Helfrich Monte Carlo Flexible Fitting: physics-based, data-driven cell-scale simulations

Helfrich Monte Carlo Flexible Fitting: physics-based, data-driven cell-scale simulations | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results Helfrich Monte Carlo Flexible Fitting: physics-based, data-driven cell-scale simulations View ORCID Profile Valentin J. Maurer , View ORCID Profile Marc Siggel , View ORCID Profile Rasmus K. Jensen , View ORCID Profile Julia Mahamid , View ORCID Profile Jan Kosinski , View ORCID Profile Weria Pezeshkian doi: https://doi.org/10.1101/2025.05.24.655915 Valentin J. Maurer 1 Structural Biology Unit, European Molecular Biology Laboratory , Notkestraße 85, 22607 Hamburg, Germany 2 Centre for Structural Systems Biology, CSSB , Notkestraße 85, 22607 Hamburg, Germany 3 Institute of Molecular Biology and Biophysics, ETH Zurich , 8092 Zurich, Switzerland Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Valentin J. Maurer Marc Siggel 1 Structural Biology Unit, European Molecular Biology Laboratory , Notkestraße 85, 22607 Hamburg, Germany 2 Centre for Structural Systems Biology, CSSB , Notkestraße 85, 22607 Hamburg, Germany Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Marc Siggel Rasmus K. Jensen 4 Molecular Systems Biology Unit, European Molecular Biology Laboratory , Meyerhofstrasse 1, 69117 Heidelberg, Germany Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Rasmus K. Jensen Julia Mahamid 4 Molecular Systems Biology Unit, European Molecular Biology Laboratory , Meyerhofstrasse 1, 69117 Heidelberg, Germany Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Julia Mahamid Jan Kosinski 1 Structural Biology Unit, European Molecular Biology Laboratory , Notkestraße 85, 22607 Hamburg, Germany 2 Centre for Structural Systems Biology, CSSB , Notkestraße 85, 22607 Hamburg, Germany 4 Molecular Systems Biology Unit, European Molecular Biology Laboratory , Meyerhofstrasse 1, 69117 Heidelberg, Germany Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Jan Kosinski Weria Pezeshkian 5 Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen , Blegdamsvej 17, 2100 Copenhagen, Denmark Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Weria Pezeshkian For correspondence: weria.pezeshkian{at}nbi.ku.dk Abstract Full Text Info/History Metrics Supplementary material Preview PDF Abstract Simulating entire cells represents the next frontier of computational biology. Achieving this goal requires methods that accurately reconstruct and simulate cellular membranes across spatial and temporal scales. Although three-dimensional electron microscopy enables detailed membrane visualization, dose limitations and restricted field-of-view often result in fragmented membrane representations incompatible with simulations. To resolve this, here we introduce Helfrich Monte Carlo Flexible Fitting (HMFF), a novel approach that integrates experimental density data into membrane simulations. Together with the accompanying Mosaic software ecosystem, HMFF translates static imaging data into complete and physically regularized simulation-ready models at micrometer scales. We demonstrate the versatility of HMFF across diverse biological systems, including influenza virus particles, Mycoplasma pneumoniae cells, and entire eukaryotic organelles. These models enable multi-scale simulations involving millions of lipids, incorporate proteins at experimentally determined positions, and allow for quantitative morphological analysis of cellular compartments. HMFF thus establishes a foundation for data-driven whole-cell simulations. I. MAIN A cornerstone of modern biological research is deciphering cellular architecture and behavior, from individual molecules to the entire cellular landscape 1 , 2 . This requires an integrative approach that merges state-of-the-art experimental methods, such as cryogenic electron microscopy (cryo-EM) 3 – 5 , with cutting-edge computational advances, including multi-scale molecular simulations, from mesoscale to all-atom molecular dynamics (MD) 6 , 7 . These approaches operate in concert, overcoming individual limitations to bring us closer to building computationally tractable models resolving both structural and dynamic aspects of the cell 8 – 12 . Within the scope of cellular architecture, biomembranes represent a particularly complex and essential component that requires specialized modeling strategies. Biomembranes play a vital role in many cellular processes such as signaling, transport, and compartmentalization across all domains of life. They form complex, curved, and branched structures, spanning from nanometer to micrometer scale 13 – 15 . Their constant turnover and reshaping occur over time scales ranging from nanoseconds to minutes. Resolving membrane architecture, however, is a challenging task, both experimentally and computationally 6 , 13 , 16 , 17 . While cryo-EM provides valuable structural details, it suffers from resolution limitations and provides only static snapshots of inherently dynamic systems 3 , 18 – 22 . On the other hand, molecular modeling faces challenges such as insufficient information about membrane composition, timescale limitations, and system parameter uncertainties 6 , 7 , 17 . A promising approach to accurately determine membrane structure is to develop an integrative modeling scheme, which combines physics-based modeling techniques with experimental data as constraints to infer structural details that remain hidden from experiments alone. To achieve this, here we introduce Helfrich Monte Carlo Flexible Fitting (HMFF), which uniquely integrates density data from three-dimensional EM (3D EM) directly into dynamically triangulated surface (DTS) mesoscale simulations of biomembranes 7 , 23 – 25 . By extending the classical Helfrich model with an energy term derived from experimental observations 26 – 29 , HMFF creates a unified framework where experimental constraints guide membrane conformations while physical principles ensure biological realism. To enable practical implementation, we developed Mosaic, a software ecosystem that streamlines the complete workflow from experimental data to simulation-ready models. Mosaic unifies membrane segmentation, mesh generation, protein identification, and multi-scale simulation, making HMFF accessible for routine use in cell-scale membrane biophysics. This integrated approach demonstrates robustness across scales, enabling simulations spanning from individual membrane proteins to complex systems with millions of lipids, as mesoscale models can be systematically backmapped 30 , 31 to molecular-level simulations and refined into all-atom structures. We demonstrate the strength of our approach by show-casing it across biological systems, length scales, and experimental methods. We generate complete membrane models of partially imaged Mycoplasma pneumoniae cells from cryo-ET data 32 , 33 , refine influenza A virus (IAV) segmentations for surface-guided glycoprotein localization from cryo-ET data 34 , 35 , and prepare simulation-ready models of entire eukaryotic organelles based on focused ion beam scanning electron microscopy (FIB-SEM) volumetric data 21 . Our approach bridges from static experimental observations to dynamic multi-scale simulations, enabling a mechanistic understanding of complex membrane systems and represents an advance toward comprehensive whole-cell modeling. II. RESULTS A. Integrating experiment and simulation using HMFF We developed HMFF to create accurate models of complex membrane systems in a data-driven manner. To achieve this, HMFF uniquely integrates 3D EM density data into membrane simulations, enabling experimental observations and physical properties to jointly determine membrane conformations. HMFF extends the DTS mesoscale simulation approach 24 , 25 , in which membranes are modeled as triangular meshes with physical properties and treated as elastic two-dimensional sheets with in-plane fluidity, embedded in three-dimensional space. The membrane geometry is coupled to the bending energy through the Helfrich model 23 , which regulates membrane shape and smoothness, with additional biologically relevant constraints such as total volume (often related to osmotic pressure), total area, and area difference between monolayers 24 , 25 . To create accurate membrane models, HMFF introduces an energy term in the DTS model that guides membrane vertices toward regions of high density in experimental data. We adapted this term from Molecular Dynamics Flexible Fitting (MDFF) of proteins 26 – 29 , implementing automatic density threshold selection and interpolation at volume boundaries. This creates a dual-constraint system where biophysical properties regularize global membrane shape while experimental density provides local structural guidance. By balancing these complementary forces, HMFF reconstructs membranes that are both physically realistic and experimentally accurate, capturing deviations from minimal energy configurations that arise from factors such as osmotic pressure, imaging grid adhesion, or protein-membrane interactions that would otherwise be infeasible to detect accurately. We implemented HMFF as part of the FreeDTS 24 mesoscale simulation method and validated the approach using a model system consisting of a planar periodic membrane mesh in a Gaussian density. For 3D EM data, we derive such membrane meshes from membrane segmentations as shown in subsequent sections. As the HMFF potential pulls vertices upward, the membrane initially curves ( Fig. 1a , Mov. M1), against its bending resistance. At equilibrium, the membrane has reached a higher elevation, returning to its preferred flat configuration, satisfying both the HMFF potential and physical membrane properties. Download figure Open in new tab Fig. 1. HMFF concept and validation with a synthetic model system. a , Planar membrane mesh with periodic-boundary conditions embedded in a Gaussian density depicted as a grayscale gradient (white=low, black=high