Methods — mesh-based membrane topology

This page documents the per-crop pipeline that produces the curvature, protrusion/indentation, and contact-gap channels rendered in the interactive 3D gallery and the static maps. It is the web companion to the manuscript draft at paper/methods_membrane_topology.tex and stays in sync with the results CSVs at results/.

What was done

For each crop a single membrane-rich cell was selected as the cell with the greatest area of cell–ECS interface (counted as the number of cell faces sharing a boundary with an ECS voxel; ties broken by cell index). Crops with finer-than-16 nm native voxel size were box-downsampled to 16 nm isotropic prior to meshing, so all crops feed an identical analysis grid regardless of acquisition resolution.

The cell's binary mask was smoothed with a 3D Gaussian of physical width σ = 1.5·vx nm (= 24 nm at the 16 nm working voxel), then surfaced by marching cubes ^[1] at iso-level 0.5 (scikit-image implementation ^[2]). Surface vertices are placed in nm coordinates accounting for both the cropping bounding-box offset and the smoothing pad. The sign convention is calibrated against a synthetic convex sphere of radius 400 nm through the same pipeline; convex membranes return positive curvature.

Signed mean curvature H (1/nm)

Computed from the cotangent Laplacian ^[3,4], a strictly local 1-ring quantity. For each interior edge of the mesh, cotangent weights of the two opposite angles enter a sparse Laplacian operator L; the mean-curvature normal vector at vertex i is Hi·ni = (LV)i / (2Ai), where Ai is the barycentric vertex area. The magnitude is the unsigned curvature; the sign is taken from sgn(Hi · nitrimesh), calibrated so convex (membrane-into-ECS) surfaces are positive.

The radius of curvature is R = 1/|H|. Representative values:

H (1/nm)	R	Feature
0.001	1000 nm	essentially flat
0.005	200 nm	gentle membrane bend
0.010	100 nm	cell-body curvature
0.020	50 nm	microvillus shaft
0.050	20 nm	sharp microvillus tip
0.100	10 nm	very sharp spike

Protrusion / indentation d (nm)

Per-vertex signed normal-projected displacement of each vertex from a smoothed reference surface generated from the same mesh. The reference is produced by random-walk Laplacian iteration on vertex coordinates (vnew = v − λ D−1 L v, λ = 0.5) ^[5] with the iteration count chosen so the effective smoothing scale is σ = 60 nm given the mesh's mean edge length: N ≈ σ²/(2h²λ). The signed deviation is the inward-normal projection of the original-minus-smoothed displacement, so d > 0 = vertex protrudes outward into ECS, d < 0 = indentation.

Scale-aware where curvature isn't. Curvature answers "is the surface bent here, and which way?"; protrusion answers "does this point stick out (or in) compared to its ~60 nm neighbourhood?". A gentle 100 nm-tall ridge gives large positive d but small |H|; a 5 nm bump on flat membrane gives small d but large |H|. The two channels agree at microvillar features (sharp tip + reaching shaft) and diverge over gently undulating surfaces.

Local contact gap g (nm)

3D Euclidean distance transform (EDT) ^[6,7] of the "not-other-cell" indicator field, sampled at the rounded voxel coordinate of each mesh vertex. gi is therefore the distance from vertex i to the nearest voxel belonging to any cell other than the one being analysed. A membrane patch was isolated by retaining only vertices within one voxel (max-norm) of an ECS-labelled voxel; the per-vertex test was dilated by two ring-neighbour iterations on the mesh edge graph to close scattered single-vertex dropouts. Faces with ≥ 50% ECS-facing vertices were included in the patch. Faces within two voxels of any volume face were dropped from the patch to avoid marching-cubes cap-face artifacts.

Volume-boundary handling

Two boundary effects are corrected in the per-channel rendering and summary statistics.

(i) Gap channel. The in-volume EDT overestimates the gap whenever the nearest neighbouring cell lies outside the crop. Formally, for every vertex gitrue ≤ min(giEDT, diwall) where diwall is the L∞ distance from vertex i to the nearest of the six volume faces. Vertices satisfying giEDT > diwall are flagged as boundary-uncertain and dropped from the gap channel (rather than silently clipped — clipping just paints a low-value rim, swapping one artifact for another). The per-patch boundary-uncertain fraction (bd-clip, shown on each gallery card) is the quality indicator.

