Edge current and pairing order transition in chiral bacterial vortices

Significance

The last two decades have seen an exciting development in the field of the active turbulence of microswimmers and their control using geometric design. While being recognized as a key feature of active turbulence, collective effects in the chiral active matter have been overlooked. We shed light on the poorly understood interplay between individual chiral swimming and axisymmetric boundary conditions mediated by polar alignment by exhibiting and controlling chiral bacterial vortices. The geometry-independent collective motion that appears near the boundary, named edge current, plays an important role and alters the geometric rule of the pairing of chiral vortices. Our findings provide a physical basis for understanding out-of-equilibrium structures in chiral microswimmers and advancing material transport engineering by microrobots in liquid.

Abstract

Bacterial suspensions show turbulence-like spatiotemporal dynamics and vortices moving irregularly inside the suspensions. Understanding these ordered vortices is an ongoing challenge in active matter physics, and their application to the control of autonomous material transport will provide significant development in microfluidics. Despite the extensive studies, one of the key aspects of bacterial propulsion has remained elusive: The motion of bacteria is chiral, i.e., it breaks mirror symmetry. Therefore, the mechanism of control of macroscopic active turbulence by microscopic chirality is still poorly understood. Here, we report the selective stabilization of chiral rotational direction of bacterial vortices in achiral circular microwells sealed by an oil/water interface. The intrinsic chirality of bacterial swimming near the top and bottom interfaces generates chiral collective motions of bacteria at the lateral boundary of the microwell that are opposite in directions. These edge currents grow stronger as bacterial density increases, and, within different top and bottom interfaces, their competition leads to a global rotation of the bacterial suspension in a favored direction, breaking the mirror symmetry of the system. We further demonstrate that chiral edge current favors corotational configurations of interacting vortices, enhancing their ordering. The intrinsic chirality of bacteria is a key feature of the pairing order transition from active turbulence, and the geometric rule of pairing order transition may shed light on the strategy for designing chiral active matter.

Turbulent flows offer a rich variety of structures at large length scales and are usually obtained by driving flows out of equilibrium (1) while overcoming viscous dampening. A peculiar class of out-of-equilibrium fluids from self-propelled colloids to microswimmers and animals, stimulated from the lower scales, also present turbulence-like structures called active turbulence (2–4). For example, a dense bacterial suspension is driven out of equilibrium by the autonomous motion of bacteria suspended therein (5–7). The collective swimming of bacteria shapes the active turbulence into vortices of similar size (8–14). However, this vortical order decays over distance, making it a long-standing issue for the development of ordered dynamics at larger scales. Hence, growing attention is paid to novel strategies to control active turbulence with simple geometric design.

Chirality, i.e., the nonequivalence of opposite handedness, is also ubiquitous across scales (15) and is commonly involved in active systems (16–19), either biological, such as bacteria (20–24), cytoskeletons (25–27), and fish (28), or nonbiological, consisting of self-propelled colloids (29–32). One of the effects of chirality is the nonequivalence of clockwise (CW) and counterclockwise (CCW) rotations. As for bacteria, broken mirror symmetry in flagellar rotation (CCW rotation around the tail-to-head direction during swimming) results in the opposite rotation of the cell body, which generates a net torque onto the solid surface the bacterium swims over, and in turn bends its trajectory circularly (20). Despite such intrinsic chirality in individual motion, active turbulence reported in the past showed CW and CCW global rotational directions have equal probability, indicating that mirror symmetry was recovered at the collective level (9–11, 13). Can microscopic chirality of bacterial motion be transferred into the macroscopic order of collective swimming? Such a question is a great challenge that would provide both fundamental understanding of active turbulence and technical applications for controlled material transport (33, 34).

In this study, we report the chiral collective swimming of a dense bacterial suspension confined in an asymmetric (different top and bottom interfaces) but achiral (perfectly circular lateral interface) boundary, with a strongly favored rotational direction. That nonequivalence between CW and CCW collective swimming reflects the interplay between the counterrotating collective bacterial motions at each of the top and bottom interfaces. The increase of bacterial density strengthens that interplay, and CCW rotation (with respect to the bottom–top direction, later referred to as “top view”) predominates. Furthermore, we found that the collective swimming of bacteria at the top and bottom interfaces appears near the lateral boundaries and takes the form of an edge current. Its robustness with respect to the shape of the lateral boundary alters the geometric constraints ruling the self-organization of interacting bacterial vortices by promoting corotational configurations. The obtained geometric rule, in excellent agreement with experiments, brings understanding of chiral active matter in order to organize larger-scale flow.

Results and Discussion

Chiral Bacterial Vortex.

A bacterial suspension of Escherichia coli RP4979 $3.5\pm 0.6$ $\mathrm{\mu }\text{m}$ in length was confined in microwells with a height of $h=20\hspace{0.17em}\mathrm{\mu }\text{m}$, made of polydimethylsiloxane (PDMS) rendered hydrophilic with a polyethylene glycol coating and sealed with an oil/water interface stabilized with a surfactant 2% SPAN80 (Fig. 1A and SI Appendix, Fig. S1) (35). Top–bottom asymmetry (later referred to as “asymmetric conditions”) presents in the water/PDMS bottom interface of the microwell and its top oil/water interface, both of which are nonslip boundary conditions (36).

Fig. 1.

