Bacteria often live in dynamic fluid environments1,2,3 and flow can affect fundamental microbial processes such as nutrient uptake1,4 and infection5. However, little is known about the consequences of the forces and torques associated with fluid flow on bacteria. Through microfluidic experiments, we show that fluid shear produces strong spatial heterogeneity in suspensions of motile bacteria, characterized by up to 70% cell depletion from low-shear regions due to ‘trapping’ in high-shear regions. Two mathematical models and a scaling analysis accurately capture these observations, including the maximal depletion at mean shear rates of 2.5–10 s−1, and reveal that trapping by shear originates from the competition between the cell alignment with the flow and the stochasticity in the swimming orientation. We show that this shear-induced trapping directly impacts widespread bacterial behaviours, by hampering chemotaxis and promoting surface attachment. These results suggest that the hydrodynamic environment may directly affect bacterial fitness and should be carefully considered in the study of microbial processes.
We investigated the effect of flow on motile bacteria by tracking them in precisely controlled laminar flows generated in a microfluidic channel (Fig. 1a). To ensure that the dominant velocity gradients occurred in the horizontal observation plane, at the channel mid-depth, we used a microchannel with aspect ratio H/W > 1 (height H = 750 μm; width W = 425 μm). In that plane, the velocity profile, u(y) = U[1 − 4(y/W)2], where U is the flow velocity at the channel centreline, is parabolic, and, thus, the shear rate S(y) = du/dy = −8yU/W2 varies linearly with distance y across the channel and is zero at the centreline (Fig. 1b). In this flow, smooth-swimming Bacillus subtilis bacteria swam unperturbed in straight paths (Fig. 1c) when near the centre of the channel, where the local shear rate was small. Conversely, in high-shear-rate regions, trajectories exhibited frequent loops resulting from rotation of the swimming bacteria by the hydrodynamic torque imparted by the local shear (Fig. 1d). The opposite handedness of the loops on either side of the channel centreline reflects the opposite sign of the shear rate (Fig. 1c, d and Supplementary Movie 1). As shown below, these looping trajectories resulted in bacteria becoming trapped in the high-shear region of the channel.
At the population scale, this trapping effect resulted in a marked depletion of cells from the central, low-shear region of the flow, and consequently, in an accumulation in the flanking regions of higher shear. When the flow was impulsively started from rest, the initially uniform distribution of cells over the imaging region across the channel width (Fig. 1e) rapidly (5–10 s) evolved into a distribution characterized by considerably fewer cells in the central part of the channel (Fig. 1f–h). The magnitude of the depletion was severe, with local cell concentrations dropping by 70% (Fig. 1h). The absence of depletion in control experiments with dead cells in flow (Fig. 1i) and with motile cells in quiescent fluid (Fig. 1e) confirmed that depletion resulted from the coupling of motility and flow. In particular, it did not require the presence of a nearby surface as evidenced by the uniform bacterial distribution in the absence of flow, in contrast to the known accumulation of motile cells arising from hydrodynamic6 or short-range steric7,8 interactions with boundaries.
Shear-induced trapping is a strong function of the shear rate and is greatest at intermediate shear levels. Over a broad range of mean shear rates, –50 s−1 (the mean of the absolute shear rate across the channel width), the intensity of the depletion grew with the shear rate up to and diminished for larger shear rates (Fig. 2a). The maximal depletion at intermediate shear rates is illustrated by the depletion index, ID (Fig. 2b), and occurred at shear rates, –10 s−1, that are typical for a wide array of environmental, physiological and engineering flows, such as those occurring in the subtidal coastal ocean9, groundwater10, mammalian reproductive tracts11 and catheters12.
Shear-induced trapping occurs in bacteria with diverse motility patterns and flagellation strategies, as demonstrated by comparing identical experiments with smooth-swimming B. subtilis, wild-type B. subtilis and Pseudomonas aeruginosa. Wild-type B. subtilis, which swim with multiple flagella (‘peritrichous’) in a random walk (‘run-and-tumble’) motility pattern by interspersing nearly straight runs with random tumbles (on average every ∼0.5 s), exhibited a broad peak in the depletion index analogous to the smooth-swimming mutant, albeit with a 50% weaker depletion (ID, MAX = 0.23 compared with 0.34; Fig. 2b). This demonstrates that rotational noise in the cells’ trajectories due to tumbling affords a moderate degree of escape from shear-induced trapping. P. aeruginosa, which in contrast to B. subtilis has a single flagellum (‘monotrichous’) and a run-and-reverse motility pattern, also exhibited strong depletion (ID, MAX = 0.28; Fig. 2b; Supplementary Fig. 1), indicating that trapping is largely independent of the details of the propulsion system and reorientation mechanism.
