Towards Multiscale Graph-based Protein Learning with Geometric Secondary Structural Motifs
Authors: Shih-Hsin Wang
Yuhao Huang
Taos Transue
Justin M. Baker
Jonathan Forstater
Thomas Strohmer
Bao Wang
Venue: NeurIPS 2025 poster ()
Keywords: Multiscale GNNs
Primary and secondary structures
Proteins
AI for Science
Primary area: machine_learning_for_sciences
TL;DR: We propose an efficient multiscale graph-based learning framework tailored to proteins.

Abstract: Graph neural networks (GNNs) have emerged as powerful tools for learning protein structures by capturing spatial relationships at the residue level. However, existing GNN-based methods often face challenges in learning multiscale representations and modeling long-range dependencies efficiently. In this work, we propose an efficient multiscale graph-based learning framework tailored to proteins. Our proposed framework contains two crucial components: (1) It constructs a hierarchical graph representation comprising a collection of fine-grained subgraphs, each corresponding to a secondary structure motif (e.g., $\alpha$-helices, $\beta$-strands, loops), and a single coarse-grained graph that connects these motifs based on their spatial arrangement and relative orientation. (2) It employs two GNNs for feature learning: the first operates within individual secondary motifs to capture local interactions, and the second models higher-level structural relationships across motifs. Our modular framework allows a flexible choice of GNN in each stage. Theoretically, we show that our hierarchical framework preserves the desired maximal expressiveness, ensuring no loss of critical structural information. Empirically, we demonstrate that integrating baseline GNNs into our multiscale framework remarkably improves prediction accuracy and reduces computational cost across various benchmarks.

Towards Multiscale Graph-based Protein Learning with Geometric Secondary Structural Motifs
Shih-Hsin Wang', Yuhao Huang', Taos Transue' Justin Baker?,
Jonathan Forstater", Thomas Strohmer' & Bao Wang
Department Mathematics and Scientific Computing and Imaging (SCI) Institute University of Utah, Salt Lake City, UT 84102, USA Department of Mathematics, UCLA, Los Angeles, CA 90095, USA Department of Mathematics, UC Davis, Davis, CA 95616, USA
Abstract
Graph neural networks (GNNs) have emerged as powerful tools for learning protein structures by capturing spatial relationships at the residue level. However, existing GNN-based methods often face challenges learning multiscale representations and modeling long-range dependencies efficiently. In this work, we propose an effi- cient multiscale graph-based learning framework tailored to proteins. Our proposed framework contains two crucial components: (1) It constructs a hierarchical graph representation comprising a collection of fine-grained subgraphs, each correspond- ing to a secondary structure motif (e.g., a-helices, 3-strands, loops), and single coarse- roarse-grained graph that connects these motifs based on their spatial arrangement and relative orientation. (2) It employs two GNNs for feature learning: the first operates within individual secondary motifs to capture local interactions, and the second models higher-level structural relationships across motifs. Our modular framework allows flexible choice of GNN in each stage. Theoretically, we show that hierarchical framework preserves the desired maximal expressiveness, ensuring no loss critical structural information. Empirically, we demonstrate that integrating baseline GNNs into multiscale framework remarkably improves prediction accuracy and reduces computational cost across various benchmarks.
Introduction
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Machine learning (ML) has transformed computational protein modeling over the past decade [20 42). The 2024 Nobel Prize in Chemistry, awarded for groundbreaking contributions to computational protein design, highlights the transformative advancements [ Among MI tech- niques, graph-based methods excel at learning to encode complex chemical interactions in three-dimensional space, effectively handling spatial information. For instance, prior works (cf. 14 [7] represent atoms as nodes and leverage edges to capture their interactions, such chem- ical bonds and hydrogen bonds. This edge-based encod- ing enables graph neural networks (GNNs) to effectively model critical chemical interactions, which are essential for understanding protein frunttie--unction relationships.
Figure 1: An example of prion proteins with identical primary structures but distinct secondary structures. The normal form, ham- ster Prpe (left), contains o-helical structures (marked in red). In contrast, its misfolded counterpart, PrPSe on the right, lacks these helices adopts a 3-sheet-rich structure (marked in yellow). This structural change leads to abnormal aggregation, ultimately re- sulting in fatal consequences.
However. representing proteins at the atomic level incurs significant computational costs due to the sheer size and
*Correspond to wangbaon Sanggbaonj@gmiil.... com
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Residue Embedding
Ist-Stage GNN
Motif Subgraphs
Motif Embedding
2nd 2nd-Stage GNN
Folded Protein
Global Features
Inter Structural Graph
Figure 2: Overview of the proposed multiscale graph-based framework. We first construct hierarchical graph representation that includes: (1) fine ane-graindd motif subgraphs, where residues within each secondary structure motif (e.g., a-helices, 3-strands, loops) are treated as nodes, and (2) a coarse-graings grained structural graph, where each motif is abstracted as a single node. The first GNN operates independently on each motif subgraph to learn local embeddings. These learned motif-level features are then used to construct the coarse-grained graph, on which second GNN performs message passing model higher-level structure and generate the final prediction.
complexity of the resulting graphs. To address this, recent methods have shifted towards residue-level representations. where each residue serves as a single node in the graph (cf. [18, 48 37 41).. This coarse-graining not only reduces the graph size but also aligns naturally with the primary structure of proteins, where residues are the fundamental building blocks that determine protein properties and overall structure. However, existing residue-leved approaches face challenges in capturing critical multiscale features. In particular, secondary structures- such as a a-helices and 3-sheets are formed by groups of residues and play a fundamental role in protein folding. Ignoring these higher-level structures can hinder the model's capacity to distinguish between biologically distinct protein states that share identical primary sequences. An illustrative example the prion protein [6. The normal cellular form of the prion protein, PrPC, is typically found on the surface of healthy neurons. However, it can misfold into the pathogenic form, PrPSc, without any change in its primary structure. The key difference lies in the spatial arrangement of residues, which are reorganized into different types of secondary structures, resulting in a markedly altered folding pattern compared to the normal form (see Section for details). This misfolded form aggregates abnormally due its 3-sheet-rich structures and induces pathogenic effects, ultimately leading to fatal neurodegenerative diseases. Figure shows hamster PrPC (PDB: 1B10) and its misfolded form PrPSe (PDB ID: 7LNA) 23 24
Some multiscale methods have been proposed to capture hierarchical features of protein struc- tures. For example, recent approaches use distance-thresholde graphs with large radial cutoffs, combined with surface modeling, to encode multiscale information [33] 47). While large-cutoff, threshccdd graphs can capture broader structural context, they often introduce significant computational overhead- both in runtime and memory footprint (see Section 5 for empirical evi- dence) and may overlook biologically meaningful relationships identified by domain experts. These limitations underscore the urgent need for a scalable framework that integrates domain knowledge to model intricate protein interactions efficiently while preserving expressive power.
1.1 Our Contributions
In this paper, we propose a new multiscale GNN framework-depice depicted in Fig. 2 with detailed discussion in Sections for learning proteins. Our key contributions are:
We introduce a hierarchical, sparse, and geometry-aware graph representation of proteins by combining domain-expert algorithms to segment sequences into secondary structure motifs and constructing multi-scale graph hierarchy: collection of fine-grained graphs that capture residue- level interactions within each secondary structure unit, and a single coarse- -ranined graph that models the spatial arrangement and relative orientation among these units. This representation preserves geometric fidelity while providing provable sparsity bound on the total number of edges, which is critical for scalability and efficiency. See Section 3 for details.
We develop a two-stage
two off-the-shelf GNNs operating in tandem to learn multiscale protein features based on the proposed hicrarchical graph representation. The-

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oretically, we characterize the maximal expressiveness of our framework, showing its ability to maintain spatial fidelity during message passing across different levels. See Section I for details.
Empirically, demonstrate that multiscale framework enables existing GNN architectures to simultaneously improve prediction accuracy and computational efficiency (in both runtime and memory footprint) across benchmark tasks. See details in Section 5
1.2 Related Works
Protein Representation Learning. A variety of deep learning approaches have been developed to model protein structures and functions by learning effective representations. Some methods leverage the sequential nature of proteins and employ convolutional neural networks (CNNs) [5] large language models (LLMs) 45
33)] to learn directly from the amino acid sequence. However, since protein function is closely tied to its 3D structure, many recent efforts have shifted toward structure- based approaches. These methods represent proteins as graphs and use GNNs to capture spatial relationships. Notable examples of methods that Iearn geometric and symmetry-awant representations 48
include [18 17
26 41]. In addition, hybrid models such DeepFRI [12] combine GNNs with sequence-level features extracted from pretrained protein language models, while ProtGO [6] integrates GNNs with Gene Ontology encoder, achieving strong results with large-scale architectures. In contrast, our work focuses on 3D structure- modeling and introduces an efficient, theoretically grounded hierarchical design that learns protein representation with substantially smaller model sizes.
37
Multiscale Graph-Based Models for Protein Representation Learning. Several recent efforts [14 K3 47 31 have explored multiscale GNN to better capture both local and global structural patterns. 47 construct large-radius radial graphs combined with surface modeling, while [31] learns residue-level clustering for hierarchical representations, However, such approaches often incur high computational costs or rely on data-driven clustering without explicit biological grounding. In contrast, [4] incorporates domain principles by applying hierarchical pooling based on different types of residue interactions to extract multiscale features. However, their final pooling stages simplify the backbone chain by clustering every two consecutive residues along the sequence, without explicitly leveraging secondary structure motifs. As a result, capturing long-range dependencies (LRDs) requires repeated pooling over many layers, leading to high computational costs. Our work shares the multiscale motivation but differs by introducing a biologically grounded hierarchical graph construction inspired by prior molecular motif-based approaches (46 43). Specifically, secondary structures as high-level motifs to build a two-level graph that captures both local geometry and long-range dependencies. This design offers provable sparsity and expressiveness guarantees while maintaining high computational efficiency.
The methods
1.3 Organization
We organize this paper follows: We recap message- saassagg-pssiingggg GNNs and the standard framework to analyze their expressiveness power, together with other necessary background materials in Section 2 We present new hierarchical graph representations for proteins and two-stage GNN architectures in Section and Section 4 respectively. We numerically validate the accuracy and efficiency of our proposed approach in Section 5 Technical proofs and additional experimental details are provided in the appendix.
2 Background
In this section, we provide background materials on point clouds, geometric graphs, local frames, and messagg-passingg GNNs and their expressiveness characterizations.
Point Clouds and Geometric Graphs. A 3D point cloud is collection of points in R3, i.e., {æ:} C RR3. Each point may have a feature vector f. E Rd, capturing additional attributes beyond the spatial coordinates. We denote such an attributed point cloud as {a,, fe ) Two point clouds J. N. and (ã,, f. are considered identical up to rigid motions if there exists a bijection [I....,N} and a rigid motion g such that fa(a) = f; and To(() g- ã, for all i. a: : NJ +
Extending this concept, a geametric graph g (V, E, F) introduces graph structure over the point cloud to model geometric relationships through edges. Here, V is the set of nodes, & the set of edges, and = [fi,...;
; fn] is the matrix of node features, which may also encode geometric attributes. When edges are equipped with features, eij E denotes both the edge and its associated attributes.
3

