CurvFlow-DTA: dual-graph discrete Ricci curvature flow for drug--target affinity prediction

Jicheng Ma; Yunyan Yang; Juan Zhao; Liang Zhao

cs.LGSep 19, 2026

CurvFlow-DTA: dual-graph discrete Ricci curvature flow for drug--target affinity prediction

Authors: Jicheng Ma, Yunyan Yang, Juan Zhao, Liang Zhao

Organizations: School of Mathematics, Renmin University of China, Beijing, 100872, China · School of Mathematical Sciences, Key Laboratory of Mathematics and Complex Systems of MOE, Beijing Normal University, Beijing, 100875, China

Abstract

Graph neural networks are widely used for drug--target affinity (DTA) prediction, and discrete Ricci curvature has recently been used to characterize molecular graph geometry. Existing curvature-aware DTA approaches mainly use static curvature on the drug graph while representing proteins primarily with sequence-derived features. This leaves pair-adaptive use of graph geometry underexplored, which may limit adaptation to unseen entities in cold-start settings relevant to practical screening. We present CurvFlow-DTA, which replaces a single static curvature representation with weighted Forman curvature flow on both molecular and protein residue--residue contact graphs. A label-independent flow trajectory is precomputed for each entity, and a pair-conditioned selector determines the horizons read by a dual-branch Flow-GINE. A frozen ESM-2 supplies residue-level representations and contact scores used to construct the protein graph. Inference requires only SMILES strings and protein sequences, without a bound complex structure. On Davis and KIBA, CurvFlow-DTA improves on the protocol-matched Ricci-GraphDTA baseline in every warm and cold-start setting. Warm-split mean squared error (MSE) decreases by $19.9\%$ on Davis and $18.9\%$ on KIBA. Across the six cold-start comparisons, MSE decreases by $14.3$ -- $27.4\%$ , with higher concordance index (CI) throughout. Within our compiled set of literature baselines, CurvFlow-DTA achieves the lowest MSE on both warm benchmarks and across four out of six cold-start evaluation settings.

Figures & tables

Figure 1: Overview of CurvFlow-DTA. (a) For each drug and target, weighted Forman curvature flow evolves the edge metrics of the molecular graph and the ESM-2–derived residue contact graph at fixed topology. The resulting trajectory and Forman profile are precomputed once per entity and reused across all pairs. (b) For a given drug–target pair, a pair-conditioned selector reads the pooled features and Forman profiles of both entities and predicts two horizons

H_{D},H_{P}

that set how far each branch reads its trajectory. (c) Drug and protein Flow-GINE encoders read the selected curvature states through a per-edge geometric gate, and their pooled representations are fused to predict affinity

\hat{y}

. Inference requires only a SMILES string and a protein sequence. Molecule and graphs are schematic. This explanatory workflow diagram was prepared with assistance from OpenAI Codex (GPT-5.6 Sol), used to help create the workflow illustration.

	Davis			KIBA
Method	MSE $\downarrow$	CI $\uparrow$	$R_{m}^{2}\uparrow$	MSE $\downarrow$	CI $\uparrow$	$R_{m}^{2}\uparrow$
DeepDTA ( Öztürk et al., 2018 )	$0.261$	$0.878$	$0.630$	$0.194$	$0.863$	$0.673$
GraphDTA (GIN) ( Nguyen et al., 2021 )	$0.229$	$0.893$	—	$0.147$	$0.882$	—
GraphDTA (GAT-GCN) ( Nguyen et al., 2021 )	$0.245$	$0.881$	—	$0.139$	$\mathbf{0.891}$	—
FusionDTA ( Yuan et al., 2022 )	$0.226$	$0.891$	$0.686$	$0.155$	$0.882$	$0.750$
MGraphDTA ( Yang et al., 2022 )	$0.228$	$0.883$	$0.679$	$0.152$	$0.884$	$0.766$

Table 1: Warm-split prediction performance on Davis and KIBA. Boldface indicates the best numerical value among the entries listed for each metric and dataset

Figure 2: Predicted versus measured affinities for CurvFlow-DTA on the Davis and KIBA warm-split test sets. Each point represents one drug–target pair, and the dashed identity line denotes perfect agreement ( $y=x$ ).

