Size-Generalizable RNA Structure Evaluation by Exploring Hierarchical Geometries
Authors: Zongzhao Li
Jiacheng Cen
Wenbing Huang
Taifeng Wang
Le Song
Venue: ICLR 2025 Poster (2025)
Keywords: RNA Structure
RNA Evaluation
Geometric Deep Learning
Graph Neural Networks
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Understanding the 3D structure of RNA is essential for deciphering its function and developing RNA-based therapeutics. Geometric Graph Neural Networks (GeoGNNs) that conform to the $\mathrm{E}(3)$-symmetry have advanced RNA structure evaluation, a crucial step toward RNA structure prediction. However, existing GeoGNNs are still defective in two aspects: 1. inefficient or incapable of capturing the full geometries of RNA; 2. limited generalization ability when the size of RNA significantly differs between training and test datasets. In this paper, we propose EquiRNA, a novel equivariant GNN model by exploring the three-level hierarchical geometries of RNA. At its core, EquiRNA effectively addresses the size generalization challenge by reusing the representation of nucleotide, the common building block shared across RNAs of varying sizes. Moreover, by adopting a scalarization-based equivariant GNN as the backbone, our model maintains directional information while offering higher computational efficiency compared to existing GeoGNNs. Additionally, we propose a size-insensitive $K$-nearest neighbor sampling strategy to enhance the model's robustness to RNA size shifts. We test our approach on our created benchmark as well as an existing dataset. The results show that our method significantly outperforms other state-of-the-art methods, providing a robust baseline for RNA 3D structure modeling and evaluation.

Published as a conference paper at ICLR 2025
SIZE-GENERALIZABE RNA STRUCTURE EVALUA- TION BY EXPLORING HIERARCHICAL GEOMETRIES
Zongzhao Li Jiacheng Cen Wenbing Huang! . Wang Le Song
Gaoling School of Artificial Intelligence, Renmin University of China
"Beijing Key Lahoratory of Big Data Management and Analysis Methods BBoMap Research
11 Lizengahac20238 ruc. eeu.edduu..... en, jiace. en@out com, hwwnbing8126....m ftaifeng, songle le cmmomap.cco
ABSTRACT
Understanding the 3D structure of RNA is essential for deciphering its function and developing RNA-based therapeuties. Geometric Graph Neural Networks (Ge- oGNNs) that conform to the msymmsyyy have advanced RNA structure eval- uation, a crucial step toward RNA structure prediction. However, existing Ge- oGNNs are still defective in two aspects: 1. inefficient or incapable of capturing the full geometries of RNA; 2. limited generalization ability when the size RNA significantly differs between training and test datasets. In this paper, we propose EquiRNA, novel equivariant GNN model by exploring the three-level hierarchical geometries of RNA. At its core, EquiRNA effectively addresses the size generalization challenge by reusing the representation of nucleotide, the com- mon building block shared across RNAs of varying sizes. Moreover, by adopting a scalarization-basee equivariant GNN as the backbone, our model maintains di- rectional information while offering higher computational efficiency compared existing GeoGNNs. Additionally, we propose fize-insensitive K K-nearest neigh- bor sampling strategy enhance the model's robustness RNA size shifts. We test our approach on our created benchmark as well as an existing dataset. The results show that our method significantly outperforms other state-of-the-an meth- ods, providing a robust baseline for RNA 3D structure modeling and evaluation.
INTRODUCTION
RNA, or ribonucleic acid, is pivotal type of molecules essential to myriad processes within biological organisms. Central to its versatility is RNA's 3D structure, which dic- tates its function and interaction with other molecules (Serganov et al., 2004; Mortimer al., 2014). Thus, predicting RNA's 3D struc- ture is imperative for advancing our under- standing of its biological roles and potential therapeutic applications. Inspired by the break- throughs achieved by AlphaFoold (Jumper et al., 2021) in the field of protein structure predic- tion, more and more researchers have started to adopt deep learning (such as Graph Neural Net- works (GNNs)) for RNA 3D structure prediction (Wang et al., 2023a: Shen et al., 2022). Moreover, designing an effective scoring function for evaluation and ranking of RNA structures key step in prediction (Townshend et al., 2021: Zhang et al., 2022b), which is the main focus of our paper. The process of RNA structure evaluation can be shown Fig.
Train (50 nodes 100)
Volld/ / Test (100 nodes 200)
Gandaara RNA Stucham
RMSD
Cardisie RNA
RMSO
RNA
12.645
758
Sau RNA
10.305
25 45.453
EQuA
10.846
EquIRINA
21.207
Select structure with the lowest predicted RMSD
Figure 1: RNA structure evaluation.
"Corresponding author: Wenbing Huang.

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Published as a conference paper at ICLR 2025
RNA structure evaluation presents several challenges: 1. The predictor we leverage should con- form to the E(3) symmetry (3D translation and rotation), as RNA structures are independent to the choice of the coordinate system. 2. Since large RNA structures are difficult to determine experimen- tally, currentlyknoown RNA structures are of limited number and mostly of small size (Wang et al., 2023b). As shown in Fig. 2, in the well-known RNAsolo dataset (Adamezyk al., 2022), among RNA structures with high resolution (5 2.0 A), RNAs longer than 100 nucleotides account for only 3.60%: while at low resolutions (> 2.0 A), RNAs longer than 100 nucleotides account for 24, 24.62%. This observation underscores the current limitations of experimental methods in determining the 3D structures of large RNAs, highlighting a significant challenge in the field of structural biology. Therefore, effectively using smaller RNAs for training a predictor that is generalizable to larger RNAS poses an intriguing research topic, which is identified as the problem of size generulization in this paper. As an initial exploration, in this paper, we use smaller RNAs of 50-100 nt for training and larger RNAs of 100-200 nt for testing. The detailed definition of size generalization as well as the detailed explanation of the significance for studying this problem are provided in Appendix B.
Several Geometric GNNs (GeoGNNs) have been proposed for RNA structure evalua- tion (Zhang et al., 2022b; Townshend et al., 2021), but they still defective in address- ing these two challenges. To be specific, PaxNet (Zhang et al., 2022b) utilizes a typical GNN on invariant input (such as bond lengths and angles), making it lose directional infor- mation during message passing and thus inef- fective in characterising the spatial geometry. ARES (Townshend et al., 2021), on the other
x10
Prohability density funetion
RNAs in different resolutions
RNA with resolution smaller fhan 2
1.5
RNA with resolution larger
1.0
0.5
0.0
200
400
600
800
1000
Length of the RNA
hand, employs an equivariant GNN that is able to retain the directional information, but is ex-
Figure 2: RNN distribution of RNAsolo.
panded through high-degree irreducible representations. leading to more computational cost. More importantly, both models are not explicitly and specifically designed to tackle RNAs of varying size, leaving the issue of size generalization less explored.
In this paper, we propose hierarchical equivariant GNN, dubbed as EquiRNA, which consists of three components: the atom-level, subunit-level, and nucleotide-leve equivariant message passing processes. Built upon the biological insight that nucleotides are common building blocks shared across RNAS of varying sizes, our model effectively generalizes to different RNA sizes by reusing nucleotides. In particular, we obtain a uniform representation of each nucleotide by undertaking the atom-level and subunit-level message passing over the atoms inside each nucleotide, and then capture the interactions between nucleotides by utilizing the nucleotide-level component. Notably, our model preserves full-atom information throughout all components, in contrast to those pooling- based hierarchical models in other domains (e.g. ProNet (Wang et al., 2022)). Moreover, by adopt- ing calaaiization-basee equivariant GNN (i.e. the EGNN model (Satorras et al., 2021)) as the backbone, our model maintains directional information while offering higher computational effi ciency compared to existing GeoGNNs. Additionally, we equip model with a size- -nzenissene K-nearest neighbor sampling strategy at the nucleotide level. This complements the aforementioned hierarchical structural design, jointly mitigating the challenge of size generalization.
In summary, this paper contributes to the following aspects:
We introduce a new dataset named rRNAsolo for assessing size generalization in RNA structure evaluation. It covers a broader range of RNA sizes, includes more RNA types, and features more recent RNA structures when compared to existing datasets.
We propose EquiRNA, a hierarchical equivariant graph neural network, tailored to address the size generalization challenge by reusing the representations of nucleotides.
Extensive experiments on new dataset rRNAsolo and the existing dataset ARES (Town- shend et al., 2021) show that our model achieves better performance across all metrics than SOTA methods, establishing its superiority.
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2 OUR METHOD
In this section, we first introduce necessary preliminaries related to RNA modeling, and then present the details of our created benchmark rRANsolo for structure evaluation. After this, we describe the architecture of EquiRNA with multi-level equivariant message passing, followed by size- insensitive K K-nearest neighbor sampling strategy for handling the imbalance of the size between the training and test RNAs. Fig. 3 shows the whole architecture of our model.
2.1 PRELIMINARIES
RNA is a polymeric molecule composed of four types of nucleotides: adenine (A), uracil (U), cy- tosine (C), and guanine (G). Each nucleotide includes three distinctive chemical sub-units: a five- carbon sugar ribose, a phosphate group, and nucleobase. The atoms that constitute all components are carbon (C), nitrogen (N), oxygen (O), phosphorus (P), and hydrogen (H). In most cases, different nucleotide shares the same ribose and phosphate group (called ribose- phoooss-hhsshhtteee backbone below), and could contain different type of nucleobase. this paper, we preserve full-atom representations to enhance the characterization of RNA's 3D structure. In particular, the ribose-phosphatee backbone of each nucleotides is denoted as a fully-connected graph G, = (Vr, & Each node in Vr encapsu- lates the information of the i-th atom, namely ,, = (Z h, with H,, E R being the equivariant 3D coordinate and hy, E I being the invariant embedding feature based on atom type. The edges &, are constructed by connecting all pairs of atoms in the ribose-phosphate backbone. Similarly, we can define a fully-connected graph for a nucleobase as 9 = (V: E To reduce computational complexity, we adopt the strategy outlined in ARES (Townshend et al., 2021) in the above graph construction process, which only considers heavy atoms without hydrogen (H). Finally, nucleotide referred to as joint graph 9 (V,VV, & UE,), which is illustrated more clearly in Appendix O.
Task formulation. Given a candidate RNA structure with N nucleotides [ our goal is to de- sign neural network f to predict the Root Mean Square Deviation (RMSD) between the candidate and native structures, which will be used to score and rank all candidates to identify the one most closely matching the native structure.
2.2 NEW DATASET: RRNASOLO
Although several datasets already exist for RNA structure evaluation (Townshend et al., 2021), they contain RNAs of either limited types or narrow size ranges. Therefore, to extensively explore the problem of size generalization, we devise novel dataset from the RNAsolo database (Adamezyk et al., 2022), a publicly available online repository that comprises diverse array of biomolecular information concerning RNA. We call this new dataset as rRNAsolo. We provide the details of the dataset construction process as follows.
