# Bacterial scattering in microfluidic crystal flows reveals giant active Taylor–Aris dispersion

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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved April 22, 2019 (received for review November 15, 2018)

## Significance

The fluid habitats of swimming microbes are characterized by heterogeneous flow, typical of groundwater and marine turbulence, which can augment cell transport in unexpected ways and affect important processes ranging from biogeochemical cycling to disease transmission. Following in the footsteps of century-old light diffraction techniques, the scattering of bacteria in microfluidic crystal flows reveals how cell shape and motility couple to hydrodynamic gradients, giving rise to observed filamentous cell density patterns. Consequently, hindered bacterial mobility transverse to the flow greatly enhances downstream dispersal. These results illustrate the stark contrast of active matter transport with passive Brownian particles and may provide insights into microbial ecosystems, biomedical device design, and the guidance of swimming microrobots.

## Abstract

The natural habitats of planktonic and swimming microorganisms, from algae in the oceans to bacteria living in soil or intestines, are characterized by highly heterogeneous fluid flows. The complex interplay of flow-field topology, self-propulsion, and porous microstructure is essential to a wide range of biophysical and ecological processes, including marine oxygen production, remineralization of organic matter, and biofilm formation. Although much progress has been made in the understanding of microbial hydrodynamics and surface interactions over the last decade, the dispersion of active suspensions in complex flow environments still poses unsolved fundamental questions that preclude predictive models for microbial transport and spreading under realistic conditions. Here, we combine experiments and simulations to identify the key physical mechanisms and scaling laws governing the dispersal of swimming bacteria in idealized porous media flows. By tracing the scattering dynamics of swimming bacteria in microfluidic crystal lattices, we show that hydrodynamic gradients hinder transverse bacterial dispersion, thereby enhancing stream-wise dispersion

Scattering experiments have long been used to successfully probe the structure and dynamics of photons, electrons, and other forms of passive matter (1). A recent extension of traditional particle-scattering concepts to living matter has provided important insights into the cell–cell (2, 3) and cell-surface interactions (4⇓⇓–7) of swimming microorganisms in simple scattering geometries and under idealized quiescent fluid conditions. By contrast, very little is known about the individual and collective behaviors of bacteria and other microbes in their natural, highly dynamic, and geometrically complex fluid habitats (8, 9). The biological and ecological importance of cell-flow and cell-surface interactions in porous media and turbulent environments is now widely recognized in the regulation of cell dispersal (10, 11), chemotaxis (11, 12), fertilization (13), biofilm formation (14), and disease transmission (15). However, due to a lack of quantitative data, no validated predictive models exist to describe such processes (16). Here, we combine experiments and simulations to determine the effects of the carrier fluid flow on the transport of swimming bacteria through a periodic microfluidic lattice, drawing inspiration from classical X-ray scattering experiments by Bragg and Laue (1).

The random walks of self-propelled cells and particles in uniformly moving fluids are often described as diffusion (17⇓–19), which enabled early theoretical progress on active transport in flow (20, 21). However, recent experiments (10, 11) showed that even simplistic, 1D flow gradients can cause local cell accumulations (22), with profound implications for microswimmer transport (23⇓–25). Notwithstanding such progress, the mechanisms by which generalized fluid flows and self-propulsion can enhance or hinder the densification and dispersion of swimming cells (11) in geometrically complex environments are not yet known (26). Identifying the underlying biophysical and hydrodynamic effects is essential for understanding the dynamics of active suspensions in porous media and turbulent flows, which are often characterized by strong kinematic mixing (27) due to heterogeneous 2D and 3D velocity gradients.

To investigate and quantify how flow gradients and self-propulsion determine the transport properties of active suspensions relative to passive Brownian solutes, we studied the scattering dynamics of swimming bacteria at single-cell resolution in a periodic microfluidic lattice (Fig. 1*A*). Similar to classical light scattering, our experimental setup allowed us to precisely control the incidence angles of the carrier flows relative to the microfluidic crystal lattice and thus disentangle the effects of hydrodynamic mixing and dispersion often found in disordered systems. In agreement with predictions from a Langevin model, our data reveal a strong dependence of the cell densification patterns and dispersion coefficients on the incident flow angles, which are two key features that we show to persist in biologically relevant random media (*SI Appendix*). While local fluid shear can describe bacterial accumulation in simple, 1D flows (11), our results below illustrate that in generalized, Lagrangian-unsteady flows, advection and the velocity gradient history of the flow regulate the topology of cell distributions. We identify cell alignment to Lagrangian fluid stretching as the primary mechanism for the observed densification. Strikingly, our data further show that cell reorientation, determined by vorticity history, hinders lateral transport and amplifies stream-wise dispersion

