Statistical Pharmacology via Kakutani Dichotomy VIII

Empirical Validation on Experimental All-Atom Protein Ensembles, Influence-Function Block Selection, and Bootstrap Coverage Calibration

Zhengyi Chen¹, Ruqing Chen²

¹ Guangxi Key Laboratory of Drug Discovery and Optimization, School of Pharmacy, Guilin Medical University, Guilin 541199, P. R. China — chenzhengyi@glmc.edu.cn ² GUT Geoservice Inc., Montreal, Québec, Canada — ruqing@hotmail.com

This part of the series Statistical Pharmacology via Kakutani Dichotomy follows Papers I–VII (VII: DOI 10.5281/zenodo.23039869; integrating article: 10.5281/zenodo.23042828). It addresses three open problems of Paper VII:

• (1) real protein ensembles;
• (4) a block length derived from the index itself;
• (2) the coverage of the bootstrap intervals.

Experiment 08 is complete and passes all acceptance tests (results/exp08_console_log.txt). The unit tests pass: 8 inherited package tests and 4 new calibration tests.

The paper is complete: paper/paper08_empirical_pdb_and_calibration.tex and .pdf, 16 pages. Its numbers come from the companion script code/paper08_compute_tables.py. The script reads the console log and data files and writes results/paper08_tables_summary.txt, the table rows results/paper08_table{1..4}_rows.tex and results/paper08_tables.json.

• Proposition 2.2, exact null for exchangeable ensembles. For NMR models the split-null excess has mean exactly zero under the null. The within-ensemble controls of both ubiquitin ensembles contain zero.
• Theorem 3.1, Isserlis bound. For a stationary Gaussian process, a centred quadratic form has integrated autocorrelation time at most half that of the slowest mode. Equality holds if and only if the form is supported on u₁u₁ᵀ. The measured ratio for PC1² is 0.455, against 0.500.
• Proposition 3.3 and Corollary 3.4, whitening collapse. The influence function of CKI weights modes by their relative change, not by their variance. Fast modes therefore dominate it, and τ(ψ)/τ_PC1 is 0.020–0.025 on adenylate kinase.
• Corollary 3.5, block rule. Take the block length b as twice the integrated autocorrelation time of the influence function.
• Proposition 4.2, interval criterion. The percentile interval contains the estimate if and only if the basic interval does. The bias-shifted interval contains it if and only if the replicate mean lies between the quantiles, whatever the bias. This is why only the bias-shifted interval contains the estimate at D = 642. The criterion holds in 2400/2400 calibration groups.

The paper states three limits explicitly:

• (a) The adenylate kinase dynamics are synthetic Ornstein–Uhlenbeck dynamics on real geometry, not molecular dynamics.
• (b) The ubiquitin difference reflects two NMR determination methods, not a ligand effect.
• (c) In high dimension, the bias-shifted interval should be reported together with the bootstrap bias.

Four entries from the RCSB Protein Data Bank are stored in data/pdb/, with SHA-256 checksums in data/pdb/SHA256SUMS.txt. The script downloads a file from files.rcsb.org only if it is absent, so the experiment runs offline.

(1A) Real all-atom NMR ensembles of ubiquitin. The analysis uses residues 1–70 and heavy atoms only.

• With 116 and 128 models against D = 210 Cα coordinates, the sample covariance gives CKI = ∞.
• Ledoit–Wolf gives CKI = 27.87, which splits into mean 5.93, commuting part 13.55 and rotation part 8.39.
• The null-corrected excess is +18.6, 95 % interval [+17.0, +22.9].
• Both within-ensemble controls are consistent with zero: 2K39 halves +0.47 [−1.25, +2.05], and 1XQQ halves −1.40 [−3.31, +0.41].
• The first-order excess is largest at residues 53, 52, 47, 39 and 37. Residues 52–53 are the D52/G53 peptide-flip region.
• Shells around the hydrophobic patch L8/I44/V70 show only +0.66, so most of the difference lies away from it.

The two ensembles come from different determination methods, so the difference is a real difference between the deposited ensembles. It should not be read as a ligand effect.

(1B) Adenylate kinase on real crystal coordinates.

• The elastic network is built on the real Cα atoms: consecutive distances are 2.99–3.88 Å, with no virtual bonds. It has dim ker H = 6 at cutoffs of 8, 10, 12 and 15 Å.
• Superimposed on the CORE domain, the real closure from 4AKE to 1AKE is 11.2 Å RMS for the NMP domain and 15.6 Å for the LID.
• Ap5A, mapped into the open structure, defines a 19-residue pocket, 18 of them in CORE.

