Nucleotide-dependent conformations of FtsZ dimers and force generation observed through molecular dynamics simulations

Abstract

The bacterial cytoskeletal protein FtsZ is a GTPase that is thought to provide mechanical constriction force via an unidentified mechanism. Purified FtsZ polymerizes into filaments with varying structures in vitro: while GTP-bound FtsZ assembles into straight or gently curved filaments, GDP-bound FtsZ forms highly curved filaments, prompting the hypothesis that a difference in the inherent curvature of FtsZ filaments provides mechanical force. However, no nucleotide-dependent structural transition of FtsZ monomers has been observed to support this force generation model. Here, we present a series of all-atom molecular dynamics simulations probing the effects of nucleotide binding on the structure of an FtsZ dimer. We found that the FtsZ-dimer structure is dependent on nucleotide-binding state. While a GTP-bound FtsZ dimer retained a firm monomer-monomer contact, a GDP-bound FtsZ dimer lost some of the monomer-monomer association, leading to a “hinge-opening” event that resulted in a more bent dimer, while leaving each monomer structure largely unaffected. We constructed models of FtsZ filaments and found that a GDP-FtsZ filament is much more curved than a GTP-FtsZ filament, with the degree of curvature matching prior experimental data. FtsZ dynamics were used to estimate the amount of force an FtsZ filament could exert when hydrolysis occurs (20–30 pN per monomer). This magnitude of force is sufficient to direct inward cell-wall growth during division, and to produce the observed degree of membrane pinching in liposomes. Taken together, our data provide molecular-scale insight on the origin of FtsZ-based constriction force, and the mechanism underlying prokaryotic cell division.

The bacterial cell-division protein FtsZ, a tubulin homolog and a GTPase, polymerizes into filaments situated underneath the cytoplasmic membrane at the division site (1–3). The ftsz gene is essential and depletion of FtsZ protein leads to cells unable to divide that instead take on a filamentous morphology (4). During division, rod-shaped bacteria such as Escherichia coli synthesize new hemispherical membrane and cell wall, requiring inward forces. While an array of proteins are needed for division to occur (5, 6), FtsZ alone has been demonstrated to be sufficient for generating inward pinching forces on liposomes in vitro (7–10), and is thought to be the source of the constriction force needed for division.

How FtsZ generates mechanical force is still unclear, but several models have been proposed (11), one of which is based on the intrinsic nucleotide-dependent bending of FtsZ filaments (12, 13). In the presence of GTP, purified FtsZ assembles into either straight or gently curved filaments with a radius of curvature of approximately 100 nm, while in the presence of GDP, highly curved filaments form with a radius of curvature of approximately 10 nm (12, 14, 15). Subsequent theoretical modeling demonstrated that the rapid transition between straight and curved FtsZ filaments during cycles of hydrolysis can generate mechanical forces larger than 8 pN (16), the amount of force needed for cell wall invagination during division (17). However, no structural transition was observed in FtsZ monomers in a comparison of all available crystallographic structures from various bacteria and in different nucleotide-binding states (18), raising the question of the molecular basis for the shift from a straight GTP-FtsZ state to a curved GDP-FtsZ state.

To understand how nucleotide hydrolysis induces a structural transition in FtsZ filaments, a method with atomic resolution that captures molecular behavior is needed. Here, we examine the conformational dynamics of FtsZ dimers in various nucleotide-binding states using unbiased all-atom molecular dynamics simulations. Molecular dynamics has been previously applied to study various eukaryotic cytoskeletal proteins (19, 20). Recently, molecular dynamics simulations revealed nucleotide-induced structural transitions in a subunit of the molecular chaperone GroEL (21) and the signaling protein Ras (22), attesting to the utility of molecular dynamics in studying nucleotide-dependent conformational shifts in proteins. Our simulations demonstrated that GTP- and GDP-bound FtsZ dimers have inherently different conformations and dynamics. While the GTP-FtsZ monomers remained fully associated with each other, GDP-FtsZ exhibited a partial loss of monomer-monomer interactions that led to a bent conformation. In both cases, minimal structural changes were observed in each FtsZ monomer. Modeling of FtsZ filaments by replicating the observed monomer-monomer interfaces demonstrated that GDP-FtsZ filaments have higher intrinsic curvature than GTP-FtsZ, with the degrees of curvature matching in vitro studies. Using the spectrum of fluctuations of each dimer, we determined the magnitude of force that could be generated via hydrolysis-induced transition from a straight GTP-FtsZ filament to a curved GDP-FtsZ filament, and found the force to be sufficient to direct inward cell-wall growth and to generate pinching of liposomes. Lastly, we identified the molecular interactions modulating the FtsZ dimer states, featuring key amino acids where mutagenesis impedes division and leads to filamentous cells without abolishing FtsZ polymerization. This work provides molecular-scale insight on the remodeling of the cell envelope during division and suggests potential pathways to disrupt the essential process of bacterial cell division.

