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Persistent homology analysis of protein structure, flexibility and folding
Kelin Xia, Guo-Wei Wei
TL;DR
Understanding protein structure, function, dynamics, and folding remains difficult, while existing models can be computationally expensive. This paper applies persistent homology to construct molecular topological fingerprints and quantitative filtrations, reporting folding-related correlations and topology-based analyses of flexibility and cutoff distances. Its stated scope is bounded by the computational cost of all-atom, long-time simulations and by solvent-model design considerations.
Problem
Protein structure–function analysis is challenging, and all-atom models with long-time integrations are prohibitively expensive for real-time protein dynamics.
Method
The study uses persistent homology, molecular topological fingerprints, multiple molecular representations, slicing, and distance-based filtrations to analyze protein structure, flexibility, and folding.
Results
A linear regression gives correlation coefficients of about 0.947 and 0.944 between folding-related accumulation-bar-length measures and total energy.
Takeaways & Limitations
The work uses persistent topological invariants to quantitatively analyze protein flexibility, cutoff distances, folding evolution, and folding stability.
Takeaways & Limitations
Real-time protein dynamics remain constrained by the computational expense of all-atom, long-time integrations, while implicit-solvent SMD models can miss friction terms and require careful water-environment design.
Abstract
from arXiv · showhide
Proteins are the most important biomolecules for living organisms. The understanding of protein structure, function, dynamics and transport is one of most challenging tasks in biological science. In the present work, persistent homology is, for the first time, introduced for extracting molecular topological fingerprints (MTFs) based on the persistence of molecular topological invariants. MTFs are utilized for protein characterization, identification and classification. The method of slicing is proposed to track the geometric origin of protein topological invariants. Both all-atom and coarse-grained representations of MTFs are constructed. A new cutoff-like filtration is proposed to shed light on the optimal cutoff distance in elastic network models. Based on the correlation between protein compactness, rigidity and connectivity, we propose an accumulated bar length generated from persistent topological invariants for the quantitative modeling of protein flexibility. To this end, a correlation matrix based filtration is developed. This approach gives rise to an accurate prediction of the optimal characteristic distance used in protein B-factor analysis. Finally, MTFs are employed to characterize protein topological evolution during protein folding and quantitatively predict the protein folding stability. An excellent consistence between our persistent homology prediction and molecular dynamics simulation is found. This work reveals the topology-function relationship of proteins.
1 Introduction
Protein structure–function analysis is central but geometry-based models can be computationally expensive and retain excessive detail. The paper develops persistent-homology fingerprints and related filtrations to characterize structure, quantify flexibility, and study folding.
- Protein structure–function understanding is a central biological challenge, with the ultimate goal of predicting functions from known structures.
- All-atom models and long-time integrations create too many degrees of freedom for real-time protein-dynamics applications.
- Geometry-based models can be computationally expensive, whereas many biomolecular functions require qualitative connectivity information rather than full geometric detail.
- Persistent homology tracks topological features across spatial scales through filtration and persistence, while retaining information about topological events.
- The study introduces molecular topological fingerprints for protein characterization, identification, classification, and topology–function analysis.
- All-atom and coarse-grained fingerprints, slicing, and a distance-based filtration are proposed for analyzing structural origins and optimal cutoff distances.
2 Theory and algorithm
The paper constructs simplicial complexes from molecular point sets and tracks their topological features across nested filtrations using persistent homology. Distance-based and correlation-based filtrations connect geometric or physical information to persistent Betti-number signatures, while examples illustrate protein-relevant topological analysis and its scope.
- Simplicial complexes: A simplicial complex combines vertices, edges, triangles, and higher-dimensional simplices while requiring every simplex face to be included.Intersections of simplices must be empty or shared faces, and the complex dimension is its maximal simplex dimension.
- Homology and Betti numbers: Betti numbers quantify isolated components, one-dimensional loops, and two-dimensional voids through the ranks of homology groups.For torsion-free homology, βk is the rank of Hk; specifically, β0, β1, and β2 describe components, circles, and voids.
