Mesoscopic modeling of bacterial flagellar microhydrodynamics

A particle-based hybrid method of elastic network model and smooth-particle hydrodynamics has been employed to describe the propulsion of bacterial flagella in a viscous hydrodynamic environment. The method explicitly models the two aspects of bacterial propulsion that involve flagellar flexibility and long-range hydrodynamic interaction of low-Reynolds-number flow. The model further incorporates the molecular organization of the flagellar filament at a coarse-grained level in terms of the 11 protofilaments. Each of these protofilaments is represented by a collection of material points that represent the flagellin proteins. A computational model of a single flexible helical segment representing the filament of a bacterial flagellum is presented. The propulsive dynamics and the flow fields generated by the motion of the model filament are examined. The nature of flagellar deformation and the influence of hydrodynamics in determining the shape of deformations are examined based on the helical filament.     

Correlation between Supercoiling and Conformational Motions of the Bacterial Flagellar Filament

The bacterial flagellar filament is a very large macromolecular assembly of a single protein, flagellin. Various supercoiled states of the filament exist, which are formed by two structurally different conformations of flagellin in different ratios. We investigated the correlation between supercoiling of the protofilaments and molecular dynamics in the flagellar filament using quasielastic and elastic incoherent neutron scattering on the picosecond and nanosecond timescales. Thermal fluctuations in the straight L- and R-type filaments were measured and compared to the resting state of the wild-type filament. Amplitudes of motion on the picosecond timescale were found to be similar in the different conformational states. Mean-square displacements and protein resilience on the 0.1 ns timescale demonstrate that the L-type state is more flexible and less resilient than the R-type, whereas the wild-type state lies in between. Our results provide strong support that supercoiling of the protofilaments in the flagellar filament is determined by the strength of molecular forces in and between the flagellin subunits.     

Correlation between Supercoiling and Conformational Motions of the Bacterial Flagellar Filament

The bacterial flagellar filament is a very large macromolecular assembly of a single protein, flagellin. Various supercoiled states of the filament exist, which are formed by two structurally different conformations of flagellin in different ratios. We investigated the correlation between supercoiling of the protofilaments and molecular dynamics in the flagellar filament using quasielastic and elastic incoherent neutron scattering on the picosecond and nanosecond timescales. Thermal fluctuations in the straight L- and R-type filaments were measured and compared to the resting state of the wild-type filament. Amplitudes of motion on the picosecond timescale were found to be similar in the different conformational states. Mean-square displacements and protein resilience on the 0.1 ns timescale demonstrate that the L-type state is more flexible and less resilient than the R-type, whereas the wild-type state lies in between. Our results provide strong support that supercoiling of the protofilaments in the flagellar filament is determined by the strength of molecular forces in and between the flagellin subunits.     

Compliance of bacterial polyhooks measured with optical tweezers.

In earlier work, a single-beam gradient force optical trap ("optical tweezers") was used to measure the torsional compliance of flagella in wild-type cells of Escherichia coli that had been tethered to glass by a single flagellum. This compliance was nonlinear, exhibiting a torsionally soft phase up to 180 degrees, followed by a torsionally rigid phase for larger angles. Values for the torsional spring constant in the soft phase were substantially less than estimates based on the rigidity determined for isolated flagellar filaments. It was suggested that the soft phase might correspond to wind-up of the flagellar hook, and the rigid phase to wind-up of the stiffer filament. Here, we have measured the torsional compliance of flagella on cells of an E. coli strain that produces abnormally long hooks but no filaments. The small-angle compliance of these cells, as determined from the elastic rebound of the cell body after wind-up and release, was found to be the same as for wild-type cells. This confirms that the small-angle compliance of wild-type cells is dominated by the response of the hook. Hook flexibility is likely to play a useful role in stabilizing the flagellar bundle.     

