Analyzing volumetric data with rotational invariance or equivariance is currently an active research topic. Existing deep-learning approaches utilize either group convolutional networks limited to discrete rotations or steerable convolutional networks with constrained filter structures. This work proposes a novel equivariant neural network architecture that achieves analytical Equivariance to Local Pattern Orientation on the continuous SO(3) group while allowing unconstrained trainable filters - EquiLoPO Network. Our key innovations are a group convolutional operation leveraging irreducible representations as the Fourier basis and a local activation function in the SO(3) space that provides a well-defined mapping from input to output functions, preserving equivariance. By integrating these operations into a ResNet-style architecture, we propose a model that overcomes the limitations of prior methods. A comprehensive evaluation on diverse 3D medical imaging datasets from MedMNIST3D demonstrates the effectiveness of our approach, which consistently outperforms state of the art. This work suggests the benefits of true rotational equivariance on SO(3) and flexible unconstrained filters enabled by the local activation function, providing a flexible framework for equivariant deep learning on volumetric data with potential applications across domains. Our code is publicly available at https://gricad-gitlab.univ-grenoble-alpes.fr/GruLab/ILPO/-/tree/main/EquiLoPO.

Designing RNA molecules that interact with specific proteins is a critical challenge in experimental and computational biology. Existing computational approaches require a substantial amount of experimentally determined RNA sequences for each specific protein or a detailed knowledge of RNA structure, restricting their utility in practice. To address this limitation, we develop RNA-BAnG, a deep learning-based model designed to generate RNA sequences for protein interactions without these requirements. Central to our approach is a novel generative method, Bidirectional Anchored Generation (BAnG), which leverages the observation that protein-binding RNA sequences often contain functional binding motifs embedded within broader sequence contexts. We first validate our method on generic synthetic tasks involving similar localized motifs to those appearing in RNAs, demonstrating its benefits over existing generative approaches. We then evaluate our model on biological sequences, showing its effectiveness for conditional RNA sequence design given a binding protein.

In this work, we introduce 6D Convolutional Neural Network (6DCNN) designed to tackle the problem of detecting relative positions and orientations of local patterns when processing three-dimensional volumetric data. 6DCNN also includes SE(3)-equivariant message-passing and nonlinear activation operations constructed in the Fourier space. Working in the Fourier space allows significantly reducing the computational complexity of our operations. We demonstrate the properties of the 6D convolution and its efficiency in the recognition of spatial patterns. We also assess the 6DCNN model on several datasets from the recent CASP protein structure prediction challenges. Here, 6DCNN improves over the baseline architecture and also outperforms the state of the art.

MOTIVATION: Effective use of evolutionary information has recently led to tremendous progress in computational prediction of three-dimensional (3D) structures of proteins and their complexes. Despite the progress, the accuracy of predicted structures tends to vary considerably from case to case. Since the utility of computational models depends on their accuracy, reliable estimates of deviation between predicted and native structures are of utmost importance. RESULTS: For the first time, we present a deep convolutional neural network (CNN) constructed on a Voronoi tessellation of 3D molecular structures. Despite the irregular data domain, our data representation allows us to efficiently introduce both convolution and pooling operations and train the network in an end-to-end fashion without precomputed descriptors. The resultant model, VoroCNN, predicts local qualities of 3D protein folds. The prediction results are competitive to state of the art and superior to the previous 3D CNN architectures built for the same task. We also discuss practical applications of VoroCNN, for example, in recognition of protein binding interfaces. AVAILABILITY AND IMPLEMENTATION: The model, data and evaluation tests are available at https://team.inria.fr/nano-d/software/vorocnn/. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Despite the progress made in studying protein-ligand interactions and the widespread application of docking and affinity prediction tools, improving their precision and efficiency still remains a challenge. Computational approaches based on the scoring of docking conformations with statistical potentials constitute a popular alternative to more accurate but costly physics-based thermodynamic sampling methods. In this context, a minimalist and fast sidechain-free knowledge-based potential with a high docking and screening power can be very useful when screening a big number of putative docking conformations. RESULTS: Here, we present a novel coarse-grained potential defined by a 3D joint probability distribution function that only depends on the pairwise orientation and position between protein backbone and ligand atoms. Despite its extreme simplicity, our approach yields very competitive results with the state-of-the-art scoring functions, especially in docking and screening tasks. For example, we observed a twofold improvement in the median 5% enrichment factor on the DUD-E benchmark compared to Autodock Vina results. Moreover, our results prove that a coarse sidechain-free potential is sufficient for a very successful docking pose prediction. AVAILABILITYAND IMPLEMENTATION: The standalone version of KORP-PL with the corresponding tests and benchmarks are available at https://team.inria.fr/nano-d/korp-pl/ and https://chaconlab.org/modeling/korp-pl. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Many proteins contain multiple folded domains separated by flexible linkers, and the ability to describe the structure and conformational heterogeneity of such flexible systems pushes the limits of structural biology. Using the three-domain protein TIA-1 as an example, we here combine coarse-grained molecular dynamics simulations with previously measured small-angle scattering data to study the conformation of TIA-1 in solution. We show that while the coarse-grained potential (Martini) in itself leads to too compact conformations, increasing the strength of protein-water interactions results in ensembles that are in very good agreement with experiments. We show how these ensembles can be refined further using a Bayesian/Maximum Entropy approach, and examine the robustness to errors in the energy function. In particular we find that as long as the initial simulation is relatively good, reweighting against experiments is very robust. We also study the relative information in X-ray and neutron scattering experiments and find that refining against the SAXS experiments leads to improvement in the SANS data. Our results suggest a general strategy for studying the conformation of multi-domain proteins in solution that combines coarse-grained simulations with small-angle X-ray scattering data that are generally most easy to obtain. These results may in turn be used to design further small-angle neutron scattering experiments that exploit contrast variation through 1H/2H isotope substitutions.

