Session U04: Low Reynolds Number Locomotion: Bacteria and Metachronal Waves

Transitioning to confined spaces impacts bacterial swimming and escape response

Symbiotic bacteria that navigate to and through specific host tissues often face tight physical confinement. The flagellated marine bacterium Vibrio fischeri, which forms a binary symbiosis with the Hawaiian bobtail squid, Euprymna scolopes, must squeeze through a bottleneck constricting to ~2 microns in width on the way to its eventual home. Using microfluidic in vitro experiments, we discovered that V. fischeri cells alter their behavior upon entry into confined space, straightening their swimming paths and promoting escape from confinement. Using a computational model, we attributed this escape response to two factors: reduced directional fluctuation and a refractory period between reversals. Additional experiments in asymmetric capillary tubes confirmed that V. fischeri quickly escape from tapered ends, even when drawn into the ends by chemoattraction. Our findings demonstrate that non-trivial distributions of swimming bacteria can emerge from simple physical gradients in the level of confinement.

    

Hydrodynamics of forward and upside-down locomotion in mysid shrimp

Mysid shrimp, Americamysis bahia, are essential to many benthic and pelagic ecosystems. Due to their abundance and sensitivity to environmental changes, they are a well-established tool for aquatic toxicology studies. But, like many planktonic crustaceans, they are also agile, efficient swimmers. Propulsion is achieved via metachronal paddling of either appendages near their mouth region, the thoracopods, or both thoracopods and pleopod appendages, which are located along their abdomen. Mysid species, A. bahia, are more efficient metachronal swimmers than other species, which makes them an attractive candidate for investigating propulsion hydrodynamics.  In this study, a high-speed tomographic particle image velocimetry (PIV) system was used to visualize and quantify a time-resolved 3D velocity field around a free-swimming, A. bahia and its wake for both thoracopod and thoracopod+pleopod swimming. When A. bahia swims with both pleopods and thoracopods, it produces a robust, well-developed 3D multi-jet system in its wake that is not seen when using only its thoracopods. The data provide novel insight into both the flow behavior at intermediate Reynolds number, in which many aquatic species reside, and the intricacies of the bio-locomotion of free-swimming zooplankton.   

Hydrodynamic Scaling of Metachronal Swimming

Metachronal swimming is a widespread locomotion mode occurring in organisms with multiple appendages that undergo sequential beating. This locomotion mode is observed in all flow regimes; viscous, inertial, and in between. Recent studies have investigated metachronal swimming in various live species with distinct morphologies and sizes. However, a relationship is absent that unifies swimming kinematics of metachronal swimmers across taxa. Here we develop a hydrodynamics-based scaling relationship linking body length L, appendage kinematics (appendage tip amplitude A and angular frequency ω), and fluid kinematic viscosity ν to swimming speed V. This scaling is given as a power law Re_b ∼ Sw that holds in all flow regimes, where Re_b is the body-based Reynolds number and Sw is the nondimensional Swimming number. This scaling is in agreement with results from experiments on metachronal swimmers including paramecia, ctenophores, tomopterids, and several species of crustaceans, ranging in body length from 0.1 mm to 0.1 m. We compare scaling laws for metachronal and undulatory swimmers, and discuss the benefits of possessing multiple appendages for locomotion in viscous flows.   

Multiflagellarity allows bacteria to maintain constant motility across cell size

We measure the swimming speed of E. coli, a model strain of multiflagellar bacteria, as a function of their body length. We find that the population-averaged swimming speed of bacteria is constant over three fold increase in their body length. We show how bacteria utilize the increasing number of flagella to regulate flagellar motor load, which results in higher rotational speeds as well as a constant swimming speed for large cell sizes. We subsequently perform simulations that reveal the role of interflagellar interactions in controlling the increase of rotational speeds. Our mechanism predicts that the swimming speed of uniflagellar species decreases with increasing cell size, which we verify directly through experiments on several strains of uniflagellar bacteria. Thus, the stark difference between the uniflagellar and multiflagellar swimming demonstrated in our study provides an insight into the crucial role of multiflagellarity in maintaining optimum motility for navigation and survival of bacteria in their native habitats.   

