Model for self-propulsive helical filaments: kink-pair propagation

Bend or Twist? What Plectonemes Reveal about the Mysterious Motility of Spiroplasma

Spiroplasma is a unique, helical bacterium that lacks a cell wall and swims using propagating helix hand inversions. These deformations are likely driven by a set of cytoskeletal filaments, but how remains perplexing. Here, we probe the underlying mechanism using a model where either twist or bend drive spiroplasma's chirality inversions. We show that Spiroplasma should wrap into plectonemes at different values of the length and external viscosity, depending on the mechanism. Then, by experimentally measuring the bending modulus of Spiroplasma and if and when plectonemes form, we show that Spiroplasma's helix hand inversions are likely driven by bending.

Swimming Motility Assays of Spiroplasma

Spiroplasma swim in liquids without the use of the bacterial flagella. This small helical bacterium propels itself by generating kinks that travel down the cell body. The kink translation is unidirectional, from the leading pole to the lagging pole, during cell swimming in viscous environments. This protocol describes a swimming motility assay of Spiroplasma eriocheiris for visualizing kink translations of the absolute handedness of the body helix with optical microscopy.

Two-dimensional localized states in an active phase-field-crystal model

The active phase-field-crystal (active PFC) model provides a simple microscopic mean field description of crystallization in active systems. It combines the PFC model (or conserved Swift-Hohenberg equation) of colloidal crystallization and aspects of the Toner-Tu theory for self-propelled particles. We employ the active PFC model to study the occurrence of localized and periodic active crystals in two spatial dimensions. Due to the activity, crystalline states can undergo a drift instability and start to travel while keeping their spatial structure. Based on linear stability analyses, time simulations, and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of resting and traveling states. We explore, for instance, how the slanted homoclinic snaking of steady localized states found for the passive PFC model is modified by activity. Morphological phase diagrams showing the regions of existence of various solution types are presented merging the results from all the analysis tools employed. We also study how activity influences the crystal structure with transitions from hexagons to rhombic and stripe patterns. This in-depth analysis of a simple PFC model for active crystals and swarm formation provides a clear general understanding of the observed multistability and associated hysteresis effects, and identifies thresholds for qualitative changes in behavior.

Lagrangian mechanics of active systems

We present a multi-scale modeling and simulation framework for low-Reynolds number hydrodynamics of shape-changing immersed objects, e.g., biological microswimmers and active surfaces. The key idea is to consider principal shape changes as generalized coordinates and define conjugate generalized hydrodynamic friction forces. Conveniently, the corresponding generalized friction coefficients can be pre-computed and subsequently reused to solve dynamic equations of motion fast. This framework extends Lagrangian mechanics of dissipative systems to active surfaces and active microswimmers, whose shape dynamics is driven by internal forces. As an application case, we predict in-phase and anti-phase synchronization in pairs of cilia for an experimentally measured cilia beat pattern.

Phase-field-crystal description of active crystallites: Elastic and inelastic collisions

The active Phase-Field-Crystal (aPFC) model combines elements of the Toner-Tu theory for self-propelled particles and the classical Phase-Field-Crystal (PFC) model that describes the transition between liquid and crystalline phases. In the liquid-crystal coexistence region of the PFC model, crystalline clusters exist in the form of localized states that coexist with a homogeneous background. At sufficiently strong activity (related to self-propulsion strength), they start to travel. We employ numerical path continuation and direct time simulations to first investigate the existence regions of different types of localized states in one spatial dimension. The results are summarized in morphological phase diagrams in the parameter plane spanned by activity and mean density. Then we focus on the interaction of traveling localized states, studying their collision behavior. As a result, we distinguish "elastic" and "inelastic" collisions. In the former, localized states recover their properties after a collision, while in the latter, they may completely or partially annihilate, forming resting bound states or various traveling states.

