Most motile bacteria swim in viscous fluids by rotating multiple helical flagellar filaments. These semi-rigid filaments repeatedly join ('bundle') and separate ('unbundle'), resulting in a two-gait random walk-like motion of the cell. In this process, hydrodynamic interactions between the filaments are known to play an important role and can be categorised into two distinct types: direct interactions mediated through flows that are generated through the actuation of the filaments themselves, and indirect interactions mediated through the motion of the cell body (i.e. flows induced in the swimming frame that result from propulsion). To understand the relative importance of these two types of interactions, we study a minimal singularity model of flagellar bundling. Using hydrodynamic images, we solve for the flow analytically and compute both direct and indirect interactions exactly as a function of the length of the flagellar filaments and their angular separation. We show (i) that the generation of thrust by flagella alone is sufficient to drive the system towards a bundled state through both types of interaction; (ii) that indirect advection dominates for long filaments and at wide separation, i.e. primarily during the early stages of the bundling process; and (iii) that, in contrast, direct interactions dominate when flagellar filaments are in each other's wake, which we characterise mathematically. We further introduce a numerical elastohydrodynamic model that allows us to compute the dynamics of the helical axes of each flagellar filament while analysing direct and indirect interactions separately. With this we show (iv) that the shift in balance between direct and indirect interactions is non-monotonic during the bundling process, with a peak in direct dominance, and that different sections of the flagella are affected by these changes to different extents.
To rotate continuously without jamming, the flagellar filaments of bacteria need to be locked in phase. While several models have been proposed for eukaryotic flagella, the synchronization of bacterial flagella is less well understood. Starting from a reduced model of flexible and hydrodynamically-coupled bacterial flagella, we rigorously coarse-grain the equations of motion using the method of multiple scales, and hence show that bacterial flagella generically synchronize to zero phase difference via an elastohydrodynamic mechanism. Remarkably, the far-field rate of synchronization is maximiz…
Trajectories and conformations of uni- and multiflagellar bacteria are studied with a coarse-grained model of a cell comprised of elastic flagella connected to a cell body. The elasticities of both the hook protein (connecting cell body and flagellum) and flagella are varied. Flexibility plays contrasting roles for uni- and multiflagellar swimmers. For a uniflagellar swimmer, hook and/or flagellar buckling occurs above a critical flexibility relative to the torque exerted by the flagellar motor. Addition of a second flagellum greatly expands the parameter regime of stable locomotion, because f…
Many species of bacteria swim through viscous environments by rotating multiple helical flagella. The filaments gather behind the cell body and form a close helical bundle, which propels the cell forward during a "run". The filaments inside the bundle cannot be continuously actuated, nor can they easily unbundle, if they are tangled around one another. The fact that bacteria can passively form coherent bundles, i.e. bundles which do not contain tangled pairs of filaments, may appear surprising given that flagella are actuated by uncoordinated motors. In this article, we establish the theoretic…
Flagellated bacteria on nutrient-rich substrates can differentiate into a swarming state and move in dense swarms across surfaces. A recent experiment measured the flow in the fluid around an Escherichia coli swarm (Wu, Hosu and Berg, 2011 Proc. Natl. Acad. Sci. USA 108 4147). A systematic chiral flow was observed in the clockwise direction (when viewed from above) ahead of the swarm with flow speeds of about $10~μ$m/s, about 3 times greater than the radial velocity at the edge of the swarm. The working hypothesis is that this flow is due to the action of cells stalled at the edge of a colony …
Bacteriophage viruses, one of the most abundant entities in our planet, lack the ability to move independently. Instead, they crowd fluid environments in anticipation of a random encounter with a bacterium. Once they land on the cell body of their victim, they are able to eject their genetic material inside the host cell. Many phage species, however, first attach to the flagellar filaments of bacteria. Being immotile, these so-called flagellotropic phages still manage to reach the cell body for infection, and the process by which they move up the flagellar filament has intrigued the scientific…
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…
The presence of a nearby boundary is likely to be important in the life cycle and evolution of motile flagellate bacteria. This has led many authors to employ numerical simulations to model near-surface bacterial motion and compute hydrodynamic boundary effects. A common choice has been the method of images for regularized Stokeslets (MIRS); however, the method requires discretization sizes and regularization parameters that are not specified by any theory. To determine appropriate regularization parameters for given discretization choices in MIRS, we conducted dynamically similar macroscopic …
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 flage…
Bacteria predate plants and animals by billions of years. Today, they are the world's smallest cells yet they represent the bulk of the world's biomass, and the main reservoir of nutrients for higher organisms. Most bacteria can move on their own, and the majority of motile bacteria are able to swim in viscous fluids using slender helical appendages called flagella. Low-Reynolds-number hydrodynamics is at the heart of the ability of flagella to generate propulsion at the micron scale. In fact, fluid dynamic forces impact many aspects of bacteriology, ranging from the ability of cells to reorie…
University of Pennsylvania（宾夕法尼亚大学）；National Institute for Mathematical Sciences（韩国数学科学国立研究所）；Chung-Ang University（中央大学）；University of Cincinnati（辛辛那提大学）
Most bacteria swim through fluids by rotating helical flagella which can take one of twelve distinct polymorphic shapes. The most common helical waveform is the "normal" form, used during forward swimming runs. To shed light on the prevalence of the normal form in locomotion, we gather all available experimental measurements of the various polymorphic forms and compute their intrinsic hydrodynamic efficiencies. The normal helical form is found to be the most hydrodynamically efficient of the twelve polymorphic forms by a significant margin - a conclusion valid for both the peritrichous and pol…
Morphological Effects on Bacterial Brownian Motion: Validation of a Chiral Two-Body Model
School of Advanced Interdisciplinary Studies, Ningxia University（宁夏大学前沿交叉学院）；School of Physics, Ningxia University（宁夏大学物理学院）；Complex Systems Division, Beijing Computational Science Research Center（北京计算科学研究中心复杂系统部）；Huanjiang Laboratory（环江实验室）；State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University（浙江大学力学系流体动力与机电系统国家重点实验室）；College of Physics, Changchun Normal University（长春师范大学物理学院）；School of Mathematics and Physics, Anqing Normal University（安庆师范大学数理学院）
Motivated by bacterial transport through porous media, here we study the swimming of an actuated, flexible helical filament in both three-dimensional free space and within a cylindrical tube whose diameter is much smaller than the length of the helix. The filament, at rest, has a native helical shape modeled after the geometry of a typical bacterial flagellar bundle. The finite length filament is a free swimmer, and is driven by an applied torque as well as a counter-torque (of equal strength and opposite direction) that represents a virtual cell body. We use a regularized Stokeslet framework …
We study the active dynamics of self-propelled asymmetrical colloidal particles (Janus particles) fueled by an AC electric field. Both the speed and the direction of the self-propulsion and the strength of attractive interaction between the particles can be controlled by tuning the frequency of the applied electric field and the ion concentration of the solution. The strong attractive force at high ion concentration give rise to chain formation of the Janus particles, which can be explained by the quadrupolar charge distribution on the particles. The chain formation is observed irrespective of…
Hydrodynamic interactions (HIs) are important in biophysics research because they influence both the collective and the individual behaviour of microorganisms and self-propelled particles. For instance, HIs at the micro-swimmer level determine the attraction or repulsion between individuals, and hence their collective behaviour. Meanwhile, HIs between swimming appendages (e.g. cilia and flagella) influence the emergence of swimming gaits, synchronised bundles and metachronal waves. In this study, we address the issue of HIs between slender filaments separated by a distance larger than their co…
