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

  1. Debasmita Mondal
  2. Ameya G Prabhune
  3. Sriram Ramaswamy
  4. Prerna Sharma  Is a corresponding author
  1. Department of Physics, Indian Institute of Science, India
  2. Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, India

Abstract

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.

Editor's evaluation

The manuscript focusses on changes in the flow fields generated by swimming microorganisms as a consequence of them being squeezed within a very narrow gap. The resulting friction with the cell body is such that the flows are dominated by the propelling appendages only, and the authors show in this regime the surprising result that there can be an enhanced nutrient flux to the microorganism.

https://doi.org/10.7554/eLife.67663.sa0

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 et al., 2020) and there are efforts to mimic these by chemical and mechanical means for applications in nano- and microtechnologies (Duan et al., 2015; Temel and Yesilyurt, 2015).

The mechanical interaction of microswimmers with confining boundaries alters their motility and flow fields (Lauga and Powers, 2009; Brotto et al., 2013; Mathijssen et al., 2016), leading to emergent self-organization in cell–cell coordination (Riedel et al., 2005; Petroff et al., 2015), spatial distribution of cells (Tsang and Kanso, 2016; Rothschild, 1963), and ecological aspects such as energy expenditure, nutrient uptake, fluid mixing, transport, and sensing (Lambert et al., 2013; Pushkin and Yeomans, 2014). It is expected that steric interactions will dominate with increasing confinement at the swimmer–wall interface and that hydrodynamic screening by the confining wall will lead to recirculating flow patterns or vortices (Persat et al., 2015; Mathijssen et al., 2016).

Among the abundant diversity of microswimmers, the unicellular and biflagellated algae Chlamydomonas reinhardtii (CR), with body diameter D10μm, are a versatile model system, widely used for understanding cellular processes such as carbon fixation, DNA repair and damage, phototaxis, ciliary beating (Sasso et al., 2018; Brumley et al., 2015; Choudhary et al., 2019; Mondal et al., 2020), and physical phenomena of biological fluid dynamics (Goldstein, 2015; Brennen and Winet, 1977; Rafaï et al., 2010). They are considered next-generation resources for wastewater remediation and synthesis of biofuel, biocatalysts, and pharmaceuticals (Hoh et al., 2016; Khan et al., 2018). Recently, extreme confinement between two hard walls has been exploited to induce stress memory in CR cells towards enhanced biomass production and cell viability (Min et al., 2014; Mikulski and Santos-Aberturas, 2021). Despite the existing and emerging contexts outlined above, knowledge about how rigid walls might modify the kinetics, kinematics, fluid flow and mixing, and theoretical description of a strongly confined microalga such as CR (or any other microswimmer) is scarce. All studies prior to ours have exclusively focused on the effect of boundaries on CR dynamics in PDMS chambers or thin fluid films of height H14μm (Jeanneret et al., 2019; Ostapenko et al., 2018; Guasto et al., 2010), that is, for weak confinement, D/H<1.

Here, we present the first experimental measurements of the flagellar waveform, motility, and flow fields of strongly confined CR cells placed in between two hard glass walls ∼10 μm apart (D/H1, denoted ‘H10 cells’), and infer from them the effect of confinement on kinetics, energy dissipation, and fluid mixing due to the cells. We also measure the corresponding quantities for weakly confined cells placed in glass chambers of height H=30μm (D/H0.3, denoted ‘H30 cells’) for comparison. We find that the cell speed decreases significantly and the trajectory tortuosity increases with increasing confinement as we go from H30 to H10 cells.

Surprisingly, the beat-cycle averaged experimental flow field of strongly confined cells has opposite flow vorticity to that expected from the screened version of bulk flow (Drescher et al., 2010; Guasto et al., 2010). This counterintuitive result comes about because the close proximity of the walls greatly suppresses the motility of the organism and, consequently, the thrust force of the flagella is balanced primarily by the non-hydrodynamic contact friction from the walls. The reason being that the flagellar thrust is largely unaffected by the walls, whereas the hydrodynamic drag on the slowly moving cell body is readily seen to be far smaller. Understandably, theoretical predictions from the source-dipole description of strongly confined swimmers do not account for this vortex flow inversion because they include only hydrodynamic stresses (Brotto et al., 2013; Mathijssen et al., 2016). We complement our experimental results with a simple theoretical description of the strongly confined microswimmer flows using a quasi-2D steady Brinkman approximation to the Stokes equation (Brinkman, 1949), instead of the complicated method of recursive images using Hankel transforms (Liron and Mochon, 1976; Mathijssen et al., 2016). Solving this equation, we demonstrate that the vortex flow inversion in strong confinement is well described as arising from a pair of like-signed force densities localized with a Gaussian spread around the approximate flagellar positions rather than the conventional three overall neutral point forces for CR (Drescher et al., 2010). We also show that under strong confinement there is enhanced fluid transport and mixing despite higher drag due to the walls.

Results

Experimental system

Synchronously grown wild-type CR cells (strain CC 1690) swim in a fluid medium using the characteristic breaststroke motion of two 11 μm long anterior flagella with beat frequency νb5060 Hz. 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=10/30μm 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 Fth and cell body drag Fhd. They balance each other completely for any swimmer in an unbounded medium (Lauga and Powers, 2009; Goldstein, 2015) 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/H1, 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 Fcf (Figure 1B). Therefore, the flagellar thrust is balanced by the combined drag due to the cell body and the strongly confining walls (Figure 1B).

Cell size affects forces acting on confined microswimmers.

Schematics of the forces exerted by a Chlamydomonas cell (green) swimming along the x-axis in between two glass plates separated by height, H under (A) weak confinement where the cell’s body diameter, D<H, and (B) strong confinement where DH. Solid arrows represent local forces exerted by the cell on the surrounding medium. Fth and Fhd are the propulsive thrust distributed equally between the two flagella and hydrodynamic drag due to the cell body, respectively. Fcf is the contact friction with the strongly confining walls (B). Time lapse images of CR cells swimming in a quasi-2D chamber of height H=10μm with (C) synchronously beating flagella with νb39Hz (D13.2μm); (D) asynchronously beating flagella (D9.9μm); and (E) paddler type flagellar beat (D9.7μm). The cell bodies in (D) and (E) wobble due to their irregular flagellar beat pattern and are called ‘Wobblers’. (F) Histogram of cell body diameter in the chamber of H=10μm (number of cells, N=70). Synchronously beating cells (N=34) typically have larger diameter than Wobblers (N=36) and thus the H10 Synchronous cells with D/H1 are strongly confined. Raw data is available in Figure 1—source data 1.

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=30μm are always in weak confinement as the cell diameter varies within D814μm<H. However, this dispersity in cell size becomes significant when CR cells are swimming within quasi-2D chambers of height, H=10μm. 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=10μm 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 xy 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 30μm, 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).

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.

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 xy 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.

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±0.94μm, number of cells, N=34) is larger than that of Wobblers (D=9.92±0.85μm, N=36). Hence, the former’s cell body is squished and strongly confined in H=10μm 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=30μm 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 (u30=122.14±31.59μm/s, N=52) to H10 Synchronous swimmers (u10=4.07±2.88μm/s, N=23). Henceforth, we equivalently refer to the H10 Synchronous CR as ‘strongly confined’ or ‘H10’ cells (D/H1) and the H30 cells as ‘weakly confined’ (D/H<1).

Cell motility in confinement.

Representative trajectories of Chlamydomonas reinhardtii (CR) cells in (A) H=30μm (N=25), (B) H=10μm, Wobblers (N=13); (C) H=10μm, Synchronous cells (N=17). All of these trajectories lasted for 8.2 s and their initial positions are shifted to origin. (D), (E) and (F) are the zoomed in trajectories of (A), (B) and (C), respectively. (G) Cell speed (circles) and tortuosity of trajectories (squares) as a function of the degree of confinement, D/H (N=52,35,23 for H30, H10 Wobbler and H10 Synchronous, respectively). The error bars in the plot correspond to standard deviation in diameter (x-axis), cell speed and tortuosity (y-axes) due to the heterogeneous population of cells. Raw data are available in Figure 2—source data 1–4.

We also note that the flagellar beat frequency of the strongly confined cells, νb1051.58±7.62Hz (averaged over 210 beat cycles for N=20) is similar to that of the weakly confined ones, νb3055.27±8.22Hz (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 (5μm) 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 xy 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 four-lobe 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).

Video 3
Video of a weakly confined Chlamydomonas cell swimming in the presence of tracers.