Tree of motility - A proposed history of motility systems in the tree of life

Motility often plays a decisive role in the survival of species. Five systems of motility have been studied in depth: those propelled by bacterial flagella, eukaryotic actin polymerization and the eukaryotic motor proteins myosin, kinesin and dynein. However, many organisms exhibit surprisingly diverse motilities, and advances in genomics, molecular biology and imaging have showed that those motilities have inherently independent mechanisms. This makes defining the breadth of motility nontrivial, because novel motilities may be driven by unknown mechanisms. Here, we classify the known motilities based on the unique classes of movement‐producing protein architectures. Based on this criterion, the current total of independent motility systems stands at 18 types. In this perspective, we discuss these modes of motility relative to the latest phylogenetic Tree of Life and propose a history of motility. During the ~4 billion years since the emergence of life, motility arose in Bacteria with flagella and pili, and in Archaea with archaella. Newer modes of motility became possible in Eukarya with changes to the cell envelope. Presence or absence of a peptidoglycan layer, the acquisition of robust membrane dynamics, the enlargement of cells and environmental opportunities likely provided the context for the (co)evolution of novel types of motility.

Coexistence of Two Chiral Helices Produces Kink Translation in Spiroplasma Swimming

The mechanism underlying Spiroplasma swimming is an enigma. This small bacterium possesses two helical shapes with opposite-handedness at a time, and the boundary between them, called a kink, travels down, possibly accompanying the dual rotations of these physically connected helical structures, without any rotary motors such as flagella. Although the outline of dynamics and structural basis has been proposed, the underlying cause to explain the kink translation is missing. We here demonstrated that the cell morphology of Spiroplasma eriocheiris was fixed at the right-handed helix after motility was stopped by the addition of carbonyl cyanide 3-chlorophenylhydrazone (CCCP), and the preferential state was transformed to the other-handedness by the trigger of light irradiation. This process coupled with the generation and propagation of the artificial kink, presumably without any energy input through biological motors. These findings indicate that the coexistence of two chiral helices is sufficient to propagate the kink and thus to propel the cell body.IMPORTANCE Many swimming bacteria generate a propulsion force by rotating helical filaments like a propeller. However, the nonflagellated bacteria Spiroplasma spp. swim without the use of the appendages. The tiny wall-less bacteria possess two chiral helices at a time, and the boundary called a kink travels down, possibly accompanying the dual rotations of the helices. To solve this enigma, we developed an assay to determine the handedness of the body helices at the single-wind level, and demonstrated that the coexistence of body helices triggers the translation of the kink and that the cell body moves by the resultant cell bend propagation. This finding provides us a totally new aspect of bacterial motility, where the body functions as a transformable screw to propel itself forward.

Dynamics of a Protein Chain Motor Driving Helical Bacteria under Stress

The wall-less, helical bacterial genus Spiroplasma has a unique propulsion system; it is not driven by propeller-like flagella but by a membrane-bound, cytoplasmic, linear motor that consists of a contractile chain of identical proteins spanning the entire cell length. By a coordinated spread of conformational changes of the proteins, kinks propagate in pairs along the cell body. However, the mechanisms for the initiation or delay of kinks and their coordinated spread remain unclear. Here, we show how we manipulate the initiation of kinks, their propagation velocities, and the time between two kinks for a single cell trapped in an optical line potential. By interferometric three-dimensional shape tracking, we measured the cells' deformations in response to various external stress situations. We observed a significant dependency of force generation on the cells' local ligand concentrations (likely ATP) and ligand hydrolysis, which we altered in different ways. We developed a mechanistic, mathematical model based on Kramer's rates, describing the subsequent cooperative and conformational switching of the chain's proteins. The model reproduces our experimental observations and can explain deformation characteristics even when the motor is driven to its extreme. Nature has invented a set of minimalistic mechanical driving concepts. To understand or even rebuild them, it is essential to reveal the molecular mechanisms of such protein chain motors, which need only two components-coupled proteins and ligands-to function.

Bending, curling, and twisting in polymeric bilayers

Polyolefin thermoplastic elastomer (POE) bilayers of varying length (L) to width (W) ratio are formed through traditional polymer processing. Each layer is completely isotropic but the bilayers have an elastic recovery mismatch such that when stretched, one layer recovers to a different extent than the other. Upon stretching bilayers from low to moderate strains and releasing the bilayer bends (curvature, κ, κ < 1/L). Stretching to moderate strain and releasing results in bilayer curling (1/L ≤ κ < 1/W). Finally, stretching to high strains and releasing such that κ ≥ 1/W results in twisting into a helix for L/W > 2π bilayers and rolling into a cylinder for L/W < 2π bilayers. Varying W can change the helical pitch, lp, of twisted bilayers. The twisted bilayer helical rise angle varies between θ = 60 and 90°. Metastability, i.e., bilayers that show a combination of the two behaviors, is observed at long absolute L or short absolute W. The bilayers are modeled using Euler-Bernoulli beam theory to show that the curvature can be predicted using the elastic recovery of the layer that recovers more.

