Motor-driven bacterial flagella and buckling instabilities

Trapping and scattering of a multiflagellated bacterium by a hard surface

Thiovulum majus, which is one of the fastest known bacteria, swims using hundreds of flagella. Unlike typical pusher cells, which swim in circular paths over hard surfaces, T. majus localize near hard boundaries by turning their flagella to exert a net force normal to the surface. To probe the torques that stabilize this hydrodynamically bound state, the trajectories of several thousand collisions between a T. majus cell and a wall of a quasi-two-dimensional microfluidic chamber are analyzed. Measuring the fraction of cells escaping the wall either to the left or to the right of the point of contact-and how this probability varies with incident angle and time spent in contact with the surface-maps the scattering dynamics onto a first passage problem. These measurements are compared to the prediction of a Fokker-Planck equation to fit the angular velocity of a cell in contact with a hard surface. This analysis reveals a bound state with a narrow basin of attraction in which cells orient their flagella normal to the surface. The escape angle predicted by matching these near field dynamics with the far-field hydrodynamics is consistent with observation. We discuss the significance of these results for the ecology of T. majus and their self-organization into active chiral crystals.

Self-buckling and self-writhing of semi-flexible microorganisms

The twisting and writhing of a cell body and associated mechanical stresses is an underappreciated constraint on microbial self-propulsion. Multi-flagellated bacteria can even buckle and writhe under their own activity as they swim through a viscous fluid. New equilibrium configurations and steady-state dynamics then emerge which depend on the organism's mechanical properties and on the oriented distribution of flagella along its surface. Modeling the cell body as a semi-flexible Kirchhoff rod and coupling the mechanics to a flagellar orientation field, we derive the Euler-Poincaré equations governing the dynamics of the system, and rationalize experimental observations of buckling and writhing of elongated swarmer cells of the bacterium Proteus mirabilis. A sequence of bifurcations is identified as the body is made more compliant, due to both buckling and torsional instabilities. These studies highlight a practical requirement for the stiffness of bacteria below which self-buckling occurs and cell motility becomes ineffective.

A minimal robophysical model of quadriflagellate self-propulsion

Locomotion at the microscale is remarkably sophisticated. Microorganisms have evolved diverse strategies to move within highly viscous environments, using deformable, propulsion-generating appendages such as cilia and flagella to drive helical or undulatory motion. In single-celled algae, these appendages can be arranged in different ways around an approximately 10 μm long cell body, and coordinated in distinct temporal patterns. Inspired by the observation that some quadriflagellates (bearing four flagella) have an outwardly similar morphology and flagellar beat pattern, yet swim at different speeds, this study seeks to determine whether variations in swimming performance could arise solely from differences in swimming gait. Robotics approaches are particularly suited to such investigations, where the phase relationships between appendages can be readily manipulated. Here, we developed autonomous, algae-inspired robophysical models that can self-propel in a viscous fluid. These macroscopic robots (length and width = 8.5 cm, height = 2 cm) have four independently actuated 'flagella' (length = 13 cm) that oscillate under low-Reynolds number conditions (Re∼O(10-1)). We tested the swimming performance of these robot models with appendages arranged two distinct configurations, and coordinated in three distinct gaits. The gaits, namely the pronk, the trot, and the gallop, correspond to gaits adopted by distinct microalgal species. When the appendages are inserted perpendicularly around a central 'body', the robot achieved a net performance of 0.15-0.63 body lengths per cycle, with the trot gait being the fastest. Robotic swimming performance was found to be comparable to that of the algal microswimmers across all gaits. By creating a minimal robot that can successfully reproduce cilia-inspired drag-based swimming, our work paves the way for the design of next-generation devices that have the capacity to autonomously navigate aqueous environments.

