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Persistent local Laplacian prediction of protein-ligand binding affinities

Jian Liu, Hongsong Feng

Abstract

Accurate prediction of protein-ligand binding affinity remains a central challenge in structure-based drug discovery. The effectiveness of machine learning models critically depends on the quality of molecular descriptors, for which advanced mathematical frameworks provide powerful tools. In this work, we employ a novel mathematical theory, termed the persistent local Laplacian (PLL), to construct molecular descriptors that capture localized geometric and topological features of biomolecular structures. The PLL framework addresses key limitations of traditional topological data analysis methods, such as persistent homology and the persistent Laplacian, which are often insensitive to local structural variations, while maintaining high computational efficiency. The resulting molecular descriptors are integrated with advanced machine learning algorithms to develop accurate predictive models for protein-ligand binding affinity. The proposed models are systematically evaluated on three well-established benchmark datasets, demonstrating consistently strong and competitive predictive performance. Computational results show that the PLL-based models outperform existing approaches, highlighting their potential as a powerful tool for drug discovery, protein engineering, and broader applications in science and engineering.

Persistent local Laplacian prediction of protein-ligand binding affinities

Abstract

Accurate prediction of protein-ligand binding affinity remains a central challenge in structure-based drug discovery. The effectiveness of machine learning models critically depends on the quality of molecular descriptors, for which advanced mathematical frameworks provide powerful tools. In this work, we employ a novel mathematical theory, termed the persistent local Laplacian (PLL), to construct molecular descriptors that capture localized geometric and topological features of biomolecular structures. The PLL framework addresses key limitations of traditional topological data analysis methods, such as persistent homology and the persistent Laplacian, which are often insensitive to local structural variations, while maintaining high computational efficiency. The resulting molecular descriptors are integrated with advanced machine learning algorithms to develop accurate predictive models for protein-ligand binding affinity. The proposed models are systematically evaluated on three well-established benchmark datasets, demonstrating consistently strong and competitive predictive performance. Computational results show that the PLL-based models outperform existing approaches, highlighting their potential as a powerful tool for drug discovery, protein engineering, and broader applications in science and engineering.
Paper Structure (22 sections, 2 theorems, 23 equations, 4 figures, 3 tables)

This paper contains 22 sections, 2 theorems, 23 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Methods
  4. Datasets
  5. Persistent local Laplacian theory
  6. Simplicial complex
  7. Simplicial homology
  8. Simplicial Laplacian
  9. Relative homology
  10. Local homology
  11. Link complex
  12. Local Laplacian
  13. Connection to link Laplacian
  14. Persistent Local Laplacian
  15. Persistent Link Laplacian
  16. ...and 7 more sections

Key Result

Theorem 3.3

Let $\mathcal{K}$ be a finite simplicial complex, and $v \in \mathcal{K}$ a vertex. Then there is a unitary equivalence between the local Laplacian and the corresponding link Laplacian at a shifted dimension. In particular, the harmonic space of $\Delta_k^{\mathcal{K},v}$ is isomorphic to the harmonic space of $\Delta_{k-1}^{\mathop{\mathrm{Lk}}\nolimits_{\mathcal{K}}(v)}$, i.e.,

Figures (4)

  • Figure 1: Conceptual diagram of the persistent local Laplacian (PLL) platform for protein–ligand binding affinity prediction. a) Construction of a Vietoris-Rips-complex–induced filtration from atoms in a protein–ligand complex. b) Generation of nested families of element-specific subcomplexes along distance-based filtration parameters. c) Formation of atom-specific local Laplacian matrices to characterize atomic interactions within protein–ligand subcomplexes. d) Spectral analysis of the local Laplacian matrices to derive discriminative features for protein–ligand subcomplexes. e) Multiscale topological features for protein–ligand complex are established by concatenating features of protein–ligand subcomplexes. f) Utilization of PLL-based features for downstream binding affinity prediction and analysis using gradient-boosting decision tree models.
  • Figure 2: Comparison of the predictive performance of the PLLML model with other published models in terms of the Pearson correlation coefficient ($R$) on three PDBbind datasets.
  • Figure 3: A comparison between the experimental and predicted binding affinities from our PLLML model on three PDBbind datasets.
  • Figure 4: Illustration of persistent local Laplacians (PLL) at one point in a point cloud. (c): a filtration process of the Vietoris-Rips complex constructed from a point cloud of 12 points. Among the 12 points, six lie on a circle of radius $1$ and the other six lie on a circle of radius $0.5$. The points on the smaller circle are obtained by a rotation of $\frac{\pi}{6}$ so that each point on the smaller circle lies on the same radial line from the origin as a corresponding point on the larger circle. (d): Analysis of zero-dimensional spectral information of local Laplacian matrices defined for the red point $v$ over the filtration process. (e): Analysis of one-dimensional persistent Laplacian spectra of PLL defined for point $v$ over the filtration process.

Theorems & Definitions (5)

  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Definition 3.4: Persistent Local Laplacian
  • Theorem 3.5