The dynamics are synthetic: Ornstein–Uhlenbeck mode dynamics on this real-geometry network, not molecular dynamics.

The exact population curve of the local perturbation is flat on the real geometry, as Corollary 5.2 of Paper VII predicts. For the cross-domain state, with 642 coordinates, the bootstrap distribution is displaced by +0.62 from the estimate. The percentile interval lies entirely above the estimate, and the basic interval, its reflection, lies entirely below it. Only the bias-shifted interval contains it. For high-dimensional shells, report the bias-shifted interval together with the bootstrap bias.

(2) Autocorrelation of the index (Isserlis theorem). For a Gaussian stationary process with independent modes, a quadratic statistic has integrated autocorrelation time at most half that of the slowest linear mode. The bound is attained only by quadratics of the slowest mode itself.

• For PC1² the measured ratio is 0.455, against the Isserlis value (1+φ²)/(1+φ)² = 0.500.
• The autocorrelation of PC1² at lag 10 is 0.520, equal to ρ(PC1)².
• The influence function of CKI is not concentrated on slow modes. The inverse covariances weight every mode comparably, so on adenylate kinase its τ is 1.0–1.3 frames, against τ_PC1 = 51. The Isserlis prediction is 1.0–1.1.

The halving is therefore a bound, not an identity. The influence function can be slow in one case: when a mean shift lies along a slow mode, its linear part inherits that mode's time.

(3) Coverage of 95 % intervals. The test uses 200 macro replicates of pairs with known CKI (D = 30, T = 2000), with B = 200 bootstrap draws.

The split-null excess is unbiased: +0.0017 under the null (z = +1.07), and +0.0012 against the truth for the perturbed pair. The block length should come from the influence function of the index, b = 2 τ(influence):

• For the null pair, τ(influence) is 4.7, so every tested block is longer than needed and the intervals are conservative.
• For the perturbed pair, τ(influence) is 16.8 because its mean shift lies along the slowest mode. The rule gives b ≈ 34, close to τ_PC1 = 40, where coverage is nominal. The shorter block, b = 20, undercovers.

.
├── kakutani_pharma/        # package (Paper VII) + calibration.py (influence function, Isserlis times, intervals)
├── tests/                  # test_kakutani_pharma.py (8), test_calibration.py (4)
├── code/
│   ├── exp08_real_pdb_and_bootstrap_calibration.py
│   ├── exp08_lib.py        # PDB files, elastic networks, OU mode dynamics
│   └── paper08_compute_tables.py   # companion script: table rows and checks for the paper
├── data/
│   ├── pdb/                # 2K39, 1XQQ, 4AKE, 1AKE and SHA256SUMS.txt
│   └── exp08_*.csv         # ubiquitin residues and shells, adenylate kinase shells, autocorrelation, calibration
├── figures/fig08_real_pdb_and_calibration.{pdf,png}
├── results/                # exp08_console_log.txt, paper08_tables_summary.txt, paper08_table*_rows.tex, paper08_tables.json
├── paper/                  # paper08_empirical_pdb_and_calibration.tex / .pdf
├── CITATION.cff, LICENSE, requirements.txt, pyproject.toml, .gitignore
└── README.md

pip install -r requirements.txt
python tests/test_kakutani_pharma.py && python tests/test_calibration.py
python code/exp08_real_pdb_and_bootstrap_calibration.py      # about 11 minutes; prints OVERALL ACCEPTANCE: PASS
python code/paper08_compute_tables.py                        # table rows and checks for the paper
cd paper && pdflatex paper08_empirical_pdb_and_calibration.tex   # run three times

https://github.com/Ruqing1963/kakutani-statistical-pharmacology-VIII

@misc{ChenChen2026VIII,
  author  = {Chen, Zhengyi and Chen, Ruqing},
  title   = {Statistical Pharmacology via Kakutani Dichotomy VIII: Empirical Validation on Experimental All-Atom Protein Ensembles, Influence-Function Block Selection, and Bootstrap Coverage Calibration},
  year    = {2026},
  publisher = {Zenodo},
  doi     = {10.5281/zenodo.23045772},
  url     = {https://doi.org/10.5281/zenodo.23045772}
}

National Natural Science Foundation of China (No. 22464010), Guangxi Natural Science Foundation of China (No. 2025GXNSFAA069294), and the 2025 Bagui Youth Top Talent Project.

Code under the MIT License; text, figures and data under CC BY 4.0 (see LICENSE). The PDB files are distributed by the RCSB Protein Data Bank under its own usage policy (CC0).