The broad occurrence of shear-induced trapping among bacteria, and potentially a host of other microorganisms, is supported by a Langevin model of cell motility in flow, which revealed that the mechanism underlying trapping is the preferential alignment of bacteria with the flow direction due to their elongated shape. We modelled bacteria as prolate ellipsoids with aspect ratio q = 10, accounting for the combined hydrodynamic resistance of the cell body and flagellar bundle in B. subtilis13, and swimming speed V directed along the long axis. A cell’s equations of motion in the same parabolic flow as in the experiments are then
where ξR models rotational noise as a Gaussian-distributed angular velocity with mean zero and variance 2DR/Δt, Δt is the elapsed time and DR is the cell’s effective rotational diffusivity. Cell concentration profiles across the channel, computed by integrating 105 trajectories in flow using measured values of V = 50 μm s−1 and DR = 1 rad2 s−1 (Supplementary Fig. 2), are in excellent agreement with experimental profiles (Fig. 2a). In particular, the magnitude of the depletion and the optimal shear rate (2.5–10 s−1) closely match those found in experiments (Fig. 2b). The conclusion that trapping results from the preferential alignment in shear caused by cell elongation is supported by simulations with spherical swimmers (Fig. 2a), which exhibit no depletion.
Intriguingly, the Langevin model predicts only marginal depletion in the absence of rotational noise (DR = 0; Fig. 3a, d). In this limit, cell trajectories are deterministic and a separatrix in y–ϕ phase space14 segregates cell trajectories into two types: cells outside the separatrix continuously perform loops (Fig. 1d) owing to the high shear, whereas cells inside the separatrix ‘swing’ back and forth across the channel centreline owing to opposite-handed vorticity on either side of the channel, never completing a full loop. Peaks in the cell concentration in phase space closely track the cusps of the separatrix (Fig. 3a), and cell concentration profiles across the channel collapse for all shear rates (Fig. 3d), when cell positions y are rescaled by the width WS of the separatrix14, given by WS/W = C0 (where C0 ≃ 3.5 is a constant), showing that the separatrix governs trapping when rotational noise is negligible. Real bacteria, however, are always subject to random fluctuations in their orientation, due to rotational Brownian motion or tumbling. The Langevin model shows that, by making cells depart from their deterministic trajectories, rotational noise causes cell trajectories in phase space to cross the separatrix, resulting in their accumulation in the high-shear region (Fig. 3b), in good agreement with observations (Fig. 3c). Shear trapping thus provides a biological realization of a counterintuitive class of systems where stochasticity suppresses transport rather than enhancing it15.
The maximum in cell depletion, observed experimentally and confirmed by the Langevin model (Fig. 2b), can be understood by considering the effect of spatial segregation by the separatrix at high shear rates, and, conversely, the competition between shear-induced alignment and rotational noise at low shear rates. When the mean-channel shear is large compared with noise—that is, for a large rotational Péclet number, —cell depletion is limited to the low-shear region defined by the distance between the separatrix cusps (Fig. 3d), WS. As the depletion index, ID, is proportional to the width of the depleted region, and the separatrix width decreases with increasing shear rate14 as , we predict that for large Pe, which is in good agreement with the Langevin model (Fig. 3e).