Results

Loss of Monomer-Monomer Association in GDP-FtsZ.

Unbiased all-atom molecular dynamics simulations were carried out for FtsZ dimers bound to GTP and GDP; an example simulation setup of the GTP-bound FtsZ dimer is shown in Fig. 1A. The GTP- and GDP-FtsZ simulations started with the same initial protein conformation defined by the crystallographic structure of the Methanococcus jannaschii FtsZ dimer with bound GTP (Fig. 1B; PDB code 1W5B) (23). An alternative GTP-FtsZ dimer structure, 1W5A, is available and contains an Mg²⁺ ion at the nucleotide-binding site (23); the two structures, 1W5A and 1W5B, show no significant conformational differences (23). Nonetheless, it should be noted that Mg²⁺ is required for GTP hydrolysis in FtsZ (24), thus it could modulate the rate of disassembly of an FtsZ filament (25).

Fig. 1.

Molecular dynamics simulations predict higher curvature for GDP-FtsZ filaments. (A) Setup of the molecular dynamics simulation of a GTP-bound FtsZ dimer, with the two FtsZ monomers colored in light and dark blue for distinction. N- and C-termini are shown as purple and pink spheres, respectively. GTP molecules are shown in green, and one GTP is highlighted in a dashed red circle. Water box is shown in transparent gray, and Na⁺ and Cl^- ions are shown as yellow and red spheres, respectively. (B–D) Snapshots of structures of the GTP-bound FtsZ dimer (blue) and GDP-bound FtsZ dimer (orange) during their respective molecular dynamics simulations at (B) 0 ns, (C) 80 ns, and (D) 320 ns; see also Movie S3. Nucleotides, water and ion molecules are not shown for clarity. (E–G) Models of FtsZ filaments constructed by replicating the monomer-monomer interface of the simulated FtsZ dimers at (E) 0 ns, (F) 80 ns, and (G) 320 ns. GTP-FtsZ filament: light and dark blue; GDP-FtsZ filament: orange and brown. Red rectangles outline the simulated FtsZ dimers.

Each simulation was conducted for approximately 330 ns, and was performed with explicit water molecules and neutralizing ions. During the simulation, GTP-bound FtsZ dimer retained its monomer-monomer contact, and maintained an upright conformation (Movie S1). In contrast, GDP-bound FtsZ dimer experienced partial loss of the monomer-monomer interaction interface within 100 ns, resulting in the top monomer tilting forward relative to the bottom monomer, leading to a bent dimer conformation (Fig. 1C; Movie S2). Notably, a separate simulation of a nucleotide-free FtsZ dimer also resulted in similar opening of the monomer-monomer interface (Fig. S1 and S2), suggesting that absence of the γ-phosphate causes partial loss of monomer-monomer contact in an FtsZ dimer. Later in the GDP-FtsZ simulation, the dimer recovered some monomer-monomer contact, but the overall dimer structure still appeared to be more bent than the GTP-FtsZ dimer (Fig. 1D).

GDP-FtsZ Forms a More Curved Filament than GTP-FtsZ.

To illustrate the effect of monomer-monomer interaction on the structure of an FtsZ filament, we constructed models of FtsZ filaments by replicating the monomer-monomer interface observed in our molecular dynamics simulations of GTP- and GDP-bound FtsZ dimers. We used nine monomers in each filament model to represent the shorter, approximately 40 nm, FtsZ filaments observed in vivo with cryoelectron microscopy (2). At the onset of the two simulations, the GTP-FtsZ and GDP-FtsZ filaments overlap because the simulations started with identical dimer structures. This initial filament was slightly curved with a radius of curvature of 58 nm (Fig. 1E). By 80 ns, the partial loss of monomer-monomer interaction in the GDP-FtsZ dimer led to a much more curved filament, which formed an open ring with a radius of 9 nm (Fig. 1F). In contrast, the GTP-FtsZ filament was less curved with a radius of curvature of 22 nm (Fig. 1F). By 320 ns, both filaments relaxed their curvature, with the GDP-FtsZ filament adopting a radius of curvature of 30 nm, and the GTP-FtsZ filament acquiring a radius of curvature of 36 nm (Fig. 1G).

