[Paper Review] Multiscale persistent functions for biomolecular structure characterization

Kelin Xia, Zhiming Li+1 more|arXiv (Cornell University)|Dec 26, 2016

Topological and Geometric Data Analysis54 references4 citations

TL;DR

This paper introduces multiscale persistent functions, particularly multiscale persistent entropy, by integrating multiscale rigidity functions with persistent homology to characterize biomolecular structures. The method enables natural clustering of protein conformations and achieves accurate classification of all-alpha, all-beta, and mixed proteins, with a novel protein structure index (PSI) quantifying structural regularity based on topological entropy.

ABSTRACT

In this paper, we introduce multiscale persistent functions for biomolecular structure characterization. The essential idea is to combine our multiscale rigidity functions with persistent homology analysis, so as to construct a series of multiscale persistent functions, particularly multiscale persistent entropies, for structure characterization. To clarify the fundamental idea of our method, the multiscale persistent entropy model is discussed in great detail. Mathematically, unlike the previous persistent entropy or topological entropy, a special resolution parameter is incorporated into our model. Various scales can be achieved by tuning its value. Physically, our multiscale persistent entropy can be used in conformation entropy evaluation. More specifically, it is found that our method incorporates in it a natural classification scheme. This is achieved through a density filtration of a multiscale rigidity function built from bond and/or dihedral angle distributions. To further validate our model, a systematical comparison with the traditional entropy evaluation model is done. It is found that our model is able to preserve the intrinsic topological features of biomolecular data much better than traditional approaches, particularly for resolutions in the mediate range. Moreover, our method can be successfully used in protein classification. For a test database with around nine hundred proteins, a clear separation between all-alpha and all-beta proteins can be achieved, using only the dihedral and pseudo-bond angle information. Finally, a special protein structure index (PSI) is proposed, for the first time, to describe the "regularity" of protein structures. Essentially, PSI can be used to describe the "regularity" information in any systems.

Motivation & Objective

• To develop a multiscale topological framework for characterizing complex biomolecular structures beyond traditional methods.
• To address the limitations of quasiharmonic and Cartesian coordinate-based entropy estimation in capturing intrinsic topological features.
• To embed natural clustering and structural regularity information into topological invariants using persistent homology.
• To propose a protein structure index (PSI) that quantifies structural regularity based on $β_0$ persistent entropy.
• To validate the method on a 900-protein database and demonstrate superior preservation of topological features in the intermediate resolution range.

Proposed method

• Combines multiscale rigidity functions—derived from bond and dihedral angle distributions—with density filtration to generate multiscale barcode representations.
• Introduces a resolution parameter in kernel functions to enable tuning across multiple scales, allowing multiscale analysis.
• Defines multiscale persistent functions, particularly multiscale persistent entropy, on the resulting barcode spaces.
• Uses Gaussian kernel with scale parameter $\eta = 5^\circ$ to compute topological entropy for PSI evaluation.
• Applies persistent homology to $\beta_0$, $\beta_1$, and $\beta_2$ Betti numbers, focusing on $\beta_0$ for PSI construction.
• Employs density filtration over multiscale rigidity functions to generate hierarchical topological features.

Experimental results

Research questions

• RQ1 Can multiscale persistent functions preserve intrinsic topological features of biomolecular data better than traditional entropy models?
• RQ2 Can the proposed method naturally classify protein conformations based on topological structure?
• RQ3 Can persistent entropy derived from $\beta_0$ barcodes serve as a reliable measure of structural regularity in proteins?
• RQ4 How does the resolution parameter influence the performance of topological characterization in intermediate-scale biomolecular features?
• RQ5 Can the protein structure index (PSI) effectively distinguish proteins with high loop/disorder content from those with stable secondary structures?

Key findings

• The proposed multiscale persistent entropy model preserves intrinsic topological features of biomolecular data significantly better than traditional methods, especially in the intermediate resolution range.
• For a test database of ~900 proteins, the method achieved clear separation between all-alpha and all-beta proteins using only dihedral and pseudo-bond angle information.
• Mixed-alpha-and-beta proteins showed persistent entropy values in the intermediate range, with only a few overlapping cases, indicating robust classification.
• The protein structure index (PSI) successfully quantifies structural regularity: proteins with more loops and disordered regions have higher PSI values, while those rich in alpha-helices and beta-sheets have lower PSI.
• PSI values ranged from 0.000 (most regular) to 2.504 (most irregular) across a 110-protein test set, confirming its sensitivity to structural organization.
• High-dimensional topological entropies from $\beta_1$ and $\beta_2$ barcodes are expected to yield richer information, though not explored in this work.