High-speed video microscopy of a weakly confined Chlamydomonas cell swimming in H = 30 μm chamber in the presence of tracer particles at 500 frames/s. This video shows the natural motility of cells in bulk where they spin about their body axis. The video starts with the cell and its flagella beating in the image plane. At ∼90–180 ms, the flagellar beat of the cell is out of the image plane, when the cell body is rotating about its axis. The flow field is calculated only when the flagellar beat of the H30 cell is in the image plane, that is, for 0–90 and 180–252 ms for this particular video.

Figure 3 with 1 supplement see all
Experimental flow fields of Chlamydomonas reinhardtii (CR) cells in weak and strong confinement.

Experimentally measured, beat-averaged flow fields in the xy plane of synchronously beating CR cells swimming in (A) H=30μm, and (C) H=10μm. Black arrows on the cell body indicate that the cells are swimming to the right. Solid black lines indicate the streamlines of the flow in lab frame. The colourbars represent flow magnitude, v. (B and D) denote the speed variation in (A and C), respectively, along anterior, posterior, and lateral to the cell (where the vortices are present). Distances along anterior and posterior are measured along horizontal lines from the cell centre (0, 0); whereas the lateral (vortex) distances are measured along the vertical line passing through (x, y) = (2, 0) for (B) and (8, 0) for (D), respectively. Raw data are available in Figure 3—source data 1 and Figure 3—source data 2.

The flow field of a representative H10 swimmer (u=5.67±1.57μm/s, νb42.67±2.24Hz) 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 3—figure 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 (u0.15μm/s) and the fastest (u14μm/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 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 Fth exerted by the flagellar motion of the cell balances the hydrodynamic drag Fhd on the moving cell body (Figure 1A). We assume this balance holds for the case of weak confinement (H30) as well. We estimate |Fhd|=3πηDu as the Stokes drag on a spherical cell body of diameter D10μm moving at speed u through a fluid of viscosity η=1mPas (Goldstein, 2015) which in the regime of weak confinement (H30), for a cell speed u30120μm/s, is Fhd3011.31pNx^, so that the corresponding thrust force is Fth3011.31pNx^.

Given that CR operates at nearly constant thrust since uη-1 (Qin et al., 2015; Rafaï et al., 2010) and that the flagella of the H10 cell are beating far from the walls (5μm) 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 Fth10Fth3011.31pNx^ as in weak confinement. This thrust is balanced by the total drag on the cell body. The cell speed, u104μm/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)/2D 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 Fcf. Force balance on the fluid element and rigid walls enclosing the CR in strong confinement requires Fth10+Fhd10+Fcf10=0 (Figure 1B). We know that the hydrodynamic drag under strong confinement is greater than 0.38pN (Stokes drag at u104μm/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 Fcf1010.93pNx^. 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)+η2v(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)=v0(x,y)cos(πz/H) (Figure 4—figure supplement 1), where v0=(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

(1) xyp(r)+η(xy2π2H2)v(r)+Fδ(r)=0,xyv(r)=0

where p and vv0 are the pressure and fluid velocity in the x-y plane and xy=xx^+yy^. We Fourier transform the above equation in 2D and invoke the orthogonal projection operator Ok=1k^k^ to annihilate the pressure term and obtain the quasi-2D Brinkman equation in Fourier space.

(2) vk=OkFη(k2+π2H2)

We perform inverse Fourier transform on Equation 2 in 2D for a Stokeslet oriented along the x-direction, F=Fx^ 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 1/2x^ each at approximate flagellar positions (xf,±yf)=(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 Figure 4—figure supplement 2A,B. The root mean square deviation (RMSD) between these two flows in vx, vy, and |v| are 20.3%, 14.2%, and 22.6%, respectively (see Materials and methods for RMSD definition).

Figure 4 with 3 supplements see all
Theoretical flow fields in strong confinement.

Theoretically computed flow fields for (A) two Stokeslets and (C) two Gaussian forces, both positioned at (6,±11)μm (red arrows) using the quasi-2D Brinkman equation for H=10μm at the z=0 plane. The colourbars represent flow magnitudes normalized by their maximum, vnor. (B, D) Comparison of normalized experimental flow of the Chlamydomonas reinhardtii (CR) in H=10μm (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 4—source data 2.

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 L11μm with high aspect ratio. We, therefore, associate a 2D Gaussian source g(r)=er2/2σ22πσ2 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,

(3) vk=OkFη(k2+π2H2)ek2σ2/2.

Superposing the inverse Fourier transform of the above equation for two sources of F=(1/2,1/2)x^ at (xf,±yf)=(6,±11)μm with σL/2=5μm, we obtain the theoretical flow shown in Figure 4C. RMSD in vx, vy, 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=30μm, where the thrust and drag forces almost balance each other. Hence, we use the conventional three-Stokeslet model for CR, but with a Gaussian distribution for each point force. We, therefore, superpose the solution of Equation 3 for three-Gaussian forces representing the cell body and two flagella in H=30μm. The resulting flow field (Figure 4—figure supplement 3) matches qualitatively with the experimental flow field of CR in weak confinement (Figure 3A). This deviation is expected in weak confinement, D/H0.3, because the quasi-2D theoretical approximation is mostly valid at D/H1, even though RMSD in vx, vy, 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=VlV/DS where V and lV are the flow-speed and diameter of the flagellar vortex, and DS 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 DS10-9m2/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 Pe300.5 and Pe102, 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=13.2μm for the strongly confined H10 flow, which is 37.5% higher than the weakly confined flow in H=30μm (λ=9.6μm), 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 vr-2 in 3D to vr-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 section). 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.

Figure 5 with 1 supplement see all
Correlation in fluid flow and tracer displacements.

(A) Normalized radial velocity–velocity correlation function, Cvv(R), of flow fields in Figures 3A, C, 4C. The dashed vertical lines denote the correlation length scales for the flows, λ=9.6μm (H30) and 13.2 μm (H10, both experiment and theory), where the correlation function decays to 1/e (horizontal dashed line). (B and C) Snapshots showing passive tracer trajectories (coloured) due to a Chlamydomonas reinhardtii (CR) cell (white) swimming along the black dashed arrow in H=30μm and H=10μm, respectively. The H30 swimmer (u=121μm/s) passes through the field of view within 1.3 s whereas the H10 cell (u=3μm/s) traces a semicircular trajectory staying in the field of view for the recording time of 8.2 s. The tracer trajectories are colour coded, according to the colourbar below, based on their maximum displacement, Δrtrcr, during a fixed lag time of Δt=0.2s (∼10 flagellar beat cycles). Scale bars, 15 μm. Raw data is available in Figure 5—source data 1.

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, Δrtrcr, during a fixed lag time of Δt=0.2s (∼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, Rad — 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 Rad15μm with the cell's swimming path as its axis (Figure 5B) and that for the H10 cell is a sphere of radius Rad35μm 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—figure supplement 1). These plots with a scaling Δrtrcr2Δtα show a higher MSD exponent in H10 (α1.55) than H30 (α1.25) 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 CO2 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/H0.7) (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 force-monopolar 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 flagella-mediated 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 [uglide1μm/s (Sasso et al., 2018) and uswim4μm/s]. We note that apart from the time-averaged flows, the oscillations produced in the flow (vosc) due to the periodic beating of the flagella can play a role in fluid transport and mixing for both the H30 (νb55 Hz, order of magnitude estimate of voscL×2πνb3450μm/s) and H10 (νb52 Hz, vosc3270μm/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

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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 Ca2+, Mg2+, SO42- which 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

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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=30μm 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

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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

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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=30μm) 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 μm2. Streamlines are plotted using the ‘streamslice’ function in MATLAB.

Trajectory tortuosity

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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 20μm. 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

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The match between experimental and theoretical flow fields is quantified by the RMSD of their velocities in the normalized scale (v/vmax). RMSD=j=1NG(vjexpt-vjth)2/NG, where vjexpt and vjth 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 vx and vy, respectively, for a comparison of the vector nature of the flow fields. This is because the signed magnitudes of vx and vy determine the vector direction of the flow. We also calculate RMSD in the flow speed (|v|=[vx2+vy2]1/2) to compare their scalar magnitudes.

