Measuring 3D membrane curvature from electron microscopy
By modeling the membrane as a continuous phase field, our approach enables direct estimation of biophysically relevant geometric quantities, including mean and Gaussian curvature.
Applied to endocytic pits in cells, 3D curvature analysis reveals anisotropic curvature at the pit neck — previously predicted to be a lower-energy pathway for constricting the membrane neck.
Volumetric electron microscopy — cryo-electron tomography, FIB-SEM — is producing large, high-quality public datasets of organelles and cellular membrane structures. To understand the physical drivers of membrane reshaping, we need accurate measurements of quantities like membrane curvature, and curvature has been hard to measure well in 3D.
Small geometric errors introduced during segmentation are amplified during numerical differentiation.
Curvature depends on higher-order derivatives of the underlying surface. Each step in the chain — surface, normal, curvature — takes another derivative, so a reconstruction can be accurate in position and still be inaccurate in curvature. Voxel-level accuracy does not guarantee curvature-level accuracy.
What a curvature-ready reconstruction has to satisfy
From the paper's model formulation: (1) the geometry should be consistent with the volumetric image data; (2) its curvature distribution should be consistent with a membrane-mechanics model; (3) it should accommodate topologically complex membranes without manual intervention; (4) it should remain robust to imaging noise and artifacts; (5) it should permit direct computation of differential geometric quantities. §2 Model formulation ↗
Represent the membrane as a continuous function — and train it against the image and membrane physics at once.
So don't label voxels. Represent the membrane as a continuous function — a small neural network that maps any point in space to a single number, the phase field, with the membrane sitting where that number crosses zero. Because it's a function rather than a grid, you can differentiate it directly. Curvature comes out of automatic differentiation, at any resolution, without needing to build a mesh.
Then train it against two things at once: the image, and membrane physics. The physics term is the Helfrich bending energy — the same elasticity that governs how a real lipid bilayer resists being bent.
Is it AI? Only a little!
The implicit neural representation is itself a neural network, just like the machine learning algorithms. Physics and smoothing loss functions condition the neural network. But it's not a generative model the way that ChatGPT is.
Why the smoothing term is the expensive one
Each loss climbs the derivative ladder of φ. The data and boundary losses need ∇φ (first derivatives). The physics loss — the phase-field Helfrich energy — needs Δφ (second). The smoothing loss penalizes |∇ₛH|², the surface gradient of mean curvature: third derivatives. And every training step differentiates its loss once more with respect to the network weights. This is also why the losses are introduced in stages — the high-order terms are poorly behaved while the field is far from the solution, so training runs data/boundary first, then adds physics, then smoothing. §6.2 Loss functions ↗
Sequential addition of the physics and smoothing losses improved the accuracy of recovered curvature.
The benchmarks have a known ground truth: membrane shapes generated with the mechanics simulator Mem3DG, voxelized and degraded into synthetic EM volumes. Surface Dice — agreement in position — remained near 1.0 at every optimization stage. Surface normals agreed closely after the first, image-only stage. Mean and Gaussian curvature did not: their correlations with ground truth were poor (r = 0.45 and 0.23 for the bud shape). Adding the physics loss (stage 2) and smoothing loss (stage 3) raised every geometric quantity to r > 0.95. §3.1 ↗ Fig 3 ↗
The division of labor is the point: the data and boundary losses reconstruct the membrane's general location in space, while the physics and smoothing terms enable accurate reconstruction of the higher-order quantities — mean and Gaussian curvature.
By the final stage, both curvatures agreed strongly with the ground truth even under high noise.
With increasing noise, surface Dice stayed high — overall morphology was preserved even when the input images were very degraded — while the image-only stage's curvature correlations fell away. Physics and smoothing restored them.