- Persistent homology: Persistent homology represents point-set topology across scales by tracking long-lasting homology features through a filtration.Filtration produces nested subcomplexes whose homology groups reveal topological features; persistence emphasizes features that survive across scales.
- Filtration construction: Distance-based filtration grows balls around atoms, and overlapping balls generate increasingly higher-dimensional simplices in nested complexes.The distance matrix supplies pairwise atom distances for constructing the filtration.
- Examples and scope: In toy and molecular examples, persistent bars capture connectivity, rings, and voids, including the icosahedron’s central void and fullerene C70’s ring structure.The icosahedron has 12 initial β0 bars and one persisting β2 bar; C70 has 70 β0 bars, 36 β1 bars, and a persisting central β2 bar.
3 Persistent homology analysis of proteins
Persistent homology is applied to protein structure, flexibility, and folding through topological invariants, with complementary all-atom and coarse-grained representations. The analysis also links topological connectivity to folding behavior and stability.
- Persistent homology analyzes protein structure, flexibility, and folding through topological invariants.
- All-atom and coarse-grained representations are compared for protein structure analysis.The comparison targets the distinct structural information captured at different representation scales.
- Cutoff distance is identified as a shared parameter across several protein flexibility and rigidity methods.The methods include NMA, GNM, ENM, ANM, MND, and FRI.
- Accumulated bar length continuously decreases during simulated protein unfolding, representing decreasing total connectivity.The analysis uses configurations generated by constant-velocity pulling in steered molecular dynamics.
- Protein folding stability is related to topological connectivity during unfolding analysis.
3.1 Topological fingerprints of proteins
Molecular topological fingerprints reveal protein structural features, while coarse-grained models simplify global patterns and slicing links bars to geometric loops and holes. Alpha helices, beta sheets, and beta barrels show interpretable topological signatures, although all-atom fingerprints can become complicated.
- Structural representations: Alpha helices and beta sheets are analyzed as basic protein structural components using all-atom and coarse-grained models.The all-atom model includes all atom types, whereas the coarse-grained model reduces degrees of freedom.
- Alpha-helix fingerprints: The all-atom alpha-helix fingerprint is complicated because many atoms surround the main-chain backbone.Consequently, its barcode does not clearly display the backbone’s loop-type pattern.
- Alpha-helix fingerprints: Slicing shows that each four Cα atoms in the alpha helix form a one-dimensional loop corresponding to a β1 bar.Adding Cα atoms sequentially creates additional loops and bars, explaining the 16-bar pattern.
- Beta-sheet fingerprints: The all-atom beta-sheet fingerprint also has a complicated pattern because of excessive residual atoms.The example contains beta sheets with eight residues each.
- Alpha-helix fingerprints: 19 Cα atoms in the coarse-grained alpha helix produce 19 β0 bars and 16 β1 bars.The β0 bar length is around 3.8 Å, corresponding to the average distance between adjacent Cα atoms.
- Beta-sheet fingerprints: Two beta-sheet strands generate 7 β1 bars, while adjacent Cα atoms across strands form the corresponding one-dimensional circles.The β0 panel contains 16 Cα atoms from 8 residue pairs.
- Topological interpretation: The beta-barrel beta-sheet fingerprint contains one long β1 bar for the global hole and 97 predicted bars overall.The persistent homology calculation shows 98 β1 bars, with one extremely short-lived bar barely visible.
3.2 Persistent homology analysis of protein flexibility
The paper uses persistent homology to analyze protein flexibility and connectivity, linking accumulated topological features with characteristic distances and elastic-network behavior.
- Framework: Persistent homology provides a topology-based framework for protein flexibility analysis alongside MND and FRI correlation-matrix filtrations.The approach targets the topology-function relationship of proteins.
- Protein folding: MND transverse stability increases during folding from disordered conformations to the well-defined natural structure, indicating progressively more unified dynamics.The folded state is associated with an intrinsically low-dimensional manifold.