density) projected to the backplane of the box. Shown are three configurations, corresponding to 0, 30, and 100 thousand simulation steps. b , Comparison of relative HMFF potential across simulation steps for different values of the interaction parameter, showcasing the propensity to optimize the HMFF potential when increasing data importance through ξ . Potentials were min-max scaled to 0-1 for each value of ξ to facilitate curve shape comparison. HMFF incorporates a hyperparameter ξ , which balances the contribution of the data against physical regularization. Increasing ξ led to faster convergence to-ward the highest-density regions, while values approaching zero resulted in essentially stationary membranes, demonstrating the direct relationship between potential coupling and membrane behavior ( Fig. 1b ). In extreme cases with ξ → ∞, the simulation is purely driven by the data, whereas ξ → 0 recapitulates the Helfrich model. Similar assessments have been made for bilayer asymmetry parameters 36 . Like in MDFF 26 – 29 , the choice of ξ in HMFF is system-specific and must be balanced and refined in a practical setting. B. The Mosaic ecosystem: Bridging the gap between 3D EM and physical simulations While HMFF provides the theoretical framework for data-driven membrane modeling, practical implementation with experimental 3D EM data requires addressing numerous technical challenges, including imaging artifacts, incomplete membrane coverage, and noise. To make HMFF accessible for routine use, we developed Mosaic, a software ecosystem that unifies 3D EM membrane segmentation, simulation-ready mesh generation for HMFF, protein identification, and multi-scale simulation, in a graphical user interface (GUI). We first show-case the Mosaic workflow and HMFF for cryo-ET data of the human pathogen M. pneumoniae ( Fig. 2 ). Download figure Open in new tab Fig. 2. Workflow enabled by Mosaic illustrated on M. pneumoniae cryo-ET data. a , Main graphical user interface with multiview display of tomogram and segmentation data. b , Progressive refinement of membrane segmentations from raw data (left) to cleaned point clouds (right). c , Generation of simulation-ready triangular meshes from refined segmentations. d , Downstream analyses enabled by Mosaic. Euclidean distance to template-matched ribosomes is presented as median ± 1.5 interquartile range. Points correspond to individual ribosomes. The violin plot shows the distribution of Gaussian curvature with median lines. Hull refers to a convex hull fitted to refined membrane segmentation, Initial to the mesh from polyharmonic deformation, HMFF to the optimized mesh, and Seg. to the refined membrane segmentation. Contact points correspond to the quantized Euclidean distance between the HMFF mesh and ribosomes. The right panels demonstrate the workflow to set up multi-scale simulations, integrating experimentally determined protein positions. Mosaic integrates deep-learning-based membrane segmentation 18 with interactive refinement tools to create membrane meshes for HMFF. Initial automated segmentations required substantial refinement ( Fig. 2b , S1a-b ). Using Mosaic’s interactive tools, we manually removed incorrectly labeled regions based on visual assessment, primarily in the cytosol and at tomogram boundaries. We thinned the membrane to single-voxel thickness, an essential step for generating simulation-ready meshes, reducing the number of labeled voxels to 4% of the initial segmentation (86,992 voxels after thinning, Fig. S1c , Mov. M2). This particular M. pneumoniae cell extended beyond the tomogram boundaries. Therefore, the segmentation exhibited a disjoint topology, with only opposite sections of the membrane visible and their connections outside the tomogram, complicating the creation of closed membrane meshes. Standard meshing approaches like Poisson reconstruction 37 – 39 failed due to insufficient normal vector information caused by missing segments ( Fig. S1f ). To address this, Mosaic implements a biologically motivated mesh completion and polyharmonic deformation procedure, which creates closed, sufficiently sized meshes while preserving smoothness and local curvature of the observed membrane ( Fig. 2c-e , S8 ). The volume of the resulting mesh was 43% (0.091 µm 3 ) larger than the convex hull of the membrane segmentation and encapsulated all template-matched ribosomes (see methods IV K). We devised a two-step equilibration approach, consisting of remeshing and hybrid MC, to regularize mesh edge lengths to ensure stability and physical validity of the subsequent simulation ( Fig. S1e ). With a mesh prepared from membrane segmentations, we used HMFF to refine it based on experimental densities ( Fig. 2d , S2 , Mov. M3). Since the tomogram did not contain the entire M. pneumoniae bacterium, careful management of boundary effects was required. Initial simulations using zero density outside the tomogram boundaries produced artifacts, where the membrane mesh adhered to the boundaries (Mov. M4). We resolved this using a dedicated padding procedure ( Fig. S1g and methods IV B). The resulting HMFF-refined mesh accurately captured the experimentally determined membrane structure, while interpolating beyond the tomogram boundaries based on physical membrane properties ( Fig. 2d , S2 , Mov. M3). Using Mosaic, we performed quantitative analyses based on the HMFF-refined membrane mesh ( Fig. 2d left). We found that membrane representation impacts protein-membrane distance distributions. Whereas raw segmentations overestimate distances, HMFF-refined meshes provide more realistic distance estimates by generating physically regularized membrane conformations. We then leveraged the HMFF-refined mesh to identify Nap complexes using template matching with PyTME (35, see methods IV J). Nap is a transmembrane protein complex that is essential for infectivity and is conserved across Mycoplasma genera 40 . We found Nap complexes to concentrate around the specialized attachment organelle, consistent with previous observations 40 ( Fig. 2d ). We combined the HMFF-refined membrane mesh with our Nap picks to generate an in silico model for equilibrium simulation of Nap protein dynamics ( Fig. 2d right). Using TS2CG 30 , 31 , embedded in Mosaic, we converted this model to a coarse-grained representation, creating a system with a complete cell membrane and 44 Nap complexes, totaling approximately 50 million particles. We then performed a short MD simulation for equilibration of lipid tails using the Martini 3 force field 41 , 42 , demonstrating that our approach enables the construction of simulatable models suitable for investigating emergent behavior up to the near-atomistic scale ( Fig. 2d ). Taken together, the Mosaic ecosystem links static 3D EM with dynamic simulations by enabling the construction of large-scale, physically regularized models directly from experimental data, allowing mechanistic analysis of cellular processes through morphometrics and simulation. C. HMFF enables accurate membrane modeling and protein localization in influenza A virus-like particles Influenza A virus (IAV) presents unique challenges for membrane modeling due to its pleomorphic nature, ranging from spherical to elongated filamentous phenotypes. While previous computational studies have focused on spherical models 43 – 45 , elongated and filamentous IAV remain unmodeled despite their clinical significance 46 . Filamentous variants are characterized by tightly packed surface proteins and a dense matrix protein layer (M1) coating the inner membrane surface, making the creation of accurate models challenging. To build a filamentous IAV model with HMFF and Mosaic, we used publicly available cryo-ET data of filamentous virus-like particles (VLPs) 34 . Fig. 3a illustrates our workflow from raw tomographic data (panel i) to HMFF-optimized mesh (panel iv). In the experimental data, the membrane and M1 protein layers are observed in close proximity, which deep-learning-based segmentation failed to differentiate accurately, primarily segmenting the M1 layer presumably due to its high density ( Fig. 3a , Mov. M5). This erroneous labeling propagated to the initial mesh derived from the segmentation, which was positioned in the mid-plane of the M1 layer. However, this was successfully resolved by HMFF, creating a smooth surface that accurately represents the viral membrane. Download figure Open in new tab Fig. 3. HMFF application for IAV modeling and glycoprotein analysis. a , Workflow from tomographic data (EMD:11075) to optimized mesh: (i) deconvolved tomogram; (ii) membrane segmentation (red) reflecting M1 layer; (iii) initial mesh; (iv) HMFF-optimized mesh accurately modeling the viral membrane. b , Density profile of the cross-section in A , highlighting higher density in the M1 layer compared to the membrane. c , Spatial distribution of template-matched glycoproteins HA (red) and NA (orange) on the HMFF-refined mesh. d , Comparison of inter-protein geodesic distances to four nearest neighbors. e , Gaussian curvature at glycoprotein locations showing higher curvature at NA vs. HA sites (One-tailed Mann-Whitney-U-Test; **** p < 0.0001). Data is presented as median ± 1.5 interquartile range. f , Equilibrated coarse-grain model from backmapped proteins and HMFF mesh. Points correspond to Martini beads: red (HA), yellow (NA), blue (membrane). The top-right region lacks proteins due to tomogram boundaries. The underlying density profile ( Fig. 3b ) shows that the M1 layer is substantially denser than the membrane. This differential creates a gradient that would drive naive density optimization to incorrectly position the mesh within the M1 layer rather than the membrane. However, by evaluating the HMFF potential within the complete physical DTS model with appropriate volume coupling (related to internal pressure), we successfully prevented this collapse ( Fig. 3a iv ). This demonstrates that HMFF is not merely a density optimizer but rather a