(ii) Curvature + protrusion channels. The cotangent Laplacian and the 60 nm smoothing kernel both reach beyond the patch rim into the marching-cubes cap face at the volume boundary, biasing values inward and painting an artificial protrusion stripe along the edge of every patch. Faces whose vertices lie within σ = 60 nm of any volume face are therefore dropped from the curvature + protrusion render. Patch geometry and per-cell statistics (face counts, ECS-facing area fraction) are reported on the unfiltered patch, so the boundary trims affect rendering and per-channel summaries but not the geometric denominator.

Per-crop statistics emitted

For each crop's patch the manifest records: patch face count, full-cell mesh face count, ECS-facing fraction, gap-channel face count, boundary-uncertain fraction, dataset and cell id, adaptive gap colormap range, and per-channel signed/unsigned percentile statistics (p10/p50/p90 of H, d, g; |H| and |d| at the same percentiles; convex/concave and protrusion/indent fractions). See results/membrane_topology_per_crop.csv for the full per-crop table.

Visualisation

The interactive 3D gallery exports each per-vertex scalar as a vertex-coloured glTF 2.0 (.glb) mesh via trimesh ^[8] coloured with the matplotlib RdBu_r (curvature, protrusion) and viridis (gap) colormaps; meshes are rendered in the browser by the <model-viewer> web component ^[9]. Each gallery card also exposes a Neuroglancer ^[10] link wired to the underlying crop with the EM, ECS silhouette mesh, and all cell meshes pre-loaded for visual verification against the raw FIB-SEM EM. The volumes themselves are hosted by the CellMap project ^[11,12]. The numerical pipeline runs on top of NumPy ^[13] and SciPy ^[7].

Results — overall

The Liver Chemical pool shows the largest membrane topology magnitudes (median |H| = 0.0048 1/nm, R ≈ 208 nm; median |d| = 3.2 nm) and the largest Chem–HPF gap (Liver HPF: |H| = 0.0024 1/nm, |d| = 1.4 nm). Heart and Kidney show smaller fixation differences. The contact-gap median on the ECS-facing patch is 25–90 nm across tissues; Cortex Chemical is the tightest (g = 16 nm) and Heart HPF the widest (g = 88 nm). bd-clip is below 0.21 in every (tissue, prep) cell, indicating the gap channel is data-driven for almost all crops.

Tissue	Fixation	n	|H| (1/nm)	|d| (nm)	g (nm)	bd-clip
Cortex	Chemical	7	0.0026 [0.0018, 0.0037]	1.87 [1.08, 2.22]	16.0 [16.0, 35.8]	0.00
Cortex	Rapid HPF	5	0.0025 [0.0019, 0.0036]	1.70 [1.18, 2.59]	27.7 [22.6, 27.7]	0.01
Heart	Chemical	4	0.0029 [0.0023, 0.0059]	1.81 [1.47, 3.04]	69.3 [29.0, 91.6]	0.14
Heart	Rapid HPF	4	0.0032 [0.0028, 0.0038]	1.91 [1.72, 2.42]	88.1 [44.4, 165.7]	0.18
Kidney	Chemical	7	0.0031 [0.0021, 0.0038]	2.30 [1.20, 2.54]	48.0 [22.6, 73.3]	0.18
Kidney	Rapid HPF	6	0.0034 [0.0024, 0.0057]	2.19 [1.36, 3.74]	51.5 [36.3, 129.8]	0.21
Liver	Chemical	12	0.0048 [0.0023, 0.0063]	3.18 [1.29, 4.12]	42.5 [23.9, 99.0]	0.11
Liver	Rapid HPF	10	0.0024 [0.0015, 0.0038]	1.44 [0.74, 2.07]	41.9 [27.7, 58.6]	0.04

Each cell reports the median across crops of the per-crop median of the absolute scalar value, with [Q1, Q3] across crops. n is the number of crops in that (tissue, prep) cell.