Chiral bacterial vortex. (A) Experimental setup: a dense bacterial suspension confined in hydrophilic-treated PDMS microwells and sealed under an oil (32 mPa$\cdot$s)/water (0.8 mPa$\cdot$s) interface stabilized with a surfactant. PEG, polyethylene glycol. (B) Ensemble of chiral bacterial vortices, in microwells with radii $R=35\hspace{0.17em}\mathrm{\mu }\text{m}$ (Top) and $R=20\hspace{0.17em}\mathrm{\mu }\text{m}$ (Bottom) and a height $h=20\hspace{0.17em}\mathrm{\mu }\text{m}$. The bacterial density is 20% vol/vol. Color map codes for the direction of the velocity field. (Scale bar, $100\hspace{0.17em}\mathrm{\mu }\text{m}$.) Schematic illustration of a bacterial vortex in a single microwell. “+” defines the positive sign of CCW rotation. CCW and CW occurrences are displayed with blue and red arrows, respectively.

Bacteria collectively move following the lateral boundary of the microwell, with a surprisingly selective CCW rotational direction (top view, Movie S1). The orientation map ${\theta }_i$ of the velocity field $\mathit{v}\left({\mathit{r}}_i\right)$ obtained from particle image velocimetry shows vortical structure maintained across the microwell of $R=20\hspace{0.17em}\mathrm{\mu }\text{m}$ (Fig. 1B). A CCW-biased vortex (called “chiral bacterial vortex” hereafter) occurs without built-in chirality of the confinement, e.g. ratchets (33, 34). CCW rotation is strongly favored at 95% probability ($N=145$; SI Appendix, Fig. S2) in the chiral bacterial vortex, while the bacterial vortex rotates at equal probability in CCW or CW in water-in-oil droplets squeezed between two glass slides (symmetric conditions; SI Appendix, Fig. S2). Flow reversal in the chiral vortices was not observed, which emphasizes that a chiral bacterial vortex is more stable than the bacterial vortices in droplets (8, 9, 11). Chiral ordered motion is also observed in larger microwells ($R=50\hspace{0.17em}\mathrm{\mu }\text{m}$; Fig. 2A) but only within the edge layer 10 $\mathrm{\mu }\text{m}$ away from the circular boundary, which is defined as an edge current.

Fig. 2.

Chiral edge current. (A) Color map of the orientation of collective motion in a chiral bacterial vortex ($R=50\hspace{0.17em}\mathrm{\mu }\text{m}$). The edge layer, defined as the area within $10\hspace{0.17em}\mathrm{\mu }\text{m}$ from the boundary, is separated from the rest of a microwell (the bulk) by a dashed white line. The height of the microwell is $20\hspace{0.17em}\mathrm{\mu }\text{m}$. The bacterial density is 20% vol/vol. (Scale bar, $20\hspace{0.17em}\mathrm{\mu }\text{m}$.) (B) Normalized azimuthal velocity in microwells of various sizes. The blue circles indicate the edge current, and the green squares indicate the motion in bulk. $v_{\theta }$ is averaged over 10 s and plotted with error bars representing SD. (C) Schematic illustration of annulus microchannels within asymmetric conditions. (D) Color map of the orientation of the velocity field in collective motion in annulus microchannels. The height of annulus microchannels is $h=20\hspace{0.17em}\mathrm{\mu }\text{m}$. The bacterial density is 20% vol/vol. (Scale bar, $50\hspace{0.17em}\mathrm{\mu }\text{m}$.) (E) Edge currents in annulus channels with different curvatures. Normalized azimuthal velocities averaged over 10 s are plotted. The blue circles and squares (red circles and squares) indicate the edge current in a CCW direction (in CW direction) that emerges in the channels with the width $w=\hspace{0.17em}50\hspace{0.17em}\mathrm{\mu }\text{m}$ and $w=\hspace{0.17em}100\hspace{0.17em}\mathrm{\mu }\text{m}$, respectively.

Edge Current in Chiral Bacterial Vortex.

We in turn examined the size dependence of the edge current in order to investigate the mechanism of such stability and selectivity in rotational direction. We define the tangential vector $\mathit{t}\left({\theta }_i\right)$ in a CCW direction along the circular boundary and the azimuthal velocity $v_{\theta }\left({\mathit{r}}_i\right)=\mathit{v}\left({\mathit{r}}_i\right)\cdot {\mathit{t}}_i$. The ordered orientation of the edge current along the boundary wall is analyzed by $⟨v_{\theta }\left({\mathit{r}}_i\right)⟩/⟨\left|\mathit{v}\right|⟩$, where $⟨\cdot ⟩$ denotes the average over all possible sites $i$. This edge current is maintained even in very large microwells ($R\ge 100\hspace{0.17em}\mathrm{\mu }\text{m}$), the size of which is much larger than the critical size of a stable bacterial vortex in the bulk ($\approx 35\hspace{0.17em}\mathrm{\mu }\text{m}$) (Fig. 2B). On the other hand, under symmetrical conditions, the collective motion along the boundary disappeared as the turbulent state developed in the large microwells (SI Appendix, Fig. S3), which indicates that the edge current appears selectively under asymmetric conditions.

A key feature in chiral many-body systems is that chiral motion with a coherent orientation is robust to changes in geometry, such as the curvature of the wall (29–31). To examine whether the edge current depends on confinement geometry, we also analyzed collective motion in annulus microchannels (with widths $w=50\hspace{0.17em}\mathrm{\mu }\text{m}$ and $100\hspace{0.17em}\mathrm{\mu }\text{m}$), where the disturbance from active turbulence can be suppressed. The microchannels, in which a bacterial suspension was confined in narrow rings within asymmetric conditions, are arranged concentrically, allowing us to explore curvatures from 0.2 $\times 10^{-3}$ $\mathrm{\mu }{\text{m}}^{-1}$ to 1.0 $\times 10^{-2}$ $\mathrm{\mu }{\text{m}}^{-1}$ (Fig. 2C). Collective motion occurs in the CCW direction near the outer channel walls seen from the center of the circle (Fig. 2D). On the other hand, the direction of collective motion is CW near the inner channel wall as seen from the center of the circle (Fig. 2D). Furthermore, the edge currents are stable in both directions and do not disappear regardless of the curvature of the microchannels (Fig. 2E). These edge currents occur in a wide range of annulus curvatures, indicating that the chiral edge current is a robust phenomenon regardless of the confinement geometry under asymmetric conditions.