Vibrio anguillarum swims with variable speeds during runs

Singly-flagellated bacteria perform run-and-reverse, or run-reverse-flick motility, in which reorientation mostly occurs during isolated reversal or flick events. During runs, the cell is thought to move with a constant velocity determined by motor torque, motor speed, and cell geometry, with changes in direction caused by Brownian diffusion. We perform experiments on

Vibrio anguillarum, which undergoes run-reverse-flick motility. Under darkfield video microscopy we observe that during runs, cells often slow down then speed up again over tenth of a second timescales, which may be due to the motor stopping entirely on timescales faster than our framerates or slower changes in motor speed. We characterize these events in terms of timescale, changes in cell speed, trajectory direction, and flagellar configuration. We use this data to estimate the change in motor speed and torque through time with numerical simulations. Our results help in understanding the mechanism of these slowdowns. These events may provide ways for bacteria to reorient during motility in addition to reversals or flicks, and impact their chemotactic ability.   

RoboKrill: the role of morphology on thrust production during metachronal swimming

Metachronal, drag-based swimming in krill (\textit{E. superba}) has been studied to understand its ecological significance and find optimal engineering solutions for underwater locomotion. Krill modulate the profile area of their appendages while maintaining a phase shift between appendage pairs resulting in efficient propulsion in the intermediate Reynolds number (Re) regime. Previous studies have informed many aspects of krill swimming. However, the connection between thrust production and vortex generation remains largely unexplored. Leveraging our recently developed Robokrill, we performed velocimetry experiments to understand the hydrodynamic effects of morphological traits during forward-swimming. In particular, we focus our study on the role of setae, which are flexible hair-like structures. By broadening the profile area of the appendages, setae promote the production of thrust by increasing the area enclosed by generated vortex structures. Exploring morphological traits allows us to understand the design parameters behind efficient propulsion at intermediate Re. Going forward, we aim to identify unifying success mechanisms of different species of drag-based swimmers to engineer a new generation of underwater robots.   

Effects of swimming environment on bacterial motility

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.   

Swimming isn't such a drag: How the coalescence and flexibility of shrimp pleopods minimize drag during metachronal swimming

Metachronal swimming is characterized by the sequential beating of closely spaced flexible swimming legs (pleopods), phase-shifted in time. The stiffness and increased surface area of pleopods during the power stroke have been shown to maximize thrust. However, their characteristic bending and associated fluid flow effects during the recovery stroke have received far less attention despite being antagonistically important in reducing drag. By combining measurements of shrimp pleopod stiffness with kinematics and particle image velocimetry, we explore the relationship between the mechanical properties of pleopods and organism-fluid interactions. Unrecognized in previous works, we show that pleopods bend almost horizontally and shed no observable tip vortices during the recovery stroke. At Re = 1000, shrimp essentially trade inertial drag forces for much weaker viscous forces. Other species exhibit similar pleopod bending, suggesting a standard stiffness coefficient for efficient metachronal swimming. Further, due to their proximity, three out of five pleopods coalesce at any time during a complete cycle, effectively reducing the drag of three legs to only one. Considering appendage stiffness opens new avenues for the design of novel, more capable multi-functional underwater robots.   

Optimal swimming of multi-flagellated bacteria

An important characteristic of motile multi-flagellated bacteria is their variable number of flagella, with some bacteria having only one, while others have a few dozen. The number of flagella in a cell is difficult to control in experiments, but it can be changed easily in simulations. This has motivated several theoretical investigations into the link between the swimming of bacteria and their number of flagella [1,2]. How does the number of flagella affect the swimming speed and efficiency of a bacterium? We revisit this open question using slender-body theory simulations, where we include the full hydrodynamic interactions inside a bundle of parallel helical filaments that rotate and translate in synchrony. In contrast to previous studies, we incorporate the full torque-speed relationship of the bacterial flagellar motor [3]. This enables us to obtain novel and surprising predictions on the swimming speed of multi-flagellated bacteria. Our observations are relevant to bacteria with a small number of flagella, such as the model organism Escherichia coli, and we hope will inspire new experiments to address this question.

