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The motility of bacteria in viscoelastic fluids plays a crucial role in diverse biological processes such as infection and biofilm formation. In such environments, the rotation of long helical flagella is often employed as a primary means of bacterial propulsion. In this paper, we present a general version of the immersed boundary method incorporating the finitely extensible nonlinear elastic (FENE) constitutive equations for viscoelastic fluids. We first verify that the method solves correctly the rotational dynamics of a helical flagellum in a FENE fluid by conducting a convergence study. We then investigate the hydrodynamic interactions of single and multiple flagella in Newtonian and FENE viscoelastic fluids. For a single flagellum at the same solvent viscosity, the fluid flux generated by the flagellum is higher in FENE fluid than in Newtonian fluid for nearly identical rotation rates, implying enhanced bacterial swimming speeds in a viscoelastic fluid. The dependence of rotational dynamics on applied motor torque, relaxation time, and polymer viscosity is systematically analyzed. For multiple flagella, the time required for bundle formation is shown to increase with hook rigidity, initial separation, and the number of flagella. In the case of two flagella rotating counterclockwise under different applied torques, bundling occurs only when the torque difference is within a limited range, producing a bundle whose axis is tilted from the flagellar axes. We further examine conditions leading to bundling failure, providing insights into the mechanical and hydrodynamic factors governing flagellar coordination. 

A swimming microorganism stirs the surrounding fluid, creating a flow field that governs not only its locomotion and nutrient uptake, but also its interactions with other microorganisms and the environment. Despite its fundamental importance, capturing this flow field and unraveling its biological implications remains a challenge. Here, we report direct, time-resolved measurements of the three-dimensional (3D) flow field generated by a single, free-swimming microalga, , a model organism for microbial locomotion and flagellar dynamics. Supported by hydrodynamic modeling and simulations, our measurements resolve how established two-dimensional (2D) flow features such as in-plane vortices and the stagnation point emerge from and shape the full algal flow in 3D. Moreover, we reveal unexpected low-Reynolds-number flow phenomena including micron-sized vortex rings and periodically recurring translating vortices and uncover topological changes in the underlying flow structure associated with the puller-to-pusher transition of an alga. Biologically, access to the 3D flow field enables rigorous quantification of the alga’s energy expenditure, as well as its swimming and feeding efficiency, improving the precision of these physiological metrics. Taken together, our study demonstrates rich vortex dynamics in inertialess flows and shows their influence on microbial motility. The work also introduces an experimental method for mapping the fluid environment sculpted by beating flagella. 

We present a mathematical model of lophotrichous bacteria, motivated by Pseudomonas putida, which swim through fluid by rotating a cluster of multiple flagella extended from near one pole of the cell body. Although the flagella rotate individually, they are typically bundled together, enabling the bacterium to exhibit three primary modes of motility: push, pull, and wrapping. One key determinant of these modes is the coordination between motor torque and rotational direction of motors. The computational variations in this coordination reveal a wide spectrum of dynamical motion regimes, which are modulated by hydrodynamic interactions between flagellar filaments. These dynamic modes can be categorized into two groups based on the collective behavior of flagella, i.e., bundled and unbundled configurations. For some of these configurations, experimental examples from fluorescence microscopy recordings of swimming P. putida cells are also presented. Furthermore, we analyze the characteristics of stable bundles, such as push and pull, and investigate the dependence of swimming behaviors on the elastic properties of the flagella. 

We present an in silico microswimmer motivated by peritrichous bacteria, E. coli, which can run and tumble by spinning their flagellar motors counterclockwise (CCW) or clockwise (CW). Runs are the directed movement driven by a flagellar bundle, and tumbles are reorientations of cells caused by some motors' reversals from CCW to CW. In a viscous fluid without obstacles, our simulations reveal that material properties of the hook and the counterrotation of the cell body are important factors for efficient flagellar bundling and that longer hooks in mutant cell models create an instability and disrupt the bundling process, resulting in a limited range of movement. In the presence of a planar wall, we demonstrate that microswimmers can explore environment near surface by making various types of tumble events as they swim close to the surface. In particular, the variation of tumble duration can lead the microswimmer to run in a wide range of direction. However, we find that cells near surface stay close to the surface even after tumbles, which suggests that the tumble motion may not promote cells' escape from the confinement but promote biofilm formation. 