This persistence of an edge current motivates us to further investigate its physical origin by analyzing the interplay between the boundary and the intrinsic chirality of bacteria. With respect to the tail-to-head direction, the flagella of the bacteria rotate in the CCW direction. Torque balance then imposes a CW rotation of the body. These two opposite rotations ultimately convert into the CCW rotation (top view) of bacteria swimming near the top interface (Fig. 3A) and CW rotation near bottom interface (Fig. 3B) (20). In order to clarify the role of the lateral boundary in the chiral collective motion in dense bacterial suspensions, we visualized individual bacteria that express fluorescent proteins swimming in the edge layer. We investigated the single bacteria that swim in a curved trajectory in a dilute suspension (0.01% vol/vol). When bacteria collide with the wall, those that swim at the top interface move along the lateral boundary in the CCW rotation (Fig. 3C), whereas the bacteria at the bottom interface swim in the CW rotation (Fig. 3D). Their rotations are opposite from the top view, meaning that there is no overall symmetry breaking.

Fig. 3.

Chiral edge current as the interplay between the top and bottom interfaces. The radius of microwells is $R=35\hspace{0.17em}\mathrm{\mu }\text{m}$. (Scale bars, $10\hspace{0.17em}\mathrm{\mu }\text{m}$.) (A and B) Schematic illustrations of chiral bacterial swimming (A) near the top oil/water interface and (B) near the bottom water/PDMS interface. Near the top interface, individual bacterial swimming is CCW-biased (blue arrow), while near the bottom interface it is CW-biased (red arrow). (C) Chiral swimming patterns near the wall of a single bacterium beneath the top interface in a dilute condition (0.01% vol/vol, Left) and in a dense condition (20% vol/vol, Right). The bacterial swimming is measured from the top view; their trajectories are indicated by solid lines. The white arrow indicates the heading angle at the end of the track. Scale bars, 10 $\mathrm{\mu }$m. (D) Swimming patterns near the wall of individual bacteria above the bottom interface in dilute (0.01% vol/vol, Left) and dense (20% vol/vol, Right) suspensions. Scale bars, 10 $\mathrm{\mu }$m. (E) Color map of the direction of collective motion at the top and bottom interfaces in microwells of different heights $h$. The bacterial density is 20% vol/vol. (Left) $h=20\hspace{0.17em}\mathrm{\mu }\text{m}$, (Middle) $h=32\hspace{0.17em}\mathrm{\mu }\text{m}$, (Right) $h=49\hspace{0.17em}\mathrm{\mu }\text{m}$. White dashed line indicates the edge layer. (F) Fraction of CCW rotation of individual bacteria within the edge layer in microwells of different heights. Triangle: top interface; inverted triangle: bottom interface. CCW and CW preferences are displayed with blue and red symbols, respectively. (G) The relative number density of bacteria (the ratio of top to bottom) in microwells of different heights. (H) Fraction of CCW rotation of individual bacteria within the edge layer at different bacterial densities. The height is $h=20\hspace{0.17em}\mathrm{\mu }\text{m}$. Sample size and azimuthal velocity of each case in F–H can be found in SI Appendix, Fig. S6. (I) A schematic model for the chiral edge current.

In contrast, in a dense suspension of 20% vol/vol, we labeled a small fraction (0.7% vol/vol) of swimming bacteria with fluorescent protein in order to visualize single bacteria. Bacteria swimming in the edge layer can turn to a chiral collective movement in CCW rotation, regardless of the interface they are at (Fig. 3 C and D). In particular, the edge current found there has the same handedness as the intrinsic chirality of bacterial swimming. Bacteria near the lateral boundary are constrained to a CCW rotation along that boundary, due to the torque-free condition (22), at the top interface (CW rotation at the bottom) (SI Appendix, Fig. S4). This generates two opposite chiral collective motions at either of the top and bottom interfaces. However, this effect alone does not explain the net CCW rotation observed in an edge current.

The Interplay of Collective Motions between Two Interfaces Selects Net Chirality.

To find the origin of the net chirality, we examined the interplay between the collective motions at both interfaces by changing the height of the microwells, $h$. In microwells with $h=20\hspace{0.17em}\mathrm{\mu }\text{m}$, a vortical pattern in CCW rotation was observed at both interfaces (Fig. 3E, Left). The height of $20\hspace{0.17em}\mathrm{\mu }\text{m}$ is close to the correlation length of bacterial collective motion $\xi \approx 18\hspace{0.17em}\mathrm{\mu }\text{m}$ (SI Appendix, Fig. S5), indicating that the bacteria at the bottom interface follow the same CCW rotation as those at the top interface. However, collective motion at the bottom interface became disordered at $h=32\hspace{0.17em}\mathrm{\mu }\text{m}$ (Fig. 3E, Center) and even at $h=49\hspace{0.17em}\mathrm{\mu }\text{m}$ such that the correlation between the two interfaces was lost; the vortical pattern on the bottom interface took a CW direction while the top interface kept a CCW rotation (Fig. 3E, Right).