physically informed simulation system that maintains biologically realistic membrane configurations even when confronted with competing density gradients (Mov. M6). The HMFF-refined membrane mesh model enabled the generation of precise membrane normals, facilitating high-fidelity localization of neuraminidase (NA) and hemagglutinin (HA) proteins and their angular orientations on the virus surface ( Fig. 3b , S3d-i , see methods IV J). We projected these proteins onto the refined membrane mesh model using Mosaic, which incorporates protein orientations for accurate projection, instead of closest-distance approaches ( Fig. S9 , see methods IV E). NA localized to regions with significantly higher Gaussian curvature compared to HA, almost exclusively occupying the tip, while HA populated the cylindrical shaft regions ( Fig. 3c ). This differential localization of NA and HA is consistent with previous cryo-ET analyses 46 , 47 . However, the NA curvature distribution presented a bimodal pattern, hinting at additional drivers of NA localization. Quantitative analysis of intra-class protein distances ( Fig. 3d , Fig. S9 ) revealed characteristic spacing patterns: HA-HA distances showed a narrow, unimodal distribution centered at ~10 nm, consistent with previous studies 48 . NA-NA spacing exhibited a broader, more variable pattern, consistent with their observed bimodal curvature preference (Mov. M7). We then used Mosaic to integrate the obtained protein-localization information with the refined membrane model, and assembled and briefly equilibrated a Martini VLP model comprising 10 million particles ( Fig. 3e-f ), comparable in size to recent models of the nuclear pore complex and other large-scale systems 43 , 45 for use in further simulation studies. These results demonstrate that our HMFF-Mosaic workflow enables the routine construction of simulation-ready models directly from cryo-ET data, supporting quantitative analysis of spatial organization and dynamics from molecular interactions to complete viral particles. D. HMFF for creating computationally tractable membrane models of entire cells To demonstrate HMFF’s versatility beyond cryo-ET data and showcase its ability to extend to complex membrane topologies in eukaryotic cells, we applied it to FIB-SEM data of an entire HeLa cell 21 , spanning more than 30 µm in diameter. HMFF is agnostic to data type, requiring only volumetric data informing on membrane structures as input to generate simulation-ready models. We processed FIB-SEM data and segmentation labels from the OpenOrganelle database ( Fig. 4a-b ), focusing on mitochondria, Golgi apparatus, nucleus, endoplasmic reticulum, and plasma membrane ( Fig. 4a-c , Mov. M10). Initial 3D U-Net segmentations from the database presented numerous topological issues, such as gaps, incorrectly merged objects, and artificial invaginations that prevented a direct conversion to simulation-ready meshes ( Fig. S4a-i ). Using Mosaic’s interactive tools, we separated and processed these segmentations into 235 computationally tractable mitochondrial meshes ( Fig. 4c ). The resulting models preserved complex morphological features while maintaining suitable mesh properties for simulation. Download figure Open in new tab Fig. 4. Creating simulation-ready meshes of an entire HeLa cell. a , Top-view of FIB-SEM data slice (OpenOrganelle:jrc hela-2). b , Side-view of FIB-SEM data with plasma membrane mesh overlay (blue). c , 3D rendering of triangulated cellular components with corresponding perpendicular slices displayed on the right: (1) endoplasmic reticulum (ER) showing complex tubular network, (2) nucleus with detailed triangulation visible in magnified insets, (3) Golgi apparatus, and (4) mitochondria with branched morphology and magnified insets. HMFF refinement improved the mitochondrial membrane model, allowing it to recapitulate the outer mitochondrial membrane contours accurately. This was particularly evident in regions of high curvature, as demonstrated in the comparison between the FIB-SEM data and optimized triangulation ( Fig. 5a-b , S6a-b , Mov. M8, M9). Quantitative analysis of the simulation trajectory showed that the HMFF energy term was minimized during system evolution, serving as a proxy for the refined membrane model now occupying regions of high density, i.e., membranes ( Fig. 5c ). During simulation, both volume and surface area of the initial mesh decreased, suggesting that these quantities were initially overestimated. Throughout the simulation, surface area stabilized more quickly while volume and HMFF potential decreased more gradually before reaching equilibrium ( Fig. 5c ). We found the simulation to be robust to the choice of simulation parameters, with comparable trajectories and HMFF potential minima across typical settings ( Fig. S5 ). Download figure Open in new tab Fig. 5. HeLa mitochondria simulation, classification, and characterization. a , Comparison of initial (yellow) and HMFF-refined (blue) mitochondrion mesh after 50,000 simulation steps. Shown are mesh vertices that intersect with the FIB-SEM volume slice. b , Selected regions annotated in ( a ) at higher magnification (left) and showing volumetric representations (right). c , HMFF simulation trajectory analysis. All features were max-scaled for comparison. d , t-SNE embedding of mitochondrial descriptors, showing nine morphological clusters. Representative meshes are rendered with consistent edge lengths for size comparison. e , Heatmap of the most discriminative mitochondrial descriptors (between/within-cluster variance ratios): area/volume ratio, eigenvalues (EV), segment counts, Zernike moments (ZN), Gaussian curvature. f , Mitochondrial surface area in contact with ribosomes. Data is presented as median ± 1.5 interquartile range. Points correspond to individual mitochondria. g , Percentage of mitochondria in contact with cellular components. Mito: mitochondria, Lys: lysosome, PM: plasma membrane, ER: endoplasmic reticulum, MT: microtubules. Based on the refined membrane mesh models, we investigated whether distinct mitochondrial subpopulations exist within the HeLa cell, as previously reported in other studies 50 . The mesh representation enabled us to generate a rich morphological feature set, comprising topological, geometric, and scale-dependent descriptors (see methods IV M). Applying t-distributed Stochastic Neighbor Embedding (t-SNE) dimensionality reduction to this feature set, followed by Louvain community detection, revealed nine distinct morphological clusters ( Fig. 5d-e , S6c-d ), each characterized by unique structural signatures. Clusters 1 and 2 predominantly contained compact, ovoid mitochondria with minimal branching; cluster 3 featured elongated tubular forms; clusters 4 and 5 displayed Y-shaped branching patterns with moderate complexity; cluster 6 exhibited complex branched networks with multiple junction points; cluster 7 contained flattened, sheet-like morphologies; cluster 8 showed asymmetric, curved structures with localized dilations; and cluster 9 comprised the most topologically complex mitochondria with extensive branching and network-like organization. These morphological distinctions strongly correlated with functional characteristics. One such characteristic was ribosome-mitochondria association, which exhibited pronounced cluster-specific patterns ( Fig. 5f ) when analyzed using ribosome segmentations provided with the original dataset 21 . Clusters 7 and 9, in particular, exhibited higher ribosomal surface coverage compared to the cluster average. Second, organelle contact mapping demonstrated cluster-dependent interaction preferences ( Fig. 5g ). Notably, ER-mitochondria contact sites were predominantly observed in clusters 8 and 9 (78% and 82% of mitochondria, respectively), while microtubule associations were markedly elevated in cluster 9 (65%), suggesting functional specialization linked to morphological characteristics. This was similarly observed for lysosome-mitochondria contact sites. In summary, this analysis advances beyond previous approaches by enabling automated, physically grounded reconstruction and quantitative morphometrics of eukaryotic cells with complex membranes, capturing curvature, topology, and spatial context with high fidelity. It allows systematic identification of structural subpopulations and their functional interactions, which were previously limited to manual or low-resolution analyses. III. DISCUSSION Here, we introduced HMFF, a method that uniquely integrates experimental 3D EM density data directly into mesoscale membrane simulations, creating a novel capability with no direct precedent in the field. This integration yields physically regularized membrane models that follow experimentally determined membrane structures, spanning from nanometer-scale features to entire cellular organelles at the micrometer scale. In the past, the transformation of experimental data into simulation-ready models was either infeasible due to the scale of the systems, missing data, limitations in the analysis toolkit, or required a plethora of specialized computational tools with substantial expert intervention at transition points, ultimately limiting applicability. To overcome these limitations, we developed Mosaic, which streamlines the creation of simulation-ready multi-scale models from experimental data through a GUI for cell-scale membrane biophysics. Rather than a monolithic solution, Mosaic is an ecosystem of interconnected tools where HMFF serves as a central component. This platform provides several key advantages demonstrated across the presented examples: (1) membrane models that balance experimental observation and physical realism, as shown in the IAV example, where the membrane model was correctly positioned despite confounding adjacent high-density protein layers; (2) direct integration of protein orientations from template matching in 3D EM with membrane models, creating in silico representation that