Appendix 1

1. Power dissipated through the flow fields

In low-Reynolds-number flows, the power P generated by a microswimmer is dissipated through the induced flow fields as P=2ηV(Γ:Γ)dV (Guasto et al., 2010). Here, η is the fluid viscosity, Γ=12[v+(v)T] is the fluid strain rate due to gradients in the flow velocity v, and the integral is over the quasi-2D chamber of height H. Roughly, for flows in bulk or in 2D fluid films, the velocity gradient along the chamber height is negligible and only the 2 × 2 part of Γ corresponding to directions in the plane perpendicular to the confinement direction has non-negligible components (Guasto et al., 2010). This is not true in our case because the rigid boundaries act as momentum sinks, imposing a significant gradient in the fluid flow along the confinement direction z. Since the flow velocity varies as v(x,y,z)=v0(x,y)cos(πz/H) (refer to Figure 4—figure supplement 1 and associated main text), the norm-squared strain rate tensor for hard-wall confined flows is given by Γ:Γ=(Γ:Γ)bulk+(πv0)22H2sin2(πzH) where (Γ:Γ)bulk=(xvx)2+12(yvx+xvy)2+(yvy)2 and v0=(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 P30=0.78fW in weak confinement and P10=1.05fW 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 (Pmean flow in Figure 4a of Guasto et al., 2010).

2. 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 viLM(r)=QSD(-δijr2+2rirjr4)Fj, which is equivalent to that of a 2D source dipole of strength QSD=3H8πηzH(1-zH) (Liron and Mochon, 1976).

Appendix 1—figure 1
Theoretically computed flow field in confinement from Liron and Mochon’s formula.

(A) Theoretically computed flow field using Liron and Mochon’s solution for two-Stokeslet model in 2D. The red arrows at (6, ±11) μm denote the position of the Stokeslets. The colourbar represents flow magnitude normalized by its maximum, vnor. Scale bar, 10 μm. (B) Comparison between normalized experimental flow of a cell swimming in H=10μm (Figure 3C) and Liron and Mochon’s theoretical flow field (A) along representative radial distances, r from the cell centre as a function of polar angle; r (μm)=7 (yellow), 13 (blue), 20 (magenta), and 30 (green). (C) Flow magnitude variation along four directions as indicated by separate colours in the middle inset (lateral to vortex [blue], lateral to cell centre [yellow], anterior [red], posterior [grey]) for the normalized experimental (symbols) and theoretical (solid lines) velocity fields in 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 vLM 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 vx, vy, 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).

3. Inverse Fourier transform of the quasi-2D Brinkman equation in Fourier space

The quasi-2D Brinkman equation in Fourier space, Equation 2 in the main text, is

(A1) vk=OkFη(k2+π2H2)

Here, the orthogonal projection operator in polar coordinates (k,θ) is

(A2) Ok=1k^k^=[1kx^2kx^ky^ky^kx^1ky^2]=[sin2θsinθcosθsinθcosθcos2θ]

where θ is the angle between wave vector k and x-axis. For Stokeslets/Gaussian forces pointing along x- direction only, as in our case, F=[F0], therefore O(k)F=[sin2θsinθcosθ]F.

To compute the velocity field in real space, we inverse Fourier transform Equation A1 in polar coordinates, by replacing the numerator as shown above

(A3) v(r)=1(2π)2ηeikr[sin2θsinθcosθ]F kdkdθ(k2+π2H2)

In polar coordinates, the field points in the x-y plane are given by (x,y)=(rcosϕ,rsinϕ), hence kr=krcos(θϕ). Thus, the fluid velocity field is

(A4) [vxvy](r,ϕ)=F4π2η02πdθ0dk[sin2θsinθcosθ]keikrcos(θϕ)(k2+π2H2)

Let us change the θ integral from (0,2π)(π/2+ϕ,π/2+ϕ), where cos(θϕ)>0. For example, the θ integral for vx changes as follows,

(A5) 02πsin2θeikrcos(θ-ϕ)dθ=-π2+ϕπ2+ϕsin2θeikrcos(θ-ϕ)dθ+π2+ϕ3π2+ϕsin2θeikrcos(θ-ϕ)dθ

Replacing θθπ in the 2nd integral, the limits change as (π/2+ϕ,3π/2+ϕ)(-π/2+ϕ,π/2+ϕ), and the integrands sinθ-sinθ, cosθ-cosθ, cos(θ-ϕ)-cos(θ-ϕ). Therefore, the second integral in the above equation changes to -π2+ϕπ2+ϕsin2θe-ikrcos(θ-ϕ)dθ. Hence, vx’s θ integral becomes

(A6) 02πsin2θeikrcos(θϕ)dθ=2π2+ϕπ2+ϕsin2θcos[krcos(θϕ)]dθ

Similarly, 02πsinθcosθeikrcos(θϕ)dθ=2π2+ϕπ2+ϕsinθcosθcos[krcos(θϕ)]dθ. Thus, the velocity field in polar coordinates is given by,

(A7) [vxvy](r,ϕ)=F2π2ηπ2+ϕπ2+ϕdθ0dk[sin2θsinθcosθ]kcos[krcos(θϕ)](k2+π2H2)

For Gaussian forces, the numerator just gets multiplied by e-k2σ2/2. We perform these 2D integrals in MATLAB for a 20 × 20 xy grid, with k integral ranging from 0 to 100 to obtain the theoretical flow fields in this article.

The above integration takes 3 hr of computational time for two Stokeslets whereas it takes only 1 min to compute the flow field for 2 Gaussian forces of σ=5μm (Processor: Intel i7-4770 CPU with clock speed 3.4 GHz). Hence, we try to write a semi-analytical expression for the case of two Stokeslets. Let us consider krcos(θ-ϕ)=p and πrcos(θ-ϕ)H=q. Then the k- integral changes from 0kcos[krcos(θϕ)](k2+π2/H2)dk0pcospp2+q2dp. We rename this integral as I(q) and calculate it using the Exponential Integral, Ei (Equation 3.723—5 of Gradshteyn and Ryzhik, 2007).

(A8) I(q)=0pcospp2+q2dp=12[eqEi¯(q)+eqEi(q)]

where,

(A9) Ei(q)=qemmdm=qemmdm,for  q<0

and to avoid the singularity for q>0, it is defined by using the principal value of the integral as

(A10) Ei¯(q)=ϵemmdm+ϵqemmdm, where ϵ>0,for  q>0

In our case q>0, so we use Equation A9 for calculating Ei(q) and Equation A10 for calculating Ei¯(q), wherein we use ϵ=10-5. So, Equation A7 reduces to

(A11) [vxvy](r,ϕ)=F2π2ηπ2+ϕπ2+ϕdθ[sin2θsinθcosθ]I(q)

This method computes the flow field for two Stokeslets in 12 min on the same processor.

4. Swimmer-based Péclet number

Generally, speed and length scales in the definition of Péclet number are given by the swimmer speed, u, and radius, R which we refer to as the swimmer-based Péclet number, Pec=uR/DS. By this definition, Pec300.6 and Pec100.02 for the weakly and strongly confined CR, respectively. However, we note that the flow field closer to the cell surface is dominated by the vortices lateral to the cell body (Figure 3A, C), whose magnitude is significantly higher than the swimmer speed for the strongly confined cell (V/u11), in contrast to that of the weakly confined cell (V/u0.3). Hence, the flow-based Péclet number is more appropriate for describing the enhancement of mass transport of solutes due to the vortical flow fields generated by the flagella, particularly for the strongly confined cell (H=10μm). This is shown below (Appendix 1—table 1) to be 100 times higher than the swimmer-based Péclet number, whereas both definitions yield almost similar Pe for the weakly confined cell (H=30μm).

Appendix 1—table 1
Flow-based Péclet number calculation from the flow fields.
H=30μmH=10μm
Vortical flow speed, V(μm/s) (Figure 3A, C)3045
(also frontal flow)
Vortical diameter, lV(μm)2 × vortex point distance
(Figure 3B, D)
2 × 8.5 = 172 × 20 = 40
tadv=lV/V(s)0.570.8
tdiff=lV2/DS(s)0.31.6
Pe=tdiff/tadv=lVV/DS0.52

5. Comparison of our theoretical model of strongly confined flow with that of Jeanneret et al., 2019

Jeanneret et al. provides an effective force-free 2D model for explaining the flow field of confined swimmers between 2 boundaries. They consider a force-free combination of 2D Brinkman Stokeslets along with a 2D source dipole to explain their experimental flows (Jeanneret et al., 2019). They use the analytical solution of Pushkin and Bees, 2016 for their 2D Stokeslets with the permeability length λ = H/√12 (for the z-averaged flow in a Hele-Shaw cell of height H). They consider the conventional three-Stokeslet model of CR where the flagellar thrust, distributed between two Stokeslets of strength −FS/2 each at (x1, ±y1), is balanced by the cell drag of strength FS at (x0, 0), all oriented along the direction of motion. Along with these force-free Stokeslets, they include the 2D source dipole of strength Id at (xd, 0). Finally, they used this model with six free parameters (FS, x0, x1, y1, Id, and xd) to fit their experimentally observed flow fields of CR in confinements ranging from 14 to 60 μm.