## Results and Discussion

### Bacterial Suspension Flow in a Microfluidic Crystal Lattice.

The microfluidic lattices (Fig. 1*A* and *Materials* and *Methods*) comprise a square, periodic array of circular pillars in an otherwise rectangular cross-section microchannel. The pillar diameter (65 μm) and lattice spacing (120 μm) are held constant, and the flow topology is modified by rotating the lattice orientation relative to the mean flow direction (Fig. 1*A*, *Inset*) in a series of five individual channels (*SI Appendix*, Figs. S1 and S2). A dilute suspension of wild-type *Bacillus subtilis* bacteria (mean swimming speed *Materials* and *Methods*) was flowed through the device using a syringe pump in a mean flow speed range of *SI Appendix*, Fig. S1). Results are presented in terms of the mean shear rate, which was shown to be a key parameter in microbial transport (11, 22). Video microscopy captures the cell motion (Fig. 1 *B* and *C*) and spatial distribution (Fig. 1 *D* and *E*) in the middepth of the microchannel, which was designed with a large channel height to pillar spacing ratio (*D* and *E* and *Materials* and *Methods*).

### Bacterial Scattering Yields Angle-Dependent Filamentous Density Patterns.

Steady flow through the microfluidic lattice results in striking, filamentous cell density patterns, which meander around pillars (Fig. 1*E*) and along streamlines (Fig. 2*A*). In contrast, nonmotile cells and particles do not exhibit such densification (*SI Appendix*, Fig. S3), and without flow, the random walks of motile bacteria (Fig. 1*B*) generate a homogeneous cell density, ρ, normalized by the mean density, *D*). Changes in the flow topology (*SI Appendix*, Fig. S1), generated by varying the incident mean flow angle, θ, with respect to the lattice, cause drastic changes in the filament topologies, which track the fluid streamlines (Fig. 2*A*). Filamentous bacterial density patterns are also prominent in biologically relevant random media (*SI Appendix*, Fig. S9). Topological changes in the filament structure with flow angle are accompanied by changes in the density contrast (Fig. 1*F*), which is defined as the SD of the streamline-averaged, normalized bacterial density, *SI Appendix*, Fig. S4). The density contrast increases continuously with mean shear rate for lattice angles *A*). Conversely, lattice angles having aperiodic streamlines (*SI Appendix*). All physical parameters including cell swimming speed, *Materials* and *Methods* and *SI Appendix*). The model accurately predicts the topology (Fig. 2*A*), magnitude (Fig. 1*F*), and angular dependence of the observed densification, and it enables us to make quantitative predictions beyond flow speeds achievable in our current experiments.

### Hydrodynamic Bacterial Alignment Drives Densification.

The coupling of not only translational but also rotational cell body dynamics to the flow is integral to swimming-cell transport (11, 30). Thus, to determine the origin of the densification, we investigate cell orientation in the vicinity of the pillars from which the density patterns spawn (Figs. 1*E* and 2*A*). The distribution of cell body orientation relative to coincident high-density streamlines (Fig. 2*B*), *D*, red curve), whereas cells on low-density streamlines are weakly aligned (Fig. 2*D*, blue curve). However, a local assessment of cell orientation reveals only a partial correlation between cell-streamline alignment and cell density (Fig. 2*C*). Local cell-streamline alignment arises from the hydrodynamic coupling of the cell body orientation to the extensional regions of the flow (31, 32), in this case emanating from two hyperbolic stagnation points on the upstream and downstream sides of the pillars, respectively (Fig. 2*C*, white arrowheads). Hydrodynamic scattering aligns elongated bacterial cells that swim into high extension zones with streamlines that coincide with the extensional manifolds. This preferential alignment provides the mechanism for accumulation (11). However, despite the lack of local cell-streamline alignment farther downstream, high bacterial density filaments persist. Advection sweeps bacteria away from extensional zones along departing streamlines, due in part to relatively weak cell swimming speed compared with the mean flow speed (