better reflect biological reality compared to idealized distributions, random placement, or geometric constraints 1 , 30 , 49 , 51 , 52 ; (3) multi-scale analysis capabilities that enable both direct DTS simulations and backmapping to coarse-grained Martini representations for near-atomistic resolution; and (4) morphological analysis through mesh-derived features. This methodological versatility enabled us to efficiently generate complex structures exceeding 40 million particles that, with advancing computational resources, are becoming increasingly tractable for simulation 1 , 24 , 30 , 49 , 53 , while facilitating downstream analysis tasks such as our systematic classification of mitochondrial morphological subtypes. Since HMFF balances experimentally observed and physical properties, precisely calibrated physical simulation parameters are required in the absence of data indicative of membrane localization to yield sensible results. For instance, we showed that the IAV VLP model needed to be pressurized to avoid collapse onto the M1 layer. This limitation is evident in cryo-ET data, where the missing wedge and limited sample thickness cause membranes oriented perpendicular to the electron beam to be poorly resolved or entirely invisible. In contrast, methods like FIB-SEM provide more complete membrane visualization across all orientations, though typically at lower resolution than cryo-ET. Such calibration could be achieved based on additional simulations or by integrating additional information sources into the density data passed to HMFF, such as verified membrane protein localizations that could serve as anchor points. It should also be noted that additional mesh vertices are required to represent the membrane accurately when increasing system size and complexity. This incurs a computational cost, which ultimately limits the set of systems that can be simulated. Although alleviated by the recent addition of parallel processing in FreeDTS 24 , further development is required to achieve whole eukaryotic cell simulations with sufficient mesh resolution to study functional membrane dynamics. Given HMFF’s demonstrated flexibility and Mosaic’s accessibility, we envision this as the foundation for a new ecosystem for processing 3D EM data. The mesh representation makes membranes mathematically tractable and suitable for machine learning applications to automate morphological analysis or protein binding site prediction. Furthermore, HMFF is particularly well-positioned to analyze emerging datasets from serial lift-out, which has been piloted to image near-complete organisms 54 , 55 . Serial lift-out uses a focused ion beam to extract thin slices of a vitrified biological sample, which are sequentially imaged through 3D EM approaches. Once established more broadly, HMFF can be used to interpolate the data between 3D EM slices and build realistic cell and organism models suitable for simulation. This capability would enable specific investigations that were previously infeasible, such as simulating how membrane-remodeling events during viral infection propagate across entire cells, where both local membrane deformation and global cellular architecture must be considered simultaneously. To summarize, HMFF establishes a methodological framework that directly connects experimental observations to dynamic membrane simulations, advancing integrative structural biology at cellular scales. The Mosaic ecosystem streamlines the associated workflow, reducing technical barriers to multi-scale modeling and broadening accessibility within the structural biology community. As computational resources expand and methods evolve, HMFF provides a robust foundation for interrogating membrane systems across hierarchical scales—from nanometer-level protein-lipid interactions to micrometer-scale cellular compartments. This integration of static experimental observations with dynamic physical simulations enables the mechanistic investigation of membrane morphology, dynamics, and emergent behaviors in complex biological systems, ultimately advancing the field toward our goal of creating foundational models of cellular systems. IV. METHODS A. Dynamically triangulated surface simulations At a scale much larger than its thickness, a membrane can be modeled as a two-dimensional surface in three-dimensional space. By the Fundamental Theorem of Surface Theory, given the metric and the curvature tensor, the surface can be reconstructed uniquely up to rigid motions, i.e., translations and rotations. Since the membrane surface is related to the number of biomolecules and is an extensive variable, the bending energy of the membrane, up to second order in curvature, can be well described by the Helfrich Hamiltonian 23 as where H, K , and C 0 are mean, Gaussian, and spontaneous curvatures, and κ and are the bending and Gaussian curvature moduli, respectively, and the integration is performed over the membrane area A . The higher-order terms in curvature are only relevant when the resolution of the description becomes comparable to the membrane thickness. It has been shown that Eq. 1 is suitable for length scales larger than 10 nm 56 . To evaluate Eq. 1 numerically, surface discretization is needed. A popular and robust method is DTS. In this approach, the membrane surface is represented as a tri-angular mesh M = ( V, F, E ), where V = { v 1 , …, v m } with v i ∈ ℝ 3 represents the vertex set, F = { f 1 , …, f k } denotes the face set, and E = { e 1 , …, e n } comprises the edge set. Each face f p = ( i, j, k ) consists of a tuple of three vertices that form a closed triangle, while each edge e q = ( j, k ) is a tuple of two vertex indices corresponding to connected vertices in the mesh. In the context of membrane modeling, each vertex represents a membrane patch containing hundreds of lipids. Mesh properties such as mean and Gaussian curvature ( Eq. 1 ) can be obtained using different approaches 57 – 59 . Here, DTS simulations were performed using FreeDTS (v2.0, 24), which internally uses the shape operator approach 60 . During simulation, both vertex positions and mesh connectivity evolve through Monte Carlo moves, standard vertex update, and Alexander move (edge flip), allowing the system to sample different membrane configurations while maintaining a physically realistic surface. Moreover, inhomogeneity can be induced into the model through the concept of inclusion, which can be used to model membrane domains and membrane proteins 24 . The acceptance probability of a proposed membrane configuration is given by the Metropolis-Hastings criterion 61 , 62 Here, Δ U is the total energy change between the proposed and current state, β = 1 /k b T is the inverse temperature, and r is a number drawn from a uniform distribution. While DTS simulations effectively model membrane physics based on theoretical principles, they traditionally lack the ability to incorporate experimental observational data. This disconnect between theory and experiment means that standard DTS simulations may not accurately capture the specific membrane configurations observed in real biological systems due to important hidden or unknown shape remodeling drivers. Our HMFF approach addresses this fundamental limitation by directly integrating experimental density data with physical simulation, creating a bridge between observed structures and the underlying biophysical principles. B. Helfrich Monte Carlo Flexible Fitting HMFF extends the mesoscale DTS framework by incorporating volumetric experimental data, such as 3D EM, directly into simulations, creating a bidirectional feedback between physical modeling and experimental evidence. In practice, HMFF is implemented as an additional energy term that guides vertices towards experimentally observed membrane configurations, specifically regions of high electron density in 3D EM data. Simultaneously, the physical membrane properties included during DTS simulation (see methods IV A) act as regularizers on the ensemble of possible membrane configurations. This integrative approach ensures that the resulting membrane models both recapitulate experimental observations and preserve physical realism, particularly important in regions where experimental data may be sparse, noisy, or entirely absent (e.g., outside the imaging field of view). The guidance energy term used in HMFF is adapted from the approach introduced for Molecular Dynamics Flexible Fitting (MDFF) of atomic structures to electron density maps 26 – 29 where θ ( r ) is density at position r , ξ controls interaction strength, θ max the maximum value of the densities, θ thr the threshold value to exclude solvent contribution. In HMFF, θ thr defaults to 0 and θ max defaults to the 0.999 quantile of the potential map, which we found to be stable in practice and leaves ξ as the only remaining tunable parameter. To extrapolate beyond boundaries of experimental volumes, we implement padding in the xy-plane using 0.4 × θ max times the moving average within a three-slice window, while z-slices are padded using moving averages of terminal slices. We implemented HMFF within FreeDTS as an additional energy term using the abstract energy class framework. Our implementation also includes a trilinear interpolator for processing CCP4/MRC format potential maps at arbitrary spatial resolution. The HMFF potential can be brought into action by using the Energy Method FreeDTS 1.0 HMFF with parameters outlined above. FreeDTS is available at https://github.com/weria-pezeshkian/FreeDTS , and the HMFF extension exists in version 2.1 and above. HMFF bridges the gap between static experimental imaging and physical simulation, enabling the creation of realistic and physically accurate membrane models. However, its practical implementation with biological data requires additional infrastructure to handle the associated complexity. C. The Mosaic software To address the challenges in practical HMFF implementation with biological data, we developed