Strong confinement of active microalgae leads to inversion of vortex flow and enhanced mixing

Microorganisms swimming through viscous fluids imprint their propulsion mechanisms in the flow fields they generate. Extreme confinement of these swimmers between rigid boundaries often arises in natural and technological contexts, yet measurements of their mechanics in this regime are absent. Here, we show that strongly confining the microalga Chlamydomonas between two parallel plates not only inhibits its motility through contact friction with the walls but also leads, for purely mechanical reasons, to inversion of the surrounding vortex flows. Insights from the experiment lead to a simplified theoretical description of flow fields based on a quasi-2D Brinkman approximation to the Stokes equation rather than the usual method of images. We argue that this vortex flow inversion provides the advantage of enhanced fluid mixing despite higher friction. Overall, our results offer a comprehensive framework for analyzing the collective flows of strongly confined swimmers.


Introduction
Fluid friction governs the functional and mechanical responses of microorganisms which operate at low Reynolds number. They have exploited this friction and developed drag-based propulsive strategies to swim through viscous fluids (Lauga and Powers, 2009;Pedley and Kessler, 1992). Naturally, many studies have elucidated aspects of the motility and flow fields of microswimmers in a variety of settings that mimic their natural habitats (Elgeti et al., 2015;Bechinger et al., 2016;Denissenko et al., 2012;Bhattacharjee and Datta, 2019). The self-propulsion of microbes in crowded and strongly confined environments is one such setting, encountered very commonly in the natural world as well as in controlled laboratory experiments. Examples include microbial biofilms, bacteria-and algae-laden porous rocks or soil (Qin et al., 2020;Hoh et al., 2016;Foissner, 1998;Bhattacharjee and Datta, 2019); parasitic infections in crowded blood streams and tissues (Heddergott et al., 2012); and biomechanics experiments using thin films and microfluidic channels (Durham et al., 2009;Denissenko et al., 2012;Jeanneret et al., 2019;Ostapenko et al., 2018;Kurtuldu et al., 2011). Confined microswimmers are also fundamentally interesting as active suspensions (Brotto et al., 2013;Maitra These cells are introduced into rectangular quasi-2D chambers (area, 18 mm × 6 mm) made up of a glass slide and coverslip sandwich with double tape of thickness H = 10/30 µm as spacer. Passive 200 nm latex microspheres are added as tracers to the cell suspension for measuring the fluid flow using particle-tracking velocimetry. We use high-speed phase-contrast imaging at ∼500 frames/s and ×40 magnification to capture flagellar waveform and cellular and tracer motion at a distance H/2 from the solid walls. The detailed experimental procedure is described in Materials and methods.