Hydrodynamics and direction change of tumbling bacteria

The bacterium Escherichia coli (E. coli) swims in viscous fluids by rotating several helical flagellar filaments, which are gathered in a bundle behind the cell during ‘runs’ wherein the cell moves steadily forward. In between runs, the cell undergoes quick ‘tumble’ events, during which at least one flagellum reverses its rotation direction and separates from the bundle, resulting in erratic motion in place and a random reorientation of the cell. Alternating between runs and tumbles allows cells to sample space by stochastically changing their propulsion direction after each tumble. The change of direction during a tumble is not uniformly distributed but is skewed towards smaller angles with an average of about 62°–68°, as first measured by Berg and Brown (1972). Here we develop a theoretical approach to model the angular distribution of swimming E. coli cells during tumbles. We first use past experimental imaging results to construct a kinematic description of the dynamics of the flagellar filaments during a tumble. We then employ low-Reynolds number hydrodynamics to compute the consequences of the kinematic model on the force and torque balance of the cell and to deduce the overall change in orientation. The results of our model are in good agreement with experimental observations. We find that the main change of direction occurs during the ‘bundling’ part of the process wherein, at the end of a tumble, the dispersed flagellar filaments are brought back together in the helical bundle, which we confirm using a simplified forced-sphere model.

Stokesian dynamics simulations of a magnetotactic bacterium

The swimming of bacteria provides insight into propulsion and steering under the conditions of low-Reynolds number hydrodynamics. Here we address the magnetically steered swimming of magnetotactic bacteria. We use Stokesian dynamics simulations to study the swimming of single-flagellated magnetotactic bacteria (MTB) in an external magnetic field. Our model MTB consists of a spherical cell body equipped with a magnetic dipole moment and a helical flagellum rotated by a rotary motor. The elasticity of the flagellum as well as magnetic and hydrodynamic interactions is taken into account in this model. We characterized how the swimming velocity is dependent on parameters of the model. We then studied the U-turn motion after a field reversal and found two regimes for weak and strong fields and, correspondingly, two characteristic time scales. In the two regimes, the U-turn time is dominated by the turning of the cell body and its magnetic moment or the turning of the flagellum, respectively. In the regime for weak fields, where turning is dominated by the magnetic relaxation, the U-turn time is approximately in agreement with a theoretical model based on torque balance. In the strong-field regime, strong deformations of the flagellum are observed. We further simulated the swimming of a bacterium with a magnetic moment that is inclined relative to the flagellar axis. This scenario leads to intriguing double helical trajectories that we characterize as functions of the magnetic moment inclination and the magnetic field. For small inclination angles ([Formula: see text]) and typical field strengths, the inclination of the magnetic moment has only a minor effect on the swimming of MTB in an external magnetic field. Large inclination angles result in a strong reduction in the velocity in direction of the magnetic field, consistent with recent observations that bacteria with large inclination angles use a different propulsion mechanism.

Numerical exploration on buckling instability for directional control in flagellar propulsion

We report a numerical method to control the swimming direction by exploiting buckling instability in uniflagellar bacteria and bio-inspired soft robots. Our model system is comprised of a spherical rigid head and a helical elastic flagellum. The rotation of the flagellum in low Reynolds environment generates a propulsive force that allows the system to swim in fluid. The locomotion is an intricate interplay between the elasticity of the flagellum, the hydrodynamic loading, and the flow generated by the moving head. We use the Discrete Elastic Rods algorithm to capture the geometrically nonlinear deformation in the flagellum, Lighthills Slender Body Theory to simulate the hydrodynamics, and Higdons model for the spherical head in motion within viscous fluid. This flagellated system follows a straight path if the angular velocity of the flagellum is below a critical threshold. Buckling ensues in the flagellum beyond this threshold angular velocity and the system takes a nonlinear trajectory. We consider the angular velocity as the control parameter and solve the inverse problem of computing the angular velocity, that varies with time, given a desired nonlinear trajectory. Our results indicate that bacteria can exploit buckling in flagellum to precisely control their swimming direction.