- Cutoff-distance analysis: Cutoff distances below 5 Å oversimplify elastic networks, whereas distances above 14 Å introduce excessive global connections and reduce prediction accuracy.The latter effect is especially noted for small proteins.
- Cutoff-distance analysis: Cutoff distances around 7–9 Å capture major global β1 features and provide reasonable predictions for the analyzed proteins.The proposed persistent-homology analysis is used to guide this choice.
- Accumulated bar lengths: Accumulation bar lengths A_j sum the lengths of persistent bars for each Betti number, with A_1 used to quantify connectivity across characteristic distances.For protein 1YZM, η was varied from 1 Å to 21 Å.
- Accumulated bar lengths: For protein 1YZM, correlation coefficients and A_1 both peak near η = 6 Å before decreasing at larger characteristic distances.The shared trend connects persistent-bar accumulation with FRI-based flexibility analysis.
3.3 Persistent homology analysis of protein folding
Persistent homology tracks protein topological changes during SMD-generated unfolding and links them quantitatively to energy and stability. As unfolding proceeds, topological connectivity and Betti-number measures decrease while total energy increases.
- 3.3 Persistent homology analysis of protein folding: Protein folding and unfolding involve substantial changes in local and global topology, which persistent homology can track through evolving topological invariants.The approach analyzes intermediate configurations extracted from constant-velocity steered molecular dynamics.
- 3.3 Persistent homology analysis of protein folding: As protein 1I2T unfolds, topological connectivity decreases and total energy increases across the SMD configuration sequence.The unfolding frames span configurations 1, 3, 5, 7, 10, 20, and 30.
- 3.3 Persistent homology analysis of protein folding: A first-Betti-number measure alone correlates with total energy at about 0.89 across 31 configurations, but the relation is not highly accurate.The authors therefore use the negative accumulated β1 bar length for a more robust quantitative model.
- 3.3 Persistent homology analysis of protein folding: 0.947 and 0.944 are the correlation coefficients between negative accumulated β1 bar length and total energy for distance-based and correlation-matrix filtrations, respectively.The correlation-matrix filtration uses an exponential kernel with optimized parameters κ = 2 and η = 7˚A.
- 3.3 Persistent homology analysis of protein folding: 0.972 and 0.971 are the correlation coefficients obtained for energy prediction using distance-based and correlation-matrix filtrations, respectively.These results confirm the linear relation between negative accumulated first-Betti-number bar length and total energy for extracted intermediate structures.
4 Concluding remarks
The paper develops persistent-homology representations and quantitative models for protein topology, flexibility, and folding. Its applications connect topological fingerprints and accumulated bar lengths with geometric features, rigidity, B-factor modeling, and folding stability.
- 4 Concluding remarks: Persistent homology provides a multiscale representation of topological features through filtration and persistence across spatial scales.The paper introduces this framework for protein analysis rather than relying only on metric-free computational homology.
- 4 Concluding remarks: The slicing method links individual topological invariants to their geometric origins in alpha helices, beta sheets, and beta barrels.This connection is used to deepen understanding of the topology-function relationship.
- 4 Concluding remarks: Molecular topological fingerprints characterize, identify, and classify proteins while representing local and global geometric features through all generated persistent bars.The authors treat short-lived and long-lived bars as informative components of a protein-specific fingerprint.
- 4 Concluding remarks: A cutoff-distance filtration connects topological diagrams with the optimal cutoff distance for Gaussian network model B-factor prediction.The construction relates protein compactness and topological connectivity to rigidity and flexibility.
- 4 Concluding remarks: The maximum accumulated first-Betti-number bar length accurately predicts the optimal characteristic distance for FRI analysis of protein temperature factors.The accumulated bar length sums all first-Betti-number bars and incorporates geometric information through correlation-matrix filtration.
- 4 Concluding remarks: Negative accumulated first-Betti-number bar length quantitatively predicts protein total energy during SMD-generated folding and unfolding.The model tracks topological evolution and protein folding stability through the relation between topology and energy.