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A Pipeline for Data-Driven Learning of Topological Features with Applications to Protein Stability Prediction

Amish Mishra, Francis Motta

arXiv:2408.04847v1stat.MLcs.LGphysics.data-an

TL;DR

The paper asks whether interpretable topological features can provide additional structural determinants for protein stability beyond previously identified biophysical and energetic factors. It learns features from protein persistence diagrams and reports 92%-99% of SME-based model performance, with important learned features correlated with important SME features.

  • Problem

    The paper seeks additional interpretable topological determinants of protein stability beyond previously exposed biophysical and energetic determinants.

    Source: §1 · pp. 1–2

  • Method

    The approach uses CDER to identify low-entropy regions distinguishing persistence diagrams of stable and unstable designs, then uses learned distributional coordinates to vectorize diagrams for prediction models.

    Source: §2.3 · p. 7, §4.1 · p. 11

  • Results

    Topological-feature models achieved 92%-99% of the performance of SME-feature models when modeling stability across four secondary-structure topologies.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The authors link high correlations between important learned topological and SME features to why topological models classify proteins nearly as well as SME-only models.

    Source: §5 · p. 22

    The study's conclusions are limited to a small number of designed topologies, and features associated with proteins having more ambiguous stability scores remain underexplored.

    Source: §5 · p. 22

Abstract

from arXiv · show

In this paper, we propose a data-driven method to learn interpretable topological features of biomolecular data and demonstrate the efficacy of parsimonious models trained on topological features in predicting the stability of synthetic mini proteins. We compare models that leverage automatically-learned structural features against models trained on a large set of biophysical features determined by subject-matter experts (SME). Our models, based only on topological features of the protein structures, achieved 92%-99% of the performance of SME-based models in terms of the average precision score. By interrogating model performance and feature importance metrics, we extract numerous insights that uncover high correlations between topological features and SME features. We further showcase how combining topological features and SME features can lead to improved model performance over either feature set used in isolation, suggesting that, in some settings, topological features may provide new discriminating information not captured in existing SME features that are useful for protein stability prediction.

1 Introduction

Protein stability matters for maintaining functional tertiary structures, and this study searches for additional interpretable topological determinants beyond previously exposed biophysical and energetic ones. It learns topological features from protein structures and reports performance near that of SME-based models, alongside strong correlations between important topological and SME features.

1 Introduction

  • Protein-stability ML had used sequence and atomic-interaction features, while topological methods had been applied to protein classification, binding pockets, and function prediction.The cited examples include persistent-homology and geometric-deep-learning approaches.Source: §1 · pp. 1–2
  • Protein stability supports functional tertiary structure, and prior work exposed biophysical and energetic determinants that motivate searching for additional interpretable topological features.The introduction frames stability determinants as relevant to structural biology, human health, and designing functionally stable proteins.Source: §1 · pp. 1–2
  • Topological-feature models achieved 92%-99% of the performance of SME-feature models in modeling stability across four protein secondary-structure topologies.The reported topologies are ααα, βαββ, αββα, and ββαββ.Source: §1 · p. 2
  • Adding topological features significantly improved classification average precision for αββα and ββαββ protein design topologies.The passage reports a statistically significant improvement but provides no numerical effect size.Source: §1 · p. 2
  • Strong correlations (|r| > 0.9) linked learned topological features with SME features independently important for classifying stable and unstable designs.The reported pattern is illustrated in Figures 6 to 8.Source: §1 · p. 2
  • The approach uses CDER to find topological structures overrepresented in stable or unstable designs, then vectorizes persistence diagrams for stability-model training.It targets interpretable topological differences between the most and least stable designed mini-proteins.Source: §1 · pp. 2–3

2 Preliminaries

The section introduces persistence diagrams as records of topological features across scale and presents CDER as a data-driven way to select discriminating diagram regions and vectorize them for supervised learning.

