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Fluid Flows Created by Swimming Bacteria Drive Self-Organization in Confined Suspensions
Enkeleida Lushi, Hugo Wioland, Raymond E Goldstein
TL;DR
The paper addresses whether swimmer-generated hydrodynamic flows, rather than steric interactions alone, drive self-organization in confined bacterial suspensions. Using a minimal particle–fluid model and experiments, it shows that hydrodynamics produces the spiral vortex and that bulk bacteria swim opposite to the collectively generated flow.
Problem
It remained unclear how much swimmer-generated hydrodynamic flow contributes to self-organization beyond purely steric interactions, including the swimming direction within the spiral vortex.
Method
The study combines a minimal two-dimensional model of oriented swimmers with direct and hydrodynamic interactions, circular confinement, simulations, and experiments tracking cell bodies and flagella.
Results
Hydrodynamics is necessary and crucial for reproducing the confined spiral vortex and double circulation, while experiments confirm that bulk bacteria point outward and swim against the collectively generated flow.
Takeaways & Limitations
The findings demonstrate that cell orientation and swimmer-generated fluid motion can oppose one another in dense confined bacterial suspensions.
Abstract
from arXiv · showhide
Concentrated suspensions of swimming microorganisms and other forms of active matter are known to display complex, self-organized spatio-temporal patterns on scales large compared to those of the individual motile units. Despite intensive experimental and theoretical study, it has remained unclear the extent to which the hydrodynamic flows generated by swimming cells, rather than purely steric interactions between them, drive the self-organization. Here we utilize the recent discovery of a spiral-vortex state in confined suspensions of \textit{B. subtilis} to study this issue in detail. Those experiments showed that if the radius of confinement in a thin cylindrical chamber is below a critical value the suspension will spontaneously form a steady single-vortex state encircled by a counter-rotating cell boundary layer, with spiral cell orientation within the vortex. Left unclear, however, was the flagellar orientation, and hence the cell swimming direction, within the spiral vortex. Here, using a fast simulation method that captures oriented cell-cell and cell-fluid interactions in a minimal model of discrete-particle systems, we predict the striking, counterintuitive result that in the presence of collectively-generated fluid motion the cells within the spiral vortex actually swim upstream against those flows. This is then confirmed by new experiments reported here, which include measurements of flagella bundle orientation and cell tracking in the self-organized state. These results highlight the complex interplay between cell orientation and hydrodynamic flows in concentrated suspensions of microorganisms.
I. MODEL AND SIMULATIONS
The model represents confined swimmers as oriented particles coupled through steric interactions and collectively generated pusher-like fluid flows. It combines swimmer motion, fluid advection, rotation, repulsion, and an approximate circular-boundary condition.
- Swimmers are modeled as oriented ellipses or disks that self-propel, advect with the fluid, rotate in flow, and interact through repulsive forces and torques.Ellipses have length ℓ=1 and width w=ℓ/6; disks use w=ℓ.
- The simulations assume planar dynamics and use a circular domain containing M pusher swimmers to represent the confined suspension.No vertical swimmer motion was observed experimentally, motivating the two-dimensional implementation.
- The swimmer equations couple translational motion to self-propulsion, fluid advection, and pairwise repulsion, while orientation dynamics include flow-induced rotation and neighbor torques.The model uses constant nondimensional swimming speed V=1.
- The fluid velocity is generated by the superposition of pusher-like dipolar flows from all swimmers and is computed from two-dimensional Stokes equations with active stress.The active stress has non-dimensional strength α ≈−1 for pusher swimmers.
- A method-of-images construction approximates confinement and a no-stress condition at the circular oil–water boundary while keeping integration restricted to actual swimmers.Mirror swimmers provide steric forces, torques, and fluid velocity near the boundary.
II. SIMULATION RESULTS
Simulations show that hydrodynamics destabilizes unconfined ordered states but produces the experimentally observed spiral vortex and double circulation under circular confinement. The large-scale organization depends on collectively generated fluid flow, while steric interactions provide local alignment.
- Hydrodynamics destabilizes low-concentration swarming and high-concentration bionematic states in unconfined periodic suspensions, producing turbulent dynamics.Without hydrodynamics, the corresponding systems display swarming or stable bionematic order.
- Under confinement, direct interactions alone yield boundary concentration or jamming with only small unidirectional circulation, whereas realistic elongated swimmers form a spiral vortex.The realistic case includes direct and hydrodynamic interactions.
- Steric interactions force local alignment, but collectively generated fluid flow produces the large-scale organization and double circulation.Disk-shaped swimmers with hydrodynamic reorientation and advection lose neighboring alignment and show only unstable layers with weak transient circulation.
- Spiral organization and double circulation persist when flow alignment is retained without steric alignment, and also arise for dilute ellipse-shaped swimmers.The latter additionally self-generates bidirectional fluid flows.
III. EMERGENCE OF ORGANIZATION
The spiral vortex emerges from the interplay of boundary-driven steric organization and collectively generated flows, with bulk cells swimming outward against opposing fluid motion.
- Origin of the spiral vortex: Swimming and advection contribute jointly to bacterial motion, leaving the direction of cell pointing unresolved in earlier PIV measurements.PIV established opposite boundary-layer and bulk circulation but could not determine whether swimming or advection dominated overall motion.
