A mechanical explanation for cytoskeletal rings and helices in bacteria

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.

Mechanical property of the helical configuration for a twisted intrinsically straight biopolymer

We explore the effects of two typical torques on the mechanical property of the helical configuration for an intrinsically straight filament or biopolymer either in three-dimensional space or on a cylinder. One torque is parallel to the direction of a uniaxial applied force, and is coupled to the cross section of the filament. We obtain some algebraic equations for the helical configuration and find that the boundary conditions are crucial. In three-dimensional space, we show that the extension is always a monotonic function of the applied force. On the other hand, for a filament confined on a cylinder, the twisting rigidity and torque coupled to the cross section are irrelevant in forming a helix if the filament is isotropic and under free boundary condition. However, the twisting rigidity and the torque coupled to the cross section become crucial when the Euler angle at two ends of the filament are fixed. Particularly, the extension of a helix can subject to a first-order transition so that in such a condition a biopolymer can act as a switch or sensor in some biological processes. We also present several phase diagrams to provide the conditions to form a helix.

Shape Selection of Surface-Bound Helical Filaments: Biopolymers on Curved Membranes

Motivated to understand the behavior of biological filaments interacting with membranes of various types, we employ a theoretical model for the shape and thermodynamics of intrinsically helical filaments bound to curved membranes. We show that filament-surface interactions lead to a host of nonuniform shape equilibria, in which filaments progressively unwind from their native twist with increasing surface interaction and surface curvature, ultimately adopting uniform-contact curved shapes. The latter effect is due to nonlinear coupling between elastic twist and bending of filaments on anisotropically curved surfaces such as the cylindrical surfaces considered here. Via a combination of numerical solutions and asymptotic analysis of shape equilibria, we show that filament conformations are critically sensitive to the surface curvature in both the strong- and weak-binding limits. These results suggest that local structure of membrane-bound chiral filaments is generically sensitive to the curvature radius of the surface to which it is bound, even when that radius is much larger than the filament's intrinsic pitch. Typical values of elastic parameters and interaction energies for several prokaryotic and eukaryotic filaments indicate that biopolymers are inherently very sensitive to the coupling between twist, interactions, and geometry and that this could be exploited for regulation of a variety of processes such as the targeted exertion of forces, signaling, and self-assembly in response to geometric cues including the local mean and Gaussian curvatures.

MreB Orientation Correlates with Cell Diameter in Escherichia coli

Bacteria have remarkably robust cell shape control mechanisms. For example, cell diameter only varies by a few percent across a given population. The bacterial actin homolog, MreB, is necessary for establishment and maintenance of rod shape although the detailed properties of MreB that are important for shape control remained unknown. In this study, we perturb MreB in two ways: by treating cells with the polymerization-inhibiting drug A22 and by creating point mutants in mreB. These perturbations modify the steady-state diameter of cells over a wide range, from 790 ± 30 nm to 1700 ± 20 nm. To determine which properties of MreB are important for diameter control, we correlated structural characteristics of fluorescently tagged MreB polymers with cell diameter by simultaneously analyzing three-dimensional images of MreB and cell shape. Our results indicate that the helical pitch angle of MreB inversely correlates with the cell diameter of Escherichia coli. Other correlations between MreB and cell diameter are not found to be significant. These results demonstrate that the physical properties of MreB filaments are important for shape control and support a model in which MreB organizes the cell wall growth machinery to produce a chiral cell wall structure and dictate cell diameter.

Collective effects of torsion in FtsZ filaments

Recent evidence points to the presence of torsion in FtsZ bonds. In addition, experiments with FtsZ mutants on surfaces resulted in new aggregates that cannot be explained by older models for FtsZ dynamics. We use an interaction model for FtsZ derived from molecular dynamics simulations and expand a fine-grained lattice model used to describe FtsZ aggregates on a surface. This new model includes different anchoring angles for the monomers and allows bond twist, two ingredients that oppose each other resulting in a more dynamic and interesting system. We study the role and importance of these conflicting elements and how the aggregates are characterized by the different interaction parameters.

An intrinsically disordered linker plays a critical role in bacterial cell division

In bacteria, animals, fungi, and many single celled eukaryotes, division is initiated by the formation of a ring of cytoskeletal protein at the nascent division site. In bacteria, the tubulin-like GTPase FtsZ serves as the foundation for the cytokinetic ring. A conserved feature of FtsZ is an intrinsically disordered peptide known as the C-terminal linker. Chimeric experiments suggest the linker acts as a flexible boom allowing FtsZ to associate with the membrane through a conserved C-terminal domain and also modulates interactions both between FtsZ subunits and between FtsZ and modulatory proteins in the cytoplasm.