When the mean-channel shear rate is small compared with rotational noise—that is, Pe ≪ 1—trapping is governed by the competition between shear-induced alignment with the flow and randomization of the cell orientation due to rotational noise. To gain physical insight into the increase in depletion with increasing Pe obtained for both experiments and Langevin simulations (Fig. 3e), we considered the steady-state Fokker–Planck equation describing the distribution p(ϕ; y) of the local cell orientation ϕ at a fixed position y in the microchannel,
The distribution p(ϕ; y) obtained by numerical integration of equation 1 is in good agreement with that obtained by quantifying cell orientation across the channel experimentally (Fig. 3f). This signifies that the fundamental mechanism underlying cell depletion is correctly captured by the Fokker–Planck formulation: where the torque imparted by the local shear overcomes rotational noise, elongated swimmers become aligned more strongly with the flow direction and their mobility across the channel plummets. As self-propelled particles are far-from-equilibrium systems, cells can accumulate in regions where the ensemble-averaged (‘effective’) swimming speed across the channel, VEFF(y) = ∫ 0πV sinϕ p(ϕ; y)dϕ, is small, resulting in a non-uniform cell concentration across the channel, B(y) = B(0)VEFF(0)/VEFF(y) (Ref. 16). In the small Pe regime, results from both the Langevin and the Fokker–Planck models predict ID ∼Pe2 (Fig. 3e), a scaling that is confirmed by a perturbation analysis for p(ϕ; y) in the limit Pe ≪ 1 (Supplementary Information). The Fokker–Planck model thus provides a simple, continuum description of shear-induced trapping and supports the physical argument that trapping is caused by preferential alignment vis-à-vis rotational noise.
Shear-induced trapping severely curtails the ability of bacteria to follow chemical signals (chemotaxis), a fundamental strategy4 used by bacteria to find nutrients17, colonize and infect hosts18, or evade noxious substances19. Focusing on oxygen as the chemical signal (‘aerotaxis’; ref. 20), we devised a microfluidic channel to simultaneously expose wild-type B. subtilis to fluid flow and to a linearly varying oxygen concentration profile (Fig. 4a). Under quiescent conditions, B. subtilis strongly accumulated in the oxygen-rich region of the microchannel (Fig. 4b). In the presence of flow, the cells’ chemotactic response to the same oxygen gradient was stifled (Fig. 4b). The chemotactic ability of the cells decreased with increasing mean shear rate, as demonstrated by the declining chemotactic index, IC (Fig. 4b, inset). An analysis of the bacterial distribution profiles relative to the case without an oxygen gradient (Supplementary Fig. 3) clearly shows that the deterioration of the chemotactic performance is primarily due to shear-induced trapping, particularly at low shear rates (≤10 s−1; Fig. 4b). A further process that may have acted in concert with trapping is the degradation of chemosensory dynamics in shear, previously predicted by mathematical modelling21. At the highest shear rate tested, , shear almost entirely suppressed chemotaxis, with IC dropping by 85%.
In contrast to its negative effect on chemotaxis, shear-induced trapping has a positive effect on bacterial surface attachment, a precursor of biofilm formation. Solid surfaces exposed to fluid flow always generate locally enhanced shear rates, which, we reasoned, can lead to cell accumulation by shear-induced trapping and, thus, to higher encounter and attachment rates of cells on the surface. We tested this hypothesis by measuring the attachment of P. aeruginosa PA14 to a glass surface in shallow (58 μm depth) microfluidic channels at five different shear rates (Fig. 4c). The surface coverage increased with increasing shear rate (Fig. 4d), causing a ∼125% increase in attachment for compared with quiescent conditions. The correspondence of the shear rates (5–25 s−1) at which both increased attachment (Fig. 4d, inset) and increased depletion (Fig. 2b) of P. aeruginosa were observed, suggests that the enhanced surface colonization was promoted by shear, which induced cell accumulation in close proximity of the surface. Whereas previous research on surface attachment focused on the effects of high shear rates (103–105 s−1; ref. 22) and post-attachment dynamics including surface residence time23 and catch-bond adhesion strength24, our results demonstrate that attachment can be enhanced by lower, commonly occurring shear rates with a mechanism virtually independent from specific adhesion properties. For example, P. aeruginosa biofilms are a frequent cause of infection in catheters3, where typical shear rates are of the order of 15 s−1 (ref. 12), indicating that flow effects must be carefully considered in the design of these and other biomedical devices.
A spatially varying shear rate of the appropriate magnitude is the only requirement for a flow to generate trapping, indicating that this phenomenon is likely to occur in a very broad range of flow environments. Thus, we expect shear-induced trapping of bacteria to occur not only in pipe, boundary layer and porous media flows, but also potentially in unbounded flows (Supplementary Fig. 5), including turbulence in lakes and oceans. A distinguishing feature of shear-induced trapping is the fast timescale of cell accumulation (Fig. 1 and Supplementary Fig. 6). In the fluid dynamical regime of bacteria, mechanisms including inertia, buoyancy and deformation that otherwise lead to the accumulation of particles25, bubbles26 or red blood cells27, are insignificant or exceedingly slow (Supplementary Information). Instead, shear-induced trapping occurs as a result of self-propulsion, which drives cell accumulations within a timescale governed by the swimming speed and the flow length scale.