2 Preliminaries

  • CDER selects discriminating persistence-diagram regions using local label entropy, letting the data determine which regions become features rather than choosing vectorization parameters by model performance.The approach targets compact regions where persistence pairs distinguish the classes of interest.Source: §2 · p. 3

2.1 Persistent Homology

  • Persistent homology tracks topological features as nested spaces grow, while persistence-diagram points encode each feature’s birth and death scales.The study uses weighted alpha filtration to incorporate both atom locations and atomic identities.Source: §2.1 · p. 4

2.2 PH and CDER

  • Figure 1 illustrates how CDER identifies diagram regions that distinguish noisy sphere and figure-8 samples while ignoring regions shared by both classes.The learned regions yield topological features for downstream machine-learning models.Source: §2.2 · pp. 4–5, Figure 1 · p. 5

2.3 Cover-Tree Differencing via Entropy Reduction (CDER)

  • CDER uses cover trees and entropy to find compact, low-entropy regions; its weighting treats labels equally and each cloud within a label as equally likely regardless of cloud size.The study uses the parsimonious version of CDER in its applications.Source: §2.3 · pp. 6–7
  • CDER converts learned regions into distributional coordinates that map a point cloud to a feature vector, which a supervised learner evaluates using all coordinates.In this study, the distributional coordinates are multidimensional Gaussians.Source: §2.3 · pp. 7–8

3 Data Collection and Preprocessing

The study uses experimentally scored synthetic mini-proteins spanning four secondary-structure topologies, then converts their atomic structures into persistence diagrams. It also examines diagram differences between stable and unstable EEHEE proteins as a preprocessing-stage visualization.

3.1 Data Sources and Background

  • The four designed-protein topologies are HHH, EHEE, HEEH, and EEHEE; the paper distinguishes these biological secondary-structure topologies from mathematical topology of protein conformations.HHH is described as relatively modest in intricacy, whereas EEHEE has four loops and mixed parallel and antiparallel beta-sheets.Source: §3.1 · p. 8
  • Stability was measured by trypsin and chymotrypsin degradation assays, with the lower stability score used to assign an ec50 value.The final stability score is the measured ec50 fold change over an inferred unfolded-state baseline, using the improved convolutional neural network baseline.Source: §3.1 · pp. 8–9

3.2 Data Preprocessing

  • The dataset contains 16,174 synthetic mini-proteins with experimentally determined stability scores and associated atomic-coordinate files.The proteins span EEHEE, EHEE, HEEH, and HHH topologies; the numbers scoring above 1 are 579, 679, 118, and 1,346, respectively.Source: §3.2 · pp. 9–10
  • Atomic coordinates and van der Waals radii were used to construct persistence diagrams, with birth-death pairs transformed to birth-persistence pairs separately by homology dimension.For H1 and H2 the transformation is (b, d) → (b, d − b); for H0 it is (b, d) → (d, 0).Source: §3.2 · p. 10
  • Unstable EEHEE proteins tend to show H1 points with small birth and high persistence, while stable proteins have more low-birth, low-persistence points.The authors interpret the former pattern as consistent with larger loop structures; other diagram regions have similar stable and unstable densities and are uninformative for discrimination.Source: §3.2 · p. 10

4 CDER for Feature Engineering

The section uses CDER to learn topological features from protein persistence diagrams, then compares topology-based stability models with SME-based models and examines feature relationships. CDER models approach SME-model performance with fewer features, while combining feature sets shows a topology-dependent benefit.

4.1 Feature Engineering

  • CDER identifies low-entropy regions distinguishing stable from unstable diagrams, then integrates each preprocessed diagram against the learned Gaussians to form features.Coordinates are learned only from training diagrams and then applied to held-out diagrams.Source: §4.1 · p. 11

4.2 Model Training

  • Each topology is split 80/20 for training and validation, with randomized hyperparameter search using 10-fold cross-validation, repeated across 10 randomized splits.Models are topology-specific random forests; mean APS is reported across the 10 runs.Source: §4.2 · p. 12
  • With about 1/10th the number of SME features, CDER-feature models achieve 92%-99% of SME-model performance; SME models still outperform CDER-only models on average.The comparison uses mean APS across 10 runs for each topology and feature set.Source: §4.2 · p. 13, §4.3 · p. 13, Table 1 · p. 14

