Ali Asghar Hakami Zanjani
Membrane shape is commonly described by curvature, area, and thickness, which underpin bilayer mechanics and protein–membrane interactions. In molecular dynamics (MD) simulations, these quantities must be inferred from discrete coordinates, and common grid-based smoothing approaches can introduce resolution-dependent artefacts and ambiguities, especially under periodic boundary conditions. Here we present CATpie, a Python framework for a unified geometric analysis of simulated membranes represented as smooth leaflet height fields over a periodic domain (from user-defined leaflet reference atoms or beads). CATpie fits real Fourier series to the upper and lower leaflets, constructs the mid-surface, and evaluates mean curvature, Gaussian curvature, and real area from analytic expressions of the fitted surfaces. It further defines a normal-line thickness as the distance between the intersections of each mid-surface normal with the two leaflets, providing a geometrically controlled measure of leaflet separation that complements the commonly used vertical separation D z z . Validation on analytic benchmark surfaces, including a sinusoidal surface with zero Gaussian curvature and a separable cosine surface with non-zero Gaussian curvature, shows high-accuracy recovery of curvature fields, real areas, and leaflet separation. Application to a mixed POPS/POPC bilayer demonstrates that mid-surface curvature and normal-line thickness quantify local deformations and thickness modulations, while the ratio T norm / D z z highlights regions where vertical separation overestimates normal-line spacing due to membrane tilt. CATpie supports atomistic and coarse-grained simulations of planar and undulated bilayers that remain representable as single-valued leaflet height fields z = Z ( x , y ) over a periodic domain (without overhangs or topological changes), enabling systematic geometric characterisation of membrane simulations.