GBNL: Graded Betti Number Learning of Complex Biological Data

TL;DR

GBNL addresses the challenge of predicting protein–nucleic acid binding by bridging algebraic topology and machine learning. It introduces persistent graded Betti numbers from Stanley-Reisner theory as nucleotide-level descriptors and combines them with transformer protein embeddings to form a two-channel representation that is fed into a gradient-boosted model. The method demonstrates strong predictive performance (e.g., $R_p=0.691$, RMSE=$1.81$ kcal/mol on S186) and provides fine-grained interpretability through grade-resolved invariants that can detect single-site mutations and their broader algebraic consequences. This approach offers a scalable, interpretable framework for sequence-based biomolecular modeling with potential for high-throughput analysis and deeper mechanistic insights.

Abstract

While persistent homology is widely used for data shape analysis, persistent commutative algebra (PCA) has seen limited adoption in machine learning and data science. Unlike persistent homology, which delivers topological invariants in the form of Betti numbers, PCA provides both algebraic invariants and graded Betti numbers. However, graded Betti numbers have seldom been applied to real-world data. In this work, we introduce the first-of-its-kind application of commutative algebra graded Betti numbers in machine learning and data science. Specifically, we present Graded Betti Number Learning (GBNL) for protein-nucleic acid binding prediction. Protein-DNA/RNA interactions are fundamental to cellular processes such as replication, transcription, translation, and gene regulation, and their understanding and prediction remain challenging. GBNL represents each nucleic acid sequence as a family of $k$-mer-specific sets and derives persistent graded Betti invariants from PCA, generating multiscale topological representations of local nucleotide organization. To incorporate cross-molecule context, these graded Betti representations are paired with transformer-based protein embeddings, linking nucleotide-level signals with global protein patterns. The proposed graded Betti representations effectively detect single-site mutations and distinguish complete mutation patterns. Operating on primary sequences with minimal preprocessing, GBNL bridges commutative algebra, reduced algebraic topology, combinatorics, and machine learning, establishing a new paradigm for comparative sequence analysis. Numerical studies using three datasets highlight the success of GBNL in protein-nucleic acid binding prediction.

Figures (6)

• Figure 1: An Illustration of GBNL workflow. (a). Representative protein-nucleic acid complex provide biological context; downstream computations use only the DNA/RNA sequence (for per-nucleotide point clouds) and the protein amino-acid sequence (for embeddings). (b). $k$-mers are extracted from the DNA sequences of the given complexes. For each $k=1$, the set of its occurrence positions within the sequence is treated as input data. (c). For each nucleotide, a Vietoris-Rips (VR) complex with maximum dimension 1 is constructed from its position-based point cloud across an $\varepsilon$-filtration. At each selected $\varepsilon$, the maximal simplices define a simplicial complex whose Stanley-Reisner (SR) ideal $I^{\varepsilon}$ is formed. The minimal free resolution of $S/I^{\varepsilon}$, where $S$ is a polynomial ring, is then computed in Macaulay2 (M2), yielding the graded Betti numbers $\beta_{i,j}$, which serve as the topological features for the corresponding nucleotide. (d). From the protein amino acid sequences of the same complexes, 2560-dimensional embedding vectors are generated using the state-of-the-art ESM2 model with 36 layers. (e). The extracted graded Betti features of DNA and the ESM2 embeddings of proteins are concatenated to form a unified algebraic-sequence representation. (f). Finally, a gradient boosted decision tree (GBDT) model is trained using these combined features for the prediction of experimental binding affinities.
• Figure 2: Prediction summary and dataset statistics for S186, S142, and S322. (a). Normalized experimental vs. predicted binding affinities; each dataset is min-max scaled to its own unit range and offset for visual separation, with marginal density plots shown along the top and right. (b). Sequence-length distribution: dots indicate the observed count at each length, and the curves show a 5-nt centered rolling average. (c). Experimental Gibbs free energy ($\Delta G$) vs. sequence length: dots mark the per-length mean $\Delta G$ values, and the curves show the corresponding 5-nt rolling average.
• Figure 3: Persistent graded Betti curves evolution across nucleotides for the RNA sequence ACAUGUUUUCUGUGAAAACGGAG. Four panels correspond to nucleotides A, C, G, and U, showing how graded Betti counts vary with the filtration threshold $\varepsilon$. Curve groups are organized by homological index $i$ and internal degree $j$, as indicated in the legends.
• Figure 4: Single mutation analysis by GBNL for N-China-F. Rows: nucleotides. Column 1: PH Betti curves; columns 2 and 3: persistent graded Betti, all shown before and after mutation (MUT).
• Figure 5: Complete mutation analysis by GBNL for N-China-F. Rows: nucleotides. Column 1: PH Betti curves; columns 2 and 3: persistent graded Betti, all shown before and after mutation (MUT).

Theorems & Definitions (2)

• Example 4.1: Tetrahedron with one missing face
• Example 4.2: Square bipyramid: topological vs. graded Betti numbers