MOTIVATION: Protein quality assessment (QA) is a crucial element of protein structure prediction, a fundamental and yet open problem in structural bioinformatics. QA aims at ranking predicted protein models to select the best candidates. The assessment can be performed based either on a single model or on a consensus derived from an ensemble of models. The latter strategy can yield very high performance but substantially depends on the pool of available candidate models, which limits its applicability. Hence, single-model QA methods remain an important research target, also because they can assist the sampling of candidate models. RESULTS: We present a novel single-model QA method called SBROD. The SBROD (Smooth Backbone-Reliant Orientation-Dependent) method uses only the backbone protein conformation, and hence it can be applied to scoring coarse-grained protein models. The proposed method deduces its scoring function from a training set of protein models. The SBROD scoring function is composed of four terms related to different structural features: residue-residue orientations, contacts between backbone atoms, hydrogen bonding and solvent-solute interactions. It is smooth with respect to atomic coordinates and thus is potentially applicable to continuous gradient-based optimization of protein conformations. Furthermore, it can also be used for coarse-grained protein modeling and computational protein design. SBROD proved to achieve similar performance to state-of-the-art single-model QA methods on diverse datasets (CASP11, CASP12 and MOULDER). AVAILABILITY AND IMPLEMENTATION: The standalone application implemented in C++ and Python is freely available at https://gitlab.inria.fr/grudinin/sbrod and supported on Linux, MacOS and Windows. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Protein model quality assessment (QA) is a crucial and yet open problem in structural bioinformatics. The current best methods for single-model QA typically combine results from different approaches, each based on different input features constructed by experts in the field. Then, the prediction model is trained using a machine-learning algorithm. Recently, with the development of convolutional neural networks (CNN), the training paradigm has changed. In computer vision, the expert-developed features have been significantly overpassed by automatically trained convolutional filters. This motivated us to apply a three-dimensional (3D) CNN to the problem of protein model QA. RESULTS: We developed Ornate (Oriented Routed Neural network with Automatic Typing)-a novel method for single-model QA. Ornate is a residue-wise scoring function that takes as input 3D density maps. It predicts the local (residue-wise) and the global model quality through a deep 3D CNN. Specifically, Ornate aligns the input density map, corresponding to each residue and its neighborhood, with the backbone topology of this residue. This circumvents the problem of ambiguous orientations of the initial models. Also, Ornate includes automatic identification of atom types and dynamic routing of the data in the network. Established benchmarks (CASP 11 and CASP 12) demonstrate the state-of-the-art performance of our approach among single-model QA methods. AVAILABILITY AND IMPLEMENTATION: The method is available at https://team.inria.fr/nano-d/software/Ornate/. It consists of a C++ executable that transforms molecular structures into volumetric density maps, and a Python code based on the TensorFlow framework for applying the Ornate model to these maps. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Thanks to the recent advances in structural biology, nowadays 3D structures of various proteins are solved on a routine basis. A large portion of these structures contain structural repetitions or internal symmetries. To understand the evolution mechanisms of these proteins and how structural repetitions affect the protein function, we need to be able to detect such proteins very robustly. As deep learning is particularly suited to deal with spatially organized data, we applied it to the detection of proteins with structural repetitions. RESULTS: We present DeepSymmetry, a versatile method based on 3D convolutional networks that detects structural repetitions in proteins and their density maps. Our method is designed to identify tandem repeat proteins, proteins with internal symmetries, symmetries in the raw density maps, their symmetry order and also the corresponding symmetry axes. Detection of symmetry axes is based on learning 6D Veronese mappings of 3D vectors, and the median angular error of axis determination is less than one degree. We demonstrate the capabilities of our method on benchmarks with tandem-repeated proteins and also with symmetrical assemblies. For example, we have discovered about 7800 putative tandem repeat proteins in the PDB. AVAILABILITY AND IMPLEMENTATION: The method is available at https://team.inria.fr/nano-d/software/deepsymmetry. It consists of a C++ executable that transforms molecular structures into volumetric density maps, and a Python code based on the TensorFlow framework for applying the DeepSymmetry model to these maps. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Motivation: The computational prediction of a protein structure from its sequence generally relies on a method to assess the quality of protein models. Most assessment methods rank candidate models using heavily engineered structural features, defined as complex functions of the atomic coordinates. However, very few methods have attempted to learn these features directly from the data. Results: We show that deep convolutional networks can be used to predict the ranking of model structures solely on the basis of their raw three-dimensional atomic densities, without any feature tuning. We develop a deep neural network that performs on par with state-of-the-art algorithms from the literature. The network is trained on decoys from the CASP7 to CASP10 datasets and its performance is tested on the CASP11 dataset. Additional testing on decoys from the CASP12, CAMEO and 3DRobot datasets confirms that the network performs consistently well across a variety of protein structures. While the network learns to assess structural decoys globally and does not rely on any predefined features, it can be analyzed to show that it implicitly identifies regions that deviate from the native structure. Availability and implementation: The code and the datasets are available at https://github.com/lamoureux-lab/3DCNN_MQA. Supplementary information: Supplementary data are available at Bioinformatics online.