However, our theoretical model consists of a 2D Brinkman Stokeslet because the strongly confined CR exerts a net force on the fluid due to the presence of strong non-hydrodynamic contact friction from the walls, unlike that of Jeanneret et al., 2019. This force-monopole is spatially distributed equally at the two flagellar positions, each with a Gaussian regularization to describe the strongly confined flow due to the H10 cell. The reason our theoretical approach is not the same as Jeanneret et al., 2019 is because there are two major differences in our experimental observations. First, we observe that the strongly confined H10 flow is mostly due to the flagellar motion with a 96% reduction in the cell’s swimming speed, thanks to the static friction from the walls (compared to H30 cells), leading to the hydrodynamic cell-drag being nearly absent. This coupling between motility and confinement is not observed by Jeanneret et al., 2019, likely due to the slightly weak confinement (D/H0.7) produced by their experimental methodology, where the stresses present in the system are mostly hydrodynamic. It is therefore appropriate for them to use the force-free three-Stokeslet theoretical model for CR (apart from the source dipole contribution) whereas in our case, the nearly absent hydrodynamic drag experienced by the cell body leads to a monopolar flow with only two Stokeslets (like-signed) localized with a Gaussian spread around the approximate flagellar positions. Second, the spinning motion of CR cells is restricted in our strongly confined H10 chambers unlike those in Jeanneret et al., 2019. They added the extra 2D source dipole in their theoretical model to account for both finite-sized effects of the cell body and spinning motion of the cells [explained in Figure 1c of Jeanneret et al., 2019].

6. Is the two-Gaussian Brinkman model applicable to a collection of strongly confined pullers?

We analyze the fluid flow due to two strongly confined H10 Synchronous cells as a preliminary test for determining the applicability of our theoretical methodology to a collection of microswimmers. Specifically, we measure the beat-averaged flow field of two synchronously beating cells which are separated by 9 body diameters and approach each other head-on (Appendix 1—figure 2A). Therefore, we linearly superpose the solution of the quasi-2D Brinkman equation for a pair of two-Gaussian forces (σ=5μm) at the approximate flagellar positions of the two cells and obtain the resultant flow field (Appendix 1—figure 2B). The position and direction of flow vortices along with the stagnation point in between the two cells match well between the experiment and theory. This suggests that linearly superposing two-Gaussian Brinkman flows might be an adequate description for the flow field of a dilute collection of CRs.

Appendix 1—figure 2
Flow fields due to two strongly confined H10 cells.

(A) Experimentally measured flow field for two synchronous cells swimming in H=10μm. This flow is averaged over ∼30 beat cycles for each cell during which the cells move merely 0.05 times their respective body diameters (D12.42μm). The centre-to-centre distance between the swimmers is 8.75D. Black arrows on the cell bodies indicate their swimming direction. Solid black lines indicate the streamlines of the flow in lab frame. The colourbar represents flow magnitude, v. (B) Theoretically computed flow field by linearly superposing two two-Gaussian Brinkman flow, one for each cell. The positions of the pair of two-Gaussian forces at approximate flagellar positions are denoted by red arrows. The streamlines, vortex flows, and stagnation point at the centre of the grid match qualitatively with the experimental one (A). The colourbar represents flow magnitudes normalized by its maximum, vnor. Scale bars, 20 μm.

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.

References

  1. Book
    1. Gradshteyn IS
    2. Ryzhik IM
    (2007)
    Table of Integrals, Series, and Products
    Elsevier/Academic Press.
  2. Book
    1. O’Malley S
    (2011)
    Bi-Flagellate Swimming Dynamics
    University of Glasgow.

Decision letter

  1. Raymond E Goldstein
    Reviewing Editor; University of Cambridge, United Kingdom
  2. Anna Akhmanova
    Senior Editor; Utrecht University, Netherlands

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.

Decision letter after peer review:

Thank you for submitting your article "Strong confinement of active microalgae leads to inversion of vortex flow and enhanced transport" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Anna Akhmanova as the Senior Editor. The reviewers have opted to remain anonymous.

The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.

Essential revisions:

1) A general conclusion of the reviewers is that while they appreciate the approach taken to the Brinkman equation, reference should be made to other recent work in this area, including Pushkin and Bees (10.1007/978-3-319-32189-9_12), Jeanneret et al., (PRL; 10.1103/PhysRevLett.123.248102) and Fortune et al., (JFM; 10.1017/jfm.2020.1112). In addition, the revision should not only discuss previous work but also improve the interpretation of the results. The reviewers felt that the discussion should avoid mystifying the physics, which they believe is straightforward – when puller cells are subject to wall friction due to being in a tight space, they exert a net force towards themselves, and the associated flow field decays more slowly.

2) Chlamydomonas is not neutrally-buoyant (page 4).

3) The Peclet number calculation on page 10 is unclear. The diffusion coefficient is an order of magnitude estimate, whereas the Peclet number is calculated to one significant figure, and value of 2 versus 0.5 is considered a change of regime. Really all this shows is that diffusion and advection are both likely to be important for small molecules. Considering the effect of flow oscillations (order of magnitude estimate 10 microns x 250 rad/s = 2500 microns/s) suggests advection is more important still.

4) More data on beat frequency and how it varies with confinement would be useful. The impression is given that maybe it is near-constant?

5) Some of the presentation is unclear, for example the azimuthal variation of the flow fields (Figure 5BD and some of the appendix figures) it is difficult to see the scale and the variation is very compressed.

6) The argument for wall contact force is essentially correct I think but is unclearly written. If there is no contact force, the correct force balance taken over the cell F_drag + F_propulsive = 0. The wall changes the drag and propulsive force, but does not directly exert a force on the cell. If there is contact, then the balance is F_drag + F_propulsive + F_contact = 0. The last two sentences about 'could in principle originate in direct frictional contact' is confusing because this is actually the conclusion of the argument, not just something 'in principle'.

7) There is a lack of clarity about measures of variation versus measures of uncertainty. For example, the 'error bars' in Figure 2G represent standard deviation in a heterogeneous population, not a measure of experimental uncertainty.

8) Line 48. Should it be "primed" instead of "prime"?

9) Line 51. "[…]ecological characteristics and theoretical description". Unclear meaning

10) Line 53 (and in the discussion). "soft PDMS". I do not think that the Youngs modulus of the PDMS in a normal microfluidic device (which by the way is several mm thick) is small enough for the channels to be bent to any significantly degree by swimming microorganisms. I have many years of experience in this field and personally I have never seen any such thing.

11) Line 60,61. "We find that the cell speed decreases significantly and the trajectory tortuosity increases". Have the author checked whether -and how much- the rotational diffusivity of the cells change in confinement? Is the increased tortuosity of the trajectory just a consequence of their smaller speed?

12) Lines 65,66. "but also to those predicted from the source dipole theory of strongly confined swimmers"

Line 339-342. "This result is contrary to the common theoretical expectation that the far-field flow of a confined microswimmer between two closely spaced solid walls is a 2D source dipole pointing along the swimmer's propulsion direction". I find these sentences misleading. The "theoretical expectations" were made with the assumption that the only stresses present in the system are hydrodynamic. Therefore it is not surprising that one finds something different when this is not the case. The statements in the paper suggest to the reader that the previous theoretical models were wrong, whereas it is just that now the cells' friction with the walls needs to be taken into account.

13) I am not convinced by that the current method used to extract the friction force for the body. The authors rely on an estimate of the drag based on the bulk drag coefficient, which is certainly an underestimate of the hydrodynamic drag of the body under confinement. I agree that in the 10um case the cells move at such a low speed that the contribution from hydrodynamic friction is much smaller than that from contact friction with the wall. But at this point one might as well just forget about hydrodynamic friction rather than removing the wrong estimate for it.

14) Line 245. It would help the reader to include that v_rad^0 is equal to \vec{v}\dot\hat{r}.

Reviewer #1:

The manuscript aims to study (through particle tracking microscopy), understand (through mathematical modelling) and interpret (through analysis of mixing) the time-averaged flow fields around swimming algae when the cells are squeezed between rigid glass surfaces.

The main strengths of the work are the acquisition and analysis of extensive flow data, and development of a rather simple and elegant mathematical model for the flow, based on depth-averaging the viscous flow equations with spatially-smoothed force terms, and subsequent Fourier solution. This approach is much simpler than known solutions of the 3D flow equations involving infinite sequences of images or Hankel transforms, and/or computational solutions of the flow problem which resolve the cell body and flagella.