Monomer-Monomer Interaction of GDP-FtsZ Is Weaker than GTP-FtsZ.

The partial loss of monomer-monomer contact observed in the GDP-FtsZ dimer during the first half of the simulation can be quantified by tracking the buried Solvent Accessible Surface Area (SASA) between the monomers (method of calculation described in SI Text and Fig. S3). Higher values of buried SASA between monomers indicate more contact and stronger interaction. Fig. 2A shows the time evolution of the buried SASA between FtsZ monomers for both simulations with GTP-FtsZ and GDP-FtsZ dimers. For the GTP-FtsZ dimer, the buried SASA increased steadily, reaching a stable value of approximately 12 nm² by 200 ns. In contrast, the GDP-FtsZ dimer experienced a period of low buried SASA for nearly 150 ns, fluctuating around 9 nm². At approximately 150 ns, the GDP-FtsZ dimer transitioned sharply into a more tightly associated conformation, corresponding to the partial recovery of the monomer-monomer interface shown in Fig. 1D. As the two states for the GDP-FtsZ dimer were well defined and separated by a clear transition at 150 ns, we hereafter denote the loosely associated state as the “GDP-Open” state (corresponding to the orange structure in Fig. 1C), and the more tightly associated state as the “GDP-Closed” state (orange structure in Fig. 1D). Only one distinct conformation was observed for the GTP-FtsZ dimer, which we hereafter refer to as the “GTP-Closed” state.

Fig. 2.

Changes in monomer-monomer interface account for different dimer structures. (A) Measurement of buried SASA between FtsZ monomers in the two simulations with GTP-FtsZ and GDP-FtsZ dimers. (B) Distribution of buried SASA values for the three states (filled circles) and their Gaussian fits (dashed curves). (C) Backbone rmsd values of each FtsZ monomer. Little structural variation was seen for any of the FtsZ monomers, regardless of nucleotide-binding state.

Although in Fig. 2A it appears that both the GTP- and GDP-FtsZ dimers reached similar buried SASA values, a closer look at their distributions revealed that the GDP-Closed state was not as tightly associated as the GTP-Closed state (Fig. 2B). Because buried SASA provides a qualitative estimate for the strength of monomer-monomer association, the consistently lower SASA values observed for the GDP-FtsZ dimer are in accordance with prior results showing that assembly of GDP-FtsZ is weaker than GTP-FtsZ, and that hydrolysis destabilizes FtsZ filaments (15, 26–28). It is likely that hydrolysis of GTP transitions a relatively straight GTP-FtsZ filament to one that can take on a highly curved conformation while decreasing the association strength between monomers, leading to a filament more prone to depolymerization. Notably, the presence of Mg²⁺ ions has been shown to alter filament stability (25). Because Mg²⁺ is required for GTP hydrolysis (24), it is possible that Mg²⁺ modulates depolymerization of FtsZ filament indirectly by promoting hydrolysis, but it remains to be determined whether Mg²⁺ has a direct role in the filament structure of FtsZ.

Importantly, while the dimer structures differ between GDP-FtsZ and GTP-FtsZ, each FtsZ monomer retained its structure in both nucleotide-binding states, as shown by the backbone rmsd of each monomer in Fig. 2C (excluding the mobile N-terminal helix, H0, defined in Fig. S4). The low rmsd values (approximately 1–1.5 Å) are consistent with a structural analysis of FtsZ crystallographic structures, which revealed little structural variation among FtsZ molecules from various bacterial species and nucleotide-binding states (18). Thus, the nucleotide-dependent difference in dimer and filament conformations we observed stemmed not from structural transitions of individual FtsZ proteins, but rather from changes in the monomer-monomer interface.

Flexibility of each amino acid in the FtsZ monomers was also tracked by measuring their rmsf values (Fig. S5), and the higher flexibility of several amino acids was also observed previously (29). Interestingly, difference in nucleotide-binding states was also associated with altered residue flexibility, with many of the most affected amino acids situated at the dimerization interface near the nucleotide-binding site (Fig. S5).