We then recorded the trajectories of individual bacteria in a dense suspension at the top and bottom interfaces to reveal why the global rotation direction is always the same as that of the CCW collective motion at the top interface. The bacterial tracking analysis clarifies that bacteria within the edge layer of $10\hspace{0.17em}\mathrm{\mu }\text{m}$ exhibit a predominantly CCW rotation near both the top and bottom interfaces at $h=20\hspace{0.17em}\mathrm{\mu }\text{m}$, although the top interface was more biased toward the CCW direction (Fig. 3F). When the height of the microwells became larger than the correlation length, bacteria beneath the top interface maintained a CCW rotation ($\approx$82% fraction) at any height, whereas bacteria swimming at the bottom interface were achiral at $h=32\hspace{0.17em}\mathrm{\mu }\text{m}$ and then changed their rotational preference from CCW to CW rotation at $h=49\hspace{0.17em}\mathrm{\mu }\text{m}$ (Fig. 3F). Although the average speeds $⟨\left|v\right|⟩$ at both interfaces were comparable (top: 10.9 $\mathrm{\mu }$m/s and bottom: 11.0 $\mathrm{\mu }$m/s at $h=49\hspace{0.17em}\mathrm{\mu }\text{m}$), bacterial density was $\simeq$1.2 times larger near the top oil interface rather than near the bottom PDMS (Fig. 3G). This nonequivalence reflects the dominant role of the top oil interface and determines the global rotational direction of the opposing chiral collective motions.

To further analyze the interplay between the two interfaces, we examined the effect of bacterial density under $h=20\hspace{0.17em}\mathrm{\mu }\text{m}$ (Fig. 3H). At 0.01% vol/vol, bacteria rotationally swim in contact with the boundary wall in CCW direction at the top but in opposite CW direction at the bottom, suggesting that the top and bottom interfaces were not fully correlated at a low density. When the density exceeds 11% vol/vol, the fraction of CCW rotation within the edge layer becomes dominant at both interfaces. Hence, under the asymmetric condition at a large density, the bacterial interaction between the two interfaces holds bacteria to follow the circular boundary within the edge layer. The collective bacterial swimming at a high density shows a symmetry breaking state, where either CW or CCW direction is chosen out of the competition between top and bottom interfaces. Thus, the overall direction can be controlled by a small perturbation. For this reason, the flow is CCW (top) and CW (bottom) when the interplay is weak between two interfaces in a deeper microwell, and no global collective motion arises. Once this interplay is increased, a slight difference in the density leads to aligned configurations of the edge current with CCW handedness (Fig. 3I).

We also investigated the effects of the oil viscosity and the boundary conditions at the top interface. The mineral oil we used in Figs. 1–3 contains surfactants and is significantly more viscous than water. The presence of surfactants should prevent the in-plane shear deformation because of counteracting Marangoni flow (37) and, together with higher viscosity, would ensure a nonslip boundary (36). To change the physical properties of the top interface, we first removed the surfactant and then we used a lower-viscosity oil. We observed the same CCW chiral vortex in the absence of surfactant in a higher-viscosity mineral oil (Table 1 and SI Appendix, Fig. S8). The bacterial density is slightly higher at the top interface, indicating that the surfactant does not contribute to the predominance of the top interface in the formation of the chiral bacterial vortex. When the top medium consists of hexadecane (2.7 mPa$\cdot$s)—10 times less viscous than the mineral oil used—the predominance of the top interface remained unaffected, and the bacterial density was also slightly higher at its vicinity (Table 1 and SI Appendix, Fig. S8). While the swimming speed is comparable at both interfaces among those conditions, the relative bacterial density is usually higher at the top interface and the global rotation of bacterial vortex was in a CCW direction. This result indicates that the viscosity of oil has little effect on the dominance of CCW collective motion. Even if the top and bottom interfaces were inverted with respect to the direction of gravity, the chiral collective motion on the oil/water interface remained the dominant one.

Table 1.

The properties of chiral swimming in various asymmetric microwells

		Swimming	Relative density	Selected
Material of the top interface	Chiral swimming	velocity $⟨v⟩$	at the top	chirality
Mineral oil with surfactant (32 mPa$\cdot$s)	Top: CCW	Top: 10.9 $\mathrm{\mu }$m/s	1.24 $\pm$ 0.12	CCW
	Bottom: CW	Bottom: 11.0 $\mathrm{\mu }$m/s
Mineral oil without surfactant (32 mPa$\cdot$s)	Top: CCW	Top: 10.2 $\mathrm{\mu }$m/s	1.30 $\pm$ 0.08	CCW
	Bottom: CW	Bottom: 11.0 $\mathrm{\mu }$m/s
Hexadecane without surfactant (2.7 mPa$\cdot$s)	Top: CCW	Top: 13.2 $\mathrm{\mu }$m/s	1.45 $\pm$ 0.40	CCW
	Bottom: CW	Bottom: 15.1 $\mathrm{\mu }$m/s
Air	Top: Equal mix of CCW and CW	Top: 8.7 $\mathrm{\mu }$m/s	1.11 $\pm$ 0.07	CW
	Bottom: CW	Bottom: 8.9 $\mathrm{\mu }$m/s

Numerical Simulation of Chirality Selection.

To determine the effect of bacterial density on this chirality selection we considered a theoretical model in which the collective motions at the two interfaces couple through polar alignment of chiral self-propelled particles (see Materials and Methods and SI Appendix).

