Tests whether extending the reconstruction formula with physical/spatial terms —
Ĉ = b0·(P·G·β·GT·PT) + b2·1/d_soma + b3·1/d_neurite + b4·overlap_volume
— improves the fit to the true connectome C beyond what the existing gene-based
model already achieves, across all 25 gene-selection × metacell-generation combos.
b0–b4 are fit per combo via non-negative-constrained weighted least squares
(b2, b3, b4 ≥ 0 — the biologically sensible direction: closer / more overlap
should predict more connectivity, never less).
| gene method | metacell method | baseline loss | b0 (gene) | b2 (1/d_soma) | b3 (1/d_neurite) | b4 (overlap) | total improvement | physical terms' own gain |
|---|---|---|---|---|---|---|---|---|
| seurat_vst_approx | 1_constrained_kmeans_local_pca | 16,923 | 1.032 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.098% | +0.0000pp |
| seurat_vst_approx | 3_graph_ward_supercell | 17,025 | 1.023 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.050% | -0.0000pp |
| seurat_vst_approx | 5_archetypal_farthest_point | 17,128 | 1.031 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.089% | +0.0000pp |
| seurat_vst_approx | 4_agglomerative_ward_full | 17,376 | 1.030 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.082% | +0.0000pp |
| dispersion_binned | 1_constrained_kmeans_local_pca | 17,465 | 1.021 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.042% | +0.0000pp |
| dispersion_binned | 5_archetypal_farthest_point | 17,526 | 1.024 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.052% | +0.0000pp |
| dispersion_binned | 3_graph_ward_supercell | 17,717 | 1.024 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.049% | +0.0000pp |
| dispersion_binned | 2_kmeans_global_pca | 17,750 | 1.019 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.031% | +0.0000pp |
| trend_residual_approx | 3_graph_ward_supercell | 17,914 | 1.027 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.064% | -0.0000pp |
| seurat_vst_approx | 2_kmeans_global_pca | 18,017 | 1.019 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.030% | +0.0000pp |
| trend_residual_approx | 4_agglomerative_ward_full | 18,103 | 1.028 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.066% | +0.0000pp |
| trend_residual_approx | 1_constrained_kmeans_local_pca | 18,226 | 1.027 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.061% | +0.0000pp |
| trend_residual_approx | 5_archetypal_farthest_point | 18,282 | 1.033 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.090% | +0.0000pp |
| dispersion_binned | 4_agglomerative_ward_full | 18,426 | 1.020 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.032% | +0.0000pp |
| trend_residual_approx | 2_kmeans_global_pca | 18,915 | 1.013 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.014% | +0.0000pp |
| binomial_deviance_approx | 2_kmeans_global_pca | 24,786 | 1.000 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.000% | +0.0000pp |
| binomial_deviance_approx | 4_agglomerative_ward_full | 24,873 | 0.992 | 0.00e+00 | 0.00e+00 | 6.46e-03 | +0.002% | +0.0023pp |
| binomial_deviance_approx | 5_archetypal_farthest_point | 24,966 | 0.988 | 0.00e+00 | 0.00e+00 | 9.78e-03 | +0.005% | +0.0053pp |
| binomial_deviance_approx | 1_constrained_kmeans_local_pca | 24,974 | 0.998 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.000% | +0.0000pp |
| binomial_deviance_approx | 3_graph_ward_supercell | 25,032 | 0.995 | 0.00e+00 | 0.00e+00 | 2.39e-03 | +0.000% | +0.0003pp |
| raw_variance | 3_graph_ward_supercell | 25,037 | 0.998 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.000% | +0.0000pp |
| raw_variance | 2_kmeans_global_pca | 25,476 | 0.998 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.000% | +0.0000pp |
| raw_variance | 1_constrained_kmeans_local_pca | 25,519 | 0.997 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.000% | +0.0000pp |
| raw_variance | 4_agglomerative_ward_full | 25,807 | 0.997 | 0.00e+00 | 0.00e+00 | 0.00e+00 | +0.000% | +0.0000pp |
| raw_variance | 5_archetypal_farthest_point | 26,258 | 0.976 | 0.00e+00 | 0.00e+00 | 1.81e-02 | +0.016% | +0.0157pp |
Finding: b2 (soma distance) is zero in 25/25 combos. b3 (neurite closest-approach distance) is zero in 25/25 combos. b4 (overlap volume) is nonzero in only 4/25 combos, and even there the effect is negligible (max +0.0157 percentage points of loss improvement). Mean unique physical-terms contribution across all 25 combos: +0.00095pp.
Why, given C itself does correlate with these physical quantities (R² up to 5.4%
for neurite distance and overlap): the gene-based reconstruction Chat_baseline
already correlates with 1/d_neurite and overlap at r ≈ 0.31–0.37 in every combo
tested, regardless of which gene-selection method built it. Gene-compatible cell types tend to
also be anatomically co-located, so the gene model implicitly re-derives most of the physical
proximity signal on its own. An earlier, unconstrained version of this same fit found large
apparent "improvements" (up to +2.6%) — but with negative b2/b3/b4 weights, meaning
the physical terms were being used to subtract away the gene model's own
proximity-correlated over-predictions, not to add genuinely new, biologically-interpretable
signal. Constraining the weights to the sensible non-negative direction removes that
artifact and reveals the physical terms add essentially nothing on top of the gene model.
Caveat: d_neurite is currently ~90% exact brute-force minimum, ~10% an exact bounding-box lower bound pending a still-running GPU computation (see the Wiring vs. Distance page) — this affects only the exact value for the remaining pairs, not the sign or overall magnitude of the correlations reported here.