Cryo-electron Microscopy Maps Research Articles

Conserved structures and dynamics in 5'-proximal regions of Betacoronavirus RNA genomes.

Betacoronaviruses are a genus within the Coronaviridae family of RNA viruses. They are capable of infecting vertebrates and causing epidemics as well as global pandemics in humans. Mitigating the threat posed by Betacoronaviruses requires an understanding of their molecular diversity. The development of novel antivirals hinges on understanding the key regulatory elements within the viral RNA genomes, in particular the 5'-proximal region, which is pivotal for viral protein synthesis. Using a combination of cryo-electron microscopy, atomic force microscopy, chemical probing, and computational modeling, we determined the structures of 5'-proximal regions in RNA genomes of Betacoronaviruses from four subgenera: OC43-CoV, SARS-CoV-2, MERS-CoV, and Rousettus bat-CoV. We obtained cryo-electron microscopy maps and determined atomic-resolution models for the stem-loop-5 (SL5) region at the translation start site and found that despite low sequence similarity and variable length of the helical elements it exhibits a remarkable structural conservation. Atomic force microscopy imaging revealed a common domain organization and a dynamic arrangement of structural elements connected with flexible linkers across all four Betacoronavirus subgenera. Together, these results reveal common features of a critical regulatory region shared between different Betacoronavirus RNA genomes, which may allow targeting of these RNAs by broad-spectrum antiviral therapeutics.

Cryo-electron microscopy of IgM-VAR2CSA complex reveals IgM inhibits binding of Plasmodium falciparum to Chondroitin Sulfate A

Placental malaria is caused by Plasmodium falciparum-infected erythrocytes (IEs) adhering to chondroitin sulfate proteoglycans in placenta via VAR2CSA-type PfEMP1. Human pentameric immunoglobulin M (IgM) binds to several types of PfEMP1, including VAR2CSA via its Fc domain. Here, a 3.6 Å cryo-electron microscopy map of the IgM-VAR2CSA complex reveals that two molecules of VAR2CSA bind to the Cµ4 of IgM through their DBL3X and DBL5ε domains. The clockwise and anti-clockwise rotation of the two VAR2CSA molecules on opposite faces of IgM juxtaposes C-termini of both VAR2CSA near the J chain, where IgM creates a wall between both VAR2CSA molecules and hinders its interaction with its receptor. To support this, we show when VAR2CSA is bound to IgM, its staining on IEs as well as binding of IEs to chondroitin sulfate A in vitro is severely compromised.

Cryo-EM structure of a plant photosystem II supercomplex with light-harvesting protein Lhcb8 and α-tocopherol.

The heart of oxygenic photosynthesis is the water-splitting photosystem II (PSII), which forms supercomplexes with a variable amount of peripheral trimeric light-harvesting complexes (LHCII). Our knowledge of the structure of green plant PSII supercomplex is based on findings obtained from several representatives of green algae and flowering plants; however, data from a non-flowering plant are currently missing. Here we report a cryo-electron microscopy structure of PSII supercomplex from spruce, a representative of non-flowering land plants, at 2.8 Å resolution. Compared with flowering plants, PSII supercomplex in spruce contains an additional Ycf12 subunit, Lhcb4 protein is replaced by Lhcb8, and trimeric LHCII is present as a homotrimer of Lhcb1. Unexpectedly, we have found α-tocopherol (α-Toc)/α-tocopherolquinone (α-TQ) at the boundary between the LHCII trimer and the inner antenna CP43. The molecule of α-Toc/α-TQ is located close to chlorophyll a614 of one of the Lhcb1 proteins and its chromanol/quinone head is exposed to the thylakoid lumen. The position of α-Toc in PSII supercomplex makes it an ideal candidate for the sensor of excessive light, as α-Toc can be oxidized to α-TQ by high-light-induced singlet oxygen at low lumenal pH. The molecule of α-TQ appears to shift slightly into the PSII supercomplex, which could trigger important structure-functional modifications in PSII supercomplex. Inspection of the previously reported cryo-electron microscopy maps of PSII supercomplexes indicates that α-Toc/α-TQ can be present at the same site also in PSII supercomplexes from flowering plants, but its identification in the previous studies has been hindered by insufficient resolution.

Structural Insights into the Assembly of the African Swine Fever Virus Inner Capsid.