Davis
	Cold-drug		Cold-target		Dual-cold
Method	MSE $\downarrow$	CI $\uparrow$	MSE $\downarrow$	CI $\uparrow$	MSE $\downarrow$	CI $\uparrow$
GraphDTA	$0.557$	$0.735$	$0.439$	$0.770$	$0.656$	$0.607$
MGPLI	$0.895$	$0.584$	$0.457$	$0.792$	$0.654$	$0.556$
GRA-DTA	$0.578$	$0.725$	$0.376$	$0.827$	$0.560$	$0.690$
GDilatedDTA	$0.500$	$0.761$	$0.382$	$0.810$	$0.718$	$0.633$

Table 2: Cold-start results for representative baselines. Boldface indicates the best numerical value among the entries listed within each dataset and scenario.

Model variant	Davis				KIBA
Model variant	Warm	Cold-drug	Cold-target	Dual-cold	Warm	Cold-drug	Cold-target	Dual-cold
No curvature	0.231	0.513	0.401	0.542	0.151	0.363	0.406	0.470
Static Forman	0.221	0.501	0.344	0.523	0.148	0.358	0.399	0.465
Fixed flow	0.204	0.487	0.323	0.507	0.143	0.352	0.392	0.454
Entity-only horizon	0.194	0.481	0.311	0.498	0.140	0.349	0.387	0.449
Metric-flow shuffle	0.226	0.526	0.360	0.560	0.153	0.374	0.403	0.483

Table 3: Ablation analysis of CurvFlow-DTA in terms of MSE (

\downarrow

) across warm, cold-drug, cold-target, and dual-cold evaluation on Davis and KIBA. The controls modify the curvature representation, trajectory readout, or graph branch. Boldface indicates the best value in each column.

Model variant	Davis				KIBA
Model variant	Warm	Cold-drug	Cold-target	Dual-cold	Warm	Cold-drug	Cold-target	Dual-cold
No curvature	0.889	0.764	0.807	0.700	0.883	0.739	0.726	0.662
Static Forman	0.895	0.771	0.817	0.708	0.884	0.742	0.733	0.667
Fixed flow	0.899	0.777	0.828	0.716	0.885	0.745	0.740	0.671
Entity-only horizon	0.901	0.780	0.831	0.720	0.886	0.746	0.743	0.672
Metric-flow shuffle	0.899	0.756	0.824	0.688	0.882	0.728	0.732	0.648

Table 4: Ablation analysis of CurvFlow-DTA in terms of CI (

\uparrow

) across warm, cold-drug, cold-target, and dual-cold evaluation on Davis and KIBA. Boldface indicates the best value in each column.

Figure 3: Protein-graph attribution on two held-out cold-target co-crystals: (A) MET–03X (PDB 3U6H) and (B) CDK2–olomoucine (PDB 1W0X), shown as PyMOL binding-pocket close-ups. The inhibitor is drawn as yellow sticks, the $4$ Å crystallographic pocket residues in blue, the model’s top- $20$ attributed residues in red, residues in both in purple, and ligand–protein hydrogen bonds as orange dashes. Each panel is annotated with the pocket-recovery AUPRC (versus the residue-degree baseline) and the top- $K$ enrichment over chance. Both crystal structures are wild-type constructs matching the model’s input sequences.

	Davis	KIBA
Drugs	68	2 111
Targets	442	229
Measured pairs	30 056	118 254
Warm train pairs (folds 0–4)	25 046	98 545
Warm test pairs	5 010	19 709

Table S1: Benchmark statistics. Counts are for the full measured matrix. The warm-split train/test sizes follow the released DeepDTA setting-1 folds used unchanged in this work.

Figure S1: Distributions of benchmark labels ( A , D ), protein sequence lengths ( B , E ), and molecular weights ( C , F ) for Davis (top row) and KIBA (bottom row). Dashed vertical lines indicate the medians. Molecular weights are computed using RDKit.