We assume that self-propelled particles move at velocity $\mathit{v}$ at the top and bottom interfaces, while being confined to their circular boundary (Fig. 4A). The particles move on one of two interfaces and not across them so that the number of particles on each plane is constant with $N_t$ on the top and $N_b$ on the bottom. The motion of self-propelled particles from the top view is chiral in the CCW direction at the top interface and CW at the bottom. In addition, the motion of the particles is biased near the boundary in the CCW direction at the top (CW direction at the bottom) (Fig. 4A, along the boundary).

Fig. 4.

Numerical simulation of chiral bacterial vortex. (A) Schematic of the orientation interactions considered in the theoretical model. The upper panel shows the top interface, and the lower panel shows the bottom interface, both from the top view. The chiral self-propelled particles in blue move in the CCW direction, and the red particles move CW. The white arrows indicate the heading angles of the particles. (Left) Along the boundary: polar alignment of the particles near the boundary. The particles move in CCW on the top interface and CW on the bottom interface along the boundary, respectively. (Center) At the same interface: polar alignment of particles moving at the same interface. The radius of the dotted circle indicates the range of the polar alignment, $\epsilon$ ($\epsilon =1$). The particles change their heading angles in the mean direction (dotted arrow). (Right) Between two interfaces: polar alignment of particles between the top and bottom interfaces. The projected area in yellow indicates the effective range of the polar alignment, ${\epsilon }_{tb}\le 1$, projected to the facing interface. The colored arrows represent the rotational velocity of the chiral motion. (B) Numerical simulation of chirality selection in the coupled collective motion of chiral self-propelled particles at various ${\epsilon }_{tb}$. (C) Fraction of CCW rotation of individual particles within the edge layer at various ${\epsilon }_{tb}$. Circle: top interface; asterisk: bottom interface. CCW and CW preferences are displayed with blue and red symbols, respectively.

For orientation interactions, the particles in the same interface have orientation interactions to align their heading angles when the interparticle distance is within the effective range of $\epsilon$ (Fig. 4A, at the same interface). In addition, the polar alignment also works between the particles on the top and bottom interfaces, and its effective distance projected to the opposite interface is ${\epsilon }_{tb}$ (Fig. 4A, between two interfaces). We consider that the effective range of polar alignment $\epsilon$ is longer than that across two interfaces ${\epsilon }_{tb}$, and numerical simulation was performed at $\epsilon =1$ and ${\epsilon }_{tb}\le 1$. Increasing ${\epsilon }_{tb}$ allows the particles to align their orientations easily over the two interfaces. In this respect, it corresponds to the experimental condition where the height of the microwell is reduced so that the collective motions on the two interfaces are correlated with each other.

We performed numerical calculations with a density ratio $N_t/N_b=1.22$ (N_t = 5,500 and N_b = 4,500; Fig. 4B), referring to the experimental values at the oil/water interface (Table 1). At a small ${\epsilon }_{tb}$ (${\epsilon }_{tb}=0.01$) the particles do not build orientation interactions in the top and bottom interfaces and show chiral motion in different directions. As ${\epsilon }_{tb}$ becomes larger (${\epsilon }_{tb}=0.25$), the collective motion at the top interface partially aligns the orientation of the particles at the bottom interface to CCW rotation, but chiral rotation in the CW direction still remains. At a sufficiently large ${\epsilon }_{tb}$ (${\epsilon }_{tb}=0.5$), the two strongly correlated populations allow the chiral motion near the boundary to become a stable CCW-oriented edge current, and the collective motion at the top completely dominates over the CW rotation at the bottom interface. We plotted the fraction of particles rotating in the CCW direction with ${\epsilon }_{tb}$ and found that only the particles at the bottom turn into the CCW rotation (Fig. 4C), consistent with our experimental results.

Notably, when the top oil/water interface was changed to the air/water interface where bacteria showed an equal mix of CW and CCW chiral swimming, possibly due to a slip boundary condition, the net chirality of the bacterial vortex was reversed in the CW direction (Table 1). The achiral collective motion at the top interface was aligned with the CW-oriented collective motion at the bottom, leading to the CW chiral vortex, because the polar alignment of opposing chiral motions between the two interfaces is also a crucial factor in determining one global direction of rotation (Fig. 4A). The numerical simulation explains that the chirality bias in the CW direction at the bottom interface can break the equal mix of CW and CCW rotations at the top and align the overall rotation in the CW direction, though the density of particles is slightly higher at the top interface (N_t = 5,300 and N_b = 4,700; SI Appendix, Fig. S9).

Edge Current-Induced Pairing Order Transition of Chiral Vortices.

Corroborating microwell experiments and numerical simulation, we have shown that chiral vortices with an edge current emerge from the interplay of collective motions with opposite chiral preferences. The next question is whether the robust nature of the overall chirality, independent of the curvature of the boundary, controls the self-organization of bacterial vortices. Although engineering the dynamics of active turbulence under geometrical constraints has been extensively studied (9, 10, 13), the chirality of vortices with edge currents remains challenging. Furthermore, it is unclear what physical factors for the edge current can be used to control the chirality of active turbulence. To this end, we address these questions by examining the ordered patterns of chiral vortices in a doublet of overlapping circular microwells.

When two bacterial vortices interact with one another via near-field interaction, they have two types of pairing order, either the same rotational direction (ferromagnetic vortices, FMV) or opposite rotational directions (antiferromagnetic vortices, AFMV) in a geometry-dependent manner (9, 10, 13) (Fig. 5A). The chiral edge current can exist stably along the boundary (Fig. 2), but the pairing order transition is more sensitive to the boundary shape. If the edge current changes the FMV–AFMV pairing order transition, then by finding the shift of its transition point we can measure the strength of the chiral edge current and evaluate its effectiveness in controlling turbulent state.