Mechanical equilibrium of confined cells
The net force and torque on microswimmers, together with the ambient medium and boundaries, can be taken to be zero as gravitational effects are negligible in the case of CR for the range of length scales considered (Drescher et al., 2010;Brennen and Winet, 1977;Pedley and Kessler, 1992;Elgeti et al., 2015;Mathijssen et al., 2016). The two local forces exerted by any dipolar microswimmer on the surrounding fluid are flagellar propulsive thrust F th and cell body drag F hd . They balance each other completely for any swimmer in an unbounded medium (Lauga and Powers, 2009;Goldstein, 2015)   Source data 1. Source data for Figure 1F. and approximately in weak confinement between two hard walls ( Figure 1A). In these regimes, CR is the classic example of an active puller where the direction of force dipole due to thrust and drag are such that the cell draws in fluid along the propulsion axis (x-axis in Figure 1A) and ejects it in the perpendicular plane (Lauga and Powers, 2009). CR is described well by three point forces or Stokeslets (Drescher et al., 2010) as in Figure 1A because the thrust is spatially extended and distributed equally between the two flagella. However, microswimmers in strong confinement between two closely spaced hard walls, D/H ≳ 1 , are in a regime altogether different from bulk because the close proximity of the cells to the glass walls results in an additional drag force F cf ( Figure 1B). Therefore, the flagellar thrust is balanced by the combined drag due to the cell body and the strongly confining walls ( Figure 1B).
Size polydispersity, confinement heterogeneity, and consequences for flagellar waveform and motility We define the degree of confinement of the CR cells as the ratio D/H of cell body diameter to chamber height. CR cells in chambers of height H = 30 µm are always in weak confinement as the cell diameter varies within D ∼ 8 − 14 µm < H . However, this dispersity in cell size becomes significant when CR cells are swimming within quasi-2D chambers of height, H = 10 µm . Here, the diameter of individual cell is crucial in determining the character -weak or strong -of the confinement and, as a consequence, the forces acting on the cell. Below, we illustrate how the cell size determines the type of confinement in this regime through measurements of flagellar waveform and cell motility.
CR cells confined to swim in H = 10 µm chambers show three kinds of flagellar waveform: (1) synchronous breaststroke and planar beating of flagella interrupted by intermittent phase slips ('H10 Synchronous', Figure 1C, Video 1); (2) asynchronous and planar flagellar beat over large time periods ( Figure 1D, Video 2); and (3) a distinctive paddling flagellar beat wherein flagella often wind around each other and paddle irregularly anterior to the cell with their beat plane oriented away from the x-y Video 1. Video of a strongly confined Chlamydomonas cell swimming with synchronous beat in the presence of tracers. High-speed video microscopy of a strongly confined swimmer (synchronously beating Chlamydomonas cell in H = 10 μm chamber) in the presence of tracer particles at 500 frames/s. This phase-contrast video clearly shows the synchronous breaststroke and planar beating of flagella with intermittent phase slips. This is the representative cell whose flow field is shown in Figure 3C. The direction of vortex flow is evident from the tracers' motion. https://elifesciences.org/articles/67663/figures#video1 Video 2. Video of wobbling Chlamydomonas cells with asynchronous or paddling flagellar beat. Flagellar waveform of Chlamydomonas cells in H = 10 μm chamber with wobbling cell body, that is, H10 Wobblers. The video is divided into three parts. The first part shows the asynchronous and planar flagellar beat of a cell which leads to a wobbling motion of the cell body. The second part shows the distinctive paddling flagellar beat of a cell, anterior to the cell body. Here, the flagellar beat plane is perpendicular to the imaging x-y plane and one of the flagella is mostly out of focus. In both these cases, the cell bodies wobble due to their irregular flagellar beat pattern. The third part shows a representative H10 Wobbler which switches from paddling beat to an asynchronous one.
https://elifesciences.org/articles/67663/figures#video2 plane ( Figure 1E, Video 2). While both synchronous and asynchronous beats are typically observed for CR in bulk (Polin et al., 2009) and weak confinement of 30 µm , the paddler beat is associated with calcium-mediated mechanosensitive shock response of the flagella to the chamber walls (Fujiu et al., 2011). The cell body wobbles for both asynchronous and paddler beat of cells ( Figure 1D, E) and often the flagellar waveform in a single CR switches between these two kinds (Video 2). Hence, we collectively call them 'H10 Wobblers' (Qin et al., 2015).
We correlate the Synchronous and Wobbler nature of cells to their body diameter ( Figure 1F). The mean projected diameter in the image plane of Synchronous cells ( D = 12.28 ± 0.94 µm , number of cells, N = 34 ) is larger than that of Wobblers ( D = 9.92 ± 0.85 µm , N = 36 ). Hence, the former's cell body is squished and strongly confined in H = 10 µm chamber in comparison with that of the latter. This leads to planar swimming of Synchronous cells, whereas Wobblers tend to spin about their body axis and trace out a near-helical trajectory which is a remnant of its behaviour in the bulk. Thus, the Wobblers likely compromise their flagellar beat into asynchrony and/or paddling over long periods, as a shock response, due to frequent mechanical interactions with the solid boundaries while rolling and yawing their cell body (Fujiu et al., 2011;Choudhary et al., 2019).
The motility of CR cells in H = 30 µm is similar to that in bulk and has the signature of back-and-forth cellular motion due to the recovery and power strokes of the flagella (Figure 2A, D). As confinement increases, the drag on the cells due to the solid walls increases and they trace out smaller distances with increasing twists and turns in the trajectory (Figure 2A-F). These phenomena can be quantitatively characterized by cell speed and trajectory tortuosity (Materials and methods) as a function of the degree of confinement of the cells ( Figure 2G). Cellular speed decreases and tortuosity of trajectories increases with increasing confinement as we go from H30 → H10 Wobblers → H10 Synchronous cells. Notably, the cell speed u decreases by 96% from H30 ( ⟨u 30 ⟩ = 122.14 ± 31.59 µm/s , N = 52 ) to H10 The online version of this article includes the following source data for figure 2: Source data 1. Source data for Figure 2A.
Source data 2. Source data for Figure 2B.
Source data 3. Source data for Figure 2C.
Source data 4. Source data for Figure 2G.
We also note that the flagellar beat frequency of the strongly confined cells, ν 10 b ≈ 51.58 ± 7.62 Hz (averaged over 210 beat cycles for N = 20 ) is similar to that of the weakly confined ones, ν 30 b ≈ 55.27 ± 8.22 Hz (averaged over 194 beat cycles for N = 20 ). This is because even in the 10 μm chamber where the CR cell body is strongly confined, the flagella are beating far from the walls ( ∼ 5 µm ) and almost unaffected by the confinement.