### Lagrangian Fluid Stretching Underpins Bacterial Density Patterns.

Bacteria remain localized on streamlines for a finite time, λ, until the combined effects of hydrodynamic rotation (32), flagellar-induced tumbling (17), and Brownian rotation facilitate their escape (*SI Appendix*, Fig. S5). During this time, bacteria may be advected up to several unit cells downstream and experience a range of flow conditions. While local shear was perceived to be an indicator of bacterial accumulation in 1D flows (11), the observed cell density patterns in our Lagrangian-unsteady 2D flows do not correlate with local velocity gradients (Figs. 1*E* and 2*A* and *SI Appendix*, Fig. S1), suggesting that nonlocal effects stemming from advection dominate. The Lagrangian fluid stretching field encapsulates the integrated extension experienced over a particle’s history in the flow. Stretching fields are known to be a good predictor of elongated particle alignment, even in chaotic flows (31), which suggests that stretching may also be indicative of densification (Fig. 2). Lagrangian stretching, *SI Appendix*). Fluid stretching is directly related to the finite-time Lyapunov exponent field and has been used to characterize transport in applications ranging from weather patterns to chemical reacting systems (34, 35). Stretching fields for the steady periodic lattice flows investigated here (Fig. 2*E* and *SI Appendix*) reveal strikingly similar topologies to the observed bacterial density patterns (Figs. 1*E* and 2*A*). Regions of high cell-streamline alignment (Fig. 2*C*) correspond to regions of high stretching (Fig. 2*E*), where local fluid deformation is strongest. However, Lagrangian stretching manifolds extend throughout the periodic lattice along streamlines due to advection (Fig. 2*A*). Bacterial density is thus strongly correlated with regions of high stretching (Fig. 2 *F* and *G* and *SI Appendix*), which critically relies on the elongated shape of the bacteria (Fig. 2*G*).

The effects of bacterial scattering are sensitive not only to the imposed mean shear rate, but also, in a further analogy with light scattering, to the incident angle of the mean flow direction, relative to the lattice (Fig. 1*E*). Mean flow angles exhibiting periodic streamlines (*SI Appendix*), the stretching field, as with the cell density, tends to a uniform spatial distribution for large mean shear rates. The emergence of bacterial densification patterns, along with their mediation by the incident angle of the flow, marks a strong departure from Brownian solutes.

### Flow Hinders Lateral Bacterial Dispersion.

On a larger scale, the dispersion of the scattered bacterial suspension depends upon the motility of the cells, which further distinguishes the transport properties of active and passive particles. Without flow, the persistent random walks, and thus effective diffusion coefficient, *SI Appendix*, Fig. S6). However, examination of the cell displacement distribution transverse to the flow, *A*). This observation is corroborated by the increasingly fast decorrelation of the cells’ swimming velocity with increasing mean shear rate (Fig. 3*B*). The velocity correlation function of the random-walking cells, *B*), which are modeled as rotational diffusion, *SI Appendix*). As the shear rate increases, the slow exponential decay of the cell velocity correlation gives way to rapidly decaying oscillations, which implicate hydrodynamic cell rotation through vorticity, ω, as the mechanism enhancing decorrelation (Fig. 3*B* and *SI Appendix*, Fig. S1). We examine changes in the transport coefficients as a function of the rotational Péclet number (11) based on the mean absolute vorticity, *C*) are obtained directly from the correlation functions of orientation through the Green–Kubo relation (36, 37), *SI Appendix*, Fig. S7). Augmenting the flow strength, and thus the vorticity, rapidly decreases *C* and *SI Appendix*).

The lateral dispersion of bacteria depends strongly on the incident angle of the scattered suspension within the microfluidic lattice, which is rationalized by a simple model. A swimmer with rotational diffusion in constant vorticity (38, 39) yields an effective diffusion coefficient *C*, black dashed line) (*SI Appendix*), setting the lower bound of *SI Appendix*)*SI Appendix*, Fig. S8), which is at the origin of the vorticity history dependence, and thus incident angle dependence, for *SI Appendix*), *C*).