Weaknesses include:

a) The terminology around 'extreme/strong/weak confinement' (borrowing from quantum physics?) perhaps gives the impression of physical complexity, whereas what has been done is quite simple – cells were examined in a chamber in which there was quite a lot of space to move, or not very much space, or they were squeezed very tightly and barely able to move. In the first two cases they could move effectively through the fluid and so cell tends to drag fluid along from behind and push it forward in front. In the squeezed case, the cell is barely moving and so the flagella pull fluid from front to back, reversing the sense of the surrounding vortices.

b) Along similar lines, perhaps more is made of the use of 'hard versus soft' boundaries. In appendix 1.1 is it claimed that soft boundaries (such as PDMS?) will not produce a significant velocity gradient. However, the correct boundary condition on a soft solid surface is still the no-slip condition. I would think the important issue is the deformability of the boundary compared with the cell, and hence the level of friction resulting from squeezing.

c) The model is good, but the reasons for its success relative to the solution of Liron and Mochon that it is compared to are perhaps simpler than suggested. The approximation for the flow field due to a force monopole in a confined domain of Liron and Mochon (which is potential-dipole in character) is (i) singular, which is responsible for the spike in appendix figure 1c, (ii) in any case is the far-field limit of the full singular solution, so cannot be expected to be accurate in the near-field. Conversely, it is relatively unsurprising that the near-field of the cilia motion is better modelled by a spatially-averaged force. Whether the force needs to be Gaussian, or some other regularization, is unclear.

d) Both the data and the model are for time-averaged flow field, which loses the (large) oscillations occurring in the near-field flow due to the flagellar beat. These oscillations may have a role in mixing. Resolving this flow is probably very difficult experimentally, but can be accomplished in silico through computational modelling. The difference between time-averaged and instantaneous flow should at least be acknowledged.

e) The slower decay of the velocity correlation in the confined case (figure 5 – supplement figure 2) could be explained by the fact that the confined case produces a force monopole, and hence a lower order of decay than a force-free swimmer? In which case we are not necessarily seeing evidence of increased mixing?

f) Apart from a brief reference to a review paper, relatively little contact is made with Chlamydomonas biology or ecology, particularly concerning the functional importance of fluid mixing. I am not a specialist in this area; my question is, under what circumstances is the ability to exchange e.g. carbon dioxide with the surrounding fluid a limiting factor in metabolism? Are we seeing something specific to CR's adaptation to certain natural habitats, or just that tethered swimmers always disturb more fluid than free swimmers due to the force monopole?

Appraisal: I believe the authors have partially achieved their aim of studying and understanding the flow fields produced by confined swimming algae.

Likely impact: I believe the paper may help to clarify understanding of the fact that the flow field changes when a swimming cell is in contact with a surface; the mathematical approach will be valuable in producing simplified mechanistic models of the flow fields produced by cells in the ubiquitous microscope slide-coverslip set up.

Reviewer #2:

The manuscript focusses on changes in the flow fields generated by swimming microorganisms as a consequence of them being squeezed within a gap that is narrower than their size. This is studied here in the context of the unicellular green microalga Chlamydomonas reinhardtii, which is a common model system for microbial motility of body sizes ranging from 8um to 14um. The authors compare their behaviour in the two cases of a 30um and 10um-thick Hele-Shaw cell. The resulting flow fields are then compared with those obtained by a superposition of quasi-2D Stokeslets, Green functions of the Brinkman equation.

The paper has three main results. Firstly, cell behaviour in the 10um-thick samples depends clearly on microorganismal size. Larger cells beat their flagella with the standard synchronous breaststroke, while smaller ones display either asynchronous beating or "paddling" (as the authors call it, but see below my comment on this). Secondly, the larger cells display an average flow field that, in the far field, is directed oppositely to what would be predicted in absence of cell squeezing by the walls. Thirdly, the paper presents a theoretical modelling for the observed flow fields in terms of a "Gaussian" Brinkman Stokeslets. This flow field is proposed to increase the flow of nutrients to the cell.

1) The confinement that the authors focus on is in a different regime that what has been addressed earlier, in the sense that it starts to probe strong non-hydrodynamic friction, which is a case that can definitely happen in nature. This is interesting, although likely to depend quantitatively very much on topographical and chemical details of the actual surfaces that are in close contact.

Given that the strongly confined cells experience a non-hydrodynamic friction that is large enough to almost halt their swimming, it is not surprising that the measured flow field is dominated by the forces imposed on the fluid by the flagella and therefore a force-free 3-Stokeslet model is inappropriate. After all, the presence of non-hydrodynamic friction means that the cell exerts a net force on the fluid. It seems to me that this situation essentially resembles that of a tethered microorganism like Vorticella and I would have liked to see more discussion on the similarities between the two cases, both experimentally and from a modelling perspective. I do not think this is developed sufficiently in the manuscript.

Besides this, the range of gaps probed by the authors is rather coarse. The 10um gap is not much smaller than the smallest gaps that were probed in ref [12] for which -however- the flow field was qualitatively different. In my opinion, within the context of the effect of confinement on flow fields, it would have been interesting to explore this range in more detail, studying the transition from the hydrodynamic to the contact-friction case.

2) Regarding the modelling, I have two main comments. Firstly, I appreciate the idea of a diffused Stokeslet. However, it does look very close to the idea of a regularised Stokeslet (Cortez 2005), which is not even mentioned in the manuscript. I think this is quite surprising. I would expect the manuscript to comment on differences between the two approaches.

Secondly, I do not understand the FT approach to finding the 2D Stokeslet for the Brinkman equation, when this is actually known analytically. It has been published in

Pushkin, D. O. and Bees, M. A. Bugs on a slippery plane: Understanding the motility of microbial pathogens with mathematical modelling. Adv. Exp. Med. Biol. 915, 193-205 (2016).

which is cited in ref [12] of the manuscript (Jeanneret et al., PRL 2019). This paper also shows that a model based on the point forces from Liron and Mochon does not perform well (see Suppl Mat), while one made of Brinkman Stokeslets with a 2D source dipole does. Therefore I do not think that it is surprising to see that the same approach works well for the current manuscript. In fact, given the similarity between the current setup and that in [12], I would have expected a comparison between the current approach and the full model from [12] at least in the Supplementary Material.

Finally, I think that comparing the magnitude of the flow fields as in Figure 5B,D is insufficient. One should instead show that both magnitude and direction of the flow fields are well captured by the model. This cannot really be grasped by comparing by eye Figure 3C and Figure 5A, C.

3) As for the question of nutrient transport, I find the claim on the enhancement for the 10um-cells not completely convincing. It is clear that the fluid fluxes for the 30um-cells and the 10um-cells are organised in an opposite way. However, Figure 4D shows that the positive part of the flux is of a very similar magnitude in both cases, in particular for the average fluxes. This is true also when one compares the negative fluxes. Given the similarities, I do not find a compelling reason why the cell should have a much higher nutrient uptake in one of the two cases. This might be the case, but there is not enough evidence in the paper to support this statement.

4) Finally, regarding the “paddling” state, I’m afraid this is the normal flagellar shock response in Chlamydomonas and not some sort of previously unknown state of the cells. It is known that the shock response can be elicited both by intense light stimuli or by mechanical stimuli. It is very likely that this is just a mechanosensitive shock, which comes from the mechanical interactions between the flagella and the upper/lower walls. In turn, these are possible due to the fact that cells of smaller size than the gap can spin around their body enough to touch the walls with their flagella. I think that the authors would need to investigate further the existing literature on flagellar shock response in Chlamydomonas and put appropriately in context the “paddling” behaviour they observe.

https://doi.org/10.7554/eLife.67663.sa1

Author response

Essential revisions:

1) A general conclusion of the reviewers is that while they appreciate the approach taken to the Brinkman equation, reference should be made to other recent work in this area, including Pushkin and Bees (10.1007/978-3-319-32189-9_12), Jeanneret et al., (PRL; 10.1103/PhysRevLett.123.248102) and Fortune et al., (JFM; 10.1017/jfm.2020.1112). In addition, the revision should not only discuss previous work but also improve the interpretation of the results. The reviewers felt that the discussion should avoid mystifying the physics, which they believe is straightforward – when puller cells are subject to wall friction due to being in a tight space, they exert a net force towards themselves, and the associated flow field decays more slowly.

We thank the editor for summarising the reviewers’ comments and suggesting the changes to the discussion. We now cite recent related works and discuss their connection to our approach in the revised manuscript. We’re sorry our text gave the impression of “mystifying” a simple but worthwhile piece of physics, and we hope our modified discussion is free of this defect.

2) Chlamydomonas is not neutrally-buoyant (page 4).

