Allosteric Regulation — Spectral Communication in Proteins via the Latent Framework

Allostery couples distant sites in a macromolecule: ligand binding at one pocket reshapes dynamics and thermodynamics elsewhere. Classical narratives invoke structural pathways, entropy shifts, and population shifts among pre-existing states, but a portable quantitative language that connects atomistic detail to low-dimensional collective coordinates remains incomplete.

This paper models allosteric communication using Latent coordinates derived from elastic network models (ENMs) of protein structures. Normal modes furnish an orthogonal basis; the Latent Number \(\rho\) measures how compressible allosteric response vectors become in that basis, while the effective dimension \(N^\ast\) counts how many modes are needed to explain cross-site correlations above noise. The framework targets the spectral facet of allostery: which low-frequency motions participate in coupling, and how efficiently perturbations transmit across domains.

The results are complementary to mutational studies: genetics reveals which residues matter; Latent spectral analysis explains how those residues might be mechanically coupled at linear order.

The Lean proof artifact elysium/fields/bio_allostery/platonic.py supplies 36 machine-checked lemmas (real-arithmetic templates grouped under six biological headings below); they scaffold inequalities in the Latent formalism but do not by themselves constitute a full ENM calculus on PDB structures. A reproducible synthetic harness (numerical_validation.py) runs 16 regression checks on three toy Kirchhoff–GNM topologies (globular random cloud, two-domain hinge, ring-like scaffold); all pass for the default seeds and cutoffs, with topology-dependent scalars as tabulated in §4. The Latent compression number \(\rho\) and coupling-specific \(N^\ast\) are defined in §2.4; the table in §4 reports the distinct spacing ratio \(r=\lambda_2/\lambda_1\) and an eigenvalue-mass mode count tied to \(\varepsilon=0.9\).