### Hindered Lateral Dispersion Enhances Longitudinal Dispersion.

We have demonstrated that the transverse dispersion of swimming bacteria is reduced (Fig. 3*C*), but we have yet to establish whether dispersion in active suspensions competes with or enhances the well-known longitudinal Taylor–Aris dispersion of passive Brownian particles (28, 29). The longitudinal dispersion coefficient, *C*, *C* and *SI Appendix*). Drawing a parallel between the relatively unidirectional flow for the *SI Appendix*). The predicted giant Taylor–Aris dispersion coefficient scaling bounds the observed stream-wise transport coefficients for the periodic *C*). Similar to Taylor–Aris dispersion (40), the observed giant longitudinal dispersion coefficient also exhibits a dependence upon the incident flow angle (Fig. 3*C*). However, active dispersion is systematically higher than passive Taylor dispersion, and it reflects the increase of the longitudinal dispersion coefficient, *C*) (41).

## Conclusions

This work translates century-old ideas of X-ray scattering from crystalline materials (1) to the transport of active matter in an idealized porous medium, bearing curious similarities in the dependence of scattering strength on incident angle. Bacterial scattering in a microfluidic lattice reveals that flow topology, coupled with self-propulsion and cell shape, modifies the microscale spatial distribution of bacteria and impacts their macroscale transport properties, which strikingly depart from the behavior of Brownian solutes. This departure is mediated by the hydrodynamic stretching and vorticity history of the swimming cells, which are two fundamental Lagrangian properties, shown here to govern densification and dispersion, respectively. Lagrangian coherent structures have proved invaluable in understanding passive fluid transport (33⇓–35): Our analysis shows that they offer the potential to not only characterize but also predict the transport properties of active matter in complex, dynamical fluid systems (42). In particular, the correlation of cell densification with Lagrangian stretching may be extended to random porous media (*SI Appendix*, Fig. S9) and unsteady flows, including marine turbulence, where predicting cell transport is both challenging and important to large-scale bio-oceanography modeling (21, 24, 43). The emergent heterogeneous cell distributions at the pore scale will inform our understanding of microbial function and biome dynamics in processes ranging from biofilm formation (44) to niche partitioning (45). Harnessing these novel transport properties could inspire new methods for cell separation (46) or tailoring dispersion (29) for applications in water filtration (47), remediation (48⇓–50), and control of active matter (51, 52).

## Materials and Methods

### Bacterial Culturing.

Wild-type *B. subtilis* bacteria (OI1085) were cultured by inoculating 5 mL of Cap Assay Minimal (CAM) motility medium with cells obtained from a frozen glycerol stock (53). Cells were grown initially for 12 h (

### Microfabrication and Microfluidic Experiments.

Polydimethylsiloxane (PDMS) microfluidic channels were fabricated through soft lithography (54) and plasma bonded to standard glass microscope slides. The 100-μm high channels had an overall length and width of 40 mm and 3.6 mm, respectively. The square lattice of circular pillars (65 μm diameter, 120 μm spacing) occupied the central 10 mm of five different microchannels, which were prepared with lattices oriented at angles

### Cell Imaging and Tracking.

Imaging was performed at middepth in the microchannel pillar array test section far from side walls and four or more lattice spacings from the end of the test section. Cells were imaged using phase-contrast microscopy on an inverted microscope (Nikon Ti-E; 10×, 0.3 NA objective). For each experimental condition, a 4,000-frame video was recorded at 45 fps (Zyla sCMOS camera; Andor Technology). Bacteria were tracked using a custom predictive particle-tracking algorithm (MATLAB; MathWorks), yielding

### Periodic Cell Density Averaging.

The full field of view comprises ∼

## Acknowledgments

We thank G. A. Voth and S. Parsa for helpful discussions on Lagrangian stretching. This work was funded by NSF Awards CBET-1511340, CAREER-1554095, and CBET-1701392 (to J.S.G.) and CBET-1510768 (to J.D.) and by a Complex Systems Scholar Award from the James S. McDonnell Foundation (to J.D.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: Jeffrey.Guasto{at}tufts.edu.

Author contributions: A.D., N.W., J.D., and J.S.G. designed research; A.D. and N.W. performed research; A.D. and N.W. analyzed data; and A.D., N.W., J.D., and J.S.G. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1819613116/-/DCSupplemental.

Published under the PNAS license.

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