We thank the editor for pointing this out. Indeed, Chlamydomonas reinhardtii (CR) is not neutrally buoyant, rather its density is 5% greater than that of water [Drescher et al., 2010]. However, CR can still be considered as a force-free swimmer because the sedimentation speed is far smaller than its swimming velocity [Ishikawa et al., 2006, Mathijssen et al., 2015]. The gravitational effect due to the excess density (Δρ50 kg/m3) results in the sedimentation speed to be given by us=Fg6πηR=2ΔρR2g9η where Fg is the buoyant weight of CR, R5μm is the cell body radius, g is the acceleration due to gravity and η=1mPa.s is the viscosity of the medium. Therefore, us2.72 μm/s which is about 2% of the swimming speed u30120 μm/s of CR in bulk. Hence, the Stokeslet component in the flow due to Fg is negligible for the near-field distances r<35R [Drescher et al., 2010] and it is appropriate to consider the swimmer as force-free over the distances where we are measuring the flow field.

We remove the word ‘neutrally buoyant’ and rephrase the statement at the end of page 4 of the revised draft as:

“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].”

3) The Peclet number calculation on page 10 is unclear. The diffusion coefficient is an order of magnitude estimate, whereas the Peclet number is calculated to one significant figure, and value of 2 versus 0.5 is considered a change of regime. Really all this shows is that diffusion and advection are both likely to be important for small molecules. Considering the effect of flow oscillations (order of magnitude estimate 10 microns x 250 rad/s = 2500 microns/s) suggests advection is more important still.

We appreciate the editor’s viewpoint and agree that our Peclet number calculation only suggests that both diffusion and advection are important for most of the biological molecules (nutrient salts, oxygen etc.) which are small. It is evident from the cell suspension videos that the tracers are advected more by the H10 cell than the H30 cell (Video 1 and 3). Hence, as a first and standard measure to characterize the relative significance of advective to diffusive transport, we calculate the Peclet number. We agree with the editor that we should not consider the value of 2 vs 0.5 as a change of regime and that all it says that both advection and diffusion are important for the confined CR cells.

We also agree that the order-of-magnitude estimates of flagella-driven flow oscillations (vosc) suggest that advection is important for both the H30 (νb55 Hz, vosc3450 μm/s) and H10 (νb52 Hz, vosc3270 μm/s) cells as their beat frequencies (νb) are similar. However, we are interested in the long-time behaviour where these flow oscillations are averaged out and the recorded videos of the H10 cell suspensions hint at enhanced advection due to beat-averaged flows. The slower decay of the velocity correlation for the strongly confined H10 flow (revised Figure 5A) already supports this observation. In addition, we now include direct evidence of enhanced mixing and transport of tracers due to the H10 flows (revised Figure 5, B and C; Figure 5 —figure supplement 1), thanks to the suggestion of the reviewers (please see our response to point (e) of Reviewer 1). This explicitly shows that the fluid is indeed advected more in the strongly confined case than the weakly confined one. Calculation of the Peclet number was simply a preamble at attempting to quantify the increased fluid transport and mixing through flow correlation length scales and mean-squared displacement of tracers averaged over several flagellar beat periods.

We acknowledge the editor’s suggestions by modifying the text related to Peclet number in our revised manuscript appropriately (pg. 18). We also add a few lines acknowledging the importance of flow oscillations in fluid mixing as suggested by the editor in the discussion of our revised manuscript (pg. 21).

4) More data on beat frequency and how it varies with confinement would be useful. The impression is given that maybe it is near-constant?

We thank the editor for this relevant suggestion. We measure the beat frequency of the strongly confined H10 Synchronous cells and weakly confined H30 cells (when their flagellar beat is in the image plane, the same data that is considered for analyzing the flow fields). The flagellar beat pattern of the H10 Wobblers is irregular and hence we cannot assign a beat period to these cells.

The beat frequency of H10 Synchronous cells (degree of confinement, DH1.2) is νb1051.58±7.62 Hz (averaged over 210 beat cycles from 20 representative cells) and that of H30 cells (DH0.35) is νb3055.27±8.22 Hz (averaged over 194 beat cycles from 20 representative cells). Therefore, the flagellar beat frequency is similar with varying confinement. 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.

We acknowledge the editor’s suggestion by including this data in our revised manuscript (page 8).

5) Some of the presentation is unclear, for example the azimuthal variation of the flow fields (Figure 5BD and some of the appendix figures) it is difficult to see the scale and the variation is very compressed.

We appreciate the editor’s concern and acknowledge it by modifying all the comparison between the azimuthal variation of the experimental and theoretical flow fields to the same normalised scale between 0 and 1, without any compression (see revised Figures 4B, 4D, Figure 4—figure supplement2, Figure 4—figure supplement3, Appendix1-Figure1B). In these plots, we have included the azimuthal variation for four representative radial distances from the cell centre, r = 7, 13, 20 and 30 μm, for clarity.

6) The argument for wall contact force is essentially correct I think but is unclearly written. If there is no contact force, the correct force balance taken over the cell F_drag + F_propulsive = 0. The wall changes the drag and propulsive force, but does not directly exert a force on the cell. If there is contact, then the balance is F_drag + F_propulsive + F_contact = 0. The last two sentences about 'could in principle originate in direct frictional contact' is confusing because this is actually the conclusion of the argument, not just something 'in principle'.

We appreciate the suggestion of the editor. We completely agree as regards the force balance equation in the absence and presence of contact force. We remove the sentence “The forces exerted by the cell…” while introducing the wall drag at the beginning of the section – Force balance on confined cells. We rephrase and re-write this whole section of force balance (pg. 12) to clearly state the contribution from different forces in our revised manuscript and it acknowledges the editor’s suggestion in the last sentence:

“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.”

7) There is a lack of clarity about measures of variation versus measures of uncertainty. For example, the 'error bars' in Figure 2G represent standard deviation in a heterogeneous population, not a measure of experimental uncertainty.

We agree with the editor and accordingly, clarify the statement in Figure 2G as:

“The error bars in the plot corresponds to standard deviation due to the heterogenous population of cells.”

8) Line 48. Should it be "primed" instead of "prime"?

We thank the editor for seeking clarification. We have now modified the word ‘stress-prime’ for better readability in the revised manuscript (line 64):

“…extreme confinement between two hard walls has been exploited to induce stress memory in CR cells towards enhanced biomass production and cell viability.”

9) Line 51. "[…]ecological characteristics and theoretical description". Unclear meaning

We replace the word ‘ecological characteristics’ with ‘fluid flow and mixing’ in the revised manuscript and the modified sentence in line 68 reads as “…how rigid walls might modify the kinetics, kinematics, fluid flow and mixing, and theoretical description of a strongly confined microalga such as CR is scarce …”

10) Line 53 (and in the discussion). "soft PDMS". I do not think that the Youngs modulus of the PDMS in a normal microfluidic device (which by the way is several mm thick) is small enough for the channels to be bent to any significantly degree by swimming microorganisms. I have many years of experience in this field and personally I have never seen any such thing.

We thank the editor for pointing this out. We agree with the editor that the elastic modulus of PDMS is too high for the microorganism to bend the microfluidic device. We, therefore, refrain from using the word ‘soft’ for PDMS chambers in the introduction and discussion of the revised manuscript.

11) Line 60,61. "We find that the cell speed decreases significantly and the trajectory tortuosity increases". Have the author checked whether -and how much- the rotational diffusivity of the cells change in confinement?

We appreciate the editor for raising this concern and seeking clarifications. We cannot measure the rotational diffusivity of the cells from the data we have used in Figure 2G (cell speed and tortuosity) because the trajectories are captured through 40X objective and hence, we have access to only short-time data. Therefore, we capture long-time trajectories of CR cells in chambers of height 10 and 30 μm through a 10X bright-field objective. The only drawback of these data is that we cannot differentiate Wobblers from Synchronous cells in H=10 μm robustly as we cannot observe their flagellar beat through bright-field microscopy. We filter the trajectories whose instantaneous speed < 0.5 μm/s to remove the cells which are stuck to the glass surfaces at some points in their trajectory through flagella or cell body, at the cost of excluding some Synchronous (non-stuck) cells from the analyses because their speed is similar to that of a stuck cell.

The rotational diffusion coefficients, DR, extracted from the mean squared angular displacement vs lag time plot (from 10X data) for the H30 cells are DR300.11 rad2/s and that of the H10 cells are DR100.76 rad2/s. That is, DR increases by almost 85% when the confinement increases from 30 μm to 10 μm. The corresponding increase in tortuosity (from 40X data shown in Figure 2G) from H30 to H10 cells is approximately 60%. Therefore, the increase in tortuosity with increasing confinement is correlated with increase in the rotational diffusivity of cells.

