Kinematics of the swimming of Spiroplasma

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.

Cytoskeletal components can turn wall-less spherical bacteria into kinking helices

Bacterial cell shape is generally determined through an interplay between the peptidoglycan cell wall and cytoplasmic filaments made of polymerized MreB. Indeed, some bacteria (e.g., Mycoplasma) that lack both a cell wall and mreB genes consist of non-motile cells that are spherical or pleomorphic. However, other members of the same class Mollicutes (e.g., Spiroplasma, also lacking a cell wall) display a helical cell shape and kink-based motility, which is thought to rely on the presence of five MreB isoforms and a specific fibril protein. Here, we show that heterologous expression of Spiroplasma fibril and MreB proteins confers helical shape and kinking ability to Mycoplasma capricolum cells. Isoform MreB5 is sufficient to confer helicity and kink propagation to mycoplasma cells. Cryoelectron microscopy confirms the association of cytoplasmic MreB filaments with the plasma membrane, suggesting a direct effect on membrane curvature. However, in our experiments, the heterologous expression of MreBs and fibril did not result in efficient motility in culture broth, indicating that additional, unknown Spiroplasma components are required for swimming.

Helical micro-swimmer: hierarchical tail design and propulsive motility

Helical micro-swimmers have markedly extended the reach of human beings in numerous fields, ranging from in vitro tasks in lab-on-a-chip to in vivo applications for minimally invasive medicine. The previous studies on the propulsive motility optimization of the micro-swimmers mainly focused on the distinct actuation principles (e.g., chemically powered, magnetic- or ultrasound energy-driven) and paid little attention to the structural design of these swimming machines themselves. The improvements of the structures can assist the externally powered motors in providing propulsion in a tiny scale and satisfy the agile locomotion demands. This paper presents the design, mechanics modeling and available experiments of a novel type of hierarchical helical swimming robot that significantly enhances the motility of the helix-based swimmers. Validated by the resistive force theory, our numerical model can well analyze the mechanical properties with a variety of geometric parameters. The motion performance of the hierarchical and conventional helical structures in low Reynolds regimes is presented, highlighting the advantages of hierarchical swimmers over the existing typical swimmers. In addition, the stability and resilience of the hierarchical swimmers can be maintained at a decent level. Moreover, the variable forward velocity resulting from the combined hierarchical structures is investigated here, which can thereby serve as a reliable design strategy. The proposed hierarchical helical design enables enticing opportunities for various device systems of medical robots and bio-integrated electronics.

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.

Propulsion by stiff elastic filaments in viscous fluids

Flexible filaments moving in viscous fluids are ubiquitous in the natural microscopic world. For example, the swimming of bacteria and spermatozoa as well as important physiological functions at organ level, such as the cilia-induced motion of mucus in the lungs, or individual cell level, such as actin filaments or microtubules, all employ flexible filaments moving in viscous fluids. As a result of fluid-structure interactions, a variety of nonlinear phenomena may arise in the dynamics of such moving flexible filaments. In this paper we derive the mathematical tools required to study filament-driven propulsion in the asymptotic limit of stiff filaments. Motion in the rigid limit leads to hydrodynamic loads which deform the filament and impact the filament propulsion. We first derive the general mathematical formulation and then apply it to the case of a helical filament, a situation relevant for the swimming of flagellated bacteria and for the transport of artificial, magnetically actuated motors. We find that, as a result of flexibility, the helical filament is either stretched or compressed (conforming previous studies) and additionally its axis also bends, a result which we interpret physically. We then explore and interpret the dependence of the perturbed propulsion speed due to the deformation on the relevant dimensionless dynamic and geometric parameters.

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.

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.

Autophoretic motion in three dimensions

Janus particles with the ability to move phoretically in self-generated chemical concentration gradients are model systems for active matter. Their motion typically consists of straight paths with rotational diffusion being the dominant reorientation mechanism. In this paper, we show theoretically that by a suitable surface coverage of both activity and mobility, translational and rotational motion can be induced arbitrarily in three dimensions. The resulting trajectories are in general helical, and their pitch and radius can be controlled by adjusting the angle between the translational and angular velocity. Building on the classical mathematical framework for axisymmetric self-phoretic motion under fixed-flux chemical boundary conditions, we first show how to calculate the most general three-dimensional motion for an arbitrary surface coverage of a spherical particle. After illustrating our results on surface distributions, we next introduce a simple intuitive patch model to serve as a guide for designing arbitrary phoretic spheres.