Experimental flow fields
We measure the beat-averaged flow fields of H30 and H10 CR cells to systematically understand the effect of strong confinement on the swimmer's flow field. We determine the flow field for H30 cells only when their flagellar beat is in the x-y plane (Video 3) for appropriate comparison with planar H10 swimmers. Figure 3A shows the velocity field for H30 cells obtained by averaging ∼178 beat cycles from 32 cells. It shows standard features of an unbounded CR's flow field (Drescher et al., 2010;Guasto et al., 2010), namely far-field fourlobe flow of a puller, two lateral vortices at 8-9 μm from cell's major axis, and anterior flow along the swimming direction till a stagnation point, 21 μm from the cell centre ( Figure 3B). These near-field flow characteristics are quite well explained theoretically by a three-bead model (Jibuti et al., 2017;Friedrich and Jülicher, 2012;Bennett and Golestanian, 2013) or a three-Stokeslet model (Drescher et al., 2010), where the thrust is distributed at approximate flagellar positions between two Stokeslets of strength ( −1/2, −1/2 ) balanced by a +1 Stokeslet due to viscous drag on the cell body ( Figure 1A).
The flow field of a representative H10 swimmer ( u = 5.67 ± 1.57 µm/s , ν b ∼ 42.67 ± 2.24 Hz ) is shown in Figure 3C, averaged over ∼328 beat cycles. Strikingly, the vortices contributing dominantly to the flow in this strongly confined geometry are opposite in sign to those in the bulk (Drescher et al., 2010) or weakly confined case (H30, Figure 3A). This two-lobed flow is distinct from expectations based on the screened version of the bulk or three-Stokeslet flow, which is four-lobed ( Figure 3figure supplement 1A). Importantly, the far-field flow resembles a 2D source dipole pointing opposite to the swimmer's motion, which is entirely different from that produced by the standard source dipole theory of strongly confined swimmers (Figure 3-figure supplement 1B; Brotto et al., 2013;Mathijssen et al., 2016;Jeanneret et al., 2019). This is because the source-dipole treatment does not consider the possibility that the cells are squeezed by the walls, or in other words, it does not account for contact friction (Brotto et al., 2013;Mathijssen et al., 2016). Other significant differences from the bulk flow include front-back flow asymmetry, opposite flow direction posterior to the cell, distant lateral vortices (20 μm) and closer stagnation point (11 μm) ( Figure 3D). All other H10 Synchronous swimmers, including the slowest ( u ∼ 0.15 µm/s ) and the fastest ( u ∼ 14 µm/s ) cells, show similar flow features. Even though the flow fields of H30 and H10 cells look strikingly different, the viscous power dissipated through the flow fields is nearly the same (Appendix 1.1).
A close examination suggests that the vortex contents of the flow fields of Figure 3A (H30) and Figure 3C (H10) are mutually compatible. The large vortices flanking the rapidly moving CR in H30 are shrunken and localized close to the cell body in H10 due to the greatly reduced swimming speed. The online version of this article includes the following source data and figure supplement(s) for figure 3: Source data 1. Source data for Figure 3A.
Source data 2. Source data for Figure 3C. The frontal vortices generated by flagellar motion now fill most of the flow field in H10. Generated largely during the power stroke of flagella, they are opposite in sense to the vortices produced by the moving cell body.

Force balance on confined cells
In an unbounded fluid, the thrust F th exerted by the flagellar motion of the cell balances the hydrodynamic drag F hd on the moving cell body ( Figure 1A). We assume this balance holds for the case of weak confinement (H30) as well. We estimate |F hd | = 3πηDu as the Stokes drag on a spherical cell body of diameter D ≃ 10 µm moving at speed u through a fluid of viscosity η = 1 mPa s (Goldstein, 2015) which in the regime of weak confinement (H30), for a cell speed u 30 ≈ 120 µm/s , is F 30 hd ≈ 11.31 pNx , so that the corresponding thrust force is F 30 th ≈ −11.31 pNx . Given that CR operates at nearly constant thrust since u ∝ η −1 (Qin et al., 2015;Rafaï et al., 2010) and that the flagella of the H10 cell are beating far from the walls ( ∼ 5 µm ) with beat frequency and waveform similar to that of the H30 cell (Videos 1 and 3), we take the flagellar thrust force in strong confinement to be F 10 th ≈ F 30 th ≈ −11.31 pNx as in weak confinement. This thrust is balanced by the total drag on the cell body. The cell speed, u 10 ≈ 4 µm/s , is down by a factor of 30, and so is the hydrodynamic contribution to the drag if we assume the flow is the same as for the H30 geometry. Even if we take into account the tight confinement, and thus assume that the major hydrodynamic drag comes (Brotto et al., 2013;Persat et al., 2015;Bhattacharya et al., 2005) from a lubricating film of thickness δ = (H − D)/2 ≪ D between the cell and each wall, the enhancement of drag due to the fluid, logarithmic in δ/D (Bhattacharya et al., 2005;Ganatos et al., 2006), cannot balance thrust for any plausible value of δ .
The above imbalance drives the vortex flow inversion observed in Figure 3C, as will be shown later theoretically, and implies that the drag is dominated by the direct frictional contact between the cell body and the strongly confining walls, which we denote by F cf . Force balance on the fluid element and rigid walls enclosing the CR in strong confinement requires F 10 th + F 10 hd + F 10 cf = 0 ( Figure 1B). We know that the hydrodynamic drag under strong confinement is greater than 0.38 pN (Stokes drag at u 10 ≈ 4 µm/s ), but lack a more accurate estimate as we do not know the thickness δ of the lubricating film. We can therefore say that the contact force F 10 cf ≲ 10.93 pNx . Thus, the flagellar thrust works mainly against the non-hydrodynamic contact friction from the walls as expected due to the extremely low speed of the strongly confined swimmer.