Mosaic, a complete software ecosystem for building simulation-ready models of entire cells. Implemented as a Python-based application with a PyQt6/VTK 63 , 64 graphical user interface (GUI), Mosaic combines existing tools with novel computational methods to enable the complete workflow from 3D EM data to physical simulations. Mosaic uses MemBrain-seg 18 for deep-learning-based membrane segmentation, the output of which is refined, clustered, and analyzed within the GUI. Mosaic converts segmentations into triangular meshes using a biologically motivated meshing approach, which are subsequently equilibrated to be suitable for simulation. Equilibrated meshes can be refined through DTS simulation using FreeDTS 24 with the HMFF potential. HMFF-refined meshes can be morphometrically characterized within Mosaic, used in equilibrium simulations, or form the basis for template matching membrane proteins using PyTME 35 . Mosaic implements a dedicated projection procedure, which allows for analyzing membrane-protein characteristics, such as distribution and curvature preference. The combined refined-mesh and protein picks can be mapped back onto coarse-grained representations using TS2CG 30 , 31 , enabling analyses from mesoscale dynamics to molecular interactions. All analyses presented in this work were performed through Mosaic, which is available at https://github.com/KosinskiLab/mosaic . D. Creating simulation-ready meshes Mosaic converts segmentations into simulation-ready models in a multi-stage process: initial construction of a mesh from segmentation data, completion of the mesh to form a closed surface representing a biologically plausible membrane, and regularization to ensure mesh properties suitable for DTS simulation. 1. Mesh construction Mosaic implements complementary methods for creating meshes from segmentations, that is, finding a mesh M that approximates the surface formed by an unordered set of points P = { p 1 , …, p n } ∈ ℝ 3 . Each approach offers distinct advantages, with overall applicability depending on data characteristics. Mosaic provides optimized default parameters for each method derived from testing across diverse biological datasets. Ball Pivoting 65 provides robust mesh construction and only requires tuning a single parameter, the pivoting radius, which determines the maximum local curvature that can be captured. The pivoting radius is limited by the sampling density of the point cloud, i.e., no sub-voxel precision mesh construction is possible. An alternative is Poisson reconstruction 66 , which fits an implicit function to generate smooth surfaces. This method can handle variable feature sizes but requires reliable normal vector estimates and more careful parameter tuning for accurate mesh construction. α -shapes 67 are suitable for segmentations with large topological gaps and can effectively bridge discontinuities while preserving the overall structure. Reducing the α parameter allows for capturing concave features such as membrane invaginations, though excessively small values can lead to disconnected components. Finally, for densely labeled volumetric data, e.g., the FIB-SEM-based segmentations of entire mitochondria as opposed to just their membranes, Marching Cubes 68 efficiently constructs meshes at grid resolution, but requires parameter tuning to avoid erroneous merges or inaccurate representation in high-curvature regions. See section VIII A for a mathematical description of the individual algorithms and Fig. S7 for an illustration of the different use-cases of each. 2. Mesh completion and deformation The mesh creation methods outlined above are not guaranteed to result in a closed surface that is a realistic representation of biological membranes. Therefore, we devise targeted completion strategies, which are outlined below and illustrated in Fig. S8 . For meshes with identifiable hole boundaries, Liepa triangulation 69 systematically closes holes by identifying boundary loops and applying iterative subdivision. The subdivision process minimizes the ratio between edge length and geodesic distance until hole closure d g ( v i , v j ) represents the geodesic, ∥ v i − v j ∥ the Euclidean distance along the boundary and L is the total boundary edge length. This process continues until each subdivision forms a triangle, effectively closing the hole while maintaining the overall shape. For meshes without a connected set of boundary vertices, we utilize α -shapes. After constructing the initial mesh, we compute distances between the original point cloud and mesh vertices. Regions, where distances exceed a threshold, by default 6 × voxel size, are identified as missing areas requiring geometric optimization ( Fig. S8a ). Both completion strategies introduce inferred vertices into the mesh. To reconcile inferred local and observed global geometries, we smooth the mesh using biologically motivated polyharmonic deformation ( Fig. S8b ) subject to Dirichlet boundary conditions v | ∂Ω = v real 70 , which fixes the positions of non-inferred vertices. Δ ∈ ℝ n×n represents the discrete Laplacian operator, M ∈ ℝ n×n is the area-mass matrix derived from Voronoi tessellation, and n v is the normal of vertex v . The weights α, β, γ ∈ ℝ n control smoothness, curvature, and pressure respectively where where w ∈ ℝ is a user-defined factor for each weight. We construct Δ and M using intrinsic Delaunay triangulation with edge length mollification, which introduces a mollification factor for numerical stability where and is the mean edge length. This ensures the triangle inequality is strictly satisfied in all mesh elements. The Laplacian is then constructed using the cotangent formula as implemented in libigl 71 where N ( i ) are the vertices adjacent to vertex i , and α ij , β ij are the angles opposite to edge i, j . The mass matrix is derived from Voronoi vertex areas, providing an appropriate inner product space for solving the polyharmonic system presented in Eq. 5 . 3. Mesh regularization Not every mesh is suitable for DTS simulations, as the vertex update scheme and edge flips can result in a mesh that no longer represents a realistic membrane surface, for example, a surface intersecting itself. This demands that the edge length falls within a specific range with a ∈ ℝ + and that the dihedral angle between two adjacent triangles satisfies cos( θ ) > −1 24 . We devised a two-step equilibration process to create meshes suitable for DTS simulation. First, isotropic explicit remeshing 72 is applied to adjust vertex positions and connectivity. For most input meshes, this step alone suffices to produce simulation-ready surfaces. In cases where additional edge length regularization is required, Mosaic uses Trimem (v0.2.2, 25) to minimize the following energy term where l j is the length of edge j, l c 1 and l c 0 being the lower-and upper-onset of penalization and r the slope of the potential well. Constraining the equilibration to preserve surface area, volume, and local curvature ensures that all edge lengths converge to the desired range while maintaining the overall shape of the mesh. E. Mesh projection In biological systems, numerous entities of interest, from individual proteins to contact sites with membranes of other organelles, exist in relation to membrane triangulations. To accurately characterize membrane-associated structures, mesh projection is required to relate spatial orientation to the continuous mesh representations. The projection process serves dual purposes: First, it enables rigorous structural analysis, e.g., quantification of protein curvature preference, geodesic distance distributions, and co-localization patterns. Second, projection ensures that in silico model systems, e.g., for equilibrium DTS or MD simulation, faithfully recapitulate the experimentally observed protein positions. Mosaic includes a set of computational methods to project objects onto membrane triangulations and quantify their geometric context, which are described below and illustrated in Fig. S9 . 1. Dual-mode projection Mosaic implements a ray-casting-based dual-mode projection algorithm for mapping structures (e.g., protein positions, organelle segmentations, membrane meshes) onto triangulated mesh surfaces. The first mode computes the closest point on the mesh by minimizing Euclidean distance where S is the surface defined by the mesh, q ∈ ℝ 3 a position on S , and p ∈ ℝ 3 is the position to be projected. The second mode accounts for structures with defined position and orientation, such as proteins relative to a membrane surface. On the level of individual points, they can be defined as a tuple ( p ∈ ℝ 3 , R SO(3)). Given the principal axis of the protein, which Mosaic internally sets as R e z , we modify the projection procedure to cast rays from the position along the principal axis where t ∈ ℝ ≥0 is the distance to the first intersection with the mesh surface along n ( Fig. S9b ). Particularly in regions of high curvature or for proteins that are bent from the mesh surface normal, the second mode is more accurate. 2. Mesh extension To obtain precise geometric properties at projected positions, we extend the mesh by incorporating these positions as new vertices. Given a set of projected points that intersect with a triangle f i formed by vertices ( v a , v b , v c ) ∈ ℝ 3 , we map them into the local 2D reference frame of f i as where e 1 and e 2 form an orthonormal basis in the triangle plane. A valid surface triangulation of this 2D space that includes all projection points is obtained using Delaunay triangulation implemented in scipy. The extended mesh preserves the original surface geometry while including exact representations of projected positions, enabling more accurate computation of local properties compared to interpolation approaches using barycentric coordinates. 