# The Nexus Framework and the Sarrus Allocation ## Expanded, locked-method write-up for the biological validation (v9.2 → v10 "Diamond")

# The Nexus Framework and the Sarrus Allocation ## Expanded, locked-method write-up for the biological validation (v9.2 → v10 “Diamond”) **Scope:** This document is the *methods+math complete* version of the biological validation arm of the Nexus Framework. It is written to be self-contained, reproducible, and “pre-registered” in the sense that the primary feature and statistical tests are fixed **before** inspecting outcomes. **Locked feature (primary):** Miyazawa–Jernigan burial scale (MJ), helix lags $[3,4]$, sheet lag $2$, shuffle null $N_{\mathrm{shuf}}=1000$, Sarrus Linkage $Z_{\mathrm{helix}}-Z_{\mathrm{sheet}}$. --- ## 1. Executive claim (what the data supports) On a curated **two-state** folding benchmark (Ivankov et al. dataset), a **sequence-only** feature derived from autocorrelation *pattern above composition* predicts folding rate: - Pearson correlation between Sarrus Linkage and $\ln(k_f)$: $r \approx 0.54$ with permutation $p \approx 0.004$ (typical locked runs). - Partial correlation controlling for length (using $\ln L$): $r_{\mathrm{partial}} \approx 0.56$. - Leave-one-out cross-validation: $\mathrm{R}^2_{\mathrm{LOO}} \approx 0.17$ (modest but non-zero generalization). **Importantly:** the same feature does *not* reliably classify kinetic mechanism (two-state vs multi-state) on its own, and it does not reliably predict **multi-state** rates; that failure is informative about what the feature measures (coherent constraint propagation vs trapped/intermediate dynamics). --- ## 2. The “Allocate” primitive (the conceptual core) **Primitive verb:** *Allocate.* A finite system allocates a bounded “budget” between: - **Exploration** (entropy/search over configuration space), and - **Collapse** (constraint satisfaction/solution formation). Let $\sigma\in[0,1]$ represent the *fraction* of budget spent on exploration. Then the remaining fraction available for collapse is: $$ \rho(\sigma)=\sqrt{1-\sigma^2}. $$ This is the minimal “Lorentz-like” remainder consistent with an isotropic quadratic budget in exploration space (a circular constraint in the exploration degrees of freedom), producing a pseudo-Riemannian signature in the $(\sigma,\rho)$ plane. Define a Lorentz-like factor: $$ \gamma(\sigma)=\frac{1}{\sqrt{1-\sigma^2}}=\frac{1}{\rho(\sigma)}. $$ **Interpretation:** as $\sigma\to 1$, the remainder $\rho\to 0$ and $\gamma\to\infty$, representing **time dilation / freezing** under maximal exploration with vanishing collapse bandwidth. > This document **does not** claim the protein data already proves the full Lorentz mapping. It claims the data supports a *sequence-only constraint-coherence observable* that is consistent with an allocation picture and predictive of rates in the cooperative (two-state) regime. --- ## 3. Operationalization in proteins (how $\sigma$ is proxied) We cannot directly observe $\sigma$ (exploration fraction) for each protein from sequence alone. Instead we measure a **constraint-coherence signature** from the sequence, and test whether it predicts folding rate. ### 3.1 Convert sequence to a numerical carrier wave Given an amino acid sequence $a_1a_2\ldots a_N$, map each residue to a scalar using the **Miyazawa–Jernigan burial energy** scale: $$ x_i = \mathrm{MJ}(a_i). $$ Let $\bar{x}$ be the mean of $\{x_i\}$, and define the centered signal: $$ s_i = x_i - \bar{x}. $$ Define the energy-normalization: $$ \|s\|^2=\sum_{i=1}^N s_i^2. $$ If $\|s\|^2\approx 0$, the metric is undefined and the protein is excluded. ### 3.2 Autocorrelation at structural lags Define normalized autocorrelation at lag $\ell$: $$ \mathrm{ACF}(\ell)=\frac{\sum_{i=1}^{N-\ell} s_i\, s_{i+\ell}}{\sum_{i=1}^N s_i^2}. $$ **Locked lags:** - Helix proxy: average of $\ell=3$ and $\ell=4$ (alpha helix periodicity $\sim 3.6$ residues/turn) $$ H \equiv \frac{1}{2}\left[\mathrm{ACF}(3)+\mathrm{ACF}(4)\right]. $$ - Sheet proxy: $\ell=2$ (alternating tendency) $$ S \equiv \mathrm{ACF}(2). $$ ### 3.3 Composition control via shuffled null (pattern above composition) To isolate **pattern** from **composition**, destroy order while preserving residue multiset by shuffling the amino acids. For shuffle $j=1,\dots,N_{\mathrm{shuf}}$, compute $H^{(j)}$ and $S^{(j)}$ on the shuffled sequence, forming null distributions. Compute z-scores: $$ Z_H = \frac{H-\mu(H^{(j)})}{\sigma(H^{(j)})},\qquad Z_S = \frac{S-\mu(S^{(j)})}{\sigma(S^{(j)})}. $$ A shuffle standard deviation of $\approx 0$ invalidates the z-score; such cases are excluded. ### 3.4 The Sarrus Linkage (primary feature) Define the **Sarrus Linkage**: $$ \boxed{ \mathcal{S} \equiv Z_H - Z_S } $$ Interpretation (operational): - $\mathcal{S}$ increases when the helix-lag structure is unusually strong relative to its composition-matched null **and** sheet-lag structure is weak, i.e., a