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Message Passing GNNs. Consider a (geometric) graph g = -V,E,, F). Starting from f. = S., message passing GNNs propagate features from iteration to t + 1 as follows:
f ((t+)) UPD (1 AGG (0) .5 egj 1j EMB)),
(1)
f readout (( | VD),
where esj represents the attribute of edge (i. .N, denotes the neighborhood of node consisting of nodes in V directly connected to i by an edge in E, f-B denotes multiset, and UPD, AGG, and readout are learnable functions, parameterized by multilayer perceptions.
Maximal Expressive GNNs. The expressiveness of GNNs is often analyzed through the lens of the which provides theoretical foundation for distinguishing non-isomorphic graphs. A GNN is said to be maximally expressive if its key components, i.e., UPD, AGG, and readout, are injective pushing a given GNN architecture to its theoretical expressive limit under ideal conditions. Under this assumption, we ask whether given GNN architecture can distinguish all non-isomorphic graph structures; that is, whether maximally expressive GNN can produce distinct readout features for non-isomorphic graphs, given sufficient number of layers T'.
Weisfeiler-Lehman (WL) graph isomorphism test 44 28
19 40)
This notion can be viewed as
This thcoretical framework has been extended to geometric graphs, particularly those derived from point clouds, where node and edge features encode spatial or geometric information. A recent study [1 investigates whether maximally expressive GNNs can distinguish point clouds-up to rigid motions- by operating on their corresponding SCHull graphs (see Appendix review), specific geometric graphs constructed from these point clouds. Since our framework adopts the SCHull graph as its underlying representation, we summarize the relevant expressiveness result below:
Theorem 2.1. [4 Let F be a maximally expressive GNN with depth T = 1. Then F can distinguish between the attributed SCHull graphs of any two non non-isomorphic generic point clouds.
Remark 2.2. The genericity of point clouds refers to the condition that the point coordinates are algebraically independent over the field of rational numbers-i.e. they do not satisfy any nontrivial polynomial equation with rational coefficients. This condition holds for most protein structures encountered in practice. Therefore, assume genericity for all structures studied in this work.
Local Frames. A local frame is an orthogonal matrix E O(3), consisting of three orthonormal vectors that define a local 3D coordinate system. Notably, local frames are equivariant under rotations and reflections: when a rotation or reflection is applied to an object's coordinates, the associated local frame transforms accordingly. Formally, if g denotes the local frame computed from a matrix of coordinates X, then for any / E 0(3), we have g(h X) h 9XX). In molecular and structural modeling, spatial units such as functional groups are often associated with such local frames to capture their orientations (9, g Given two spatial units with associated local frames 9: and gj. the product g: 9j represents the rotation (or reflection) that aligns one frame with the other, thereby encoding their relative orientation [8]. This formulation provides principled way to compare geometric configurations and serves a role analogous to transition maps in differential geometry, enabling accurate geometric information to pass across local reference systems.
3 Hierarchical Graph Representations for Proteins
In this section, we present a new multiscale hierarchical graph representation for proteins. We begin by discussing the rationale for hierarchical graph construction, designed to capture protein structures at multiple levels of granularity. Next, we describe the graph construction process, detailing how it represents structural information within and across secondary structures and preserves critical spatial features at each level.
3.1 Protein Hierarchical Structures
Proteins exhibit an inherently hierarchical organization, structured across multiple scales. At the most fundamental level, they can be considered as linear sequences of amino acids, known as the primary structure. Current trends in graph- protein modeling leverage this residue-level information by representing each amino acid node, with edges capturing chemical bonds and spatial proximities. This graph-based representation effectively models local interactions, reflecting the biological principles underlying protein folding.
However, the primary structure is only the first step in the hierarchical organization of proteins. As folding progresses, contiguous sequences of residues self-assemble into more complex geometric mo-

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Protein Sequence
DSSP
SS Tokens
$10
814
Segmentation
02
V/
07
V10
1/1
0/2
U14
S²
S3
S4
Figure 3: A visual illustration of the identification and segmentation process for protein secondary structures. Each residue ug is assigned secondary structure (SS) token &g by DSSP, and consecutive residues with the same token are grouped into subsequences St.
tifs-such a-helices and B-strands-formme the secondary structure. These recurring geometric patterns are stabilized through hydrogen bonding and are essential not only for maintaining the overall structure of the protein but also for determining functions and interactions with other molecular units. This multiscale complexity underscores the necessity for hierarchical modeling approach that can effectively capture both local and global structural features, as well as LRDs inherent in protein folding.
3.2 Identification and Segmentation of Protein Secondary Structures
The identification of secondary structures critical step in protein modeling, providing insights into the folding patterns and spatial organization of amino acid sequences. Various methods have been developed for this purpose, especially the
Token
Secondary Structure
B
Isolated G-bridge
Strand (all other 3-ladder residues)
310
DSSP (Define Secondary Structure of Proteins) algorithm [2] being one of the most widely used due to its robustness and accuracy. DSSP analyzes a protein's 's backbone conformation by first identifying mackbone-backbog bonds) based on geometric criteria and hydrogen-be calculations. It then uses these H-bonds to detect structural motifs, including turns, bridges, a-helices, and 3-sheets. A detailed description of the DSSP algorithm is provided in Appendix
T--ellix
34
T
x-belix (puly proline Il helix)
Turn
Bend
hydrogen bonds (H- energy
None
Table I: Secondary structure tokens and their corresponding types.
For protein with amino acid sequence represented {1x)a= where each ey denotes a residue, DSSP assigns secondary structure token 8% to each residue, indicating its structural type (e.g.. a-helix, 3-strand, none). Table summarizes the complete list of secondary structure tokens and their corresponding motifs. This process yields an annotated sequence {(2), Sx) 1 k=1 where each residue is paired with its secondary structure token. We then use this annotation to segment
1 " H Each subsequence S; consists of consecutive residues that share the token of the secondary structure 8k- Figure illustrates the process of identifying and segmenting the protein sequence based on secondary structure annotations.
sequence into a set structurally coherent subsequences, denoted by {S; = { j==n,
The resulting set of subsequences {S. (_S))_, grouping residues by their secondary structure types, forms the foundation for constructing the hierarchical graphs. Each jubsequence-rrepeesmnine distinct secondary structural motif-will serve as node in the higher-level graph representation.
3.3 Construction of Hierarchical Geometric Graphs
Building upon segmenting a protein sequence into secondary structure- subsequences {S:)= we construct a hierarchical geometric graph representation that captures both line-grained and coarse- grained relationships. Specifically, we construct a collection of intra-structural graphs, each modeling residue-level interactions within secondary structure unit by treating residues as nodes, and single inter-structural graph that captures the spatial organization and relative orientation among these units, treating each unit as a node. This multiscale design enables the framework to preserve detailed geometric features within motifs while capturing LRDs across the overall protein structure. An illustrative example of our hierarchical graph construction is provided in Fig. 4 To ensure geometric completeness and computational efficiency, adopt the SCHull graph construction method [4]
5

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S1
S1
g
S2
S2
PY
FLO
Sa
S4
Intra-Structural Graphs
Inter-Structural Graph
Figure 4: Hierarchical geometric graph construction. Left: synthetic protein-like structure composed 14 Intra-structural graphs G, capture local information within each subsequence St using SCHull. Right: The inter-structural graph & is formed by connecting the geometric centers of each S,, modcling higher-level structural relationships between secondary structural motifs.
residues [ox 14 grouped into four secondary structure subsequences {SS) j== Middle:
(see also Appendix C. which constructs sparse yet rigid graphs based on node coordinates. We now describe how this method is applied at both levels of our hierarchical construction:
Intra-Structural Graph Gi- For each secondary structure unit S,, we build an intra-structural graph G, where nodes represent residues within S,, and edges are computed based on the 3D coordinates of their a-carbon atoms using the SCHull method. Node and edge features include geometric features generated by SCHull, combined with -ssiuue-specific attributes (e.g., amino acid type).
Inter-Structural Graph g. The inter-structural graph G is built by treating each secondary structure unit S; as a node, with its coordinate defined by the geometric center of its residues. SCHull is again applied determine edges between these structural units based the spatial arrangement of their centers. In addition to geometric features on SCHull, we incorporate an additional edge feature, 9: 9j. where 9i F(G:) denotes local frame computed from G.. The product g; 9j captures the relative 3D orientation between two secondary structure units, following the orientation encoding proposed in {8] (see also Section for details). This orientation encoding is the key ensuring the expressiveness guarantee proved in Section The construction of frames is discussed in Appendix B
Remark 3.1. Dihedral angles widely used capture relative orientations between adjacent residues 200 or local structural motifs [ 38). In contrast, our approach leverages local frames to encode relative orientations between secondary structure units. Notably, shown in [8), dihedral angles are inherently contained within the product 9, 9j. which encodes richer geometric features.
Beyond geometric fidelity, our hierarchical construction also ensures strong sparsity, which is crucial for scaling GNNs to long protein sequences. The following proposition provides upper bound the total number of edges in the hierarchical graph:
Proposition 3.2. Let N be the total number of residues in protein, denoted by {uk]R= and let {S. represent its segmentation into secondary structure units. For each unit S,, let G. denote the intra-structural graph, and let G be the inter inter-structural graph connecting the structural units. Let E, and e denote the sets of edges in Gi and G, respectively. Then the total number of edges in the full hierarchical representation satisfies:
18
1 < 3N.
1=1
In Section 5 we report the total number of edges created in our framework, along with the average runtime and memory usage, and compare these metrics against existing methods across benchmarks to highlight the efficiency gains enabled by our sparse hierarchical design.
Two-Stage GNN Architecture for Multiscale Protein Modeling
Finally, we introduce a two-stage GNN framework that leverages our multiscale graph representation for efficient and expressive protein learning. Figure 2 illustrates the overall architecture.

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The first-stage GNN operates independently on each intra-structural graph Ge, where each graph corresponds single secondary structure unit. It encodes local geometric and chemical interactions among residues and generates embeddings that summarize each unit's internal feature. These embeddings are then passed the second-stage GNN, which treats each secondary structure unit as node in the inter-structural graph g. This graph captures spatial and functional relationships between structural motifs, enabling the second GNN to model LRDs and global features of the protein. Figure 2 provides overview of this multiscale learning framework.
Message Passing within Secondary Structure Units. The first-stage GNN applies message passing to each intra-structural graph Ge, capturing local geometric and chemical interactions and producing a compressed embedding s; for each secondary structure unit:
fk (1+1) = UPD, ( AGG,(E , f/0,ew eu (UJR ( (Ti) | V(G.)))
for = 1,.,...,,
, 1,
(2)
8; = readout
where f( denotes the initial node feature of residue uy (e.g., amino acid type) and es denotes the attribute of edge (k. 7) on G. (9c) is the neighborhood of node h on Gi. UPD, updates node features, AGG1 aggregates neighbor features, and readout; produces the final embedding.
Message Passing across Secondary Structure Units. The second-stage GNN operates on the inter-structural graph G. where each node represents secondary structure unit. Node features are initialized using the embeddings produced by the first-stage GNN. This stage then performs message passing over G and ultimately outputs the global feature vector Sglobal. which serves as the final output of our framework:
((+1) UPD2 (s,",AGG, (H(S)),.f.e.. |9g E &N,. for t = -,,,.... T2 -1, 8
,99. s (T)) l V(G)D),
glohal readout
where 8; (0)
si eiz denotes the attribute of edge (i.j) on G, 8global represents the final output feature of our framework. The functions UPD2, AGG2, and readoutz denote the update, aggregation, and readout operations, respectively.
Maximal Expressiveness Two-Stage GNN Framework. We now provide a theoretical characteri- zation of the expressiveness of our proposed multiscale hierarchical learning framework. This analysis builds on the notion of maximal expressiveness introduced in Section
2 (see also Theorem 2.1]
standard approach for evaluating the expressiveness of GNNs 44 28
19.. To formalize maximal
expressiveness in our setting, we begin with the following assumption:
Assumption 4.1. UPD,, UPD2, AGG,, AGG2, readout, and readoutz are injective.
This assumption commonly adopted in theoretical analyses of GNNs characterize the best possible representational power of an architecture 44 28 19 40) Crucially, it serves as theoretical tool rather than a requirement for practical implementations, illustrating what the model could achieve under ideal conditions. Under this framework, we can formally state the following result on the maximal expressiveness of our model:
Theorem 4.2. Let F denote the two-stage GNN architecture defined in Section with depths Ti, T2 > and using the hierarchical graph construction described in Section 4 Under Assumption 4.1 F can distinguish any pair of protein structures that are not identical under rigid motions.
Remark 4.3. In practice, we do strietly enforce the injectivity required by Assumption 4.1 instead relying on sufficiently expressive MLPs with ReLU activations. Even when using non- injective pooling operations (e.g., mean pooling), models integrated with the SSHG framework still demonstrate consistent performance improvements, as shown in Section[5] Incorporating more expressive or injective aggregation schemes remains promising direction for future work.
5 Numerical Experiments
We evaluate the effectiveness and efficiency of our proposed Secondary Structure-based Hierarchical Graph (SSHG) learning framework on two benchmark protein modeling tasks: enzyme reaction
7