We have demonstrated a common yet hitherto neglected interplay between cell motility, cell elongation—a nearly universal feature of flagellated cells including bacteria and spermatozoa—and hydrodynamic shear, which results in the trapping and accumulation of bacteria in high-shear regions. Our results suggest that flow, ubiquitous in microbial habitats yet rarely considered in biological and physical studies of microbes, encourages sessile over free-swimming lifestyles, because the pursuit of chemical signals by planktonic microbes is hampered by flow, whereas attachment to surfaces is favoured. This fundamental interaction between motility and shear, and the subtle role of rotational noise, are thus expected to affect microbial transport in a myriad of applications, from groundwater remediation10, to microfiltration28, to the design of micro-robots as therapeutic agents for drug delivery29.
The bacterial strains used in this work were Bacillus subtilis wild-type strain OI1085 and smooth-swimming strain OI4139, and Pseudomonas aeruginosa strain PA14, and were cultured using standard protocols (Supplementary Information). To ensure a high percentage of motile bacteria in experiments, non-motile and dead cells were gently removed from bacterial suspensions using sterile cell culture inserts incorporating a 3 − μm-pore-size membrane.
For shear-induced trapping and chemotaxis experiments, we fabricated a microchannel with a 532-mm-long serpentine pattern (Fig. 1a) to ensure a minimum of 60 s exposure time to shear of the bacteria for all flow rates. For chemotaxis experiments, two parallel channels flanking either side of the original (‘test’) channel, and separated from it by 350 μm-thick polydimethylsiloxane (PDMS) walls, carried flowing oxygen and nitrogen, respectively, to generate a steady, linear oxygen concentration profile (2.5 mM mm−1; Fig. 4a) within the test channel through diffusion. For surface attachment experiments, we fabricated a microfluidic device consisting of 7 separate channels (58 μm high, ranging from 0.25 to 5.0 mm in width), to measure the surface coverage of cells for 5 shear rates simultaneously (plus 2 no-flow controls) on a single chip (Fig. 4c). Potential confounding factors stemming from cell growth, cell–cell signalling, extracellular matrix production and variability among cultures were minimized by focusing on early attachment (<1 h), performing experiments at different shear rates in parallel using a single cell culture, and ensuring that the same number of cells per unit surface area flowed in each microchannel.
Cell imaging and tracking.
Long bacterial trajectories (Fig. 1c, d), were acquired using dark-field microscopy (×6 magnification, 70.57 frames per second) and by moving the microscope stage in synchrony with the mean flow in the channel, at 500 μm s−1. Spatial distributions of bacteria in flow were obtained using phase-contrast microscopy (×15 magnification, 60 frames per second) with a stationary stage. All image analysis was performed in Matlab (the Mathworks) using in-house cell tracking and identification algorithms.
The equations of motion (1) were integrated numerically for 105 cells using a fourth-order Runge–Kutta scheme implemented in Matlab. We used anti-symmetric periodic boundary conditions at y = ±W/2, so that the parabolic flow profile repeats periodically with alternating sign and modelling cell–surface interactions becomes unnecessary. We tested that the integration time step, Δt = 4 ms, was sufficiently small to ensure independence of the solution from further reductions in Δt. Physical parameters controlling simulated cell dynamics including flow speed, cell swimming speed, elongation and rotational noise were all measured experimentally or calculated for the same conditions as in the experiments (Supplementary Information).
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We thank G. Boffetta, V.I. Fernandez, G.L. Miño and N.T. Ouellette for discussions and comments on the manuscript, and acknowledge support by NSF grants OCE-0744641-CAREER, IOS-1120200, CBET-1066566, CBET-0966000 and a Gordon and Betty Moore Marine Microbial Initiative Investigator Award (award number 3783) (to R.S.).
The authors declare no competing financial interests.
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Rusconi, R., Guasto, J. & Stocker, R. Bacterial transport suppressed by fluid shear. Nature Phys 10, 212–217 (2014). https://doi.org/10.1038/nphys2883
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