Theoretical model of strongly confined flow
We begin by using the well-established far-field solution of a parallel Stokeslet between two plates by Liron and Mochon in an attempt to explain the strongly confined CR's flow field (Liron and Mochon, 1976). However, the theoretical flow of Liron and Mochon decays much more rapidly than the experimental one and does not capture the vortex positions and flow variation in the experiment (Appendix 1.2 and Appendix 1-figure 1). This is because the Liron and Mochon approximation to the confined Stokeslet flow is itself singular and also the far-field limit of the full analytical solution, so it cannot be expected to accurately explain the near-field characteristics of the experimental flow (Liron and Mochon, 1976).
We therefore start afresh from the incompressible 3D Stokes equation, −∇p(r) + η∇ 2 v(r) = 0, ∇ · v(r) = 0 , where p and v are the fluid pressure and velocity fields, respectively. Next, we formulate an effective 2D Stokes equation and find its point force solution. In a quasi-2D chamber of height H , we consider an effective description of a CR swimming in the z = 0 plane of the coordinate system with the first Fourier mode for the velocity profile along z , satisfying the no-slip boundary condition on the solid walls, v(x, y, z = ±H/2) = 0 (Figure 4-figure supplement  1). Therefore, the flow velocity varies as v(x, y, z) = v 0 (x, y) cos(πz/H) (Figure 4-figure supplement  1), where v 0 = (vx, vy) is the flow profile in the swimmer's x − y plane that is experimentally measured in Figure 3 (Fortune et al., 2021). Substituting this form of velocity field in the Stokes equation we obtain its quasi-2D Brinkman approximation (Brinkman, 1949), which for a point force of strength F at the z = 0 plane, is where p and v ≡ v 0 are the pressure and fluid velocity in the x − y plane and ∇xy = ∂x x + ∂ y y . We Fourier transform the above equation in 2D and invoke the orthogonal projection operator O k = 1 − k k to annihilate the pressure term and obtain the quasi-2D Brinkman equation in Fourier space.
We perform inverse Fourier transform on Equation 2 in 2D for a Stokeslet oriented along the x-direction, F = F x to obtain its flow field v(r) at the z = 0 plane (Appendix 1.3). This solution is identical to the analytical closed-form expression of Pushkin and Bees, 2016. We have already shown that superposing our Brinkman solution for the conventional three point forces at cell centre and flagellar positions of CR, which leads to the effective three-Stokeslet model in 2D, is an inappropriate description of the strongly confined flow (Figure 3-figure supplement 1A). This is not surprising at this point because the force imbalance between the flagellar thrust and hydrodynamic cell drag suggests that the cell is nearly stationary compared to the motion of its flagella. We utilize this experimental insight by superposing only two Stokeslets of strength −1/2 x each at approximate flagellar positions (x f , ±y f ) = (6, ±11) µm to find qualitatively similar streamlines and vortex flows ( Figure 4A) as that of the experimental flow field ( Figure 3C). However, this theoretical 'two-Stokeslet Brinkman flow' ( Figure 4A) decays faster than the experiment as shown in the quantitative comparison of these two flows in Figure 4B and With the experimental streamlines and vortices well described by a two-Stokeslet Brinkman model, we now explain the slower flow variation in experiment. Strongly confined experimentally observed flow is mostly ascribed to the flagellar thrust, as described above. Clearly, a delta-function point force will not be adequate to describe the thrust generated by flagellar beating as they are slender rods of length L ∼ 11 µm with high aspect ratio. We, therefore, associate a 2D Gaussian source g(r) = e −r 2 /2σ 2 2πσ 2 of standard deviation σ , to Equation 1 instead of the point source δ(r) , in a manner similar to the regularized Stokeslet approach (Cortez et al., 2005). Thus, the quasi-2D Brinkman equation in Fourier space (Equation 2) for a Gaussian force Fg(r) becomes, Superposing the inverse Fourier transform of the above equation for two sources of F = (−1/2, −1/2)x at (x f , ±y f ) = (6, ±11) µm with σ ∼ L/2 = 5µm , we obtain the theoretical flow shown in Figure 4C. RMSD in v x , v y , and v between this theoretical flow and those of the experimental one ( Figure 3C) are 7.8%, 9%, and 8.3%, respectively. Comparing these two flows along representative radial distances from the cell centre as a function of polar angle show a good agreement ( Figure 4D and Figure 4-figure  supplement 2C, D). Notably, Figure 4C, that is, the 'two-Gaussian Brinkman flow', has captured the flow variation and most of the experimental flow features accurately. Specifically, these are the lateral vortices at 20 μm and an anterior stagnation point at 13 μm from cell centre. The only limitation of this theoretical model is that it cannot account for the front-back asymmetry of the strongly confined flow, as is evident from Figure 4D for the polar angles 0 or 2π and π which correspond to front and back of the cell, respectively. This deviation is more pronounced in the frontal region as the cell body squashed between the two solid walls mostly blocks the forward flow from reaching the cell posterior. Thus, the no-slip boundary on the cell body needs to be invoked to mimic the front-back flow asymmetry, which is a more involved analysis due to the presence of multiple boundaries and can be addressed in a follow-up study. Now that we have explained the flow field of CR in strong confinement, we test our quasi-2D Brinkman theory in weak confinement, H = 30 µm , where the thrust and drag forces almost balance each other. Hence, we use the conventional three-Stokeslet model for CR, but with a Gaussian  Figure 3C) with theoretical flow fields (A) and (C), respectively, along representative radial distances, r, from the cell centre as a function of polar angle. Inset of (B) shows the convention used for polar angle. Plots for each r denote the flow magnitudes for those grid points which lie in the radial gap (r, r + 1) μm; r (μm) = 7 (yellow), 13 (blue), 20 (magenta), and 30 (green). Raw data are available in Figure 4-source data 1 and Figure 4source data 2.  Figure 3A). This deviation is expected in weak confinement, D/H ∼ 0.3 , because the quasi-2D theoretical approximation is mostly valid at D/H ≳ 1 , even though RMSD in v x , v y , and |v| remain in the low range at 11.4%, 11.2%, and 13.8%, respectively.
Together, the experimental and theoretical flow fields show that the contact friction from the walls reduces the force-dipolar swimmer in bulk or weak confinement (H30) to a force-monopole one in strong confinement (H10).