3. Mesh property computation Mesh properties such as curvatures, geodesic distances can be computed on the extended mesh. Curvature computation uses the local vertex n-ring neighborhood with radius r to estimate principal curvatures k 1 and k 2 at each vertex 71 . Based on principal curvatures, the Gaussian curvature K and the mean curvature H can be computed as Geodesic distances between vertices on the mesh surface are computed using an exact algorithm 73 , which provides accurate surface distances even in regions of high curvature where Euclidean measurements would be inaccurate. F. Mycoplasma pneumoniae data acquisition and processing M. pneumoniae cryo-ET data was collected and processed as previously described 32 . In brief, the bacterium was cultured on gold EM grids with a holey carbon film (Quantifoil) before being vitrified by plunge freezing. The sample was imaged on a Titan Krios G3 microscope (Thermo Fischer) equipped with a K2 direct electron detector (Gatan), dose symmetrically between −60 and 60 degrees with 3-degree increments at a pixel size of 1.7 Å. Tilt series processing was performed using Warp, and the initial tilt series alignment was determined using IMOD 74 , 75 . A ribosome reference was created from manually picked ribosomes in RELION at a voxel size of 6.082 Å. It was subsequently low-pass filtered to 30 A and used directly as a template in PyTOM 76 . The 400 highest-scoring cross-correlation peaks were extracted and filtered by visual inspection. Ribosomes were subtomogram averaged using RELION, and used as fiducials for improving tilt series alignment in M 77 , 78 . The updated alignments were used to reconstruct the final tomograms in Warp. G. Membrane segmentation Membranes of M. pneumoniae and IAV VLPs (EMD:11075, 34) tomograms were segmented in Mosaic using MemBrain-seg (v0.05, 18), with Mem-Brain seg v10 alpha weights, which were downloaded from https://github.com/teamtomo/membrain-seg . The input tomograms were resampled to a voxel size of 12 Å using the utilities provided with MemBrain-seg. Segmentation was performed using 8-fold test time augmentation and a 160-voxel sliding window on an NVIDIA RTX 4070 Super GPU. Connected component labeling was activated to differentiate disjoint components. H. Initial mesh generation The initial mesh for the planar membrane simulation in a potential field was generated using the GEN script provided within FreeDTS (v2.0, 24). The corresponding potential map was generated as an isotropic Gaussian with σ = 4, centered ten voxels above the planar membrane mesh using PyTME (v0.3.0, 35). Initial meshes for M. pneumoniae were generated using Mosaic. Membrane segmentations were cleaned by removing erroneous segments and thinning the segmentations to the outer cloud. The cleaned segmentations were triangulated using α -shapes with an elastic weight of 1.0 and a curvature weight of 10.0. We achieved a more realistic estimate of cell volume by performing 100 Å equidistant sampling from the created mesh, manually removing undesirable samples, and repeating the α -shape triangulation with a pressure of 0.1. The resulting triangulation was remeshed to an average edge length 170 Å and equilibrated using default parameters bending energy coefficient: 300.0, area conservation coefficient: 1.010 6 , volume conservation coefficient: 1.010 6 , edge tension coefficient: 1.010 5 , surface repulsion coefficient: 1.010 3 , with volume and area fraction targets of 1.1, ensuring that overall shape is maintained while creating a mesh suitable for dynamic triangulated surface simulations. Initial meshes of IAV were created analogously to M. pneumoniae but remeshed to an average edge length of 110 Å. Initial meshes of cellular organelles were generated using publicly available segmentations 21 . EM data, organelle segmentations, and ribosome positions for jrchela2 were downloaded from https://open.quiltdata.com/b/janelia-cosem-datasets/tree/jrc_hela-2/ using custom Python scripts. Initial meshes were generated from the unbinned s0 data layer, providing a resolution of (4 nm, 4 nm, 5.24 nm) along x, y, z . One-time binned data was used specifically for the ER, as it yielded visually higher quality meshes. Volumes were converted into triangular meshes by marching cubes. Briefly, volumes were split into non-overlapping subvolumes with a box size of 448 and meshed separately using zmesh (v1.8.0, https://github.com/seung-lab/zmesh ). The initial meshes were obtained using a reduction factor on a triangle count of 100 and a maximum simplification error of 40 nm. Following simplification, meshes were merged and simplified again using pyfqmr (0.3.0, https://github.com/Kramer84/pyfqmr-Fast-Quadric-Mesh-Reduction ), primarily to reduce elevated triangle count at volume boundaries. The second simplification was performed using an aggressiveness of 5.5 and a decimation factor of 2.0. These settings are the default in Mosaic, which uses an implementation inspired by Igneous 79 . Meshes of cellular organelles were imported into Mosaic and assessed for quality. Meshes were repaired by equidistant sampling from the surface, closing gaps using face-projection and equidistant sampling via α -shapes, and manually removing erroneous segmentations. Subsequently, cleaned point clouds were meshed using Poisson reconstruction, using default Mosaic parameters but varying the depth between 9 and 14 depending on the complexity of the point cloud. Mitochondria meshes were prepared for DTS simulation analogous to previous examples but remeshed to 20 nm. I. HMFF simulation Reconstructed tomograms of M. pneumoniae and IAV VLPs were obtained at a voxel-size of 13.6 Å and 6.8 Å, respectively. Through Mosaic, they were bandpass filtered to 900 Å and 50 Å for IAV and 900 Å and 140 Å for M. pneumoniae . The latter were further normalized by dividing each z-slice by its maximum density value. For jrc-hela2, we used a volume with voxel size of (16 nm, 16 nm, 20.96 nm) and no further preprocessing. HMFF simulations were run on an AMD Ryzen 5 7600 using 32GB of DDR5 RAM. An overview of key parameters used for HMFF simulation is presented in Tab. 1 . View this table: View inline View popup Download powerpoint TABLE 1. Overview of key HMFF simulation parameters. J. Constrained template matching Seed points for constrained template matching were created from HMFF-refined meshes using Poisson disk sampling 66 with five-fold oversampling in Mosaic. M. pneumoniae seed points had an average Euclidean distance of 80 Å and were offset by the surface normal of the mesh by 80 Å. IAV seed points had an average Euclidean distance of 40 Å and were offset by 100 Å. These parameters reflect the expected glycoprotein spacing and positioning of membrane surface protein centers. PDB:8pbz was used as a template to pick Nap complexes in M. pneumoniae 40 . The corresponding structure was aligned to the z-axis, converted into a density map, low-pass filtered to 13.604 Å, and resampled using cubic spline interpolation. A spherical mask centered on the extracellular head group with a radius of 81.6 Å and smoothing sigma decay of 1.0 was used. For IAV, structural models of HA and NA from AlphaFold 2 multimer were used as templates 80 , 81 . HA models were based on the sequence of A/Hong-Kong/1/1968 H3N2 (UniProt: P11134), NA models on the sequence of A/California/04/2009 H1N1 (UniProt: C3W5S3). AlphaFold 2 multimer was run with default parameters except for increasing refinement cycles to 6. Structural models were converted to density maps analogously to M. pneumoniae . A cylindrical mask with a height of 251.6 Å, radius of 68.0 Å, and smoothing sigma decay of 2.0 was used for both templates. All templates and masks were created using PyTME 35 . Template matching was performed using the FLC score, continuous wedge masks reflecting on the respective tilt range, and an angular sampling rate of 7 degrees. M. pneumoniae and IAV tomograms used for template matching had a voxel size of 6.80 Å. Matches were accepted when deviating no more than 15° from the nearest seed point normal and falling within an ellipse of radii (10,10,20) voxels centered around that seed point and oriented along its normal. These cutoffs were changed for IAV to 15° and radii of (6,6,10) and (6,6,12) for HA and NA, respectively. In M. pneumoniae , template matches were identified using PeakCallerMaximumFilter and a minimum peak distance of 20 voxels, a score cutoff of 0.09, and a distance to the HMFF-refined mesh between 70.0 Å and 120.0 Å. For IAV, a minimum peak distance of 10 voxels and a score cutoff of 0.135 were used for HA, and a 7 voxel distance and 0.12 score cutoff were used for NA. Peaks were manually refined and filtered to be within 100.0 Å and 150.0 Å of the HMFF-refined mesh. NA picks were filtered to include the top 97% quantile of scores, and HA picks falling within 7 voxels of remaining NA picks were removed to avoid clashes. K. Unconstrained template matching M. pneumoniae ribosomes were picked using EMD:17132 33 as template and a spherical mask with radius 142.8 Å. The initial template was lowpass filtered to 27.2 Å and resampled using cubic spline interpolation. Template matching was performed as outlined above, but on a tomogram with voxel size 13.60 Å. Peaks were identified using PeakCallerScipy, a minimum peak distance of 15 voxels, and a score cutoff of 0.21. Picks were subsequently manually refined. L. Backmapping of DTS to the coarse-grained Martini model TS2CG (v1.2.2, 30,31) was used to backmap DTS onto coarse-grained models using Mosaic. Briefly, HMFF-refined meshes of M. pneumoniae were remeshed to 120 Å and IAV meshes to 20 Å. Picked proteins were projected onto the mesh and uniquely mapped to the closest vertex. Atomic structures were coarse-grained using the martinize2 (v0.13.0, 82) tool with default parameters and using an elastic network with standard cutoffs to maintain the tertiary structure of NA, HA, and the Nap complex. TS2CG PCG was run assuming a bond length of 0.1 and a CG lipid library downloaded from https://github.com/weria-pezeshkian/TS2CG-v2.0 83 . For simplicity, a pure POPC bilayer was