Modeling FtsZ ring formation in the bacterial cell—anisotropic aggregation via mutual interactions of polymer rods

Chiral-filament self-assembly on curved manifolds

Rod-like and banana-shaped proteins, like BAR-domain proteins and MreB proteins, adsorb on membranes and regulate the membrane curvature. The formation of large filamentous complexes of these proteins plays an important role in cellular processes like membrane trafficking, cytokinesis and cell motion. We propose a simplified model to investigate such curvature-dependent self-assembly processes. Anisotropic building blocks, modeled as trimer molecules, which have a preferred binding site, interact via pair-wise Lennard-Jones potentials. When several trimers assemble, they form an elastic ribbon with an intrinsic curvature and twist, controlled by bending and torsional rigidity. For trimer self-assembly on the curved surface of a cylindrical membrane, this leads to a preferred spatial orientation of the ribbon. We show that these interactions can lead to the formation of helices with several windings around the cylinder. The emerging helix angle and pitch depend on the rigidities and the intrinsic curvature and twist values. In particular, a well-defined and controllable helix angle emerges in the case of equal bending and torsional rigidity. The dynamics of filament growth is characterized by three regimes, in which filament length increases with the power laws t^z in time, with z≃ 3/4, z = 1/2, and z = 0 for short, intermediate, and long times, respectively. A comparison with the solutions of the Smoluchowski aggregation equation allows the identification of the underlying mechanism in the short-time regime as a crossover from size-independent to diffusion-limited aggregation. Thus, helical structures, as often observed in biology, can arise by self-assembly of anisotropic and chiral proteins.

A stochastic model for dynamics of FtsZ filaments and the formation of Z -ring

Understanding the mechanisms responsible for the formation and growth of FtsZ polymers and their subsequent formation of the Z -ring is important for gaining insight into the cell division in prokaryotic cells. In this work, we present a minimal stochastic model that qualitatively reproduces in vitro observations of polymerization, formation of dynamic contractile ring that is stable for a long time and depolymerization shown by FtsZ polymer filaments. In this stochastic model, we explore different mechanisms for ring breaking and hydrolysis. In addition to hydrolysis, which is known to regulate the dynamics of other tubulin polymers like microtubules, we find that the presence of the ring allows for an additional mechanism for regulating the dynamics of FtsZ polymers. Ring breaking dynamics in the presence of hydrolysis naturally induce rescue and catastrophe events in this model irrespective of the mechanism of hydrolysis.

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.

Protein Recruitment through Indirect Mechanochemical Interactions

Some of the key proteins essential for important cellular processes are capable of recruiting other proteins from the cytosol to phospholipid membranes. The physical basis for this cooperativity of binding is, surprisingly, still unclear. Here, we suggest a general feedback mechanism that explains cooperativity through mechanochemical coupling mediated by the mechanical properties of phospholipid membranes. Our theory predicts that protein recruitment, and therefore also protein pattern formation, involves membrane deformation and is strongly affected by membrane composition.

Minimal coarse-grained models for molecular self-organisation in biology

The molecular machinery of life is largely created via self-organisation of individual molecules into functional assemblies. Minimal coarse-grained models, in which a whole macromolecule is represented by a small number of particles, can be of great value in identifying the main driving forces behind self-organisation in cell biology. Such models can incorporate data from both molecular and continuum scales, and their results can be directly compared to experiments. Here we review the state of the art of models for studying the formation and biological function of macromolecular assemblies in living organisms. We outline the key ingredients of each model and their main findings. We illustrate the contribution of this class of simulations to identifying the physical mechanisms behind life and diseases, and discuss their future developments.

Mutations on FtsZ lateral helix H3 that disrupt cell viability hamper reorganization of polymers on lipid surfaces

FtsZ filaments localize at the middle of the bacterial cell and participate in the formation of a contractile ring responsible for cell division. Previous studies demonstrated that the highly conserved negative charge of glutamate 83 and the positive charge of arginine 85 located in the lateral helix H3 bend of Escherichia coli FtsZ are required for in vivo cell division. In order to understand how these lateral mutations impair the formation of a contractile ring,we extend previous in vitro characterization of these mutants in solution to study their behavior on lipid modified surfaces. We study their interaction with ZipAand look at their reorganization on the surface. We found that the dynamic bundling capacity of the mutant proteins is deficient, and this impairment increases the more the composition and spatial arrangement of the reconstituted system resembles the situation inside the cell: mutant proteins completely fail to reorganize to form higher order aggregates when bound to an E.coli lipid surface through oriented ZipA.We conclude that these surface lateral point mutations affect the dynamic reorganization of FtsZ filaments into bundles on the cell membrane, suggesting that this event is relevant for generating force and completing bacterial division.

