Constricting force of filamentary protein rings evaluated from experimental results

The dynamics of shapes of vesicle membranes with time dependent spontaneous curvature

We study the time evolution of the shape of a vesicle membrane under time-dependent spontaneous curvature by means of phase-field model. We introduce the variation in time of the spontaneous curvature via a second field which represents the concentration of a substance that anchors with the lipid bilayer thus changing the local curvature and producing constriction. This constriction is mediated by the action on the membrane of an structure resembling the role of a Z ring. Our phase-field model is able to reproduce a number of different shapes that have been experimentally observed. Different shapes are associated with different constraints imposed upon the model regarding conservation of membrane area. In particular, we show that if area is conserved our model reproduces the so-called L-form shape. By contrast, if the area of the membrane is allowed to grow, our model reproduces the formation of a septum in the vicinity of the constriction. Furthermore, we propose a new term in the free energy which allows the membrane to evolve towards eventual pinching.

Bacterial cell division: modeling FtsZ assembly and force generation from single filament experimental data

The bacterial cytoskeletal protein FtsZ binds and hydrolyzes GTP, self-aggregates into dynamic filaments and guides the assembly of the septal ring on the inner side of the membrane at midcell. This ring constricts the cell during division and is present in most bacteria. Despite exhaustive studies undertaken in the last 25 years after its discovery, we do not yet know the mechanism by which this GTP-dependent self-aggregating protein exerts force on the underlying membrane. This paper reviews recent experiments and theoretical models proposed to explain FtsZ filament dynamic assembly and force generation. It highlights how recent observations of single filaments on reconstituted model systems and computational modeling are contributing to develop new multiscale models that stress the importance of previously overlooked elements as monomer internal flexibility, filament twist and flexible anchoring to the cell membrane. These elements contribute to understand the rich behavior of these GTP consuming dynamic filaments on surfaces. The aim of this review is 2-fold: (1) to summarize recent multiscale models and their implications to understand the molecular mechanism of FtsZ assembly and force generation and (2) to update theoreticians with recent experimental results.

Modeling the Mechanics of Cell Division: Influence of Spontaneous Membrane Curvature, Surface Tension, and Osmotic Pressure

Many cell division processes have been conserved throughout evolution and are being revealed by studies on model organisms such as bacteria, yeasts, and protozoa. Cellular membrane constriction is one of these processes, observed almost universally during cell division. It happens similarly in all organisms through a mechanical pathway synchronized with the sequence of cytokinetic events in the cell interior. Arguably, such a mechanical process is mastered by the coordinated action of a constriction machinery fueled by biochemical energy in conjunction with the passive mechanics of the cellular membrane. Independently of the details of the constriction engine, the membrane component responds against deformation by minimizing the elastic energy at every constriction state following a pathway still unknown. In this paper, we address a theoretical study of the mechanics of membrane constriction in a simplified model that describes a homogeneous membrane vesicle in the regime where mechanical work due to osmotic pressure, surface tension, and bending energy are comparable. We develop a general method to find approximate analytical expressions for the main descriptors of a symmetrically constricted vesicle. Analytical solutions are obtained by combining a perturbative expansion for small deformations with a variational approach that was previously demonstrated valid at the reference state of an initially spherical vesicle at isotonic conditions. The analytic approximate results are compared with the exact solution obtained from numerical computations, getting a good agreement for all the computed quantities (energy, area, volume, constriction force). We analyze the effects of the spontaneous curvature, the surface tension and the osmotic pressure in these quantities, focusing especially on the constriction force. The more favorable conditions for vesicle constriction are determined, obtaining that smaller constriction forces are required for positive spontaneous curvatures, low or negative membrane tension and hypertonic media. Conditions for spontaneous constriction at a given constriction force are also determined. The implications of these results for biological cell division are discussed. This work contributes to a better quantitative understanding of the mechanical pathway of cellular division, and could assist the design of artificial divisomes in vesicle-based self-actuated microsystems obtained from synthetic biology approaches.

Beyond force generation: Why is a dynamic ring of FtsZ polymers essential for bacterial cytokinesis?

We propose that the essential function of the most highly conserved protein in bacterial cytokinesis, FtsZ, is not to generate a mechanical force to drive cell division. Rather, we suggest that FtsZ acts as a signal-processing hub to coordinate cell wall synthesis at the division septum with a diverse array of cellular processes, ensuring that the cell divides smoothly at the correct time and place, and with the correct septum morphology. Here, we explore how the polymerization properties of FtsZ, which have been widely attributed to force generation, can also be advantageous in this signal processing role. We suggest mechanisms by which FtsZ senses and integrates both mechanical and biochemical signals, and conclude by proposing experiments to investigate how FtsZ contributes to the remarkable spatial and temporal precision of bacterial cytokinesis.