Data Collection and Purification. To better investigate the intrinsic properties of RNA molecules, exclusively select data from the solo RNA category with resolution higher than 4 Å. We engage in a detailed cleaning process of the collected data, which is unfolded through two levels. At the atomic level: First, scrutinize all atoms within every RNA molecule to ensure the legitimacy of their valency, eliminating any RNA that contained illicit atoms. Second, we conduct a compre- hensive count of atoms within each nucleotide of all RNAs, excluding those with the number of atoms deviated from the normal range. At the nucleotide level: First, we analyze the sequencing of nucleotide numbers across all RNAs, discarding those with discontinuous numbering. This dis- continuity might be due to insufficient data from solution nuclear magnetic resonance experiments needed to determine their structures. Second, we assess the number of chains contained within the RNA, and exclude any instance with more than two chains, which possibly are redundant chains as result of experimental techniques. Third, we verify that all RNAs consisted solely of nucleotides and rule out any containing additional residues, such as amino acids. Fourth, aligning with the method- ology presented in paper (Townshend et al., 2021), we climinate RNAs with non-canonical base pairs. Finally, we examine every RNA molecule reduce conformational ambiguity, specifically for atoms that could occupy possible positions A or B. We choose the configuration with higher occurrence frequency, removing all other configurations.
Data Partitioning and Clustering. We employ USAlign (Zhang et al., 2022a) to calculate the TM- score between any two RNA samples. USAlign is metric used to evaluate the structural similarity
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Input
Atom-Level
Subunil-Level
Nucleolide-Level
Output
Nachecoone
Nacsobbase
RMSD
Nuxdectiie
Tau
Fa
Huu Devattee Srace
Phosprat
Prosphale
xL Layera
Figure 3: The architecture of EquiRNA. Each layer comprises three levels: atom-level, subunit- level, and nucleotide-level. (hr,, and (he, 2b) denote the features and coordinates of the atom in ribose-phospe phooss-p and nucleobase, respectively. (h, X) denotes the features and coordinates each nucleotide. KNN represents our size-insee ineennssensttve K -nneerest neighbor sampling strategy. EquiRNA predicts RMSD from native structure and selects structure with the lowest predicted RMSD.
of two RNAs. A TM-score greater than 0.45 indicates that the two RNAs belong the same RNA family. We then apply the clustering method proposed in USAlign based on the TM-scores to cluster all RNA structures. Specifically, order. Then, starting with the largest RNA as the cluster representative. the TM-score is computed with the rest using the USAlign algorithm. If the score higher than 0.45, it's categorized into the same cluster as the largest RNA; otherwise, the largest RNA forms its own cluster. Repeat the process with the second largest RNA until RNAs are clustered. Given that RNAs with over 200 nucleotides require considerable time for candidate conformation generation, our validation and test sets primarily focus on RNAs with 100-200 nucleotides, using a training set composed of RNAs with 50-100 100 nucleotides. Since the number of RNAs within the 100-200 range is smaller than those within 1001 we first create the validation and test sets by enumerating all clusters and randomly sampling RNAs within the 100-200 range, ensuring maximum coverage of all potential clusters. After the completion of the validation and test sets, we remove the clusters that contain any sample in the validation leakage. To be specific, we enumerate the remaining clusters and randomly sample a certain number of RNAS from each cluster to form the training set.
first sort RNAS by the number of nucleotides in descending
set, and construct the training using the remaining clusters to prevent data
Table
The detailed comparisons between our rRNAsolo and ARES.
Method
Range of sizes
Numbers
Release date
Train
Valid
Test
Train
Valid
Test
ARES
(10, 50)
50)
(20, 150)
18
4
16
"2020
RRNAsolo
(50, 100)
(100, 200)
(100, 200)
200
15
15
"2023
Table 1 compares the statistic of rRNAsolo against the existing benchmark ARES (Townshend et al., 2021). Our rRNAsolo exhibits the following advantages: Wider range of RNA sizes: the train- ing set includes RNAs of 50-100 nucleotides. whereas ARES covers only RNAs less than 50 nu- cleotides. Our validation and test consists of RNAs of 100-200 nucleotides, while ARES has only three entities over 100 nucleotides. 2. A greater number of RNAs: our dataset comprises RNAs approximately times more than ARES. 3. More recently released data: our dataset contains all the eligible RNA structures up to the latest release, opposed to ARES which includes RNAs published between 1994-2006 for the training set. Overall, our dataset possesses higher quality and thus enables a more comprehensive evaluation of the current methods. More detailed information about our rRNAsolo can be found in Appendix D.
2.3 THE PROPOSED EQUIRNA MODEL
EquiRNA consists of the three-level E(3)-equivariant processes: atom-level, subunit-level and nucleotide-level. In this way, EquiRNA largely reuses the common building block (i.e. the nu- cleotide) for RNAs of different sizes, better addressing size generalization. Specifically, the atom-
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level and subunit-levell modules able obtain uniform representation of each nucleotide, while the nucleotide-len message passing can capture the interaction between different nucleotides.
(1) Atom-Level Equivariant Message Passing.
Previous works like (Zhang et al., 2022b) initially calculate E(3) -invariant scalars from atomic co- ordinates, such as planar angles, dihedral angles, and bond lengths, and then encode and propagate these invariant information through a typical GNN. However, the explicit calculations of scalars suf- fer from suboptimal computational efficiency and insufficient representational capability. Here, employ an equivariant GNN to process the atomic coordinates straightforwardly, which is able to retain the orientation information in each layer by propagating geometric information beyond local neighborhoods (Joshi et al., 2023), thus enhancing the modeling of RNAs.
Our model is built upon a prevailing equivariant GNN model, i.e., EGNN (Satorras al., 2021), by further drawing inspiration from the biological properties of RNA. In particular, for each nucleotide, we conduct atom-level equivariant message passing over the atoms within the ribose-phosphate backbone G. and the nucleobase Gb. separately. We denote these two processes as Equivariant Ribose-phosphate Network (ERN) and Equivariant nucleoBase Network (EBN).
ERN. For ribose-phosphate graph Gr comprising Nr nodes (atoms) denoted as (कt h 1 at the 1-th layer, the formula for updating each node within the graph is given by:
m r,r, = Pm (IbE(1 Jt,,11 (D,1), 11' (r.,tt,,)
1+1 att; + N 1#1
1
(o% ox (
(1)
h41 n + (h Σ m
where Pm, Pz and Phh represent different Multi-Layer Perceptrons (MLPs), rbf(.) is the Radial basis function on the relative distance | o and m'.ry is the invariant message from node
j to To derive z++1 we first multiply the ID invariant scalar gpx (mm(...) E R with the relative coordinate (x - to recover the orientation information and maintain orthogonality equivari- ance, and then aggregate all such terms from all neighbors, followed by an addition of the original coordinate म to ensure translation equivariance. The update of the hidden embedding h4t1 follows the conventional message passing strategy. For simplicity, we abstract Eq. (1) as g. ERN(G)).
p; E.
EBN. Given a nucleobase graph g ( h1, J the I-th layer, the updates of all atoms are similar to ERN detailed in Eq. (1). We denote this message passing process G/+11 (R)(
(2) Subunit-Level Equivariant Massage Passing.
The atom-level message passing above only models the geometry of each subunit, without charac- terizing their interactions. Biologically, interactions between the phoose-phosphate backbone and the nucleobase involve complex interplay of forces, e.g., covalent bonds, hydrogen bonds, and so on. To involve interactions, we develop subunit-level equivariant message passing across the atoms in the ribose-phosphate backbone and the nucleobase. We first compute the center of V, and V., as - - ΣM' a!+1 and a//1 N, E. a/+1 Subsequently, we achieve translation invari- ance by center normalization as 3/+1 _ at+1 and a/+1 al+1 att1 respectivell. Then,
the update process for each atom in G. is given by:
q: 4g h+41
=bf
aij = Softmax
att2 -ttt 1 + + Σ anj Par par
al
k; sPk (n/t
1+1
rbf
rbf(
(2)
JEV,
h44 = h + Phr h
E anj,)) JEV.
v, so (n (b
where Pg. ik. Pes Par and Phr all MLPs, rbf(-) is the Radial basis function. We apply the attention mechanism to depict the interaction between each atom in G, and Gb- The invariant quantities qi k, and vy refer to the query, key and value features, respectively, and ai signifies the
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attention weights, with higher values indicating more pronounced interactions between atom and The aforementioned procedure can be succinetly expressed as G EB2R( g
Following this, we employ an analogous approach to model the interactions exerted by the ribose- phosphate backbone on the nucleobase, leading to g ER2B(G) ,
(3) Nueleotide-L.evl Equivariant Message Passing.
Upon completing the atom-level and subunit-level modeling, we advance message passing at the nucleotide level. We aggregate the features of all atoms within the the same nucleotide, which have been updated as described previously, deriving the nucleotide feature by summation: h/+2 = -i 1+2 r.
2==1 1+2 and the coordinate matrix by concatenating all atomic coordi- nates: X1+2 E (3) To distinguish between nucleotides, we denote different nucleotide via subscript i: (x 2222) with N, being the number of atoms in this nucleotide.
It is challenging design function X since different nucleobase could have a distinet number of atoms, implying that the dimension of x!+2 varies. Here, we address this challenge by using templates. First, we define atom templates {We E Rd 1==1 as there are 4 types of atoms used in our model. Each atom template is learnable with the number of channels as dg. The atom-level templates are engineered to capture the intrinsic characteristics of all atoms. We further define nucleotide templates ( E N... [ = where 4 reflects the 4 types of nucleotides, and d, reflects the dimension of the nucleotide template. The nucleotide-level templates aim to capture the unique attributes of the same atom when situated in different categories of nucleotides. In summary, W and W are two learnable templates that store information about each type of atom and nucleotide, respectively. In W different row represents the characteristics of different atom, and therefore does not share the same representation. We apply these templates to X to derive fixed-size message from nucleotide j and i.
In detail, we begin by calculating the distances between any pair of atoms in nucleotide and j. yielding the distance matrix Dig INNNN We then compute the center of cach nucleotide, de- No 2/ The coordinates and features of each nucleotide
noted as 2xt² = N.+N. (EN, xl - ===
are updated as follows:
m = (We, @ W
(PrPPT) 3 Dis) (we, @ WT:).