Quantifying the Relative Rotations Between FtsZ Monomers.

To quantitatively describe the dynamics of monomer-monomer motion observed in the GTP-FtsZ and GDP-FtsZ dimer simulations, we defined a coordinate system for the FtsZ protein based on its principal axes of inertia (SI Text). Rotations around each axis, denoted by θ₁, θ₂, and θ₃, represent orthogonal relative monomer-monomer motions, as depicted on the right-hand side of Fig. 3 A–C. The coordinate system adopted here defines θ₁ to be the “twisting” motion between the two monomers, and θ₂ and θ₃ to be “bending” motions in orthogonal directions.

Fig. 3.

Measurement of the relative rotation between FtsZ monomers in simulations of GTP-FtsZ and GDP-FtsZ dimers. (A) Time evolution of θ₁, which corresponds to the “twisting” motion between the top and bottom monomers. (B) Time evolution of θ₂, which corresponds to a “bending” motion between the top and bottom monomers; this motion is the main source of filament curvature. (C) Time evolution of θ₃, which describes bending motion between the monomers orthogonal to θ₂. (D) Distribution of the bending angle θ₂ for the conformations GDP-Open (brown), GDP-Closed (orange), and GTP-Closed (blue). Raw data are shown as filled circles, and Gaussian fits are shown as dashed curves, with the means and variances σ² given in units of degrees and degrees², respectively.

Throughout the simulations, both GTP- and GDP-bound FtsZ dimers exhibited noticeable amounts of twisting (Fig. 3A). We speculate that the monomer-monomer twisting is exaggerated by simulating only two monomers. In a filament, the twisting motion between each monomer is constrained by two neighboring units, and is likely not as prominent as observed here, where each FtsZ monomer only had one neighboring unit. In a separate simulation with four GTP-bound FtsZ molecules, the twisting motion was reduced but still evident (Fig. S6), although it should be noted that this larger molecular system might take longer than the 40 ns performed to exhibit its full bending and twisting motions. The bending motion θ₂ displayed an interesting trend for the GDP-FtsZ dimer, with a high degree of bending (centered around 26°) during the first half of the simulation corresponding to the GDP-Open state, and a smaller degree of bending (centered around 7°) for the GDP-Closed state (Fig. 3B). In contrast, the GTP-FtsZ dimer showed little bending as its θ₂ value fluctuated near 0 (Fig. 3B). For the orthogonal bending direction, both GTP-FtsZ and GDP-FtsZ dimers had small values of θ₃, indicating that FtsZ has a strongly preferred bending direction around θ₂ (Fig. 3C). Purified FtsZ proteins fused to membrane-binding domains assemble into filaments that attach to liposomes, and generate bending forces on the membrane with a preferred direction of bending (such that the C-terminal ends of FtsZ are on the outer curvature and the N-terminal ends are on the inner curvature) (7, 8, 10); rotation around θ₂ matches this direction (Fig. S7).

Estimating the Force Generated by FtsZ Hydrolysis.

The stiffness with which an FtsZ filament resists bending around θ₂ can be determined from the fluctuations around the mean in each state. The distributions of θ₂ for the GDP-Open, GDP-Closed, and GTP-Closed states each resemble a Gaussian (Fig. 3D),

[1]

with the mean and variance σ² of each fit provided in Fig. 3D. Measurements of the means and variances allow estimation of the force an FtsZ filament generates when it transitions from a gently curved GTP-bound state to a more curved GDP-bound state, as each Gaussian distribution is related to a harmonic energy, U(θ₂), via P(θ₂) ∝ exp[-U(θ₂)/k_BT], where k_B is the Boltzmann factor and T is temperature (T = 300 K in the following calculation). Therefore, the bending motion around θ₂ can be described by the harmonic energy

[2]

where the effective force constant of monomer-monomer bending in each state is k = k_BT/σ².