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classification [13] and protein-ligand binding affinity (LBA) prediction 39 27 the following:
Our goal is to assess
Efficiency: The hierarchical design of SSHG (Figure 2 enables integrated GNNs to operate on sparse graphs, leading to a significant reduction in training time.
Efficacy: Despite operating on sparser graphs, SSHG-based models achieve superior accuracy,
consistently matching or even exceeding state-of-thh-ar (SOTA) models. Full model configurations
and dataset statistics are provided in Appendix E.2 and E.T
Experiment Setup: All models are implemented using PyTorch Geometric [10] and trained on NVIDIA RTX 3090 GPUs. To mitigate overfitting, we follow [37] and apply Gaussian noise (std 0.1) and anisotropic scaling in the range [0.9, 1.1] to the node coordinates in both the original graph framework and SSHG framework. Additionally, we randomly mask amino acid types and secondary structure types with probabilities of 0.1 or 0.2. We apply the SCHull graph construction method [4] to construct intra-structural and inter-structural graphs. Specific training setups, architectures, and hyperparameters for different tasks are available in Appendix B
Baseline and Metries:
We integrate our SSHG framework with several backbone models, including GVP-GNN 18) FroNet-Backbood [37, and Mamba [33; see Appendix E.3 for implementation
details. Models enhanced with SSHG are denoted by appending "+SSHG" to the original model
name (e.g., MSmba+SSHG, ProNet+SSHG). We compare these SSHG-un -Sggeeegeeteed models against a range of established baselines, including GCN [22], IEConv [4), DWNN (6),, GearNet [48],
HoloProt 33 GVP-GNN [8], and ProNet-Backbone 37 across two benchmark tasks: enzyme reaction classification (React) [5) and protein-ligand binding affinity prediction (L.BA) 27).
Performance is evaluated using classification accuracy for EC reaction classification, and standard regression metrics, including root mean square error (RMSE), Pearson correlation, and Spearman correlation for LBA. To further highlight the efficiency and scalability of our framework, we also report additional metrics in the ablation study, including training time per epoch (s/epoch), memory usage, number of model parameters, and the average total number of edges in the graph representations used by the GNNs.
5.1 EC Reaction Classification
Enzymes, which catalyze bi- ological reactions, are catego- rized using Enzyme Commis- sion (EC) numbers based on the types of reactions they fa-
Method
Test Ace
Ave. Time (slepoch)
# params
GCN
186
GCN+SSHG (ours)
71.2
150
IEComy [14
87.2
9.8M
DWNN 26
76.7
GearNet
cilitate (2)]. this task, we evaluate the performance of our SSHG-based models-ProNNet- SSHG and Mamba-SSHG-oo enzyme reaction classification to demonstrate the benefits of incor- porating secondary structure in- formation and encoding geomet- ric relationships within/across structural motifs. We follow the same dataset and experimental setup in (7 (15). Details of the dataset splits and training settings provided in Appendix [E.T] Notice that the baseline GVP-GNN in [18] uses radius cutoff of 4.5 A, achieving an accuracy of 65.5%, while we increase the cutoff to 10 A, which improves accuracy to 68.5%
HoloProt [3]
78.9
300
1.4M
GVP-GNN [8]]
68.5-10..1
334
LOM
GVP GNN+SSIIG (ours)
73.6..0.1
236
1.0M
ProNet-Backbone [77
86.4+-0.2
1.3M
ProNat-SSHG (ours)
87.2-1:0.2
140
1.3M
Mamba [13]
85.9110.2
236
SSH (ours)
88.41:0.3
157
1.5M
Table Results of protein reaction classification. Here, "Ave,Time" denotes the average time for training one epoch.
As shown in Table 2 the use of SSHG consistently improves performance across bascline models, including GCN [22]. GVP-GNN [18), ProNet (37), and Mamba [13]. In particular, ProNet-SSHG significantly reduces training time compared the original ProNet-Backbont
while matching the best- test-prrfoomingggg baseline (IEConv) with far fewer parameters (1.3M compared to 9.8M). Mamba- SSHG further improves prediction accuracy, highlighting the benefits of integrating sequence mod- eling into our hierarchical framework. This task confirms the advantage of SSHG in boosting both computational efficiency and classification performance. For baseline comparisons, we adopt results reported in prior works and omit training time when not provided in the original papers, except for models integrated with SSHG, which we re-evaluate using our setup to ensure fair comparison. Reported bascline results are consistent across all tasks.
8

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Method
RMSE (1)
Peurson (1)
Spearman (1)
Ave. Time (siepoch) ($)
TAPE [2
1.890
0.338
0.286
IBConv [1]]
1,554
0.414
0.428
Holoprot-Full Surface 15
1.464
0.509
0.500
45
GCN [2]
1.925
0.322
0.287
28
GCN+SSHG (ours)
1.788
0.392
0,359
23
GVP-GNN [8]
1,529
0.441
0.432
49
GVP-GNN SSHG (ours)
0.477
ProNet-Buckbone (37)
1.458
0.546
0,550
32
ProNot+ SSHG (ours)
1.435#0.004 004
0.579=0.004
0.591 003
24
Mambe [13
1.457 0.000
0.565 0.003
0.554 0,004
27
Mambe+SSHG (ours)
1.399=000003
0.614=0.003
0.610= 000
Table 3: Results of LBA prediction task. Herc, "Ave.Time" denotes the average time for training one epoch.
5.2 Ligand Binding Affinity
We further demonstrate the effectiveness of our SSHG framework on the benchmark task of protein- ligand binding affinity (LBA) prediction. Accurate LBA prediction plays critical role in drug discovery by guiding the selection of promising drug candidates and minimizing the need for costly and time-intensive experiments. We evaluate our Gadels-GGN+SSHEE GVP-GNN+SSHG, ProNet- SSHG, and Mania-SSHGG-usnng the PDBbind dataset [39 27), following the experimental protocol established by 188 which includes 30% sequence identity threshold assess model generalization to unseen proteins. More details on the dataset and experimental setup are provided in Appendix I To quantify how geometric features and secondary structure information enhance the predictive capacity and generalization ability of GNNs, we evaluate model performance on the test set using standard regression metrics: RMSE, Pearson correlation, and Spearman correlation. As shown in Table 3 ProNet-SSHG outperforms all baseline models, including the best baselind"] ProNet- Backbone, in terms of both predictive accuracy and computational efficiency. Mamba-SS SSHG further improves performance across all three metrics while maintaining competitive training speed. These results confirm the advantages integrating SSHG into existing architectures, enabling both higher accuracy and improved scalability.
5.3 Ablation Studies
In this section, we conduct ablation studies to investigate the impact of key components in our SSHG framework. Specifically, we examine: (1) the performance and efficiency trade-offs between existing dense sparse radial graphs versus SSHG-baser construction, (2) architectural variations in the two-stage GNN design within SSHG under comparable parameter budgets, (3) the role of the hierarchical strategy, (4) the contribution of geometrie features 9, 9j. and (5) the effect of incorporating secondary structure (SS) information. Tables and factors. Due space constraints, analysis of the remaining components is deferred Appendix E.5 (Table B) All experiments are performed on the EC reaction classification task, with each model trained for 300 epochs using batch size of 16. We report test accuracy alongside training efficiency metrics, including average time per epoch and peak memory usage.
5 present results for the first two
Table/coompar training efficiency and resource usage across different graph construction strategies. For baseline models like ProNet and GVP-GNN, increasing the radial cutoff improves accuracy but incurs substantial computational costs. Raising the cutoff from 4 to 16 increases the average number of edges from ~1 to ~ leading to much higher memory usage and training time. While denser graphs enhance expressiveness, they less practical for large-scale applications. In contrast, SSHG-hased models achieve equal or better accuracy with far fewer edges and significantly lower computational overhead. This efficiency stems from the hierarchical design, which decouples local and global interactions. On both ProNet and GVPGNN, SSHG attains up to a 2x speedup in training time and a 90% reduction in memory usage while still improving accuracy.
evalluaes the robustness of SSHG to architectural variations, specifically how parameters are distributed between the two GNN stages. All configurations perform well, showing the framework's 's flexibiliity. Notably, allocating more capacity to the first-stage GNN slightly improves accuracy, suggesting that richer local (residue-level) representations are more beneficial than larger global inter-structural stage alone in our SSHG framework.
For these protein tasks, Mamba is implemented in work rather than using prior implementations.

---

Model
+SIHG
CutolT
Avg. Num Edges
Time (slepach)
Mem (MiB))
Test Ace (%i)
ProNet
1,034,5
1,290
78.1
6
4,755,2
165
7,760
82.
B
8.013,9
185
9,580
85.6
11,316.8
16
14,881
247
17,768
87.0
1,593.3
140
1,818
87.2
GVP-GNN
4
1,344,5
216
1,558
65.5
6
4,755.2
254
3,828
66.9
6,286
10
11.316.8
334
8,930
68.5
16
14,881
354
11,248
69.2
1,593.3
236
1,416
73.6
Table 4: Efficiency comparison. Training efficiency and accuracy of different GNNs with and without SSHG
across varying cutoff radii. SSHG achieves higher accuracy while substantially reducing runtime and memory usage.
MPGNNI
MPGNN2 params
Ave. Time (s/epach)
Mem(MiR)
Test Acc
ProNet-SSHG
0.69M
0.69M
140
1818
87.2
2656
87.4
0.34M
1.03M
142
1720
87.1
GYPONN-SSIIG
0.53M
0.53M
236
1416
73.6
0.79M
0,27M
232
1451
75.3
0.27M
0.79M
228
1372
71.6
Table Architecture Comparison: Two-stage GNNs with varying size ratios between the first- (MPGNNI) and second-stage (MPGNN2) graph networks.
Concluding Remarks
In summary, we propose multiscale and scalable GNN-based framework for protein representation and learning by leveraging hierarchical graph construction that aligns naturally with biological structures. By combining domain knowledge of secondary motifs with multiscale graph design, our approach captures both fine-grained residue-level interactions and coarse- coarse-grained structural rela- tionships through a collection intra-structural graphs, each corresponding a secondary structure motif, and single inter-structural graph that encodes their spatial arrangement and relative orien- tation. Theoretically, we establish that our framework preserves maximal expressiveness, ensuring no loss of critical geometric information. Empirically, we demonstrate consistent improvements in both predictive accuracy and computational efficiency across standard benchmarks. These results highlight the potential of our method as general and flexible foundation for protein- learning tasks, opening up new avenues for integrating biological priors into geometric deep learning.
In future work, we plan to extend our investigation beyond the current experiments on enzyme classification and bigand-binding affinity prediction. We aim to evaluate the framework on additional tasks such as fold classification and protein-protein protein interaction prediction. Moreover, we will explore architectural enhancements through injective aggregation schemes, more expressive pooling mechanisms, alternative motif definitions, and integration with pretrained protein language models to further improve the framework' generality and performance.
Societal Impacts: Our paper presents new efficient and accurate machine learning model for learning biomolecules, which can impact structural biology and life sciences. We do see additional negative societal impact compared to existing approaches due to our work.
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Acknowledgement
This material based on research sponsored by NSF grants DMS- DMS-2208361, DMS- 2219956. DMS-2208332 and and DE-SC0025801. This work also supported by NIH grant RO1HLI6351.
and DOE grants DE-SC0023490, DE-SC0025589,
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Appendices
A Missing proofs
Proposition 3.2. Let be the total number of residues protein, denoted by {vx and ler {S:)=1 represent its segmentation into secondary structure units. For each unit S let Gs denote the intra-structural graph, and let 9 be the inter-structural graph connecting the structural units. Let & and & denote the sets of edges in 9 and 9, respectively. Then the total number of edges in the full hierarchical representation satisfies:
1
Σ 1=1
& < 3N.
Recall that for a point cloud with m 2 points, the SCHull algorithm constructs geometric graph with at most 3m 6 edges [41]. For the remaining cases, when = 1, no edges can formed, and when m 2, there exactly one edge connecting the two points.
Proof of Proposition 3.2
Now, consider the following partition of {1, ,
13:
J1 = [i||S,] = 1}.
J2 = [i||S:] = 2}.
(4)
J23 = [i]|s;] 2 3).
According to the construction of St, we have
J1 + [22 + |2al = 7 and |1| + 2/ +
Σ iEJ>a
|S = N. (5)
Then have for each intra-structural graph Gi, constructed over the residues in secondary structure unit St, the number of edges satisfics:
if E JI if E J2 3|S.| -
| <
(6)
if E J23
Summing over all intra-structural graphs gives:
Σ
1
Σ iEJ1
|8 +
Σ
iEJ2
iEJos
0. | + 1J2 + Σ
3|S 6
KEJs3
|J2l + 3N - |41 - 2/12() - 6|.2s1
< 3(N |1 1 J22 - 122al)
< 3(N- - I).
For the inter-structural graph G, which connects the secondary structure units, SCHull yields:
|3| S 37 - 6.
Combining both bounds:
| +
Σ 3 (3I - 6) 3(N - -)) 3N - 3N.
1=1
Therefore, the total number of edges in the hierarchical representation is strictly less than 3N, completing the proof.
0
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Before proving Theorem 4.2
we first recall Theorem 2.1 which will be applied in the argument:
Theorem 2.1. FT Let be a maximally expressive GNN with depth II' 1. Then F can distinguish between the attributed SCHull graphs of any two non-isomorphic generic point clouds.
Theorem 4.2. Let F denote the two-stage GNN architecture defined d in Section44 with depths T 1, and using the hierarchical graph construction described in Section 4 Under Assumption 4.1 F
T2
can distinguish any pair of protein structures that are not identical under rigid motions.
Suppose the model F assigns identical outputs to the hierarchical graphs of two protein structures represented by the point clouds (X, F) and (X', F²). Let 9, 9" denote their respective hierarchical graphs, and let GJ GJ denote the intra-structural graphs corresponding to their secondary structure units. Denote the first- and second-stage GNNs in F by Fi and F2, respectively.
Proof of Theorem 4.2
By construction, the inter-structural graph is defined over the set of tuples [FF( x:)]== and ((F))))) = where 2; and zj are the geometric centers of the secondary structure units Gi and 9j. respectively. According to Theorem 2.1 F2 assigns identical outputs to 9 and g' and only (G)) =) and ((()(9)), z})} are identical up to rigid motion. That is, there exists bijection b {l...., + {1....,J. J/ and a rigid transformation g such that for all i,
(G.) = (9%))) and 2; =
Reindexing the units according to the bijection, we assume: Fi Fi(G.) Fi (G;) for all i. Addition- ally, our construction includes the relative frame feature 9. 9j as edge attribute. Since these are preserved between G and G', we have
9 9; 96 9j for all i.j. where Si denotes the local frame of G. constructed using the method described Appendix B Specifically, each frame Si comprises vector specifying the geometric center. By Lemma A.1 the equality (G.)). Fi (9)),,) 9, 93) = (F)) Fi (g)): : s. σÿ) implies that the underlying point clouds of G. Ug, and G UgJ are identical up an isometry. Using an inductive argument, we conclude that the union of all point clouds across the units G. and G must also be identical up to an isometry. Thus, the full point clouds (X, F) and (X', F') must be identical up global isometry.
orthogonal matrix representing the orientation and
Lemma A.1. The 3-tuple (Fi FI) 93) uniquely determines the union of the underlying point clouds of Gi, G3 up to an isometry.
Proof. Let = ((Fi, J(x) and P, = {( f5,}] denote the underlying point clouds of G and Gj. respectively. Let FF be the function used to generate the equivariant local frame for each secondary structure unit, as described Appendix B That is, 9i F(G.) = F(P,) for any Our goal is to show that the union P. U P3 is uniquely determined, to an isometry, by the 5-tuple (9.), z,, Fi (9)), 3j 9. 9;).
We first observe that the function c, which maps the point cloud P. to F(G.) P. = {FF())) 2i,k, f(x)) }. is invariant under any rigid motion applied to {x,,k)- To see this, con- sider a rigid motion g E(3), where E(3) denotes the Euclidean group of isometries. By the equivariance of F, we have:
(Flg (.)) -1 = (F(G.) T :ii F(G.) L.
which confirms that is invariant.
By Theorem 2.1 there exists a function h such that ((.:) h (Fi(G.)) for any point cloud. Now consider the following construction that combines the information the 3-tuple.
1 ((F) (oT - (FF(()))) = c(P.) (g) 91 .p(p))))))))))))))))
= 9. P.. u (o 9, P,)
(8)
= 9. (P.UP)).
This final expression shows that the union P; UPP uniquely determined up to a global isometry by the 3-tuple data, completing the proof.
D
23