Enhancement of fluid mixing in strong confinement
The photosynthetic alga CR feeds on dissolved inorganic ions/molecules such as phosphate, nitrogen, ammonium, and carbon dioxide from the surrounding fluid in addition to using sunlight as the major source of energy (Tam and Hosoi, 2011;Kiørboe, 2008). Importantly, nitrogen and carbon are limiting macronutrients to algal growth and metabolism (Khan et al., 2018;Short et al., 2006;Kiørboe, 2008). For example, dissolved carbon dioxide in the surrounding fluid contains the carbon source essential for photosynthesis and acts as pH buffer for optimum algal growth. It is widely known that flagella-generated flow fields help in uniform distribution of these dissolved solute molecules through fluid mixing and transport which have a positive influence on the nutrient uptake of osmotrophs like CR (Kiørboe, 2008;Tam and Hosoi, 2011;Ding et al., 2014;Short et al., 2006;Leptos et al., 2009;Kurtuldu et al., 2011). This is even more important for the strongly confined CR cells as they cannot move far enough to outrun diffusion of nutrient molecules because of slow swimming speed.
We first calculate the flow-field-based Péclet number, Pe = Vl V /D S where V and l V are the flowspeed and diameter of the flagellar vortex, and D S is the solute diffusivity in water, as the standard measure to characterize the relative significance of advective to diffusive transport. Using the experimentally measured flow data from Figure 3 and D S ≈10 −9 m 2 /s (Shapiro et al., 2014;Kiørboe, 2008;Tam and Hosoi, 2011), we compute the Péclet numbers for the weakly and strongly confined cell to be Pe 30 ≈ 0.5 and Pe 10 ≈ 2 , respectively (see Appendix 1-table 1 and Appendix 1.4). These numbers suggest that flow-field-mediated advection does not completely dominate, but nevertheless can play a role in nutrient uptake for small biological molecules along with diffusion-mediated transport, especially for the strongly confined cell. However, it is evident from the recorded videos of weakly and strongly confined cell suspensions that the tracers are advected more in the H10 than in the H30 chamber (Videos 1 and 3). Hence, we attempt to quantify the observed differences in fluid mixing through correlation in flow velocity and displacement of passive tracers by the swimmers.
We calculate the normalized spatial velocity-velocity correlation function of the flow fields,