Computer Simulation of Flagellar Movement: I. Demonstration of Stable Bend Propagation and Bend Initiation by the Sliding Filament Model

A program has been developed for digital computer simulation of the movement of a flagellar model consisting of straight segments connected by joints at which bending occurs. The program finds values for the rate of bending at each joint by solving equations which balance active, viscous, and elastic bending moments at each joint. These bending rates are then used to compute the next position of the model. Stable swimming movements, similar to real flagellar movements, can be generated routinely with a 25-segment model using 16 time steps/beat cycle. These results depend on four assumptions about internal flagellar mechanisms: (a) Bending is generated by a sliding filament process. (b) The active process is controlled locally by the curvature of the flagellum. (c) Nonlinear elastic resistances stabilize the amplitude of the movement. (d) Internal viscous resistances stabilize the wavelength of the movement and explain the relatively low sensitivity of flagellar movement to changes in external viscosity.     

A microstructurally informed model for the mechanical response of three-dimensional actin networks

We propose a class of microstructurally informed models for the linear elastic mechanical behaviour of cross-linked polymer networks such as the actin cytoskeleton. Salient features of the models include the possibility to represent anisotropic mechanical behaviour resulting from anisotropic filament distributions, and a power law scaling of the mechanical properties with the filament density. Mechanical models within the class are parameterized by seven different constants. We demonstrate a procedure for determining these constants using finite element models of three-dimensional actin networks. Actin filaments and cross-links were modelled as elastic rods, and the networks were constructed at physiological volume fractions and at the scale of an image voxel. We show the performance of the model in estimating the mechanical behaviour of the networks over a wide range of filament densities and degrees of anisotropy.     

Coarse-grained molecular dynamics simulations of a rotating bacterial flagellum

Many types of bacteria propel themselves using elongated structures known as flagella. The bacterial flagellar filament is a relatively simple and well-studied macromolecular assembly, which assumes different helical shapes when rotated in different directions. This polymorphism enables a bacterium to switch between running and tumbling modes; however, the mechanism governing the filament polymorphism is not completely understood. Here we report a study of the bacterial flagellar filament using numerical simulations that employ a novel coarse-grained molecular dynamics method. The simulations reveal the dynamics of a half-micrometer-long flagellum segment on a timescale of tens of microseconds. Depending on the rotation direction, specific modes of filament coiling and arrangement of monomers are observed, in qualitative agreement with experimental observations of flagellar polymorphism. We find that solvent-protein interactions are likely to contribute to the polymorphic helical shapes of the filament.     

A microstructurally informed model for the mechanical response of three-dimensional actin networks

We propose a class of microstructurally informed models for the linear elastic mechanical behaviour of cross-linked polymer networks such as the actin cytoskeleton. Salient features of the models include the possibility to represent anisotropic mechanical behaviour resulting from anisotropic filament distributions, and a power law scaling of the mechanical properties with the filament density. Mechanical models within the class are parameterized by seven different constants. We demonstrate a procedure for determining these constants using finite element models of three-dimensional actin networks. Actin filaments and cross-links were modelled as elastic rods, and the networks were constructed at physiological volume fractions and at the scale of an image voxel. We show the performance of the model in estimating the mechanical behaviour of the networks over a wide range of filament densities and degrees of anisotropy.     

Role of two flagellin genes in Campylobacter motility.

Campylobacter coli VC167 T2 has two flagellin genes, flaA and flaB, which share 91.9% sequence identity. The flaA gene is transcribed from a o-28 promoter, and the flaB gene from a o-54 promoter. Gene replacement mutagenesis techniques were used to generate flaA+ flaB and flaA flaB+ mutants. Both gene products are capable of assembling independently into functional filaments. A flagellar filament composed exclusively of the flaA gene product is indistinguishable in length from that of the wild type and shows a slight reduction in motility. The flagellar filament composed exclusively of the flaB gene product is severely truncated in length and greatly reduced in motility. Thus, while both flagellins are not necessary for motility, both products are required for a fully active flagellar filament. Although the wild-type flagellar filament is a heteropolymer of the flaA and flaB gene products, immunogold electron microscopy suggests that flaB epitopes are poorly surface exposed along the length of the wild-type filament.     

