Biology

The claim that essentiality is set by degree over partner redundancy rather than by degree alone — the strongest-looking biological claim in the set.

Are hub proteins essential because their partners are redundant?

Hypothesis 314, Volume XVI · Biology · verdict: Not supported

Result

The claim fails on P2 and P3. Degree strongly correlates with essentiality — the centrality–lethality baseline is present — but adding partner redundancy to a degree-only model raises out-of-sample AUC by 0.0056 (95% CI 0.0025–0.0109) in the full model and 0.0002 in the ratio model, against a pre-specified 0.03 threshold. A label permutation rerunning the cross-validation gives p = 0.726 and 0.985; such gains are ordinary under the null (mean +0.0049, SD 0.0149). The within-degree association runs the other way: the highest redundancy tercile is more likely to be essential, not less (Mantel–Haenszel OR 1.37, 95% CI 1.15–1.63, p = 0.0005; clustering OR 1.45, 1.24–1.71, p = 0.001).

What was asked

"The probability that a protein is essential is set by the ratio of its interaction degree to the redundancy of its partners, not by degree alone."

P1: degree predicts essentiality (the baseline). P2: redundancy terms (clustering, mean partner degree) improve out-of-sample prediction. P3: within degree strata, higher-redundancy proteins are less likely to be essential. Support required an AUC gain of at least 0.03 (CI excluding zero, permutation p < 0.05) and a within-stratum OR below 1 (CI excluding 1). Pre-specification: hypothesis.md.

Design

STRING v12.0 S. cerevisiae links were thresholded at combined_score >= 700, with self-loops dropped and 4932. stripped. The file lists every interaction twice; collapsing unordered pairs leaves 104,188 edges and 5,791 nodes. SGD null-allele phenotypes define essentiality: any inviable null allele means essential; otherwise any viable null allele means non-essential; neither is excluded. That gives 6,266 labelled genes (1,245 essential); the graph intersection is 5,403 proteins, 1,188 essential (22.0%).

Features, computed on all graph nodes then subset to labelled proteins: log10 degree, local clustering, mean partner degree, and log10(degree / mean partner degree). Three hand-written IRLS logistic models (M1 degree; M2 degree + clustering + mean partner degree; M3 degree + log ratio) used in-fold standardisation and 20 repeats of stratified 10-fold CV, folds shared across models. Gains were paired-bootstrapped over proteins (2,000 resamples) and tested with 200 global label permutations rerunning the full CV. For P3, proteins were ranked into degree deciles and then within-decile terciles of each redundancy score; the Mantel–Haenszel OR compares top and bottom terciles across the ten strata, with a stratified bootstrap CI and 2,000 within-decile permutations.

Prediction verdicts

P1 — supported (baseline only). Degree alone reaches AUC 0.7569 (repeat SD 0.0002) and Spearman rho 0.369 (permutation p = 0.0005, the 2,000-permutation resolution floor). The essential fraction rises from 0.061 in the lowest degree decile to 0.487 in the highest. This is the rival explanation, not evidence for it.

P2 — refuted. The M2 gain is 0.0056 and the M3 gain 0.0002, far below 0.03. Permutation p = 0.726 and 0.985; the null gain distributions have mean +0.0049 (SD 0.0149) and +0.0047 (SD 0.0130). The protein bootstrap CI for M2 excludes zero, but it is conditional on the fitted scores and omits the label/refitting variability the permutation test measures; the permutation test is decisive. BH q = 0.968 and 0.985.

P3 — refuted, opposite direction. MH OR 1.37 (1.15–1.63, p = 0.0005) for mean-partner-degree redundancy and 1.45 (1.24–1.71, p = 0.001) for clustering, both q = 0.002. The claim requires OR < 1. The reverse association is concentrated in degree deciles D5–D9; the ratio score flips again in D10 (0.47 in T1 versus 0.41 in T3).

Sensitivity

At >= 400 (281,178 edges, n = 5,783) the picture is unchanged: M2 gain 0.0119 (0.0080–0.0168), p = 0.323; ratio OR 1.35 (1.15–1.58), clustering OR 1.75 (1.48–2.06). At >= 900 (53,176 edges, n = 4,430) the gains are −0.0035 and −0.0004 (p = 0.841, 0.955) and the ORs attenuate: ratio 1.18 (0.99–1.40), clustering 1.02 (0.86–1.21). The claim is refuted at every threshold; the reverse association is present at 400 and 700 but not at 900.

What the test cannot see

STRING is a confidence-scored association network, not a complete interactome; degree is measured with error, and missing edges attenuate redundancy signals. SGD labels are observational and heterogeneous across strains and experiments; 202 genes with both inviable and viable null alleles were labelled essential under the "any inviable" rule, adding noise. Clustering is zero-inflated at low degree. The permutation null shuffles labels globally, so it asks whether the gain exceeds chance, not whether redundancy adds value conditional on a correct degree model. Nothing here is causal.

What would change the answer

A threshold-robust AUC gain of at least 0.03 with p < 0.05 would support P2. A within-stratum OR below 1 (CI excluding 1), stable across 400/700/900, would support P3. A single-strain essentiality definition, a more complete interactome, or a different operationalisation of redundancy could change the result — but any such test run now would be post hoc.

Numbers and where they live

Disclosure

AI-assisted (DeepSeek coding agent), not peer reviewed. Observational data: "associated with" is the strongest verb available. Not medical, clinical or safety advice.

Figures