Cryo-electron tomography reveals structural insights into the membrane remodeling mode of dynamin-like EHD filaments

Eps15-homology domain containing proteins (EHDs) are eukaryotic, dynamin-related ATPases involved in cellular membrane trafficking. They oligomerize on membranes into filaments that induce membrane tubulation. While EHD crystal structures in open and closed conformations were previously reported, little structural information is available for the membrane-bound oligomeric form. Consequently, mechanistic insights into the membrane remodeling mechanism have remained sparse. Here, by using cryo-electron tomography and subtomogram averaging, we determined structures of nucleotide-bound EHD4 filaments on membrane tubes of various diameters at an average resolution of 7.6 Å. Assembly of EHD4 is mediated via interfaces in the G-domain and the helical domain. The oligomerized EHD4 structure resembles the closed conformation, where the tips of the helical domains protrude into the membrane. The variation in filament geometry and tube radius suggests a spontaneous filament curvature of approximately 1/70 nm−1. Combining the available structural and functional data, we suggest a model for EHD-mediated membrane remodeling.

G-domain that mediates stable dimerization via an EHD family-specific interface (interface-1) ( Supplementary Fig. 1a, b) 22 . Residues at the Nand C-terminal ends of the G-domain form a composite helical domain. In the crystal structure of the EHD2 dimer, the two helical domains protrude in parallel away from the G-domains. This orientation was termed the 'closed' conformation of EHDs. The tip of the helical domain was shown to constitute the primary membrane-binding site 23 .
The C-terminal EH domains interact with linear peptide sequences containing Asn-Pro-Phe (NPF) motifs 26 that are present in binding partners, such as MICAL-L1 27,28 , Rabenosyn-5 9,29 , EHBP1 30 and PACSIN1/ 2 13,15,31 . In the EHD2 dimeric structure, the EH domains bind back to a Gly-Pro-Phe (GPF) motif in the opposing monomer ( Supplementary  Fig. 1b). In this orientation, the C-terminal tails of the EH domains are positioned in the nucleotide-binding site and block the G-interface. The EH domains may thereby auto-inhibit EHD assembly 22 .
The crystal structure of an N-terminal deletion variant of EHD4 in the presence of a nonhydrolysable ATP analogue revealed a 50°rotation of the helical domains compared to the EHD2 crystal structure ( Supplementary Fig. 1b) 32 . In this conformation, the two helical domains in the dimer protrude away from each other. This arrangement was termed the 'open' conformation of EHDs. Spectroscopic experiments indicated that EHD2 binds in the open conformation to flat membrane bilayers 33 . Furthermore, the EH domains were displaced from the G-domain in the EHD4 structure, suggesting that the open conformation is not compatible with EH-domain mediated autoinhibition 32 .
An N-terminal sequence stretch folds back into a conserved hydrophobic pocket of the G-domain in the EHD2 structure 23 (Supplementary Fig. 1). In the presence of membranes, the N-terminal residues were shown to insert into the lipid bilayer 23 . In turn, a flexible loop at the periphery of the G-domain, the 'KPF loop', inserts into the hydrophobic pocket of the G-domain and serves as an oligomerization interface 32 . Deletion of N-terminal residues is therefore expected to stabilize the KPF loop in the G-domain pocket and promote oligomerization. Accordingly, enhanced membrane recruitment and oligomerization was observed for EHD2 and EHD4 variants lacking the N-terminal residues 23,32 .
Based on the two available crystal structures, a nucleotide-driven activation model of EHDs has been proposed 32 . However, since membranes were lacking in any of the reported structures, the detailed conformation of EHDs on membranes and, consequently, their membrane remodeling and oligomerization mode have remained unknown.
In the present work, we reconstitute an N-terminally truncated EHD4 variant on tubular membrane templates and determine these structures by cryo-electron tomography (cryo-ET) and subtomogram averaging (STA). We thereby clarify the oligomerization mode, reveal how EHD4 interacts with membranes, demonstrate how the EHD4 oligomer adapts to various membrane curvatures and propose a model for EHD-mediated membrane remodeling.

Results
Cryo-ET and subtomogram averaging to elucidate the structure of membrane-bound EHD4 To understand the molecular mechanisms of EHDs assembly on membranes, EHD4-coated membrane tubes were reconstituted in vitro. To this end, we used a previously described N-terminal deletion construct of mouse EHD4 (EHD4 ΔN , corresponding to amino acids 22-541) that forms a regular protein coat on membranes and displays enhanced membrane recruitment when overexpressed in eukaryotic cells compared to full-length EHD4 32 . In contrast to EHD4 ΔN , we did not obtain soluble full-length EHD4 with the same expression protocol.
As reported earlier 32 and similar to EHD1 25 and EHD2 23 , EHD4 ΔN remodeled liposomes in an ATP-dependent fashion ( Supplementary  Fig. 2a) and showed liposome-stimulated ATPase activity ( Supplementary Fig. 2b). Accordingly, our in vitro reconstitutions were done in the presence of the nonhydrolysable ATP analogue, adenylylimidodiphosphate (AMPPNP). Liposomes containing 50% Folch extract from bovine brain, 40% phosphatidylethanolamine and 10% cholesterol were chosen as template since they reproducibly yielded densely coated lipid tubes. These tubes were highly heterogeneous, with luminal diameters ranging between 30 to 100 nm (Fig. 1a) and a variety of different shapes (Supplementary Fig. 2a and Supplementary Movie 1). The protein coat had a thickness of~12 nm (Fig. 1a) without and of~16 nm including the lipid bilayer (Fig. 1b, c). In contrast to EHD4 ΔN , EHD2 and an EHD2 variant lacking its N-terminus deformed liposomes into membrane tubes of much smaller diameters (Supplementary Fig. 3).
To determine the structure of the EHD4 coat, we used cryo-ET and reference-free subtomogram averaging (STA) due to the heterogeneity of tube diameters and helical families (Fig. 1b). For this, we collected 56 tilt-series using dose-symmetric tilt scheme, from −60°to 60°with 3°o f increment. We divided the data into two half-datasets, which were processed independently ( Supplementary Fig. 4a). The structures were compared by Fourier Shell Correlation (FSC) and averaged to generate a final structure from 23,813 subtomograms. The resulting cryo-electron microscopy (cryo-EM) density map had an average resolution of 7.6 Å and 8.5 Å, as determined by FSC and FSCindependent analyses, respectively (Supplementary Table 1, Supplementary Fig. 4). Consistently, α-helices of EHD4 could be discerned in the best-resolved parts of the map (Fig. 1c, Supplementary Fig. 5, Supplementary Movie 2). The overall architecture of membranebound EHD4 was highly similar to the closed EHD2 dimer (see below), which allowed us to confidently place and orient the G-, helical and EH domains in the density maps by fitting them as rigid bodies (Fig. 1d). This yielded a pseudo-atomic structure of the assembled EHD4 coat, which was further refined using a flexible fitting strategy (see Methods and Supplementary Movie 2).
Structural determinants of the EHD4 filaments EHD4 formed right-handed helical filaments wrapping around the lipid tubes (Fig. 1e). The EH domains are located furthest from the membrane, the G-domains at the center and the helical domains closest to the membrane (Figs. 1e, 2a). The unit particle used for subtomogram averaging contained three parallel filaments of oligomeric EHD4 ΔN . The center filament (F i ) is composed of three EHD4 homodimers, and the adjacent filaments (F i+1 and F i-1 ) of three EHD4 monomers each (Figs. 1d, e and 2a). As in the previously reported crystal structures, dimerization of EHD4 is mediated by helix α6 in the G-domain. It forms a two-fold symmetric interface, to which we refer as interface-1 (Fig. 2b, Supplementary Movie 3). A point mutation of a conserved tryptophan in this interface (W238A) renders the protein insoluble (Fig. 3) 22 .
Each EHD4 dimer interacts in the filament through two additional interfaces (Fig. 2b, Supplementary Movie 3). Interface-2 is formed between the helical domain of one protomer and the G-domain of the adjacent protomer along the filament. It involves the KPF loop in the G-domain and helices α8 and α12 of the adjacent helical domain. The density of the KPF loop at the periphery of the G-domain dimer is consistent with the open EHD4 crystal structure (Fig. 2b, Supplementary Fig. 5). Point mutations in this loop were previously shown to disrupt oligomerization (Fig. 3) 32 .
Interface-3 is formed between G-domains of two adjacent dimers across the filament (Fig. 2b) and corresponds to the archetypal G-interface that is conserved in all members of the dynamin family. Dimerization via this interface induces nucleotide hydrolysis in dynamin-related proteins 34 . Highly conserved residues in the switch I, switch II, the EHD signature motif and the N-terminal part of α6 are involved in this contact close to the active site. Accordingly, point mutations in these elements in EHD2 were previously shown to abrogate stimulated ATP hydrolysis (Fig. 3) 22 .
In the EHD2 crystal structure, the EH domains bind back to the opposing G-domains 22 . Similar to this closed conformation, the EH domains were also located on top of the G-domain in our membranebound structure ( Supplementary Fig. 6a, b). However, compared to the EHD2 structure, they were 15 Å shifted away to the periphery ( Supplementary Fig. 6c) and stabilized by contacts with EH domains of adjacent dimers ( Supplementary Fig. 6b, d). Accordingly, in this orientation, the C-terminal auto-inhibitory tail may not reach into the active site so that the G-interface can be formed.

Membrane-binding mode of EHD4
We compared the EHD4 structure to the reported crystal structures of EHD2 and EHD4. Membrane-bound EHD4 adopted the closed conformation of the helical domain, akin to the reported EHD2 conformation (Fig. 4a). In this conformation, the long central helix α8 from each monomer protrudes towards the membrane. The rootmean-square deviation of Cα atoms between the closed crystal structure and the membrane-bound EHD4 conformation, excluding the EH domains, is only 2.4 Å for the monomer, attesting to the high similarity of the two structures. Thus, surprisingly, the closed, supposedly autoinhibited EHD2 dimeric crystal structure represents a good model for the membrane-bound EHD4 state determined in this study.
The membrane bilayer was well-defined in our map and had a thickness of about 45 Å (Fig. 4b). The outer and inner layers were clearly separated. The strong density at the periphery of the bilayer likely corresponds to the phosphate headgroups, which have greater signal in comparison to the fatty acid tails, as seen already in other cryo-ET structures of membrane-bound protein structures 35 the lack of bilayer structures and models, we refrained from adding a bilayer model to our map. By electron paramagnetic spin resonance experiments and mutagenesis, EHD2 residues at the tip of the helical domain were shown to insert into the membrane (Fig. 3, Fig. 4c) 23 . These findings are consistent with the structure of membrane-bound EHD4. Thus, helices α8 and α9 in the helical domain engage with the membrane bilayer by inserting hydrophobic residues at the connecting loop into the membrane outer leaflet (Fig. 4b, c). We employed the helical domains from the EHD4 crystal structure to analyze the side chain positions in relation to the membrane outer layer. In the membrane-bound structure, the Cα atoms of residues K331 and R332 are 1.5-3 Å above the outer leaflet of the bilayer, whereas residues E328, K330 were 0.5-2.5 Å below (Fig. 4c). Residues N323, M324 and F325 at the α8-α9 connecting loop deeply insert into the membrane, with their Cα atom 4, 6, and 8 Å below the membrane density, respectively (Fig. 4c). Charged residues in helix α9, such as K327, K330 and E333 were not inserted into the membrane but were close enough for interaction with the polar lipid head groups (Fig. 4c). Residues in the membrane binding region are highly conserved amongst EHD paralogues and across different species (Fig. 4d). However, each EHD paralogue has a unique combination of membrane-interacting residues (Fig. 4c, d). Thus, EHD proteins bind to membranes through charged residues at helices α8 and α9, which likely confer lipid specificity, and conserved hydrophobic residues at the connecting loop 22,37 . Notably, EHD4 membrane interaction resulted in a local bending of the outer membrane monolayer (Fig. 4b, insets).

EHD4 oligomers assemble on membranes of different curvature
EHD4 ΔN -coated tubes had a wide range of radii (Fig. 5a). Cryo-ET and STA allowed us to probe the architecture of the EHD4 ΔN coat on individual tubes, and therefore, the geometry of individual filaments to be discerned. The EHD4 ΔN coat was governed by a set of related helical families, which varied in pitch, rise, and subunits per turn (Figs. 1b and 5b). By tracing the position of the filaments with the refined orientations in the tomograms, we measured the helical angle of several filaments as a function of the underlying tube radius (Fig. 5a, b, Supplementary Fig. 7a, b).
The helical angle, i.e. the deviation of the helical filament from a simple ring around the tube, tended to increase as the tube radius decreased. This relationship is expected for an elastic filament with a spontaneous curvature less than the curvature of the wrapped tube. Similar behavior is seen for highly constricted dynamin-coated tubes 38,39 . A best fit suggests that the spontaneous curvature of the AMPPNP-bound-EHD4 ΔN filament is 1/68 nm −1 (Fig. 5a, bottom). Fitting to more complicated elastic models suggests that the twist stiffness of the filament is significantly weaker than the curvature stiffness and that the angle of each EHD4 ΔN dimer relative to the tube axis plays little to no role in determining the helical angle ( Supplementary Fig. 7c).
Since it requires energy to generate membrane curvature, the ability of EHD4 ΔN to produce membrane tubes smaller than its spontaneous curvature suggest some additional factor(s) promoting curvature. An obvious candidate is the membrane interaction of EHD4 ΔN . An alternative is the interaction of helices α1 and α12 from neighboring   EHD4 monomers (interface-4), which mediates the packing of neighboring filaments into a continuous coat (Fig. 5c). Interface-4 shows variation as the radius of the tube varies, and preferences toward a particular orientation would influence the underlying tube radius. While EHD4 forms ordered oligomers on membrane tubes with radii ranging from 15-70 nm (Fig. 5a), it can also bind to very highly curved tubes (e.g. radius <10 nm), but apparently cannot form ordered oligomers (Fig. 5d).

Discussion
Recent advances in cryo-EM have facilitated the structural analysis of membrane-bound protein scaffolds. Helical reconstructions requiring highly homogeneous samples (reviewed in 40 ) have allowed, amongst others, medium to high-resolution structural elucidation of the acetylcholine receptor 41 , BAR domain proteins 42,43 , endosomal sorting complexes required for transport (ESCRT) 36 , light-dependent protochlorophyllide oxidoreductase (LPOR) 35 and dynamin 44 assembled on membrane tubes. Structures of highly heterogeneous membranebound protein coats cannot be determined by helical reconstructions. However, recent advances in image processing have facilitated the structure solution of such specimens by cryo-ET analysis combined with subtomogram averaging. Although the average resolution of such reconstruction is often lower compared to those derived from singleparticle cryo-EM analyses (see Supplementary Table 2), pseudo-atomic models can be obtained by fitting available higher-resolution crystal structures into the density. Examples for cryo-ET structures include the membrane-bound COPI 45 and COPII coats 46 , the N-BAR protein Bin1 bound to membranes 47  Article https://doi.org/10.1038/s41467-022-35164-x bound to membrane tubes were highly heterogeneous 32 , necessitating the use of cryo-ET for structural analysis. By tracking individual tubes from the refined subtomogram averages, this analysis allowed us not only to determine the structure of EHD4 within one filament but also to determine the EHD4 filament structures on various membrane curvatures. Our analysis has important implications for understanding how the ATPase cycle of EHDs is coupled to membrane recruitment, filament assembly and disassembly and how EHD4 generates membrane curvature. A resulting working model for the ATPase-dependent membrane cycle is outlined in the following.
Previous X-ray crystallographic analyses identified two conformations of EHDs. In the reported EHD2 crystal structure 22 , the protein adopts a closed conformation, whereas the crystal structure of EHD4 features an open conformation 32 (Supplementary Fig. 1). Spectroscopic studies suggest that EHDs are recruited to flat bilayers in an open conformation 33 . Since the membrane interaction in the open conformation involves mostly polar interactions in the helical domain 32 , EHDs may be in a rapid exchange with the cytosol. The G-domain is close towards the membrane in the open conformation so that the N-terminus can switch from its hydrophobic G-domain pocket into the membrane bilayer (Fig. 6a). The release of the N-terminus allows the KPF loop to enter the hydrophobic pocket to create oligomerization interface-2. Furthermore, our previous crystallographic study on EHD4 indicated that the G-interface (interface-3 in this manuscript) cannot be formed between EHD4 dimers in the open conformation due to steric constraints 32 .
The transition of the open to the closed conformation, as observed in our study in the membrane-bound form, appears to be driven by the assembly of EHD oligomers on curved membranes (Fig. 6a, Supplementary Movie 4). By bilayer coupling 50,51 , the insertion of the hydrophobic helical tip region into the membrane is expected to generate membrane buckling, in line with the undulating appearance of the outer membrane layer of the EHD4-coated tubes in our cryo-ET reconstructions (Fig. 4b) and previous molecular dynamics simulations 25 . Similar to FYVE and ENTH domains 52 , the EHD membrane-binding site is composed of charged residues and a hydrophobic membrane-penetrating protrusion. In turn, curved membranes may facilitate the insertion of hydrophobic residues and therefore promote the transition from the open to the closed conformation in the EHD filament.
Upon initial curvature generation by this wedging mechanism, EHD filaments then assemble into ring-like or helical oligomers via interfaces-2 and -3 (Fig. 6a), representing our membrane-bound structure. ATP-binding stabilizes the switch regions which in turn promotes the assembly 34 . The involvement of the G-interface also explains the strict ATP dependence of regular filament assembly on membranes ( Supplementary Fig. 2) 23,25 . Accordingly, in the absence of nucleotide, EHDs can still interact with membranes ( Supplementary  Fig. 2) 23 , but not assemble into a membrane-remodeling filament.
G-interface formation is associated with the displacement of the EH domain tail from the G-interface by the movement of the EH domains towards the periphery of the EHD filament Tube radius R (nm)

Number ot tubes
Tube radius R (nm) Helix angle (˚)    6). In this orientation, the EH domains may bind to NPF-motif containing partner proteins ( Supplementary Figs. 6e, f). Formation of the G-interface is accompanied by a stimulation of the slow ATP hydrolysis reaction in EHDs 21,22 . In this way, ATP hydrolysis may act as an intrinsic timer to disassemble the EHD scaffold: In the ADP-bound state, the switch regions are destabilized and the interaction of switch I with the KPF loop is reduced 32 . Accordingly, interfaces-2 and -3 are therefore weakened in the ADP-bound state, likely leading to dissociation of the oligomer (Fig. 6b). The ADP-bound EHD dimer may convert back to the open conformation and eventually dissociate from the membrane, therefore completing the ATPase cycle. The EHD4 filament coat differs from the canonical oligomer architecture shared by several dynamin-related proteins, e.g. dynamin, DRP1, and Mgm1/OPA1. There, stalk interactions define the filament and nucleotide-dependent interactions between G-domains stabilize the inter-filament packing. EHDs on the other hand, orient the nucleotide-dependent interface along the filament direction and pack neighboring filaments with a stalk-stalk-like interaction. This change in architecture allows the intrinsic filament curvature to be modulated by nucleotide, or perhaps more interesting, the nucleotide state to be affected by the geometry of the filament. Additionally, whereas the pitch during active dynamin constriction is essentially fixed by the strong cross-filament interaction 53 , EHD4 architecture may allow continuous reorientation of the filament as the underlying membrane curvature changes.
As the helix angle of the filament changes, the orientation of the individual EHD4 dimers adapts. This orientation has been previously suggested to be meaningful 22 as the membrane binding surface of EHD2 dimers in the closed conformation appears curved. Our structure shows that the dimer curvature is approximately 60°out of phase with the filament, meaning that when the helix angle is 60°, the dimer's curvature is aligned with the tube curvature. In this orientation, any curvature generation by the dimer should be maximized. Interestingly, the maximum helical angles observed are roughly 60°, where the tube radius is about 15 nm. This state, with a large pitch, may represent the maximum curvature that the EHD4 filament can stabilize. Further constriction of the tubes, for example by pulling forces and/or other membrane remodeling proteins 54 , would be expected to destabilize the EHD filament on the membrane tube, leading to disassembly (Fig. 6b). Previous mutagenesis data (Fig. 3) 22,32 are consistent with the formation of membrane-remodeling EHD filaments in the cell, although such scaffolds have not yet been unequivocally visualized in a cellular context. The low spontaneous membrane curvature of the EHD4 filament is in line with the reported cellular localization of EHD4 on macroendosomes 18 or early endosomes 55 , which possess low membrane curvatures. In contrast to EHD4, ring-like assemblies around membrane tubes of higher curvatures were demonstrated for EHD1 25 and EHD2 22 (Supplementary Fig. 3), in agreement with their reported cellular localization on more highly curved cellular membranes, such as membrane tubes 5 and the neck of caveolae 14 , respectively. Thus, the filaments' curvature preference of different EHD paralogues may be adapted to the architecture of the cellular membrane compartments they are acting on. Future super-resolution light microscopy and in situ electron microscopy studies are needed to characterize the detailed integration of the EHD filaments into their cellular membrane environment, their interaction with cellular partners and the cellular requirements for the formation and proposed ATPase-driven disassembly of the filaments.
Taken together, our structural analyses of the membrane-bound EHD4 scaffold elucidates novel insights into the coordination of the ATPase cycle with membrane recruitment, assembly and disassembly of the protein scaffold, and provides experimental insights into how membrane curvature is generated by EHD scaffolds.

Liposome preparation
Liposomes were prepared by adding 50 µL of 50% Folch extract from bovine brain fraction I, 40% phosphatidylethanolamine and 10% cholesterol to 200 μL of a chloroform/methanol (1:0.3 v/v) solution, which was then dried on a glass surface under an argon stream. Lipids were resuspended in liposome buffer (20 mM Hepes/NaOH (pH 7.5), 150 mM NaCl and 2.5 mM β-ME), sonicated in a water bath for 30 sec and extruded through a 1 μm filter.

Tubulation assay
For membrane tubulation assays, 10 μM EHD4 ΔN , EHD2 or EHD2 ΔN in tubulation buffer were incubated at room temperature for 20 min with 1 mg/ml liposomes in the absence or presence of 1 mM of the indicated nucleotide.
ATPase assay ATP hydrolysis of EHD4 ΔN was studied at 30°C in 20 mM Hepes/NaOH (pH 7.5), 150 mM NaCl, 2.5 mM β-ME and 0.2 mM MgCl 2 using 10 μM protein and 100 μM ATP as a substrate, in the presence and absence of 1 mg/ml liposomes (see before). Reactions were started by adding the protein. At any given timepoint, aliquots from the reaction were 6 times diluted and flash-frozen in liquid nitrogen. Hydrolysis was measured using HPLC measurement. The nucleotides were separated with a reversed-phase column (C18 100 mm × 4.6 mm) and detected by absorption at 254 nm. Quantification was done by integrating the corresponding nucleotide peaks (see Source Data). GraphPad Prism 7.05 was used to plot the curve.

Negative-stain EM
Liposomes at a concentration of 1 mg/ml were incubated for 20 min at room temperature with 10 µM protein in the presence of 1 mM ATP. Samples were applied for 30 sec to carbon-coated formvar films mounted on copper grids, excess liquid was removed, and the sample stained for 30 sec with 2% uranyl acetate. Four areas on three different grids were monitored on a Talos L120C (Thermo Fisher Scientific) at 120 kV equipped with a Ceta Detector to obtain representative images.

Cryo-electron microscopy and image processing
Complexes formed of EHD4 ΔN and liposome were diluted with a buffer containing 10 nm colloidal gold. 4 µl of this mixture was applied on a glow-discharged Quantifoil R2/2 grid (Quantifoil Micro Tools GmbH) and flash-frozen in liquid ethane using a Vitrobot Mark II device (Thermo Fisher Scientific). The grids were stored under liquid nitrogen conditions until usage. Initial data was recorded on a Talos L120C microscope (Thermo Fisher Scientific) at 120 kV. The final set of 56 tilt series was collected on a Titan Krios G3 electron microscope (Thermo Fisher Scientific) operated at 300 kV equipped with a Gatan Quantum energy filter (slit with 20 eV) and a Gatan K2 detector using SerialEM 4.0 (ref. 56). The nominal magnification was 53,000× resulting in a pixel size of 2.628 Å. The data was acquired at a tilt range from −60 to 60 degrees using a dose-symmetric tilt scheme 57 at 3°increment. Tilt series were recorded as movies of 12 frames in counting mode and a dose rate of 2.3 e − /Å 2 at defocus range of −3 to −6 um resulting in a total dose of 94 e − /Å 2 per tilt series. Motion correction was carried out using MotionCor2 1.1.0 (ref. 58). The initial contrast transfer function defocus value for each image of the tilt series was estimated using CTFFind 4.1. 10 (ref. 59). CTF correction was carried out by phase flipping using the program 'ctf phase flip' of the IMOD 4.11 software package 60 . Two copies of each tomogram were reconstructed using weighted back projection and seven rounds of the Simultaneous Iterative Reconstruction Technique (SIRT) in IMOD 4.11 (ref. 60).
For comparison, two grids of EHD4 ΔN in the apo form were prepared and data collected on a Titan Krios G3i transmission electron microscope (ThermoFisher Scientific, Server Version 2.15.3) operated at 300 kV equipped with an extra bright field-emission gun (XFEG), a BioQuantum post-column energy filter (Gatan) and a K3 direct electron detector (Gatan Digital Micrograph Version 3.32.2403.0). Images were recorded in low-dose mode as dose-fractionated movies using SerialEM 4.0 in energy-filtered zeroloss (slit width 20 eV), nano-probe mode at a nominal magnification of 53,000x (resulting in a calibrated pixel size of 0.84 Å/px on the specimen level) in super-resolution mode with a 70 µM objective aperture.

Subtomogram averaging
The workflow described uses a combination of Dynamo 1.1.157 and bespoke scripts from MATLAB version r2020b. Initial particle picking was done using a filament tracer by assigning the center of each tube. A first round of alignments was performed using SIRT-filtered reconstructed tomograms binned 3 times (7.884 Å/pixel). Particles for each tube were extracted using a box size of 128 pixels, randomized along its azimuth, averaged and low-pass filtered to 40 Å to generate the initial template for 3 rounds of coarse alignments. Oversampled particles converging onto the same coordinate were removed using Dynamo's separation in the tomogram parameter. Each tube was aligned individually and sub-boxed along the membrane to generate a section of the tube (Supplementary Fig. 4a). Averaged sections were merged, aligned to the template and low-pass filtered to 40 Å. Multireference analysis (MRA) was used to eliminate bad particles from the dataset. Next, CTF-corrected subtomograms were extracted and aligned to low-pass filtered references and the dataset was further divided into two half datasets, even and odd, for independent processing. Iterations were carried out starting from binned 3X data, using a low-pass filter of 20 Å, angular sampling of 12°, allowing shifts of 47 Å, and refinements were gradually improved by decreasing the binning factor, using less stringent low-pass filters and finer angular sampling. Final refinement steps were carried out on unbinned data extracted in 128 voxel boxes, using a low-pass filter set at 8 Å, angular sampling of 4°, and shift limits of 10 Å. A total of 23,813 subtomograms (11,

Flexible fitting
The fitting procedure is summarized in Supplementary Movie 2. An atomic model consistent with the cryo-EM map was generated using MDfit with Gromacs 4.5.5 64 . MDfit uses the cryo-EM map as an umbrella potential to bias (i.e. deform) an underlying structure-based model (SBM) 65 in order to maximize the cross-correlation between the experimental density and the simulated electron density. An SBM is a molecular force field that is explicitly, albeit not rigidly, biased toward a certain native structure. The SBM for fitting was the EHD2 crystal structure (pdb code 4CID) with the sequence homology modeled by SWISS-MODELL 66 to that of EHD4 (residues 22-535). The portion of the SBM for the KPF loop (residue 114-137), which is missing from the EHD2 structure, is based on the EHD4 crystal structure (pdb code 5MVF). Building the SBM from the crystal structure ensured that the resulting model was maximally consistent with the crystal conformation. This entailed no significant changes in structure as the sequences are highly similar and included a missing loop in the crystal structure (residues 424-442). A preprocessing step was then necessary to move the EH domains within the dimer into a cis positioning because 4CID placed the EH domains in trans. This involved only the reorientation of the 424-442 loop, no other residue positions were changed. We refer to this dimeric structure as EHD4-init. Since the EH domain is missing, the SBM for the EH domain is generated from the EHD2 crystal structure (residues 443-538). An SBM using EHD4-init as the input structure was then generated using SMOGv2.3beta 65 with the template "SBM_AA" meaning all nonhydrogen atoms were explicitly represented.
The density corresponding to the central two dimers within the cryo-EM map was chosen as the constraint for MDfit, since this region had the best resolution. Relaxation of the SBM under the influence of the cryo-EM map is performed by molecular dynamics (MD), and, thus, requires an initial condition. Two EHD4-init were rigid body fit into the map using the "Fit in Map" tool of Chimera 1.14 63 . In order to compensate for the missing neighbors on either side of two dimers, the translational symmetry of the filament was exploited. Two additional copies of EHD4-init were added, positioned on either side, placed such that each dimer-dimer interface was identical. Technically, this was performed by 1) measuring the transformation X between the two central dimers in Visual Molecular Dynamics (VMD) 1.9.3, 2) duplicating the central dimers, and 3) applying X or -X to the duplicates. This four-dimer system served as the initial condition for MD. During MD, the duplicates were given strong position restraints, while the only constraint on the central dimers was the MDfit umbrella potential based on the cryo-EM map. Every 10 4 MD steps, the duplicate dimers were repositioned. Through this iterative process, the structure converged within 3×10 5 steps. The middle two dimers were taken as the atomic model. Note that even though the filament's local C2 rotational symmetry was not explicitly enforced by us during MD, the fact that the SBM was based on a C2 symmetric structure ensured that this symmetry was included.

Elastic model fitting
The fitted helical angle as a function of radius was defined by where the bit in brackets is an elastic energy and the min returns θ such that the radius is r and elastic energy is minimized. κ is the curvature, τ is the twist, η is the dimer orientation with respect to the tube axis, θ is the helix angle, and r is the radius of the tube. κ 0 and τ 0 are the spontaneous curvature and twist. Figure 5a (see also Source Data) shows the case for k κ = 1, k τ = k η = 0, where the twist elasticity and the dimer orientation are both negligible compared to filament curvature. Supplementary Fig. 7c shows two alternatives, the first with k κ = k τ = 1, k η = 0, which is typical for continuous filaments, and the second with k κ = 1, k τ = 0, k η = 0:2, which accounts for a preference for the dimer curvature to align perpendicular to the tube axis. For a constant helix, κ = r r 2 + h 2 , and τ = h r 2 + h 2 , and θ = tan À1 h r , where r is the radius and 2πh is the pitch. In our EHD4 filament, the dimer is oriented approximately 30°relative to the helix angle, defining η = θ + 30°. Therefore, when η = 90 θ = 60°, the dimer's footprint curvature is optimally oriented with respect to the membrane tube. The two best fits with k η = 0 are performed using the Python library scipy.optimize.least_squares 1.7.1 with κ 0 and τ 0 as fitting parameters. The curve with k η = 0:2 uses κ 0 = 1 68 nm À1 and has no free parameters to be fitted. Note that there is an analytic form for the minimum energy line with k κ = 1, k τ = k η = 0, which is given by Reporting summary Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Data availability
The data that support this study are available from the corresponding authors upon reasonable request. The cryo-EM map has been deposited in the Electron Microscopy Data Bank (EMDB) under accession code EMD-25362 (EHD4 filaments). Coordinates have been submitted to the Protein Data Bank (PDB) under accession code 7SOX (EHD4 asymmetric unit).
Coordinates of the previously published crystal structures used can be accessed via 4CID (EHD2) and 5MVF (EHD4 ΔN ). The source data underlying Fig. 5a and Supplementary Figs. 2b and 7c are provided as a Source Data file. Source data are provided with this paper.
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