Evidence that subcellular flagellin pools in Caulobacter crescentus are precursors in flagellum assembly

To study the assembly of the Caulobacter crescentus flagellar filament, we have devised a fractionation protocol that separates the cellular flagellin into three compartments: soluble, membrane, and assembled. Radioactive labeling in pulse-chase and pulse-labeling experiments has demonstrated for the first time that both soluble and membrane-associated flagellin pools are precursors in the assembly of the flagellar filament. The results of these experiments also indicate that flagellar filament assembly occurs via the translocation of newly synthesized flagellins from the soluble pool to the membrane pool to the assembled flagellar filaments. It is not possible to conclude whether the soluble flagellin fraction is synthesized cytoplasmically or as a loosely associated membrane intermediate which is released during lysis. It is clear, however, that the soluble and membrane flagellins are in physically and functionally distinct pools. The implications of these findings for the study of protein secretion from cells and the invariant targeting of flagellar proteins to the stalk-distal pole of the dividing cell during flagellum morphogenesis are discussed.     

A family of six flagellin genes contributes to the Caulobacter crescentus flagellar filament

The Caulobacter crescentus flagellar filament is assembled from multiple flagellin proteins that are encoded by six genes. The amino acid sequences of the FljJ and FljL flagellins are divergent from those of the other four flagellins. Since these flagellins are the first to be assembled in the flagellar filament, one or both might have specialized to facilitate the initiation of filament assembly.     

MotD of Sinorhizobium meliloti and related alpha-proteobacteria is the flagellar-hook-length regulator and therefore reassigned as FliK

The flagella of the soil bacterium Sinorhizobium meliloti differ from the enterobacterial paradigm in the complex filament structure and modulation of the flagellar rotary speed. The mode of motility control in S. meliloti has a molecular corollary in two novel periplasmic motility proteins, MotC and MotE, that are present in addition to the ubiquitous MotA/MotB energizing proton channel. A fifth motility gene is located in the mot operon downstream of the motB and motC genes. Its gene product was originally designated MotD, a cytoplasmic motility protein having an unknown function. We report here reassignment of MotD as FliK, the regulator of flagellar hook length. The FliK gene is one of the few flagellar genes not annotated in the contiguous flagellar regulon of S. meliloti. Characteristic for its class, the 475-residue FliK protein contains a conserved, compactly folded Flg hook domain in its carboxy-terminal region. Deletion of fliK leads to formation of prolonged flagellar hooks (polyhooks) with missing filament structures. Extragenic suppressor mutations all mapped in the cytoplasmic region of the transmembrane export protein FlhB and restored assembly of a flagellar filament, and thus motility, in the presence of polyhooks. The structural properties of FliK are consistent with its function as a substrate specificity switch of the flagellar export apparatus for switching from rod/hook-type substrates to filament-type substrates.     

The archaeabacterial flagellar filament: a bacterial propeller with a pilus-like structure

Common prokaryotic motility modes are swimming by means of rotating internal or external flagellar filaments or gliding by means of retracting pili. The archaeabacterial flagellar filament differs significantly from the eubacterial flagellum: (1) Its diameter is 10-14 nm, compared to 18-24 nm for eubacterial flagellar filaments. (2) It has 3.3 subunits/turn of a 1.9 nm pitch left-handed helix compared to 5.5 subunits/turn of a 2.6 nm pitch right-handed helix for plain eubacterial flagellar filaments. (3) The archaeabacterial filament is glycosylated, which is uncommon in eubacterial flagella and is believed to be one of the key elements for stabilizing proteins under extreme conditions. (4) The amino acid composition of archaeabacterial flagellin, although highly conserved within the group, seems unrelated to the highly conserved eubacterial flagellins. On the other hand, the archaeabacterial flagellar filament shares some fundamental properties with type IV pili: (1) The hydrophobic N termini are largely homologous with the oligomerization domain of pilin. (2) The flagellin monomers follow a different mode of transport and assembly. They are synthesized as pre-flagellin and have a cleavable signal peptide, like pre-pilin and unlike eubacterial flagellin. (3) The archaeabacterial flagellin, like pilin, is glycosylated. (4) The filament lacks a central channel, consistent with polymerization occurring at the cell-proximal end. (5) The diameter of type IV pili, 6-9 nm, is closer to that of the archaeabacterial filament, 10-14 nm. A large body of data on the biochemistry and molecular biology of archaeabacterial flagella has accumulated in recent years. However, their structure and symmetry is only beginning to unfold. Here, we review the structure of the archaeabacterial flagellar filament in reference to the structures of type IV pili and eubacterial flagellar filaments, with which it shares structural and functional similarities, correspondingly.     

