Ten numbers from the Laplace-Beltrami spectrum facilitate training-free classification of protein structures based on surface shape

Abstract

The comparison of protein structures is necessary to determine molecular functions and evolutionary relationships and can also facilitate drug design and the prediction of separation options. The number of available protein structures is rapidly increasing, driven by experimental determination but also prediction methods such as AlphaFold. In this context, traditional structure comparison based on atomic superposition becomes computationally inefficient. To enable scalable analysis and surface shape comparison, alternative approaches represent protein structures using compact, fixed-length vectors known as descriptors. Current vectors often contain hundreds of entries, such as three-dimensional Zernike descriptors composed of 121 entries, or combinations of molecular and geometric properties. Here, we propose to use the Laplace-Beltrami spectrum, a mathematical representation of object surfaces derived from the eigenvalues of the Laplace-Beltrami operator, as an efficient option to capture protein structural properties and facilitate structural classification. Specifically, protein surfaces are encoded using only the first 10 non-zero eigenvalues of this spectrum. We used the SHREC 2025 dataset as a benchmark and achieved 85.8% accuracy on the original 97 protein structure classes and 97.7% accuracy on the homology-grouped 45 classes. Our method is fast, requiring ~75 min computation time for the 11,555 protein surface meshes of the SHREC 2025 dataset on a conventional AMD Ryzen 7 5700X 8-Core processor with 32 GB RAM. The approach can easily be applied to large protein structure databases because it is readily parallelizable and allows the incorporation of other descriptors, including surface features such as charge. This rapid screening tool can therefore be used to identify proteins with related surface shapes independent of sequence homology.