Large variability in the motility of spiroplasmas in media of different viscosities

Spiroplasmas are bacteria that do not possess flagella and their motility is linked to kink propagation coupled to changes in the cell body helicity. While the motility of bacteria with flagellar motion has been studied extensively, less work has been devoted to the motility of spiroplasmas. We first show that the motility of such bacteria has large variability from individual to individual as well as large fluctuations in time. The Brownian motion of such bacteria both in orientation and translation is also highlighted. We propose a simple model to disentangle the different components of this motility by examining trajectories of single bacteria in different viscosity solvents. The mean velocity of the bacteria turns out to depend on the viscosity of the medium as it increases with viscosity. Further, the temporal fluctuations of the bacteria motility turn out to be very strong with a direct link to tumbling events particular to this bacteria.

The swimming of a deforming helix

Many microorganisms and artificial microswimmers use helical appendages in order to generate locomotion. Though often rotated so as to produce thrust, some species of bacteria such Spiroplasma, Rhodobacter sphaeroides and Spirochetes induce movement by deforming a helical-shaped body. Recently, artificial devices have been created which also generate motion by deforming their helical body in a non-reciprocal way (A. Mourran et al. Adv. Mater. 29, 1604825, 2017). Inspired by these systems, we investigate the transport of a deforming helix within a viscous fluid. Specifically, we consider a swimmer that maintains a helical centreline and a single handedness while changing its helix radius, pitch and wavelength uniformly across the body. We first discuss how a deforming helix can create a non-reciprocal translational and rotational swimming stroke and identify its principle direction of motion. We then determine the leading-order physics for helices with small helix radius before considering the general behaviour for different configuration parameters and how these swimmers can be optimised. Finally, we explore how the presence of walls, gravity, and defects in the centreline allow the helical device to break symmetries, increase its speed, and generate transport in directions not available to helices in bulk fluids.

Resting and traveling localized states in an active phase-field-crystal model

The conserved Swift-Hohenberg equation (or phase-field-crystal [PFC] model) provides a simple microscopic description of the thermodynamic transition between fluid and crystalline states. Combining it with elements of the Toner-Tu theory for self-propelled particles, Menzel and Löwen [Phys. Rev. Lett. 110, 055702 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.055702] obtained a model for crystallization (swarm formation) in active systems. Here, we study the occurrence of resting and traveling localized states, i.e., crystalline clusters, within the resulting active PFC model. Based on linear stability analyses and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of periodic and localized, resting and traveling states in a one-dimensional active PFC model. This allows us, for instance, to explore how the slanted homoclinic snaking of steady localized states found for the passive PFC model is amended by activity. A particular focus lies on the onset of motion, where we show that it occurs either through a drift-pitchfork or a drift-transcritical bifurcation. A corresponding general analytical criterion is derived.

Cross-kymography analysis to simultaneously quantify the function and morphology of the archaellum

In many microorganisms helical structures are important for motility, e.g., bacterial flagella and kink propagation in Spiroplasma eriocheiris. Motile archaea also form a helical-shaped filament called the ‘archaellum’ that is functionally equivalent to the bacterial flagellum, but structurally resembles type IV pili. The archaellum motor consists of 6–8 proteins called fla accessory genes, and the filament assembly is driven by ATP hydrolysis at catalytic sites in FlaI. Remarkably, previous research using a dark-field microscopy showed that right-handed filaments propelled archaeal cells forwards or backwards by clockwise or counterclockwise rotation, respectively. However, the shape and rotational rate of the archaellum during swimming remained unclear, due to the low signal and lack of temporal resolution. Additionally, the structure and the motor properties of the archaellum and bacterial flagellum have not been precisely determined during swimming because they move freely in three-dimensional space. Recently, we developed an advanced method called “cross-kymography analysis”, which enables us to be a long-term observation and simultaneously quantify the function and morphology of helical structures using a total internal reflection fluorescence microscope. In this review, we introduce the basic idea of this analysis, and summarize the latest information in structural and functional characterization of the archaellum motor.