Redefining the roles of the FtsZ-ring in bacterial cytokinesis

In most bacteria, cell division relies on the functions of an essential protein, FtsZ. FtsZ polymerizes at the future division site to form a ring-like structure, termed the Z-ring, that serves as a scaffold to recruit all other division proteins, and possibly generates force to constrict the cell. The scaffolding function of the Z-ring is well established, but the force generating function has recently been called into question. Additionally, new findings have demonstrated that the Z-ring is more directly linked to cell wall metabolism than simply recruiting enzymes to the division site. Here we review these advances and suggest that rather than generating a rate-limiting constrictive force, the Z-ring's function may be redefined as an orchestrator of septum synthesis.

Defining the rate-limiting processes of bacterial cytokinesis

Bacterial cytokinesis is accomplished by the essential 'divisome' machinery. The most widely conserved divisome component, FtsZ, is a tubulin homolog that polymerizes into the 'FtsZ-ring' ('Z-ring'). Previous in vitro studies suggest that Z-ring contraction serves as a major constrictive force generator to limit the progression of cytokinesis. Here, we applied quantitative superresolution imaging to examine whether and how Z-ring contraction limits the rate of septum closure during cytokinesis in Escherichia coli cells. Surprisingly, septum closure rate was robust to substantial changes in all Z-ring properties proposed to be coupled to force generation: FtsZ's GTPase activity, Z-ring density, and the timing of Z-ring assembly and disassembly. Instead, the rate was limited by the activity of an essential cell wall synthesis enzyme and further modulated by a physical divisome-chromosome coupling. These results challenge a Z-ring-centric view of bacterial cytokinesis and identify cell wall synthesis and chromosome segregation as limiting processes of cytokinesis.

Phase-field modelling of the dynamics of Z-ring formation in liposomes: Onset of constriction and coarsening

We propose a model for the dynamics of the formation of rings of FtsZ on tubular liposomes which produce constriction on the corresponding membrane. Our phase-field model is based on a simple bending energy that captures the dynamics of the interplay between the protein and the membrane. The short-time regime is analyzed by a linear dispersion relation, with which we are able to predict the number of rings per unit length on a tubular liposome. We study numerically the long-time dynamics of the system in the non-linear regime where we observe coarsening of Z-rings on tubular liposomes. In particular, our numerical results show that, during the coarsening process, the number of Z-rings decreases as the radius of tubular liposome increases. This is consistent with the experimental observation that the separation between rings is proportional to the radius of the liposome. Our model predicts that the mechanism for the increased rate of coarsening in liposomes of larger radius is a consequence of the increased interface energy.

Analytical results for cell constriction dominated by bending energy

Analytical expressions are obtained for the main magnitudes of a symmetrically constricted vesicle. These equations provide an easy and compact way to predict minimal requirements for successful constriction and its main magnitudes. Thus, they can be useful for the design of synthetic divisomes and give good predictions for magnitudes including constriction energy, length of the constriction zone, volume and area of the vesicle, and the stability coefficient for symmetric constriction. The analytical expressions are derived combining a perturbative expansion in the Lagrangian for small deformations with a cosine ansatz in the constriction region. Already the simple fourth-order (or sixth-order) approximation provides a good approximation to the values of the main physical magnitudes during constriction, as we show through comparison with numerical results. Results are for vesicles with negligible effects from spontaneous curvature, surface tension, and pressure differences. This is the case when membrane components generating spontaneous curvature are scarce, membrane trafficking is present with low energetic cost, and the external medium is isotonic.

Torsion and curvature of FtsZ filaments

FtsZ filaments participate in bacterial cell division, but it is still not clear how their dynamic polymerization and shape exert force on the underlying membrane. We present a theoretical description of individual filaments that incorporates information from molecular dynamic simulations. The structure of the crystallized Methanococcus jannaschii FtsZ dimer was used to model a FtsZ pentamer that showed a curvature and a twist. The estimated bending and torsion angles between monomers and their fluctuations were included in the theoretical description. The MD data also permitted positioning the curvature with respect to the protein coordinates and allowed us to explore the effect of the relative orientation of the preferred curvature with respect to the surface plane. We find that maximum tension is attained when filaments are firmly attached and oriented with their curvature perpendicular to the surface and that the twist serves as a valve to release or to tighten the tension exerted by the curved filaments on the membrane. The theoretical model also shows that the presence of torsion can explain the shape distribution of short filaments observed by Atomic Force Microscopy in previously published experiments. New experiments with FtsZ covalently attached to lipid membranes show that the filament on-plane curvature depends on lipid head charge, confirming the predicted monomer orientation effects. This new model underlines the fact that the combination of the three elements, filament curvature, twist and the strength and orientation of its surface attachment, can modulate the force exerted on the membrane during cell division.