Gaussian-curvature correlation with ground truth vs. noise. The image-only stage 1 (dotted) collapses as noise rises; stage 3 (solid) holds. Fig 4C ↗
The robustness has a boundary, and the paper states it: under a simulated missing wedge, when entire regions of the membrane are absent from the input, the physics and smoothing constraints reduced surface roughness but were insufficient to recover the missing global structure. The framework is robust to local degradation, not to large-scale missing information. Fig S2 ↗
Anisotropic voxels, and where the numbers come from
Voxel anisotropy up to 8× along z produced step-like artifacts after stage 1; physics (stage 2) and smoothing (stage 3) reduced them and improved curvature estimates. Fig S1 ↗
Correlation values shown in the stage chart are the printed Pearson r for the Ω-shaped bud in Fig 3B. The biconcave-disk values follow the same pattern (θ: 0.989 → 0.996 → 0.998; H: 0.392 → 0.913 → 0.995; K: 0.222 → 0.689 → 0.958). Lab notebook: RES — sequential loss addition, RES — noise robustness (access required).
The same framework reconstructed the Golgi apparatus and a mitochondrion from experimental EM data.
Two public datasets, two imaging modalities: the Golgi apparatus of Chlamydomonas reinhardtii from in situ cryo-electron tomograms, and a neuronal mitochondrion from serial-section tomograms of mouse cerebellum. After point-signal extraction, the remaining reconstruction was automated — no topology specification, no mesh repair. The reconstructions preserved the stacked cisternae and fenestrations of the Golgi and the nested double-membrane architecture of the mitochondrion, and the continuous representation let mean and Gaussian curvature be computed across both. §3.3 ↗
The overlay (second half) is the optimized phase field on the same slices — labels are burned into the frames.
The biological application: clathrin-mediated endocytosis, where the membrane invaginates and forms a narrow neck before a vesicle pinches off. From FIB-SEM volumes of COS-7 cells, five endocytic pits were reconstructed. As expected for a budding geometry, mean curvature increased toward the bud tip and Gaussian curvature was negative at the neck. But unlike axisymmetric idealizations, the curvature maps varied along the azimuthal direction — around the neck. §3.4 ↗
The reconstructed buds were azimuthally heterogeneous in curvature at the neck — particularly at intermediate and late stages.
Result statement from the lab notebook, and the finding of Fig 6 ↗. Explore it on the real reconstructions:
What these objects are: the five endocytic-pit reconstructions of Fig 6 and, drawn on them, the first author's own contour-ring analysis data (the stack-bud files behind Fig 6C–E). Each bud is shown in its analysis frame — fitted bud axis vertical, neck plane at z = 0 — and the traces and SD readouts are that analysis, replotted in your browser: the neck-ring Gaussian-curvature SDs here are the Fig 6C values. An earlier version of this page recomputed the same pattern independently from the raw meshes; the analysis-vs-recomputation comparison is recorded in the project notes. (The example mesh shipped with the public code is Bud-04 — confirmed by the first author.)
In other words: at sites of endocytosis, the necks of endocytic pits can be "squished" like a coke can at intermediate and late stages of endocytosis. In the paper's terms, such coke-can-like buckling is thought to reduce the energy barrier for constricting the neck prior to vesicle scission.
The shape was predicted before it was measured. In prior simulation work, uniform constriction forces on a 3D membrane tube led to asymmetric, pinched-tube deformation (Vasan et al. 2020) — no asymmetric squeeze required. Non-axisymmetric buckling can lower the energy barrier associated with neck constriction, and these measurements provide geometric evidence supporting that prediction. EVD — uniform constriction ↗
Scope, stated as in the paper: the analysis covers a small number of reconstructed pits (n = 5, one cell line, one dataset), so the biological conclusions are preliminary until the approach is applied across multiple datasets.
Clone and run — or ask your AI to run — github.com/ctleelab/pinn-exploration-model. The curvature-analysis notebook runs from the committed checkpoint; JAX with GPU acceleration recommended for training.
Is it easy to run yet? That's what we're working on next.
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