When a filament transitions from a gently curved to a highly curved state (Fig. 4), the mean value of θ₂ changes by . The work done during the transition is given by

[3]

where σ² is the variance of the final filament state. The work ΔU can then be used to bend each monomer around θ₂, applying an average torque on each monomer given by . Because torque is related to the inward radial force F on each monomer via τ = ℓF (Fig. 4), where ℓ is the dimension of each FtsZ monomer (approximately 4 nm), the average force produced by the hydrolysis of a monomer is

[4]

Based on the present simulation data, GTP-bound FtsZ dimer only has one relatively straight conformation (GTP-Closed), while hydrolysis shifts the conformational preference to a highly curved (GDP-Open) or a gently curved (GDP-Closed) configuration. The transition from the GTP-Closed to the GDP-Open state results in (Fig. 3D), while the transition from the GTP-Closed to the GDP-Closed state produces a smaller . As the GDP-Open state has a σ² of 28.3 deg², and the GDP-Closed state has a smaller σ² of 14.0 deg², the GTP-Closed to the GDP-Open transition produces an inward radial force of 30 pN per monomer, and the GTP-Closed to the GDP-Closed transition produces a comparable 20 pN of force. Interestingly, as the GTP-Closed conformation is actually slightly bent, a GTP-FtsZ filament is expected to also generate bending forces of approximately 10 pN on an originally flat membrane (). Therefore, membrane bending by FtsZ does not require hydrolysis (10).

Fig. 4.

Transitioning from a less curved GTP-FtsZ filament to a more curved GDP-FtsZ filament generates a radially inward force. Blue and orange rectangles represent GTP-FtsZ and GDP-FtsZ, respectively, while red lines between monomers represent monomer-monomer interactions.

Molecular Interactions Between FtsZ Monomers.

The contact surface between the two FtsZ monomers was maintained by many molecular interactions. In our simulations we observed specific changes in the interactions in different FtsZ dimer conformations. One major interaction stabilizing the FtsZ dimer originated from the H0 helix (defined in Fig. S4) (18, 23) of the top monomer and residues within H2 and S3 of the bottom monomer (Fig. S8). This set of interactions was observed consistently in all three FtsZ dimer conformations (GDP-Open, GTP-Closed, GDP-Closed), forming several intermonomer salt bridges (Fig. S8) and containing several highly conserved amino acids (Gln74, Leu76, Leu86, and Leu95). We reasoned that the H0 helix contributes to the formation and stabilization of the FtsZ filament; the stabilizing effect of the H0 helix was noted previously (23).

We detected another three sets of molecular interactions that resided at the monomer-monomer interface (Fig. 5). Fig. 5A shows a set of interactions present in all three states, formed between Glu298, Arg301, Glu302, and Ala305 in H10 from the top FtsZ and Pro203 to Lys205, Phe208, and Lys209 in H7 from the bottom FtsZ. With the exception of residue Phe208, all of these amino acids are poorly conserved across prokaryotes (Fig. S4). The contacts formed between these hydrophobic and charged amino acids possibly do not need to be specific, and provide a cluster of secondary filament-stabilizing interactions.

Fig. 5.

Major molecular interactions between the FtsZ monomers. (A) A cluster of interactions observed in all conformations (GDP-Open, brown; GDP-Closed, orange; GTP-Closed, blue) is located between residues 298, 301, 302, and 305 from the top FtsZ (pink spheres) and residues 203–205, 208, and 209 from the bottom FtsZ (red spheres). (B) A cluster of interactions, seen only in closed conformations (GDP-Closed and GTP-Closed) and not in the GDP-Open state, occurs between residues 235 to 238 and 241 of the top FtsZ (pink spheres) and residues 95 to 98 from the bottom FtsZ (red spheres). (C) Another cluster of interactions, only present in closed conformations, although slightly different between the GTP-Closed and GDP-Closed states, is established by residues 301 to 314 in H10 from the top FtsZ (pink spheres) and residues 164 to 168 from the bottom FtsZ (red spheres). The location of H10 is shifted comparing the GDP-Closed and GTP-Closed states, denoted by the arrows. (D) Locations of the clusters of monomer-monomer interaction. Shown here is the top surface of the bottom FtsZ molecule, viewed along the dimer axis. GTP molecule is shown in green, and the interaction clusters from (A–C) are highlighted with red spheres.