Fig. 5.

Edge current favors corotational vortex pairing. (A) The corotational vortex pairing (FMV pattern, Top) and the antirotational pairing (AFMV pattern, Bottom). (B) Illustration and definition of relevant geometric parameters. (C) FMV pattern (Left) and AFMV pattern (Right) with edge current. The edge current deviates the orientation angle of bacteria $\theta$ around the tip. (D) Vorticity map of chiral bacterial vortex pairs at various $\mathrm{\Delta }$ with $R=19\hspace{0.17em}\mathrm{\mu }\text{m}$ and $h=20\hspace{0.17em}\mathrm{\mu }\text{m}$. The density of bacteria is 20% vol/vol. (E) The absolute value of the average vorticity of interacting bacterial vortices, with a CCW edge current (inverted black triangles) and without edge current (gray circles), against $\mathrm{\Delta }/R$. Error bar represents SD. Experimental data curves were tested against a sigmoidal function. The transition point of FMV–AFMV patterns is evaluated by the threshold $\left|⟨\mathbf{\Omega }⟩\right|=0.1$.

We construct a theoretical model of interacting chiral bacterial vortices in doublets of overlapping identical circular boundaries (a full description is given in SI Appendix). Two identical overlapping circular microwells, with a radius $R$ and an intercenter $\mathrm{\Delta }$, offer the means for a systematic investigation of the pairing order transition from FMV to AFMV (Fig. 5B) (10). The ratio $\mathrm{\Delta }/R=2\cos \mathrm{\Psi }$ is a geometric parameter characterizing this pairing order transition: If $\mathrm{\Delta }$/R is small enough, two vortices align in a corotational FMV state. In the absence of edge current, transition from FMV to AFMV occurs at ${\mathrm{\Delta }}_c/R=2\cos 45°=\sqrt{2}$ because that is the only configuration at which two pairing patterns are equiprobable (10). How does chiral edge current affect the previously established geometric rule?

To answer this question, the orientational dynamics of bacteria with a heading angle $\theta$ is considered at the vicinity of the sharp areas of the boundaries (“tip”). We consider the interacting chiral vortices where the collective motions at the top and bottom interfaces are fully correlated (${\epsilon }_{tb}\simeq \epsilon$). The CCW edge current correlated at the top and bottom interfaces reorients the bacteria at the tip. The effective torque for reorientation ${\tau }_e=-\partial U_e/\partial \theta$ comes from a geometry-dependent potential $U_e=-{\gamma }_e\cos \theta \sin \mathrm{\Psi }$ with the edge current’s coefficient ${\gamma }_e$. This reorientation favors the CCW rotation of the edge current in both FMV (Fig. 5C, Left) and AFMV (Fig. 5C, Right). In addition, vortex pairing is also affected by collisions near the tip between bacteria coming from different parts of the circle. Bacterial collision is ruled by a polar alignment as a source of geometry dependence (38). By considering the most probable configurations, the orientation near the tip is decided by the potential of FMV pairing $U_p^{FMV}=-{\gamma }_p\sin \mathrm{\Psi }$ at $\theta =0$ or of AFMV pairing $U_p^{AFMV}=-{\gamma }_p\cos \mathrm{\Psi }$ at $\theta =\pi /2$, with the strength of the polar alignment ${\gamma }_p>0$ (Fig. 5C). The respective sums of the potentials coincide at the transition point, i.e., $U_p^{FMV}+U_e{|}_{\theta =0}=U_p^{AFMV}+U_e{|}_{\theta =\frac{\pi }{2}}$, which leads to

$$\left({\gamma }_p+{\gamma }_e\right)\sin {\mathrm{\Psi }}_c={\gamma }_p\cos {\mathrm{\Psi }}_c,$$ [1]

where ${\gamma }_p+{\gamma }_e$ indicates the enhancement of FMV pairing by chiral edge current. Hence, ${\mathrm{\Delta }}_c/R=2\cos {\mathrm{\Psi }}_c$ at which FMV and AFMV pairings occur at equal probability is

$$\frac{{\mathrm{\Delta }}_c}{R}=\frac{2}{\sqrt{1+{\left(\frac{{\gamma }_p}{{\gamma }_p+{\gamma }_e}\right)}^2}}\simeq \sqrt{2}\left(1+\frac{{\gamma }_e}{2{\gamma }_p}\right).$$ [2]

The FMV pattern is stabilized in $\mathrm{\Delta }/R\ge \sqrt{2}$ and the relative strength of the edge current ${\gamma }_e/{\gamma }_p$ determines the shift of the transition point.

To test those chirality effects by evaluating the shift of the transition point we examined the vorticity $\mathbf{\Omega }\left({\mathit{r}}_i\right)=\nabla \times \mathit{v}\left({\mathit{r}}_i\right)$ of the doublet of chiral vortices with $R=19\hspace{0.17em}\mathrm{\mu }\text{m}$ and $h=20\hspace{0.17em}\mathrm{\mu }\text{m}$ within the range of $0\le \mathrm{\Delta }<2R=38\hspace{0.17em}\mathrm{\mu }\text{m}$. FMV pairing of chiral vortices is dominant in $0\le \mathrm{\Delta }/R\le 1.9$ and exhibits CCW rotation (Fig. 5D). The transition to AFMV pairing is also analyzed by using the absolute value of the average vorticity inside the microwell, $\left|⟨\mathrm{\Omega }⟩\right|$, which goes down to 0 for AFMV. Under asymmetric conditions, the absolute value of average vorticity shows a transition to 0 at ${\mathrm{\Delta }}_c/R\simeq 1.8$ to 1.9, while it occurs at ${\mathrm{\Delta }}_c/R\simeq 1.3$ to 1.4 under symmetric conditions (Fig. 5E). FMV pairing is favored in the presence of edge current, while one of the CCW rotations can be reversed by AFMV pairing of chiral vortices.