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B Construction of Local Frames
In 9 g the coordinates of nodes a; and 2; on edge (i,j) are used define equivariant local frames, with the following transformation:
X 3jj
( 2.y)
|
|
This transformation creates a local frame for the edge, converting equivariant features within node's neighborhood into invariant features (independent of orientation). or vice versa, through multiplication by the frame or its inverse. It can also be used to compute the crucial transition information g. gj for neighborhoods NG and Ng
However, this method is limited to pairs of nodes and cannot be directly extended to structural units, which typically consist of more than two unordered nodes. To address this, adopt the approach in (1.. which constructs equivariant frames for point clouds (or their corresponding geometric graphs) of arbitrary size. Specifically, the algorithm in 2 constructs a frame F X E(3) over the set of point clouds X' This mapping produces not only an orthogonal matrix representing the orientation but also the geometric center of the point cloud, jointly forming complete frame Euclidean space. Importantly, the frame construction satisfies equivariance with respect to isometries: for any rigid motion g E E(3) and any point cloud E X, we have
Flg P) = g g(P),
where denotes the group product in E(3).
Remark B.I. While this construction applies all point clouds, it may not be fully equivariant but rather relaxed-equie equivvraani for symmetric inputs. This does not impact the distinguishability of the framework, as shown in Theorem 4.2] but it could reduce the framework's strict invariance. Fortunately, symmetric inputs are rare in practice-par patticcapaat for individual protein structures- and this issue be mitigated by introducing small perturbations to break the symmetry.
C SCHull Graphs
In this section, we provide brief review of the SCHull algorithm proposed in 41 for constructing a sparse, connected, and rigid for given point cloud (X, F) be a point cloud, where [æt...., .N ) F denote the point coordinates and features, respectively. Let V denote the point index set. In particular, the SCHull graph for the point cloud (V, X) is constructing in the following three steps:
Step 1: Project points onto the unit sphere. Let
N Σ x2
be the center of point cloud. Consider the projection
P
3
3
where S² = fx E R² ||ق||| 1} is the unit sphere. That is, Ps projects points onto the unit sphere centered at Applying this projection to all points, we obtain new point cloud (V.Pa()) on S?, where
P X
(x; )). Par (x))..... (=)
Step 2: Construct the convex hull of the projected point cloud. Next, SCHull constructs
convex hull hull-using the QuickHull algorithm [3] 33l-for for the projected point cloud (V, (7,Pp(()) (X)). Notice that this step is very efficient with computational complexity O(N log N).
Step 3: Construct the SCHull graph. The SCHull graph for the given point cloud (X, F) is then defined as G = (V, E, F"). Specifically, nodes i, are connected by an edge in E if and only if their projected points the unit sphere par (x, Pa (w(x,) are connected by an
24

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edge on the convex hull. In addition, the graph incorporates geometric attributes as follows. Each node feature in F' includes the original feature from F, augmented with scalar node attribute defined below. Similarly, each edge in & is associated with the following attributes:
the edge attributes of (i,5) : (IIæ, zy ll. T.y) for any (i,j) E E, and the node attributes : ||a;; æ:| for any EEW V.
SCHull graph has two remarkable properties with provable guarantees: (1) The graph is sparse and connected with edges that satisfy |8] 3.V -6 6 when N = | 2 3 (cf. [4] Proposition 3.1]). (2) SCHull graphs of any two non-isomorphic generie point clouds can be distinguished by maximally expressive GNN with depth (see Theorem 2.1 i.e., Theorem 3.6]).
D DSSP Algorithm
DSSP is both database of secondary structure assignments for all protein entries in the Protein Data Bank (PDB) 4 as well as a program that calculates DSSP entries from PDB entries [4]] The algorithm first detects the presence of backbone- backbone-backbone hydrogen bonds (H-bonds). An H-bond between amino acids is considered to be present if the electrostatic interaction energy, E, between the carboxyl group of one and the amino group of another is calculated be less than -0.5 kcal/mol.
Specifically, E- = 0.084 FONNT + FICHT FIHHT P(CN) 332 kcal/mol where r(AB) is the interatomic distance between A and B. Following (2n. Hbond(i,) j) denotes that an H-Bond is present between the carboxyl group of residue i and the amino group of residue j.
Once H-bond presence is decided, the algorithm determines the presence of elementary H-bond patterns: n-turns (where n=3, 4, or 5) and bridges (which can be parallel or antiparallel). An n-turn considered to exist at residue if Hhonddii + n)) is present. A bridge may exist between two non- overlapping stretches of three residues each. A parallel bridge is said to be present if Hbond(i - 1, and Hbond(j, + 1) or if Hbond(j 1, i) and Hbond(i, j + 1). An antiparallel bridge is said to be present if Hbond(i, j) and Hbond(j i) if Hbond(i 1, + 1) and Hbond(j - 1, + 1).
These patterns are then used to identify cooperative H-bond patterns: helices, 3-ladders, and 8-sheets. Helices consist of two consecutive n-turns for fixed n, e.g., 4-helix is present from residue i to i + 3 there a 4-turn at residue i 1 and another residue Note that a 3-helix is commonly called a 310-helix, 4-helix an a-helix, and a 5-helix a т -hellix. B-ladders consist of one or more consecutive bridges of identical type and (S-sheets consist of one or more ladders connected by shared residues. group of five residues with high curvature is known bend. The curvature at residue i is calculated as the angle between the backbone direction of the first three and last three residues of this group of five. Specifically. if Co is the position vector of the a-carbon of residue j. a bend is considered to exist if the angle between C C₁-2 and C C is greater than 70%.
Although it is possible for an amino acid to belong to more than one of these structures, each residue assigned single letter from the list in Table T The original algorithm assigned letters in the following order from left to right: H. B, E, G, I, T, S; once a residue is assigned a letter, it is not changed. More recent versions assign N -heelcess before a-helices [34] and also detect another type of II (PPII) helix (30].
helix known as x-hheix or
E
Additional Experiments Details
E.1
Datasets and Experiment Overview
E.1.1
Datasets
Reaction dataset. For the reaction classification task, 3D structures of 37,428 proteins corresponding to 384 enzyme commission (EC) numbers are obtained from the Protein Data Bank, with EC annotations for each protein retrieved from the SIFTS database [7]. The dataset is divided into 29,215 proteins for training. 2,562 for validation, and 5,651 for testing. Each EC number is represented across all three splits, and protein chains sharing more than 50% sequence similarity are grouped.
LBA dataset. Following [8] we perform ligand binding affinity predictions on a subset of the commonly-usee PDBbind refined set [39 27.. The curated dataset of 3,507 complexes is split into
25

---

train/val/test splits based on 30% sequence identity threshold to verify the model generalization ability for unseen proteins. For protein-ligamd complex, we predict the negative log- transformedd binding affinity pk = 10 K) in molar units.
E.1.2 Experiment Overview
We primarily follow the GNN architectures, training setups, and hyperparameter search spaces used in the baseline models GVP-GNN [8] and ProNet [37] Our SSHG model adopts nearly identical feature embedding functions, message passing blocks, and readout functions for the hierarchical geometric graphs-namely the Intra-Structural Graph and inter-Structural Graph. Furthermore, we integrate Mamba [13] to capture the sequential information of the tokens in the Attrr-Structural Graph.
See and E.3 for details.
E.2 Model Configuration
We integrate our SSHG framework with two models tailored for protein tasks, one is GVPNet and the other is ProNet (37). Moreover, to capture the sequential information of the secondary structure tokens, we integrate Mamba [13] into our SSHG model. See E.3.1 for further details. The
illustration of the architectures of our SSHG model is shown in Figure Below some details:
Message Passing Blocks: We use the same message-passing GNN (MPGNN) architectures
as those in the baseline models 18 37.
Edge Feature Function: We use the same edge feature construction methods as those the baseline models 18 37
Scatter: The tensor output from the message passing blocks contains embedding vectors for all nodes across all graphs in the batch. We use the Py Pyrcch scatter function to aggregate these node embeddings into tensor with one embedding per graph, corresponding the number of graphs in the batch. Similarly, we apply scatter to aggregate intra intra-structural graph node embeddings into a tensor with one embedding per unit in the inter-structural graph.
E.3
Implementation Details
E.3.1 Integrate Mamba into SSHG
To capture the sequential dependencies among secondary structure tokens in our SSHG model, we consider using Mamba. Mamba [33 special type of state space model defined by the following ODE system:
dh(t) A (x()) h(t) (=x(t))((t) B (x(t)))=((t xtt)
dt
(10)
y(t) W.h(t)
where:
denotes the time step (discrete or continuous).
c² denotes the feature vector time step in the input feature sequence.
(t) denotes the state vector, where the IC ((0)) is often learnable vector.
Atx(t)) = diag (W)x((t) dt is the input-conditioned decay/filter coefficient, where W, time-invariant learnable parameter and o denotes the sigmoid function.
(a+())) = W Wnz(1)/dl where Wa a a time-invariant learnable parameter.
W, is
learnable parameter.
has been increasingly adopted in large language models (LLMs) 25 35 due to its efficiency
and competitive performance compared to traditional Transformer architectures.
MSHG+Mamba: Let x; = s
(0)
V(G))) be the initial node readout, (f (Ti)
26