4.3 Analysis

  • Combining CDER and SME features improves mean APS by about 3% for HEEH; evidence is inconclusive for other topologies, and results may differ without downsampling.The HEEH result has p-value 0.0015; EEHEE has p-value 0.0516, while the experiments do not establish improvement for other topologies.Source: §4.2 · p. 13, §4.3 · p. 14
  • Across topologies, important CDER features are highly correlated with important SME features, while correlated predictors can obscure individual feature-importance estimates.The top three CDER features per topology have correlations ranging from 0.493 to 0.947; distinguishing signals span connectedness, loops, and voids.Source: §4.3 · pp. 14–15
  • The three most important CDER features correspond to persistence-diagram regions identified with stable proteins and are highly positively correlated with n hydrophobic noA (0.676 < r < 0.745).Region A is associated with loop structures born and dying at small length scales (0–1 Å), while Region B refers to small-scale (0–1 Å) void structures.Source: §4.3 · p. 17
  • For HEEH, stable-associated regions A–C correlate positively with buried np, while unstable-associated D and E correlate negatively; D captures larger-scale loops.Region D corresponds to loops at about 3 Å, compared with smaller-scale structures represented by B and C.Source: §4.3 · p. 17
  • Across topologies, unstable proteins are associated with high-persistence H1 regions, while stable proteins are associated with small-scale H1 and H2 loops and voids.For EEHEE, important CDER features also correlate strongly with SME features, although some SME features correlate less strongly in HEEH; combined models show topology-dependent gains.Source: §4.3 · pp. 17, 19

5 Discussion

The paper finds that learned topological features can approach SME-feature performance in stability classification, and that combining feature sets sometimes improves prediction. The discussion connects these results to patterns in stable and unstable protein structures while identifying limits in the topologies and stability labels studied.

5 Discussion

  • Topological-feature models achieved 92%-99% of SME-model performance across four protein topologies, while combining features improved prediction in some cases.Source: §5 · p. 19
  • CDER automatically learns discriminating persistence-diagram regions and outputs distributional coordinates that can train downstream classifiers.The discussion contrasts this data-driven vectorization with methods using fixed transformations or limited parameterization.Source: §5 · p. 22
  • Across protein secondary-structure types, CDER distinguished stable and unstable proteins by topological-structure scale; important CDER and SME features were highly correlated.Stable proteins showed more small-scale void and/or loop structures, while unstable proteins in certain topologies were associated with high-persistence loops.Source: §5 · p. 22
  • The conclusions cover only a small set of designed topologies, and the range of features associated with ambiguous stability scores remains underexplored.Applying CDER to all available proteins produced qualitatively similar coordinates without meaningfully altering the conclusions.Source: §5 · p. 22
  • CDER's usefulness for regression is unestablished because it requires class labels; this study instead thresholded stability scores into binary labels and compared the most and least stable designs.The authors suggest refined stability categories or using learned features in downstream regression as possible alternatives.Source: §5 · pp. 22–23
  • Feature elimination or selection could make correlation and feature-importance analyses clearer.Source: §5 · p. 23
  • The engineered features capture simple structural characteristics, which aids interpretation but may limit models to simple proteins or easier prediction tasks.Identifying the atoms responsible for discriminating features and investigating topological generators remain open directions.Source: §5 · p. 23

Appendix A. Persistent Homology with Data

The appendix outlines how topological structure is inferred from point-cloud data and summarized across scale. It introduces simplicial complexes and persistent homology as tools for characterizing shape.

Appendix A. Persistent Homology with Data

  • The point-cloud approach assumes that points sample a topological space of interest whose structure helps distinguish spaces.Source: Appendix A · p. 23
  • A simplicial complex represents point-cloud data using vertices and higher-order sets of vertices, providing a structured framework for shape.Source: Appendix A · p. 23
  • Persistent homology characterizes invariants such as components, holes, and voids across scales and represents them as persistence pairs in a diagram.Source: Appendix A · p. 23

A.1 Simplicial Complexes and Homology

A simplicial complex organizes data points into simplices, while homology summarizes its holes through cycles and boundaries. Betti numbers count holes by dimension and are topological invariants.