Cvv(R) = ⟨v(r) · v(r + R)⟩ ⟨v(r) · v(r)⟩
to estimate the enhancement of fluid mixing in strong confinement ( Figure 5A). The fluctuating flow field has a correlation length, λ = 13.2 µm for the strongly confined H10 flow, which is 37.5% higher than the weakly confined flow in H = 30 µm ( λ = 9.6 µm ), even though the cell is swimming very slowly in strong confinement. This observation is complementary to the experiments of Kurtuldu et al., 2011 where enhanced mixing is observed for active CR suspensions in 2D soap films compared to those in 3D unconfined fluid (Leptos et al., 2009). In their case, the reduced spatial dimension leads to long-ranged flow correlations due to the stress-free boundaries (the force-dipolar flow reduces from v ∼ r −2 in 3D to v ∼ r −1 in 2D). In our case, strong confinement reduces the force-dipolar swimmer in H30 to a force-monopole one in H10 (as shown in the previous The online version of this article includes the following source data and figure supplement(s) for figure 4: Source data 1. Source data for Figure 4A.
Source data 2. Source data for Figure 4C.    . This leads to longer correlation length scales in the flow velocity, which implies an increased effective diffusivity (scaling ∼ Vrmsλ for a velocity field with RMS value Vrms ) of the fluid particles on time scales ≫ λ/Vrms , in strong confinement. Next, we measure the displacement of the passive tracer particles when a single swimmer passes through the field of view (179 μm × 143 μm) in our experiments. The H30 swimmers are fast and therefore pass through this field of view in ∼1-1.4 s ( Figure 5B), whereas the slow-moving H10 swimmers stay in the field of view for the maximum recording time of ∼8 s ( Figure 5C). As the swimmer moves within the chamber, it perturbs the tracer particles. The trajectories of these tracer particles involve both Brownian components and large jumps induced by the motion and flow field of these swimmers. We colour code the tracer trajectories based on their maximum displacement, ∆rtrcr , during a fixed lag time of ∆t =0.2 s (∼10 flagellar beat cycles) ( Figure 5B,C). The tracer trajectories close to the swimming path of the representative H30 swimmer (black dashed arrow) are mostly advected by the flow whereas those far away from the cell involve mostly Brownian components ( Figure 5B). However, a majority of the tracers in the full field of view are perturbed due to the H10 flow, those in the close vicinity being mostly affected ( Figure 5C). Their advective displacements are larger than that of the tracers due to H30 flow (see the colourbar below).
We define the spatial range to which a swimmer motion advects the tracers -radius of influence, R ad -to be approximately equal to the lateral distance from the cell's swimming path (black dashed arrow) where the tracer displacements decrease to ∼20% of their maxima (dark orange trajectories). The region of influence for the H30 cell is a cylinder of radius R ad ≈ 15 µm with the cell's swimming path as its axis ( Figure 5B) and that for the H10 cell is a sphere of radius R ad ≈ 35 µm centred on the slow swimming cell's trajectory ( Figure 5C). That is, the radius of influence of the H10 flow is higher than the H30 one, which corroborates the longer velocity correlation length scale in strong confinement. We also measure the mean-squared displacement (MSD) of the tracers to quantify the relative increment in the advective transport of the H10 flow with respect to the H30 one. We calculate the MSD of approximately 500 tracers in the whole field of view for each video where a single cell is passing through it and then ensemble average over six such videos (  Figure 5-source data 1. The online version of this article includes the following source data and figure supplement(s) for figure 5: Source data 1. Source data for Figure 5A. 1). These plots with a scaling ⟨∆r 2 trcr ⟩ ∝ ∆t α show a higher MSD exponent in H10 ( α ≃ 1.55 ) than H30 ( α ≃ 1.25 ) indicating enhanced anomalous diffusion in strong confinement. Together, Figure 5 shows that the fluid is advected more in strong confinement leading to enhanced fluid mixing and transport. In other words, the opposite vortical flows driven by flagellar beating in strong confinement help in advection-dominated dispersal of nutrients, air and CO 2 in the surrounding fluid, thereby aiding the organism to avail itself of more nutrients for growth and metabolism.

Discussion
Our results show that a prototypical puller-type of microswimmer like CR, when squeezed between two solid walls with a gap that is narrower than its size, has a remarkedly different motility and flow field from those of a bulk swimmer. In this regime of strong confinement, the cells experience a non-hydrodynamic contact friction that is large enough to decrease their swimming speed by 96%. Consequently, their effect on the fluid is dominantly through the flagella, which pull the fluid towards the organism and therefore, the major vortices in the associated flow field have vorticity opposite to that observed in bulk or weak confinement. This leads to an increased mixing and transport through the flow in strong confinement. These experimental results, which arise due to mechanical friction from the walls and not due to any behavioural change, establish that confinement not only alters the hydrodynamic stresses but also modifies the swimmer motility which in turn impacts the fluid flows. This coupling between confinement and motility is typically ignored in theoretical studies because the focus tends to be on the effect of confining geometry on flow fields induced by a given set of force generators (Brotto et al., 2013;Mathijssen et al., 2016), which is appropriate for weak confinement, whereas strong confinement alters the complexion of forces generating the flow. Recent experimental reports have not observed the effect we discuss because they confine CR in chambers of height greater than the cell size ( D/H ≲ 0.7 ) (Jeanneret et al., 2019) where the stresses are mostly hydrodynamic and therefore their theoretical model is force free and different from ours (Appendix 1.5).
Our theoretical approach of using two like-signed Brinkman Stokeslets localized with a Gaussian spread on the propelling appendages can also be easily utilized to analyze flows of a dilute collection of strongly confined swimmers (Appendix 1.6 and Appendix 1- figure 2). Notably, the forcemonopolar flow field of the strongly confined CR is similar to that of tethered microorganisms like Vorticella within the slide-coverslip experimental setup (Pepper et al., 2009;O'Malley, 2011). Therefore, our effective 2D theoretical model involving Brinkman Stokeslet is applicable to these contexts as well. However, one needs to account for the differences in ciliary beating (two-ciliary flow for CR whereas multi-ciliated metachronal waves for Vorticella) for a comprehensive description of the flow field closer to the organism (Pepper et al., 2009;Ryu et al., 2016).
We note that even though CR is known to glide on liquid-infused solid substrates through flagellamediated adhesive interactions (Sasso et al., 2018), it has recently been shown that the strength of flagellar adhesion is sensitive to and switchable by ambient light (Kreis et al., 2017). Consequently, it is likely that CR in its natural habitat of rocks and soils would also utilize swimming in addition to gliding. Our quantitative analysis shows that despite the higher frictional drag due to the strongly confining walls, there is enhanced fluid mixing due to the H10 flow field. That is, the inverse vortical flows driven by the flagellar propulsive thrust help in advection-mediated transport of nutrients to the strongly confined microswimmer. This suggests that swimming is more efficient than gliding for CR under strong confinement (especially in low-light conditions), even though CR speeds are of the same order in both these mechanisms [ u glide ∼ 1 µm/s (Sasso et al., 2018) and u swim ∼ 4 µm/s ]. We note that apart from the time-averaged flows, the oscillations produced in the flow ( v osc ) due to the periodic beating of the flagella can play a role in fluid transport and mixing for both the H30 ( ν b ∼ 55 Hz , order of magnitude estimate of v osc ∼ L × 2πν b ∼ 3450 µm/s ) and H10 ( ν b ∼ 52 Hz , v osc ∼ 3270 µm/s ) cells (Guasto et al., 2010;Klindt and Friedrich, 2015).
Finally, our experimental and theoretical methodologies are completely general and can be applied to any strongly confined microswimmer, biological or synthetic from individual to collective scales. Specifically, our robust and efficient description using point or Gaussian forces in a quasi-2D Brinkman equation is simple enough to implement and analyze confined flows in a wide range of active systems. We expect our work to inspire further studies on biomechanics and fluid mixing due to hard-wall confinement of concentrated active suspensions (Kurtuldu et al., 2011;Pushkin and Yeomans, 2014;Jin et al., 2021). These effects can be exploited in realizing autonomous motion through microchannel for biomedical applications and in microfluidic devices for efficient control, navigation and trapping of microbes and synthetic swimmers (Park et al., 2017;Karimi et al., 2013;Temel and Yesilyurt, 2015).