1+3
Phn ( hij, Im
m.;;
mij
x143 xt+2 + W6]l JEN(1)
Σ Pool (Am (m) (0))(xxx (xl+2 x442
(3)
h,4 3+3_ = h! + Phn h 1+2
Σ JEN(()
143
represents concatenations along the channel dimension; t, E (1,...,4) 4) are vectors concatenated by all atom types in nucleotide and n e (1..... 4} represents its nucleotide type; the dimension of W and W are N.x. and R Nox,, respectively; P., and pr, are two learnable vectors used to model the atom-wise correlation in the distance matrix Dij: (i)) denotes all neighboring nucleotides of nucleotide i. The dimension of the derived message mi in the first equation keeps unchanged regardless of the value of N₂-
where xl42 represents the center of nucleotide
The message miy is matrix with dimension into vector of size +dd)) before feeding to the update function Phn. The pooling operation Pcol(-) with adaptive stride ensures the output of (m 1+-3) to have the same dimension as X:!
By default, we will flatten mi;
size-insensitive K-nearest neighbor sampling strategy. To further alleviate the size- generalization issue, we propose a size-insensitive K-nearest neighbor sampling strategy, designed to leverage the abundant smaller RNAs for training and enable the model to generalize effectively to larger RNAs. In detail, we first perform M-nearest neighbor sampling for all RNAs, then randomly select K out of these M neighbors (K < .M) with equal probability. Consequently, the model has been exposed to M neighhors'' information during training, allowing it to transfer the spatial infor- mation of M --niihhorrs learned from the training set to larger RNAs in the test set. This process is

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analogous randomly masking nodes and dropping edges in the graph during each training period, which renders the model less sensitive to variations in the size of RNAs and enhances its robustness.
Learning objective. After implementing L layers of computations using the modules described in $ 2.3, we perform the final calculation on the output h from Eq. (3) to predict the RMSD between the candidate structure and the native structure. Specifically, we first sum up the features of all nucleotides, and then feed them through MLP to predict scalar. This scalar represents the predicted RMSD by the model. Ulimmaell, we employ Smooth & loss as our loss function. This process is delineated as follows:
(v. MLP h)t))
smooth
(4)
where y is scalar, calculated by FARFAR2 when generating the candidate structure, representing the ground truth label. Nu represents the number of nucleotide RNA. A pivotal characteristic of our model is its -33--equivariance, with proofs provided in Appendix C.
3 EXPERIMENT
Datasets. We conducted exhaustive experiments on two datasets comprising wide range of RNA structures. 1. rRNAsolo: In our meticulously designed dataset, we employs candidate structures of RNAs with 50-100 nt as training set and candidate structures with 100-200 nt as validation and test sets. The dataset rRNAsolo consists of 80k/6k/6k candidate structures generated from 200/1 5/15 RNAs for training, validation, and test sets, respectively. 2. ARES: Within ARES dataset (Town- shend et al., 2021), 14 RNAS with 19-47 nt form training set and RNAS with 17-41 nt form vali- dation set, each generating Ik candidate structures. Additionally, 16 RNAs (only three of which are larger than 100 nt) with 27-147 nt form test set, with each RNA generating 5k candidate structures. The training, validation, and test sets include 14k/4k/80k candidate structures, respectively.
Baselines and Implementation. We compare our method with the following baselines: the invariant GNNs including PaxNet (Zhang al., 2022b) and RDesign (Tan al., 2024), and the equivariant GNNs including ARES (Townshend et al., 2021), EGNN (Satorras et al., 2021), dyMEAN (Kong al., 2023b), and GET (Kong et al., 2024). We use the default configurations in the corresponding source codes for all baselines. Details of the configuration settings are provided in Appendix E and Appendix P.
Metrics. For each RNA in the validation and test sets, the models are required to score all related candidate structures and selects the best-scoring one. Subsequently, the RMSD between the selected best structure and the native structure is utilized as the criterion for assessing the model's ' evaluation ability. The metrics include: 1.Median RMSD: Following the evaluation metries in Zhang et al. (2022b); Townshend et al. (2021), we calculate the median RMSD across RNAs; 2.Mean RMSD: We also calculate the mean RMSD across RNAs to further evaluate the model's scoring capability comprehensively. 3.Relative Error of Median/Mean RMSD: For each RNA, there is theoretical minimum RMSD value among all the candidate structures. Thus, we calculate the relative error of the median/mean RMSD with respect to the median/mean of the minimum RMSD.
3.1 MAIN RESULTS
Table 2 provides a detailed comparison of various methods' performance on rRNAsolo. Within this dataset, our model shows significant improvements all evaluation metrics compared to the current SOTA methods for both validation and test sets. We can observe that: 1. Our model achieves the greatest performance gains by 2.00 and 1.55 across the Mean RMSD and Medium RMSD metrics on both validation and test sets. Furthermore, as indicated by Relative Error of Mean RMSD and Relative Error of Medium RMSD, our model yields improvements of up to 17% and 13% on the validation and test sets, respectively. Given the small performance differences among SOTA models, this demonstrates the efficiency and superiority of our method in RNA 3D structure modeling. 2. Analyzing from the perspective of equivariance, the PaxNet, which retains only invariant feature, underperforms equivariant models like dyMEAN and EquiRNA. This underscores the importance of considering physical symmetry in RNA 3D structure modeling. 3. As a method in protein domain, dyMEAN still exhibits a certain gap in performance compared to EquiRNA. This suggests that despite structural similarities, the intricacy and flexibility of RNA
Results on rRNAsolo.
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Table All four metries on the validation and sets of rRNAsolo. The four metrics in the table, ER, DR, R-ER, and R-DR, correspond to the abbreviations of the previously mentioned metrics Mean RMSD, Median RMSD, Relative Error of Mean RMSD, and Relative Error of Median RMSD, respectively.
Validation Set
Test Set
ER 4
DR
R-ER
R-DR 4
ER 4
DR
R-ER 4
R-DR
ARES (Townshend et al., 2021)
22.13
22.45
1.12
1.22
20.98
20.89
0.84
0.84
EGNN (Satorras et al., 2021)
22.12
23.12
1.01
1.34
20.99
20.42
0.71
0.69
PaxNet (Zhang et al., 2022b)
22.71
24.00
1.12
1.43
21.64
21.53
0.76
0.78
dyMEAN (Kong et al., 2023b)
21.34
20.94
0.99
1.11
20.91
20.52
0.71
0.70
RDegisn (Tan al., 2024)
21.99
20.79
1.05
1.11
20.22
19.34
0.65
0.60
GET (Kong al., 2024)
21.61
21.68
1.01
1.20
20.03
19.19
0.63
0.59
EquiRNA
19.77
19.62
0.84
0.99
18.22
17.79
0.48
0.47
(100, 110)
(110, 120)
(120, 150)
(150, 200)
dyMEAN
IMSIE: 23.66
IMSU: 20J8
RMSSL 22.9
RMSD: 26.67
RDesign
RMSD:21.35
RMSDD
EquiRNA
RMSSI
RMSD. 17.20
Figure 4: Native structures (gray) versus predicted structures (beige) by EquiRNA and other models.
structures places higher demands on model's s ability to capture RNA characteristics. Inferior performance is observed of GET, attributable its lack of specialized mechanisms for addressing unique structural features of RNAs. This limitation likely stems from GET's general-purgee design, rather than RNA-specific optimization.
To comprehensively compare our model's abil- ity assess RNAs of various sizes with cur- rent SOTA models, Fig. further analyzes each model's evaluation capability across all RNAs
RDsigg
PaxNet
dyMFAN
EqERNA
Avenge RMSD
in the validation and test sets. We first divide the RNAs into size intervals of 100-110, 110- 120, 120-150, and 150-200, based on the dis- tribution of RNA sizes. Then we calculate the average prediction values for RNAs in each in- terval. Fig. clearly shows our method's supe- rior predictive results across various RNA sizes, indicating our approach effectively addresses the size generalization problem, transferring the knowledge nucleotide interactions learned from small RNAs to larger RNAs. Moreover, we selecte four RNAs from the size intervals discussed previously. Fig. displays the native struc- tures, compared with the predicted structures of the current SOTA models and our model.
16
RNA Nucleotide Size Intervals
Figure 5: Comparison of different methods in the validation and test sets.
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Table 3: Results across four metrics on the test set of ARES dataset.
Table The Relative Ranking of all the models on rRNAsolo and ARES datasets.
Method
ER
DR
R-ER
R-DR
Method
rRNAsolo
ARES
ARES
12.33
11.22
1.02
0.90
ARES
0.591
0.294
EGNN
13.47
12.12
1.21
1.05
EGNN
0,435
0.325
PaxNet
13.15
10.28
1.16
0.74
PaxNet
0.628
0.343
dyMEAN
13.56
11.30
1.23
0.91
dyMEAN
0.603
0.347
RDegisn
13.37
12.96
1.20
1.20
RDesign
0.476
0.351
GET
13.29
11.62
1.18
0.97
GET
0.423
0.343
EquiRNA
11.74
9.57
0.93
0.62
EquiRNA
0.280
0.238
Results on ARES. To further assess the versatility and generalization capabilities of our model, conducted experiments the public dataset ARES, As shown in Table 3, our method achieves optimal results on all four metrics. Despite the smaller size and fewer training samples in this dataset, model still outperforms the current SOTA method ARES, demonstrating its capability to encode essential structural information within and between mucleotides. Additionally, following the ARES paper, we present the results of all methods on the rRNAsolo and ARES datasets using the 1 best-scoring model and 10 best-scoring models in Table of Appendix F.1. Morever, we also provide detailed analyses of the experimental results in Appendix F.2.
High-level RMSD values presented in rRNAsolo. The RMSD values observed in our rRNAsolo dataset are higher than those in ARES dataset. We briefly elucidate the underlying factors contribut- ing to this discrepancy and introduce a more appropriate metric for evaluating model performance. Comprehensive analyses of this phenomenon are provided in Appendix G. There are primarily two reasons: (1) Our rRNAsolo is more challenging with wider range of RNA sizes and greater number of large RNAs. (2) The reported RMSD values are predominantly contingent upon the quality of candidate structures; however, for large RNAs, existing predictive methods often struggle to generate high-quality candidates with low RMSD values. To provide a more direct understanding of model's s evaluation capability, we have reported the Relative Ranking metric of all models rRNAsolo and ARES datasets in Table 4. The calculation of Relative Ranking is as follows: the evaluation model first ranks all candidate structures' ground-truth RMSD values in an ascending order out of N candi- date structures in total. Then we identify the order of evaluation model's selected structure as E, and Relative Ranking is calculated uation models on these two datasets. Moreover, our method shows consistent performance across both rRNAsolo and ARES datasets, while the performance of other methods declines noticeably on rRNAsolo. Compared to metrics we originally used, Relative Ranking provides more scientific and intuitive assessment of the model's 's's ability to rank candidate structures. One can directly use ranking metric to evaluate whether evaluation model offers reliable guidance for RNA selection.