generated using a thickness of 3.8 nm and an area per lipid of 0.64 and 0.5 for M. pneumoniae and IAV, respectively. Proteins were inserted into the bilayer with a normal offset of 6.5 nm for the Nap complex, 7.5 nm for HA, and 11.5 nm for NA. Since TS2CG does not allow for including perprotein orientations, HA and NA particles identified by template matching clashed. Inter-protein clashes were identified, and those with a distance of < 0.3 nm were removed, followed by membrane rebuilding to avoid holes in the regions with removed proteins. Subsequent energy minimization and equilibration were performed using the GROMACS simulation package 42 . The system was simulated as-is in a vacuum to keep the computational load manageable 30 . In general, the system could be solvated using conventional tools, however. The structures were first minimized using the steepest descent algorithm with soft core potentials (init-lambda = 0.001, then 0.0005) to alleviate remaining steric clashes, followed by a 1000-step minimization using regular potential with reaction-field electrostatics. The minimized structure was then equilibrated with protein and lipid headgroup position restraints using a stochastic dynamics integrator with a 1 fs timestep and Berendsen thermostat set to 310 K. Pressure coupling remained deactivated throughout the equilibration to account for the missing solvent, while maintaining the Verlet cut-off scheme with periodic boundary conditions in all dimensions. For M. pneumoniae , soft core potentials were disabled, and the equilibration step was omitted. Mdp files with a complete list of settings can be found in the supporting files. M. Mitochondria classification Three complementary feature sets capturing topological and geometrical properties were derived to differentiate latent mitochondria classes. Topological information was captured through the spectral properties of the Laplace operator by solving the generalized eigenvalue problem Δ ϕ i = λ i Mϕ i for the first 15 eigenvalues with the smallest magnitude. These eigen-values encode the intrinsic topology of the mesh, which is independent of the spatial embedding. The sparse system was solved using Scipy’s sparse eigenvalue solver (v1.14.1, 84) with shift-inversion σ = 10 −8 . Geometric information was captured by 3D Zernike moments as rotation-invariant shape descriptors. The position of each mitochondrial vertex v i ∈ V was normalized as ( v i − µ V ) /r max , where and r max = max i ∥ v i − µ V ∥ 2 , ensuring unit sphere containment. 3D Zernike moments were computed from normalized vertices as where R nl ( r ) are radial polynomials of order n with n − l being even and non-negative, Y lm ( θ, ϕ ) are spherical harmonics of degree l and order m with | m | ≤ l, f ( r ) is the point cloud density function defined within the unit sphere, and r is the position vector with magnitude r = | r |. In spherical coordinates, the volume element dV = d 3 r is expressed as r 2 sin θdrdθdϕ with θ ∈ [0, π ] and ϕ ∈ [0, 2 π ]. Zernike moments were computed using a public implementation https://github.com/kheyer/3D-Zernike-Descriptors . The rotation-invariant descriptors , were calculated with l ranging from l 0 = n mod 2 to n in steps of 2, yielding ( n + 1)( n + 2) / 2 descriptor coefficients. The descriptors were computed up to order n = 8. Topological and geometric descriptors were supplemented with scale-dependent features: surface area, volume, skeleton branch count, and curvature statistics. Mosaic calculated surface area using Open3D (0.19.0, 85), volume using tetrahedron decomposition, and mean and Gaussian curvature using a radius of ten. Topological complexity was quantified through segment counts using skeletor (v1.3.0, https://github.com/navis-org/skeletor ). A wavefront-based skeletonization algorithm was applied with 5 waves, and segments longer than 5 nodes were counted. All features were standardized to a zero mean and unit variance. Principal component analysis was applied to the standardized features, followed by t-SNE dimensionality reduction using Rtsne (v0.17, 86) with parameters perplexity 10, early exaggeration 24, and max iterations 15,000. Louvain community detection was performed using igraph (v2.0.3, 87) with a resolution parameter of 0.5 to determine classes. N. Contact point analysis Contact points were determined using distance-based detection. Geometric proximity was computed using Mosaic, which internally relies on the Open3D 85 raycasting scene implementation. For each mitochondrion mesh and target organelle segmentation, spatial relationships were analyzed. A contact was defined when any part of the organelle segmentation was located within 14 nm of the mitochondrial surface, similar to the threshold used in previous analyses of this dataset 21 . The contact area was calculated by summing the tri-angular face areas from the mitochondrial mesh that fell within the contact threshold. The percentage of mitochondria within each morphological cluster forming contacts with other organelles was determined by counting mitochondria with non-zero contact area, then dividing by the total number of mitochondria in each respective cluster. V. DATA AVAILABILITY Mosaic sessions and cryo-ET data of Mycoplasma pneumoniae will be made available. VI. CODE AVAILABILITY HMFF is available in FreeDTS from https://github.com/weria-pezeshkian/FreeDTS . Mosaic is available from https://github.com/KosinskiLab/mosaic . Analysis scripts and parameter files used in this study are available upon request. VIII. EXTENDED DATA A. Meshing Algorithms Several established algorithms exist for generating tri-angular meshes from point clouds, each with distinct mathematical foundations and operational characteristics. The paragraphs below outline the approaches implemented in Mosaic and used throughout this work. The Marching Cubes algorithm discretizes ℝ 3 into a regular grid of cubes and constructs the surface through local triangulation. For each cube intersected by the implicit surface f ( x, y, z ) = 0, the algorithm determines the surface-edge intersections based on the sign of f at the cube vertices. The local triangle configuration is selected from a predefined set of 2 8 possible cases, ensuring consistent topology between adjacent cubes. This produces a triangulation that approximates the level set of the implicit function 68 . Ball Pivoting Algorithm (BPA) operates through a geometric rolling process. Starting from a seed triangle, a ball of radius ρ pivots around an active edge until it touches another point, forming a new triangle. Mathematically, this process identifies points p satisfying where v i , v j are the vertices of the active edge. This generates a subset of the ρ -regular triangulation of the point set 65 . α -shapes provide a formal mathematical framework for shape reconstruction through a filtration of the Delaunay triangulation. For a given α ∈ ℝ, the alpha complex consists of all Delaunay simplices whose dual Voronoi cells intersect an α -ball centered at each vertex. The resulting shape captures topological features at the scale determined by α , with the family of α -shapes forming a hierarchical representation of the point set 67 . Poisson Reconstruction estimates the indicator function χ of the shape by solving the Poisson equation where n represents the oriented normal field sampled at the input points. The solution is efficiently computed using a multi-scale approach in an adaptive octree, with the final surface extracted as an appropriate level set of the resulting scalar function 37 . Download figure Open in new tab Fig. S1. Setting up M. pneuomoniae HMFF simulations. a , Tomographic slice showing a M. pneuomoniae cell. b , Initial membrane segmentation. c , Cleaned segmentation following processing in Mosaic. d , Creating a cell membrane mesh model suitable for HMFF simulation by polyharmonic deformation and subsequent equilibration of an α -shape fitted to refined membrane segmentation. e , Mesh edge lengths stratified by stage shown ( d ). f , Poisson reconstruction based on refined membrane segmentation. g , Padding values used by HMFF for the particular tomogram shown in ( a ), compared to the raw tomographic densities. Tomogram boundaries are indicated by vertical lines. Scaled average density corresponds to the per-slice average, divided by the maximum average value over all slices. Download figure Open in new tab Fig. S2. M. pneuomoniae HMFF simulation. a , Simulation trajectory showing the evolution of the membrane mesh during HMFF optimization at 0, 75, and 150 thousand simulation steps. Top and side views demonstrate how the mesh progressively adapts to the underlying tomographic data while maintaining physical membrane properties. The mesh extends beyond the tomogram boundaries (gray box), where its shape is determined solely by physical membrane properties. b , Comparison between initial (yellow) and HMFF-refined (blue) meshes in key regions. The left column shows vertices projected onto tomogram slices, highlighting how the refined mesh more accurately follows membrane densities. The right column displays the corresponding complete mesh structures. Download figure Open in new tab Fig. S3. Building a filamentous IAV VLP model. a , Tomographic slice showing a filamentous Influenza A VLP (EMD:11075). b , Initial membrane segmentation with visible artifacts. c , Membrane segmentation cleaned using Mosaic. d , HMFF-refined membrane mesh. e , Seed points drawn from the HMFF-refined mesh ( d ) used for template matching. f , Distance distribution of seed points, and relative to the HMFF-refined mesh. g , HA (red), and NA (orange) structures from Alphafold and corresponding template densities (grey)(see methods IV J). h - i , Central slice of template matching cross-correlations (CC) for HA (left) and NA (right). Download figure Open in new tab Fig. S4. Mitochondrial segmentation quality issues and resolution through applying Mosaic. a, d, g , Raw FIB-SEM data with original mitochondrial segmentations from Heinrich et al . 