Furthermore, the variance in the orientation angle in AFMV pattern was ${\sigma }_{AFMV}^2=0.30$ at $\cos \mathrm{\Psi }=0.96$, which is given by ${\sigma }_{AFMV}^2=\frac{D}{{\gamma }_p\cos \mathrm{\Psi }}$ because the angular fluctuation of individual bacteria $D=0.12\hspace{0.17em}{\text{s}}^{-1}$ is changed at the tip of angle $\mathrm{\Psi }$ by the polar alignment ${\gamma }_p$ (SI Appendix, Figs. S14 and S15). In addition, the variance in the orientation angle in FMV pattern was ${\sigma }_{FMV}^2=0.21$ at $\sin \mathrm{\Psi }=0.61$, which is written by $\frac{D}{\left({\gamma }_p+{\gamma }_e\right)\sin \mathrm{\Psi }}$. With the obtained coefficients ${\gamma }_p=0.40\hspace{0.17em}{\text{s}}^{-1}$ and ${\gamma }_e=0.51\hspace{0.17em}{\text{s}}^{-1}$, the transition point obtained without approximation by Eq. 2 is ${\mathrm{\Delta }}_c/R=1.83$, which is consistent with the value found in the geometric control (Fig. 5E). This quantitative agreement indicates that chiral edge current can suppress orientation fluctuations to the same extent as polar orientation, and it eventually determines pairing order transition.

Concluding Remarks

In conclusion, we revealed that confining a dense bacterial suspension in microwells with asymmetric condition but achiral circular lateral boundaries stabilizes a chiral vortex without changing the rotational direction. Alignment with the lateral boundary aids in inducing chiral bacterial swimming at the top oil/water interface (or bottom water/PDMS interface) in a CCW direction (or CW direction) from the top view. The interplay of chiral collective motions at the opposite interfaces selects the preference for CCW rotation owing to the preponderance in density at the top, and it is thus possible to generate the broken mirror symmetry through the coupling between both planes, without using built-in chiral ratchet (13, 33, 34, 39). The strength with which the edge current controls the active turbulence is important to understand the effect of chirality on collective motion. Moreover, counterrotating layers at lateral boundaries do not form in a suspension of shorter E. coli bacteria (8, 9, 11). Thanks to these properties, chiral bacterial vortices can rotate a rigid rod much longer than a bacterial body in a CCW direction over multiple rounds at 0.5 rad/s (SI Appendix, Fig. S5), offering simple and fast material transport without built-in chirality (39, 40). We note that the density of bacteria is usually higher at the top interface (oil/water and air/water) irrespective of its viscosity, slip conditions, and even gravity direction, implying that the slight density increase may be attributed to the biological effects of the proximity to an oxygen source (41, 42). Dissecting the key factors underlying chiral collective motions between top and bottom interfaces is also a future challenge to further develop the control of microswimmers and microrobots in liquid.

The chiral bacterial vortex with an edge current opens new directions for the tailoring of collective motion. Asymmetric but achiral boundaries are also involved in confined flow of bacterial colonies (22) and the flocking of swimming droplets (43). The edge current with a similar collective effect may advance the generic understanding of chiral collective behavior observed in such achiral condition (44, 45). Even in triplets of identical overlapping circular microwells, the pairing order transition from FMV to (frustrated) AFMV patterns is also shifted to higher values ${\mathrm{\Delta }}_c/R\simeq 1.8$ to 1.9 (SI Appendix, Fig. S10), suggesting that chiral bacterial vortex has fewer limitations from geometric frustration. Furthermore, stabilized corotational pairing order is a key to clarifying how a broad class of active matter amplifies microscopic chirality. As such pairing order is also relevant to chiral spinners (46) and programmable chiral states (47), controlling chirality-induced order with a simple geometric rule would verify the validity of the geometric approach.

Materials and Methods

Bacterial Strain.

We used bacterial strain E. coli RP4979, which lacks tumbling ability. For the quantitative recording of the trajectories of individual bacterial swimming in either dilute or dense suspensions, we used a mixture (29:1 ratio) of two genetically modified bacteria that constitutively express fluorescent protein (either yellow fluorescent protein or dTomato). Bright-field video microscopy was performed by using an inverted microscope (IX73, Olympus) with a charge-coupled device (CCD) camera (DMK23G445, Imaging Source) to record collective motion. The velocity field of bacterial collective motion $\mathit{v}\left(r,t\right)$ was analyzed by particle image velocimetry with the Wiener filter method. In addition, the recording of individual bacteria was done with a confocal microscope (IX73, Olympus, and confocal scanning unit CSU-X1 from Yokogawa Electric Co. Ltd., iXon-Ultra electron-multiplying CCD camera from Andor Technologies). All the recordings were done at 33 frames per second.

Device Microfabrication.