Getting into shape: How do rod-like bacteria control their geometry?

Rod-like bacteria maintain their cylindrical shapes with remarkable precision during growth. However, they are also capable to adapt their shapes to external forces and constraints, for example by growing into narrow or curved confinements. Despite being one of the simplest morphologies, we are still far from a full understanding of how shape is robustly regulated, and how bacteria obtain their near-perfect cylindrical shapes with excellent precision. However, recent experimental and theoretical findings suggest that cell-wall geometry and mechanical stress play important roles in regulating cell shape in rod-like bacteria. We review our current understanding of the cell wall architecture and the growth dynamics, and discuss possible candidates for regulatory cues of shape regulation in the absence or presence of external constraints. Finally, we suggest further future experimental and theoretical directions which may help to shed light on this fundamental problem.

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.

Methods for modeling cytoskeletal and DNA filaments

This review summarizes the models that researchers use to represent the conformations and dynamics of cytoskeletal and DNA filaments. It focuses on models that address individual filaments in continuous space. Conformation models include the freely jointed, Gaussian, angle-biased chain (ABC), and wormlike chain (WLC) models, of which the first three bend at discrete joints and the last bends continuously. Predictions from the WLC model generally agree well with experiment. Dynamics models include the Rouse, Zimm, stiff rod, dynamic WLC, and reptation models, of which the first four apply to isolated filaments and the last to entangled filaments. Experiments show that the dynamic WLC and reptation models are most accurate. They also show that biological filaments typically experience strong hydrodynamic coupling and/or constrained motion. Computer simulation methods that address filament dynamics typically compute filament segment velocities from local forces using the Langevin equation and then integrate these velocities with explicit or implicit methods; the former are more versatile and the latter are more efficient. Much remains to be discovered in biological filament modeling. In particular, filament dynamics in living cells are not well understood, and current computational methods are too slow and not sufficiently versatile. Although primarily a review, this paper also presents new statistical calculations for the ABC and WLC models. Additionally, it corrects several discrepancies in the literature about bending and torsional persistence length definitions, and their relations to flexural and torsional rigidities.

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.

Negative-stain electron microscopy of inside-out FtsZ rings reconstituted on artificial membrane tubules show ribbons of protofilaments

FtsZ, the primary cytoskeletal element of the Z ring, which constricts to divide bacteria, assembles into short, one-stranded filaments in vitro. These must be further assembled to make the Z ring in bacteria. Conventional electron microscopy (EM) has failed to image the Z ring or resolve its substructure. Here we describe a procedure that enabled us to image reconstructed, inside-out FtsZ rings by negative-stain EM, revealing the arrangement of filaments. We took advantage of a unique lipid that spontaneously forms 500 nm diameter tubules in solution. We optimized conditions for Z-ring assembly with fluorescence light microscopy and then prepared specimens for negative-stain EM. Reconstituted FtsZ rings, encircling the tubules, were clearly resolved. The rings appeared as ribbons of filaments packed side by side with virtually no space between neighboring filaments. The rings were separated by variable expanses of empty tubule as seen by light microscopy or EM. The width varied considerably from one ring to another, but each ring maintained a constant width around its circumference. The inside-out FtsZ rings moved back and forth along the tubules and exchanged subunits with solution, similarly to Z rings reconstituted outside or inside tubular liposomes. FtsZ from Escherichia coli and Mycobacterium tuberculosis assembled rings of similar structure, suggesting a universal structure across bacterial species. Previous models for the Z ring in bacteria have favored a structure of widely scattered filaments that are not in contact. The ribbon structure that we discovered here for reconstituted inside-out FtsZ rings provides what to our knowledge is new evidence that the Z ring in bacteria may involve lateral association of protofilaments.

The molecular origins of chiral growth in walled cells

Cells from all kingdoms of life adopt a dizzying array of fascinating shapes that support cellular function. Amoeboid and spherical shapes represent perhaps the simplest of geometries that may minimize the level of growth control required for survival. Slightly more complex are rod-shaped cells, from microscopic bacteria to macroscopic plants, which require additional mechanisms to define a cell's longitudinal axis, width, and length. Recent evidence suggests that many rod-shaped, walled cells achieve elongated growth through chiral insertion of cell-wall material that may be coupled to a twisting of the cell body. Inspired by these observations, biophysical mechanisms for twisting growth have been proposed that link the mechanics of intracellular proteins to cell shape maintenance. In this review, we highlight experimental and theoretical work that connects molecular-scale organization and structure with the cellular-scale phenomena of rod-shaped growth.