Thrust and Hydrodynamic Efficiency of the Bundled Flagella

The motility mechanism of prokaryotic organisms has inspired many untethered microswimmers that could potentially perform minimally invasive medical procedures in stagnant fluid regions inside the human body. Some of these microswimmers are inspired by bacteria with single or multiple helical flagella to propel efficiently and fast. For multiple flagella configurations, the direct measurement of thrust and hydrodynamic propulsion efficiency has been challenging due to the ambiguous mechanical coupling between the flow field and mechanical power input. To address this challenge and to compare alternative micropropulsion designs, a methodology based on volumetric velocity field acquisition is developed to acquire the key propulsive performance parameters from scaled-up swimmer prototypes. A digital particle image velocimetry (PIV) analysis protocol was implemented and experiments were conducted with the aid of computational fluid dynamics (CFD). First, this methodology was validated using a rotating single-flagellum similitude model. In addition to the standard PIV error assessment, validation studies included 2D vs. 3D PIV, axial vs. lateral PIV and simultaneously acquired direct thrust force measurement comparisons. Compatible with typical micropropulsion flow regimes, experiments were conducted both for very low and higher Reynolds (Re) number regimes (up to a Re number = 0.01) than that are reported in the literature. Finally, multiple flagella bundling configurations at 0°, 90° and 180° helical phase-shift angles were studied using scaled-up multiple concentric flagella thrust elements. Thrust generation was found to be maximal for the in-phase (0°) bundling configuration but with ~50% lower hydrodynamic efficiency than the single flagellum. The proposed measurement protocol and static thrust test-bench can be used for bio-inspired microscale propulsion methods, where direct thrust and efficiency measurement are required.

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.

Microswimmer Propulsion by Two Steadily Rotating Helical Flagella

Many theoretical studies of bacterial locomotion adopt a simple model for the organism consisting of a spheroidal cell body and a single corkscrew-shaped flagellum that rotates to propel the body forward. Motivated by experimental observations of a group of magnetotactic bacterial strains, we extended the model by considering two flagella attached to the cell body and rotating about their respective axes. Using numerical simulations, we analyzed the motion of such a microswimmer in bulk fluid and close to a solid surface. We show that positioning the two flagella far apart on the cell body reduces the rate of rotation of the body and increases the swimming speed. Near surfaces, we found that swimmers with two flagella can swim in relatively straight trajectories or circular orbits in either direction. It is also possible for the swimmer to escape from surfaces, unlike a model swimmer of similar shape but with only a single flagellum. Thus, we conclude that there are important implications of swimming with two flagella or flagellar bundles rather than one. These considerations are relevant not only for understanding differences in bacterial morphology but also for designing microrobotic swimmers.

Spatial arrangement of several flagellins within bacterial flagella improves motility in different environments

Bacterial flagella are helical proteinaceous fibers, composed of the protein flagellin, that confer motility to many bacterial species. The genomes of about half of all flagellated species include more than one flagellin gene, for reasons mostly unknown. Here we show that two flagellins (FlaA and FlaB) are spatially arranged in the polar flagellum of Shewanella putrefaciens, with FlaA being more abundant close to the motor and FlaB in the remainder of the flagellar filament. Observations of swimming trajectories and numerical simulations demonstrate that this segmentation improves motility in a range of environmental conditions, compared to mutants with single-flagellin filaments. In particular, it facilitates screw-like motility, which enhances cellular spreading through obstructed environments. Similar mechanisms may apply to other bacterial species and may explain the maintenance of multiple flagellins to form the flagellar filament.

Dynamic instability in the hook-flagellum system that triggers bacterial flicks

Dynamical bending, buckling, and polymorphic transformations of the flagellum are known to affect bacterial motility, but run-reverse-flick motility of monotrichous bacteria also involves the even more flexible hook connecting the flagellum to its rotary motor. Although flick initiation has been hypothesized to involve either static Euler buckling or dynamic bending of the hook, the precise mechanism of flick initiation remains unknown. Here, we find that flicks initiate via a dynamic instability requiring flexibility in both the hook and flagellum. We obtain accurate estimates of forces and torques on the hook that suggest that flicks occur for stresses below the (static) Euler buckling criterion, then provide a mechanistic model for flick initiation that requires combined bending of the hook and flagellum. We calculate the triggering torque-stiffness ratio and find that our predicted onset of dynamic instability corresponds well with experimental observations.

