Sequence Constraint Coherence Predicts Two-State Protein Folding Rates: The Sarrus Linkage and a Lorentz-Form Latency Law

Sequence Constraint Coherence Predicts Two-State Protein Folding Rates: The Sarrus Linkage and a Lorentz-Form Latency Law Driven by Dean Kulik February 2026 Abstract We introduce the Sarrus Linkage, a sequence-only observable that predicts two-state protein folding rates from amino acid arrangement beyond composition. The feature computes the differential between helix-lag and sheet-lag autocorrelation z-scores, measured against a composition-preserving shuffle null. On a benchmark of 30 two-state folders from the Ivankov dataset (all proteins included, zero skipped), the Sarrus Linkage correlates with ln(kf) at r = 0.54 (permutation p = 0.002, n = 10,000). The correlation is robust: partial r = 0.57 controlling for sequence length, jackknife stability = 3.6% relative variation with no influential proteins, and leave-one-out cross-validated R² = 0.19. The same predictor applied to multi-state folders yields r ≈ 0.002, confirming selectivity for cooperative folding. We further show that a Lorentz-form latency function, ln(kf) ∼ ½ln(1 − σ²), fits the data better than a linear model by every metric: AIC (61.4 vs 63.5), LOO R² (0.24 vs 0.19), and in-sample r (0.59 vs 0.54). We interpret this geometry through a budget-allocation framework in which a protein’s folding rate is governed by how it partitions a finite constraint budget between exploration (entropy) and collapse (structure). The Lorentz form emerges naturally when this budget obeys an isotropic quadratic constraint. The framework requires no structural databases, molecular dynamics, or machine learning. It runs on any hardware in milliseconds per protein and produces a deterministic, auditable result. 1. Introduction The protein folding problem has two faces. The first—predicting the three-dimensional structure from sequence—has been substantially addressed by deep learning approaches such as AlphaFold, which reconstruct coordinates from evolutionary covariance patterns in multiple sequence alignments. The second—predicting folding kinetics—remains largely open. Why do some proteins fold in microseconds while others require seconds? Why do some fold cooperatively (two-state) while others populate intermediates? The most successful empirical predictor of two-state folding rates is relative contact order (CO), which requires knowledge of the native structure. CO correlates with ln(kf) at |r| ≈ 0.7–0.8 across standard benchmarks. However, CO is not a sequence-only predictor: it requires a solved or predicted structure. A purely sequence-derived predictor of folding rates would be both practically useful and theoretically informative, revealing what information about the folding process is encoded directly in the linear sequence. Here we demonstrate that a simple quantity—the differential between helix-lag and sheet-lag autocorrelation of a hydrophobicity signal, z-scored against composition-preserving shuffles—contains statistically significant information about two-state folding rates. We call this quantity the Sarrus Linkage. We further show that its relationship to folding rate follows a Lorentz-form latency law, consistent with a finite budget allocation between entropic exploration and structural collapse. 2. Methods 2.1 Feature Definition (Pre-registered) All parameters were fixed before examining outcomes. Given an amino acid sequence, we map each residue to a scalar using the Miyazawa–Jernigan (MJ) inter-residue contact energy scale. The centered signal si = MJ(ai) − mean is used to compute normalized autocorrelation at structural lags. The helix observable H is the mean of ACF at lags 3 and 4 (bracketing the 3.6 residues/turn of α-helices). The sheet observable S is ACF at lag 2 (alternating pattern of β-strands). Autocorrelation uses total-energy normalization: ACF(ℓ) = Σ si si+ℓ / Σ si². To isolate arrangement from composition, we generate 1,000 composition-preserving shuffles per protein. Each shuffle permutes the amino acid list (not the signal array) and recomputes both autocorrelation values. Shuffles are deterministically seeded using MD5(sequence) mod 232 with NumPy’s default_rng, ensuring reproducibility across platforms. Z-scores are computed using population standard deviation (ddof = 0): ZH = (H − μH) / σH and analogously for ZS. The Sarrus Linkage is defined as S = ZH − ZS. 2.2 Dataset and Domain Enforcement We use the 30 two-state proteins from the Ivankov et al. benchmark, supplemented by 16–18 multi-state folders for selectivity testing. For each protein, the analyzed sequence must match the kinetic construct used to measure kf. Where PDB entries contain extra domains, fusion tags, or chain fragments that differ from the experimental construct by more than 10% in length, we apply curated domain overrides (13 of 30 proteins). The remaining 17 sequences are fetched from RCSB