Results — region-matched

Eight region groups have both Chemical and HPF representation in the crop set. Within these matched pools:

• Liver Hepatocyte lateral — Chemical ≈ HPF. Both report a tight, smooth interface (Chem |H| = 0.0023 vs HPF 0.0016; |d| = 1.3 vs 0.8 nm; g = 25 vs 30 nm).
• Liver Bile canaliculus — Chemical shows substantially sharper, more protrusive membranes than HPF (Chem |H| = 0.0075, |d| = 4.4 nm vs HPF 0.0035, 2.0 nm). Chemical fixation appears to over-resolve / sharpen the microvillar brush border.
• Heart Intercalated disc — HPF higher than Chemical (HPF |H| = 0.0034, |d| = 2.2 vs Chem 0.0026, 1.7) — opposite of the Bile canaliculus trend. Consistent with HPF preserving the interdigitating microvilli that Chemical collapses.
• Kidney Glomerular — HPF higher than Chemical (|H| = 0.0047 vs 0.0037; g = 144 vs 87 nm).
• Other region groups have small n per prep — read trends off the per-crop CSV rather than the aggregate.

Tissue	Region group	n (Chem/HPF)	|H| (1/nm)		|d| (nm)		g (nm)
			Chem	HPF	Chem	HPF	Chem	HPF
Heart	Cardiac interstitial	2/2	0.0049 [0.0029, 0.0068]	0.0031 [0.0028, 0.0033]	2.54 [1.68, 3.40]	1.82 [1.67, 1.96]	91.2 [90.5, 91.9]	88.1 [81.6, 94.7]
Heart	Intercalated disc	2/2	0.0026 [0.0021, 0.0030]	0.0034 [0.0030, 0.0039]	1.67 [1.39, 1.94]	2.22 [1.86, 2.58]	35.3 [22.6, 48.0]	110.7 [32.0, 189.3]
Kidney	DCT base	1/3	0.0021 [0.0021, 0.0021]	0.0025 [0.0022, 0.0029]	1.20 [1.20, 1.20]	1.41 [1.19, 1.77]	22.6 [22.6, 22.6]	39.2 [27.7, 45.3]
Kidney	Glomerular	2/2	0.0037 [0.0036, 0.0038]	0.0047 [0.0039, 0.0055]	2.50 [2.46, 2.54]	3.14 [2.60, 3.68]	87.3 [73.3, 101.2]	144.5 [57.7, 231.3]
Kidney	PCT lateral	1/1	0.0016 [0.0016, 0.0016]	0.0063 [0.0063, 0.0063]	0.78 [0.78, 0.78]	3.92 [3.92, 3.92]	22.6 [22.6, 22.6]	96.0 [96.0, 96.0]
Liver	Bile canaliculus	3/4	0.0075 [0.0053, 0.0078]	0.0035 [0.0031, 0.0042]	4.41 [3.28, 4.54]	1.97 [1.84, 2.16]	53.1 [32.0, 59.9]	51.8 [48.6, 69.6]
Liver	Hepatocyte lateral	6/6	0.0023 [0.0014, 0.0034]	0.0016 [0.0014, 0.0023]	1.33 [0.72, 2.00]	0.84 [0.66, 1.39]	25.2 [22.6, 49.9]	29.9 [26.4, 51.2]

Figures

The membrane-topology and effect-size graphs are shown below. The full set of per-metric Chemical-vs-HPF plots (volume fraction, ECS width, Voronoi gap, SA:V, both native and resolution-matched) plus region vignettes and 3D renders is in the figures gallery.

Headline observations

Reading these as observations from the 55-crop set, not yet biological conclusions:

1. Chemical fixation has opposite effects on two microvillar interfaces. Liver Bile canaliculus is sharper and more protrusive under Chemical than under HPF (|H| ratio 2.11×, |d| ratio 2.24×) — consistent with chemical fixation rigidifying or over-resolving the canalicular brush border. Heart Intercalated disc runs the other way: HPF is sharper than Chemical (|H| ratio 0.74, |d| ratio 0.75) — consistent with chemical fixation collapsing the interdigitating microvilli at the disc. The same fixation condition produces opposite topology changes at two microvillar interfaces. A single “Chemical smooths” / “Chemical sharpens” narrative doesn't fit the data.
2. Kidney Glomerular: HPF preserves a substantially larger interstitial gap. Median g = 144 nm under HPF vs 87 nm under Chemical — a 1.66× preservation. HPF also shows higher curvature and protrusion (1.27× each), tracking the genuine podocyte / endothelium / basal-membrane morphology that chemical fixation collapses into a flatter, narrower interface.
3. Liver Hepatocyte lateral is the calmest baseline, but also the noisiest peer group. Median values (|H| 0.0023 vs 0.0016, g 25 vs 30 nm) suggest a real tight apposition with weak fixation effect. But all four outlier candidates land in this group: crop1044 (Chemical) and crop1071 (HPF) both at >2× the group median on |H| and |d|. The peer-group label may be too coarse (Kupffer / sinusoidal interfaces likely confounded with hepatocyte–hepatocyte appositions), or the cell-selection heuristic may have picked a non-representative cell. The flagged crops are linked on the gallery for review.
4. Cortex Chemical patches are tightly packed. Median g = 16 nm — the smallest in the dataset — with median bd-clip = 0.00 across the 7-crop pool, indicating the reading is data-driven, not boundary-affected. Consistent with the historically reported severe ECS reduction in chemically-fixed cortex (Korogod et al. 2015 territory).
5. Outlier exclusion does not flip any Chem-vs-HPF conclusion. Excluding the 4 candidate outliers from Hepatocyte lateral leaves every direction unchanged; the Chemical/HPF |H| and |d| ratios actually shift slightly more extreme (1.44 → 1.56 on H, 1.58 → 1.87 on d). The matched-region story does not depend on the suspicious crops.
6. The Bile-canaliculus vs Intercalated-disc divergence is a finding the Voronoi-only gap metric (Metric 5) cannot see. Metric 5 reports gap and only gap; it cannot distinguish “sharpened brush border” from “collapsed brush border” if their gap distributions happen to look similar. The shape channels in Metric 8 are what make observation 1 visible.

Outlier candidates within annotated peer groups

For each annotation group ((tissue, region group, fixation) with n ≥ 3 crops), the per-crop median curvature |H| and median protrusion |d| are compared against the within-group median. Any crop whose ratio to the group median exceeds 1.7× or falls below 0.6× on either metric is flagged. Singleton and n=2 groups are skipped (the median is ill-defined for a peer of one or two). These are candidates for re-annotation or sub-grouping, not automatic rejections — the gallery cards surface a ⚠ outlier badge so the candidates aren't forgotten. Re-annotation decisions belong to the annotator (Kayvon, Wei-Ping). The flagged list:

Crop	Tissue	Region group	Prep	n	|H| vs group	|d| vs group	Reason
crop1044	Liver	Hepatocyte lateral	Chemical	6	2.60×	2.70×	|H| 0.0061 is 2.60× the Hepatocyte lateral Chemical median (0.0023, n=6); |d| 3.60 nm is 2.70× the Hepatocyte lateral Chemical median (1.33 nm, n=6)
crop1118	Liver	Hepatocyte lateral	Chemical	6	0.64×	0.59×	|d| 0.79 nm is 0.59× the Hepatocyte lateral Chemical median (1.33 nm, n=6)
crop1121	Liver	Hepatocyte lateral	Chemical	6	0.45×	0.38×	|H| 0.0011 is 0.45× the Hepatocyte lateral Chemical median (0.0023, n=6); |d| 0.50 nm is 0.38× the Hepatocyte lateral Chemical median (1.33 nm, n=6)
crop1071	Liver	Hepatocyte lateral	Rapid HPF	6	2.35×	2.81×	|H| 0.0038 is 2.35× the Hepatocyte lateral Rapid HPF median (0.0016, n=6); |d| 2.36 nm is 2.81× the Hepatocyte lateral Rapid HPF median (0.84 nm, n=6)

Heuristic rationale: within a correctly-annotated region group the per-crop topology should cluster (the membrane biology should be the same modulo crop-level variation). A >1.7× or <0.6× spread on either |H| or |d| is well outside the typical within-group MAD observed across the other 6 groups, so an outlier here usually indicates either (a) the cell selected by the most-ECS-facing-surface heuristic happens to capture a non-representative part of the tissue (e.g.~a Kupffer cell rather than a hepatocyte in the Hepatocyte lateral pool), (b) the underlying anatomical substructure is heterogeneous and the region group label is too coarse, or (c) the crop genuinely sits at the tail of the biological distribution.

Sensitivity of Chem-vs-HPF to outlier exclusion

To check whether the candidate outliers drive the Chem-vs-HPF comparison in their region groups, every aggregate was recomputed with the flagged crops removed. The table below shows the per-region Chemical-to-HPF ratio of the per-crop median values (so values > 1 mean Chemical is higher than HPF on that channel) for every region with both prep arms, computed twice: with all crops and with the candidate outliers excluded. Only region groups whose n changed under exclusion are highlighted; the others reproduce verbatim (the outliers all sit in one region group in the current data).

Tissue	Region group	All crops				Outliers excluded
		n (C/H)	H	d	g	n (C/H)	H	d	g
Heart	Cardiac interstitial	2/2	1.59	1.40	1.04	2/2	1.59	1.40	1.04
Heart	Intercalated disc	2/2	0.74	0.75	0.32	2/2	0.74	0.75	0.32
Kidney	DCT base	1/3	0.82	0.85	0.58	1/3	0.82	0.85	0.58
Kidney	Glomerular	2/2	0.79	0.80	0.60	2/2	0.79	0.80	0.60
Kidney	PCT lateral	1/1	0.26	0.20	0.24	1/1	0.26	0.20	0.24
Liver	Bile canaliculus	3/4	2.11	2.24	1.02	3/4	2.11	2.24	1.02
Liver	Hepatocyte lateral	6/6	1.44	1.58	0.84	3/5	1.56	1.87	1.00

Reading the table: a "1.50" in the H column means the per-region median |H| of the Chemical crops is 1.5× that of the HPF crops; a value < 1 means HPF higher. In the current data the only region group whose ratio changes is Liver Hepatocyte lateral, and the direction of the Chem-vs-HPF effect is preserved when the outliers are excluded — H and d ratios drift slightly higher (Chem ≫ HPF becomes a touch more pronounced), and the gap ratio shifts from 0.84 (HPF wider) to 1.00 (parity). No flagged outlier flips the sign of any Chem-vs-HPF comparison.

Methods comparison vs Metric 5 (Voronoi gap)

The mesh-based contact gap channel measures the same physical quantity as the manuscript's Voronoi-tessellation Metric 5, but the two operate on different supports and disagree in instructive ways.

	Metric 5 — Voronoi gap	Metric 8 — Mesh gap (this analysis)
Support	every ECS Voronoi-boundary voxel (population of all cell–cell appositions in the crop)	vertices of one ECS-facing membrane patch (the most ECS-rich cell, per-vertex distribution)
Voxel floor	~14 nm at 8 nm voxels — drives a documented Chem-vs-HPF artifact at thresholds < 20 nm	same EDT floor, but every crop downsamples to a uniform 16 nm working voxel — floor is shared between Chem and HPF, eliminating the prep×voxel confound
Open-space inflation	large ECS pools (vessel lumen, bile canaliculus) widen the Voronoi boundary and inflate the population median	EDT sampled at the membrane surface, so the reading is "distance from this membrane point to the nearest other cell" — open pools register as correctly high gap on the relevant patch vertices
Boundary handling	boundary overestimates absorbed into the crop-level median	per-vertex boundary uncertainty made explicit and dropped from the gap channel; per-crop bd-clip quantifies the loss
Co-located channels	gap only	gap colocated with curvature and protrusion on the same support — supports per-vertex queries like "what is the gap at the microvillus tip vs base"

The two metrics answer complementary questions: Metric 5 characterises the population of cell–cell appositions per crop; Metric 8 characterises the shape and contact landscape of one representative cell's ECS-facing surface. Consilience across both metrics (e.g. Liver Hepatocyte lateral Chem ≈ HPF, Heart Intercalated disc HPF-preserved, Liver Bile canaliculus Chem-sharpens) is taken as confirmation; disagreements are usually traceable to support / scale.

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