Variation in bacterial flagellins: from sequence to structure

Bacterial motility relies chiefly on the rotation of a molecular propeller, the flagellar filament, which is constructed from the protein flagellin. Here, flagellin sequence conservation and diversity is examined in the light of the recently determined flagellar filament structure. As expected, the surface-exposed domains are not conserved. However, the sequences that mediate filament assembly show remarkable conservation, which indicates that all bacterial flagellins are likely to pack into filaments in a similar manner. Flagellins provide a striking illustration of the twin evolutionary themes of conservation and variability.     

Polymorphic transition of the flagellar polyhook from Escherichia coli and Salmonella typhimurium

Bacterial flagellar polyhook fibers were reversibly transformed into a set of helical forms depending on pH, ionic strength and temperature. Electron microscopy with formalin fixation and freeze-drying was useful for observing three-dimensional shapes of various polyhook helices and determining their helical handedness. A Cartesian plot of curvature against twist for these polyhook helices gave a sinusoidal curve as in the case of the polymorphic forms of flagellar filament. In the study on the polymorphism of flagellar filaments. Calladine (1976, 1978) and Kamiya et al. (1979) pointed out that such a relation in the polymorphic forms could be derived from the assumption that the subunits on the near-longitudinal (11-start) helical lines should work as elastic fibers (protofilaments) having two distinct states of conformation. In contrast, the observed twist for the polyhook helices is too large to be explained by the same assumption. Instead, we must assume that subunits on the strongly twisted, 16-start helical line should work as the co-operative protofilament.     

Three-dimensional mechanics of eukaryotic flagella.

Equations are derived that account for the contribution of internal structure of cilia and flagella to motion in three dimensions according to a sliding filament model of the motile system. It is shown that for reasonable amounts of bending and twisting, the bending properties of an axoneme can be described by a linear elastic bending resistance, and approximate values for the bending and twisting resistances are computed. Expressions for the shear moments contributed by purely elastic or pinned links between filaments are also derived. It is shown that within the confines of a strict sliding filament model such internal structures cannot by themselves produce twist. Thus planar bending will occur if the internal shear force lies in a plane. Application of an external force, however, will in general produce twisting. Computer simulations of flagellar shape in response to a constant external force applied to the distal end of the axoneme are presented. It is shown that a small amount of twist may arise because of acylindrical bend resistance. Large twists, however, result when the external force is applied to an axoneme with internal shear resistant links.     

Bacterial flagellar microhydrodynamics: Laminar flow over complex flagellar filaments,analog archimedean screws and cylinders,and its perturbations

The flagellar filament, the bacterial organelle of motility, is the smallest rotary propeller known. It consists of 1), a basal body (part of which is the proton driven rotary motor), 2), a hook (universal joint-allowing for off-axial transmission of rotary motion), and 3), a filament (propeller-a long, rigid, supercoiled helical assembly allowing for the conversion of rotary motion into linear thrust). Helically perturbed (so-called "complex") filaments have a coarse surface composed of deep grooves and ridges following the three-start helical lines. These surface structures, reminiscent of a turbine or Archimedean screw, originate from symmetry reduction along the six-start helical lines due to dimerization of the flagellin monomers from which the filament self assembles. Using high-resolution electron microscopy and helical image reconstruction methods, we calculated three-dimensional density maps of the complex filament of Rhizobium lupini H13-3 and determined its surface pattern and boundaries. The helical symmetry of the filament allows viewing it as a stack of identical slices spaced axially and rotated by constant increments. Here we use the closed outlines of these slices to explore, in two dimensions, the hydrodynamic effect of the turbine-like boundaries of the flagellar filament. In particular, we try to determine if, and under what conditions, transitions from laminar to turbulent flow (or perturbations of the laminar flow) may occur on or near the surface of the bacterial propeller. To address these questions, we apply the boundary element method in a manner allowing the handling of convoluted boundaries. We tested the method on several simple, well-characterized cylindrical structures before applying it to real, highly convoluted biological surfaces and to simplified mechanical analogs. Our results indicate that under extreme structural and functional conditions, and at low Reynolds numbers, a deviation from laminar flow might occur on the flagellar surface. These transitions, and the conditions enabling them, may affect flagellar polymorphism and the formation and dispersion of flagellar bundles-factors important in the chemotactic response.     