Reconstructing the gradient source position from steady-state fluxes to small receptors

Recovering the position of a source from the fluxes of diffusing particles through small receptors allows a biological cell to determine its relative position, spatial localization and guide it to a final target. However, how a source can be recovered from point fluxes remains unclear. Using the Narrow Escape approach for an open domain, we compute the diffusion fluxes of Brownian particles generated by a steady-state gradient from a single source through small holes distributed on a surface in two dimensions. We find that the location of a source can be recovered when there are at least 3 receptors and the source is positioned no further than 10 cell radii away, but this condition is not necessary in a narrow strip. The present approach provides a computational basis for the first step of direction sensing of a gradient at a single cell level.

Empirical resistive-force theory for slender biological filaments in shear-thinning fluids

Many cells exploit the bending or rotation of flagellar filaments in order to self-propel in viscous fluids. While appropriate theoretical modeling is available to capture flagella locomotion in simple, Newtonian fluids, formidable computations are required to address theoretically their locomotion in complex, nonlinear fluids, e.g., mucus. Based on experimental measurements for the motion of rigid rods in non-Newtonian fluids and on the classical Carreau fluid model, we propose empirical extensions of the classical Newtonian resistive-force theory to model the waving of slender filaments in non-Newtonian fluids. By assuming the flow near the flagellum to be locally Newtonian, we propose a self-consistent way to estimate the typical shear rate in the fluid, which we then use to construct correction factors to the Newtonian local drag coefficients. The resulting non-Newtonian resistive-force theory, while empirical, is consistent with the Newtonian limit, and with the experiments. We then use our models to address waving locomotion in non-Newtonian fluids and show that the resulting swimming speeds are systematically lowered, a result which we are able to capture asymptotically and to interpret physically. An application of the models to recent experimental results on the locomotion of Caenorhabditis elegans in polymeric solutions shows reasonable agreement and thus captures the main physics of swimming in shear-thinning fluids.

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.

Chirality-specific lift forces of helix under shear flows: Helix perpendicular to shear plane

Chiral objects in shear flow experience a chirality-specific lift force. Shear flows past helices in a low Reynolds number regime were studied using slender-body theory. The chirality-specific lift forces in the vorticity direction experienced by helices are dominated by a set of helix geometry parameters: helix radius, pitch length, number of turns, and helix phase angle. Its analytical formula is given. The chirality-specific forces are the physical reasons for the chiral separation of helices in shear flow. Our results are well supported by the latest experimental observations.

Helical and rod-shaped bacteria swim in helical trajectories with little additional propulsion from helical shape

It has frequently been hypothesized that the helical body shapes of flagellated bacteria may yield some advantage in swimming ability. In particular, the helical-shaped pathogen Helicobacter pylori is often claimed to swim like a corkscrew through its harsh gastric habitat, but there has been no direct confirmation or quantification of such claims. Using fast time-resolution and high-magnification two-dimensional (2D) phase-contrast microscopy to simultaneously image and track individual bacteria in bacterial broth as well as mucin solutions, we show that both helical and rod-shaped H. pylori rotated as they swam, producing a helical trajectory. Cell shape analysis enabled us to determine shape as well as the rotational and translational speed for both forward and reverse motions, thereby inferring flagellar kinematics. Using the method of regularized Stokeslets, we directly compare observed speeds and trajectories to numerical calculations for both helical and rod-shaped bacteria in mucin and broth to validate the numerical model. Although experimental observations are limited to select cases, the model allows quantification of the effects of body helicity, length, and diameter. We find that due to relatively slow body rotation rates, the helical shape makes at most a 15% contribution to propulsive thrust. The effect of body shape on swimming speeds is instead dominated by variations in translational drag required to move the cell body. Because helical cells are one of the strongest candidates for propulsion arising from the cell body, our results imply that quite generally, swimming speeds of flagellated bacteria can only be increased a little by body propulsion.

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.

Junctional angle of a bihanded helix

Helical filaments having sections of reversed chirality are common phenomena in the biological realm. The apparent angle between the two sections of opposite handedness provides information about the geometry and elasticity of the junctional region. In this paper, the governing differential equations for the local helical axis are developed, and asymptotic solutions of the governing equations are solved by perturbation theory. The asymptotic solutions are compared with the corresponding numerical solutions, and the relative error at second order is found to be less than 1.5% over a range of biologically relevant curvature and torsion values from 0 to 1/2 in dimensionless units.

Bending, twisting and beating trunk robot bioinspired from the '3 + 0' axoneme

The axoneme is the skeleton and motor axis of flagella and cilia in eukaryotic organisms. Basically it consists of a series of longitudinal fibers (outer doublets of microtubules) that design a cylinder and whose sliding, due to the coordinated activities of dedicated molecular motors (the dynein arms), is converted into a bending because outer doublets pairs are stabilized by elastic links (the nexine molecules). In spite of these interesting mechanical properties, mechanical and robotics engineers have never considered this amazing molecular machinery as a model. The aim of this paper is to propose the robotic design and the kinematic modeling of the '3 + 0' axoneme that makes motile the flagellum of Diplauxis hatti, the simplest that exists. The model that we propose bends and twists and combines the two movements. It is able to propagate wave trains that could be involved in the development of biomimetic actuators of various mechanisms such as (sub)aquatic robotic propellers as well as robotic trunks.