and pass the 10% length tolerance. A complete audit table with status (FETCH vs OVERRIDE), sequence lengths, and z-score diagnostics accompanies every run. 2.3 Statistical Tests Four locked tests: (1) Pearson correlation between S and ln(kf); (2) permutation p-value for |r| (10,000 permutations of ln(kf), preserving marginal distributions); (3) partial correlation controlling for ln(sequence length); (4) leave-one-out cross-validation R² for linear regression of ln(kf) on S. 2.4 Lorentz Bridge We test whether the relationship between S and ln(kf) is better described by a Lorentz-form latency function than a linear model. The Sarrus values are mapped to σ ∈ (0,1) via rank normalization (assumption-free, monotone). The Lorentz term is ½ln(1 − σ²). We compare linear and Lorentz models by AIC, in-sample r, and LOO-CV R². 3. Results 3.1 Primary Validation Metric Value Significance Pearson r (S vs ln(kf)) 0.5436 p = 1.9 × 10⁻³ Permutation p (|r|, 10,000 perms) 0.0019 < 0.01 Partial r (controlling ln L) 0.5714 p = 9.7 × 10⁻⁴ LOO-CV R² (linear) 0.188 Lorentz r (½ln(1−σ²) vs ln(kf)) 0.5851 p = 6.8 × 10⁻⁴ LOO-CV R² (Lorentz) 0.239 AIC linear / Lorentz 63.5 / 61.4 Lorentz wins Multi-state r 0.002 p = 0.99 (flat) Contact order r (benchmark) −0.746 p = 2.2 × 10⁻⁶ Jackknife stability ±3.6% No influential proteins Table 1. Summary statistics for the Sarrus Linkage on the Ivankov two-state benchmark (n = 30). The Sarrus Linkage predicts two-state folding rates at r = 0.54 (Table 1). The permutation test (p = 0.0019) rules out compositional artifact: the correlation arises from amino acid arrangement, not mere amino acid content. The partial correlation increases when controlling for sequence length (0.57 vs 0.54), indicating that length was partially masking the true signal. The jackknife analysis confirms that no single protein drives the result: removing any one protein changes r by less than 0.05 (3.6% relative variation). 3.2 Selectivity for Cooperative Folding When applied to 16 multi-state folders from the same benchmark, the Sarrus Linkage yields r = 0.002 (p = 0.99). The predictor is not simply weak for multi-state proteins; it is entirely flat. This selectivity is informative: two-state folders have a single dominant barrier (one “stack trace” in the computational analogy), making a single scalar sufficient. Multi-state folders have branched pathways with intermediates, requiring multiple scalars to describe their kinetics. The Sarrus Linkage captures the coherence of the dominant constraint, which exists only when folding is cooperative. 3.3 The Lorentz Bridge The relationship between folding rate and constraint coherence is better described by a Lorentz-form function than a linear model. Using rank-based normalization to map S to σ ∈ (0,1), the Lorentz term ½ln(1 − σ²) achieves higher correlation (r = 0.585 vs 0.543), lower AIC (61.4 vs 63.5), and higher out-of-sample prediction accuracy (LOO R² = 0.239 vs 0.188, a 27% improvement). The Lorentz form wins every metric. This functional form has a natural interpretation. If a protein allocates a finite budget between entropic exploration (σ) and structural collapse (ρ), subject to an isotropic quadratic constraint σ² + ρ² = 1, then the folding rate scales as ρ = √(1 − σ²) and the log-rate as ½ln(1 − σ²). This is formally identical to the Lorentz factor of special relativity, arising from the same mathematical structure: a finite capacity split between competing demands under rotational symmetry. We emphasize that this is an analogy grounded in shared geometry, not a claim about relativistic physics in proteins. Figure 1. Six-panel diagnostic. (A) Primary: Sarrus Linkage vs ln(kf) for 30 two-state folders. (B) Lorentz bridge: rank-based σ mapping with Lorentz curve overlay. (C) LOO-CV: linear vs Lorentz out-of-sample prediction. (D) Spectrum: two-state (blue), multi-state (orange) overlaid. (E) Contact order benchmark. (F) Cross-domain γ curve. 4. Discussion 4.1 What the Sarrus Linkage Measures The Sarrus Linkage is not a propensity score. It measures the differential between helix-period and sheet-period autocorrelation in the hydrophobicity signal, z-scored against a null model that preserves composition but destroys arrangement. Positive values indicate that helix-lag coherence exceeds what composition alone would predict, relative to sheet-lag coherence. Negative values indicate the reverse. Near-zero values indicate that the observed autocorrelation is explained entirely by composition. The shuffle null is the methodological core. Without it, the autocorrelation would confound arrangement with composition: proteins rich in hydrophobic residues would show high autocorrelation at all lags simply because their signal has large amplitude. By z-scoring against shuffles, we isolate the contribution of residue ordering—the “verb” (how residues are arranged) rather than the “noun” (which residues are present). This distinction matters: