Quantification and demonstration of the collective constriction-by-ratchet mechanism in the dynamin molecular motor

Nucleotide binding and dissociation rates to this construct were evaluated using stopped-flow experiments with fluorescently labeled 2′(3′)-O-(N-methylanthraniloyl) (mant)-nucleotides (Fig. 1 B and C and SI Appendix, Fig. S1A). The affinities for GTP and GDP were both in the low micromolar range (Fig. 1D). Therefore, at physiological nucleotide concentrations of GTP (300 $\mathrm{\mu }\text{M}$) and GDP (30 $\mathrm{\mu }\text{M}$) (27), about 90% of monomeric MMs in the cell are bound to GTP. Previous measurements (28, 29) reporting even higher GTP affinity were performed using full-length dynamin, thus also including possible contributions from nucleotide-induced assembly that enhance nucleotide occupancy.

A low basal rate of GTPase hydrolysis and a linear dependence of the specific hydrolysis rate $k_{\mathrm{\text{hyd}}}^{\mathrm{\text{spec}}}$ on MM concentration were observed (Fig. 1 E and F and SI Appendix, section S1.A.1), confirming that MM dimerization should precede the hydrolysis and is required for it. These GTPase measurements additionally indicated that the GTP-bound dimer has a weak affinity ($K_{\mathrm{d}}^{\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{r}}\gg 50\hspace{0.17em}\mathrm{\mu }\text{M}$).

Running the GTPase assay beyond the linear regime allows GDP to compete with GTP for the active site. Therefore, the GTPase kinetics of such an experiment contain information on the relative affinity between GTP and GDP for the MM. A kinetic model was devised to fit experimental traces and determine unknown parameters (SI Appendix, section S1.A.2). Varying this setup with different initial conditions (SI Appendix, Fig. S1 B and D) allowed us to estimate an apparent GTP on rate of 6.6 ${\text{s}}^{-1}\cdot \mathrm{\mu }\text{M}{\hspace{0.17em}}^{-1}$, which was not straightforward to determine with mant-GTP (SI Appendix, Fig. S1A).

Measuring nucleotide exchange kinetics within the dimer is complicated by the low MM dimerization affinity. To resolve this difficulty, we engineered a ${\mathrm{Z}\mathrm{n}}^{2+}$-dependent metal bridge by introducing histidines into the G interface (MM-HH). X-ray crystallography confirmed that ${\mathrm{Z}\mathrm{n}}^{2+}$ stabilized a similar dimer arrangement as previously observed in the GDP-bound MM dimer (24) (SI Appendix, Fig. S2 and Table S1). In line with our predictions, addition of ${\mathrm{Z}\mathrm{n}}^{2+}$ to this double histidine (MM-HH) construct induced dimerization in gel-filtration (SI Appendix, Fig. S3A) and FRET experiments (SI Appendix, Fig. S3B). Of note, addition of $\text{GTP}\gamma \text{S}$ reduced mutant dimer formation (SI Appendix, Fig. S3B).

Dimer dissociation rates were measured by FRET experiments and stopped-flow. Addition of GDP slowed the dimer dissociation rate of the ${\mathrm{Z}\mathrm{n}}^{2+}$-stabilized MM-HH dimer sixfold relative to the apo state (SI Appendix, Fig. S3 D and E), consistent with a long-lived posthydrolysis dimeric state. Addition of GTP greatly accelerated dimer dissociation, suggesting that the GTP-bound state is not compatible with the engineered ${\mathrm{Z}\mathrm{n}}^{2+}$ bridge. Relevant to the motor cycle, mant-GDP did not bind to empty ${\mathrm{Z}\mathrm{n}}^{2+}$-stabilized MM-HH dimers. Furthermore, mant-GDP dissociation from MM-HH dimers was 100-fold slower in the presence of ${\mathrm{Z}\mathrm{n}}^{2+}$ (SI Appendix, Fig. S3 G–J). These observations imply that nucleotide exchange is greatly reduced in the MM dimer, therefore simplifying dynamin’s motor cycle (see below).

To characterize nucleotide-dependent conformational changes in the MM, single-molecule FRET (smFRET) was employed. We monitored the relative orientation of the G domain and BSE by introducing a pair of FRET dyes, one in BSE and the other in the G domain (Fig. 2 A and B and SI Appendix, Fig. S4). The experiments were performed on freely diffusing MMs.

In the presence of GTP, three populations could be discerned (Fig. 2A and SI Appendix, Fig. S4C) by fitting to Gaussian distributions: a dominant open state with a FRET efficiency peak ($E$) at $E=0.18$, an intermediate state at $E=0.39$, and a closed state at $E=0.77$. The peak values of the open and closed states were in excellent agreement with the predicted mean FRET efficiencies from simulations based on the crystal structures of 0.19 and 0.79, respectively (Fig. 2C). The GTP analogs (GMPPCP and $\text{GTP}\gamma \text{S}$) showed bimodal distributions. Interestingly, the low FRET peak for $\text{GTP}\gamma \text{S}$ coincided with the intermediate peak found in the presence of GTP. Note that the nonhydrolyzable GTP analogs produce different structural ensembles from those found with GTP. Care has therefore to be taken when interpreting experiments using GTP analogs. Remarkably, in absence of nucleotides or in presence of GDP, the conformational distributions were strongly shifted toward the closed state. For the apo protein, the position of the FRET peak coincided with that of the GTP-bound closed state. For GDP, the closed state was shifted slightly higher to $E=0.8$, consistent with crystal data (SI Appendix, Fig. S4D).

Ensemble stopped-flow measurements were used to obtain relaxation rates to the equilibrium distributions. The rate of opening, i.e., of a transition from apo (closed) to the GTP-bound state, was fitted to a single exponential with a time constant of approximately $300\hspace{0.17em}{\text{s}}^{-1}$ (Fig. 2 D and E). This rate contains both nucleotide binding and the conformational change. However, since nucleotide binding at 1 mM GTP is much faster than opening, the measured rate must correspond to the conformational change. Because 300 ${\text{s}}^{-1}$ is large compared to the overall turnover rate of the GTPase cycle, GTP binding and opening are combined in our summary of the GTPase cycle (Fig. 3). The rate of closing in solution, induced by GTP dissociation, is larger than $105\hspace{0.17em}{\text{s}}^{-1}$ (SI Appendix, Fig. S4E). However, this parameter is of little relevance for dynamin’s function since, in the filament, closing takes place within a dimer and under load.

The kinetic and conformational results are summarized in the diagram of the GTPase cycle (Fig. 3). The GTP-bound MM monomers are predominantly open and must dimerize (GTP-D) to enable hydrolysis. Hydrolysis leads to a strong preference in the MM dimer (GDP-D) for the closed state. Using the MM-HH mutant, we confirmed that the GDP-bound dimer is strongly closed in a similar fashion to the GDP-bound monomer (SI Appendix, Fig. S3C). The MM dimer in its GDP$\cdot$inorganic phosphate (Pi) or GDP-bound states dissociates only slowly, consistent with the high stability of the dimers bound to GDP$\cdot$Al${\mathrm{F}}_4^{-}$ (23) and with our kinetic results with GDP (SI Appendix, Fig. S3E). Nucleotide exchange within the MM dimer is slow and can therefore be neglected in the scheme (SI Appendix, Fig. S3 I and J). Once monomeric, GDP exchanges in favor of GTP, thus completing the cycle.

Note that hydrolysis ($k_{\mathrm{h}\mathrm{y}\mathrm{d}}$) and posthydrolysis MM dimer dissociation ($k_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}$) rates, as well as the dimerization ($k_{\mathrm{o}\mathrm{n}}^{\mathrm{D}}$) and dissociation ($k_{\mathrm{o}\mathrm{ff}}^{\mathrm{D}}$) rates of MM monomers in the GTP-bound state, remain yet unknown, meaning the entire GTPase cycle could not be precisely specified. Therefore, in our model simulations, various values of the effective hydrolysis rate $k_{\mathrm{h}\mathrm{y}\mathrm{d}}^{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}}$ and of $k_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}$ were chosen and explored (see SI Appendix, section S1.B for details). Measurements of GTPase activity in the presence of membranes set a lower limit of 4 ${\text{s}}^{-1}$ (20) for its turnover rate. The slowest measured rate in the forward direction is that of GDP dissociation at 120 ${\text{s}}^{-1}$, yielding an upper limit for the overall turnover rate.

In the helical filament, a pair of GTP-bound MMs belonging to adjacent rungs can dimerize to form a cross-bridge (Fig. 4A). Upon hydrolysis in both MMs, the equilibrium conformation of the dimer shifts from the open to the closed ensemble, thus shortening its equilibrium bridging length and creating a contracting force along it.

The length of a bridging MM dimer is quantified by the end-to-end length distributions $P_{\mathrm{G}\mathrm{T}\mathrm{P}}\left(\ell \right)$ and $P_{\mathrm{G}\mathrm{D}\mathrm{P}}\left(\ell \right)$. Although distance distributions can be yielded by smFRET, our FRET experiments are limited to MM monomers. Therefore, probability distributions over $\ell$ for the two nucleotide states were obtained from MD simulations of a dual-basin structure-based model (31) (see SI Appendix, Fig. S5 and section S1.C for a detailed discussion). The MM dimer representation was built by connecting two MM monomers, each described by the dual-basin model, at their G interface (Fig. 4B). Importantly, the simulations incorporated the experimentally observed energetics by enforcing that the simulated FRET distribution in each MM monomer match the observed smFRET distribution of open and closed states (SI Appendix, Fig. S5).

The free energy profile of a GTP-bound MM dimer along $\ell$ shows a single broad basin (Fig. 4C). This implies that initial cross-bridges can form over a wide range of distances. This property is enabled by the heterogeneity of the GTP-bound conformations (Fig. 2A). In contrast, the free energy $G_{\mathrm{G}\mathrm{D}\mathrm{P}}$ for GDP-bound MM dimers has a single narrow minimum at a shorter $\ell$ (Fig. 4C). The effective contracting force along $\ell$ is given by $F\left(\ell \right)=-d{G}_{\mathrm{G}\mathrm{D}\mathrm{P}}\left(\ell \right)/d\ell$. In the simulations of membrane constriction, this instantaneous force, produced by a MM dimer when it is in the GDP-bound state and has length $\ell$, is employed. Remarkably, the force remains approximately constant (Fig. 4C) at $F_0$ = 1.3 ${\mathrm{k}}_{\mathrm{B}}$T/nm over the range of $\ell$ from 8 to 14 nm.

To transform the cyclic changes of the GTPase cycle (Fig. 3) into a directed force acting on the helical filament, a ratchet mechanism must be employed. The principal feature of the ratchet is a differential response to the power stroke and the recovery stroke. The transition from an open to a closed conformation, representing a power stroke, takes place when the two MMs from the neighboring rungs are dimerized and forming a cross-bridge. The recovery stroke, corresponding to the reverse transition to the open state, however, occurs only in the monomeric MM state, i.e., when the bridge is absent. Therefore, force is applied to the filament only in one part of the GTPase cycle.

There is, however, an additional component to the ratchet effect. If it were equally probable that the cross-bridges slanted to the right and to the left, shortening of such links could not generate a net force. Hence, a directional bias in slanting is needed. For dynamin’s right-handed helix, preferentially slanting the upper MM to the right and the lower MM to the left with respect to the stalk filament would generate a constricting torque (i.e., as in Fig. 4A). Indeed, the comparison of all available dynamin structures has shown that hinge 1 is biased in this way, and our molecular dynamics (MD) simulations have indicated that the biased orientation is robust to thermal fluctuations (SI Appendix, Fig. S6).

One can calculate the mean force generated by a pair of MMs over many GTPase cycles by averaging the force applied between two immobilized stalk filaments (under stall conditions). Note that time averaging is equivalent to statistical averaging over an ensemble. Using the distributions for $\ell$ from MD simulations and the geometry of Fig. 4C, the component of the mean force acting along the filament is $⟨F_s⟩=\omega {\int }_{{\ell }_{\mathrm{m}\mathrm{i}\mathrm{n}}}^{{\ell }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{P}_{\mathrm{G}\mathrm{T}\mathrm{P}}\left(\ell \right)F\left(\ell \right)\sin \theta \left(\ell \right)d\ell$, where $\theta$ is the slant angle and $\omega$ is the probability (often referred to as the duty ratio) (32) to find a MM in the dimerized state. Because of the ratchet property of hinge 1, only configurations with positive displacements $\mathrm{\Delta }s$ are included into the average. The transverse component $⟨F_t⟩$ is given by the same equation with $\sin \theta$ replaced by $-\cos \theta$.

The mean forces $⟨F_s⟩$ and $⟨F_t⟩$ are shown as functions of the rung separation $h$ in Fig. 4D. Note that the derived estimates are sensitive to the relative weights of the open and closed conformations in a GDP-bound MM (SI Appendix, Fig. S5 C and D). Since the observed frequency of the open conformation in smFRET is likely enhanced by experimental noise, they should be viewed as providing the lower bounds for the force.

The torque, locally applied to the filament by a MM dimer motor, is $⟨F_s⟩R$, where $R$ is the filament radius. Assuming that a MM is found predominantly in the dimerized state, i.e., $\omega$ is near unity, $⟨F_s⟩$ is 2.9 pN (0.7 ${\mathrm{k}}_{\mathrm{B}}$T/nm) and the mean torque generated by a single motor in the nonconstricted state (with $h=8$ nm and $R=25$ nm as in Fig. 4A) is 72 pN$\cdot$nm. When many MMs are simultaneously cooperating, large torques on the order of 1 nN$\cdot$nm can be produced, similar in size to what has been experimentally observed (10). These collective effects are discussed in the next section.

The MM dimers, connecting neighboring rungs, produce forces that tend to rotate them with respect to one another. Collectively, they generate the torque applied to the filament. We directly demonstrate this via computer simulations of a mesoscopic model. It was obtained by augmenting the recently developed coarse-grained description of a dynamin polymer filament on a deformable membrane tube (16) to additionally account for the GTPase motor activity of dynamin. In the model, MM dimers are introduced as elastic cross-bridges that connect the polymer beads and apply forces between them. Since each bead corresponds to a dynamin stalk dimer, it has two MMs associated with it. The nucleotide state of each MM is tracked and the GTPase cycle is implemented within it. Dimerization, i.e., formation of a cross-bridge, can occur if both MMs are GTP bound. After the hydrolysis-induced conformational change, the cross-bridge becomes strained and produces, as outlined above, a contracting force $F\left(\ell \right)$ for a cross-bridge of instantaneous length $\ell$. The MM dimers dissociate (and thus cross-bridges disappear) at rate $k_{\mathrm{\text{diss}}}$. See Materials and Methods and SI Appendix, Fig. S7 for the details.

In endocytosis, constriction and cleavage of membrane necks are performed by short dynamin filaments with only a few tens of MMs (33, 34). Filaments with 28 or 40 beads were therefore probed in our simulations. They were carried out for various MM dimer dissociation rates $k_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}$ (Movie S1), thus yielding the regimes with different duty ratios $\omega$ (SI Appendix, Eq. S8). Fig. 5A gives example snapshots of the initial and final states. The dependence of the final steady-state inner lumen radius (ILR) of the membrane tube on the duty ratio is displayed in Fig. 5B. Tight constriction from 10.5 to $2\text{to}3$ nm ILR is found above $\omega =0.65$. Increasing the filament length from 28 to 40 significantly decreases the ILR; however, increasing the length beyond 40 beads does not lead to further decreases in ILR.

Imaging of single endocytic pits during CME suggests that dynamin is active from a few seconds to tens of seconds before scission occurs (33, 34). Consistent with this, simulations showed that ILR below 3 nm could be reached within about 1 s for the duty ratios of $0.6\text{to}0.7$ and within 15 s for $\omega =0.94$ (Fig. 5C). Examination revealed that constriction speed was limited by $k_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}}$. When MM dimer dissociation is slow (corresponding to large duty ratios), cross-bridges continue to stay after the power stroke and are driven along with the sliding filament to reverse their slants (Fig. 5D). The misoriented cross-bridges get oppositely stretched and block the constriction.

In the actomyosin system, key kinetic steps are sensitive to strain (35–38) and similar sensitivity may be present in dynamin. In our model, strain control could be implemented by introducing immediate dissociation of MM dimers once their slant is reversed and they become stretched. For a 40-bead filament and $\omega =0.94$, this reduced the time of constriction to 3 nm from 15 s down to 300 ms (Fig. 5C and Movie S2). This effect could underlie the reported fast endocytosis within hundreds of milliseconds (39). Note that strain control affects only the kinetics, since the standing filament in the steady state cannot drive cross-bridges to alter their slants (and therefore the same ILR dependence as in Fig. 5B holds).

In in vitro experiments, scission of membrane tubes by long dynamin filaments can be observed. Surprisingly, such experiments find that breaking of the tube takes place not in the middle of the dynamin coat, but rather at its flanks (10, 40). In line with this finding, our simulations show that constriction by a long filament begins near its ends (Fig. 6 A and B and Movie S3) and only later propagates toward the middle. Fission may thus occur at a flank before the steady state has been reached. A simple explanation is that, in the middle of a long filament, a rung is pulled in opposite directions by its two neighboring rungs (Fig. 6 C and D). Such cancellation is absent for the terminal turns and, thus, only the motors on the flanks of a long helical filament apply a net torque and contribute to constriction of the membrane. Hence, the work produced by a long dynamin filament becomes independent of its length. Note that, in the employed model, dynamin-coated membrane tubes are assumed to be straight. Therefore, the effects of supercoiling, reported for very long tubes (8, 9), could not be considered here.

The pET-M11 vector was used to express a codon-optimized His6-tagged GTPase-BSE (MM) construct (Eurofins; residues 6 to 322 fused to residues 712 to 746 with a GSGSGS linker) in BL21 (DE3) Rosetta 2. Cells were grown at 37 $°$C to an OD600 of 0.4 and the protein was expressed in the presence of 0.15 mM Isopropyl $\mathrm{\beta }$-d-1-thiogalactopyranoside (IPTG) overnight at 25 $°$C in terrific broth (TB) medium, supplemented with 34 $\mathrm{\mu }$g/mL of chloramphenicol and 50 $\mathrm{\mu }$g/mL of kanamycin. Cells were harvested by centrifugation, resuspended in buffer A (20 mM HEPES (4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid)-NaOH, pH 7.5, 500 mM NaCl, 15 mM imidazole, 5% glycerol) and lysed in a microfluidizer. The cleared lysate after centrifugation was incubated for 30 min with 5 mL Ni-NTA resin at 4 $°$C on a rocking platform. Afterward, Ni-NTA beads were washed in batch three times with buffer A and transferred into a gravity-flow 20-mL column. Column material was further washed with 10 column volumes (CV) of buffer A supplemented with 2 mM ATP and 5 mM ${\mathrm{M}\mathrm{g}\mathrm{C}\mathrm{l}}_2$ and then with 2 CV of buffer A containing 50 mM imidazole. Eventually, the protein was eluted with 5 CV of buffer A containing 250 mM imidazole. The His6 tag was cleaved by incubation with Tobacco Etch Virus nuclear-inclusion-a endopeptidase during overnight dialysis against buffer A without imidazole. Subsequently, the protein was reapplied to the Ni-NTA column, collected in the flow-through fraction, and further purified via size-exclusion chromatography (SEC) on an S200 26/600 Superdex column with running buffer C (20 mM HEPES-NaOH, pH 7.5, 150 mM NaCl). Fractions containing the pure protein were pooled, concentrated to 30 to 50 mg/mL, aliquoted, and shock frozen in liquid nitrogen. The MM wild-type construct as well as various mutant variants were purified following the same protocol. For protein labeling with Alexa fluorophores or for experiments exploring metal-mediated dimerization of the MM constructs, the following mutations were introduced individually or in combination: T165C, R318C, N112C, K142H, N183H. Point mutations in the MM construct were introduced using QuikChange site-directed mutagenesis (Agilent). Cysteine mutations were used for covalent attachment of Alexa Fluor dyes and their locations were chosen to avoid any obvious functional disruption. See SI Appendix for methods specific to the crystallization and characterization of the ${\mathrm{Z}\mathrm{n}}^{2+}$-stabilized dimer.

Thiol-reactive fluorescent dyes (ThermoFisher Scientific) were used for site-specific labeling of cysteines in MM variants. Using Alexa Fluor 488 as donor and Alexa Fluor 594 as acceptor provides a Föster distance of 5.4 nm (54, 55). The T165C variant was used to determine the dissociation constants of the artificial dimer and the R318C/N112C double mutant for smFRET measurement. To reduce cysteines, 250 $\mathrm{\mu }$L of a protein solution at 30 to 40 mg/mL was incubated with 20 mM dithiothreitol (DTT) for 1 h on ice, following DTT removal by gel filtration on an S200 10/300 column equilibrated with buffer F (20 mM HEPES-NaOH, pH 7.5, 5 mM ${\mathrm{M}\mathrm{g}\mathrm{C}\mathrm{l}}_2$, 150 mM NaCl, 10 mM KCl). The concentrated proteins were mixed with the corresponding thiol-reactive dye in a 1:2 M ratio (protein:dye) and incubated for 1 h at room temperature (RT). In the case of double labeling by Alexa Fluor 488 and 594, the reaction was started with addition of Alexa Fluor 488 to the protein solution in a 1:0.66 (protein:dye) molar ratio. After 40 min of incubation at RT, Alexa Fluor 594 dye was added in a 1:2 (protein:dye) molar ratio and the reaction was continued for 1 h at RT. The labeling process was terminated by the addition of 10 mM DTT. After 5 min incubation at RT, insoluble material was removed by centrifugation and the sample was applied to a second round of SEC to remove unreacted dye and DTT. Labeling efficacy was calculated according to the manual of Thermo Fisher Scientific; a typical range was 90 to 100%. All fluorescence-based functional and GTP hydrolysis assays for the monomeric form of the MM constructs were performed in buffer 1 (20 mM HEPES-NaOH, pH 7.5, 150 mM KCl, 4 mM ${\mathrm{M}\mathrm{g}\mathrm{C}\mathrm{l}}_2$). Experiments with the artificially dimerized form of the MM-HH variant were performed in buffer 2 (20 mM HEPES-NaOH, pH 7.5, 150 mM KCl, 4 mM ${\mathrm{M}\mathrm{g}\mathrm{C}\mathrm{l}}_2$, 160 $\mathrm{\mu }$M ${\mathrm{Z}\mathrm{n}\mathrm{S}\mathrm{O}}_4$). All experiments were carried out at RT if not otherwise stated.

Single-molecule FRET experiments were performed with a MicroTime 200 confocal system (PicoQuant) equipped with an Olympus IX73 inverted microscope and two pulsed excitation sources (40 MHz) controlled by a PDL 828-L “Sepia II” (PicoQuant). Pulsed interleaved excitation (PIE) (56) was used to identify molecules with active acceptor and donor dyes. Two light sources, a 485-nm diode laser (LDH-D-C-485; PicoQuant) and a white-light laser (Solea; PicoQuant) set to an excitation wavelength of 585 nm, were used to excite the donor and the acceptor dyes alternatingly. The laser intensities were adjusted to 100 $\mathrm{\mu }$W at 485 nm and 30 $\mathrm{\mu }$W at 585 nm (Pm100D; Thor Laboratories). The excitation beam was guided through a major dichroic mirror (ZT 470-491/594 rpc; Chroma) to a 60×, 1.2-NA water objective (Olympus) that focuses the beam into the sample. The sample was placed in a homemade cuvette with a volume of 50 $\mathrm{\mu }$L (quartz 25-mm diameter round coverslips [Esco Optics], borosilicate glass 6-mm diameter cloning cylinder [Hilgenberg], Norland 61 optical adhesive [Thorlabs]). All measurements were performed at a laser power of 100 $\mathrm{\mu }$W (485 nm) and 30 $\mathrm{\mu }$W (585 nm) measured at the back aperture of the objective. Photons emitted from the sample were collected by the same objective and after passing the major dichroic mirror (ZT 470-491/594 rpc; Chroma), the residual excitation light was filtered by a long-pass filter (BLP01-488R; Semrock) and sent through a 100-$\mathrm{\mu }$m pinhole. The sample fluorescence was detected with two channels. Donor fluorescence and acceptor fluorescence were separated via a dichroic mirror (T585 LPXR; Chroma) and each color was focused onto a single-photon avalanche diode (SPAD) (Excelitas) with additional bandpass filters: FF03-525/50 (Semrock) for the donor SPAD and FF02-650/100 (Semrock) for the acceptor SPAD. The arrival time of every detected photon was recorded with a HydraHarp 400M time correlated single-photon counting module (PicoQuant) and stored with a resolution of 16 ps.

Photons from individual molecules, separated by less than 100 $\mathrm{\mu }$s, were combined into bursts if the total number of photons exceeded 30. Photon counts were corrected for background, acceptor direct excitation, and different detection efficiencies of the individual detectors. A PIE stoichiometry ratio $S<0.7$ was used to identify molecules with an active acceptor. Identified bursts were corrected for background, differences in quantum yields of donor and acceptor, different collection efficiencies in the detection channels, cross-talk, and direct acceptor excitation. FRET histograms were fitted with a combination of Gaussian functions.

The labeled MM construct was diluted to a concentration of approximately 50 pM in 20 mM HEPES-NaOH, pH 7.5, 150 mM KCl, 4 mM ${\mathrm{M}\mathrm{g}\mathrm{C}\mathrm{l}}_2$ in the absence or presence of increasing amounts of GTP, GTP$\gamma$S, GMPPCP, and GDP. To prevent surface adhesion of the protein and to maximize photon emission, 0.001% Tween 20 (Pierce) was included in the buffer. All measurements were performed at 23 $°$C.

In the presence of saturating concentrations of GTP and GTP analogs ($>1$ mM; SI Appendix, Fig. S4C), a significant closed population remained, which could not be fully explained by unbound MM (predicted to $<2$% based on the determined affinities) or GDP contamination. The purity of GTP was 99.4% and that of GTP$\gamma$S was 95%, as determined by high-performance liquid chromatography (HPLC).

All fast kinetic measurements were carried out on a Chirascan stopped-flow accessory unit equipped with an independent light-emitting diode power source (Applied Photophysics). For nucleotide-binding studies, fluorescent mant-substituted nucleotides were used. The dye fluorescence was excited at 350 nm and the fluorescence change monitored through a Schott 395-nm cutoff filter. To measure $k_{\mathrm{o}\mathrm{n}}$ values, increasing concentrations of MM construct (2, 3, 4.5, 7, 10.5, 15, and 20 $\mathrm{\mu }$M) were titrated to 1 $\mathrm{\mu }$M mant-nucleotide. For each concentration, at least five traces originated from the same sample were averaged. A linear regression was performed and the slope provided $k_{\mathrm{o}\mathrm{n}}$ with error given by the standard deviation in the slope. For quenching experiments to measure $k_{rmoff}$, 20 $\mathrm{\mu }$M protein and 1 $\mathrm{\mu }$M mant-nucleotide were mixed with 500 $\mathrm{\mu }$M GTP$\gamma$S or GDP. Three independent experiments were performed. Rates were determined by single exponential fits and the presented rate represents the average over the three experiments and the error the standard deviation. Error in $K_{\mathrm{d}}$ is determined by propagating the errors in $k_{\mathrm{o}\mathrm{n}}$ and $k_{\mathrm{o}\mathrm{ff}}$.

To detect conformational changes in the MM construct, FRET between Alexa Fluor 488 and 596 fluorescent pairs was recorded. Signals from both dyes were simultaneously detected using two photomultiplier tubes and a Chroma ET525/50m bandpass and Schott 645-nm cutoff filter. G-domain/BSE opening was performed by mixing 5 $\mathrm{\mu }$M of protein and 500 $\mathrm{\mu }$M of GTP$\gamma$S. Experiments for measuring G-domain/BSE closure were performed as follows: A total of 5 $\mathrm{\mu }$M protein and 100 $\mathrm{\mu }$M GTP$\gamma$S were mixed with 2 mM GDP, where GDP will outcompete GTP$\gamma$S for binding to the protein. Concentrations of reactants in both syringes of all stopped-flow experiments were two times higher compared to the measurement cell since they were diluted upon mixing in 1:1 ratio. Throughout this paper, indicated concentrations are those after mixing.

GTPase assays were performed in 20 mM HEPES-NaOH, pH 7.5, 150 mM KCl, 4 mM ${\mathrm{M}\mathrm{g}\mathrm{C}\mathrm{l}}_2$ containing 5 $\mathrm{\mu }$M of the MM construct, and 1 mM GTP. The reaction was carried out at 37 $°$C. Upon starting the reaction, 2-$\mathrm{\mu }$L reaction aliquots were withdrawn at different time points and mixed with 2 $\mathrm{\mu }$L of 1 M HCl to stop the reaction and then diluted with 26 $\mathrm{\mu }$L of HPLC buffer (100 mM potassium phosphate buffer, pH 6.5, 10 mM tetrabutylammonium bromide, 7.5% acetonitrile). Reaction products were quantified on an HPLC system from Agilent equipped with a Hypersil ODS-2 C18 column (250 $\times$ 4 mm). Each time-point measurement (five time points) was done in triplicate with the same sample. The specific hydrolysis rate for each protein concentration was computed by linear regression with error given by the standard deviation of the linear fit. To investigate the effect of ${\mathrm{Z}\mathrm{n}}^{2+}$ on artificial dimer activity, 2.5, 10, 40, 80, 160, and 400 $\mathrm{\mu }$M ${\mathrm{Z}\mathrm{n}\mathrm{S}\mathrm{O}}_4$ were added to the standard reaction. For testing concentration-dependent GTPase activity, reactions were performed in standard buffer at RT with 5, 15, 30, and 50 $\mathrm{\mu }$M enzyme. The competition GTPase assay was performed by adding 0.21 or 0.85 mM of GDP to the reaction. See SI Appendix, section S1.A.2 for details of the kinetic modeling of the competition assay.

The conformational dynamics of the MM were studied by simplified structure-based models (SBMs) (57, 58), a method commonly employed to investigate protein dynamics. The dual-basin SBM was constructed in such a way that both the known open and closed MM structures were explicit energetic minima of it. The open structure was defined by chain A of the GMPPCP-bound MM (${\mathrm{S}}_{\mathrm{O}}$) (6) and the closed structure by chain A of the GDP-bound MM (${\mathrm{S}}_{\mathrm{C}}$) (24). The G domains between the two structures are nearly identical. The structural differences mainly reside in the positioning of the first BSE helix relative to the G domain (SI Appendix, Fig. S5). The weighting $\lambda$ between open and closed in the simulation was determined by matching the simulated open/closed distributions to the smFRET data. See SI Appendix for details of the energy potential. Simulations were performed using GROMACS 4.5.3 (59) containing code edits implementing Gaussian contact interactions (available at ). Single-basin simulation topologies were generated using the SMOG2 software (https://smog-server.org) (31, 60) with the forcefield “SBM_calpha + gaussian” (61) and combined using an in-house script. The temperature (T = 0.92 in reduced units) was chosen such that the average ${\mathrm{C}}_{\alpha }$ atom rmsd in the closed MM agreed between a 100-ns explicit solvent simulation using AMBER99SB-ILDN at 310 K and the SBM. The temperature was maintained with stochastic dynamics with coupling constant 0.1 and the time step was 0.0005.

To model more closely the distance measured during smFRET, explicit dyes were attached at T165C and R318C. The dye consisted of a linear chain of six ${\mathrm{C}}_{\alpha }$ beads connected by bonds 3.8 Å in length (as in Fig. 2C). The FRET efficiency for each snapshot was calculated based on $E={\left(1+{\left(d/d_0\right)}^6\right)}^{-1}$ using the distance $d$ between the two terminal dye beads and $d_0=5.4$ nm. To compute the estimated FRET expected in either the open or the closed ensemble including thermal fluctuations (Fig. 2C) a $\lambda$ of 0.5 (always open) and 1.2 (strongly closed) was used. $\lambda$ of 0.98 and 1.1 matched the smFRET histograms for GTP bound or GDP bound, respectively (SI Appendix, Fig. S5D). Note that the photon counting statistics in the experimental FRET system are sensitive to the timescale of moving between the open and closed states. If the conformational motion time is comparable to the time a single molecule dwells in the laser point ($\sim$1 ms), the FRET distribution will become unimodal. Since separated peaks were observed in the experimental data, the dynamics observed must have been slower than 1 ms. Nonetheless, since averaging always tends to compress the peaks, the simulation data, which were not averaged, were expected to have wider separation between the peaks.

We note that the experimental smFRET histogram for GTP-bound MM appears to possess an intermediate state (Fig. 2A), although a two-state fit with skewed Gaussians is reasonable too. We have not included this intermediate state into the model because there is no crystal structure of it. For our purposes here, such an intermediate within GTP-bound dimers would have little effect, serving only to further flatten the already flat $\ell$ distribution.

The flexibility of hinge 1 (SI Appendix, Fig. S6) was estimated by a ${\mathrm{C}}_{\alpha }$ SBM simulation equivalent to that just described except single basin, meaning that any term in the energy function containing the open structure was removed and $\lambda =1$. A full monomer (chain A) taken from the apo (closed) crystal structure (20) was used to build the SBM. The PH domain along with stalk domain residues farther than 2 nm from hinge 1 were frozen during the simulation, mimicking its filament-bound state. The temperature was the same as above (T = 0.92) and the simulation was run for ${\mathrm{10}}^8$ time steps. For analysis of the structural fluctuations in terms of the helical coordinates, the frozen part of the stalk was fitted to a stalk domain in the helical reconstruction of Fig. 4A. Hinge 1 was identified by Proline 322 and hinge 2 was identified by Proline 27.

The probability distribution for the end-to-end lengths $\ell$ of cross-bridges is determined by the equilibrium distance distribution in prehydrolysis GTP-bound dimers, $P\left(\ell \right)=P_{\mathrm{G}\mathrm{T}\mathrm{P}}\left(\ell \right)$. On the other hand, the contraction force is $F\left(\ell \right)=-d{G}_{\mathrm{G}\mathrm{D}\mathrm{P}}\left(\ell \right)/d\ell =k_{B}T\left(d\ln P_{\mathrm{G}\mathrm{D}\mathrm{P}}\left(\ell \right)/d\ell \right)$, where $G_{\mathrm{G}\mathrm{D}\mathrm{P}}\left(\ell \right)$ is the Helmholtz free energy of a posthydrolysis GDP-bound MM dimer. The distance distributions for GTP- and GDP-bound MM dimers are presented in Fig. 4C. Thus, the mean force acting along the filament under the stall condition can be determined aswhere $\theta$ is the slant angle and $\omega$ is the probability (often referred to as the duty ratio) (32) to find a MM in the dimerized state. Refer to Fig. 4A for the definition of the geometry. The transverse component $⟨F_t⟩$ is given by the same equation with $\sin \theta$ replaced by $-\cos \theta$.

The integral in Eq. 1 was computed by using the distribution $P\left(\ell \right)=A\exp \left[-G_{\mathrm{G}\mathrm{T}\mathrm{P}}\left(\ell \right)/{\mathrm{\text{k}}}_{\mathrm{\text{B}}}\mathrm{\text{T}}\right]$ and the force $F\left(\ell \right)=-d{G}_{\mathrm{G}\mathrm{D}\mathrm{P}}/d\ell$ with the free energies $G_{\mathrm{G}\mathrm{T}\mathrm{P}}$ and $G_{\mathrm{G}\mathrm{D}\mathrm{P}}$ obtained in MD simulations, where $A$ is a normalization constant for the probability distribution $P\left(\ell \right)$. In Eq. 1, we have $\sin \theta =\mathrm{\Delta }s/\ell$ with $\mathrm{\Delta }s=\sqrt{{\ell }^2-h^2}$. Because of the ratchet property of hinge 1, only configurations with positive displacements $\mathrm{\Delta }s$ should be included when estimating the force. The minimal possible length of a link is ${\ell }_{\text{min}}=h$. Functions $P\left(\ell \right)$ and $F\left(\ell \right)$ were discretized in 0.5-nm steps and it was assumed that the filament radius is the same in the two rungs.

The code used to implement the constriction model (written in Java) is publicly available at https://bitbucket.org/jknoel/constrictionsimulation. A detailed formulation and analysis of the model for a dynamin filament on a membrane tube, in absence of the motor activity, was given in a previous publication (16). See SI Appendix for a full description. The dynamin filament was represented as an elastic polymer with each bead corresponding to a dynamin dimer. The membrane tube description was based on an axially symmetric continuous Helfrich elastic membrane with stiffness $\chi$ and under tension $\gamma$. The membrane stiffness $\chi =24$ ${\mathrm{k}}_{\mathrm{B}}$T and the tension $\gamma =0.03$ ${\mathrm{k}}_{\mathrm{B}}$T/${\mathrm{n}\mathrm{m}}^2$ were chosen. At such a stiffness, the elastic energy cost of constricting the filament/membrane system is dominated by the membrane energy (SI Appendix, Fig. S7E). It should be noted that membrane composition and protein insertion can affect the resulting stiffness (62). As a point of reference a pure 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) membrane has a stiffness around 30 ${\mathrm{k}}_{\mathrm{B}}$T. While the true stiffness at the scission neck remains unknown, the known active remodeling of the membrane composition and protein insertions should presumably act to facilitate the scission by shifting the membrane stiffness as low as possible.

In the current study, the effects of motor GTPase activity were additionally incorporated into the above-described model. This entailed including explicit cross-bridges representing MM dimers and resolving the kinetics of GTPase cycles. Again, each simulation bead corresponds to a stalk dimer and, thus, has two MMs associated with it. A MM undergoes stochastic transitions between the five ligand states of the GTPase cycle, i.e., Apo and with the nucleotides GDP, GTP, GTP-D, or GDP-D (Fig. 3). The transitions between three monomeric states are characterized by the nucleotide on/off rates displayed in Fig. 1D. The bulk concentrations of GTP and GDP are chosen as 300 $\mathrm{\mu }\text{M}$ and 30 $\mathrm{\mu }\text{M}$, respectively (these are typical physiological concentrations) (27). The Apo and GDP states cannot form cross-dimers. Binding of GTP by a MM alters the conformational state from the predominantly closed to the predominantly open at the rate of approximately 300 ${\text{s}}^{-1}$ (Fig. 2 D and E). Since opening transitions are thus relatively fast, they are not explicitly resolved in the model. Instead, all MMs in the GTP state are assumed to immediately go to the equilibrium open/closed distribution.

Two MM monomers in the GTP state can form the dimeric GTP-D state that represents a weak prehydrolysis dimeric state. The GTP state of a monomer transits to a GTP-D state at rate $k_{\mathrm{\text{on}}}^{\mathrm{\text{D}}}{\left[\mathrm{M}\mathrm{M}\right]}_{\mathrm{e}\mathrm{ff}}$, provided that a valid MM partner for dimerization exists. A partner MM is valid if it is also in the GTP state and if the generated cross-dimer would satisfy the conditions $\ell <{\ell }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $\mathrm{\Delta }s>0$, with the latter condition taking into account the ratchet asymmetry of hinge 1. ${\ell }_{\mathrm{m}\mathrm{a}\mathrm{x}}=15$ nm is the natural extension limit of the dimer as can be seen in the GTP-bound free energy $G_{\mathrm{G}\mathrm{T}\mathrm{P}}\left(\ell \right)$ (Fig. 4C). The geometry is explained in SI Appendix, Fig. S7A. Additionally, due to the excluded volume of the MM dimer, a valid cross-dimer cannot intersect any existing cross-dimers. The broad free energy basin for GTP-bound dimers, found in MD simulations, implies high flexibility (Fig. 4C). Therefore, no forces are generated by a GTP-D dimer in the model. The GTP-D dimer dissociates into two GTP MMs at rate $k_{\mathrm{\text{off}}}^{\mathrm{\text{D}}}$.

The transition from a prehydrolysis GTP-D dimer to the posthydrolysis GDP-D dimer takes place at rate $k_{\mathrm{\text{hyd}}}$. Note that hydrolysis is chosen to occur simultaneously in both monomers; i.e., high cooperativity is assumed. Such cooperativity is suggested by our kinetic measurements showing that the hydrolysis is equally fast for the dimers containing either two GTP molecules or one GTP and one GDP (SI Appendix, section S1.A.2). The GDP-D dimer generates a force $\overrightarrow{F}\left(\ell \right)$ with the magnitudeand directed along the vector connecting the two MM attachment points (SI Appendix, Fig. S7A). The Helmholtz free energy $G_{\mathrm{G}\mathrm{D}\mathrm{P}}\left(\ell \right)$ for the GDP-bound dimer is shown in Fig. 4C. As stated above, ${\ell }_{\mathrm{m}\mathrm{a}\mathrm{x}}=15$ nm. The parameter $\eta$ is set to 3 ${\mathrm{k}}_{\mathrm{B}}$T/${\mathrm{n}\mathrm{m}}^3$, so that the force steeply increases when $\ell$ exceeds ${\ell }_{\mathrm{m}\mathrm{a}\mathrm{x}}$.

The transition from a GTP-D state to the GDP-D state is irreversible due to hydrolysis. The GDP-D state can be left through dissociation into two MM monomers, each in the GDP state, at rate $k_{\mathrm{\text{diss}}}$. When the strain dependence of this rate was taken into account, this was done by choosingThus, an oppositely strained dimer dissociates immediately. In the model without strain dependence, $k_{\mathrm{\text{diss}}}$ remains constant regardless of the displacement $\mathrm{\Delta }s$.

In kinetic simulations involving transitions between the states, the Gillespie algorithm is often used (63). In the present study, explicit stochastic simulations of the GTPase cycle were, however, carried out, because most of the computer time was already needed for integrating the equations of motion for the membrane and the beads. The weighting times of transitions obeyed an exponential distribution with a single rate parameter; i.e., the probability of a transition with rate $k$ within time $\mathrm{\Delta }t$ is $1-\exp \left(-k\mathrm{\Delta }t\right)\approx k\mathrm{\Delta }t$ for $k\mathrm{\Delta }t\ll 1$. The transition times (inverse of transition rates) between states were all much longer than the integration time step $\delta t$ of 10 ns. Therefore, the probability of a transition occurring during a single time step was taken to be $k\delta t$. The small time step was dictated by the membrane description that had to capture the lipid flows. As a characteristic example, a 1-s simulation (i.e., ${\mathrm{10}}^8$ time steps) for a 200-bead filament on a 600-nm-long membrane tube required 5 h on a single core of the Intel Xeon E5-2695 v3 2.30-GHz central processing unit.

The nucleotide association/dissociation rates are given in Fig. 1D. Some GTPase-cycle parameters could not be directly measured (Fig. 3). Unless otherwise noted, $k_{\mathrm{\text{on}}}^{\mathrm{\text{D}}}{\left[\mathrm{\text{MM}}\right]}^{\mathrm{\text{eff}}}=3\times 10^5\hspace{0.17em}{\text{s}}^{-1},k_{\mathrm{\text{hyd}}}=175\hspace{0.17em}{\text{s}}^{-1}$, and $k_{\mathrm{\text{off}}}^{\mathrm{\text{D}}}=10^5$. The duty ratio $\omega$ was varied between 0.37 and 0.94 by changing $k_{\mathrm{\text{diss}}}$ between 100 ${\text{s}}^{-1}$ and 3.3 ${\text{s}}^{-1}$. See SI Appendix, section S1.B for further discussion of the choice of the GTPase-cycle kinetic parameters.