Dataset	MSE $\downarrow$	RMSE $\downarrow$	CI $\uparrow$	$R_{m}^{2}\uparrow$	Pearson $\uparrow$
Davis	$0.185\pm 0.016$	$0.430\pm 0.019$	$0.902\pm 0.004$	$0.711\pm 0.019$	$0.861\pm 0.004$
KIBA	$0.137\pm 0.002$	$0.370\pm 0.002$	$0.887\pm 0.002$	$0.790\pm 0.007$	$0.893\pm 0.001$

Table S2: Extended warm-split results for CurvFlow-DTA. Values are mean

\pm

sample standard deviation over five training seeds.

Dataset		MSE $\downarrow$	RMSE $\downarrow$	CI $\uparrow$	$R_{m}^{2}\uparrow$	Pearson $\uparrow$
Davis	Cold-drug	$0.475\pm 0.018$	$0.689\pm 0.013$	$0.782\pm 0.010$	$0.275\pm 0.009$	$0.546\pm 0.015$
	Cold-target	$0.300\pm 0.029$	$0.547\pm 0.027$	$0.833\pm 0.010$	$0.329\pm 0.037$	$0.638\pm 0.014$
	Dual-cold	$0.492\pm 0.033$	$0.701\pm 0.023$	$0.723\pm 0.047$	$0.071\pm 0.018$	$0.305\pm 0.052$
KIBA	Cold-drug	$0.347\pm 0.032$	$0.588\pm 0.027$	$0.748\pm 0.013$	$0.449\pm 0.032$	$0.682\pm 0.026$
	Cold-target	$0.383\pm 0.015$	$0.619\pm 0.012$	$0.746\pm 0.006$	$0.454\pm 0.022$	$0.706\pm 0.011$
	Dual-cold	$0.445\pm 0.059$	$0.666\pm 0.044$	$0.674\pm 0.022$	$0.182\pm 0.067$	$0.431\pm 0.142$

Table S3: Extended cold-start results for CurvFlow-DTA. Values are mean

\pm

sample standard deviation over five training seeds.

Figure S2: Training and validation Huber loss versus epoch during warm-split model selection for CurvFlow-DTA on (A) Davis and (B) KIBA (seed 0). The dashed line marks the epoch selected by early stopping.

Component	Setting
Frozen ESM-2 backbone	esm2_t30_150M_UR50D
Hidden dimension	256
Drug / protein encoder layers	5 / 3
Dropout	0.1
Flow steps $T$	8
Flow step size $\eta$	0.1

Table S4: Main hyperparameters of CurvFlow-DTA.

Case	Pocket prevalence	Degree AUPRC	Model AUPRC	Enrichment
MET–03X	0.075	0.179	0.198	$3.2\times$
CDK2–olomoucine	0.043	0.122	0.171	$3.9\times$

Table S5: Case-level attribution summaries. The table reports the prevalence, degree-control AUPRC, model AUPRC, and pocket-size-based enrichment.

Source	Publication and result location
D18	DeepDTA, Tables 3–6 ( Öztürk et al., 2018 )
G21	GraphDTA, Tables 2 and 3 ( Nguyen et al., 2021 )
L20	DeepGS, Tables 2 and 3 ( Lin et al., 2020 )
Z24	GDilatedDTA, Tables 3–5 ( Zhang et al., 2024 )
W22	MGPLI, Table 4 (full-model row) ( Wang et al., 2022 )
T24	GRA-DTA, Tables 2 and 4 ( Tang et al., 2024 )

Table S6: Numerical-source identifiers used in the complete comparison. The source of a result may differ from the original publication of the method.