To swim through a viscous fluid, a flagellated bacterium must overcome the fluid drag on its body by rotating a flagellum or a bundle of multiple flagella. Because the drag increases with the size of bacteria, it is expected theoretically that the swimming speed of a bacterium inversely correlates with its body length. Nevertheless, despite extensive research, the fundamental size–speed relation of flagellated bacteria remains unclear with different experiments reporting conflicting results. Here, by critically reviewing the existing evidence and synergizing our own experiments of large sample sizes, hydrodynamic modeling, and simulations, we demonstrate that the average swimming speed ofEscherichia coli, a premier model of peritrichous bacteria, is independent of their body length. Our quantitative analysis shows that such a counterintuitive relation is the consequence of the collective flagellar dynamics dictated by the linear correlation between the body length and the number of flagella of bacteria. Notably, our study reveals how bacteria utilize the increasing number of flagella to regulate the flagellar motor load. The collective load sharing among multiple flagella results in a lower load on each flagellar motor and therefore faster flagellar rotation, which compensates for the higher fluid drag on the longer bodies of bacteria. Without this balancing mechanism, the swimming speed of monotrichous bacteria generically decreases with increasing body length, a feature limiting the size variation of the bacteria. Altogether, our study resolves a long-standing controversy over the size–speed relation of flagellated bacteria and provides insights into the functional benefit of multiflagellarity in bacteria. 

Abstract Lophotrichous bacteria swim through fluid by rotating their flagellar bundle extended collectively from one pole of the cell body. Cells experience modes of motility such as push, pull, and wrapping, accompanied by pauses of motor rotation in between. We present a mathematical model of a lophotrichous bacterium and investigate the hydrodynamic interaction of cells to understand their swimming mechanism. We classify the swimming modes which vary depending on the bending modulus of the hook and the magnitude of applied torques on the motor. Given the hook’s bending modulus, we find that there exist corresponding critical thresholds of the magnitude of applied torques that separate wrapping from pull in CW motor rotation, and overwhirling from push in CCW motor rotation, respectively. We also investigate reoriented directions of cells in three-dimensional perspectives as the cell experiences different series of swimming modes. Our simulations show that the transition from a wrapping mode to a push mode and pauses in between are key factors to determine a new path and that the reoriented direction depends upon the start time and duration of the pauses. It is also shown that the wrapping mode may help a cell to escape from the region where the cell is trapped near a wall. 

Swimming trajectories of bacteria can be altered by environmental conditions, such as background flow and physical barriers, that limit the free swimming of bacteria. We present a comprehensive model of a bacterium that consists of a rod-shaped cell body and a flagellum which is composed of a motor, a hook, and a filament. The elastic flagellum is modeled based on the Kirchhoff rod theory, the cell body is considered to be a rigid body, and the hydrodynamic interaction of a bacterium near a wall is described by regularized Stokeslet formulation combined with the image system. We consider three environmental conditions: (1) a rigid surface is placed horizontally and there is no shear flow, (2) a shear fluid flow is present and the bacterium is near the rigid surface, and (3) while the bacterium is near the rigid surface and is under shear flow, an additional sidewall which is perpendicular to the rigid surface is placed. Each environmental state modifies the swimming behavior. For the first condition, there are two modes of motility, trap and escape, whether the bacterium stays near the surface or moves away from the surface as we vary the physical and geometrical properties of the model bacterium. For the second condition, there exists a threshold of shear rate that classifies the motion into two types of paths in which the bacterium takes either a periodic coil trajectory or a linear trajectory. For the last condition, the bacterium takes upstream motility along the sidewall for lower shear rates and downstream motility for larger shear flow rates.