(E/N). Clearly, our method significantly outperforms other eval-
Further exploratory experiments. We also conduct a series of exploratory experiments, the de- tails of which are presented in the appendices, including: the incorporation of molecular chirality in the model architecture (Appendix H); the influence of the noise (Appendix N); the model per- formance under more general scenario where size generalization is not constraint (Appendix J); performance comparisons with other leading models (Appendix L); the performance of the model after fine-tuning on dataset constructed using the recently published RhoFold (Shen et al., 2024) as a novel candidate structure generator (Appendix Q); and the analysis of some error cases (Ap- pendix M).
Complexity analyses. We now proceed to conduct a complexity analysis of the different models. The efficiency analysis and comparison are provided in Table 5. In this table, No. nla Ka, Km. C. and L denote number of atoms in each RNA, number of atoms in each nucleotide, number of nearest atom neighbors, number of nearest nucleotide neighbors, channel size of high-degree tensors, and angular order, respectively. The Ka and K2 in PaxNet denotes the the number of nearest atom neighbors in two different settings, the d, and dma in EquiRNA denotes the embedding size of the atom template and the nucleotide template. Our EquiRNA costs much less inference time than the methods that using high-degree irreducible representations (ARES, SE(3)-Transfommer (Fuchs et al., 2020), Equiformer (Liao & Smidt, 2022) and SE(3)-Hyena (Moskalev et al., 2024)), and is even faster than EGNN since EquiRNA employs the hierarchical modeling and can fulfil desirable performance with smaller neighbor size K than EGNN in the atom-level message passing.

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Table Complexity and inference time.
Method
Complexity
Inference Time
ARES
O(N,K,09LL0)
0.095s
SE(3)-Transformm
Equiformer
SE(3)-Hyena
O(N,(K, log.
EGNN
O(N,K))
0.042s
PaxNet
O(Na(Kal + K(z))
0.035s
EquiRNA
O(n (ng na(d, + dna) )) N
0.038s
3.2
ABLATION STUDIES
Table 6: Ablation studies on rRNAsolo.
Comprehensive ablation studies are carried out
evaluation the impact of each component of the EquiRNA model on performance, with specific results presented in Table 6. The At, Su, Nu, Sa, and Eq correspond the abbreviations of the atom-level, subunit-level, mucleotide-level. size- insensitive Kenearest neighbor sampling strategy, and equivariance, respectively. We have the fol- lowing observations: 1. The experiments indicate that climinating any one of atom-level, subunit- level, or nucleotide-level submodule will lead to a decrease in performance. This phenomenon firming that EquiRNA effectively encodes critical information of RNA 3D structures, such as dihe- dral angles, and facilitates message passing within and between these three levels. 2. The removal of the nucleotide-level results in a significant decline performance, revealing the significance of nucleotide interactions. 3. The ablation of the size-insensitive K K-nearest neighbor sampling strategy also leads diminished performance, demonstrating that this strategy allows the model to engage with a more extensive range of local neighbors during training, which is beneficial when coping with the large RNAs. 4. From the results presented in Row 5, the introduction of equivariance enhances the model's ability to encode physical symmetry, resulting in stronger capacity for 3D structure representation. 5. Removing atom template W or nucleotide template WT hinders the performance, indicating their ability to capture the biology information. More ablation studies subunit-level and nucleotide-level can be found in Appendix I.
ER
DR
R-ED
R-DE
w/o At
19.38
19.40
0.58
0.61
w/o Su
19.53
19.57
0.59
0.62
w/o Nu
20.25
19.41
0.65
0.61
w/o Sa
19.40
17.94
0.58
0.49
w/o Eq
20.53
19.42
0.67
0.61
w/o W 20.03
19.64
0.63
0.63
w/o W 18.69
17.97
0.52
0.49
EquiRNA
18.22
17.79
0.48
0.47
4 CONCLUSION
We introduce EquiRNA, a hierarchical equivariant model designed for RNA structure evaluation. The innovation of this model lies in deconstructing RNA structures into multiple levels. It performs message passing and aggregation from the bottom-up, starting at the atom-level and progressing through subunit-level and nucleotide-level. while preserving full-atom details. Additionally, we incorporate a size-insensitive K K-nearest neighbor sampling strategy to mitigate challenges associ- ated with generalizing to large RNAs. The entire architecture EquiRNA guarantees the critical E(3) -qqivarrriee to properly encode physical symmetries and enhance its expressive capabilities. Comprehensive experiments on two datasets with a broad range of RNA sizes demonstrate that the selected structures of our model have significantly lower RMSD values, showcasing its superior performance. We hope that EquiRNA will serve as a robust baseline for the field and foster further advancement.
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ACKNOWLEDGMENTS
This work was jointly supported by the following projects: the National Natural Science Foundation of China (No. 62376276); Beijing Nova Program (No. 20230484278); Beijing Outstanding Young the Fundamental Research Funds for the
Scientist Program (No. 100WZYY0012011000000 Central Universities, and the Research Funds of Renmin University of China (23XNKI19). We thank Jiahui Su for her insightful suggestions that improved this paper.
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A
RELATED WORK
RNA Structure Prediction and Evaluation. Due to the scarcity of RNA structural data and the inherent variability of RNA structures, predicting RNA 3D structures has long been a challenge in the academic community. Furthermore, key subtask of 3D structure prediction, which involves ac- curately evaluating and scoring the predicted set of RNA 3D candidate conformations, also presents substantial challenge, largely because many features closely related to RNA structural energy have yet to be fully understood. Traditional approaches often employ physics-based methods (Li methods (Das & Baker, 2007: Watkins et al., 2020) for struc-
& Chen, 2023) knowledge- ture prediction and leverage statistical potentials (Samudrala & Moult, 1998; Wang et al., 2015) for conformation evaluation. Recently, deep learning- based methodologies have begun to emerge, with DeepFoldRNA (Pearce et al., 2022) and trRosettaRNA (Wang et al., 2023a) employing Trans- former networks as their main architecture to predict 3D structures. ARES (Townshend et al., 2021) and PaxNet (Zhang et al., 2022b) modeling and scoring candidate conformations at the atomic level with notable success. Our research, drawing upon RNA's biological nature, introduces a hierarchical modeling approach to enhance the accuracy of structural scoring.
Geometric GNNs. In scientific scenarios such as molecular property prediction (Jiao et al., 2023; 2024; Liu et al., 2025), dynamics simulation (Han et al., 2022: Zhang et al., 2024; Wu et al., 2023; Xu al., 2024b). and protein design (Kong et al., 2023a; Yu al., 2024; Yue al., 2025), large amount of 3D structured data is contained, which is often modeled as geometric graphs. In or- der to efficiently process this type of special data structure, geometric GNNs came into being and achieved great success (Han et al., 2024). Different from invariant GNNs (e.g. SchNet (Schütt et al., 2018), DimeNet (Gasteiger et al., 2020), DisGNN (Li et al., 2024b), GeoNGNN (Li et al., 2025a), QMP (Yue et al., 2024)) with limited expressiveness (Joshi et al., 2023) and high-degree steerable models (e.g. TFN (Thomas et al., 2018), SEGNN (Brandstetter et al., 2021), MACE (Batatia et al., 2022)) which incur lot of computational overhead on tensor products, ccalarization-base mod- els (e.g. EGNN (Satorras et al., 2021), GMN (Huang et al., 2022), PAINN (Schütt et al., 2021)) utilize scalars as weights to combine geometric vector information has achieved remarkable re- sults in both accuracy and efficiency. Motivated by this spherrcallsscalaiization approaches (e.g.. SO3KRATES (Frank et al., 2024), HEGNN (Cen et al., 2024), GotenNet (Aykent & Xia, 2025)) employ invariant information-sutt sentations to strengthen model expressivity. More recently, ETNN (Batriloro et al., 2025) and Equi- LLM (Li et al., 2025b) enhance model capabilities by injecting information obtained through deep topological learning and large language models into the scalar, respectively. In the RNA scoring task, ARES (Townshend et al., 2021) and PaxNet (Zhang et al., 2022b) adopt high-degree steer- able GNNs and invariant GNNs, respectively, revealing inherent limitations as described above. In this paper, we introduce EGNN (Satorras et al., 2021) the backbone that preserves the model's equivariant characteristics while achieving computational efficiency.
as inner products or moduli-derived from high-degree repre-
Size Generalization. In the realm of graph neural networks (GNNs), particularly in biological con- texts, the size generalization capability of GNNs remains underexplored. Existing literature has noted models trained on small-scale graphs often fail retain their performance when generalized to larger graphs (Ji et al., 2022; Chu et al., 2023; Zhou et al., 2022; Chen et al., 2022). In pursuit of a more comprehensive understanding of this phenomenon, Yehudai et al. (2021) identify discrepan- cies in local structural d d-patterns as the cause for the reduction in generalization capacity. Buffelli al. (2022) introduce graph coarsening to simulate size shift, thereby augmenting the model's 's ro- bustness. Additionally, Yang et al. (2022) advocate for the identification of invariant substructures within graphs, such as molecular substructures, to ascertain key (bio)chemical properties. To solve this problem, our work presents a high-quality dataset specifically curated to validate model general- izability over varying sizes of RNA data. Additionally, we introduce a hierarchical equivariant GNN as well as size-insensitive K -narestt neighbor sampling strategy that enhances model's adaptabil- ity to lcage-scale RNAs.
B DEFINITION AND SIGNIFICANCE OF SIZE GENERATION
The definition of size generation:
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We define the size generalization problem follows: Given two datasets composed of different RNA structures, one designated as the training set and the other as the test set. The training set and test set have the following two characteristics: 1. The structural similarity (measured by TM-score) between any RNA in the training set and any RNA in the test set is less than 0.45 to prevent data leakage; 2. All RNA sequences in the test set longer than any RNA sequence in the training set. Our research question is: After training on the training set, how well does the model perform on the test set? We measure the model's generalization ability by evaluating its accuracy in assessing the 3D structures of RNA in the test set.
The significance of studying size generation:
During our experiments, we observe that the average length of RNA in the existing ARES dataset (Adamezyk al., 2022) is significantly shorter during training compared testing (26.42 nt vs. 78. 18.12 nt). This pattern is not unique to the ARES dataset; similar statistics from other RNA structure datasets indicate a consistent trend: the proportion of large RNAs much smaller than that of smaller ones. Specifically:
(1) According to statisties from the RCSB PDB (Berman et al., 2000), the early RNA structure datasets ( (1978-2010), RNAs longer than 100 nucleotides accounted for only 9.81 % of the total, with the remaining being shorter RNAs.
(2) Based statistics from the RNAsolo dataset (Adamezyk et al., 2022), currently, RNAs longer than 100 nucleotides account for only 16.79% of all experimentally determined RNA 3D structures. In contrast, according to RNAcentral data consisting of ID sequences (Sweeney et al., 2019), RNAs longer than 100 nucleotides account for 53.36% of all sequences. This indicates that while RNA sequences longer than 100 nucleotides widely present, there is significant scarcity of 3D struc- tural data for them.
