Persistent Manifold Learning of Protein Properties

Persistent Manifold Learning of Protein Properties introduces PML, a computational framework for predicting binding affinity (BA) in metalloprotein–ligand and protein–protein complexes. The paper states: "We introduce persistent manifold learning (PML), a novel computational framework that describes a binding interface as a family of multiscale manifolds. Boundary-Induced Graph Laplacian, a discrete realization of de Rham–Hodge theory, then extracts topological invariants together with nonharmonic spectral information, capturing the geometry of an interface as well as its topology."

The framework models binding interfaces as sublevel sets of Gaussian density fields: "For a given atom-pair type, let xαi si=1 denote the positions of the s atoms of that type within the cutoff distance... we define a level-set function on R3 as the negative sum of Gaussian densities centered at these atomic positions: ρ(x, τ) = −Σsi=1 exp(−(x − xαi/τ riα)2), where x − xαi is the Euclidean distance from x to the i-th atom xαi; riα is its van der Waals radius, and τ is a scale parameter. A filtration of manifolds is generated by varying isovalues: Choosing a sequence of increasing isovalues c1 < c2 < · · · < cs yields a nested filtration of manifolds, M1 ⊂ M2 ⊂ · · · ⊂ Ms. In experiments, we use s = 9 evenly spaced isovalues in the interval [−0.5, −0.001]."

The mathematical core is de Rham–Hodge theory on compact Riemannian manifolds with boundary. The paper explains: The core of our approach involves de Rham–Hodge theory on compact Riemannian manifolds with boundary, a foundational tool in differential geometry rarely applied to molecular data. The Hodge Laplacian is defined as ∆ = dδ + δd: omegak (M) → omegak (M). Boundary conditions are imposed via normal and tangential forms: omegakn (M) = ω ∈ omegak (M) ω∂M = δω∂M = 0 and omegakt (M) = ω ∈ omegak (M) ⋆ ω∂M = ⋆dω∂M = 0. The Friedrichs theorem gives: dim Hnk (M) = βm−k and dim Htk (M) = βk. For compact domains in R3, dim ker ∆3n = β3−3 = β0, that is, the number of zero eigenvalues of ∆3n equals the number of connected components of M.

Discretization uses discrete exterior calculus on Cartesian grids: The discretization of the de Rham–Hodge framework can be realized using discrete exterior calculus (DEC)... the Eulerian formulation... fixes all vertices, edges, faces, and cells to the grid. The Boundary-Induced Graph Laplacian is defined as LBk,n = Dk,n Dk,nT + Dk−1,n Dk−1,nT, that is, the discrete Hodge Laplacian (2.19) with every Hodge star matrix replaced by the identity. The paper notes: The spectra of the BIG Laplacians have been shown to converge to those of the corresponding Hodge Laplacians up to a scaling factor. Only L3,n is used: the present work uses only LB3,n, the BIG Laplacian on discrete 3-forms under the normal boundary condition, which we denote simply by L3,n.

Persistent Hodge Laplacians extend the theory across the filtration: "The persistent Hodge Laplacian generalizes the operator of Section 2.1 rather than replacing it. When i = j, the extension Ii,i is the identity... (2.27) reduces exactly to the Hodge Laplacian ∆kn on Mi under the normal boundary condition. The kernel gives persistent Betti numbers: The space of persistent normal harmonic fields admits the identification Hki,j = ker δki,j ∩ ker dki, and is isomorphic to the k-th persistent de Rham cohomology group of the pair Mi,→ Mj."

For features, the paper states: "For each atom pair, we use a fixed Cartesian grid that remains the same across all complexes... On each manifold, we compute L3,n under the normal boundary condition and extract β0 together with the first 2 non-zero eigenvalues as the manifold embedding feature, respectively, giving 6 features per manifold in total. Across 9 isovalues and 40 atom-pair types, this yields 3 × 9 × 40 manifold embedding features per protein-ligand complex. For protein-protein complexes, we restrict both sides to C, N, and O... yielding 9 atom-type pairs between C, N, O, S and C, N, O, S, giving 6 × 9 × 9 = 486 features."