21 overlaid in green. b, e, h , Initial meshes obtained from marching cubes applied to the segmentations on the left, showing separate entities in the original segmentation in different colors. c, f, i , Refined meshes after processing in Mosaic, showing properly connected structures. The three rows represent distinct mitochondrial morphologies: tubular with bulbous ending (top row), branched structure (middle row), and complex network with multiple branches (bottom row). Download figure Open in new tab Fig. S5. Systematic analysis of HMFF parameter optimization for mitochondrial mesh refinement Individual facets show the relative HMFF potential over 50,000 simulation steps for varying combinations of membrane bending rigidity κ and HMFF interaction coefficient ξ . Download figure Open in new tab Fig. S6. Topological analysis of mitochondrial meshes. a , Representative mitochondrion mesh prior to HMFF refinement. b , The same mitochondrion after 50,000 steps of HMFF simulation, showing preserved biological structure with local constrictions to adapt to the experimental observation. c , Hierarchical clustering and Pearson correlation heatmap of mitochondria based on extracted feature sets, revealing distinct morphological clusters (labeled 1-9). d , Unrooted neighbor-joining tree constructed from the centroids of the nine mitochondrial clusters in t-SNE space, visualizing the morphological relationships between different mitochondrial subtypes. Branch lengths reflect the degree of morphological dissimilarity between clusters, with closely related morphologies positioned near each other in the tree. Download figure Open in new tab Fig. S7. Comparison of key mesh construction methods implemented in Mosaic. Illustrated are meshing approaches, showing well-suited examples where a method excels and poorly-suited examples where it faces limitations. Points shown in grey indicate the input used for triangulation. α -shape, Poisson reconstruction, and Ball pivoting operate on point clouds, while marching cubes processes volumetric data. Mosaic integrates these complementary approaches, allowing users to select the optimal method based on specific membrane morphologies and data quality. Download figure Open in new tab Fig. S8. Mesh completion strategies for generating biological membrane models. a , Two complementary approaches for completing partial membrane structures: when boundary vertices are identifiable (top), Liepa triangulation preserves local geometry while filling missing regions; for disconnected structures (bottom), distance-based remeshing with α -shapes reconstructs closed surfaces. In both approaches, triangulations are based on observed points (green), with fixed vertices (blue) originating from the observation and inferred vertices (red) from the completion strategy. b , Progressive deformation of inferred vertices guided by biophysical principles. Inferred vertices (red) undergo optimization to assimilate into the geometry dictated by fixed vertices (blue). This procedure integrates elasticity, smoothness, curvature, and pressure terms (labeled 2-4, see methods IV D 2), ensuring the completed regions seamlessly connect with observed structures while maintaining properties required for HMFF simulation. Download figure Open in new tab Fig. S9. Protein-membrane projection and analysis workflow in Mosaic. a , Mosaic GUI with highlighted projection dialog showing customizable parameters for protein-membrane mapping. b , Visualization of the projection procedure: membrane proteins (red/orange) are raycasted along their normal vectors onto the membrane surface (blue), creating new vertices and edges (white) at intersection points that preserve orientation information from template matching. Mesh projection enables accurate determination of geodesic distances and curvatures, indicated as principal curvature planes k 1 and k 2. Protein orientation can be omitted in favor of the shortest distance projection through the dialog in ( a ). c , Property analysis interface providing access to spatial relationships, surface properties, and topological features through an interactive panel with mesh coloring, graph visualization, and statistics options (see Fig. 3 ). Extended Data Movie https://oc.embl.de/index.php/s/HewEk8DS7xb9gLu?path=%2Fplane#/ Mov. M1. Planar membrane simulations with varying HMFF coupling constants ( ξ ). HMFF simulations of a planar membrane in a Gaussian density with different coupling constants. As ξ increases from 0 (no coupling) to 10 (strong coupling), the membrane progressively shows stronger and more rapid adaptation to the density, demonstrating how the coupling parameter can be tuned to achieve desired fitting behavior. Extended Data Movie https://oc.embl.de/index.php/s/HewEk8DS7xb9gLu?path=%2Fmycoplasma Mov. M2. M. pneumoniae membrane segmentation process. The video shows a flythrough of the raw cryo-ET data (left), the deep-learning-based membrane segmentation (middle), and the result of manual refinement and thinning from Mosaic (right). Extended Data Movie https://oc.embl.de/index.php/s/HewEk8DS7xb9gLu?path=%2Fmycoplasma Mov. M3. HMFF simulation trajectory of M. pneumoniae membrane model. The video shows the dynamic refinement process as the triangulated mesh evolves according to the combined Helfrich-Hamiltonian and density-derived potentials, progressively adapting to the experimental data while maintaining biophysically realistic membrane properties. Extended Data Movie https://oc.embl.de/index.php/s/HewEk8DS7xb9gLu?path=%2Fmycoplasma Mov. M4. HMFF simulation trajectory of M. pneumoniae membrane model, without padding beyond tomogram boundaries, leading to a flattening of the mesh and attachment to the z-boundaries. Extended Data Movie https://oc.embl.de/index.php/s/HewEk8DS7xb9gLu?path=%2Fiav Mov. M5. IAV VLP membrane segmentation. The video shows a fly-through of the raw cryo-ET data (left), the deep-learning-based membrane segmentation (middle), and the result of manual refinement and thinning from Mosaic (right). Extended Data Movie https://oc.embl.de/index.php/s/HewEk8DS7xb9gLu?path=%2Fiav Mov. M6. HMFF simulation trajectory of IAV VLP membrane model. The video shows the dynamic refinement process as the triangulated mesh evolves under the influence of the combined Helfrich-Hamiltonian and density-derived potentials, adapting to the experimental data while maintaining the characteristic filamentous morphology of the virus. Extended Data Movie https://oc.embl.de/index.php/s/HewEk8DS7xb9gLu?path=%2Fiav Mov. M7. Annotated flythrough of the final IAV VLP model. The video showcases the HMFF-refined membrane mesh with mapped glycoproteins, highlighting the distinct spatial distribution of hemagglutinin (HA) and neuraminidase (NA) across the viral surface. This visualization demonstrates how the HMFF-refined membrane provides an accurate framework for studying glycoprotein organization and curvature preferences. Extended Data Movie https://oc.embl.de/index.php/s/HewEk8DS7xb9gLu?path=%2Fmito Mov. M8. HMFF simulation trajectory of mitochondrial membrane model. The video demonstrates the dynamic evolution of the triangulated mitochondrial mesh as it adapts to the 3D EM data while maintaining biophysical membrane properties. Extended Data Movie https://oc.embl.de/index.php/s/HewEk8DS7xb9gLu?path=%2Fmito Mov. M9. Flythrough of the HMFF-refined mitochondrial model. The video provides a 3D visualization of the final mitochondrial mesh, demonstrating how HMFF successfully captures the characteristic features of mitochondrial morphology from FIB-SEM data. Extended Data Movie https://oc.embl.de/index.php/s/HewEk8DS7xb9gLu?path=%2Fcell Mov. M10. Flythrough visualization of the HeLa cell with all refined organelle meshes. The video showcases the integrated cellular architecture with individually rendered compartments: nucleus (orange), endoplasmic reticulum (purple), Golgi apparatus (blue), and mitochondria (green). This multi-component visualization demonstrates the spatial relationships between organelles and highlights how Mosaic enables accurate reconstruction of cellular ultrastructure from FIB-SEM data at the micrometer scale. VII. ACKNOWLEDGEMENTS We thank the EMBL IT and HPC resources for providing essential computational infrastructure. VM and JK acknowledge funding from the CSSB flagship project Plasmofraction. MS acknowledges support from a research fellowship from the EMBL Interdisciplinary Postdoc (EIPOD) Programme under Marie Curie Cofund Actions MSCA-COFUND-FP (grant agreement number: 847543). RKJ was supported by the Independent Research Fund Denmark (grant number 0164-00010A). JM acknowledges the support from the EMBL, an EMBL Infection Biology Transversal Theme Synergy grant, and a Chan Zuckerberg Initiative grant for Visual Proteomics (grant No. 2021-234620). JK and JM were supported by the ERC (TransFORM, 101119142). WP acknowledges support from the Novo Nordisk Foundation (grant No. NNF18SA0035142 and NNF22OC0079182) and Independent Research Fund Denmark (grant No. 10.46540/2064-00032B). Funder Information Declared Centre for Structural Systems Biology, https://ror.org/04fhwda97 European Molecular Biology Organization, https://ror.org/04wfr2810 Independent Research Fund Denmark , 0164-00010A , 10.46540/2064-00032B EMBL Infection Biology Transversal Theme Synergy Chan Zuckerberg Initiative (United States) , 2021-234620 European Research Council , 101119142 Novo Nordisk Foundation , NNF18SA0035142 , NNF22OC0079182 Footnotes ↵ a Electronic mail: valentin.maurer{at}embl-hamburg.de ↵ b Electronic mail: jan.kosinski{at}embl.de REFERENCES ↵ J. A. Stevens , F. Grünewald , P. A. M. van Tilburg , M. König , B. R. Gilbert , T. A. Brier , Z. R. Thornburg , Z. Luthey-Schulten , and S. J. Marrink , Frontiers in Chemistry 11 ( 2023 ) , doi: 10.3389/fchem.2023.1106495 . OpenUrl CrossRef ↵ C. Bunne , Y. 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