Materials and methods
Surface modification of microspheres and glass surfaces CR cells are synchronously grown in 12:12 hr light:dark cycle in Tris-Acetate-Phosphate (TAP + P) medium. This culture medium contains divalent ions such as Ca 2+ , Mg 2+ , SO 4 2which decrease the screening length of the 200 nm negatively charged microspheres, thereby promoting inter-particle aggregation and sticking to glass surfaces and CR's flagella. Therefore, the sulfate latex microspheres (S37491, Thermo Scientific) are sterically stabilized by grafting long polymer chains of polyethylene glycol (mPEG-SVA-20k, NANOCS, USA) with the help of a positively charged poly-l-lysine backbone (P7890, 15-30 kDa, Sigma) (Mondal et al., 2020). In addition, the coverslip and slide surfaces are also cleaned and coated with polyacrylamide brush to prevent non-specific adhesion of microspheres and flagella to the glass surfaces, prior to sample injection (Mondal et al., 2020).

Sample imaging
Cell suspension is collected in the logarithmic growth phase within the first 2-3 hr of light cycle and re-suspended in fresh TAP + P medium. After 30 min of equilibration, the cells are injected into the sample chamber. The sample chamber containing cells and tracers is mounted on an inverted microscope (Olympus IX83/IX73) and placed under red light illumination (>610 nm) to prevent adhesion of flagella (Kreis et al., 2017) and phototactic response of CR (Sineshchekov et al., 2002). We let the system acclimatize in this condition for 40 min before recording any data. All flow field data, flagellar waveform and cellular trajectory (except for Figure 2A) are captured using a ×40 phase objective (Olympus,0.65 NA,Plan N,Ph2) coupled to a high-speed CMOS camera (Phantom Miro C110, Vision Research, pixel size = 5.6 μm) at 500 frames/s. As CR cells move faster in H = 30 µm chamber, a 8.2 s long trajectory cannot be captured at that magnification. So we used a ×10 objective in bright field (Olympus, 0.25 NA, PlanC N) connected to a high-speed camera of higher pixel length (pco.1200hs, pixel size = 12 μm) at 100 frames/s to capture 8.2 s long trajectories of H30 cells (Figure 2A).
Our observations are consistent across CR cultures grown on different days and cultures inoculated from different colonies of CR agar plates. We have prepared at least 15-18 samples of dilute CR suspensions from eight different days/batches of cultures, each for chambers of height 10 and 30 μm. Our imaging parameters remain same for all observations. We also use the same code, which is verified from standard particle-tracking videos, for tracking all the cells. We modify the cell tracking code to track the tracer motion for calculating the flow-field data.

Height measurement of sample chamber
We use commercially available double tapes of thickness 10 and 30 μm (Nitto Denko Corporation) as spacer between the glass slide and coverslip. To measure the actual separation between these two surfaces, we stick 200 nm microspheres to a small strip (18 mm × 6 mm) on both the glass surfaces by heating a dilute solution of microspheres. Next, we inject immersion oil inside the sample chamber to prevent geometric distortion due to refractive index mismatch between objective immersion medium and sample. The chamber height is then measured by focusing the stuck microspheres on both surfaces through a ×60 oil-immersion phase objective (Olympus, 1.25 NA). We find the measured chamber height for the 10 μm spacer to be 10.88 ± 0.68 μm and for the 30 μm spacer to be 30.32 ± 0.87 μm, from eight different samples in each case.