---

Original Node Features: Amino acid type, Residue atoms features etc.
Edges from original protein structure or radial cutoff method;
Node coordinates
Node Feature Fimbedding Layers (A) (MLP)
Edge Attribute Function: embed distances und angles
SSHG
Nodes i V and Coordinates (2.) CR
Node Feutures: F (f1..... fn)
Stira-Structural Graph
Edies:: ji E Edge Attribute E
Ey for (
Message Passing Blocks (Stage
Graph
Edges constructed SCHull with rudial cutoff method;
Original Node Features: Token infomaaions: Geometric features SCHull
Unit coordinates
Scatter: Compress node feuture embeddings for each unit of the secondary structure into one feature vector
Edgo Attribute Function: embed distances and angles
Node Feature Embedding Layers (B) (MLP)
Feature Fushion Layers F
8
81
Unit Nodes V(g) and Unit Coordinates (af)
Unit Node Features: ps [s [ srl
Edges: (i, E E
Edge Aitributes: 5 for E
Mamba for Sequential Feature Embedding (Alternative)
Message Passing
Blocks (Stage 2)
Embedding Fushion Function (Scatter MLP)
Output Function (Scatter + MLP)
Figure 5: Illustration of the architectures of our SSHG model, with and without the integration of Mamba. The red arrows and red dashed arrows indicate the input-output dependencies in the SSHG and SSHG+Mamba models, respectively.
features for node of the inter-structural graph as defined in equation 3 equation 10 into
Then we discretise
h++1 A(x,),, B(x))a
(1I)
Vi+1 h(
and then obtain the outputs 8:am = yi+1 for V(G).
Back to the message passing across structural units, we obtain the node features of the inter-structural graph Sylabal readout2 (f ( ((() V(G) in equation 3 Then we input stam and &ylobat into output function tailored for the task's target final output. Figure shows the architecture of our model integrated with Mamba.
27

---

E.4 Discussion on Mamba
We observe that directly applying Mamba to the node sequences of the original protein graph increases the sequence length by over threefold, significantly slowing down the model. Despite this, the Mamba-only model achieves performance comparable to GNN-based methods such as ProNet, even without leveraging geometric information. This efficiency is largely due to mamba_ _amm, a highly optimized CUDA C++ implementation that enables fast training of complex sequence models. In contrast, most GNN implementations are primarily in Python, which introduces overhead due to slower data processing and iterative computation.
To illustrate the difference, consider typical pipeline structures:
GNN-based models (e.g., ProNet):
Feature embedding + Python
Iteration over blocks + Python
Message passing Python
Output layer + Python
Mamba Mamba_ssm-base ssm-baaed models:
Feature embedding Python
Iteration over time steps - CUDA C++ (fused kernel)
Selective SSM operations - CUDA C++
Output layer + Python
The expensive recurrent and state-space computations in Mamba are fused into CUDA kernels, bypassing Python's loop overhead. As result, mamba_ narrows the performance gap without relying on structural priors. While it achieves results comparable to other baselines, we still observe significant improvements when incorporating SSHG.
E.4.1 Architecture and Experimental Setup
The number of message passing blocks, hidden channels, and dropout rates used for training SSHG on different tasks are listed Table 6 The implementation of our methods based on Py PyTorch and Pytorch Geometric, and all models are trained with the Adam optimizer. All are conducted on single NVIDIA GeForce RTX 3090 24 GB. The hyperparameter searching space for training is shown in Table
Values/Search Space
Hyperparameter
React
LBA
Number of layers (Ist)
1, 2
Number of layers (2nd)
2, 3
2,
Hidden channels
64, 128, 256
128, 192, 256
Dropout
0.2, 0.3, 0.5
0.2, 0.3
Mamba Blocks
4
Table 6: Model hyperparameters for SSHG
Values/Search Space
Hyperparameter
React
LBA
Epochs
500, 1000
300, 500
Batch size
16, 32
8, 16, 32
Learning rate
le-4, Se-4 steplr
5e-5, le-4, 2e-4 steplr
Learning rate scheduler
Learning rate decay factor
0.5
0.5
Learning rate decay epochs
50, 100
50, 100
Table Training hyperparameters search space.
28

---

E,5
Additional Ablation Studies
w/ SS
hierarchical
w/ goometry
Test Acc
ProNet
X
86.4
87.0
87.5
GVPGNN IK
X
X
68.5
X
X
66.7
71.5
Table 8: Feature Selection: Different GNNs with (w/) without (w/o) hierarchical geometrie graphs, geometric
features 9i gj. or secondary structure tokens.
Table 8 shows that simply appending SS tokens as a feature to the original GNNs does not necessarily improve performance. In contrast, combining SS information and geometric features through dedicated hierarchical mechanism leads to consistent improvements.
29