(3) In the RNAsolo dataset, among RNA structures with high resolution (S 2.0 Å), RNAs longer than 100 nucleotides account for only 3.60%; while at resolutions > 2.0 A, RNAS longer than 100 nucleotides account for 24.62%. Table 7 shows detailed statistical information about the distribution.
All these findings indicate that experimental determination of large RNA structures presents signif- icant challenges, leading to much smaller number of large RNA structures than the small ones. Several papers published in top-tier journals have also highlighted this issue (Ma et al., 2022; Li al., 2024a).
Given the difficulty in experimentally determining large RNA structures, we naturally consider using deep learning to predict RNA 3D structures. In this context, the evaluation task aims to select the candidate structure that most closely matches the experimentally determined native structure from the predicted RNA 3D structure ensembles. However, due the limited number of available large RNA structures, it is not feasible to train model on large dataset composed of such RNAs to achieve strong evaluation performance. Therefore, we propose dataset that leverages small RNA structures to train the evaluation model, then tests whether the evaluation capability successfully generalize to larger RNA structures. This is precisely the size generalization problem we propose. We believe this problem is worthy of in-depth exploration.
Table 7: The distribution of RNAs of different sizes in the RNAsolo dataset.
RNA Length Range
RNA < 2.0 Á
RNA >2.0
0-100
610
648
100-200
11
118
200-300
19
300-400
0
53
400-500
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C PROOF OF EQUVVARIANCE/VVARINNC
In this section, will prove that, the whole EquiRNA model is B(3)-invariant) And first, will prove that, such a conclusion can be drawn with three sub-conclusion as follow:
1. The modules EBN(-) and ERN(.) in are -(3)eequivariant;
2. The modules EB2R(.) and ER2B(-) are -quiivaaiv,
3. The module ENN(.) is -qqiivarann;;
Theorem C.1. The whole EquiRNA model's E(3))invariance can be drawn with the three sub- conclusion mentioned above.
Proof. Consider sequence composed of functions {ip; x(-11 + 200 equivariant same group G, the equivariance lead to an interesting property that
PN (i++ pxto (g)
Φ1 pN . 0941 Px (g)
P1
holds for all i,j = 1,2..... N and g E G. which means that the group elements can be freely exchanged in the composite sequence of equivariant functions. In particular, if one of the equivariant functions (e.g. (Px) is replaced by an invariant function, the group element will be absorbed, that means
N
Up o .............. o PP+1 o Pxrco (g)p, o 0p N 0 P11
holds for all g but only 2.... ........ ... Although yN o...0 o (pk is still equivariant, because the group elements must be input starting from 101, the overall
0 is still an invariant function.
Noted that the loss function in Eq. (4) only uses invariant feature h, the whole model's invariance can be implied by the equivariance of all layers just as the three sub-conclusion.
Since EBN(-) and ERN(-) are all based on EGNN (Satorras et al., 2021), and the equivariance is obvious. We only need to prove the equivariance of Eqs. (2) and (3). Theorem C.2. The modules EB2R(-) and ERZB(-) are E(3) -qquvvariat.
Proof. The key point is to prove the attention mechanism is E(3)-invarian)) Since at+1 at+ gl+1 and a/+1 x/+1 a(+1 all resluts based on them will be manslaaion-invaiianne And with the norm operator || we can build orthogonal-invarian features since ||O=|| = || and i here
h.
can be replaced by - or a That means the attention mechanism is E(3)-invariant.
Other operators only linearly combine all Cartesian vectors without changing their own symmetries, so the module is E(3)- eqqivvaiaat.
Theorem C.3. The module ENN(-) is equiivaria.
Proof. The key point is prove the m. is E(3)-invariant. Since W., WR. WE, & WT P.P. P all invariant embeddings from atom type ,nucleotide type and position (not coordinates), only to show that the distance matrix Diy is O(3) -i))-invariant. The distance 0(3) -ivvarraa can also be proved by unitarity of the norm. that means ENN(.) will not change the symmetry of the vector input. Since 2l+2 is the center of nucleotide j. x:t22 al+2 outputs the relative position difference, ey
which translation-invann and O(3) ((3)_cuuivariant, and that means the whole ENN(.) module is
D RRNASOLO DATASET
In this section, we provide details not covered in the main paper about rRNAsolo dataset.
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D.1 GENERATION OF CANDIDATE STRUCTURES
Following the description of data collection, purification, and partitioning in the main paper, first predict the secondary structure of all RNAs using (Zok et al., 2018). Then, with the primary and fragment assembly method
predicted secondary structures inputs, FARFAR2 (Watkins et al., 2020) to generate candidate tertiary structures. Finally, we calculate the RMSD between these candidate structures and the native structure as the label for task.
the knowledge-basse
D.2 THE CHOICE OF FARFAR2
Here, we chose FARFAR2 for dataset construction primarily for two reasons:
Firstly, the paper (Townshend et al., 2021) published in Science '21 uses FARFAR2 to construct their dataset, demonstrating that its generation quality has been practically verified and has undergone rigorous peer review, ensuring reliable quality.
Secondly, FARFAR2, classic RNA 3D structure prediction method. is widely recognized in the field. For instance, "Childs-Disnee et al., 2022) published in Nature Reviews Drug Discovery '22 compares FARFAR2 to ROSETTA: "currently available programs, including FARFAR2 (RNA ana- logue of ROSETTA for protein prediction)" The paper (Wong et al., 2023) published in Science '23 lists FARFAR2 alongside AlphaFold as representative sequence-to-struetie
models: "We anticipate that sequence-to--truuctuee models, such AlphaFood for proteins or FARFAR2 for RNAs."
Given FARFAR2's widespread recognition the field and its use in constructing datasets peer- reviewed papers, decide to employ FARFAR2 to generate RNA candidates and build our rRNA- solo datasct. Additionally, other methods could also be used as long as they produce high-quality RNA candidate structures. In the future, plan to further explore and optimize the rRNAsolo dataset to enhance the quality of candidate structures.
E CONFIGURATIONS
Table 8 presents the hyper-paramettes of EquiRNA used in two experiments of this paper. Addi- tionally, the results reported in our paper for other baselines are all obtained by retraining using publicly available code on rRNAssol dataset. Each layer here consists of the Eq. (1), Eq. (2), and Eq. (3). Both our approach and all other baseline methods are trained and tested on a single NVIDIA A100-80G GPU.
Table Hyper- paremetrrssts of EquiRNA. The mucleotide.template..ae denotes the size of trainable template for each type of nuclcotide, the atom template size denotes the size of trainable template for each type of atom, the hidden size denotes the size of hidden states our EquiRNA, and the n_layers denotes the number of layers in EquiRNA.
Hyperparameter
rRNAsolo dataset
ARES dataset
Learning Rate Epochs
le-4
1e-4
20
20
nucleotide, nucleotide.templatte
16
16
atom_teme coom_template_tee
16
16
hidden_size
128
128
n_layers
3
K
16
16
M
26
26
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F MORE COMPREHENSIVE EXPERIMENTS AND ANALYSIS
F.1 MORE EXPERIMENTS USING THE I BEST-SCORING MODEL AND 10 BEST-SCORING MODELS
To provide more comprehensive analysis of our method compared to other methods, we present the results of all methods on the rRNAsolo and ARES datasets using the best-scoring model and 10 best-scoring models in Table 9. As shown in the table, method still achieves the best performance on the rRNAsolo dataset for the 10 best-scoring models, demonstrating the superior performance and robustness of our model. On the ARES dataset, our 10 best-scoring model is slightly inferior to the ARES method, but still achieves the second-best results, surpassing most of the other baseline methods. The 10 best-scoring setting evaluates the models' performance within the top 10 candidates, but it does not identify which candidate represents the best prediction. In contrast, the best-scoring setting assumes that the top candidate the best prediction, which better aligns with practical applications.
Table Results all methods on the rRNAsolo and ARES datasets using the best-scoring model and 10 best-scoring models. Bold indicates the best method, while underline represents the second- best method.
rRNAsolo
ARES
best-scoring
10 best-scoring
1 best-scoring
10 best-scoring
ER +
R-ER +
ER J
R-ER 4
ER 4
R-ER 4
ER
R-ER
ARES
22.57
0.84
16.19
0.32
12.33
1.02
8.42
0.43
EGNN
20.99
0.71
15.69
0.28
13.47
1.21
10.00
0.70
PaxNet
21.64
0.76
15.77
0.29
13.15
1.16
10.78
0.83
dyMEAN
20.91
0.71
15.95
0.30
13.56
1.23
9.35
0.59
RDesign
20.22
0.65
15,86
0.29
13.37
1.20
9.51
0.61
EquiRNA
18.22
0.48
15.30
0.25
11.74
0.93
8.99
0.52
F.2 DETAILED ANALYSIS OF THE EXPERIMENTAL RESULTS
In this section, we endeavor to elucidate the potential factors contributing the notable performance degradation of ARES the rRNAsolo dataset.
(1) The strong performance of ARES reported in their paper might be due to the careful tuning of hyperparameters specific to their dataset. In the supplementary materials of the ARES paper, the authors stated, "We optimized several hyperparameters that specify characteristies of the training process: batch size, number of epochs, learning rate, and number of candidate structural models per RNA to feed in. We considered 100 sets of hyperparameter values. For each of these sets, trained the network parameters." When applied to rRNAsolo dataset using the same set of hyperparameters, ARES performs significantly worse, indicating reduced robustness across diverse scenarios. In contrast, our method, without extensive hyperparameter tuning and using the same hyperparameters across both datasets, consistently achieves superior performance.
(2) As mentioned in the final paragraph of $ 2.2 and shown in Table 1, our rRNAsolo dataset features a broader range of RNA sizes, a larger number of RNAs, and more recently released data compared the ARES dataset. Notably, the RNA size range overlaps between the training and sets in the ARES dataset, whereas in our rRNA dataset, these ranges are distinct. Therefore, we believe our dataset presents more challenging study for evaluating size generalization.
G REGARDING THE HIGH-LEVEL RMSD VALUES
Regarding the high-level RMSD values presented in the experimental tables, they are primarily due to the following two reasons:
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Table 10: The results of rRNAsolo when considering molecular chirality.
Validation Set
Test Set
ER /
DR |
R-ER
R-DR
ER L
DR 4
R-ER 4
R-DR
EquiRNA with SBDD
20.66
21.02
0.92
1.13
18.85
19.28
0.53
0.60
EquiRNA
19.77
19.62
0.84
0.99
18.22
17.79
0.48
0.47
Table 11: Ablation studies on rRNAsolo.