Efficient Multiscale Models of Polymer Assembly

Protein polymerization and bundling play a central role in cell physiology. Predictive modeling of these processes remains an open challenge, especially when the proteins involved become large and their concentrations high. We present an effective kinetics model of filament formation, bundling, and depolymerization after GTP hydrolysis, which involves a relatively small number of species and reactions, and remains robust over a wide range of concentrations and timescales. We apply this general model to study assembly of FtsZ protein, a basic element in the division process of prokaryotic cells such as Escherichia coli, Bacillus subtilis, or Caulobacter crescentus. This analysis demonstrates that our model outperforms its counterparts in terms of both accuracy and computational efficiency. Because our model comprises only 17 ordinary differential equations, its computational cost is orders-of-magnitude smaller than the current alternatives consisting of up to 1000 ordinary differential equations. It also provides, to our knowledge, a new insight into the characteristics and functioning of FtsZ proteins at high concentrations. The simplicity and versatility of our model render it a powerful computational tool, which can be used either as a standalone descriptor of other biopolymers' assembly or as a component in more complete kinetic models.

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.

Organization of FtsZ filaments in the bacterial division ring measured from polarized fluorescence microscopy

Cytokinesis in bacteria is accomplished by a ring-shaped cell-division complex (the Z-ring). The primary component of the Z-ring is FtsZ, a filamentous tubulin homolog that serves as a scaffold for the recruitment of other cell-division-related proteins. FtsZ forms filaments and bundles. In the cell, it has been suggested that FtsZ filaments form the arcs of the ring and are aligned in the cell-circumferential direction. Using polarized fluorescence microscopy in live Escherichia coli cells, we measure the structural organization of FtsZ filaments in the Z-ring. The data suggest a disordered organization: a substantial portion of FtsZ filaments are aligned in the cell-axis direction. FtsZ organization in the Z-ring also appears to depend on the bacterial species. Taken together, the unique arrangement of FtsZ suggests novel unexplored mechanisms in bacterial cell division.

Cell shape can mediate the spatial organization of the bacterial cytoskeleton

The bacterial cytoskeleton guides the synthesis of cell wall and thus regulates cell shape. Because spatial patterning of the bacterial cytoskeleton is critical to the proper control of cell shape, it is important to ask how the cytoskeleton spatially self-organizes in the first place. In this work, we develop a quantitative model to account for the various spatial patterns adopted by bacterial cytoskeletal proteins, especially the orientation and length of cytoskeletal filaments such as FtsZ and MreB in rod-shaped cells. We show that the combined mechanical energy of membrane bending, membrane pinning, and filament bending of a membrane-attached cytoskeletal filament can be sufficient to prescribe orientation, e.g., circumferential for FtsZ or helical for MreB, with the accuracy of orientation increasing with the length of the cytoskeletal filament. Moreover, the mechanical energy can compete with the chemical energy of cytoskeletal polymerization to regulate filament length. Notably, we predict a conformational transition with increasing polymer length from smoothly curved to end-bent polymers. Finally, the mechanical energy also results in a mutual attraction among polymers on the same membrane, which could facilitate tight polymer spacing or bundling. The predictions of the model can be verified through genetic, microscopic, and microfluidic approaches.

Polymorphism of FtsZ filaments on lipid surfaces: role of monomer orientation

FtsZ is a bacterial cytoskeletal protein involved in cell division. It forms a ringlike structure that attaches to the membrane to complete bacterial division. It binds and hydrolyzes GTP, assembling into polymers in a GTP-dependent manner. To test how the orientation of the monomers affects the curvature of the filaments on a surface, we performed site-directed mutagenesis on the E. coli FtsZ protein to insert cysteine residues at lateral locations to orient FtsZ on planar lipid bilayers. The E93C and S255C mutants were overproduced, purified, and found to be functionally active in solution, as well as being capable of sustaining cell division in vivo in complementation assays. Atomic force microscopy was used to observe the shape of the filament fibers formed on the surface. The FtsZ mutants were covalently linked to the lipids and could be polymerized on the bilayer surface in the presence of GTP. Unexpectedly, both mutants assembled into straight structures. E93C formed a well-defined lattice with monomers interacting at 60° and 120° angles, whereas S255C formed a more open array of straight thicker filament aggregates. These results indicate that filament curvature and bending are not fixed and that they can be modulated by the orientation of the monomers with respect to the membrane surface. As filament curvature has been associated with the force generation mechanism, these results point to a possible role of filament membrane attachment in lateral association and curvature, elements currently identified as relevant for force generation.