Method	Source	MSE $\downarrow$	CI $\uparrow$	$R_{m}^{2}\uparrow$
SimBoost	D18	$0.282$	$0.872$	$0.644$
KronRLS	D18	$0.379$	$0.871$	$0.407$
DeepDTA	D18	$0.261$	$0.878$	$0.630$
GraphDTA (GIN)	G21	$0.229$	$0.893$	—
GraphDTA (GAT-GCN)	G21	$0.245$	$0.881$	—
DeepGLSTM	Z24	$0.294$	$0.867$	$0.624$

Table S7: Complete warm-split comparison on Davis. Numerical sources are defined in Table S6 .

Method	Source	MSE $\downarrow$	CI $\uparrow$	$R_{m}^{2}\uparrow$
SimBoost	D18	$0.222$	$0.836$	$0.629$
KronRLS	D18	$0.411$	$0.782$	$0.342$
DeepDTA	D18	$0.194$	$0.863$	$0.673$
GraphDTA (GIN)	G21	$0.147$	$0.882$	—
GraphDTA (GAT-GCN)	G21	$0.139$	$\mathbf{0.891}$	—
DeepGLSTM	Z24	$0.185$	$0.855$	$0.705$

Table S8: Complete warm-split comparison on KIBA. Numerical sources are defined in Table S6 .

Method	Source	MSE $\downarrow$	CI $\uparrow$
GraphDTA	T24	$0.557$	$0.735$
MGPLI	T24	$0.895$	$0.584$
GRA-DTA	T24	$0.578$	$0.725$
GDilatedDTA	Z24	$0.500$	$0.761$
GLFA	N22	$0.861$	$0.670$
GEFA	N22	$0.847$	$0.709$

Table S9: Complete Davis cold-drug comparison. Sources are defined in Table S6 .

Method	Source	MSE $\downarrow$	CI $\uparrow$
GraphDTA	T24	$0.439$	$0.770$
MGPLI	T24	$0.457$	$0.792$
GRA-DTA	T24	$0.376$	$0.827$
GDilatedDTA	Z24	$0.382$	$0.810$
GLFA	N22	$0.453$	$0.780$
GEFA	N22	$0.434$	$0.795$

Table S10: Complete Davis cold-target comparison. Sources are defined in Table S6 .

Method	Source	MSE $\downarrow$	CI $\uparrow$
GraphDTA	T24	$0.656$	$0.607$
MGPLI	T24	$0.654$	$0.556$
GRA-DTA	T24	$0.560$	$0.690$
GDilatedDTA	Z24	$0.718$	$0.633$
GLFA	N22	$1.144$	$0.636$
GEFA	N22	$0.990$	$0.639$

Table S11: Complete Davis dual-cold comparison. Sources are defined in Table S6 .

Method	Source	MSE $\downarrow$	CI $\uparrow$
GraphDTA	T24	$0.349$	$0.761$
MGPLI	T24	$0.500$	$0.725$
GRA-DTA	T24	$\mathbf{0.337}$	$\mathbf{0.765}$
FusionDTA	H25	$0.429$	$0.748$
MGraphDTA	H25	$0.425$	$0.746$
DMFF	H25	$0.408$	$0.753$

Table S12: Complete KIBA cold-drug comparison. Sources are defined in Table S6 .

Method	Source	MSE $\downarrow$	CI $\uparrow$
GraphDTA	T24	$0.519$	$0.651$
MGPLI	T24	$0.531$	$0.673$
GRA-DTA	T24	$0.435$	$0.692$
FusionDTA	H25	$0.439$	$0.685$
MGraphDTA	H25	$0.435$	$0.674$
DMFF	H25	$0.410$	$\mathbf{0.748}$

Table S13: Complete KIBA cold-target comparison. Sources are defined in Table S6 .

Method	Source	MSE $\downarrow$	CI $\uparrow$
GraphDTA	T24	$0.473$	$0.647$
MGPLI	T24	$0.640$	$0.611$
GRA-DTA	T24	$\mathbf{0.441}$	$0.657$
FusionDTA	H25	$0.587$	$0.641$
MGraphDTA	H25	$0.590$	$0.626$
DMFF	H25	$0.567$	$0.667$

Table S14: Complete KIBA dual-cold comparison. Sources are defined in Table S6 .

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