ER
DR
R-ED
R-DE
EGNN-like at Subumit-Level
18.86
19.75
0.53
0.64
Averaging Nucleotide-Level
19.87
18.66
0.62
0.55
EquiRNA
18.22
17.79
0.48
0.47
(1) The inherent difficulty of the task: As RNA size increases from small large, the complexity and flexibility of RNA structures increase significantly, making the task more challenging.
(2) The quality of the RNA candidates: Since our task is RNA structure evaluation, our reported RMSD values largely depend on the candidate structures generated by FARFAR2 (Watkins et al., 2020) (FARFAR2 is a widely recognized method in the field, and for the rationale behind using FARFAR2, please refer to Appendix D.2). Here, would like to further clarify the computation process the RMSD metric for better understanding. For each RNA the test set, we generate multiple candidate structures using FARFAR2. The task of the evaluation model is to score these candidate structures based on the model's predicted RMSD values, then select the one with the minimum predicted value, and finally report the ground-truth RMSD between the selected candidate structure and the native structure. Thus, the theoretical minimum of this value is NOT necessarily zero; instead, it is the lowest RMSD between the best candidate and the native structures. As our constructed dataset rRNAsolo is more challenging than the ARES dataset, the generated candidates by FARFAR2 (the same method used in the ARES paper to generate candidates) generally exhibit higher-level RMSD than those in the ARES dataset, leading to high-level RMSD reported in our experiments.
H CONSIDERING MOLECULAR CHIRALITY
We attempt the approach from SBDD (Schneuing al., 2022) by incorporating Eq. (6) and Eq. (7) of SBDD into our model to further account for molecular chirality. The results are shown in the Table 10. As can seen, including chirality actually led slight performance drop. This is quite reasonable. In fact, our task is to predict the RMSD between candidate RNA structures and stable structures, which is an F(3)-invariant quantity and insensitive to structural flipping. Therefore, for our task, an E(3)-invariant model is more appropriate than an SE(3)- model. Of course, we agree that if the task involves predicting molecular properties sensitive to chirality, such as optical activity, then an SE(3)-invariant model would indeed be more suitable.
I THE ABLATION STUDIES ON SUBUNIT-LEVEL. AND NUCLEOTIDE-LLEVEL
We conduct an experiment by replacing the attention mechanism with a simpler EGNN-like message-passing scheme subunit-level. The results, presented Table 11, show that the EGNN- like scheme performs worse compared to original strategy. These findings highlight the effec- tiveness of the attention mechanism capturing the intricacies of subunit-level interactions. More- over, conduct experiment by averaging the atomic coordinates within each nucleotide. The experimental results, as shown in Table 11, indicate that this approach adversely affects the model's performance.
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Table 12: The results of all four metrics the first split.
Validation Set
Test Set
ER L
DR +
R-ER +
R-DR
ER 4
DR +
R-ER +
R-DR
RDesign EquiRNA
13.04
9.56
0.92
0.46
16.36
16.12
0.86
1.44
11.56
10.05
0.71
0.53
15.00
14.15
0.70
1.14
Table 13: The results of all four metrics on the second split.
Validation Set
Test Set
ER 4
DR 4
R-ER +
R-DR
ER 4
DR 4
R-ER +
R-DR
RDesign
15.86
13.84
1.22
1.78
13.01
11.16
1.07
0.83
EquiRNA
14.08
12.46
0.97
1.50
11.47
9.74
0.82
0.59
J MORE SPLITS ON RRNASOLO
We extend our study to general scenario where size generalization is not constraint, Based on the clustering approach described in $ 2.2, we randomly split the training, validation, and test sets the cluster level. Due time and resource limitations, rations, with each split containing 20k, 6k, and 6k samples in the training, validation, and test sets, respectively. The results, presented in Table 12, Table 13 and Table 14, demonstrate that our model consistently achieves superior performance across nearly all three splits, further underscoring its robustness and effectiveness. Furthermore, we also list the PDB IDs of the RNAs included in each split.
three random split configu-
J.1 FIRST SPLIT
Training: 4MGN_I.D 2B57_1_A, 7EQJ_LA, 7KVT.I_B, 3SKW_IA, 4YB0_I_A, STPY.LA.
7KVV_1_D, 6UC9.1_B, 70AV_1_B, 2EET.1 70AX.IBB, 70AX.1., 6TFF_1_, 2GDLLY,, 6CC3 1 3DS7 A, 3D2G B, 4YBO R, 6UET A, 6UFH B, 6Q57
7KJU A,
3OXE B, 6PRV 1 C, 5C7U B, 4FEL. 1 B, 30W1 A, 6QN3 A-B, 4FE5 1 B, 7D7W 1 A, 6UES_1LA, 6TFE.1_A, 7U4A_1AA, 5FKI_L,A 30WI.1_B, 7D82.1AA, 6UFL1_A, 3GER.LA, 6DMC.L_B 70AV_LD, 6WLQ.3.A, 2EEW_LA, 4ENB.LA. 3GAO_ 2G9C.1L, 4FEO.I.B, 2EES_LLA, 2QUW_ 6TF3.I_A
Validation: 4PLX_I LA, 5OB3_I 4JRC_LA, 4K27_1_U, 2YIF_I 3R4F_LA, 4WFM.IBB, X-IZ 6SVS.1.B, 7JRT.LA, 4JRC.LB, 6UFM.IBB, 3SLM.I 6SVS.LA, 7L0Z.1-G
Test: SDSL.LA, 7JRR_LA, 6DN2-LY-X, 5V3LLA, 4JF2-LA, 6DNIJZ-XX, 7LYJ_LA, P.I GWTL.1 7IJU_LA--B, B-A..L.-A, SDSL. 4P5J_LA, 6WTR.LA, 5_SL..IB
J.2 SECOND SPLIT
Training: 70AV_1 1_B, L.K. 6DME.LA, 3IVN.IB, 3IVN.I_A, 2G9C.1AA, 4FE5_
4LVW ILA, 4WFM_LA, 70AV I.C. IDNIII-X. 4WFM.IB. 6UC9 7D7Y
3SKL.1LA, 4FEL_1_,, 6DMD_I_B, 4YB0_ SAKF_LA, 1Y27_1_X, 6Q57_1A, 30WW 3FO6_1_A, 4YBO.I.R, 6DN2.1.Y-X, 4TZX.IXX 3SUX.IXX, 6TB7_LA, 3OXE.IBB 2 X. 6UC8.IBB, 2GDL_LX, 4LVZ.LA 3SKL.IBB, 2XNZ.LA, 6UFK_LL, 4WFL.IA, 3SKLLA, 6UFJ C. 7D81 1 A, 3SKI B, 3SKT A, 4MGN I C, 7KVU 1 G. 6UFK C, 5FK6 A,,
3G0G 1 A, 6SVS A, 2YTE. X-Z, 30XD
Validation: 6NSN_14 4JF2.1A,, 5TPY_1-A, 7JRR_LA, 5DSL. 4PQV 1.A, 3FWO.1 L.B, 6CC3 7K16_1.P, 7U4A 7JJU A-BB 5D5L. B, 201U_1_Q 5DSL. D. 4R4V_14 1.A
Test: 7LOZ G, 4EN5 A, 3R4F 1 A, 6V9D B-E, 7LYJ A, 4ENB I A, 4ENA A,,
4ENC A, 50B3 1 A, 4P5I 1 A, 1KXK SUNE A-B, 4K27 U, 6JQ5
B-A, 14.300371 1 A
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Table 14: The results of all four metrics on the third split.
Validation Set
Test Set
ER 4
DR /
R-ER +
R-DR
ER 4
DR +
R-ER +
R-DR
RDesign EquiRNA
15.61
15.41
1.05
1.14
16.68 14.89
15.16
1.03
1.24
14.15
14.36
0.86
1.00
13.63
0.81
1.02
Table 15: The comparison with more leading models.
Validation Set
Test Set
ER 4
DR
R-ER L
R-DR
ER J
DR
R-ER 4
R-DR
ProNet
21.33
22.16
0.99
1.24
20.56
20.45
0.67
0.69
Equiformer
20.76
20.70
0.93
1.10
19.22
18.20
0.56
0.51
EquiRNA
19.77
19.62
0.84
0.99
18.22
17.79
0.48
0.47
J.3 THIRD SPLIT
Training: 6WTR_1 L,, 3DIR_LA, 3IVN_ L.B, 6.,,, 7D7Y_1.A 4FEL. 6DMC_I I.C. IBB, 70AX I LB, 4PLX 1 A, 3DIY A,, 4LX5 A, 30XB I. 7KVU G. 7L0Z. 6UEY BEEWII_A 30WW _WW_WWA _AA, 3, 3GOT_1_A, 6N5N_L_, 7EQIL_A, 6IQ5_1_B-A, 1,,
3SKW_1.B ESKW-ILA, 6N5K.IAA 6UFG.I.B, 3OXE_LA, AN5P_I_A, 6NST.1_A, 3SKT.LA, 3SKL 1 B. 3SLQ A,, 3SKL A, 6N50 A. 7OAV C, 6N5Q 1 A, 3SD3 A.A, 7KVT B, 4TZX.1 1.X, 4FEN 1. B, ASKR.1.A, 6U,,.1.,, 3D2V. 1.B. 4FE5. 1.B. 6UFJ_1 A,, 4FEO.
2CKYU, AA 6N5
Validation: 5FK1 1 A, 201U I.P. 2QUW_1 D. 4K27_1_U 2QUW IB, 3Q3Z.1., 6DN1 Z-X, SFKF_IAA A,, 3Q37.1 1 A, 2QUS 1 SFJC. I.C. 6V9D A.D.1.B-A B-E, 2QUS B, 4P8Z. A, IKXK LA
Test: 6DN2 Y-X, 3SLM B. 3VRS 1 A, 4WFL A, 4ENB A, 4YB0 1 A, 4YB0 R, 7JJU_1A--B, 4ENC_ 4PQV _1A, 4WFM_ LA, JEN5_1_A, 4ENA_1 LA, 4R4V_ __A, 6WIQQ3.A
K DISCUSSION ON THE RECENT SCALABLE EQUIVARIANCE METHODS
SE(3)- ((eenHHyna (Moskalev et al., 2024), VN-EGNN (Sestak al., 2024), and Ponita (Bekkers al., 2023) employ global context (the SE(3)-Hyena operator, virtual nodes, and global group actions, respectively) to enhance the scalability of equivariant neural networks. These approaches represent excellent strategies for further improving general- purpose models. In contrast, our EquiRNA is specialized design tailored to leverage the prior knowledge of RNA nucleotide structures, enabling the effective application of equivariant neural networks to large RNA molecules. This specialization allows EquiRNA to address the unique challenges of modeling RNA, distinguishing from the more general approaches discussed.