net “helical constraint flow” relative to “sheet constraint flow.” - This is a *constraint propagation differential*, not a simple propensity. --- ## 4. Determinism (reproducible null model) Shuffles must be reproducible *per protein*. Let $\mathrm{md5}(seq)$ be the MD5 digest of the raw amino acid string. Define an integer seed: $$ \mathrm{seed} = \mathrm{md5}(seq)\bmod 2^{32}. $$ Each protein uses its own deterministic RNG initialized by this seed, producing identical null distributions across runs and machines (given the same RNG algorithm). --- ## 5. Dataset integrity (“domain match” requirement) A sequence-only predictor is meaningless if the analyzed sequence does not match the kinetic construct used to measure $k_f$. ### 5.1 Domain alignment rule For each protein entry: 1. Prefer **curated domain overrides** when PDB entries contain extra domains, fusions, or fragments. 2. Otherwise fetch FASTA from RCSB. 3. If the fetched length differs from the experimental construct length by more than a tolerance (default 10%), **skip** (unless overridden). Formally, if expected length is $L_{\mathrm{exp}}$ and used sequence length is $L_{\mathrm{used}}$, require: $$ \left|L_{\mathrm{used}}-L_{\mathrm{exp}}\right| \le \tau\, L_{\mathrm{exp}},\qquad \tau=0.10, $$ unless the sequence is explicitly overridden with a curated construct. ### 5.2 Audit trail requirement Each run must output an **audit table** showing, for every entry: - PDB ID, expected length, used length - status: `FETCH_MATCH`, `OVERRIDE`, `SKIP`, reason - shuffle standard deviations (sanity check for z-scoring) This ensures transparency and guards against “silent” dataset drift. --- ## 6. Primary statistical tests (locked) Let $x_i=\mathcal{S}_i$ be the Sarrus Linkage for protein $i$ and $y_i=\ln(k_{f,i})$. ### 6.1 Pearson correlation $$ r = \frac{\sum_i (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_i (x_i-\bar{x})^2}\sqrt{\sum_i (y_i-\bar{y})^2}}. $$ ### 6.2 Permutation test (distribution-free) Compute observed $|r_{\mathrm{obs}}|$. For $m=1,\dots,N_{\mathrm{perm}}$: - permute $y$ to $y^{(m)}$ (break association, preserve marginal distributions) - compute $|r^{(m)}|$ Permutation $p$: $$ p_{\mathrm{perm}}=\frac{1+\sum_{m=1}^{N_{\mathrm{perm}}}\mathbf{1}\left(|r^{(m)}|\ge |r_{\mathrm{obs}}|\right)}{1+N_{\mathrm{perm}}}. $$ (Using the +1 correction avoids $p=0$ under finite permutations.) ### 6.3 Partial correlation controlling for length Let the covariate be $c_i=\ln(L_i)$ where $L_i$ is the used sequence length. Compute residuals from linear regression: $$ \tilde{x}_i = x_i - (\alpha_x + \beta_x c_i),\qquad \tilde{y}_i = y_i - (\alpha_y + \beta_y c_i). $$ Then: $$ r_{\mathrm{partial}} = \mathrm{corr}(\tilde{x},\tilde{y}). $$ ### 6.4 Leave-one-out cross-validation (generalization) For each held-out index $i$: - fit a linear model on all $j\neq i$: $y = a_{-i}+b_{-i}x$ - predict $\hat{y}_i=a_{-i}+b_{-i}x_i$ Compute: $$ R^2_{\mathrm{LOO}} = 1 - \frac{\sum_i (y_i-\hat{y}_i)^2}{\sum_i (y_i-\bar{y})^2}. $$ Also report $r_{\mathrm{LOO}}=\mathrm{corr}(y,\hat{y})$ for interpretability. > **Why this matters:** A decent in-sample $r$ can be spurious on $n\sim 25$ if it doesn’t generalize. $R^2_{\mathrm{LOO}}>0$ is the minimal “good faith” check. --- ## 7. Secondary analyses (informative, not the primary claim) ### 7.1 Two-state vs multi-state mechanism A separate validation tests whether $\mathcal{S}$ differs systematically between: - two-state folders - multi-state folders This can be evaluated by a Mann–Whitney U test or similar. Current runs show **no significant separation** and poor threshold classification accuracy. This implies: - $\mathcal{S}$ is primarily a **rate predictor** in the cooperative regime, - not a robust classifier of kinetic order. ### 7.2 The “Diamond” spectrum (cooperative ↔ trapped ↔ IDP) An exploratory visualization plots $(\mathcal{S},\ln k_f)$ for: - two-state proteins (cooperative) - multi-state proteins (intermediate-trapped) - IDPs (controls) Observed means in one locked run: - Two-state mean $\mathcal{S}\approx 0.06$ - Multi-state mean $\mathcal{S}\approx 0.82$ - IDP mean $\mathcal{S}\approx 0.77$ This supports the idea that $\mathcal{S}$ captures *constraint-style differences*, but these are not yet a “law” without external datasets and tighter IDP ground truth. --- ## 8. What must be true (the “Good” checklist) The analysis is “good” only if **all** the following are true: 1. **Domain match:** each analyzed sequence corresponds to the experimental construct for which $k_f$ was measured (override/chain-select/skip; no silent mismatches). 2. **Composition control:** z-scores are computed against shuffled nulls; shuffle standard deviations are non-zero. 3. **Pre-registered feature:** scale, lags, shuffle count fixed *before* outcome inspection. 4. **Determinism:** shuffles seeded deterministically per sequence (e.g., MD5-based). 5. **Generalization:** report