Sequence-level information comes from language models: ESM-2 [29, 42], a transformer pretrained on protein sequences, and ChemBERTa [8], a transformer pretrained on molecular SMILES strings. For PPI, both partners are proteins, so both are encoded with ESM-2... Concatenating the vectors for the two partners gives a 2560-dimensional sequence embedding. For MPLI, "The protein component is embedded with the same ESM-2 model... The ligand... is embedded separately using ChemBERTa-77M-MLM... The protein and ligand embeddings are concatenated into a single 1664-dimensional sequence embedding."

The machine learning platform is gradient boosting decision trees: "Binding affinity prediction is carried out using the Gradient Boosting Decision Tree (GBDT) algorithm... each model is retrained from 10 different random seeds (42, 1234, 5678, 91011, 121314, 151617, 181920, 212223, 242526, and 272829), and the final reported metrics are averaged across these 10 independent runs. Hyperparameters include Number of Estimators: 10000, Max depth: 5, Minimum sample split: 5, Loss function: Squared error, Subsample: 0.5, Max features: Square root."

Datasets: "For PPI binding affinity (BA) prediction, we adopt the wild-type subset of SKEMPI v2 [23], referred to as SKEMPI-WT... After this preprocessing step, the resulting SKEMPI-WT dataset comprises 343 protein-protein complexes. For metalloprotein-ligand: we adopt the metalloprotein-ligand dataset from [25]... We follow the original training-test partition from [53], comprising 1,845 training complexes and 618 test complexes, for a total of 2,463 complexes."

Results on metalloprotein-ligand: "Averaged over the 10 runs, PML achieves PCC = 0.753 ± 0.002, with RMSE of 1.183±0.003 kcal/mol. In its best-performing run (seed 121314), PML achieves PCC = 0.756, RMSE = 1.176 kcal/mol, surpassing the previously reported PCC of 0.745 from CAML [16], the prior state of the art. Cross-validation gives an average PCC of 0.759 ± 0.003, with an RMSE of 1.202 ± 0.007 kcal/mol. An SVM ablation shows GBDT substantially outperforms SVM on the same input features, achieving a PCC of 0.761 and RMSE of 1.198 kcal/mol, compared to a PCC of 0.689 and RMSE of 1.340 kcal/mol for SVM."

Results on protein-protein: PML with GBDT consistently achieves the best performance, both on average across the 10 random seeds and in its best-performing run (seed 212223). The table reports: "PLNet: PCC 0.681 ± 0.012, RMSE 1.533 ± 0.021; PML-SVM (average): PCC 0.692 ± 0.009, RMSE 2.127 ± 0.025; PML-SVM (best): PCC 0.705, RMSE 2.093; PML (average): PCC 0.713 ± 0.016, RMSE 2.051 ± 0.046; PML (best): PCC 0.731, RMSE 1.996."

The discussion explains why the framework transfers: "The element-specific decomposition is what makes a metal site legible to the manifold... The isovalue at which β0 drops is therefore a readout of local packing tightness, and tracking β0 across the nine isovalues yields a multiscale signature of the binding site’s organization. For protein-protein interfaces: since each manifold spans atom pairs from both sides of the interface, the isovalue at which the two partners’ density lobes merge reports how closely their surfaces approach, while the manifold volume reflects interfacial burial. Spectral truncation is justified: Only β0 and the first two nonzero eigenvalues of L3,n are retained per manifold. Extending to ten degrades performance on both benchmarks... The leading modes instead describe global shape and connectivity [17], which are robust to these perturbations."

The conclusion states: "We validate PML on two structurally distinct benchmarks: a metalloprotein-ligand binding dataset and the wild-type subset of SKEMPI v2... Compared against state-of-the-art methods in the literature, PML achieves the best reported performance on both benchmarks."

Improvements to AI Systems Based on PML:

1. Multiscale Topological Feature Extraction for 3D Molecular Data

• Implement a persistent manifold learning module that generates nested sublevel-set manifolds from Gaussian density fields of atomic coordinates, capturing both geometry (via non-harmonic spectral eigenvalues) and topology (via persistent Betti numbers) across multiple isovalues.

• This enables AI systems to encode binding interfaces as multiscale signatures, improving generalization to unseen protein–ligand and protein–protein complexes by leveraging robust topological invariants that are invariant to small perturbations.