Mechanics of constriction during cell division: a variational approach

During symmetric division cells undergo large constriction deformations at a stable midcell site. Using a variational approach, we investigate the mechanical route for symmetric constriction by computing the bending energy of deformed vesicles with rotational symmetry. Forces required for constriction are explicitly computed at constant area and constant volume, and their values are found to be determined by cell size and bending modulus. For cell-sized vesicles, considering typical bending modulus of [Formula: see text], we calculate constriction forces in the range [Formula: see text]. The instability of symmetrical constriction is shown and quantified with a characteristic coefficient of the order of [Formula: see text], thus evidencing that cells need a robust mechanism to stabilize constriction at midcell.

Estimating the bending modulus of a FtsZ bacterial-division protein filament

FtsZ, a cytoskeletal protein homologous to tubulin, is the principle constituent of the division ring in bacterial cells. It is known to have force-generating capacity in vitro and has been conjectured to be the source of the constriction force in vivo. Several models have been proposed to explain the generation of force by the Z ring. Here we re-examine data from in vitro experiments in which Z rings formed and constricted inside tubular liposomes, and we carry out image analysis on previously published data with which to better estimate important model parameters that have proven difficult to measure by direct means. We introduce a membrane-energy-based model for the dynamics of multiple Z rings moving and colliding inside a tubular liposome and a fluid model for the drag of a Z ring as it moves through the tube. Using this model, we estimate an effective membrane bending modulus of 500-700 pN nm. If we assume that FtsZ force generation is driven by hydrolysis into a highly curved conformation, we estimate the FtsZ filament bending modulus to be 310-390 pN nm(2). If we assume instead that force is generated by the non-hydrolysis-dependent intermediate curvature conformation, we find that B(f)>1400 pN nm(2). The former value sits at the lower end of the range of previously estimated values and, if correct, may raise challenges for models that rely on filament bending to generate force.

Inside-out Z rings--constriction with and without GTP hydrolysis

The bacterial tubulin homologue FtsZ forms a ring-like structure called the Z ring that drives cytokinesis. We showed previously that FtsZ-YFP-mts, which has a short amphipathic helix (mts) on its C terminus that inserts into the membrane, can assemble contractile Z rings in tubular liposomes without any other protein. Here we study mts-FtsZ-YFP, where the membrane tether is switched to the opposite side of the protofilament. This assembled 'inside-out' Z rings that wrapped around the outside surface of tubular liposomes. The inside-out Z rings were highly dynamic, and generated a constriction force that squeezed the tubular liposomes from outside. This is consistent with models where the constriction force is generated by curved protofilaments bending the membrane. We used this system to test how GTP hydrolysis by FtsZ is involved in Z-ring constriction. Without GTP hydrolysis, Z rings could still assemble and generate an initial constriction. However, the constriction quickly stopped, suggesting that Z rings became rigidly stabilized in the absence of GTP hydrolysis. We propose that remodelling of the Z ring, mediated by GTP hydrolysis and exchange of subunits, is necessary for the continuous constriction.

FtsZ in bacterial cytokinesis: cytoskeleton and force generator all in one

FtsZ, a bacterial homolog of tubulin, is well established as forming the cytoskeletal framework for the cytokinetic ring. Recent work has shown that purified FtsZ, in the absence of any other division proteins, can assemble Z rings when incorporated inside tubular liposomes. Moreover, these artificial Z rings can generate a constriction force, demonstrating that FtsZ is its own force generator. Here we review light microscope observations of how Z rings assemble in bacteria. Assembly begins with long-pitch helices that condense into the Z ring. Once formed, the Z ring can transition to short-pitch helices that are suggestive of its structure. FtsZ assembles in vitro into short protofilaments that are ∼30 subunits long. We present models for how these protofilaments might be further assembled into the Z ring. We discuss recent experiments on assembly dynamics of FtsZ in vitro, with particular attention to how two regulatory proteins, SulA and MinC, inhibit assembly. Recent efforts to develop antibacterial drugs that target FtsZ are reviewed. Finally, we discuss evidence of how FtsZ generates a constriction force: by protofilament bending into a curved conformation.