Reconfigurable Microscale Frameworks from Concatenated Helices with Controlled Chirality

The utility of helical structures in driving motion of microorganisms and plants has inspired efforts to develop synthetic stimuli-responsive helical architectures for self-motile and shape-morphing systems. While several approaches to responsive helices based on hydrogels and liquid crystalline polymers have been reported, they have so far been limited to macroscopic (cm scale) dimensions, and have not been applied to concatenated helices with more than two segments. Here, a robust method for microfabrication of helices inspired by Bauhinia seedpods, based on trilayer samples consisting of rigid plastic stripes sandwiching a swellable temperature-responsive hydrogel, is reported and the formation of responsive shape-controlled frameworks from concatenated multiple helices (multihelices) with controlled chirality is demonstrated. The block angle at each helical junction is controlled by the change in stripe direction, while the torsion angle defined by each segment of three helices is prescribed by the net twist of the middle segment, providing simple geometric design rules for the fabrication of complex 3D structures. This work opens new directions in programming 3D shapes by providing new insight into helical segments as building blocks, with potential applicability to the fabrication of scaffolds for cell culture, reconfigurable microfluidic channels, and microswimmers.

Transformation of crustacean pathogenic bacterium Spiroplasma eriocheiris and expression of yellow fluorescent protein

Spiroplasma eriocheiris, the cause of crab trembling disease, is a wall-less bacterium, related to Mycoplasmas, measuring 2.0-10.0 μm long. It features a helical cell shape and a unique swimming mechanism that does not use flagella; instead, it moves by switching the cell helicity at a kink traveling from the front to the tail. S. eriocheiris seems to use a novel chemotactic system that is based on the frequency of reversal swimming behaviors rather than the conventional two-component system, which is generally essential for bacterial chemotaxis. To identify the genes involved in these novel mechanisms, we developed a transformation system by using oriC plasmid harboring the tetracycline resistant gene, tetM, which is under the control of a strong promoter for an abundant protein, elongation factor-Tu. The transformation efficiency achieved was 1.6 × 10^-5 colony forming unit (CFU) for 1 μg DNA, enabling the expression of the enhanced yellow fluorescent protein (EYFP).

Chemotaxis without Conventional Two-Component System, Based on Cell Polarity and Aerobic Conditions in Helicity-Switching Swimming of Spiroplasma eriocheiris

Spiroplasma eriocheiris is a pathogen that causes mass mortality in Chinese mitten crab, Eriocheir sinensis. S. eriocheiris causes tremor disease and infects almost all of the artificial breeding crustaceans, resulting in disastrous effects on the aquaculture economy in China. S. eriocheiris is a wall-less helical bacterium, measuring 2.0 to 10.0 μm long, and can swim up to 5 μm per second in a viscous medium without flagella by switching the cell helicity at a kink traveling from the front to the tail. In this study, we showed that S. eriocheiris performs chemotaxis without the conventional two-component system, a system commonly found in bacterial chemotaxis. The chemotaxis of S. eriocheiris was observed more clearly when the cells were cultivated under anaerobic conditions. The cells were polarized as evidenced by a tip structure, swimming in the direction of the tip, and were shown to reverse their swimming direction in response to attractants. Triton X-100 treatment revealed the internal structure, a dumbbell-shaped core in the tip that is connected by a flat ribbon, which traces the shortest line in the helical cell shape from the tip to the other pole. Sixteen proteins were identified as the components of the internal structure by mass spectrometry, including Fibril protein and four types of MreB proteins.

Structural mechanics and helical geometry of thin elastic composites

Helices are ubiquitous in nature, and helical shape transition is often observed in residually stressed bodies, such as composites, wherein materials with different mechanical properties are glued firmly together to form a whole body. Inspired by a variety of biological examples, the basic physical mechanism responsible for the emergence of twisting and bending in such thin composite structures has been extensively studied. Here, we propose a simplified analytical model wherein a slender membrane tube undergoes a helical transition driven by the contraction of an elastic ribbon bound to the membrane surface. We analytically predict the curvature and twist of an emergent helix as functions of differential strains and elastic moduli, which are confirmed by our numerical simulations. Our results may help understand shapes observed in different biological systems, such as spiral bacteria, and could be applied to novel designs of soft machines and robots.