Reviews:
[{"review_id": "UBpdtEINTI", "reviewer": "Reviewer_5Kv4", "summary": "This paper introduces a multiscale Graph Neural Network (GNN) framework that constructs protein representations using secondary structure motifs annotated by DSSP as building blocks. The model employs a two-stage GNN that first learns local interactions within each motif and then models higher-level relationships between the motifs using geometric orientation features.", "questions": "1. Regarding Table 8 in Section E.4, it is unclear if SCHull graphs are still applied to construct the whole protein graph in the ablation studies. Clarifying this would provide stronger evidence for the usefulness of the hierarchical framework.\n2. The result for GVP-GNN [5] reported in Table 2 is 68.5%, but it appears to be 65.5% in the ProNet paper (\"Learning Hierarchical Protein Representations via Complete 3D Graph Networks\") [1]. Could the authors please clarify this discrepancy?\n3. Could you provide some support for assumption 4.1?\n\n\n### References\n[1] Z. Zhang, Y. Du, et al. (2023). \"Learning Hierarchical Protein Representations via Complete 3D Graph Networks.\"\n\n[2] S. Wang, et al. (2024). \"A Theoretically-Principled Sparse, Connected, and Rigid Graph Representation of Molecules.\" arXiv preprint arXiv:2404.04021.\n\n[3] Z. Zhang, et al. (2024). \"ProtGO: Function-Guided Protein Modeling for Unified Representation Learning.\"\n\n[4] S. Long, et al. (2023). \"Clustering for Protein Representation Learning.\"\n\n[5] B. Jing, S. Eismann, P. Suriana, R. Townshend, & R. Dror. (2021). \"Learning from Protein Structure with Geometric Vector Perceptrons.\"", "limitations": "Since SSHG does not consistently perform better than some recent baselines, it would be beneficial to emphasize the simple framework and its computational efficiency, which allow it to achieve comparable results.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n\n1. The paper is clearly written with well-explained assumptions and methodology.\n1. The method demonstrates promising, improved results on EC reaction classification and ligand binding affinity benchmarks when compared to state-of-the-art methods like ProNet [1].\n1. The adoption of SCHull graphs [2] maintains computational efficiency during training after constructing hierarchical representations for proteins.\n\n**Weaknesses**\n1. The technical contribution represents incremental novelty, as the approach primarily integrates established methods—including SCHull graphs [2], DSSP secondary structure prediction, and two-stage message passing—without fundamental innovation.\n2. The authors did not include results from ProtGO [3] or a related clustering paper [4], which have reported better EC classification performance than Mamba+SSHG. The work in [3] shares a similar concept of clustering proteins to form a multiscale representation, making a comparison worthwhile.\n3. It would be beneficial to benchmark the model on a broader range of protein learning tasks, such as the fold classification and protein-protein interaction tasks discussed in [1], to provide a more comprehensive evaluation of its capabilities. Other tools have provided comprehensive benchmarks for this setting: https://pmc.ncbi.nlm.nih.gov/articles/PMC11213157/, https://proceedings.neurips.cc/paper_files/paper/2023/hash/b6167294ed3d6fc61e11e1592ce5cb77-Abstract-Datasets_and_Benchmarks.html", "was_revised": true, "final_justification": "Score raised from 3 to 4 after author response.", "rebuttal": "We thank the reviewer for this insightful comment. We would like to clarify the key novelties of our framework and highlight the significant improvements it offers beyond prior work. Below, we address your concerns and questions.\n\n---- \n**Q1. The technical contribution represents incremental novelty, as the approach primarily integrates established methods—including SCHull graphs [2], DSSP secondary structure prediction, and two-stage message passing—without fundamental innovation.**\n\n**Response:** \n\nWhile our work builds upon established components such as DSSP annotations and SCHull graphs [2], we respectfully disagree that our contribution is merely incremental. Our key innovation lies in the principled **integration of biological hierarchy and geometric structure into a unified multiscale GNN framework**, which to our knowledge, is the first to: \n\n- Use **secondary structure motifs** as hierarchical building blocks to define coarse-grained graph nodes, replacing radius-based heuristics with biologically meaningful segmentation—substantially improving efficiency and scalability.\n- Introduce orientation-aware edges between secondary structures using relative frame encodings (e.g., $g_i^\\top g_j$), extending prior definitions based on local neighborhoods of individual residues to capture geometric relationships between structural motifs. This is non-trivial, as existing methods for computing such elements do not apply to structural motifs.\n- Provide formal theoretical guarantees, including expressiveness (Theorem 4.2) and sparsity bounds (Proposition 3.2). This contrasts with prior motif-based models, which often lack such guarantees.\n\nThis is not a simple stacking of existing tools, but a biologically aligned architectural design that supports both rigid intra-structural and coarse-grained inter-structural modeling, enabling accurate and interpretable learning of long-range dependencies. \n\nWe also emphasize the practical impact of our method: \n\n- Our SSHG framework consistently improves accuracy and efficiency across tasks like enzyme classification and ligand binding affinity, validated on multiple backbones (ProNet, GVP-GNN, Mamba). We have added further ablation studies; key results are summarized below. Notably, on ProNet for EC classification, SSHG reduces the average number of edges from 14,881 to 1,593, resulting in nearly a **2× speedup in training time** per epoch (247s → 140s) and a **90% memory reduction** (17,768 MiB → 1,818 MiB), while slightly improving accuracy (87.0% → 87.2%). This marks a significant breakthrough in balancing efficiency and performance.\n\n**Ligand Binding Affinity (LBA)**\n\n| Model                  | RMSE           ↓     | Pearson        ↑     | Spearman       ↑     |     Ave.Time  ↓    |\n|---|-----|--|--|--|\n| GVP-GNN                | 1.529                | 0.441                | 0.432                |   49\n| GVP-GNN + SSHG     | **1.488**                | **0.512**                | **0.477**                |  **35**\n| | | |\n| Mamba                  | 1.457 ± 0.004        | 0.565 ±  0.003    | 0.554 ±  0.004    | **27**\n| Mamba + SSHG           | **1.399 ± 0.003**        | **0.614 ±  0.003**    | **0.610 ±  0.003**    | 29\n\n**Extended Ablation Study on EC**\n| Model   | +SSHG   | Cutoff | Avg. Num Edges | Time (s/epoch) ↓ | Mem (MiB) ↓ | Test Acc (%) ↑ | Acc/Time (×100) ↑ |\n| - | - | - | - | - | -- | -- | ---- |\n| ProNet  | No      | 4      | 1,034.5        | 138              | 1,290       | 78.1           | 56.6              |\n|         | No      | 6      | 4,755.2        | 165              | 7,760       | 82.1           | 49.8              |\n|         | No      | 8      | 8,013.9        | 185              | 9,580       | 85.6           | 46.3              |\n|         | No      | 10     | 11,316.8       | 210              | 14,548      | 86.4           | 41.1              |\n|         | No      | 16     | 14,881.1       | 247              | 17,768      | 87.0           | 35.2              |\n|         | **Yes** | —      | 1,593.3        | 140              | 1,818       | **87.2**       | **62.3**          |\n|         |         |        |                |                  |             |                |                   |\n\n\n- Compared to SCHull, our hierarchical design significantly reduces reliance on dense radius-based connections, resulting in faster training times and lower memory usage while maintaining or even improving predictive performance. We have now included th following table for a more comprehensive comparison addressing this into the revised paper.\n\n| Model                  | RMSE           ↓     | Pearson        ↑     | Spearman       ↑     |     Ave.Time  ↓    |\n|------|---|-----|-----|---|\n| ProNet-Backbone        | 1.458                | 0.546                | 0.550                | 32.1\n| ProNet-Backbone + SCHull| **1.321**        | **0.581 ± 0.001**         | 0.578    | 34.4\n| ProNet + SSHG| 1.435 ± 0.004    | **0.579 ± 0.004**    | **0.591 ± 0.003**    | **24**\n\n\n--- \n\n**Q2. The authors did not include results from ProtGO [3] or a related clustering paper [4], which have reported better EC classification performance than Mamba+SSHG. The work in [3] shares a similar concept of clustering proteins to form a multiscale representation, making a comparison worthwhile.**\n\n**Response:** \n\nWe thank the reviewer for highlighting these important works and will include detailed discussions and comparisons of ProtGO [3] and the clustering method [4] in the revised manuscript. Briefly: \n- ProtGO focuses on Gene Ontology prediction using transformers on full protein sequences, differing from our 3D structure-based EC classification and secondary structure motif modeling. \n- While [4] uses clustering of residues for multiscale representation, we leverage **biologically grounded secondary structure** motifs with provable sparsity and expressiveness guarantees. \n- Although these methods show strong performance, they rely on much larger and more computationally intensive architectures (e.g., ProtGO uses channel sizes up to 2048), whereas our model is significantly more lightweight (channel size up to 256), highlighting efficiency benefits. \n\nWe will include this discussion and are happy to add empirical results. However, we found that these models are not yet publicly available. Nevertheless, our work addresses a distinct problem using a compact and efficient architecture that achieves competitive performance. We appreciate the reviewer’s valuable suggestions and will cite these works appropriately.\n\n---\n**Q3. It would be beneficial to benchmark the model on a broader range of protein learning tasks, such as the fold classification and protein-protein interaction tasks discussed in [1], to provide a more comprehensive evaluation of its capabilities.** \n\n**Response:** \n\nWe appreciate the reviewer’s thoughtful suggestion and agree that evaluating across a broader range of protein learning tasks would further demonstrate the generality of our framework. Our current focus on enzyme classification and ligand binding affinity prediction was motivated by their biological relevance and the availability of high-quality benchmark datasets with 3D structural annotations. We are currently working on the Protein–Protein Interaction (PPI) task and training models from scratch, which makes obtaining meaningful results within a week challenging. We plan to include these results in the future and will discuss this ongoing extension in the revised manuscript.\n\n---\n**Q4. Regarding Table 8 in Section E.4, it is unclear if SCHull graphs are still applied to construct the whole protein graph in the ablation studies. Clarifying this would provide stronger evidence for the usefulness of the hierarchical framework.**\n\n**Response:** \n\nWe thank the reviewer for highlighting the need for clarification. In the experiments reported in Table 8, baseline models without our hierarchical framework (SSHG) do not use SCHull graphs; instead,they rely on the default graph construction in these frameworks, which uses radius graphs with a large cutoff. This design clearly isolates the impact of introducing SSHG. Additionally, to directly compare SCHull and SSHG, we have included a new table, as presented in our response to Q1.\n\n---\n**Q5. The result for GVP-GNN [5] reported in Table 2 is 68.5%, but it appears to be 65.5% in the ProNet paper (\"Learning Hierarchical Protein Representations via Complete 3D Graph Networks\") [1]. Could the authors please clarify this discrepancy?**\n\n**Response:** \n\nThank you for catching this important point. As reported in Table 4 (Section 5.3), the baseline GVP-GNN uses a radius cutoff of 4.5 Å, achieving 65.5% accuracy. Increasing the cutoff to 10 Å improves accuracy to 68.5%, as also listed in Table 2 for comparison. We have now clarified this point in the revised paper.\n\n---\n**Q6. Could you provide some support for assumption 4.1?**\n\n**Response:** \n\nAs introduced in the Background section, Assumption 4.1 is necessary for leveraging the theoretical framework of the Weisfeiler-Lehman (WL) test to analyze the maximal expressiveness of GNN architectures—that is, to characterize the best possible representational power an architecture can achieve. This framework is widely adopted across the literature, e.g., [16, 25, 37, 39] (cited in our paper). However, as is common in practice, we do not enforce this assumption strictly. Instead, we follow the standard approach of employing sufficiently expressive multilayer perceptrons (MLPs) with appropriate nonlinearities and sufficient width, as supported by prior works mentioned above. Our implementation adheres to this principle by using MLPs with ReLU activations and sufficient capacity to approximate favorable mappings over the relevant input domains. We have refined the discussion of maximal expressiveness and its underlying assumptions in the revised paper to enhance clarity."}, {"review_id": "6XpAmbPRUe", "reviewer": "Reviewer_obWc", "summary": "The paper proposes a multi-scale protein learning strategy via GNNs. They apply a two-level strategy to learn protein embeddings. The first is to split the proteins into a set of motifs (where each one contains rich domain-specific knowledge), and encode each motif individually. Then, they connects the motifs to form an additional graph and use another GNN to learn the final protein embedding.", "questions": "1. Can the authors clearly demonstrate the novelty of the paper? \n2. Can the authors provide additional ablation studies to demonstrate the superiority of the method?", "limitations": "See weaknesses.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths: \n1. The paper is easy to follow. \n2. The idea is straightforward and make sense. \n3. The proposed method achieves good performance. \n\nWeaknesses: \n1. The work seems to be incremental. The idea of using motifs to represent a protein/molecule is widely applied in many works [1,2,3]. The authors use a widely applied technique to split a protein sequence into a set of groups, where each one represents a motif, and treat each group as a fully connected subgraph. The novelty only lies on the side of how to encode each motif. \n\n[1] Representation for Discovery of Protein Motifs, ISMB 1993\n\n[2] Molecular Representation Learning via Heterogeneous Motif Graph Neural Networks, ICML 2022. \n\n[3] Molformer: Motif-based transformer on 3d heterogeneous molecular graphs, AAAI 2023.\n\n2. The experiments are not sufficient. The authors did experiments over two benchmark datasets and also analyze the model efficiency and architecture. However, there is no clear understanding/demonstration on whether learning the motifs can facilitate the model performance and what are the trade-offs of the method.", "was_revised": true, "final_justification": "The authors provide a simple yet effective method for pretein learning using secondary motifs. Despite the authors provide some of evidences to show the superiority of the method, I still have concerns on the novelty. I think the paper is still below the bar of NeurIPS.", "rebuttal": "We appreciate the reviewer’s insightful comments on our work. We would like to clarify the key novelties of our framework and highlight the substantial improvements in both efficiency and performance that our method achieves compared to prior approaches.\n\n---- \n\n**Q1. The work seems to be incremental. The idea of using motifs to represent a protein/molecule is widely applied in many works [1,2,3]. The authors use a widely applied technique to split a protein sequence into a set of groups, where each one represents a motif, and treat each group as a fully connected subgraph. The novelty only lies on the side of how to encode each motif. Can the authors clearly demonstrate the novelty of the paper?