Is the increased tortuosity of the trajectory just a consequence of their smaller speed?

The increase in tortuosity is not because of the decrease in speed with increasing confinement. It depends on the nature of the walk i.e., the statistics of angular displacement, rather than the speed magnitude. In principle, a slow swimmer can still move along straight lines without frequent turning events which will lead to a greater persistence and thereby low tortuosity in the trajectories (as well as low rotational diffusivity of the cells).

12) Lines 65,66. "but also to those predicted from the source dipole theory of strongly confined swimmers"

Line 339-342. "This result is contrary to the common theoretical expectation that the far-field flow of a confined microswimmer between two closely spaced solid walls is a 2D source dipole pointing along the swimmer's propulsion direction". I find these sentences misleading. The "theoretical expectations" were made with the assumption that the only stresses present in the system are hydrodynamic. Therefore it is not surprising that one finds something different when this is not the case. The statements in the paper suggest to the reader that the previous theoretical models were wrong, whereas it is just that now the cells' friction with the walls needs to be taken into account.

We thank the editor for pointing this out. We modify these sentences in the text of our revised manuscript to clearly state the conditions in which previous theoretical models are applicable.

13) I am not convinced by that the current method used to extract the friction force for the body. The authors rely on an estimate of the drag based on the bulk drag coefficient, which is certainly an underestimate of the hydrodynamic drag of the body under confinement. I agree that in the 10um case the cells move at such a low speed that the contribution from hydrodynamic friction is much smaller than that from contact friction with the wall. But at this point one might as well just forget about hydrodynamic friction rather than removing the wrong estimate for it.

We thank the editor for raising this concern. We agree that the contribution from the zeroth-order Stokes drag is an underestimate of the net hydrodynamic drag of the cell body under strong confinement. We, therefore, remove this estimate as the total hydrodynamic cell drag for the strongly confined cells in the revised manuscript and re-write the section ‘Force balance on confined cells’ (pg. 12, purple coloured text) to acknowledge this concern.

14) Line 245. It would help the reader to include that v_rad^0 is equal to \vec{v}\dot\hat{r}.

We thank the editor for this suggestion. In the revised manuscript, we would have included vrad0(r,ϕ)=v0r^=vxcosϕvysinϕ which denotes the inwards radial flow in the z=0 plane i.e., the flow towards the cell situated at the origin of the coordinate system and hence the negative sign. However, we have replaced this fluid flux analysis with direct evidence of enhanced mixing in revised Figure 5 and associated text in response to the suggestion (3) of reviewer 2. Therefore, this text is not present is the revised manuscript.

Reviewer #1:

The manuscript aims to study (through particle tracking microscopy), understand (through mathematical modelling) and interpret (through analysis of mixing) the time-averaged flow fields around swimming algae when the cells are squeezed between rigid glass surfaces.

The main strengths of the work are the acquisition and analysis of extensive flow data, and development of a rather simple and elegant mathematical model for the flow, based on depth-averaging the viscous flow equations with spatially-smoothed force terms, and subsequent Fourier solution. This approach is much simpler than known solutions of the 3D flow equations involving infinite sequences of images or Hankel transforms, and/or computational solutions of the flow problem which resolve the cell body and flagella.

We sincerely thank the reviewer for appreciating our work and providing us positive feedback.

Weaknesses include:

a) The terminology around 'extreme/strong/weak confinement' (borrowing from quantum physics?) perhaps gives the impression of physical complexity, whereas what has been done is quite simple – cells were examined in a chamber in which there was quite a lot of space to move, or not very much space, or they were squeezed very tightly and barely able to move. In the first two cases they could move effectively through the fluid and so cell tends to drag fluid along from behind and push it forward in front. In the squeezed case, the cell is barely moving and so the flagella pull fluid from front to back, reversing the sense of the surrounding vortices.

We thank the reviewer for raising this concern. Our use of the terminology wasn’t inspired by quantum physics but rather relied on the fact that these terms are commonly used in the existing literature concerning microswimmers. For example, Brotto et al., 2013; Mathijssen et al., 2016 and Jeanneret et al., 2019 frequently use terms such as “strongly confined geometries”, “strong confinement”, “weak confinement” in the main text to describe the experimental/theoretical analysis conditions for microswimmers. Hence, in accordance with the existing literature we use the term ‘strong/extreme’ confinement when the ratio of average diameter of cells to chamber height, DH1.2 and ‘weak’ confinement when DH0.3. We would like to re-draw the attention of the reviewer to the existing statement in our manuscript (pg. 8) “Henceforth, we equivalently refer to the H10 Synchronous CR as ‘strongly confined’ or H10 cells (DH1) and the H30 cells as ‘weakly confined' (DH<1).” We feel this sentence defines the terminology explicitly for the reader’s convenience and addresses the concern sufficiently well.

b) Along similar lines, perhaps more is made of the use of 'hard versus soft' boundaries. In appendix 1.1 is it claimed that soft boundaries (such as PDMS?) will not produce a significant velocity gradient. However, the correct boundary condition on a soft solid surface is still the no-slip condition. I would think the important issue is the deformability of the boundary compared with the cell, and hence the level of friction resulting from squeezing.

We thank the reviewer for pointing this out. Indeed, any microfluidic chamber with a solid surface will induce a no-slip boundary condition. However, in appendix 1.1, we mean that the soft boundaries are that due to a freely-suspended fluid film such as a soap film which has stress-free film boundaries at the fluid-air interface (e.g., the experiments of [Guasto et al., 2010; Wu and Libchaber, 2000]) and not a chamber made of PDMS. In such a quasi-2D fluid film with no solid boundaries, there is no velocity gradient along the height of the film (Supplementary materials of [Guasto et al., 2010]). But we completely agree with the reviewer that the elastic modulus of PDMS is too high for the microorganism to deform the microfluidic device. We, therefore, use the term ‘soft’ carefully and modify ‘soft 2D film’ to ‘thin fluid film’ in appendix 1.1 as well as refrain from using the word ‘soft’ for PDMS chambers in introduction and discussion.

c) The model is good, but the reasons for its success relative to the solution of Liron and Mochon that it is compared to are perhaps simpler than suggested. The approximation for the flow field due to a force monopole in a confined domain of Liron and Mochon (which is potential-dipole in character) is (i) singular, which is responsible for the spike in appendix figure 1c, (ii) in any case is the far-field limit of the full singular solution, so cannot be expected to be accurate in the near-field. Conversely, it is relatively unsurprising that the near-field of the cilia motion is better modelled by a spatially-averaged force. Whether the force needs to be Gaussian, or some other regularization, is unclear.

We thank the reviewer for this pertinent suggestion. We agree that the solution of Liron and Mochon cannot be expected to match our experimental flow data in the near-field because of the reasons (i) and (ii) mentioned above. We have now added and modified texts to acknowledge these reasons in our revised manuscript (see the first paragraph of section ‘Theoretical model of strongly confined flow’ and Appendix 1.2).

We agree with the reviewer that it may be relatively unsurprising that the near-field flow characteristics due to the flagellar motion are better modelled by a spatially averaged force. Unfortunately, the evidence of using such regularization for comparison with the experimentally observed flagellar flows is elusive in the literature. Mostly, the very near-field flow is theoretically described using a line distribution of Stokeslets along the flagella, which also works well. But, on the spatial scale that we are observing the flow, use of a single Gaussian force is a neat trick to explain the coarse-grained flagellar flow easily.

We now acknowledge the method of regularized Stokeslets [Cortez, 2001; Cortez et al., 2005] which is in a manner similar to our approach, thanks to the suggestion of Reviewer 2. Our convolution approach differs from this method in its implementation – our method gives numerical solution to any form of regularization to the Stokeslet whereas Cortez et al., approach gives analytical closed-form solutions to the same. The reason we choose a Gaussian form of the regularization is because it is the simplest example of a radially symmetric function with the same functional form in the Fourier space. Any other form of radially symmetric smooth function would have worked equally well, but we need not attempt to find such regularization because the theoretical flow obtained from the Gaussian spread to the force agrees very well with our experimentally observed one.

d) Both the data and the model are for time-averaged flow field, which loses the (large) oscillations occurring in the near-field flow due to the flagellar beat. These oscillations may have a role in mixing. Resolving this flow is probably very difficult experimentally, but can be accomplished in silico through computational modelling. The difference between time-averaged and instantaneous flow should at least be acknowledged.