The microwells are fabricated using standard soft lithography techniques. The PDMS film is cut around a single array of microwells and peeled off to be bonded on its unpatterned surface to the glass cover slide. PDMS microwells are then treated hydrophilic with a solution of polyethylene glycol-poly-l-lysine (Nanocs, PG2k-PLY). Once a flow cell is ready, it is filled with 20 $\mathrm{\mu }$L of a bacterial suspension. After filling the microwell with bacteria, the oil (Sigma-Aldrich, light mineral oil) with surfactant (Nacalai, SPAN80) at 2.0 wt % is injected from the same side, while the excess bacterial suspension over the microwells is flushed out and absorbed with filter paper from the other side of the flow cell. This seals the microwells under an oil/water interface. To suppress unwanted flow in the flow cell, both of its ends are sealed with epoxy glue (Huntsman, Ltd.). The array of microwells is then ready to be observed under a microscope.

Numerical Simulation.

To examine the chirality selection in the numerical simulation we consider the theoretical model for chiral self-propelled particles that move at the same velocity $\mathit{v}$ at the top and bottom interfaces, while confined to their circular boundary with radius $R$. The m-th particle is defined with a position ${\mathit{r}}_m^z\left(t\right)=\left(x_m^z\left(t\right),y_m^z\left(t\right)\right)=r_m^z\left(\cos {\phi }_m^z,\sin {\phi }_m^z\right)$ and the orientation $\mathit{d}\left({\theta }_m^z\right)=\left(\cos {\theta }_m^z,\sin {\theta }_m^z\right)$ under a circular boundary. The superscript $z$ indicates whether the particle is at the top (i.e., ${\theta }_m^t\left(t\right)$) or bottom (i.e., ${\theta }_m^b\left(t\right)$). Here, only the orientation dynamics of the particles on the top interface is shown. See SI Appendix for a full description including parameter values. The rotational velocity of the particle is $\omega$ at the top interface. The particle moves along the boundary with a chiral bias at the coefficient ${\gamma }_e^t$, while the coefficient of nematic interaction with boundary is ${\gamma }_w$. The coefficient of polar alignment among the particles is given by ${\overline{\gamma }}_p$. The orientation dynamics of the particle at the top interface is given by the following equation:

$$\begin{array}{cc}\hfill & {\dot{\theta }}_m^t={\omega }_0^t-{\overline{\gamma }}_p\left[\sum _{r_{mn}^t<\epsilon }\sin \left({\theta }_m^t-{\theta }_n^t\right)+\sum _{r_{mn}^{tb}<{\epsilon }_{tb}}\sin \left({\theta }_m^t-{\theta }_n^b\right)\right]\hfill \\ \hfill & -\left[{\gamma }_e^t\sin \left({\theta }_m^t-\phi -\frac{\pi }{2}\right)+{\gamma }_w\sin 2\left({\theta }_m^t-\phi -\frac{\pi }{2}\right)\right]\mathrm{\Theta }\left(r_m^t-R\right)+{\eta }_m^t,\hfill \end{array}$$ [3]

where ${\eta }_m^t$ is angular fluctuation and $\mathrm{\Theta }\left(x\right)$ represents the Heaviside step function. The distance between two particles in different interfaces is defined as $r_{mn}^{tb}=|{\mathit{r}}_m^t-{\mathit{r}}_n^b|$.

Theoretical Model for Pairing Order Transition.

We consider a mean-field theoretical model of orientational dynamics by considering the motion of self-propelled particles. This model is simplified by considering the two interfaces to be well-correlated as ${\epsilon }_{tb}\approx \epsilon$ in Eq. 3, and also neglecting the nematic alignment that does not affect geometric dependence in the pairing order transition (10, 38). The dynamics of particle orientation at the tip of doublet circular boundary is given by

$${\dot{\theta }}_m=-{\overline{\gamma }}_p\sum _{r_{mn}<\epsilon }\sin \left({\theta }_m-{\theta }_n\right)-{\overline{\gamma }}_e\sum _{r_{mw}<\epsilon }\sin \left({\theta }_m-{\theta }_w\right)+{\eta }_m\left(t\right),$$ [4]

where $r_{mn}=|{\mathit{r}}_m-{\mathit{r}}_n|$ is the distance between two particles $\mathrm{m}$ and $n$, $\epsilon$ is the effective radius of particle interaction, ${\theta }_w$ is the tangential direction of the wall where the bacteria reach the boundary, and $r_{mw}$ represents the distance from nearest boundary wall. The first term governs the polar alignment and the second represents the alignment by chiral edge current at the tip of the doublet circular boundary with ${\gamma }_e={\sum }_{r_{mn}<\epsilon }{\overline{\gamma }}_e$. The third term ${\eta }_m\left(t\right)$ represents the random fluctuation of the rotational direction. By considering collisions of bacteria at the tip, the change of bacterial orientation under CCW edge current is described by $\frac{{\gamma }_e}{2}\left(\sin \left({\theta }_m-\frac{\pi }{2}+\mathrm{\Psi }\right)+\sin \left({\theta }_m+\frac{\pi }{2}-\mathrm{\Psi }\right)\right)$.

$${\dot{\theta }}_m=-{\overline{\gamma }}_p\sum _{\left|{\mathit{r}}_{mn}\right|<\epsilon }\sin \left({\theta }_m-{\theta }_n\right)-{\gamma }_e\sin {\theta }_m\sin \mathrm{\Psi }+{\eta }_m\left(t\right).$$ [5]

Solving Eq. 5 with ${\gamma }_p={\sum }_{\left|{\mathit{r}}_{mn}\right|<\epsilon }{\overline{\gamma }}_p$ and finding the condition where the probability of taking FMV and AFMV patterns is equal, we obtain Eq. 1. The detailed calculation is described in SI Appendix.

Data Availability

All study data and procedures are included in this article and/or supporting information.