Depolymerization dynamics of individual filaments of bacterial cytoskeletal protein FtsZ

We report observation and analysis of the depolymerization filaments of the bacterial cytoskeletal protein FtsZ (filament temperature-sensitive Z) formed on a mica surface. At low concentration, proteins adsorbed on the surface polymerize forming curved filaments that close into rings that remain stable for some time before opening irreversibly and fully depolymerizing. The distribution of ring lifetimes (T) as a function of length (N), shows that the rate of ring aperture correlates with filament length. If this ring lifetime is expressed as a bond survival time, (T(b) ≡ NT), this correlation is abolished, indicating that these rupture events occur randomly and independently at each monomer interface. After rings open irreversibly, depolymerization of the remaining filaments is fast, but can be slowed down and followed using a nonhydrolyzing GTP analogue. The histogram of depolymerization velocities of individual filaments has an asymmetric distribution that can be fit with a computer model that assumes two rupture rates, a slow one similar to the one observed for ring aperture, affecting monomers in the central part of the filaments, and a faster one affecting monomers closer to the open ends. From the quantitative analysis, we conclude that the depolymerization rate is affected both by nucleotide hydrolysis rate and by its exchange along the filament, that all monomer interfaces are equally competent for hydrolysis, although depolymerization is faster at the open ends than in central filament regions, and that all monomer-monomer interactions, regardless of the nucleotide present, can adopt a curved configuration.

FtsZ polymers bound to lipid bilayers through ZipA form dynamic two dimensional networks

Bacteria divide by forming a contractile ring around their midcell region. FtsZ, a cytoskeletal soluble protein structurally related to tubulin, is the main component of this division machinery. It forms filaments that bundle at the inner side of the cytoplasmic membrane. These FtsZ bundles do not attach to bare lipid surfaces. In Escherichia coli they remain near the membrane surface by attaching to the membrane protein ZipA and FtsA. In order to study the structure and dynamics of the ZipA-FtsZ bundles formed on a lipid surface, we have oriented a soluble form of ZipA (sZipA), with its transmembrane domain substituted by a histidine tag, on supported lipid membranes. Atomic force microscopy has been used to visualize the polymers formed on top of this biomimetic surface. In the presence of GTP, when sZipA is present, FtsZ polymers restructure forming higher order structures. The lipid composition of the underlying membrane affects the aggregation kinetics and the shape of the structures formed. On the negatively charged E. coli lipid membranes, filaments condense from initially disperse material to form a network that is more dynamic and flexible than the one formed on phosphatidyl choline bilayers. These FtsZ-ZipA filament bundles are interconnected, retain their capacity to dynamically restructure, to fragment, to anneal and to condense laterally.

Timing of Z-ring localization in Escherichia coli

Bacterial cell division takes place in three phases: Z-ring formation at midcell, followed by divisome assembly and building of the septum per se. Using time-lapse microscopy of live bacteria and a high-precision cell edge detection method, we have previously found the true time for the onset of septation, τ(c), and the time between consecutive divisions, τ(g). Here, we combine the above method with measuring the dynamics of the FtsZ-GFP distribution in individual Escherichia coli cells to determine the Z-ring positioning time, τ(z). To analyze the FtsZ-GFP distribution along the cell, we used the integral fluorescence profile (IFP), which was obtained by integrating the fluorescence intensity across the cell width. We showed that the IFP may be approximated by an exponential peak and followed the peak evolution throughout the cell cycle, to find a quantitative criterion for the positioning of the Z-ring and hence the value of τ(z). We defined τ(z) as the transition from oscillatory to stable behavior of the mean IFP position. This criterion was corroborated by comparison of the experimental results to a theoretical model for the FtsZ dynamics, driven by Min oscillations. We found that τ(z) < τ(c) for all the cells that were analyzed. Moreover, our data suggested that τ(z) is independent of τ(c), τ(g) and the cell length at birth, L(0). These results are consistent with the current understanding of the Z-ring positioning and cell septation processes.