Hypothesis

Biological aging drives a progressive loss of higher‑order topological features (specifically persistent H2 voids) in the manifold of gene‑expression states, reflecting a collapse of multi‑way chromatin interactions that maintain transcriptional fidelity. This topological simplification can be quantified via persistent Laplacian spectra and will outperform conventional geometry‑based embeddings (UMAP/PCA) in predicting chronological age and correlating with molecular hallmarks of aging.

Mechanistic Rationale

Persistent H2 cavities in a point cloud of transcriptomic profiles arise when three or more expression patterns interact in a coordinated, non‑linear fashion across conditions, analogous to higher‑order chromatin loops that bring together enhancers, promoters, and insulators. As cells age, accumulating DNA damage, altered CTCF/cohesin dynamics, and histone‑modification drift weaken these multi‑way contacts, flattening the expression landscape and eliminating voids. Persistent Laplacians, which encode both topological and geometric information via eigen‑values of the up‑ and down‑Laplacian operators, are sensitive to the disappearance of such cavities while retaining sensitivity to local gene‑co‑expression clusters captured by H0 and H1. Thus, a spectral shift toward higher eigenvalues in the persistent Laplacian—indicating reduced void volume—should mirror the decline in chromatin‑loop complexity measured by Hi‑C.

Testable Predictions

1. Age‑related decline in persistent Betti number β₂: Across human GTEx tissues, the average persistence‑weighted β₂ will decrease monotonically with donor age (Spearman ρ < –0.4, p < 0.001).
2. Superior age prediction: A regression model using the first ten persistent Laplacian eigenvalues will achieve a lower mean absolute error (MAE) in predicting chronological age than models built on the top 10 UMAP or PCA components (ΔMAE ≥ 0.5 years).
3. Correlation with chromatin void loss: In a subset of GTEx samples with matched Hi‑C data (e.g., from the ENCODE project), the persistent Laplacian spectral signature of void loss will positively correlate with the normalized Hi‑C contact‑distance exponent (α) derived from decay curves (r > 0.5, p < 0.01).
4. Experimental perturbation: Acute depletion of CTCF in young murine myoblasts (using auxin‑inducible degron) will recapitulate the aging‑associated Laplacian spectral shift, increasing the first non‑zero eigenvalue by ≥20 % within 48 h, without substantially altering overall expression variance.

Experimental Approach

• Data: Process GTEx v8 RNA‑seq (TPM) across 30 tissues, filtering for genes with median TPM > 1. Normalize and log‑transform.
• Topological pipeline: Construct Vietoris–Rips filtrations on the expression vectors (distance = 1 – Pearson correlation). Compute persistence diagrams for H0–H2, derive persistence landscapes, and calculate the persistent Laplacian (k‑=0,1,2) using the algorithm of [4]. Extract the first ten eigenvalues for each sample.
• Statistical analysis: Fit linear mixed‑effects models with age as fixed effect, tissue as random effect; test predictions 1–3. For prediction 4, generate scRNA‑seq from treated vs. control myoblasts, compute Laplacian spectra, and perform a two‑sample Kolmogorov‑Smirnov test on eigenvalue distributions.
• Validation: Compare age‑prediction MAE against baseline models (UMAP, PCA, elastic net) using 5‑fold cross‑validation.

Expected Outcomes and Falsifiability

If aging truly erodes higher‑order topological structure, we will observe a significant, tissue‑consistent reduction in β₂ and a Laplacian‑based age estimator that outperforms geometry‑only methods, with accompanying Hi‑C validation. Failure to detect a monotonic β₂ decline, or to find Laplacian features predictive of age beyond chance, would falsify the hypothesis and suggest that transcriptomic aging is captured sufficiently by lower‑order topology or linear covariance structures, redirecting focus toward alternative topological descriptors (e.g., persistent homology of gene‑regulatory networks).