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.

In vivo structure of the E. coli FtsZ-ring revealed by photoactivated localization microscopy (PALM)

The FtsZ protein, a tubulin-like GTPase, plays a pivotal role in prokaryotic cell division. In vivo it localizes to the midcell and assembles into a ring-like structure-the Z-ring. The Z-ring serves as an essential scaffold to recruit all other division proteins and generates contractile force for cytokinesis, but its supramolecular structure remains unknown. Electron microscopy (EM) has been unsuccessful in detecting the Z-ring due to the dense cytoplasm of bacterial cells, and conventional fluorescence light microscopy (FLM) has only provided images with limited spatial resolution (200-300 nm) due to the diffraction of light. Hence, given the small sizes of bacteria cells, identifying the in vivo structure of the Z-ring presents a substantial challenge. Here, we used photoactivated localization microscopy (PALM), a single molecule-based super-resolution imaging technique, to characterize the in vivo structure of the Z-ring in E. coli. We achieved a spatial resolution of ∼35 nm and discovered that in addition to the expected ring-like conformation, the Z-ring of E. coli adopts a novel compressed helical conformation with variable helical length and pitch. We measured the thickness of the Z-ring to be ∼110 nm and the packing density of FtsZ molecules inside the Z-ring to be greater than what is expected for a single-layered flat ribbon configuration. Our results strongly suggest that the Z-ring is composed of a loose bundle of FtsZ protofilaments that randomly overlap with each other in both longitudinal and radial directions of the cell. Our results provide significant insight into the spatial organization of the Z-ring and open the door for further investigations of structure-function relationships and cell cycle-dependent regulation of the Z-ring.

Constricting force of filamentary protein rings evaluated from experimental results

We present a model of Z -ring constriction in bacteria based on different experimental in vitro results. The forces produced by the Z ring due to lateral attraction of its constituent parts, estimated in previous studies that were based on FtsZ filaments observed by atomic force microscopy, are in good agreement with an estimation of the force required for recently found deformations in liposomes caused by FtsZ. These forces are calculated within the usual Helfrich energy formalism. In this context, we also explain the apparent attraction of multiple Z rings in the liposomes initially separated by small distances, as well as the stable distribution of rings separated by distances greater than approximately twice the diameter of the cylindrical liposomes. We adapted the model to the in vivo conditions imposed by the bacterial cell wall, concluding that the proposed mechanism gives a qualitative explanation for the force generation during bacterial division.

Sculpting the bacterial cell

Prokaryotes come in a wide variety of shapes, determined largely by natural selection, physical constraints, and patterns of cell growth and division. Because of their relative simplicity, bacterial cells are excellent models for how genes and proteins can directly determine morphology. Recent advances in cytological methods for bacteria have shown that distinct cytoskeletal filaments composed of actin and tubulin homologs are important for guiding growth patterns of the cell wall in bacteria, and that the glycan strands that constitute the wall are generally perpendicular to the direction of growth. This cytoskeleton-directed cell wall patterning is strikingly reminiscent of how plant cell wall growth is regulated by microtubules. In rod-shaped bacilli, helical cables of actin-like MreB protein stretch along the cell length and orchestrate elongation of the cell wall, whereas the tubulin-like FtsZ protein directs formation of the division septum and the resulting cell poles. The overlap and interplay between these two systems and the peptidoglycan-synthesizing enzymes they recruit are the major driving forces of cylindrical shapes. Round cocci, on the other hand, have lost their MreB cables and instead must grow mainly via their division septum, giving them their characteristic round or ovoid shapes. Other bacteria that lack MreB homologs or even cell walls use distinct cytoskeletal systems to maintain their distinct shapes. Here I review what is known about the mechanisms that determine the shape of prokaryotic cells.

Simple modeling of FtsZ polymers on flat and curved surfaces: correlation with experimental in vitro observations

FtsZ is a GTPase that assembles at midcell into a dynamic ring that constricts the membrane to induce cell division in the majority of bacteria, in many archea and several organelles. In vitro, FtsZ polymerizes in a GTP-dependent manner forming a variety of filamentous flexible structures. Based on data derived from the measurement of the in vitro polymerization of Escherichia coli FtsZ cell division protein we have formulated a model in which the fine balance between curvature, flexibility and lateral interactions accounts for structural and dynamic properties of the FtsZ polymers observed with AFM. The experimental results have been used by the model to calibrate the interaction energies and the values obtained indicate that the filaments are very plastic. The extension of the model to explore filament behavior on a cylindrical surface has shown that the FtsZ condensates promoted by lateral interactions can easily form ring structures through minor modulations of either filament curvature or longitudinal bond energies. The condensation of short, monomer exchanging filaments into rings is shown to produce enough force to induce membrane deformations.PACS codes: 87.15.ak, 87.16.ka, 87.17.Ee.