 

[1] Kanehl, P. and Ishikawa, T. (2014). Phys. Rev. E, 89:042704

[2] Nguyen, F. T. M. and Graham, M. D. (2018). Phys. Rev. E, 98:042419

[3] Xing, J., Bai, F., Berry, R. M., and Oster, G. (2006). PNAS, 103:1260–1265   

Reynolds number scalability of metachronal paddling

Metachronal paddling is a form of drag-based locomotion that is used by numerous aquatic organisms of sizes varying on the order of 0.01 mm to 100 mm. The paddling rhythm remains similar despite a wide variation in appendage-based Reynolds number on the orders of 10^-2 to 10³. We examined the effect of varying Reynolds number on the metachronal paddling wake and swimming performance. We conducted time-resolved 2D-2C PIV measurements on a dynamically similar metachronal paddling robotic model. Stroke frequency was varied from 1.5 to 3 Hz, while fluid viscosity was varied from 1 to 800 cSt, to obtain Reynolds numbers ranging from 35 to 54,000. Changing the fluid viscosity was found to greatly affect the direction and structure of the wake, while changing the stroke frequency was found to affect cycle to cycle interactions in the wake. For a given viscosity, steady swimming speed was found to vary linearly with stroke frequency. At steady swimming, the ratio of thrust to drag coefficients was found to decrease slightly with increasing Reynolds number. The Strouhal number was found to remain nearly constant in the range of 50 < Re < 54,000, indicating that metachronal paddling is a robust strategy that can function across size, stroke frequency and viscosity scales.   

Rowing with a spatiotemporally asymmetric paddle at intermediate Reynolds numbers

Metachronal rowing is a locomotive strategy widely used by animals that swim at intermediate Reynolds numbers (e.g., krill, shrimp, etc.). It consists of creating thrust by the sequential beating of a row of closely spaced paddles. This thrust is obtained by two kinematic mechanisms: spatial asymmetry (Sa, i.e., differing paddle flow-normal area between power vs. recovery strokes) and temporal asymmetry (Ta, i.e., differing duration between power vs. recovery strokes). However, the combined effect of Sa-Ta on flow production at intermediate Reynolds numbers (Re) is not fully understood, particularly for highly deformable structures. We designed a dynamically scaled compliant propulsor capable of mimicking the spatiotemporal asymmetries of the beat cycle observed in ctenophores. A flexible paddle is connected to a  motor at the base, which controls the stroke's Ta and the overall Re. The Sa of the stroke is encoded into the structure and geometry of the paddle. This construction allows us to explore the flows generated by spatiotemporally asymmetric, flexible paddles across a range of intermediate Re. Improving our understanding of metachronal rowing at intermediate Re could lead to the design of bioinspired sensors, pumps, or mixers for micro-fluidic platforms.

    

Hydrodynamics of Metachronal Rowing at Low-to-Intermediate Reynolds Numbers

Metachronal rowing is commonly found among small swimming invertebrates. Animals that locomote via this mechanism feature rows of appendages that perform propulsive strokes sequentially with a constant phase lag from their neighbor. In this study, ctenophores (comb jellies, the largest animals in the world to locomote via cilia) are used to explore the effects of varying propulsive configuration and Reynolds numbers on hydrodynamics. We reconstruct the beating motion of ctenophore appendages based on high-speed video recordings. To model the metachronal wave and the movement of appendages, beating kinematics are represented by the truncated Fourier series. An in-house immersed-boundary-method-based computational fluid dynamics solver is used to simulate the flow field and associated hydrodynamic performance. A parametric study is conducted to investigate the vortex structures and propulsive characteristics of metachronal rowing over a range of key parameters, including the number of appendages, phase lag, space between neighbor appendages, and Reynolds number. Our simulation results aim to provide fundamental fluid dynamic principles for guiding the design of bio-inspired miniaturized robots swimming in the low-to-intermediate Reynolds number regime.