Dynamics of a bacterial flagellum under reverse rotation

To initiate tumbling of an E. coli, one of the helical flagella reverses its sense of rotation. It then transforms from its normal form first to the transient semicoiled state and subsequently to the curly-I state. The dynamics of polymorphism is effectively modeled by describing flagellar elasticity through an extended Kirchhoff free energy. However, the complete landscape of the free energy remains undetermined because the ground state energies of the polymorphic forms are not known. We investigate how variations in these ground state energies affect the dynamics of a reversely rotated flagellum of a swimming bacterium. We find that the flagellum exhibits a number of distinct dynamical states and comprehensively summarize them in a state diagram. As a result, we conclude that tuning the landscape of the extended Kirchhoff free energy alone cannot generate the intermediate full-length semicoiled state. However, our model suggests an ad hoc method to realize the sequence of polymorphic states as observed for a real bacterium. Since the elastic properties of bacterial flagella are similar, our findings can easily be extended to other peritrichous bacteria.

Getting drowned in a swirl: Deformable bead-spring model microswimmers in external flow fields

Deformability is a central feature of many types of microswimmers, e.g., for artificially generated self-propelled droplets. Here, we analyze deformable bead-spring microswimmers in an externally imposed solvent flow field as simple theoretical model systems. We focus on their behavior in a circular swirl flow in two spatial dimensions. Linear (straight) two-bead swimmers are found to circle around the swirl with a slight drift to the outside with increasing activity. In contrast to that, we observe for triangular three-bead or squarelike four-bead swimmers a tendency of being drawn into the swirl and finally getting drowned, although a radial inward component is absent in the flow field. During one cycle around the swirl, the self-propulsion direction of an active triangular or squarelike swimmer remains almost constant, while their orbits become deformed exhibiting an "egglike" shape. Over time, the swirl flow induces slight net rotations of these swimmer types, which leads to net rotations of the egg-shaped orbits. Interestingly, in certain cases, the orbital rotation changes sense when the swimmer approaches the flow singularity. Our predictions can be verified in real-space experiments on artificial microswimmers.

[Morphology and motility of the spirochetes]

Spirochetes have flagella within the cell body and swim by wriggling the spiral cell body. Besides they have been known to be critical agents causing various infectious diseases, their eccentric appearances and motilities have been attracting many scientists in a wide variety of fields other than bacteriologists. Unlike externally flagellated bacteria that swim by using flagella as a screw propeller, spirochetes progress in a liquid by changing their cell shapes. To understand the unique motion mechanism of spirochetes, many experiments and theoretical studies are being carried out. In this review, I will summarize morphological and motile properties of various species of spirochete, such as Borrelia, Treponema and Brachyspira. I will also expound on the motion mechanism of Leptospira with our latest results obtained by high-resolution optical photometry.

Confotronic dynamics of tubular filaments

Tubular lattices are ubiquitous in nature and technology. Microtubules and nanotubes of all kinds act as important pillars of biological cells and the man-made nano-world. We show that when prestress is introduced in such structures, localized conformational quasiparticles emerge and govern the collective shape dynamics of the lattice. When coupled via cooperative interactions these quasiparticles form larger-scale quasipolymer superstructures exhibiting collective dynamic modes and giving rise to a hallmark behavior radically different from semiflexible beams.

Active crystals and their stability

A recently introduced active phase field crystal model describes the formation of ordered resting and traveling crystals in systems of self-propelled particles. Increasing the active drive, a resting crystal can be forced to perform collectively ordered migration as a single traveling object. We demonstrate here that these ordered migrating structures are linearly stable. In other words, during migration, the single-crystalline texture together with the globally ordered collective motion is preserved even on large length scales. Furthermore, we consider self-propelled particles on a substrate that are surrounded by a thin fluid film. We find that in this case the resulting hydrodynamic interactions can destabilize the order.

Bacterial microsystems and microrobots

Microorganisms and specifically motile bacteria have been recently added to the list of micro-actuators typically considered for the implementation of microsystems and microrobots. Such trend has been motivated by the fact these microorganisms are self-powered actuators with overall sizes at the lower end of the micrometer range and which have proven to be extremely effective in low Reynolds number hydrodynamic regime of usually less than 10(-2). Furthermore, the various sensors or taxes in bacteria influencing their movements can also be exploited to perform tasks that were previously considered only for futuristic artificial microrobots. Bacterial implementations and related issues are not only reviewed, but this paper also proposes many techniques and approaches that can be considered as building blocks for the implementations of more sophisticated microsystems and microrobots.