Pulling helices inside bacteria: imperfect helices and rings

We study steady-state configurations of intrinsically-straight elastic filaments constrained within rod-shaped bacteria that have applied forces distributed along their length. Perfect steady-state helices result from axial or azimuthal forces applied at filament ends, however azimuthal forces are required for the small pitches observed for MreB filaments within bacteria. Helix-like configurations can result from distributed forces, including coexistence between rings and imperfect helices. Levels of expression and/or bundling of the polymeric protein could mediate this coexistence.

Force generation by a dynamic Z-ring in Escherichia coli cell division

FtsZ, a bacterial homologue of tubulin, plays a central role in bacterial cell division. It is the first of many proteins recruited to the division site to form the Z-ring, a dynamic structure that recycles on the time scale of seconds and is required for division to proceed. FtsZ has been recently shown to form rings inside tubular liposomes and to constrict the liposome membrane without the presence of other proteins, particularly molecular motors that appear to be absent from the bacterial proteome. Here, we propose a mathematical model for the dynamic turnover of the Z-ring and for its ability to generate a constriction force. Force generation is assumed to derive from GTP hydrolysis, which is known to induce curvature in FtsZ filaments. We find that this transition to a curved state is capable of generating a sufficient force to drive cell-wall invagination in vivo and can also explain the constriction seen in the in vitro liposome experiments. Our observations resolve the question of how FtsZ might accomplish cell division despite the highly dynamic nature of the Z-ring and the lack of molecular motors.

In vitro reconstitution of the initial stages of the bacterial cell division machinery

Fission of many prokaryotes as well as some eukaryotic organelles depends on the self-assembly of the FtsZ protein into a membrane-associated ring structure early in the division process. Different components of the machinery are then sequentially recruited. Although the assembly order has been established, the molecular interactions and the understanding of the force-generating mechanism of this dividing machinery have remained elusive. It is desirable to develop simple reconstituted systems that attempt to reproduce, at least partially, some of the stages of the process. High-resolution studies of Escherichia coli FtsZ filaments' structure and dynamics on mica have allowed the identification of relevant interactions between filaments that suggest a mechanism by which the polymers could generate force on the membrane. Reconstituting the membrane-anchoring protein ZipA on E. coli lipid membrane on surfaces is now providing information on how the membrane attachment regulates FtsZ polymer dynamics and indicates the important role played by the lipid composition of the membrane.

Reconstitution of contractile FtsZ rings in liposomes

FtsZ is a tubulin homolog and the major cytoskeletal protein in bacterial cell division. It assembles into the Z ring, which contains FtsZ and a dozen other division proteins, and constricts to divide the cell. We have constructed a membrane-targeted FtsZ (FtsZ-mts) by splicing an amphipathic helix to its C terminus. When mixed with lipid vesicles, FtsZ-mts was incorporated into the interior of some tubular vesicles. There it formed multiple Z rings that could move laterally in both directions along the length of the liposome and coalesce into brighter Z rings. Brighter Z rings produced visible constrictions in the liposome, suggesting that FtsZ itself can assemble the Z ring and generate a force. No other proteins were needed for assembly and force generation.

FtsZ bacterial cytoskeletal polymers on curved surfaces: the importance of lateral interactions

A recent theoretical article provided a mechanical explanation for the formation of cytoskeletal rings and helices in bacteria assuming that these shapes arise, at least in part, from the interaction of the inherent mechanical properties of the protein polymers and the constraints imposed by the curved cell membrane (Andrews, S., and A. P. Arkin. 2007. Biophys. J. 93:1872-1884). Due to the lack of experimental data regarding the bending rigidity and preferential bond angles of bacterial polymers, the authors explored their model over wide ranges of preferred curvature values. In this letter, we present the shape diagram of the FtsZ bacterial polymer on a curved surface but now including recent experimental data on the in vitro formed FtsZ polymers. The lateral interactions between filaments observed experimentally change qualitatively the shape diagram and indicate that the formation of rings over spirals is more energetically favored than estimated in the above-mentioned article.