African swine fever virus (ASFV), the cause of a highly contagious hemorrhagic and fatal disease of domestic pigs, has a complex multilayer structure. The inner capsid of ASFV located underneath the inner membrane enwraps the genome-containing nucleoid and is likely the assembly of proteolytic products from the virally encoded polyproteins pp220 and pp62. Here, we report the crystal structure of ASFV p150△NC, a major middle fragment of the pp220 proteolytic product p150. The structure of ASFV p150△NC contains mainly helices and has a triangular plate-like shape. The triangular plate is approximately 38 Å in thickness, and the edge of the triangular plate is approximately 90 Å long. The structure of ASFV p150△NC is not homologous to any of the known viral capsid proteins. Further analysis of the cryo-electron microscopy maps of the ASFV and the homologous faustovirus inner capsids revealed that p150 or the p150-like protein of faustovirus assembles to form screwed propeller-shaped hexametric and pentametric capsomeres of the icosahedral inner capsids. Complexes of the C terminus of p150 and other proteolytic products of pp220 likely mediate interactions between the capsomeres. Together, these findings provide new insights into the assembling of ASFV inner capsid and provide a reference for understanding the assembly of the inner capsids of nucleocytoplasmic large DNA viruses (NCLDV). IMPORTANCE African swine fever virus has caused catastrophic destruction to the pork industry worldwide since it was first discovered in Kenya in 1921. The architecture of ASFV is complicated, with two protein shells and two membrane envelopes. Currently, mechanisms involved in the assembly of the ASFV inner core shell are less understood. The structural studies of the ASFV inner capsid protein p150 performed in this research enable the building of a partial model of the icosahedral ASFV inner capsid, which provides a structural basis for understanding the structure and assembly of this complex virion. Furthermore, the structure of ASFV p150△NC represents a new type of fold for viral capsid assembly, which could be a common fold for the inner capsid assembly of nucleocytoplasmic large DNA viruses (NCLDV) and would facilitate the development of vaccine and antivirus drugs against these complex viruses.

Protein–protein interactions in the Mla lipid transport system probed by computational structure prediction and deep mutational scanning

The outer membrane (OM) of Gram-negative bacteria is an asymmetric bilayer that protects the cell from external stressors, such as antibiotics. The Mla transport system is implicated in the Maintenance of OM Lipid Asymmetry by mediating retrograde phospholipid transport across the cell envelope. Mla uses a shuttle-like mechanism to move lipids between the MlaFEDB inner membrane complex and the MlaA-OmpF/C OM complex, via a periplasmic lipid-binding protein, MlaC. MlaC binds to MlaD and MlaA, but the underlying protein-protein interactions that facilitate lipid transfer are not well understood. Here, we take an unbiased deep mutational scanning approach to map the fitness landscape of MlaC from Escherichia coli, which provides insights into important functional sites. Combining this analysis with AlphaFold2 structure predictions and binding experiments, we map the MlaC-MlaA and MlaC-MlaD protein-protein interfaces. Our results suggest that the MlaD and MlaA binding surfaces on MlaC overlap to a large extent, leading to a model in which MlaC can only bind one of these proteins at a time. Low-resolution cryo-electron microscopy (cryo-EM) maps of MlaC bound to MlaFEDB suggest that at least two MlaC molecules can bind to MlaD at once, in a conformation consistent with AlphaFold2 predictions. These data lead us to a model for MlaC interaction with its binding partners and insights into lipid transfer steps that underlie phospholipid transport between the bacterial inner and OMs.

Automatic and accurate ligand structure determination guided by cryo-electron microscopy maps

Advances in cryo-electron microscopy (cryoEM) and deep-learning guided protein structure prediction have expedited structural studies of protein complexes. However, methods for accurately determining ligand conformations are lacking. In this manuscript, we develop EMERALD, a tool for automatically determining ligand structures guided by medium-resolution cryoEM density. We show this method is robust at predicting ligands along with surrounding side chains in maps as low as 4.5 Å local resolution. Combining this with a measure of placement confidence and running on all protein/ligand structures in the EMDB, we show that 57% of ligands replicate the deposited model, 16% confidently find alternate conformations, 22% have ambiguous density where multiple conformations might be present, and 5% are incorrectly placed. For five cases where our approach finds an alternate conformation with high confidence, high-resolution crystal structures validate our placement. EMERALD and the resulting analysis should prove critical in using cryoEM to solve protein-ligand complexes.