**\n\n[1] Representation for Discovery of Protein Motifs, ISMB 1993 \n\n[2] Molecular Representation Learning via Heterogeneous Motif Graph Neural Networks, ICML 2022. \n\n[3] Molformer: Motif-based transformer on 3d heterogeneous molecular graphs, AAAI 2023.\n\n**Response:** \n\nWe thank the reviewer for pointing out these important and relevant motif-based works. While our framework does share the high-level idea of leveraging motifs, we respectfully clarify that our contribution goes beyond this general idea. Our key novelty lies in the **biologically grounded, hierarchical, and geometrically expressive graph construction**, as well as the theoretically supported and **scalable GNN design**. Specifically:\n\n**1. Biologically Grounded Hierarchical Graph Construction**:\nPrior works (e.g., [2][3]) represent molecules using low-level semantic units such as bonds, rings, or functional groups (e.g., carboxyl groups) as motifs. In contrast, our framework adopts biologically meaningful secondary structure elements (e.g., α-helices, β-strands), which better reflect the hierarchical structures of proteins, similar to [1].\nUnlike [1], which encodes only the sequential order of motifs, we build a two-level hierarchical graph capturing both local geometry and long-range 3D dependencies:\n- Each secondary structure motif (e.g., α-helix, β-strand) is modeled as a rigid intra-structural graph using SCHull, rather than a fully connected graph, preserving fine-grained geometric details while maintaining sparsity.\n- A coarse-grained inter-structural graph is constructed over motifs, incorporating both geometric centers and relative 3D orientations (via local frames), to capture geometric relationships and long-range dependencies among structural units.\n\nTo the best of our knowledge, this is the first framework to incorporate secondary structures into motif-level graph construction for downstream GNN models.\n\n**2. Theoretical Guarantees of Expressiveness:**\nWe provide a **formal expressiveness guarantee** (Theorem 4.2) showing that our two-stage GNN can **distinguish non-isomorphic protein structures up to rigid motions**—a key property of the graph learning framework not addressed in [1], [2], or [3]. Such guarantees are typically offered only in non-motif-based frameworks [16, 25, 37, 39] (cited in our paper).\n\n**3. Sparse and Scalable Design with Provable Bounds:**\nWe provide a provable sparsity bound (Proposition 3.2) on the total number of edges in the hierarchical graph, which is critical for scaling to large proteins. This contrasts with prior motif-based models, which often lack such guarantees due to their design or depend on dense or heuristic graph constructions. \n\n**4. General-Purpose Framework with Empirical Gains:**\nOur SSHG framework is modular and model-agnostic, and we demonstrate consistent improvements in both accuracy and efficiency across diverse tasks (enzyme classification, ligand binding affinity) and backbones (ProNet, Mamba, GVP-GNN). This generality and empirical robustness go beyond the scope of prior motif-based models.\n\nIn summary, this is the **first work** to combine:\n* Secondary structure-defined motifs,\n* Hierarchical geometric graphs with orientation-aware edges,\n* Expressiveness guarantees via WL analysis, and\n* Scalable, sparse modeling for large protein structures.\nThese provide **interpretability**, **biological alignment**, and **geometric expressiveness** not seen in prior motif-based GNNs. We will revise the manuscript to more clearly articulate these distinctions and cite \\[1–3] in our related work section.\n---\n\n**Q2. The experiments are not sufficient. The authors did experiments over two benchmark datasets and also analyze the model efficiency and architecture. However, there is no clear understanding/demonstration on whether learning the motifs can facilitate the model performance and what are the trade-offs of the method. Can the authors provide additional ablation studies to demonstrate the superiority of the method?**\n\n**Response:** \n\nOur experimental design rigorously evaluates both predictive performance and computational efficiency of the proposed SSHG framework against existing methods. In response to the reviewers’ requests, we have conducted more comprehensive ablation studies, some of which are summarized below. \n\nImportantly, on the ProNet model for EC classification, integrating SSHG drastically reduces the number of edges from 14,881 to 1,593—yielding nearly a **2× speedup in training time** per epoch (247s down to 140s) and a **90% reduction in memory usage** (17,768 MiB down to 1,818 MiB). This is achieved while improving accuracy slightly from 87.0% to 87.2%. Such results represent a substantial and meaningful breakthrough in balancing efficiency and accuracy, clearly demonstrating the practical value and impact of SSHG.\n\n**Ligand Binding Affinity (LBA)**\n\n| Model                  | RMSE           ↓     | Pearson        ↑     | Spearman       ↑     |     Ave.Time  ↓    |\n|------------------------|----------------------|----------------------|----------------------|---------------|\n| GVP-GNN                | 1.529                | 0.441                | 0.432                |   49\n| GVP-GNN + SSHG     | **1.488**                | **0.512**                | **0.477**                | **35**\n| | | |\n| Mamba                  | 1.457 ± 0.004        | 0.565 ±  0.003    | 0.554 ±  0.004    | **27**\n| Mamba + SSHG           | **1.399 ± 0.003**        | **0.614 ±  0.003**    | **0.610 ±  0.003**    | 29\n| | | |\n| ProNet-Backbone        | 1.458                | 0.546                | 0.550                | 32.1\n| ProNet-Backbone + SCHull| **1.321**        | **0.581 ± 0.001**         | 0.578    | 34.4\n| ProNet + SSHG| 1.435 ± 0.004    | **0.579 ± 0.004**    | **0.591 ± 0.003**    | **24**\n\n**Extended Ablation Study on EC**\n| Model   | +SSHG   | Cutoff | Avg. Num Edges | Time (s/epoch) ↓ | Mem (MiB) ↓ | Test Acc (%) ↑ | Acc/Time (×100) ↑ |\n| ------- | ------- | ------ | -------------- | ---------------- | ----------- | -------------- | ----------------- |\n| ProNet  | No      | 4      | 1,034.5        | 138              | 1,290       | 78.1           | 56.6              |\n|         | No      | 6      | 4,755.2        | 165              | 7,760       | 82.1           | 49.8              |\n|         | No      | 8      | 8,013.9        | 185              | 9,580       | 85.6           | 46.3              |\n|         | No      | 10     | 11,316.8       | 210              | 14,548      | 86.4           | 41.1              |\n|         | No      | 16     | 14,881.1       | 247              | 17,768      | 87.0           | 35.2              |\n|         | **Yes** | —      | 1,593.3        | 140              | 1,818       | **87.2**       | **62.3**          |\n|         |         |        |                |                  |             |                |                   |\n| GVP-GNN | No      | 4      | 1,034.5        | 216              | 1,558       | 65.5           | 30.3              |\n|         | No      | 6      | 4,755.2        | 254              | 3,828       | 66.9           | 26.3              |\n|         | No      | 8      | 8,013.9        | 298              | 6,286       | 68.1           | 22.9              |\n|         | No      | 10     | 11,316.8       | 334              | 8,930       | 68.5           | 20.5              |\n|         | No      | 16     | 14,881.1       | 354              | 11,248      | 69.2           | 19.5              |\n|         | **Yes** | —      | 1,593.3        | 236              | 1,416       | **73.6**       | **31.2**          |\n\nWhile our experiments focus on two widely used benchmarks—enzyme classification and ligand binding affinity—we agree that a deeper analysis of motif learning and its trade-offs would further strengthen the work. We are currently extending our study to the PPI (Protein–Protein Interaction) dataset; however, training models from scratch makes it challenging to obtain meaningful results within a week. We will update the rebuttal with any results we obtain and include them in the camera-ready version if possible."}, {"review_id": "OnlsYBnaDu", "reviewer": "Reviewer_Ckw3", "summary": "This paper proposes a framework for graph-based protein learning, SSHG (Secondary Structure-based Hierarchical Graph). It bases protein secondary structures (α-helices, β-strands) to build hierarchical graph representations. The method constructs two-level graphs: fine-grained subgraphs within each secondary motif and a coarse-grained graph connecting motifs. A two-stage GNN processes these hierarchical graphs to capture both local and global features. Experiments on enzyme classification and ligand binding affinity show improved accuracy and reduced training time compared to baselines.", "questions": "What specific GNN architecture is used in Mamba+SSHG, and how are the sequential dependencies handled?\nCould you add more evaluation tasks, like Protein-Protein Interface (PPI) in ATOM3D? This task is also used by ProNet and GVP-GNN.\nHow does your method compare to simpler long-range modeling methods like adding virtual nodes or fully-adjacent layers?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\nClear biological motivation: Using secondary structures as hierarchical units is intuitive and biologically meaningful\nComputational efficiency: Reduces edge complexity while maintaining performance\nModular design: Framework can integrate with different backbone GNNs, like GVP-GNN and ProNet\nWeaknesses:\nOverly complex theoretical analysis: Sections 2-3 contain extensive expressiveness theory that makes the paper hard to follow, while the core idea is simple and intuitive\nQuestionable theoretical assumptions: Assumption 4.1 is unrealistic with standard GNN components like commonly used pooling operations; for example, GVP-GNN uses mean pooling, which is not injective\nLimited baselines: Missing comparisons with basic GNNs, like GCN or GAT. Recent work shows that these simple, basic GNNs can work quite well.\nIncomplete experimental tables: Table 2 and Table 3 compared ProNet-Backbone and ProNet+ SSHG. However, only GVP-GNN and Mamba+SSHG are listed in the tables, and the corresponding GVP-GNN+SSHG and Mamba are missing for comparison.\n\nMinor issues:\nMissing equation number at line 119\nAppendix C skips step 3\nIn the checklist, they claim that \"in the conclusion section, we have listed a few potential future works\", but in fact, there are none.", "was_revised": true, "final_justification": "The authors have answered my most significant requests for clarifications and I also appreciate their efforts of answering the other reviews. I think the paper has gained in clarity and have updated my score accordingly", "rebuttal": "We appreciate the reviewer’s insightful comments. Below, we address each of your concerns one by one:\n\n---\n**Q1. Weaknesses: Overly complex theoretical analysis: Sections 2-3 contain extensive expressiveness theory that makes the paper hard to follow, while the core idea is simple and intuitive.**\n\n**Response:** \n\nWe thank the reviewer for this valuable feedback. We agree that the core idea—leveraging secondary structure motifs in a multiscale GNN—is intuitive and biologically motivated. Our goal in Sections 2–3 was to provide a rigorous theoretical foundation for our hierarchical framework, particularly to support our claims of expressiveness and geometric fidelity—properties that are critical for graph learning but have not been addressed in existing motif-based approaches. Such theoretical guarantees are typically provided only in non-motif-based frameworks [16, 25, 37, 39] (as cited in our paper).\n\nTo improve clarity and accessibility, we will revise the manuscript to:\n- Add a high-level intuition and summary at the beginning of Section 2 to explain the motivation and practical implications of the theory in plain language.\n- Streamline the theoretical exposition by moving detailed proofs and formal definitions (e.g., SCHull graph properties, local frame transformations) to the appendix\n- Clarify the connection between the theoretical results and the empirical benefits observed in our experiments (e.g., improved accuracy and efficiency).\n\nWe believe this balance will preserve the rigor for interested readers while making the main narrative more approachable.\n\n**Q2. Questionable theoretical assumptions: Assumption 4.1 is unrealistic with standard GNN components like commonly used pooling operations; for example, GVP-GNN uses mean pooling, which is not injective.**\n\n**Response:** \n\nWe appreciate the reviewer’s thoughtful observation. Indeed, Assumption 4.1—requiring injective message and aggregation functions—does not strictly hold in many practical GNN architectures, such as GVP-GNN, which employs mean pooling. We would like to clarify the motivation and scope of this assumption in our theoretical analysis.\n\n**Theoretical Motivation:**\nAs discussed in the Background section, Assumption 4.1 is adopted to align with the expressive power analysis based on the Weisfeiler–Lehman (WL) test, a widely used framework for evaluating the representational capacity of GNNs [16, 25, 37, 39]. Our goal with this assumption is to characterize the best possible representational power an architecture can achieve, but not to impose a constraint on practical implementations. Similar assumptions are standard in the literature for theoretical completeness.\n\n**Practical Implementation:**\nIn practice, like many prior works, we do not strictly enforce injectivity. Instead, we rely on sufficiently expressive multilayer perceptrons (MLPs) with ReLU activations and adequate width to approximate desirable mappings, as supported by prior studies. Our models—including those using non-injective pooling like mean pooling—still benefit from the SSHG framework and achieve consistent performance gains, demonstrating its practical effectiveness.\n\n**Future Directions:**\nWe agree that incorporating more expressive or injective aggregation schemes (e.g., PNA) could further enhance performance. We will mention this as a promising direction for future work in the revised paper.\n\nTo enhance clarity, we have revised the discussion of Assumption 4.1 in the updated manuscript.\n\n**Q3. Limited baselines: Missing comparisons with basic GNNs, like GCN or GAT. Recent work shows that these simple, basic GNNs can work quite well.**\n\n**Response:** Thank you for the helpful comment. To address your concerns, we have now included a comparison between GCN models with and without the SSHG framework on both the EC and LBA benchmarks. We observe consistent improvements in terms of both accuracy and average runtime, demonstrating the general applicability and efficiency of our method.\n\n**EC**\n|   | Test Acc | Ave.Time |\n|-|-|-|\n| GCN|68.5|186|\n| GCN + SSHG|**71.2**| **150**|\n\n**LBA**\n| |  RMSE ↓|Pearson  ↑ | Spearman  ↑ | Ave.Time |\n|-|--|-|-|-|\n| GCN |  1.925 | 0.322  | 0.287 | 28|     \n| GCN + SSHG |**1.788** | **0.392**|**0.359**| **23** | \n\n---\n\n**Q4. Incomplete experimental tables: Table 2 and Table 3 compared ProNet-Backbone and ProNet+ SSHG. However, only GVP-GNN and Mamba+SSHG are listed in the tables, and the corresponding GVP-GNN+SSHG and Mamba are missing for comparison.**\n\n**Response:** \n\nWe agree that including both the original and SSHG-enhanced versions of each backbone model is essential for a complete and fair comparison. We have now complete the table for the comprehsen comparison as below. Importantly, on the GVP-GNN model for EC classification, integrating SSHG reduces the average number of edges from 14,881 to 1,593—leading to  a **~87% reduction in memory usage** (from 11,248 MiB to 1,416 MiB) and a **33% reduction in training time** per epoch (from 354s to 236s). At the same time, test accuracy improves from 69.2% to 73.6%. This demonstrates that SSHG not only brings significant efficiency gains  but also  improves accuracy.