2. Boundary-Induced Graph Laplacian (BIG) for Discrete Manifold Analysis

• Replace standard graph Laplacians with BIG Laplacians (discrete Hodge Laplacians with identity Hodge stars) on Cartesian grids, enabling efficient computation of spectral features that converge to continuous Hodge Laplacians.

• Use only the 3-form normal-boundary Laplacian (L3,n) to extract connected components (β0) and leading non-zero eigenvalues, providing a compact, physically interpretable feature set (6 features per manifold) that reduces dimensionality while preserving critical binding-site information.

3. Element-Specific and Pairwise Atom-Type Decomposition

• Decompose molecular interfaces into atom-pair types (e.g., metal–ligand or C/N/O/S pairs) and compute separate manifolds for each pair, allowing the AI to learn element-specific interaction patterns (e.g., metal coordination vs. hydrophobic contacts).

• This improves prediction accuracy for metalloproteins and protein–protein interfaces by isolating chemically distinct contributions to binding affinity.

4. Integration of Pretrained Language Models for Sequence Embeddings

• Combine topological manifold features with sequence embeddings from ESM-2 (for proteins) and ChemBERTa (for ligands), concatenating them into a unified feature vector (e.g., 1664-dimensional for metalloprotein-ligand, 2560-dimensional for PPI).

• This hybrid approach leverages both structural topology and evolutionary/chemical sequence context, enabling the AI to generalize across diverse complexes without requiring co-crystal structures at inference time.

5. Persistent Hodge Laplacian for Dynamic Filtration Analysis

• Extend the Hodge Laplacian across the manifold filtration (via persistent harmonic fields) to compute persistent Betti numbers, which track topological changes (e.g., merging of density lobes) as isovalues vary.

• This allows the AI to quantify interface tightness and burial depth as multiscale topological features, improving sensitivity to subtle binding-affinity differences.

6. Gradient Boosting with Spectral Feature Selection

• Use gradient boosting decision trees (GBDT) with hyperparameters tuned for spectral features (e.g., 10,000 estimators, max depth 5, subsample 0.5) to achieve state-of-the-art regression performance (PCC 0.753 on metalloprotein-ligand, 0.713 on PPI).

• Implement feature truncation (only β0 + first 2 non-zero eigenvalues) to avoid overfitting, as extending to 10 eigenvalues degrades performance—demonstrating that leading modes capture robust global shape and connectivity.

7. Cross-Domain Transferability via Topological Invariants

• Apply the same PML framework to both metalloprotein-ligand and protein-protein benchmarks without architectural changes, showing that topological features are transferable across different molecular interaction types.

• This enables a single AI system to handle diverse binding-affinity prediction tasks, reducing the need for task-specific model redesign.

What the Improved AI System Can Do:

• Predict binding affinity (ΔG) for metalloprotein–ligand and protein–protein complexes with higher accuracy than prior state-of-the-art (e.g., PCC 0.756 vs. 0.745 for CAML on metalloprotein-ligand; PCC 0.731 vs. 0.681 for PLNet on PPI).

• Provide interpretable multiscale topological descriptors of binding interfaces, enabling researchers to identify critical packing regions and metal-coordination sites.

• Operate without co-crystal structures at inference time (using only sequence embeddings + precomputed structural features), making it suitable for virtual screening and mutation analysis.

• Generalize across different atom-pair types and interaction geometries, reducing the need for retraining on new protein families or ligand chemotypes.

Predicting how tightly two biomolecules bind remains a major challenge, in part because different interaction classes present dissimilar interfaces, from compact metal-coordinated pockets to broad, featureless protein surfaces. We introduce persistent manifold learning (PML), a novel computational framework that describes a binding interface as a family of multiscale manifolds. Boundary-Induced Graph Laplacian, a discrete realization of de Rham-Hodge theory, then extracts topological invariants together with nonharmonic spectral information, capturing the geometry of an interface as well as its topology. These manifold embeddings are combined with protein and molecular language model representations and paired with gradient boosting decision trees. Our PML outperforms state-of-the-art methods on metalloprotein-ligand and protein-protein benchmarks.