Amino acids responsible for flagellar shape are distributed in terminal regions of flagellin

The shape of the flagellar filaments of the bacterium Salmonella typhimurium under ordinary conditions is a left-handed helix. In addition to the normal wild-type filament, non-helical (i.e. straight), right-handed helical (early), or circular (semi-coiled and coiled) filaments and filament with small amplitude (fl-type) have been found in mutants or in filaments reconstituted in vitro. We analysed wild-type flagellin and flagellins from 17 flagellar-shape mutants (6 with straight filaments, 6 with curly filaments, 4 with coiled filaments and 1 with fl-type filament) by amino acid sequencing to identify the mutational sites. All mutant flagellins except that of the fl-type filament had single mutations; the fl-type flagellin had two mutations in the molecule. The sites of these mutations were localized in alpha-helical segments of the terminal regions of flagellin. A possible mechanism of the polymorphism of the flagellar filament is discussed.     

Refining the structure of the Halobacterium salinarum flagellar filament using the iterative helical real space reconstruction method: insights into polymorphism

The eubacterial flagellar filament is an external, self-assembling, helical polymer approximately 220 A in diameter constructed from a highly conserved monomer, flagellin, which polymerizes externally at the distal end. The archaeal filament is only approximately 100 A in diameter, assembles at the proximal end and is constructed from different, glycosylated flagellins. Although the phenomenology of swimming is similar to that of eubacteria, the symmetry of the archebacterial filament is entirely different. Here, we extend our previous study on the flagellar coiled filament structure of strain R1M1 of Halobacterium salinarum. We use strain M175 of H.salinarum, which forms poly-flagellar bundles at high yield which, under conditions of relatively low ionic-strength (0.8 M versus 5 M) and low pH ( approximately 2.5 versus approximately 6.8), form straight filaments. We demonstrated previously that a single-particle approach to helical reconstruction has many advantages over conventional Fourier-Bessel methods when dealing with variable helical symmetry and heterogeneity. We show here that when this method is applied to the ordered helical structure of the archebacterial uncoiled flagellar filament, significant extensions in resolution can be obtained readily when compared to applying traditional helical techniques. The filament population can be separated into classes of different morphologies, which may represent polymorphic states. Using cryo-negatively stained images, a resolution of approximately 10-15 A has been achieved. Single alpha-helices can be fit into the reconstruction, supporting the proposed similarity of the structure to that of type IV bacterial pili.     

Elastic properties of bacterial flagellar filaments. II. Determination of the modulus of rigidity

Elongation of a helical bacterial flagellar filament subjected to fluid flow was calculated on the assumption that one end of the filament is firmly attached to a substratum. It was found that the quantity [E(d/2 pi r)2 + 2 mu] could be determined by measuring the elongation at various flow rates, where E is Young's modulus, mu the modulus of rigidity, r the radius of the helix, and d the helical pitch. Experiments were carried out to determine the above quantity for Salmonella flagellar filaments assuming a close-coil form. Because the above quantity is almost equal to 2 mu for a helical form with a large radius/pitch ratio, we were able to determine the modulus of rigidity for this kind of flagellar filament from plots of elongation vs. flow rates. The modulus of rigidity was determined to be about 1 X 10(11) dyn/cm2, i.e., 2 orders of magnitude larger than the previously estimated value.