\n\n**LBA**\n| Model   | RMSE ↓  | Pearson ↑ | Spearman ↑    | Ave.Time  ↓    |\n|--|----|--|-|---|\n| GVP-GNN                | 1.529  | 0.441| 0.432  |   49\n| GVP-GNN + SSHG     | **1.488**  | **0.512**   | **0.477** | **35**\n| | | |\n| Mamba  | 1.457 ± 0.004 | 0.565 ±  0.003    | 0.554 ±  0.004    | **27**\n| Mamba + SSHG  | **1.399 ± 0.003**        | **0.614 ±  0.003**    | **0.610 ±  0.003**    | 29\n\n**EC**\n| Model  | Acc |  Ave.Time   |\n| -|- | -|\n|  Mamba |85.9  |  236    |\n|  Mamba + SSHG   |  **88.4**| **157** |\n\n**Extended Ablation Study on EC**\n| Model   | +SSHG   | Cutoff | Avg. Num Edges | Time (s/epoch) ↓ | Mem (MiB) ↓ | Test Acc (%) ↑ | Acc/Time (×100) ↑ |\n| --| -| -| - | - | -- | - | - |\n| GVP-GNN | No | 4      | 1,034.5  | 216 | 1,558 | 65.5  | 30.3  |\n|  | No | 6 | 4,755.2   | 254  | 3,828       | 66.9  | 26.3   |\n|  | No | 8| 8,013.9   | 298  | 6,286       | 68.1  | 22.9  |\n|  | No | 10| 11,316.8    | 334  | 8,930       | 68.5   | 20.5 |\n|  | No | 16 | 14,881.1  | 354 | 11,248      | 69.2 | 19.5  |\n|   | **Yes** | —      | 1,593.3  | 236   | 1,416       | **73.6** | **31.2**    |\n\nWe appreciate the reviewer’s attention to detail and will ensure that all tables are updated in the revised manuscript to include complete comparisons across all model variants.\n\n**Q5. Minor issues: Missing equation number at line 119 Appendix C skips step 3 In the checklist, they claim that \"in the conclusion section, we have listed a few potential future works\", but in fact, there are none.**\n\n**Response:** \n\nWe will address each point as follows:\n- Missing equation number at line 119: We acknowledge that the message-passing equation in Section 2 was not numbered. We will assign it a proper equation number (e.g., Eq. (1)) in the revised manuscript for clarity and consistency.\n\n- Appendix C skips step 3: Thank you for catching this. The numbering in Appendix C inadvertently jumps from Step 2 to Step 4. We will correct this to include Step 3 (which was implicitly described) and renumber the steps appropriately.\n\n- Checklist inconsistency regarding future work: You're right—the conclusion currently lacks a discussion of future work despite the checklist claim. We will revise it to briefly outline directions such as adopting injective aggregation, alternative motif definitions, and integration with protein language models.\n\nWe appreciate the reviewer’s attention to detail and will ensure these corrections are made in the final version.\n\n**Q6. What specific GNN architecture is used in Mamba+SSHG, and how are the sequential dependencies handled? Could you add more evaluation tasks, like Protein-Protein Interface (PPI) in ATOM3D? This task is also used by ProNet and GVP-GNN. How does your method compare to simpler long-range modeling methods like adding virtual nodes or fully-adjacent layers?**\n\n**Response:**\n\nWe observe that directly applying Mamba to the node sequences of the original protein graph increases the sequence length by over threefold, significantly slowing down the model. Despite this, the Mamba-only model achieves performance comparable to GNN-based methods such as ProNet, even without leveraging geometric information. This surprising efficiency is largely due to the use of `mamba_ssm`, a highly optimized CUDA C++ implementation that enables fast training of complex sequence models. In contrast, most GNN implementations are primarily in Python, which introduces overhead due to slower data processing and iterative computation.\n\nTo illustrate the difference, consider the following typical pipeline structures:\n\n**GNN-based models (e.g., ProNet):**\n\n* Feature embedding → Python\n* Iteration over blocks → Python\n* Message passing → Python\n* Output layer → Python\n\n**Mamba\\_ssm-based models:**\n\n* Feature embedding → Python\n* Iteration over time steps → CUDA C++ (fused kernel)\n* Selective SSM operations → CUDA C++\n* Output layer → Python\n\nThe expensive recurrent/state-space computations in Mamba are fused into CUDA kernels, bypassing Python’s loop overhead. As a result, Mamba\\_ssm narrows the performance gap without relying on structural priors. While it achieves results comparable to other baselines, we still observe significant improvements when incorporating SSHG. We are currently working on the PPI task and comparing our approach with virtual node and fully connected layers. Training all models from scratch makes it difficult to obtain meaningful results within a week. For fully connected layers, we suspect that any accuracy gains come at the significant cost of efficiency. As shown in the table above, increasing the cutoff to 16 in GVPNN significantly increases memory usage (nearly 10×) without outperforming our model. We plan to include these results in future work and will mention this ongoing extension in the revised manuscript."}, {"review_id": "APAZ7DTlar", "reviewer": "Reviewer_kpTM", "summary": "The authors propose a hierarchical method for modeling proteins by grouping residues into contiguous units that share the same secondary structure. They develop an architecture which first models each secondary structure unit with a GNN, then models the relationship between units with another GNN. The method is applied to two problems using two different GNN architectures and shows improvements over the baseline in two benchmark tasks.", "questions": "1. Fix the set of models included in the benchmarks (GVP-GNN w/ SSHG, Mamba-backbone).\n2. Include more results or analysis in the paper. In particular, capability improvements on multiple would improve the significance score, new capabilities that were out of reach due to memory issues would improve the significance score, and evaluating different ways creating hierarchical graphs on proteins would improve the originality score. Other options also exist, if the authors have ideas, I again don't want to be too prescriptive.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "## Strengths\nThe paper presents a simple, widely applicable idea (break a problem into hierarchical graphs where the breaks are defined by prior domain knowledge).\n\nIt then demonstrates the performance improvements not only on different tasks, but across multiple backbone architectures (caveat: the performance of Mamba-backbone is not shown, I think it needs to be).\n\n## Weaknesses\nFirst - there appear to be some missing comparisons in the experiments. The authors say they applied SSHG to the GVP-GNN, but no results are shown in the table (these seem to appear only in the appendix ablations, which is a bit confusing). They authors also show Mamba-SSHG but not Mamba-backbone, so it is not possible to say whether Mamba-backbone would be more performant or less.\n\nSecond - the paper overall seems just a bit light on content. It's a pretty simple idea, with a pretty straightforward implementation (which I'm not opposed to! I like simple ideas that work well). Still, I'd expect this to be made up for by more of something - more benchmarks, more ablations, more analysis. As a comparison, one paper the authors build on, ProNet, has 4 benchmark tasks with multiple splits (in comparison to the 2 from this paper).\n\nI don't want to be hugely prescriptive of what the authors could do, but here are a few ideas:\n\n* Include more benchmarks, at least get parity with what similar papers report\n* Ablate the graph construction: what happens if you use 3-class secondary structure instead of 8-class? What happens if you assign classes based on foldseek structure discretization? Is it possible to construct the graph based on TERMs (from the Grigoriyan Lab: https://grigoryanlab.org/terms/)\n* Analyze what is learned by the model that *isn't* learned by backbone models. Are there particular structures it performs really well on? Do you observe increased generalization with lower sequence identity cutoffs?\n* Show capabilities on proteins that weren't previously possible. Does the decreased memory enable processing very large protein structures?", "was_revised": true, "final_justification": "See comment on author submission.", "rebuttal": "We sincerely appreciate the reviewer’s acknowledgment, insightful comments, and detailed suggestions for improving our paper.\nWe have carefully followed your feedback and addressed your concerns as outlined below:\n\n---\n**Q1. First - there appear to be some missing comparisons in the experiments. The authors say they applied SSHG to the GVP-GNN, but no results are shown in the table (these seem to appear only in the appendix ablations, which is a bit confusing). They authors also show Mamba-SSHG but not Mamba-backbone, so it is not possible to say whether Mamba-backbone would be more performant or less.**\n\n**Response:** \n\nThank you for highlighting this important point. We have now included a complete set of results for both GVP-GNN and Mamba backbones, with and without SSHG, in the revised paper. This ensures a more comprehensive and transparent comparison. Importantly, we emphasize the impact of integrating SSHG into the GVP-GNN model for EC classification. SSHG reduces the average number of edges from 14,881 to 1,593—leading to  a **~87% reduction in memory usage** (from 11,248 MiB to 1,416 MiB) and a **33% reduction in training time** per epoch (from 354s to 236s). At the same time, test accuracy improves from 69.2% to 73.6%. This demonstrates that SSHG not only brings significant efficiency gains  but also  improves accuracy.\n\n**LBA**\n| Model   | RMSE ↓  | Pearson ↑ | Spearman ↑    | Ave.Time  ↓    |\n|--|----|--|-|---|\n| GVP-GNN                | 1.529  | 0.441| 0.432  |   49\n| GVP-GNN + SSHG     | **1.488**  | **0.512**   | **0.477** | **35**\n| | | |\n| Mamba  | 1.457 ± 0.004 | 0.565 ±  0.003    | 0.554 ±  0.004    | **27**\n| Mamba + SSHG  | **1.399 ± 0.003**        | **0.614 ±  0.003**    | **0.610 ±  0.003**    | 29\n\n**EC**\n| Model  | Acc |  Ave.Time   |\n| -|- | -|\n|  Mamba    |85.9  |  236    |\n|  Mamba + SSHG   |  88.4| 157    |\n\n**Extended Ablation Study on EC**\n| Model   | +SSHG   | Cutoff | Avg. Num Edges | Time (s/epoch) ↓ | Mem (MiB) ↓ | Test Acc (%) ↑ | Acc/Time (×100) ↑ |\n| --| -| -| - | -- | -- | - | - |\n| GVP-GNN | No      | 4      | 1,034.5  | 216 | 1,558       | 65.5  | 30.3  |\n|  | No      | 6      | 4,755.2   | 254  | 3,828       | 66.9  | 26.3   |\n|  | No      | 8      | 8,013.9   | 298  | 6,286       | 68.1  | 22.9  |\n|  | No      | 10     | 11,316.8    | 334  | 8,930       | 68.5   | 20.5 |\n|  | No      | 16     | 14,881.1  | 354 | 11,248      | 69.2 | 19.5  |\n|   | **Yes** | —      | 1,593.3  | 236   | 1,416       | **73.6** | **31.2**    |\n\nWe appreciate the reviewer’s attention to detail and will ensure that all tables are updated in the revised manuscript to include complete comparisons across all model variants.\n\n---\n**Q2. Second - the paper overall seems just a bit light on content. It's a pretty simple idea, with a pretty straightforward implementation (which I'm not opposed to! I like simple ideas that work well). Still, I'd expect this to be made up for by more of something - more benchmarks, more ablations, more analysis. As a comparison, one paper the authors build on, ProNet, has 4 benchmark tasks with multiple splits (in comparison to the 2 from this paper).**\n\n**Response:** We thank the reviewer for the detailed and constructive feedback. We greatly appreciate the recognition of the simplicity and potential of our approach, and we agree that further content and analysis would strengthen the manuscript. Below, we summarize our additional experiments:\n\n**Ablation Studies:**\nWe have included comparisons with Mamba-backbone and GVP-GNN w/ SSHG to clarify the performance improvements gained through SSHG as mentioned above. Moreover, we have expanded our experiments by including more ablation studies for ProNet and GVP-GNN models on the EC benchmark. These results demonstrate that integrating SSHG consistently improves both accuracy and efficiency. For example, on the ProNet model, integrating SSHG drastically reduces the number of edges from 14,881 to 1,593—yielding nearly a **2× speedup in training time** per epoch (247s down to 140s) and a **90% reduction in memory usage** (17,768 MiB down to 1,818 MiB). This is achieved while improving accuracy slightly from 87.0% to 87.2%. Such results represent a substantial and meaningful breakthrough in balancing efficiency and accuracy, clearly demonstrating the practical value and impact of SSHG.\n\n**Extended Ablation Study on EC**\n| Model   | +SSHG   | Cutoff | Avg. Num Edges | Time (s/epoch) ↓ | Mem (MiB) ↓ | Test Acc (%) ↑ | Acc/Time (×100) ↑ |\n| ------- | ------- | ------ | -------------- | ---------------- | ----------- | -------------- | ----------------- |\n| ProNet  | No      | 4      | 1,034.5        | 138              | 1,290       | 78.1           | 56.6              |\n|         | No      | 6      | 4,755.2        | 165              | 7,760       | 82.1           | 49.8              |\n|         | No      | 8      | 8,013.9        | 185              | 9,580       | 85.6           | 46.3              |\n|         | No      | 10     | 11,316.8       | 210              | 14,548      | 86.4           | 41.1              |\n|         | No      | 16     | 14,881.1       | 247              | 17,768      | 87.0           | 35.2              |\n|         | **Yes** | —      | 1,593.3        | 140              | 1,818       | **87.2**       | **62.3**          |\n\n**Other Baselines:** to address additional reviewers' concerns,\nwe have now included a comparison between GCN models with and without the SSHG framework on both the EC and LBA benchmarks. We observe consistent improvements in terms of both accuracy and average runtime, demonstrating the general applicability and efficiency of our method.\n\n**EC**\n|   | Test Acc | Ave.Time |\n|-|-|-|\n| GCN|68.5|186|\n| GCN + SSHG|**71.2**| **150**|\n\n**LBA**\n| |  RMSE ↓|Pearson  ↑ | Spearman  ↑ | Ave.Time |\n|-|--|-|-|-|\n| GCN |  1.925 | 0.322  | 0.287 | 28|     \n| GCN + SSHG |**1.788** | **0.392**|**0.359**| **23** |  \n\nWe are currently training SSHG-enhanced models for the Protein–Protein Interaction (PPI) task in ATOM3D. As these models are being trained from scratch and require extensive time and tuning, obtaining high-quality results within a week is challenging. We plan to include PPI evaluation in future versions and will mention this ongoing extension in the revised manuscript’s conclusion. In addition, we agree that studying the impact of different motif definitions is valuable. We are particularly interested in the reviewer’s suggestions of using 3-class secondary structure. However, implementing and validating these alternatives would require significant additional work, especially around preprocessing pipelines, so we plan to explore them as part of our future work. We will explicitly state this in the discussion section.\n\n\nWe appreciate the reviewer’s thoughtful suggestions and will continue building on them to improve both the clarity and depth of the work."}]

Author rebuttal:

Meta-review: The paper proposes a simple yet effective neural network design for protein representation learning. The method proposes to apply GCN layers on two distinct graphs. The first one is constructed as independent subgraphs of residue groups, grouped by secondary structures, while the second one is constructed by representing each secondary structure as a node and edges between those representing spatial proximity. The paper received initial mixed reviews, with concerns regarding limited evaluation. During the reviewer-author discussion, most of these concerns were addressed, and most of the reviewers provided a positive assessment of the work. After the AC-reviewer discussion, reading the reviews, and the paper, I agree with the overall assessment of the paper and therefore recommend accepting the paper. I encourage the authors to include all the additional baselines and experiments provided during the reviewer-author discussion into the final paper, and the additional protein-protein interaction experiment promised during this period.
Decision: Accept (poster)