Fig. 5B displays a set of interactions that was present in the two closed states, but absent in the GDP-Open state. This set of interactions occurred between Leu95 to Gly98 from the bottom FtsZ and Asp235 to Asp238 and Ala241 in H8 from the top FtsZ. All of these residues (except Ala241) are highly conserved (Fig. S4), and possibly modulate the opening and closing between FtsZ monomers, and, subsequently, the curvature of the FtsZ filament and its ability to generate constriction forces. Another set of interactions only present in the GTP-Closed and GDP-Closed states is shown in Fig. 5C, arising between residues Arg301 to Asn314 in H10 from the top FtsZ and residues Met164 to Val168 from the bottom FtsZ. While residues in H10 are poorly conserved, Met164, Glu165, and Gly166 are highly conserved (Fig. S4). Interestingly, this set of interactions is not identical in the GTP-Closed and GDP-Closed states. In the GDP-Closed state, the H10 helix moved inward (towards the direction of inner curvature) relative to the GTP-Closed state, as denoted by yellow arrows in Fig. 5C. We speculate that this set of interactions also controls the transition from the open to closed FtsZ states, and further fine-tunes the degree of curvature of the FtsZ filament after hydrolysis.

The three sets of monomer-monomer interactions shown in Fig. 5 A–C are located across the top surface of the bottom FtsZ, forming a triangle around the nucleotide-binding site (Fig. 5D). The presence of all three clusters of interactions ensures a firm monomer-monomer association and a stable FtsZ filament.

Discussion

Employing all-atom molecular dynamics simulations, we observed that the FtsZ dimer structure is nucleotide dependent, with GTP-FtsZ projected to form relatively straight filaments and GDP-FtsZ to form much more curved filaments (Fig. 1). In vitro, purified FtsZ molecules assemble into filaments with varying curvature (9). Three filament structures have been observed, one of which is the highly curved, ring-like conformation with a radius of approximately 10 nm in the presence of GDP (12, 14); this conformation is very similar to the highly curved GDP-FtsZ filament in Fig. 1F. In the presence of GTP in vitro, FtsZ either forms straight filaments with little curvature (14), or filaments with intermediate curvature with radii on the order of 100 nm (30–33). The modeled GTP-FtsZ filament in Fig. 1 E–G displayed a radius of curvature of 22–58 nm during the simulation, and is therefore more representative of the intermediate curvature conformation. We did not observe a straight GTP-FtsZ filament conformation, possibly because our simulations contained only an FtsZ dimer. In an FtsZ filament, all subunits (except the two at the ends) are restricted in motion by two neighboring proteins, whereas in a dimer conformation, the lack of additional restraints could allow the subunits to be more dynamic and adopt a higher curvature. Alternatively, the straight FtsZ filaments observed in experiments could be stabilized by weak lateral interactions with nearby FtsZ filaments.

By tracking the relative rotations between the FtsZ monomers, we found that FtsZ has a preferred bending direction in agreement with prior studies (Fig. 3B) (7, 8, 10), and the distribution of the major bending angle θ₂ for each FtsZ state provided an estimate of the magnitude of force FtsZ can generate during a hydrolysis cycle (Fig. 4). Previously, computational modeling has shown that for a rod-shaped, Gram-negative bacterium such as E. coli, at least 8 pN is needed to pull the cell wall inward at the division site to initiate and sustain the growth of new hemispherical end caps, and forces in the range of 8 to 80 pN all could lead to division with a reasonably accurate septal morphology (17); the forces we calculated here that FtsZ can generate when GTP is hydrolyzed, 20–30 pN per polymerized monomer, fall within this range (Fig. 4). Notably, the inherently slightly curved GTP-FtsZ filament is capable of supplying a smaller but still significant force of approximately 10 pN, in line with previous experiments indicating that occasional division events could still take place in vivo in mutants with reduced FtsZ hydrolysis activity (10, 34), and that force generation of FtsZ does not require hydrolysis in vitro (10).

In addition, FtsZ also exhibits a noticeable amount of relative twisting motion (Fig. 3A, Fig. S6). The bending motion between FtsZ monomers combined with twisting would lead to the assembly of helical FtsZ filaments. Many imaging studies have reported that FtsZ filaments are highly dynamic in living cells and appear to form helices (3, 9, 35–38), indicating that both bending and twisting motions between monomers are inherent in FtsZ.