Particle-tracking velocimetry
The edge of a CR cell body appears as a dark line ( Figure 1C-E) in phase-contrast microscopy and is detected using ridge detection in ImageJ (Wagner and Hiner, 2017). An ellipse is fitted to the pixelated CR's edge and the major axis vertex in between the two flagella is identified through custom-written MATLAB codes (refer to Source code 1). The cell body is masked and the tracers' displacement in between two frames (time gap, 2 ms) are calculated in the lab frame using standard MATLAB tracking routines (Blair and Dufresne, 2008). The velocity vectors obtained from multiple beat cycles are translated and rotated to a common coordinate system where the cell's major axis vertex is pointing to the right ( Figure 3A, C). Outliers with velocity magnitude more than six standard deviations from the mean are deleted. The resulting velocity vectors from all beat cycles (including those from different cells in H = 30 µm ) are then placed on a mesh grid of size 2.24 μm × 2.24 μm and the mean at each grid point is computed. The gridded velocity vectors are then smoothened using a 5 × 5 averaging filter. Furthermore, for comparison with theoretical flow, the x and y components of the velocity vectors are interpolated on a grid size of 1 × 1 μm 2 . Streamlines are plotted using the 'streamslice' function in MATLAB.

Trajectory tortuosity
Tortuosity characterizes the number of twists or loops in a cell's trajectory. It is given by the ratio of arclength to end-to-end distance between two points in a trajectory. We divide each trajectory into segments of arc-length ≈ 20 µm . We calculate the tortuosity for individual segments and find their mean for each trajectory. We consider the trajectories of all cells whose mean speed >1 μm/s and are imaged at 500 frames/s through ×40 objective for consistency. There were 52 H30 cells, 35 H10 Wobblers, and 23 H10 Synchronous cells which satisfied these conditions and the data from these cells constitute Figure 2G.

Root mean square deviation
The match between experimental and theoretical flow fields is quantified by the RMSD of their veloci- and v th j are the experimental and theoretical values of the velocity fields at the jth grid point, respectively, and NG is the total number of grid points. We calculate RMSD in the x and y components of the flow velocity, that is, in v x and v y , respectively, for a comparison of the vector nature of the flow fields. This is because the signed magnitudes of v x and v y determine the vector direction of the flow. We also calculate RMSD in the flow speed ( |v| = [v 2 x + v 2 y ] 1/2 ) to compare their scalar magnitudes.

Data availability
All data generated or analyzed during this study are included in the manuscript and supporting files. Separate source data files containing source data for each subfigure have been provided. A source code file containing the custom-written MATLAB codes has also been provided.
where (Γ : Γ) bulk = (∂xvx) 2 + 1 2 (∂yvx + ∂xvy) 2 + (∂yvy) 2 and v 0 = (vx, vy) is the flow profile in the swimmer's x − y plane that is experimentally measured in Figure 3. We calculate the viscous power dissipation from the beat-averaged flow fields of CR to be P 30 =0.78 fW in weak confinement and P 10 =1.05 fW in strong confinement. These values are of the same order for both types of confinement and also to that measured for CR in thin fluid films ( P mean flow in Figure 4a of Guasto et al., 2010).

Comparison of our experimental flow data in strong confinement with Liron and Mochon's theoretical solution
The far-field solution of Liron and Mochon for a parallel Stokeslet, F located midway between two no-slip plates is given by (Liron and Mochon, 1976).  Figure 3C and (A), respectively. Except for the theoretical speed along the vortex direction (blue), others are negligible compared to the experiment as shown in the rightmost inset, which is a semilog plot of (C) in the y-axis.
As we have shown that the hydrodynamic cell drag is negligible to the flagellar thrust, the cell-body drag is insignificant and the observed flow field is mostly due to flagellar thrust. We, therefore, superpose Liron and Mochon's solution for two flagellar forces and obtain the flow in Appendix 1- figure 1A. The streamlines of the 'two-Stokeslet Liron and Mochon flow' are qualitatively similar to that of the experiment ( Figure 3C). However, the two-Stokeslet theoretical flow of Liron and Mochon decays much more rapidly than the experimental one and does not capture the experimental flow variation as shown in Appendix 1-figure 1B,C. Notably, there is no signature of vortex position lateral to the forcing point, that is, no minimum in the blue solid curve in Appendix 1- figure 1C because v LM is singular. Therefore, this far-field limit of the theoretical model is insufficient to describe the near-field flow variation, positions of vortices and other flow features of the strongly confined flow accurately. The root mean square deviation (RMSD) in v x , v y , and |v| between the experimental flow of a H10 cell ( Figure 3C) and two-Stokeslet Liron and Mochon's flow is 25.9%, 16.8%, and 30.8%, respectively (see Materials and methods for RMSD definition).

A B
Experiment (