A.1 Simplicial Complexes and Homology

  • A simplicial complex is a collection of point subsets closed under taking subsets, with pairwise intersections either empty or in the collection.Its elements are simplices, and a p-simplex contains p + 1 points.Source: §A.1 · pp. 23–24
  • The p-th chain group is generated by p-simplices, and the boundary map sends p-simplices to combinations of lower-dimensional simplices.The boundary operator satisfies ∂p ◦ ∂p+1 = 0.Source: §A.1 · p. 24
  • Homology is formed from p-cycles and p-boundaries, with cycles as the kernel of the boundary map and boundaries as its image.The homology group is a vector space when the appropriate scalar multiplication is defined over F.Source: §A.1 · p. 24
  • Betti numbers count holes by dimension: β0 counts connected components, β1 loops, and β2 voids; they are topological invariants.The p-th Betti number is the dimension of the p-th homology group.Source: §A.1 · p. 24

A.2 Filtrations, Persistent Homology, and Persistence Diagrams

Because the appropriate complex for data is not known in advance, the paper studies topological invariants across a filtration. Persistent homology tracks features across scales, with births and deaths summarized in persistence diagrams.

A.2 Filtrations, Persistent Homology, and Persistence Diagrams

  • A filtration is a nested sequence of subcomplexes across increasing scales, enabling the study of topological invariants across a family of complexes.The need arises because the appropriate complex for data is not clear beforehand.Source: §A.2 · p. 24
  • Persistent homology tracks how homology groups evolve across the filtration by examining induced maps between associated groups.Source: §A.2 · p. 24
  • A persistence diagram records each feature’s birth and death scales; longer-lived features are treated as valuable topological information.Each point (b, d) represents a feature born at b and dying at d, with persistence d − b.Source: §A.2 · pp. 24–25
  • Persistence diagrams are stable under small perturbations, supporting robust comparisons of shapes.Features further from the diagonal y = x are more persistent.Source: §A.2 · p. 25

A.3 Weighted Alpha Complexes

The paper constructs a filtration using weighted alpha complexes, adapting the complex construction to the local density of the point-cloud data.

A.3 Weighted Alpha Complexes

  • In protein modeling, atoms are represented as balls weighted by the extent of their van der Waals interactions, determined by atom type.Source: §A.3 · p. 25
  • The weighted alpha complex is a subcomplex of the Delaunay triangulation, including simplices whose weighted Delaunay circumspheres contain no other data points in their interiors.Source: §A.3 · p. 25
  • The weighted alpha complex adapts to local data density, and the paper uses it to construct a complex and filtration on protein data.Source: §A.3 · p. 25

Appendix B. Table of Feature Descriptions

Appendix B catalogs 109 SME features used in the models, spanning sequence, structural, energetic, and fragment-based descriptors.

Appendix B. Table of Feature Descriptions

  • The table describes 109 SME features used in the models, including sequence profiles, structural measurements, energies, and fragment metrics.Source: Table 2 · p. 25
  • The ABEGO res profile scores amino-acid over- or underrepresentation relative to its overall frequency, conditioned on the local three-residue ABEGO structure.The table also lists a variant summing only positions with negative profile scores.Source: Appendix B · p. 26
  • Fragment-based descriptors summarize RMSD across selected fragments, including averages and high-RMSD fragment measures.Source: Appendix B · pp. 26, 29
  • Sequence and hydrophobic-pattern descriptors include residue counts and sizes of contiguous regions lacking large hydrophobic residues.Source: Appendix B · pp. 26–28
  • Structural descriptors include sidechain contacts, local neighbor counts, exposed surface area, and buried polar atoms without hydrogen bonds.Source: Appendix B · pp. 26–27
  • Energy and geometry descriptors include Rosetta interaction and solvation energies, hydrogen-bond counts, and shape complementarity.Source: Appendix B · p. 27
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