Propulsion efficiency of a dynamic self-assembled helical ribbon

We study the dynamic self-assembly and propulsion of a ribbon formed from paramagnetic colloids in a dynamic magnetic field. The sedimented ribbon assembles due to time averaged dipolar interactions between the beads. The time dependence of the dipolar interactions together with hydrodynamic interactions cause a twisted ribbon conformation. Domain walls of high twist connect domains of nearly constant orientation and negligible twist and travel through the ribbon. The particular form of the domain walls can be controlled via the frequency and the eccentricity of the modulation. The flux of twist walls-a true ribbon property absent in slender bodies-provides the thrust onto the surrounding liquid that propels this biomimetic flagellum into the opposite direction. The propulsion efficiency increases with frequency and ceases abruptly at a critical frequency where the conformation changes discontinuously to a flat standing ribbon conformation.

Surface traction and the dynamics of elastic rods at low Reynolds number

Molecular and cell biological processes often use proteins and structures that are significantly longer in one dimension than they are in the other two, for example, DNA, actin, and bacterial flagella. The dynamics of these structures are the consequence of the balance between the elastic forces from the structure itself and viscous forces from the surrounding fluid. Typically, the motion of these filamentary objects is described using variations of the Kirchhoff rod equations with resistive forces from the fluid treated as body forces acting on the centerline. In reality, though, these forces are applied to the surface of the filament; however, the standard derivation of the Kirchhoff equations ignores surface traction stresses. Here, we rederive the Kirchhoff rod equations in the presence of resistive traction stresses and determine the conditions under which treating the drag forces as body forces is reasonable. We show that in most biologically relevant cases the standard implementation of resistive forces into the Kirchhoff rod equations is applicable; however, we note one particular biological system where the Kirchhoff rod formalism may not apply.

Motor-driven bacterial flagella and buckling instabilities

Many types of bacteria swim by rotating a bundle of helical filaments also called flagella. Each filament is driven by a rotary motor and a very flexible hook transmits the motor torque to the filament. We model it by discretizing Kirchhoff's elastic-rod theory and develop a coarse-grained approach for driving the helical filament by a motor torque. A rotating flagellum generates a thrust force, which pushes the cell body forward and which increases with the motor torque. We fix the rotating flagellum in space and show that it buckles under the thrust force at a critical motor torque. Buckling becomes visible as a supercritical Hopf bifurcation in the thrust force. A second buckling transition occurs at an even higher motor torque. We attach the flagellum to a spherical cell body and also observe the first buckling transition during locomotion. By changing the size of the cell body, we vary the necessary thrust force and thereby obtain a characteristic relation between the critical thrust force and motor torque. We present a elaborate analytical model for the buckling transition based on a helical rod which quantitatively reproduces the critical force-torque relation. Real values for motor torque, cell body size, and the geometry of the helical filament suggest that buckling should occur in single bacterial flagella. We also find that the orientation of pulling flagella along the driving torque is not stable and comment on the biological relevance for marine bacteria.

Collective motion of binary self-propelled particle mixtures

In this study, we investigate the phenomenon of collective motion in binary mixtures of self-propelled particles. More precisely, we consider two particle species, each of which consisting of pointlike objects that propel with a velocity of constant magnitude. Within each species, the particles try to achieve polar alignment of their velocity vectors, whereas we analyze the cases of preferred polar, antiparallel, as well as perpendicular alignment between particles of different species. Our focus is on the effect that the interplay between the two species has on the threshold densities for the onset of collective motion and on the nature of the solutions above onset. For this purpose, we start from suitable Langevin equations in the particle picture, from which we derive mean field equations of the Fokker-Planck type and finally macroscopic continuum field equations. We perform particle simulations of the Langevin equations and linear stability analyses of the Fokker-Planck and macroscopic continuum equations, and we numerically solve the Fokker-Planck equations. Both spatially homogeneous and inhomogeneous solutions are investigated, where the latter correspond to stripelike flocks of collectively moving particles. In general, the interaction between the two species reduces the threshold density for the onset of collective motion of each species. However, this interaction also reduces the spatial organization in the stripelike flocks. The case that shows the most interesting behavior is the one of preferred perpendicular alignment between different species. There a competition between polar and truly nematic orientational ordering of the velocity vectors takes place within each particle species. Finally, depending on the alignment rule for particles of different species and within certain ranges of particle densities, identical and inverted spatial density profiles can be found for the two particle species. The system under investigation is confined to two spatial dimensions.