Structural and DNA end resection study of the bacterial NurA-HerA complex

BackgroundThe nuclease NurA and the ATPase/translocase HerA play a vital role in repair of double-strand breaks (DSB) during the homologous recombination in archaea. A NurA-HerA complex is known to mediate DSB DNA end resection, leading to formation of a free 3′ end used to search for the homologous sequence. Despite the structures of individual archaeal types of NurA and HerA having been reported, there is limited information regarding the molecular mechanisms underlying this process. Some bacteria also possess homologs of NurA and HerA; however, the bacterial type of this complex, as well as the detailed mechanisms underlying the joining of NurA-HerA in DSB DNA end resection, remains unclear.ResultsWe report for the first time the crystal structures of Deinococcus radiodurans HerA (drHerA) in the nucleotide-free and ADP-binding modes. A D. radiodurans NurA-HerA complex structure was constructed according to a low-resolution cryo-electron microscopy map. We performed site-directed mutagenesis to map the drNurA-HerA interaction sites, suggesting that their interaction is mainly mediated by ionic links, in contrast to previously characterized archaeal NurA-HerA interactions. The key residues responsible for the DNA translocation activity, DNA unwinding activity, and catalytic activities of the drNurA-HerA complex were identified. A HerA/FtsK-specific translocation-related motif (TR motif) that guarantees the processivity of double-stranded DNA (dsDNA) translocation was identified. Moreover, a mechanism for the translocation-regulated resection of the 5′ tail of broken dsDNA and the corresponding generation of a recombinogenic 3′ single-stranded DNA tail by the drNurA-HerA complex was elucidated.ConclusionsOur work provides new insights into the mechanism underlying bacterial NurA-HerA-mediated DSB DNA end resection, and the way this complex digests the 5′ tail of a DNA duplex and provides long 3′ free end for strand invasion in the bacterial homologous recombination process.

Equilibrium Dynamics of a Biomolecular Complex Analyzed at Single-amino Acid Resolution by Cryo-electron Microscopy

The biological function of macromolecular complexes depends not only on large-scale transitions between conformations, but also on small-scale conformational fluctuations at equilibrium. Information on the equilibrium dynamics of biomolecular complexes could, in principle, be obtained from local resolution (LR) data in cryo-electron microscopy (cryo-EM) maps. However, this possibility had not been validated by comparing, for a same biomolecular complex, LR data with quantitative information on equilibrium dynamics obtained by an established solution technique. In this study we determined the cryo-EM structure of the minute virus of mice (MVM) capsid as a model biomolecular complex. The LR values obtained correlated with crystallographic B factors and with hydrogen/deuterium exchange (HDX) rates obtained by mass spectrometry (HDX-MS), a gold standard for determining equilibrium dynamics in solution. This result validated a LR-based cryo-EM approach to investigate, with high spatial resolution, the equilibrium dynamics of biomolecular complexes. As an application of this approach, we determined the cryo-EM structure of two mutant MVM capsids and compared their equilibrium dynamics with that of the wild-type MVM capsid. The results supported a previously suggested linkage between mechanical stiffening and impaired equilibrium dynamics of a virus particle. Cryo-EM is emerging as a powerful approach for simultaneously acquiring information on the atomic structure and local equilibrium dynamics of biomolecular complexes.

Structural Insights into Common and Host-Specific Receptor-Binding Mechanisms in Algal Picorna-like Viruses.

Marnaviridae viruses are abundant algal viruses that regulate the dynamics of algal blooms in aquatic environments. They employ a narrow host range because they merely lyse their algal host species. This host-specific lysis is thought to correspond to the unique receptor-binding mechanism of the Marnaviridae viruses. Here, we present the atomic structures of the full and empty capsids of Chaetoceros socialis forma radians RNA virus 1 built-in 3.0 Å and 3.1 Å cryo-electron microscopy maps. The empty capsid structure and the structural variability provide insights into its assembly and uncoating intermediates. In conjunction with the previously reported atomic model of the Chaetoceros tenuissimus RNA virus type II capsid, we have identified the common and diverse structural features of the VP1 surface between the Marnaviridae viruses. We have also tested the potential usage of AlphaFold2 for structural prediction of the VP1s and a subsequent structural phylogeny for classifying Marnaviridae viruses by their hosts. These findings will be crucial for inferring the host-specific receptor-binding mechanism in Marnaviridae viruses.

The insect perspective on Z-disc structure and biology.

Myofibrils are the intracellular structures formed by actin and myosin filaments. They are paracrystalline contractile cables with unusually well-defined dimensions. The sliding of actin past myosin filaments powers contractions, and the entire system is held in place by a structure called the Z-disc, which anchors the actin filaments. Myosin filaments, in turn, are anchored to another structure called the M-line. Most of the complex architecture of myofibrils can be reduced to studying the Z-disc, and recently, important advances regarding the arrangement and function of Z-discs in insects have been published. On a very small scale, we have detailed protein structure information. At the medium scale, we have cryo-electron microscopy maps, super-resolution microscopy and protein-protein interaction networks, while at the functional scale, phenotypic data are available from precise genetic manipulations. All these data aim to answer how the Z-disc works and how it is assembled. Here, we summarize recent data from insects and explore how it fits into our view of the Z-disc, myofibrils and, ultimately, muscles.

Model building of protein complexes from intermediate-resolution cryo-EM maps with deep learning-guided automatic assembly

Advances in microscopy instruments and image processing algorithms have led to an increasing number of cryo-electron microscopy (cryo-EM) maps. However, building accurate models into intermediate-resolution EM maps remains challenging and labor-intensive. Here, we propose an automatic model building method of multi-chain protein complexes from intermediate-resolution cryo-EM maps, named EMBuild, by integrating AlphaFold structure prediction, FFT-based global fitting, domain-based semi-flexible refinement, and graph-based iterative assembling on the main-chain probability map predicted by a deep convolutional network. EMBuild is extensively evaluated on diverse test sets of 47 single-particle EM maps at 4.0–8.0 Å resolution and 16 subtomogram averaging maps of cryo-ET data at 3.7–9.3 Å resolution, and compared with state-of-the-art approaches. We demonstrate that EMBuild is able to build high-quality complex structures that are comparably accurate to the manually built PDB structures from the cryo-EM maps. These results demonstrate the accuracy and reliability of EMBuild in automatic model building.

Dyclonine inhibition of TRPV3: From functional discovery to structural insight

• Yao and colleagues recently reported that the clinically approved local anesthetic dyclonine acts as a potent inhibitor of TRPV3, and identified several key residues in TRPV3 pore region that could toggle the inhibitory efficiency. The study by Neuberger et al. sheds light on the cryo-electron microscopy (cryo-EM) map of TRPV3–dyclonine complex, which reveals binding of the inhibitor within the channel pore. The allosteric binding site of dyclonine emerges as a potentially leverageable template for designing TRPV3-targeting drugs.

AI-based structure prediction empowers integrative structural analysis of human nuclear pores.

INTRODUCTION The eukaryotic nucleus pro-tects the genome and is enclosed by the two membranes of the nuclear envelope. Nuclear pore complexes (NPCs) perforate the nuclear envelope to facilitate nucleocytoplasmic transport. With a molecular weight of ∼120 MDa, the human NPC is one of the larg-est protein complexes. Its ~1000 proteins are taken in multiple copies from a set of about 30 distinct nucleoporins (NUPs). They can be roughly categorized into two classes. Scaf-fold NUPs contain folded domains and form a cylindrical scaffold architecture around a central channel. Intrinsically disordered NUPs line the scaffold and extend into the central channel, where they interact with cargo complexes. The NPC architecture is highly dynamic. It responds to changes in nuclear envelope tension with conforma-tional breathing that manifests in dilation and constriction movements. Elucidating the scaffold architecture, ultimately at atomic resolution, will be important for gaining a more precise understanding of NPC function and dynamics but imposes a substantial chal-lenge for structural biologists. RATIONALE Considerable progress has been made toward this goal by a joint effort in the field. A synergistic combination of complementary approaches has turned out to be critical. In situ structural biology techniques were used to reveal the overall layout of the NPC scaffold that defines the spatial reference for molecular modeling. High-resolution structures of many NUPs were determined in vitro. Proteomic analysis and extensive biochemical work unraveled the interaction network of NUPs. Integra-tive modeling has been used to combine the different types of data, resulting in a rough outline of the NPC scaffold. Previous struc-tural models of the human NPC, however, were patchy and limited in accuracy owing to several challenges: (i) Many of the high-resolution structures of individual NUPs have been solved from distantly related species and, consequently, do not comprehensively cover their human counterparts. (ii) The scaf-fold is interconnected by a set of intrinsically disordered linker NUPs that are not straight-forwardly accessible to common structural biology techniques. (iii) The NPC scaffold intimately embraces the fused inner and outer nuclear membranes in a distinctive topol-ogy and cannot be studied in isolation. (iv) The conformational dynamics of scaffold NUPs limits the resolution achievable in structure determination. RESULTS In this study, we used artificial intelligence (AI)-based prediction to generate an exten-sive repertoire of structural models of human NUPs and their subcomplexes. The resulting models cover various domains and interfaces that so far remained structurally uncharac-terized. Benchmarking against previous and unpublished x-ray and cryo-electron micros-copy structures revealed unprecedented accu-racy. We obtained well-resolved cryo-electron tomographic maps of both the constricted and dilated conformational states of the hu-man NPC. Using integrative modeling, we fit-ted the structural models of individual NUPs into the cryo-electron microscopy maps. We explicitly included several linker NUPs and traced their trajectory through the NPC scaf-fold. We elucidated in great detail how mem-brane-associated and transmembrane NUPs are distributed across the fusion topology of both nuclear membranes. The resulting architectural model increases the structural coverage of the human NPC scaffold by about twofold. We extensively validated our model against both earlier and new experimental data. The completeness of our model has enabled microsecond-long coarse-grained molecular dynamics simulations of the NPC scaffold within an explicit membrane en-vironment and solvent. These simulations reveal that the NPC scaffold prevents the constriction of the otherwise stable double-membrane fusion pore to small diameters in the absence of membrane tension. CONCLUSION Our 70-MDa atomically re-solved model covers >90% of the human NPC scaffold. It captures conforma-tional changes that occur during dilation and constriction. It also reveals the precise anchoring sites for intrinsically disordered NUPs, the identification of which is a prerequisite for a complete and dy-namic model of the NPC. Our study exempli-fies how AI-based structure prediction may accelerate the elucidation of subcellular ar-chitecture at atomic resolution. [Figure: see text].