L THE COMPARISON WITH MORE LEADING MODELS
There are various strong baselines like SphereNet (Liu et al., 2022), Equiformer (Liao & Smidt, 2022), MACE (Batatia et al., 2022), and ProNet (Wang al., 2022). We conduct experiments with Equiformer and ProNet. The results are shown in Table 15. Equiformer demonstrates strong performance, likely due to its incorporation higher-order information. ProNet, however, does not perform well. We suspect that the original ProNet paper is for proteins and it computes Euler angles using alpha carbon, carbon, and nitrogen atoms, which are not strictly corresponding to the atoms in RNA. In reproduction, have influenced ProNet's performance to some extent. Overall, our EquiRNA still achieves the best performance, substantiating claims.
substituted these with P, C4', and C3' atoms, which might
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Ground Truth
EquiRNA
4R4V
6JQ5
Figure 6: The visualization bad cases.
Table 16: The results of rRNAsolo when considering noise.
Validation Set
Test Set
ER
DR 4
R-ER
R-DR 4
ER J
DR J
R-ER 4
R-DR
EquiRNA w noise
20.44
20.68
0.90
1.09
19.51
18.94
0.59
0.57
EquiRNA
19.77
19.62
0.84
0.99
18.22
17.79
0.48
0.47
M THE ANALYSIS OF SOME ERROR CASES
We have presented two RNA samples Fig. where EquiRNA performs poorly. We observe that both samples exhibit significant structural symmetry, which we hypothesize may contribute to the suboptimal performance. Symmetry could result in a phenomenon akin to "systematic extinction" in crystallography (Janner & Janssen, 1977), where information in certain directions is weakened or lost. Although RNA structures are not perfectly symmetric, this might affect the model's ability to effectively learn directional information. Indeed, prior literature (Joshi et al., 2023; Lawrence et al., 2024) suggests that equivariant neural networks may encounter challenges when dealing with symmetric structures.
N CONSIDERING NOISE
Since noise is common in experimental structure determinaation/sqqueennn (Xu et al., 2024a), conduct the experiments to evaluate our model when considering noise. We apply Gaussian noise (00, 0.01 B[ æell) I) to the coordinates of all training RNAs, where represents the 3D coordinate each atom in RNA, and x represents the mean coordinate RNA. The results on rRNAsolo decline in model performance, Excitingly, our EquiRNA with noise is still better than other methods that are trained from clean data.
all the atoms in one shown in the Table 16. is observed that adding noise leads to
O THE FIGURE FOR THE GRAPH STRUCTURE
We illustrate the graph structures in 2.1 clearly with the Fig.
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(M,,, Ex)
Nucleobase
(N
Do day))
(ho, Mo..
Carbon
A Seu
Hydrogen
(leyy F
Oxygen
Nitrogen
RiboseTM... Phosphate
Phosphorus
G.(VA.
Figure 7: RNA distribution of RNAsolo.
P THE DETAILED EXPLANATION OF THE IMPLEMENTATION OF OTHER BASELINES
ARES and PaxNet were originally designed for RNA structure evaluation. We use their official code for our experiments.
As for EGNN, the original paper models small molecules at the atomic level. When adapting it to task, we model RNA the full-atom level and directly use their official code.
For dyMEAN, the original paper models proteins using amino acids as basic units. When adapting our task, we replaced amino acids with nucleotides and modeled RNA using nucleotides basic units, employing their official code.
For RDesign, the original paper is developed for RNA inverse folding tasks and considers only backbone atoms. Since RNA representations in the original method primarily rely on backbone dihedral angles and distance information, we retain the original settings by using backbone atoms to represent RNA and employ the invariant feature h calculated from the method for the final RMSD prediction. We use their official code for our experiments.
For GET, the original paper models proteins hierarchically by capturing intra-amino acid and inter- amino acid interactions nucleotides and use their official code.
blocks. When adapting our task, we replace amino acids with
Q FINETUNED RESULTS A NEW DATASET GENERATED BY RHOFOLD.
In this section, we attempt utilize the recently published RhoFold (Shen et al., 2024) method new candidate structure generator to construct a new dataset and finetune the models on this dataset.
Predicting 3D structures with RhoFold requires both ID fasta sequences and MSA data. How- ever, the RNAsolo dataset used in our experiments does not provide the corresponding MSA data. To overcome this, we first search for similar sequences each RNA using RNAcentral (https:/mmaceetral.orgg) and BLAST multiple sequence alignment with Infernal (Nawrocki Eddy, 2013) and MAFFT (Katoh et al., 2002) to generate the required MSA data files. Following the methodology outlined in the origi- nal paper (Shen et al., 2024), we randomly sample from the MSA data to ensure the uniqueness of each generated candidate structure. Finally, compute the RMSD values between the candidate structures and their native counterparts using the pair fit function in PyMOL, which is based on the Kabsch algorithm (Kabsch, 1976).
We then perform
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Published as a conference paper at ICLR 2025
We randomly select 20 RNAs from the training set in rRNAsolo and generate 400 candidate structures for each RNA using RhoFold, following the process described above. This results in dataset containing 8,000 samples. We then finetune both RDesign and our EquiRNA model on this new dataset. The experimental results are presented in Table 17.
Table 17: The finetuned results on the new dataset generated by RhoFold.
Validation Set
Test Set
ER +
DR 4
R-ER +
R-DR
ER
DR 4
R-ER 4
R-DR
RDegisn 21.99
20.79
1.05
1.11
20.22
19.34
0.65
0.60
RDegisn+finetune 20.86
21.02
0.94
1.31
21.53
20.45
0.75
0.69
EquiRNA
19.77
19.62
0.84
0.99
18.22
17.79
0.48
0.47
EquiRNA+finetune 18.87
18.54
0.76
0.88
18.78
18.25
0.53
0.51
The results show that after finetuning, our EquiRNA model achieves improved performance of all four metrics the validation set, while RDesign shows better performance only in terms of the ER and R-ER metric. However, EquiRNA showed a slight decline in test set performance, possibly due to bias introduced when using only FARFAR2 to generate candidate structures. Since the structures generated by RhoFold follow a different distribution from those generated by FARFAR2, the model's performance on the test could be partially influenced by this bias. Notably, EquiRNA exhibits much smaller performance drop compared to RDesign on the test set (e.g.. 0.56 vs. 1.5 in terms of ER), suggesting that our model more robust across different candidate structure generators.
R BROADER IMPACTS
Our in-thpth investigation into more accurate methods for modeling and evaluating the 3D structures of RNA positively impacts both our understanding of the fundamental mechanisms of biological activities and the discovery RNA-targeted drugs.
LIMITATIONS
Relative to the abundance of data for proteins, extant RNA datasets are substantially inadequate to train highly accurate prediction and scoring models, especially for large RNAs where the intricate structural variations often lead to higher RMSD values. is our aspiration to delve deeper into prediction and scoring tasks of large RNAs in our subsequent research endeavors.
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Reviews:
[{"review_id": "168gO2j4Wp", "reviewer": "Reviewer_uusj", "summary": "The paper proposes EquiRNA, a hierarchical equivariant neural network designed to evaluate RNA 3D structures and generalize across varying RNA sizes, along with a new benchmark dataset rRNAsolo. The key strength is its practical approach to scaling equivariant neural networks for RNA data through multi-level equivariant message passing and size-insensitive k-NN sampling. Additionally, the paper introduces a new curated RNA dataset for structure fitness prediction task which is an important contribution given the scarcity of RNA datasets. However, the choice of E(3) over SE(3) equivariance ignores molecular chirality, the subunit and nucleotide level message-passing mechanisms lack clear motivation, and the empirical validation would benefit from multiple dataset splits given the relatively small number of samples in the datasets.", "questions": "I suggest authors address weaknesses for the rebuttal.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper tackles important practical limitations of the scalability of equivariant neural networks applied to real biological data.\n2. The paper focuses on RNA which is a much less saturated domain compared to other biomolecular modalities such as proteins. Extending a machine learning toolkit for RNA analysis is an important contribution.\n3. The paper introduces a new dataset rRNAsolo. Authors make great effort in collecting the data and purifying it on multiple levels. This is an important contribution given the scarcity of RNA datasets for machine learning.\n4. The subunit-level message passing is novel and the ablation study underlines its importance for the fitness prediction task.\n5. The k-NN sampling strategy is a simple approach for that potentially robustifies a model against an input size distribution shift.", "weaknesses": "1. The method focuses on E(3)-equivariance while in reality SE(3)-equivariance is the most relevant due to the molecular chirality. The E(3)-invariant/equivariant method does not discriminate between RNA molecules and their reflection which often comes with drastically different properties. This severely limits the practical applicability of the proposed method. Authors may consider the method presented in [1] (Eq.6-7) as a relatively easy way to break the symmetry to reflections and achieve SE(3)-equivariance.\n2. The size generalization problem requires further elaboration. RNAs of different lengths usually correspond to different types, and hence different functions and structures. With this, it is hard to see how a network could generalize to another type of RNA with possibly completely different biological function/structure without observing it during training.\n3. The subunit-level message passing (lines 273-300) is not explained clearly, and the particular form of message-passing in equation (2) is not well-motivated. Without this clear motivation, I fail to see why this form of message-passing will bring any advantage compared to a simple EGNN-like message-passing scheme.\n4. Similarly, the particular form of nucleotide-level message passing is not well-motivated. The framework of learnable templates and matrix-valued message function in Eq. 3 appears as unnecessary over-complication. I am wondering why the simplest approach of averaging the nucleotide coordinates (similar to how it is done with invariant nucleotide features) is not enough here. At least, I would expect an ablation study with respect to simple averaging.\n5. The experiments are performed only on one split of the datasets. Given the relatively small number of samples in the datasets, it is good to run the methods over several splits and provide mean and standard deviation results. Further, the benchmarking on the proposed rRNAsolo dataset will be much more reliable if the dataset comes with several provided train\\val\\test splits. This will significantly increase the weight of the rRNAsolo contribution.\n6. The related work on Geometric GNNs misses elaboration on recent scalable equivariance methods: [2,3,4], etc. This is important since the paper claims to tackle the practical limitations of the scalability of equivariant neural networks\n\n[1]. A. Schneuing et al. Structure-based drug design with equivariant diffusion models.  \n[2]. A. Moskalev et al. SE(3)-Hyena Operator for Scalable Equivariant Learning. GRaM@ICML 2024. \\\n[3]. F. Sestak et al. VN-EGNN: Equivariant Graph Neural Networks with Virtual Nodes Enhance Protein Binding Site Identification. ICML 2024. \\\n[4]. E. Bekkers et al. Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Space. ICLR 2024. \n\nMinor:\n- line 346 typo: size-generation -> size-generalization", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wewAUPFytS", "reviewer": "Reviewer_Pevo", "summary": "The paper introduces EquiRNA, a model which has three-level hierarchical architecture—atom-level, subunit-level, and nucleotide-level—inspired by the biological composition of RNAs. The model is designed to efficiently model RNA structures while handling RNAs of varying sizes, referred to as the size generalization problem in the paper. The paper introduces a new dataset curated from public database and benchmarks the proposed model against this dataset and an existing ARES dataset outperforming existing methods for the task of predicting RMSD between candidate and native structures, used for scoring and ranking all candidates to identify the one most\nclosely matching the native structure.", "questions": "I have posed my questions/concerns in the weaknesses section already. Here is a summary and few suggestions which may help address some of my concerns.\n1. Can the authors show how does the model handle variations in RNA structures due to experimental noise or biological variability (performance with different structural resolutions)? Also since noise is common in experimental structure determination/sequencing, can we consider testing the model's performance on RNA structures with simulated noise or structural variability to assess its robustness?\n2. How sensitive is the model’s performance to the choice of templates? Is there a risk that template selection introduces bias or limits the generalizability of the model?\n3. Has the model been evaluated on a variety of RNA structures, particularly those with unusual or rare structural motifs, to ensure generalizability? \n4. Is it possible to run experiments showcasing the model's ability to truly generalize on large RNA sequences common in drug discovery (many hundreds to thousands of nucleotides), such as those derived from high-throughput sequencing or large RNA viruses?\n5. I am not sure how K-nearest neighbor sampling strategy guarantees the size generalization of the model. Can this section be expanded upon?\n6. Is it possible to showcase some error cases where the model fails and if that reveals some insights?\n7. For the model to be usable and further developments to happen in this space, releasing code and curated datasets will be useful. Can the authors release them?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper proposes three-level E(3)-equivariant message-passing design for modeling RNA 3D structures that captures atom, subunit, and nucleotide interactions effectively. This is inspired by the biology of RNA and sensible contribution.\n2. The paper showcases an application of EGNN architecture to model the interactions within different levels of RNA. Although it is a straightfoward application of EGNN, I find it addresses a meaningful modeling aspect respecting the prior knowledge from biology. Size-insensitive K-nearest neighbor sampling is also intended to boost the model’s performance on varied RNA sizes, addressing a common challenge in RNA modeling.\n3. Introducing rRNAsolo benchmark is useful for future development in this area.\n5. The paper is well-written for most parts, with ideas explained in a structured manner.", "weaknesses": "Although the paper tackles an important problem, there exist several avenues for improvement. \n1. Given that the paper is meant to address the size generalization aspect of RNA structure evaluation, the experiments are carried out mostly on relatively smaller RNA sequences (~200 nucleotides). In most practical applications, the RNA sequences can easily range from hundreds to even thousands of nucleotides. To strengthen the claim that the proposed model can generalize to larger size RNAs, is it possible to showcase performance on larger RNA sizes or to discuss any technical limitations that prevented testing on very large RNAs?\n\n2. Though the K-nearest neighbor sampling helps, there might still be overfitting to smaller RNAs, as training primarily relies on these samples. It is possible that I may have misunderstood but I am not sure how K-nearest neighbor sampling strategy guarantees the size generalization of the model. Given that this is claimed as the core contribution of the paper, it is remarkably brief and covered within 1 small paragraph. A detailed explanation around this will help improve readability. \n\n3. The design, being based on EGNN, might face issues with large RNAs common in drug discovery due to increasing complexity in both memory and computation. Famously, SE(3)-Transformer (Fuchs et al., 2020), Equiformer (Liao et al., 2022) has been utilized for modeling large molecules such as proteins in models such as AlphaFold, there also have been recent works such as SE(3)-Hyena (Moskalev et al. 2024) using long-convolutional models for equivariant modeling of large-scale systems. Such architectures may directly model all-atom resolution thus providing an alternative to hierarchical modeling at backbone level as described in the paper. Potentially, the authors may consider an ablation study comparing their hierarchical approach to an all-atom model for the largest RNAs in their dataset or may provide a complexity analysis comparing their approach to SE(3)-Transformer, Equiformer and SE(3)-Hyena for large RNA modeling to justify architectural choice.\n\n4. The paper does not seem to explore cases where EquiRNA performs poorly or where errors arise, missing an opportunity to analyze model limitations and provide insights into potential biases or weaknesses in real-world RNA structures. Can the authors provide specific examples where EquiRNA performs poorly compared to other methods or any patterns in the types of RNA structures or features that lead to higher errors?\n\n5. The use of learned atom and nucleotide templates is innovative, but I wonder if this can also hamper generalizability. I imagine that generalizability is partly tied to specific template choices, which might not always capture structural diversity in RNAs. If new RNA structures or variations appear, the model may not capture these well unless it’s retrained or the templates are updated. If the model is reliant on pre-learned templates, it may struggle with RNAs that exhibit novel conformations or atypical atom interactions.\n\n6. Experimental mehods like cryo-EM, X-ray crystallography, or NMR spectroscopy—for obtaining RNA structures—can produce data with resolution limits. This has mentioned in the introduction but I seem to have missed how the model performance changes when applied to   more/less noisy data (in terms of structural resolution).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "g0DHUIwz9Y", "reviewer": "Reviewer_egW8", "summary": "The paper presents EquiRNA, a hierarchical Geometric Graph Neural Network model designed for size-generalizable RNA structure evaluation. Addressing challenges in capturing full RNA geometries and generalizing across different RNA sizes, EquiRNA uses a three-level architecture—atom, subunit, and nucleotide—to model RNA’s structural complexity effectively. It introduces a size-insensitive K-nearest neighbor sampling strategy to enhance robustness when generalizing from small to large RNAs, and a new dataset, rRNAsolo, for thorough evaluation. Experimental results show EquiRNA outperforms SOTA methods, providing a robust new baseline for RNA 3D structure prediction and evaluation.", "questions": "1.\tBetter to have a figure when explaining the graph structrure in 3.1\n2.\tIn line 53, what’s the definition of “resolution”? Is it the threshold of deciding if there is an edge between two atoms?\nHow and why to select “resolution higher than 2 A”?\n3.\tWhy don’t use average for $\\overrightarrow{\\boldsymbol{X}}^{l+2} \\in \\mathbb{R}^{3 \\times\\left(N_r+N_b\\right)}$, instead of a concatenation?\n\n4.\tFor atom-level and subunit-level layers, do they apply only on 4 types of nucleotides (AUGC), or each nucleotide respectively?\n\n5.\tLine 878 shows the number of layers for both datasets are 3. If I stack Eq. 1, Eq.2, and Eq.3, are they together 1 layer or 3 layers?\n6.\tEquation 4 is confusing, and I’m not clear with the overall task.  \n\n(1)\tThe task is to predict RMSD between candidate and native structures. So on RNAsolo’s website (https://rnasolo.cs.put.poznan.pl), the provided structures are ground-truth (native) structures, right? Then what’s the candidate structures and how did you get them? Based on my understanding, there are multiple 3D strucure prediction tools for one RNA sequence, then what are the prediction tools? And, how many candidate structures are compared to the ground-truth structure?  \n\n(2)\tIn Eq.4, is $\\hat{\\boldsymbol{h}}_i^{l+3}$ the ground truth label for each nucleotide? How did you get that? I understand the ground truth structure is given by a PDB file, but how was it converted into the ground truth for each nuclotide?   \n\n(3) Why is $\\hat{h}_i$ ground truth? Isn't it the feature? If it is ground truth here, then the notation is duplicated.  \n\n(4) Why to use features h to predict final RMSD? Why not X? What’s the function of row 3 in Eq.3 if X is not used?  \n\nI'd appreciate it if the authors can describe the overall task, candidate structures, ground truth labels comprehensively.\n\n7. Eq. 1 and Eq.2 have the similar form with EGNN. The difference is just on different levels (atom level, sub-unit level). Is there any intuition why to use RBF?\n\n8.\tCode is not available.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1.\tHierarchical Architecture for Size Generalization: By modeling RNA structure at atom, subunit, and nucleotide levels, EquiRNA effectively captures detailed RNA geometries and adapts well to RNAs of varying sizes, addressing the common challenge of generalizing from small to large RNAs.  \n\n2.\tInnovative Sampling Strategy: The size-insensitive K-nearest neighbor sampling enhances robustness by exposing the model to diverse structural variations, allowing better generalization to larger RNAs not seen during training.  \n\n3.\tComprehensive New Dataset: The rRNAsolo dataset, covering a broader range of RNA sizes and types, provides a more rigorous benchmark for size generalization, demonstrating EquiRNA’s superior performance over state-of-the-art methods in real-world scenarios.", "weaknesses": "See Questions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pbv9r8bZRm", "reviewer": "Reviewer_sFcQ", "summary": "The paper introduces a three-level hierarchical equivariant graph neural network (EquiRNA) designed to efficiently capture the geometries of RNA, enabling it to generalize effectively across a wide range of RNA sizes. EquiRNA demonstrates promising results in evaluating RNA 3D structures.", "questions": "1.\tThe paper presents comparisons with several models, but there are other models, such as SphereNet[1], Equiformer[2], MACE[3], and ProNet[4], that show better performance. A more extensive comparison with existing methods would enhance the paper. Although the experimental results highlight the proposed method's effectiveness, a more thorough comparison with other leading models could further substantiate the paper's claims.\n2.\tA detailed explanation of the implementation of several baselines is needed. For example, RDesign was originally developed for RNA inverse folding tasks and only includes backbone atoms. How was RDesign adapted for RNA structure evaluation in this paper?\n3.\tThe author emphasizes the importance of the size-generalization task but should include an experiment that evaluates dataset splitting independent of size.\n\n[1] Liu, Yi, et al. \"Spherical message passing for 3d molecular graphs.\" International Conference on Learning Representations (ICLR). 2022.\n[2] Liao, Yi-Lun, and Tess Smidt. \"Equiformer: Equivariant graph attention transformer for 3d atomistic graphs.\" arXiv preprint arXiv:2206.11990 (2022).\n[3] Batatia, Ilyes, et al. \"MACE: Higher order equivariant message passing neural networks for fast and accurate force fields.\" Advances in Neural Information Processing Systems 35 (2022): 11423-11436.\n[4] Wang, Limei, et al. \"Learning hierarchical protein representations via complete 3d graph networks.\" arXiv preprint arXiv:2207.12600 (2022).", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1.\tThe proposed hierarchical equivariant graph for RNA structures is novel.\n2.\tThe results on the size generalization dataset are compelling.", "weaknesses": "1.\tThe assessment is not informative (see Q1).\n2.\tThe baselines compared with EquiRNA are used across various tasks, but their adaptation for structure evaluation remains unclear (see Q2).\n3.\tWhile EquiRNA performs well on datasets of RNAs with unseen sizes, its performance on the datasets splitting irrespective of size remains unclear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)