Buckling Instabilities and Complex Trajectories in a Simple Model of Uniflagellar Bacteria

Observations of uniflagellar bacteria show that buckling instabilities of the hook protein connecting the cell body and flagellum play a role in locomotion. To understand this phenomenon, we develop models at varying levels of description with a particular focus on the parameter dependence of the buckling instability. A key dimensionless group called the flexibility number measures the hook flexibility relative to the thrust exerted by the flagellum; this parameter and the geometric parameters of the cell determine the stability of straight swimming. Two very simple models amenable to analytical treatment are developed to examine buckling in stationary (pinned) and moving swimmers. We then consider a more detailed model incorporating a helical flagellum and the rotational degrees of freedom of the cell body and flagellum, and we use numerical simulations to map out the parameter dependence of the buckling instability. In all models, a bifurcation occurs as the flexibility number increases, separating equilibrium configurations into straight or bent, and for the full model, separating trajectories into straight or helical. More specifically for the latter, the critical flexibility marks the transition from periodicity to quasi-periodicity in the behavior of variables determining configuration. We also find that for a given body geometry, there is a specific flagellar geometry that minimizes the critical flexibility number at which buckling occurs. These results highlight the role of flexibility in the biology of real organisms and the engineering of artificial microswimmers.

Flagella bending affects macroscopic properties of bacterial suspensions

To survive in harsh conditions, motile bacteria swim in complex environments and respond to the surrounding flow. Here, we develop a mathematical model describing how flagella bending affects macroscopic properties of bacterial suspensions. First, we show how the flagella bending contributes to the decrease in the effective viscosity observed in dilute suspension. Our results do not impose tumbling (random reorientation) as was previously done to explain the viscosity reduction. Second, we demonstrate how a bacterium escapes from wall entrapment due to the self-induced buckling of flagella. Our results shed light on the role of flexible bacterial flagella in interactions of bacteria with shear flow and walls or obstacles.

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.

Bacteria exploit a polymorphic instability of the flagellar filament to escape from traps

Many bacterial species swim by rotating single polar helical flagella. Depending on the direction of rotation, they can swim forward or backward and change directions to move along chemical gradients but also to navigate their obstructed natural environment in soils, sediments, or mucus. When they get stuck, they naturally try to back out, but they can also resort to a radically different flagellar mode, which we discovered here. Using high-speed microscopy, we monitored the swimming behavior of the monopolarly flagellated species Shewanella putrefaciens with fluorescently labeled flagellar filaments at an agarose-glass interface. We show that, when a cell gets stuck, the polar flagellar filament executes a polymorphic change into a spiral-like form that wraps around the cell body in a spiral-like fashion and enables the cell to escape by a screw-like backward motion. Microscopy and modeling suggest that this propagation mode is triggered by an instability of the flagellum under reversal of the rotation and the applied torque. The switch is reversible and bacteria that have escaped the trap can return to their normal swimming mode by another reversal of motor direction. The screw-type flagellar arrangement enables a unique mode of propagation and, given the large number of polarly flagellated bacteria, we expect it to be a common and widespread escape or motility mode in complex and structured environments.

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.

Deformation of a soft helical filament in an axial flow at low Reynolds number

We perform a numerical investigation of the deformation of a rotating helical filament subjected to an axial flow, under low Reynolds number conditions, motivated by the propulsion of bacteria using helical flagella. Given its slenderness, the helical rod is intrinsically soft and deforms due to the interplay between elastic forces and hydrodynamic loading. We make use of a previously developed and experimentally validated computational tool framework that models the elasticity of the filament using the discrete elastic rod method and the fluid forces are treated using Lighthill's slender body theory. Under axial flow, and in the absence of rotation, the initially helical rod is extended. Above a critical flow speed its configuration comprises a straight portion connected to a localized helix near the free end. When the rod is also rotated about its helical axis, propulsion is only possible in a finite range of angular velocity, with an upper bound that is limited by buckling of the soft helix arising due to viscous stresses. A systematic exploration of the parameter space allows us to quantify regimes for successful propulsion for a number of specific bacteria.

Torque-induced precession of bacterial flagella

The bacterial flagellar motor is an ion-driven rotary machine in the cell envelope of bacteria. Using a gold nanoparticle as a probe, we observed the precession of flagella during rotation. Since the mechanism of flagella precession was unknown, we investigated it using a combination of full simulations, theory, and experiments. The results show that the mechanism can be well explained by fluid mechanics. The validity of our theory was confirmed by our full simulation, which was utilized to predict both the filament tilt angle and motor torque from experimental flagellar precession data. The knowledge obtained is important in understanding mechanical properties of the bacterial motor and hook.