Progressive assembly of multi-domain protein structures from cryo-EM density maps.

Progress in cryo-electron microscopy has provided the potential for large-size protein structure determination. However, the success rate for solving multi-domain proteins remains low because of the difficulty in modelling inter-domain orientations. Here we developed domain enhanced modeling using cryo-electron microscopy (DEMO-EM), an automatic method to assemble multi-domain structures from cryo-electron microscopy maps through a progressive structural refinement procedure combining rigid-body domain fitting and flexible assembly simulations with deep-neural-network inter-domain distance profiles. The method was tested on a large-scale benchmark set of proteins containing up to 12 continuous and discontinuous domains with medium- to low-resolution density maps, where DEMO-EM produced models with correct inter-domain orientations (template modeling score (TM-score) >0.5) for 97% of cases and outperformed state-of-the-art methods. DEMO-EM was applied to the severe acute respiratory syndrome coronavirus 2 genome and generated models with average TM-score and root-mean-square deviation of 0.97 and 1.3 Å, respectively, with respect to the deposited structures. These results demonstrate an efficient pipeline that enables automated and reliable large-scale multi-domain protein structure modelling from cryo-electron microscopy maps.

LPTD: a novel linear programming-based topology determination method for cryo-EM maps.

Topology determination is one of the most important intermediate steps toward building the atomic structure of proteins from their medium-resolution cryo-electron microscopy (cryo-EM) map. The main goal in the topology determination is to identify correct matches (i.e. assignment and direction) between secondary structure elements (SSEs) (α-helices and β-sheets) detected in a protein sequence and cryo-EM density map. Despite many recent advances in molecular biology technologies, the problem remains a challenging issue. To overcome the problem, this article proposes a linear programming-based topology determination (LPTD) method to solve the secondary structure topology problem in three-dimensional geometrical space. Through modeling of the protein's sequence with the aid of extracting highly reliable features and a distance-based scoring function, the secondary structure matching problem is transformed into a complete weighted bipartite graph matching problem. Subsequently, an algorithm based on linear programming is developed as a decision-making strategy to extract the true topology (native topology) between all possible topologies. The proposed automatic framework is verified using 12 experimental and 15 simulated α-β proteins. Results demonstrate that LPTD is highly efficient and extremely fast in such a way that for 77% of cases in the dataset, the native topology has been detected in the first rank topology in <2 s. Besides, this method is able to successfully handle large complex proteins with as many as 65 SSEs. Such a large number of SSEs have never been solved with current tools/methods. The LPTD package (source code and data) is publicly available at https://github.com/B-Behkamal/LPTD. Moreover, two test samples as well as the instruction of utilizing the graphical user interface have been provided in the shared readme file. Supplementary data are available at Bioinformatics online.