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 of microorganisms by a helical flagellum

The swimming of a bacterium or a biomimetic nanobot driven by a rotating helical flagellum is often interpreted using the resistive force theory developed by Gray and Hancock and by Lighthill, but this theory has not been tested for a range of physically relevant parameters. We test resistive force theory in experiments on macroscopic swimmers in a fluid that is highly viscous so the Reynolds number is small compared to unity, just as for swimming microorganisms. The measurements are made for the range of helical wavelengths λ, radii R, and lengths L relevant to bacterial flagella. The experiments determine thrust, torque, and drag, thus providing a complete description of swimming driven by a rotating helix at low Reynolds number. Complementary numerical simulations are conducted using the resistive force theories, the slender body theories of Lighthill and Johnson, and the regularized Stokeslet method. The experimental results differ qualitatively and quantitatively from the predictions of resistive force theory. The difference is especially large for and/or , parameter ranges common for bacteria. In contrast, the predictions of Stokeslet and slender body analyses agree with the laboratory measurements within the experimental uncertainty (a few percent) for all λ, R, and L. We present code implementing the slender body, regularized Stokeslet, and resistive force theories; thus readers can readily compute force, torque, and drag for any bacterium or nanobot driven by a rotating helical flagellum.

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.

Biomechanical analysis of gait adaptation in the nematode Caenorhabditis elegans

To navigate different environments, an animal must be able to adapt its locomotory gait to its physical surroundings. The nematode Caenorhabditis elegans, between swimming in water and crawling on surfaces, adapts its locomotory gait to surroundings that impose approximately 10,000-fold differences in mechanical resistance. Here we investigate this feat by studying the undulatory movements of C. elegans in Newtonian fluids spanning nearly five orders of magnitude in viscosity. In these fluids, the worm undulatory gait varies continuously with changes in external load: As load increases, both wavelength and frequency of undulation decrease. We also quantify the internal viscoelastic properties of the worm's body and their role in locomotory dynamics. We incorporate muscle activity, internal load, and external load into a biomechanical model of locomotion and show that (i) muscle power is nearly constant across changes in locomotory gait, and (ii) the onset of gait adaptation occurs as external load becomes comparable to internal load. During the swimming gait, which is evoked by small external loads, muscle power is primarily devoted to bending the worm's elastic body. During the crawling gait, evoked by large external loads, comparable muscle power is used to drive the external load and the elastic body. Our results suggest that C. elegans locomotory gait continuously adapts to external mechanical load in order to maintain propulsive thrust.

Engineering Nanorobots: Chronology of Modeling Flagellar Propulsion

Nanorobots are propitious to swim or fly compared with crawling and walking because of issues with desirable characteristics of high velocity, efficiency, specificity, controllability, and a simple propagation mechanism that can be realized with miniaturized parts. Inspired by the fact that microorganisms existing in nature function expeditiously under these circumstances, researchers have shown a great interest to conceptualize, model, analyze, and make micro-/nanosized swimmers (nanorobots) that can move in body fluids for applications such as targeted drug delivery, nanomedication, and in-viscera nanosurgery. The present work compiles modeling of physics as investigated since 1951 of flagellar propulsion in engineering nanorobots. Existing theories in flagellar propulsion such as resistive force theory, slender body theory, Kirchhoff rod theory, bead model, and boundary element method as well as progress in designing the propulsion system of a nanorobot are summarized, and various interdisciplinary aspects of realizing nanorobots and issues in moving nanorobots have been presented chronologically.

Parasites in motion: flagellum-driven cell motility in African trypanosomes

Motility of the sleeping sickness parasite, Trypanosoma brucei, impacts disease transmission and pathogenesis. Trypanosome motility is driven by a flagellum that harbors a canonical 9+2 axoneme, together with trypanosome-specific elaborations. Trypanosome flagellum biology and motility have been the object of intense research over the last two years. These studies have led to the discovery of a novel form of motility, termed social motility, and provided revision of long-standing models for cell propulsion. Recent work has also uncovered novel structural features and motor proteins associated with the flagellar apparatus and has identified candidate signaling molecules that are predicted to regulate flagellar motility. Together with earlier inventories of flagellar proteins from proteomic and genomic studies, the stage is now set to move forward with functional studies to elucidate molecular mechanisms and investigate parasite motility in the context of host-parasite interactions.

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.