It has been demonstrated that FtsZ fused with a membrane-anchoring domain can provide pinching forces on multilamellar liposomes, reducing the radius of liposomes from an initial radius R_i = 2.5 μm to a smaller radius R_f = 1.5 μm (estimated from Fig. 4 of ref. 7). Inducing curvature in an elastic lipid bilayer requires energy, and the amount of energy per unit area required can be calculated from the Helfrich equation (39),

[5]

where κ is the bending modulus [κ = 20k_BT for a typical phospholipid bilayer (40)], and R₁ and R₂ are the local curvature radii. The pinching observed in ref. 7 requires at least approximately 150k_BT for each bilayer (details of the calculation in SI Text and Fig. S9). Based on the thickness of the multilamellar liposome and a bilayer thickness of approximately 4 nm, there are approximately 200 layers of membrane, requiring at least 3 × 10⁴k_BT of total bending energy. On the other hand, transitioning from the GTP-Closed to the GDP-Open or GDP-Closed states provides work that can be used to bend the liposomes. Using Eq. 3, each monomer-monomer interface of an FtsZ filament can generate 4–18k_BT of work, and, for a single FtsZ ring with radius R_i = 2.5 μm, there are approximately 4,000 FtsZ molecules, so the work an FtsZ ring can provide is at most 2–7 × 10⁴k_BT, assuming all 4,000 FtsZ molecules are polymerized and have not hydrolyzed. Thus, at least 0.4–2 FtsZ rings are necessary to generate the energy needed to deform the liposome as previously observed (7). Judging from the fluorescence signal of FtsZ rings in the liposome in Figs. 3A and 4 of ref. 7, multiple rings can be identified at each pinching site.

We also identified amino acids at the monomer-monomer interface that stabilize FtsZ filaments and modulate filament curvature (Fig. 5). Several of these key residues have been subjected to prior mutagenesis studies (Table S1) (41–43). M. jannaschii Gln74, Phe208, Asp235, Phe236, Asp238, and Glu165 correspond to Gln47, Phe182, Asp209, Phe210, Asp212, and Glu138 in E. coli FtsZ, respectively, and mutations of these residues led to the loss of division function, resulting in filamentous cells (42, 43). Interestingly, in most of these mutants, FtsZ filaments with varying helicity were still visible (42, 43), indicating that FtsZ still polymerizes despite the single mutations and the loss of FtsZ division function. We showed that the FtsZ dimer is stabilized by several clusters of interactions (Fig. 5D), suggesting that mutation of one key amino acid would not completely abolish FtsZ polymerization. However, changing a key amino acid at the monomer-monomer interface likely results in the assembly of an FtsZ filament with different structural and mechanical properties, which disrupts FtsZ force generation and inhibits cell division.

In summary, FtsZ dimers were observed through molecular dynamics simulations to have inherently different conformations depending on their nucleotide-binding states, which led to assembly of filaments with different degrees of curvature. Our data suggest that during hydrolysis, FtsZ filaments can generate a significant amount of force, providing a molecular basis for the previously proposed “hydrolyze and bend” model for the force generation of FtsZ (11–13, 16). In the absence of hydrolysis, FtsZ can still produce a smaller mechanical force, which could be the constriction mechanism in the occasional in vivo division events observed in cells with a mutant FtsZ that exhibits very low hydrolysis activity (10, 34). Future work with longer simulations probing the dynamics of each FtsZ conformation, larger simulations with more FtsZ monomers, incorporation of other FtsZ-binding cell division proteins such as the membrane anchor FtsA, and inclusion of active-site divalent cations such as Mg²⁺ (23, 25), will further elucidate how the complex process of bacterial cell division is achieved. Our results also suggest that cells could modulate the structure and function of other similar cytoskeletal filaments, such as microtubules in eukaryotes, by slight alterations of molecular properties at the polymerization interfaces through hydrolysis of bound nucleotides, or through other means such as posttranslational modification of specific residues.

Materials and Methods

All simulations were performed using the molecular dynamics package NAMD (44) with the CHARMM27 force field (45, 46), including CMAP corrections (47). Water molecules were described with the TIP3P model (48). Setup, analysis, and rendering of the simulation systems were performed with the software VMD (49). Sequence alignment of FtsZ (Fig. S4) was performed with MultiSeq (50), a plugin of VMD (49). Four molecular dynamics simulations were performed: Simulation-GTP-FtsZx2, Simulation-GDP-FtsZx2, Simulation-APO-FtsZx2, and Simulation-GTP-FtsZx4. For GTP-FtsZ systems, the M. jannaschii GTP-bound FtsZ dimer structure (PDB code: 1W5B) (23) was used. For Simulation-GDP-FtsZx2, the two GTP molecules in Simulation-GTP-FtsZx2 were replaced with GDP, and for Simulation-APO-FtsZx2, no nucleotide was placed in the binding pockets. Details on simulation parameters and simulated systems are described in SI Text.