- Origin of the spiral vortex: As density increases, boundary-sliding clusters merge into an outer circulating layer, while pusher-generated flow moves oppositely to swimmer circulation.Cells point outward at a curvature-dependent angle, and their backward-pushed fluid produces bulk flow opposite to swimmer circulation.
- Bulk organization: In dense drops, steric interactions create additional angled layers, while almost all cells point outward and swim together rather than inward.The outward-pointing bulk cells generate a fluid flow opposite their orientation, strong enough in the inner drop to counterbalance swimming speed.
- Bulk organization: Central cells can therefore move counterclockwise while pointing clockwise, demonstrating upstream swimming against the collectively generated flow.This predicted arrangement reproduced the observed macroscopic suspension dynamics and motivated experiments to determine actual cell orientation.
IV. EXPERIMENTS
Fluorescent labeling and imaging distinguish bacterial orientation from motion, while simulations and experiments compare swimmer and fluid circulation across configurations.
- Simulation–experiment comparison: Figure 2 compares suspension organization for steric-only ellipses, hydrodynamic swimmers, circular pushers, flow-aligned pushers, dilute ellipsoids, and experimental bacterial flow.Insets indicate swimmer circulation and mean motion, while lower-left insets show fluid velocity for the confined swimmer cases.
- Orientation measurements: At the interface, a bacterium points and moves toward the top-left, whereas a bulk bacterium points top-left but moves toward the lower right.The bulk example directly demonstrates backward motion relative to the flagella-defined swimming direction.
- Experimental confirmation: Sampling more than 20 cells confirmed outward pointing throughout the vortex, with bulk motion opposite to the boundary-layer motion.The observations confirm the simulation prediction of backward bulk microswimmer motion.
V. QUANTIFICATION OF SPATIAL ORDER
The vortex order parameter quantifies how confinement size and suspension density organize circulation, while orientation angles show systematic size dependence and model–experiment differences.
- Vortex order parameter: The vortex order parameter Φ measures azimuthal organization: Φ = 1 denotes purely azimuthal flow, Φ = 0 disorder, and Φ < 0 mostly radial flow.It is evaluated for drop diameters from 4ℓ ≈ 20 µm to 25ℓ ≈ 125 µm in dense and semi-dilute suspensions.
- Vortex order parameter: A transition from random motion to a vortex state occurs around d = 7ℓ.This transition is identified in the simulations’ diameter-dependent order measurements.
- Vortex order parameter: Φ > 0.7 occurs for dense simulated drops with d = 7−16ℓ, corresponding to approximately 30−70 µm in experiments.The experimental and simulated diameter ranges are compared for highly ordered single-vortex states.
- Density and confinement: Turbulence appears in the centers of the largest drops, whereas cell-depleted centers in dilute suspensions permit ordered states beyond d > 14ℓ.Density therefore changes the diameter range over which ordered circulation persists.
- Swimmer orientation: Boundary-layer orientation decreases with drop size, ranging from θm ≃35° to 10° experimentally and from θm ≃42° to 36° in simulations.The model reproduces the qualitative size trend but not the experimental angles quantitatively because of simplifying assumptions.
VI. DISCUSSION
The minimal model reproduces key features of confined bacterial organization and shows that hydrodynamics are crucial for large-scale order. Experiments confirm that bulk cells swim against the collectively generated fluid flow, while the model remains simplified in important physical respects.
- The minimal model includes direct cell-cell interactions, cell-fluid interactions, and swimmer-generated flows.
- Hydrodynamic interactions are crucial for reproducing the organization and double circulation observed experimentally.
- Under circular confinement, large-scale order appears only when swimmer motion is coupled to fluid dynamics.
- Bulk cells swim against the collectively generated fluid flow, producing a net backward motion that experiments confirm by tracking cell bodies and flagella.
- Closer experimental comparison requires three-dimensional simulations, more accurate swimmer-generated flows, and potentially swimmer geometry and flagella.
A. Experimental Protocol
The experiments prepare dense, flattened B. subtilis drops, deliberately disorganize them with blue light, and image their collective motion and cell–flagella orientation.
- Wild-type and flagella-labelable mutant B. subtilis strains are grown in Terrific Broth at 35°C through exponential growth.
- Flagella are stained with Alexa Fluor 488, while bacterial membranes are stained with FM4-64.
- Dense bacterial suspensions are mixed into mineral oil containing DiPhyPC and placed between coated coverslips to form flattened drops.
- Bright-field movies are recorded at 125 fps, and blue-light-induced tumbling is used to disorganize drops before observing order emergence.
- Confocal images acquired every 0.1 s resolve stained cell bodies and flagella for measuring swimming and motion directions.
B. Simulations
The simulations model steric interactions between discretized swimmer beads with a capped Lennard-Jones potential whose smoothing permits larger integration timesteps.
- Each swimmer is discretized into beads, and beads belonging to different swimmers interact through a capped Lennard-Jones potential.
- The interaction is active when bead-center distance r is at most rc and zero when r exceeds rc.
- The cutoff is rc = 2rb, with α defined as the capping or smoothing factor.
- Smoothing the potential allows larger integration timesteps but permits swimmer overlap or possible escape from confinement.