from experimental data, or apply tabular machine learning to hand-crafted features that discard the relational structure of biological knowledge. We present Canopy, a heterogeneous graph foundation model that integrates ten public and proprietary data sources into a unified knowledge graph (KG) of 6.9M nodes across 13 types and 34 edge types, covering genes, proteins, metabolites, reactions, pathways, strains, and fermentation experiments. Node features are encoded through domain-specific foundation models (ESM-2 for protein sequences, MoLFormer for chemical SMILES, and PubMedBERT for biomedical text), yielding a multi-modal representation within a single graph. We pretrain a Heterogeneous Graph Transformer (HGT) augmented with SignNet positional encodings, Jumping Knowledge aggregation, and virtual nodes using four self-supervised objectives (link prediction, masked node modelling, distance prediction, and contrastive experiment clustering), balanced via learned homoscedastic uncertainty weighting. On the downstream task of fermentation titer prediction, frozen Canopy embeddings achieve

R^{2} = 0.41

with a lightweight probe, outperforming tabular baselines (best

R^{2} = 0.24

) and homogeneous GNN variants.

Jake Bowden, Laurence Legon, Satnam Surae

Twig Bio, London, United Kingdom

Heterogeneous GNNs Graph SSL Graph Foundation Models Multimodal Graph Learning Biomedical KGs

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Mar 20, 2026

G2DR: A Genotype-First Framework for Genetics-Informed Target Prioritization and Drug Repurposing

Human genetics offers a promising route to therapeutic discovery, yet practical frameworks translating genotype-derived signal into ranked target and drug hypotheses remain limited, particularly when matched disease transcriptomics are unavailable. Here we present G2DR, a genotype-first prioritization framework propagating inherited variation through genetically predicted expression, multi-method gene-level testing, pathway enrichment, network context, druggability, and multi-source drug--target evidence integration. In a migraine case study with 733 UK Biobank participants under stratified five-fold cross-validation, we imputed expression across seven transcriptome-weight resources and ranked genes using a reproducibility-aware discovery score from training and validation data, followed by a balanced integrated score for target selection. Discovery-based prioritization generalized to held-out data, achieving gene-level ROC-AUC of 0.775 and PR-AUC of 0.475, while retaining enrichment for curated migraine biology. Mapping prioritized genes to compounds via Open Targets, DGIdb, and ChEMBL yielded drug sets enriched for migraine-linked compounds relative to a global background, though recovery favoured broader mechanism-linked and off-label space over migraine-specific approved therapies. Directionality filtering separated broadly recovered compounds from mechanistically compatible candidates. G2DR is a modular framework for genetics-informed hypothesis generation, not a clinically actionable recommendation system. All outputs require independent experimental, pharmacological, and clinical validation.

Muhammad Muneeb, David B. Ascher

School of Chemistry and Molecular Biology, The University of Queensland, Queen Street, 4067, Queensland, Australia · Computational Biology and Clinical Informatics, Baker Heart and Diabetes Institute, Commercial Road, 3004, Victoria, Australia

Drug Discovery Scientific Hypothesis Generation Genomics

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Jul 21, 2026

BRIDGE: Bottleneck-Aware Regulator-Set Inference and Diagnosis for Cooperative Gene Regulatory Recovery

Cooperative gene regulation often depends on groups of regulators acting jointly, but most gene regulatory network (GRN) inference methods output pairwise regulator-target rankings. We introduce Bottleneck-Aware Regulator-Set Inference and Diagnosis (BRIDGE), a framework for complete regulator-set recovery, and Targeted Recovery Attribution for Cooperative Evaluation (TRACE), a diagnostic suite that attributes failures to retrieval, set-level scoring, decoding, and evaluation bottlenecks. TRACE includes a leak-free mechanism-mismatch cooperativity stress test in which cooperative targets are generated by random nonlinear mechanisms rather than product interactions. This design avoids feature-mechanism circularity: Residual higher-order set scoring (Residual HOS2) operates on raw expression vectors without handcrafted product-correlation features. Across 30 matched seed-cooperativity settings, Residual HOS2 improves Jaccard similarity from 0.382 to 0.460, recall from 0.522 to 0.597, and exact recovery from 0.053 to 0.113 over a decomposable pairwise set scorer (PairS2), although exact recovery remains low. On SERGIO DS3, oracle retrieval and TRACE show that candidate coverage is necessary but insufficient because set-level misranking remains the dominant source of exact-recovery failure. PairS2 proposal followed by Residual HOS2 reranking reduces HOS2-scored candidate sets by 94-97% while largely preserving exact-recovery behavior. These results distinguish edge ranking, candidate retrieval, set-level scoring, and exact cooperative regulator-set recovery as separate objectives.

Maryam Rahimimovassagh, Clayton Thomas Barham, Ivan Garibay +1

University of Central Florida

Feature Interaction Modeling Gene Regulatory Network Inference Genomics

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Abstract

Learning to Rank

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Canopy: A Heterograph Foundation Model for Metabolic Engineering

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