Force-extension curves of bacterial flagella

Bacterial flagella assume different helical shapes during the tumbling phase of a bacterium but also in response to varying environmental conditions. Force-extension measurements by Darnton and Berg explicitly demonstrate a transformation from the coiled to the normal helical state (N.C. Darnton, H.C. Berg, Biophys. J. 92, 2230 (2007)). We here develop an elastic model for the flagellum based on Kirchhoff's theory of an elastic rod that describes such a polymorphic transformation and use resistive force theory to couple the flagellum to the aqueous environment. We present Brownian-dynamics simulations that quantitatively reproduce the force-extension curves and study how the ratio Γ of torsional to bending rigidity and the extensional rate influence the response of the flagellum. An upper bound for Γ is given. Using clamped flagella, we show in an adiabatic approximation that the mean extension, where a local coiled-to-normal transition occurs first, depends on the logarithm of the extensional rate.

Deformable self-propelled particles with a global coupling

We have proposed a model of deformable self-propelled particles in which the time-evolution equations are given in terms of the center-of-mass velocity and a nematic order parameter representing the motion-induced deformation [T. Ohta and T. Ohkuma, Phys. Rev. Lett. 102, 154101 (2009)]. We investigate its many-body problem applying a global orientational coupling. Depending on the strength of the interaction, the self-propelled particles exhibit various kinds of collective dynamics and chaotic behavior: a ballistic procession state, a scattered state, a coherently phase synchronized state, two types of in-phase synchronized state, and an anomalously diffusive state. The phase reduction method for the weak coupling regime reveals the bifurcations between the secular collective motions. The phase boundary among the chaos regime and the synchronized regimes is determined by the linear stability analysis of the synchronized states.

Enhanced low-Reynolds-number propulsion in heterogeneous viscous environments

It has been known for some time that some microorganisms can swim faster in high-viscosity gel-forming polymer solutions. These gel-like media come to mimic highly viscous heterogeneous environment that these microorganisms encounter in-vivo. The qualitative explanation of this phenomena first offered by Berg and Turner [Nature (London) 278, 349 (1979)], suggests that propulsion enhancement is a result of flagellum pushing on quasi-rigid loose polymer network formed in some polymer solutions. Inspired by these observations, inertia-less propulsion in a heterogeneous viscous medium composed of sparse array of stationary obstacles embedded into a incompressible Newtonian liquid is considered. It is demonstrated that for prescribed propulsion gaits, including propagating surface distortions and rotating helical filament, the propulsion speed is enhanced when compared to swimming in purely viscous solvent. It is also shown that the locomotion in heterogenous viscous media is characterized by improved hydrodynamic efficiency. The results of the rigorous numerical simulation of the rotating helical filament propelled through a random sparse array of stationary obstructions are in close agreement with predictions of the proposed resistive force theory based on effective media approximation.

Propulsion of African trypanosomes is driven by bihelical waves with alternating chirality separated by kinks

Trypanosoma brucei, a parasitic protist with a single flagellum, is the causative agent of African sleeping sickness. Propulsion of T. brucei was long believed to be by a drill-like, helical motion. Using millisecond differential interference-contrast microscopy and analyzing image sequences of cultured procyclic-form and bloodstream-form parasites, as well as bloodstream-form cells in infected mouse blood, we find that, instead, motility of T. brucei is by the propagation of kinks, separating left-handed and right-handed helical waves. Kink-driven motility, previously encountered in prokaryotes, permits T. brucei a helical propagation mechanism while avoiding the large viscous drag associated with a net rotation of the broad end of its tapering body. Our study demonstrates that millisecond differential interference-contrast microscopy can be a useful tool for uncovering important short-time features of microorganism locomotion.