Motivation: The root mean square deviation (RMSD) is one of the most used similarity criteria in structural biology and bioinformatics. Standard computation of the RMSD has a linear complexity with respect to the number of atoms in a molecule, making RMSD calculations time-consuming for the large-scale modeling applications, such as assessment of molecular docking predictions or clustering of spatially proximate molecular conformations. Previously, we introduced the RigidRMSD algorithm to compute the RMSD corresponding to the rigid-body motion of a molecule. In this study, we go beyond the limits of the rigid-body approximation by taking into account conformational flexibility of the molecule. We model the flexibility with a reduced set of collective motions computed with e.g. normal modes or principal component analysis. Results: The initialization of our algorithm is linear in the number of atoms and all the subsequent evaluations of RMSD values between flexible molecular conformations depend only on the number of collective motions that are selected to model the flexibility. Therefore, our algorithm is much faster compared to the standard RMSD computation for large-scale modeling applications. We demonstrate the efficiency of our method on several clustering examples, including clustering of flexible docking results and molecular dynamics (MD) trajectories. We also demonstrate how to use the presented formalism to generate pseudo-random constant-RMSD structural molecular ensembles and how to use these in cross-docking. Availability and implementation: We provide the algorithm written in C++ as the open-source RapidRMSD library governed by the BSD-compatible license, which is available at http://team.inria.fr/nano-d/software/RapidRMSD/. The constant-RMSD structural ensemble application and clustering of MD trajectories is available at http://team.inria.fr/nano-d/software/nolb-normal-modes/. Supplementary information: Supplementary data are available at Bioinformatics online.

MOTIVATION: Docking prediction algorithms aim to find the native conformation of a complex of proteins from knowledge of their unbound structures. They rely on a combination of sampling and scoring methods, adapted to different scales. Polynomial Expansion of Protein Structures and Interactions for Docking (PEPSI-Dock) improves the accuracy of the first stage of the docking pipeline, which will sharpen up the final predictions. Indeed, PEPSI-Dock benefits from the precision of a very detailed data-driven model of the binding free energy used with a global and exhaustive rigid-body search space. As well as being accurate, our computations are among the fastest by virtue of the sparse representation of the pre-computed potentials and FFT-accelerated sampling techniques. Overall, this is the first demonstration of a FFT-accelerated docking method coupled with an arbitrary-shaped distance-dependent interaction potential. RESULTS: First, we present a novel learning process to compute data-driven distant-dependent pairwise potentials, adapted from our previous method used for rescoring of putative protein-protein binding poses. The potential coefficients are learned by combining machine-learning techniques with physically interpretable descriptors. Then, we describe the integration of the deduced potentials into a FFT-accelerated spherical sampling provided by the Hex library. Overall, on a training set of 163 heterodimers, PEPSI-Dock achieves a success rate of 91% mid-quality predictions in the top-10 solutions. On a subset of the protein docking benchmark v5, it achieves 44.4% mid-quality predictions in the top-10 solutions when starting from bound structures and 20.5% when starting from unbound structures. The method runs in 5-15 min on a modern laptop and can easily be extended to other types of interactions. AVAILABILITY AND IMPLEMENTATION: https://team.inria.fr/nano-d/software/PEPSI-Dock CONTACT: [email protected].