NMMD: Efficient Cryo-EM Flexible Fitting Based on Simultaneous Normal Mode and Molecular Dynamics atomic displacements

Atomic models of cryo electron microscopy (cryo-EM) maps of biomolecular conformations are often obtained by flexible fitting of the maps with available atomic structures of other conformations (e.g., obtained by X-ray crystallography). This article presents a new flexible fitting method, NMMD, which combines normal mode analysis (NMA) and molecular dynamics simulation (MD). Given an atomic structure and a cryo-EM map to fit, NMMD simultaneously estimates global atomic displacements based on NMA and local displacements based on MD. NMMD was implemented by modifying EMfit, a flexible fitting method using MD only, in GENESIS 1.4. As EMfit, NMMD can be run with replica exchange umbrella sampling procedure. The new method was tested using a variety of EM maps (synthetic and experimental, with different noise levels and resolutions). The results of the tests show that adding normal modes to MD-based fitting makes the fitting faster (40% in average) and, in the majority of cases, more accurate.

Deep learning-based local quality estimation for protein structure models from cryo-EM maps

In the recent decade, an increasing number of cryo-electron microscopy (cryo-EM) maps were determined to model the tertiary structures of biomolecules. Although the resolution of determined cryo-EM maps is improving in general, there are still many cases where amino acids of a protein were assigned to the map with different confidence levels, including those with relatively high ambiguity. A common error, particularly when side-chain density is not well defined and there are poorly ordered loops between secondary structures, is for the sequence of the model to become out of register.

A comprehensive landscape of 60S ribosome biogenesis factors.

SUMMARYEukaryotic ribosome biogenesis is facilitated and regulated by numerous ribosome biogenesis factors (RBFs). High-resolution cryoelectron microscopy (cryo-EM) maps have defined the molecular interactions of RBFs during maturation, but many transient and dynamic interactions, particularly during early assembly, remain uncharacterized. Using quantitative proteomics and crosslinking coupled to mass spectrometry (XL-MS) data from an extensive set of pre-ribosomal particles, we derive a comprehensive and time-resolved interaction map of RBF engagement during 60S maturation. We localize 22 previously unmapped RBFs to specific biogenesis intermediates and validate our results by mapping the catalytic activity of the methyltransferases Bmt2 and Rcm1 to their predicted nucleolar 60S intermediates. Our analysis reveals the interaction sites for the RBFs Noc2 and Ecm1 and elucidates the interaction map and timing of 60S engagement by the DEAD-box ATPases Dbp9 and Dbp10. Our data provide a powerful resource for future studies of 60S ribosome biogenesis.

De novo identification of mammalian ciliary motility proteins using cryo-EM

SUMMARYDynein-decorated doublet microtubules (DMTs) are critical components of the oscillatory molecular machine of cilia, the axoneme, and have luminal surfaces patterned periodically by microtubule inner proteins (MIPs). Here we present an atomic model of the 48-nm repeat of a mammalian DMT, derived from a cryoelectron microscopy (cryo-EM) map of the complex isolated from bovine respiratory cilia. The structure uncovers principles of doublet microtubule organization and features specific to vertebrate cilia, including previously unknown MIPs, a luminal bundle of tektin filaments, and a pentameric dynein-docking complex. We identify a mechanism for bridging 48- to 24-nm periodicity across the microtubule wall and show that loss of the proteins involved causes defective ciliary motility and laterality abnormalities in zebrafish and mice. Our structure identifies candidate genes for diagnosis of ciliopathies and provides a framework to understand their functions in driving ciliary motility.

Approximating deformation fields for the analysis of continuous heterogeneity of biological macromolecules by 3D Zernike polynomials.

Structural biology has evolved greatly due to the advances introduced in fields like electron microscopy. This image-capturing technique, combined with improved algorithms and current data processing software, allows the recovery of different conformational states of a macromolecule, opening new possibilities for the study of its flexibility and dynamic events. However, the ensemble analysis of these different conformations, and in particular their placement into a common variable space in which the differences and similarities can be easily recognized, is not an easy matter. To simplify the analysis of continuous heterogeneity data, this work proposes a new automatic algorithm that relies on a mathematical basis defined over the sphere to estimate the deformation fields describing conformational transitions among different structures. Thanks to the approximation of these deformation fields, it is possible to describe the forces acting on the molecules due to the presence of different motions. It is also possible to represent and compare several structures in a low-dimensional mapping, which summarizes the structural characteristics of different states. All these analyses are integrated into a common framework, providing the user with the ability to combine them seamlessly. In addition, this new approach is a significant step forward compared with principal component analysis and normal mode analysis of cryo-electron microscopy maps, avoiding the need to select components or modes and producing localized analysis.