Hydrodynamics of helical-shaped bacterial motility

To reveal the underlying hydrodynamic mechanism for the directed propulsion of the bacterium Spiroplasma, we formulate a coarse-grained elastic polymer model with domains of alternating helicities along the contour. Using hydrodynamic simulations and analytic arguments, we show that the propagation of helical domain walls leads to the directed propulsion of the cell body opposite to the domain-wall traveling direction. Several key features of Spiroplasma motility are reproduced by our model. We in particular show that the helical pitch angle observed for Spiroplasma meliferum, psi=35 degrees , is optimized for maximal swimming speed and energy-conversion efficiency. Our analytic theory based on the slender-body hydrodynamic approximation agrees very well with our numerical data demonstrating how the chirality switch propagating along the helical cell body is converted to a translational thrust for the cell body itself. We in detail consider thermal effects on the propulsion efficiency in the form of orientational fluctuations and conformational fluctuations of the helix shape. The body length dependence of the cell motility is studied numerically and compared to our approximate analytic theory. For fixed pitch angle psi=35 degrees , the swimming speed is maximized at a ratio of cell-body length to domain length of about 2-3, which are typical values for real cells. We also propose simple analytic arguments for an enhancement of the swimming velocity with increasing solution viscosity by taking into account the effects of transient confinement of a helical cell body in a polymeric meshwork. Comparison with a generalized theory for the swimming speed of flagellated bacteria in polymeric meshworks shows that the presence of a finite-sized bacterial head gives rise to a maximal swimming speed at a finite solution viscosity, whereas in the absence of a head the swimming speed monotonically increases with increasing viscosity.

The eukaryotic cell originated in the integration and redistribution of hyperstructures from communities of prokaryotic cells based on molecular complementarity

In the "ecosystems-first" approach to the origins of life, networks of non-covalent assemblies of molecules (composomes), rather than individual protocells, evolved under the constraints of molecular complementarity. Composomes evolved into the hyperstructures of modern bacteria. We extend the ecosystems-first approach to explain the origin of eukaryotic cells through the integration of mixed populations of bacteria. We suggest that mutualism and symbiosis resulted in cellular mergers entailing the loss of redundant hyperstructures, the uncoupling of transcription and translation, and the emergence of introns and multiple chromosomes. Molecular complementarity also facilitated integration of bacterial hyperstructures to perform cytoskeletal and movement functions.

Kinematics of the swimming of Spiroplasma

Spiroplasma swimming is studied with a simple model based on resistive-force theory. Specifically, we consider a bacterium shaped in the form of a helix that propagates traveling-wave distortions which flip the handedness of the helical cell body. We treat cell length, pitch angle, kink velocity, and distance between kinks as parameters and calculate the swimming velocity that arises due to the distortions. We find that, for a fixed pitch angle, scaling collapses the swimming velocity (and the swimming efficiency) to a universal curve that depends only on the ratio of the distance between kinks to the cell length. Simultaneously optimizing the swimming efficiency with respect to interkink length and pitch angle, we find that the optimal pitch angle is 35.5 degrees and the optimal interkink length ratio is 0.338, values in good agreement with experimental observations.

Simulation of a model microswimmer

We discuss the modelling of a microswimmer that operates in a 'squirmer' mode, by means of stochastic rotation dynamics. The squirmer that we model can easily be tuned between a 'pusher' and a 'puller'. We examine the flows produced by the squirmer and find that there is good agreement between both the predicted and simulated velocities of locomotion and the resulting flow field.

A semiflexible polymer ring acting as a nano-propeller

The dynamics of a rotating elastic nano-ring driven in a viscous fluid by an externally applied torque about a specific axis is studied using elasto-hydrodynamic simulations. We show that a helical deformation of the ring filament is excited, and that this leads to directed propulsion which is independent of the direction of rotation. It is found that the propulsive force and efficiency initially increase as the torque is increased, and then decrease discontinuously at a buckling transition at a critical torque. This unique propulsive behavior at the shape transition arises due to its specific geometry, i.e., circularity of an elastic filament. The implications of the behavior for artificial microscopic devices are discussed.

Swimming in circles: motion of bacteria near solid boundaries

Near a solid boundary, Escherichia coli swims in clockwise circular motion. We provide a hydrodynamic model for this behavior. We show that circular trajectories are natural consequences of force-free and torque-free swimming and the hydrodynamic interactions with the boundary, which also leads to a hydrodynamic trapping of the cells close to the surface. We compare the results of the model with experimental data and obtain reasonable agreement. In particular, the radius of curvature of the trajectory is observed to increase with the length of the bacterium body.