Zipping and entanglement in flagellar bundle of E. coli: Role of motile cell body

The course of a peritrichous bacterium, such as E. coli, crucially depends on the level of synchronization and self-organization of several rotating flagella. However, the rotation of each flagellum generates countermovements of the body which in turn affect the flagellar dynamics. Using a detailed numerical model of an E. coli, we demonstrate that flagellar entanglement, besides fluid flow relative to the moving body, dramatically changes the dynamics of flagella from that compared to anchored flagella. In particular, bundle formation occurs through a zipping motion in a remarkably rapid time, affected little by initial flagellar orientation. A simplified analytical model supports our observations. Finally, we illustrate how entanglement, hydrodynamic interactions, and body movement contribute to zipping and bundling.

Modelling the mechanics and hydrodynamics of swimming E. coli

The swimming properties of an E. coli-type model bacterium are investigated by mesoscale hydrodynamic simulations, combining molecular dynamics simulations of the bacterium with the multiparticle particle collision dynamics method for the embedding fluid. The bacterium is composed of a spherocylindrical body with attached helical flagella, built up from discrete particles for an efficient coupling with the fluid. We measure the hydrodynamic friction coefficients of the bacterium and find quantitative agreement with experimental results of swimming E. coli. The flow field of the bacterium shows a force-dipole-like pattern in the swimming plane and two vortices perpendicular to its swimming direction arising from counterrotation of the cell body and the flagella. By comparison with the flow field of a force dipole and rotlet dipole, we extract the force-dipole and rotlet-dipole strengths for the bacterium and find that counterrotation of the cell body and the flagella is essential for describing the near-field hydrodynamics of the bacterium.

Propulsion and Instability of a Flexible Helical Rod Rotating in a Viscous Fluid

We combine experiments with simulations to investigate the fluid-structure interaction of a flexible helical rod rotating in a viscous fluid, under low Reynolds number conditions. Our analysis takes into account the coupling between the geometrically nonlinear behavior of the elastic rod with a nonlocal hydrodynamic model for the fluid loading. We quantify the resulting propulsive force, as well as the buckling instability of the originally helical filament that occurs above a critical rotation velocity. A scaling analysis is performed to rationalize the onset of this instability. A universal phase diagram is constructed to map out the region of successful propulsion and the corresponding boundary of stability is established. Comparing our results with data for flagellated bacteria suggests that this instability may be exploited in nature for physiological purposes.

Flexibility of bacterial flagella in external shear results in complex swimming trajectories

Many bacteria use rotating helical flagella in swimming motility. In the search for food or migration towards a new habitat, bacteria occasionally unbundle their flagellar filaments and tumble, leading to an abrupt change in direction. Flexible flagella can also be easily deformed by external shear flow, leading to complex bacterial trajectories. Here, we examine the effects of flagella flexibility on the navigation of bacteria in two fundamental shear flows: planar shear and Poiseuille flow realized in long channels. On the basis of slender body elastodynamics and numerical analysis, we discovered a variety of non-trivial effects stemming from the interplay of self-propulsion, elasticity and shear-induced flagellar bending. We show that in planar shear flow the bacteria execute periodic motion, whereas in Poiseuille flow, they migrate towards the centre of the channel or converge towards a limit cycle. We also find that even a small amount of random reorientation can induce a strong response of bacteria, leading to overall non-periodic trajectories. Our findings exemplify the sensitive role of flagellar flexibility and shed new light on the navigation of bacteria in complex shear flows.

Rotation-induced polymorphic transitions in bacterial flagella

Bacteria propel themselves with the help of rotating helical flagella. They change their swimming direction during tumbling events in order to increase, for example, their supply of nutrients (chemotaxis). During tumbling a bacterial flagellum assumes different polymorphic states. Based on a continuum model for the motor-flagellum system, we demonstrate that a changing motor torque can initiate these polymorphic transformations. In particular, we investigate the run-and-stop tumble strategy of Rhodobacter sphaeroides which uses a coiled-to-normal transition in its single flagellum. We also show that torque reversal in single-flagellated Escherichia coli generates a normal-to-curly I transition as observed for tumbling E. coli that swim with a bundle of several flagella.

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.