We thank the reviewer for raising this important point. We completely agree with the reviewer that flow oscillations can indeed play a role in fluid advection and mixing, as also suggested by the reviewing editor. An order of magnitude estimates of flagella-driven flow oscillations (vosc) suggest that advection is important for both the H30 (νb55 Hz, vosc3450 μm/s) and H10 (νb52 Hz, vosc3270 μm/s) cells as their beat frequencies (νb) are similar. However, in this study, we try to understand the long-time behaviour of the flows where the oscillations are averaged out and we observe that even these time-averaged flows of swimming CR cells have interesting flow structures and transport properties when strongly confined. Nevertheless, we understand and appreciate the spirit of the concern raised by the reviewer and acknowledge it by including the following sentences in the discussion of our revised manuscript (page 21).

“We note that apart from the time-averaged flows, the oscillations produced in the flow (vosc) due to the periodic beating of the flagella can play a role in fluid transport and mixing for both the H30 (νb55 Hz, order of magnitude estimate of voscL×2πνb3450 μm/s) and H10 (νb52 Hz, vosc3270 μm/s) cells [Guasto et al., 2010; Klindt and Friedrich, 2015].”

e) The slower decay of the velocity correlation in the confined case (figure 5 – supplement figure 2) could be explained by the fact that the confined case produces a force monopole, and hence a lower order of decay than a force-free swimmer? In which case we are not necessarily seeing evidence of increased mixing?

We thank the reviewer for raising this pertinent concern. We agree with the reviewer on this point that the slower decay of velocity correlation (revised Figure 5A) in the confined case can be ascribed to lower order of decay in the swimmer’s flow field. This is because strong confinement reduces the force-free swimmer in H30 (weakly confined force-dipole with 1r3 decay) to a force-monopole one in H10 (1r2 decay). However, we respectfully differ with the concluding remark of the reviewer that a slower decay rate in fluid flow does not necessarily imply increased mixing. Rather, it can be one of the reasons contributing to increased mixing in microswimmer suspensions. That is, a velocity field with RMS value V and correlation length λ (equivalently, persistence time τ=λ/V) will contribute an amount Vλ=V2τ to the diffusivity, on time scales τ. Thus, the more persistent the velocity, the larger the diffusivity enhancement of the fluid particles on long timescales. For example, Kurtudulu et al., 2011 observe enhanced mixing in active CR suspensions in freely-suspended 2D soap films compared to those in 3D unconfined fluid [Leptos et al., 2009]. They observe higher effective diffusivity of passive tracers in 2D and attribute this increase to the long-range hydrodynamic disturbances of the swimmers due to the reduced spatial dimension in their experiments (the force-dipolar flow reduces from v1r2 in 3D to v1r in 2D) and also due to increased swimmer-tracer interactions.

Nevertheless, we understand the spirit of the concern raised by the reviewer and acknowledge it by supporting our claim of enhanced mixing with an additional analysis, in a manner similar to those of [Kurtudulu et al., 2011], which directly shows that diffusivity of passive tracers increases due to the strongly confined (yet slow-swimming) CR’s flow field. We measure the displacement of the passive tracer particles (200 nm microspheres) 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 these field of view in ~ 1-1.4 sec (revised Figure 5B) whereas the slow-moving H10 swimmers stay in the field of view for the maximum recording time of ~ 6-8 sec (revised 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, Δr, during a fixed lag time of 0.2 second (10 flagellar beat cycles) (revised 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 (revised Figure 5B). However, most of tracers in the full field of view are perturbed/advected due to the H10 flow, those in the close vicinity being mostly affected (revised Figure 5C). Their advective displacements are larger than that of the tracers due to H30 flow. Together both these representative images (revised Figure 5B,C) show that the spatial range to which a swimmer motion advects the tracers – radius of influence, Rad – is higher in the case of H10 flow (Rad35 μm) when compared to the H30 one (Rad15 μm). We define the radius Rad 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 Rad15 μm with the cell’s swimming path as its axis and that for the H10 cell is a sphere of radius Rad35 μm centred on the slow swimming cell’s trajectory.

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 swimming through the field of view and then ensemble average over 6 such videos (revised Figure 5 — figure supplement 1). These plots with a scaling Δr2(Δt)Δtα show a higher MSD exponent in H10 (α1.55) than H30 (α1.25), indicating enhanced anomalous diffusion in strong confinement.

To summarize, these plots directly show that fluid is indeed advected more in the strongly confined case (H10) than the weakly confined one (H30) leading to enhanced mixing and transport.

We acknowledge the reviewer’s concern by re-writing the section “Enhancement of fluid mixing in strong confinement” with these additional plots and analyses (revised Figure 5 and Figure5—figure supplement 1).

f) Apart from a brief reference to a review paper, relatively little contact is made with Chlamydomonas biology or ecology, particularly concerning the functional importance of fluid mixing. I am not a specialist in this area; my question is, under what circumstances is the ability to exchange e.g. carbon dioxide with the surrounding fluid a limiting factor in metabolism? Are we seeing something specific to CR's adaptation to certain natural habitats, or just that tethered swimmers always disturb more fluid than free swimmers due to the force monopole?

We thank the reviewer for this opportunity to make clarifications. Algal growth and metabolism require exchange, between organisms and water, of small molecules and ions such as phosphate, carbon dioxide, nitrogen etc. Specifically, nitrogen and carbon are limiting macronutrients to algal growth [Short et al., 2006; Khan et al., 2018]. For example, dissolved carbon dioxide in the surrounding fluid contains the carbon source essential for photosynthesis. Carbon dioxide further buffers the water against pH changes as a result of CO2/HCO3- balance, which helps in maintaining the pH between 7 and 9, optimum for algal growth [Khan et al., 2018]. Therefore, fluid mixing helps in uniform distribution of nutrients, air and CO2 in algal cultures which have a positive influence on the nutrient uptake of these organisms, especially for the strongly confined cells as they cannot move far enough to outrun diffusion of nutrient molecules because of slow swimming speed.

However, we note that this increased fluid mixing helping the organism to avail itself of more nutrients is NOT an adaptation of the organism for being in strongly confined spaces. The inverse vortical flow field is a purely physical effect due to mechanical interaction of the cell body with the solid walls and not due to any behavioural change of CR in confined space, as mentioned in multiple places in our manuscript. Even though the walls restrict the cell body from swimming freely, they hardly affect the flagellar motion as the flagella are slender rods of diameter 0.5 μm beating far from these solid boundaries (∼ 5 μm) in the 10 μm chamber. As a result, the flagellar waveform and beat frequency in strong confinement are similar to those in the bulk. This implies that the strongly confined CR flow, that is mostly ascribed to the flagellar motion, and consequently the enhancement in fluid transport appears not to be an adaptation of the organism but a mechanical effect of being in strongly confined spaces. On the other hand, we agree with the reviewer that our observations are in line with those of tethered filter feeders like Vorticella whose ciliary beating produces a vortical flow field (similar to ours) that draws in fluid with dispersed bacteria towards the organism. These organisms being tethered to a substrate exert a net force towards themselves like our force-monopole swimmer in strong confinement and therefore, the flow field decays slowly than a free-swimming one [Christensen-Dalsgaard and Fenchel, 2003; Pepper et al., 2010].

We acknowledge the reviewer’s concern by modifying and adding text at appropriate places in our revised manuscript (pgs. 16 and 18 and 21).

Reviewer #2:

The manuscript focusses on changes in the flow fields generated by swimming microorganisms as a consequence of them being squeezed within a gap that is narrower than their size. This is studied here in the context of the unicellular green microalga Chlamydomonas reinhardtii, which is a common model system for microbial motility of body sizes ranging from 8um to 14um. The authors compare their behaviour in the two cases of a 30um and 10um-thick Hele-Shaw cell. The resulting flow fields are then compared with those obtained by a superposition of quasi-2D Stokeslets, Green functions of the Brinkman equation.

The paper has three main results. Firstly, cell behaviour in the 10um-thick samples depends clearly on microorganismal size. Larger cells beat their flagella with the standard synchronous breaststroke, while smaller ones display either asynchronous beating or "paddling" (as the authors call it, but see below my comment on this). Secondly, the larger cells display an average flow field that, in the far field, is directed oppositely to what would be predicted in absence of cell squeezing by the walls. Thirdly, the paper presents a theoretical modelling for the observed flow fields in terms of a "Gaussian" Brinkman Stokeslets. This flow field is proposed to increase the flow of nutrients to the cell.

1) The confinement that the authors focus on is in a different regime that what has been addressed earlier, in the sense that it starts to probe strong non-hydrodynamic friction, which is a case that can definitely happen in nature. This is interesting, although likely to depend quantitatively very much on topographical and chemical details of the actual surfaces that are in close contact.

Given that the strongly confined cells experience a non-hydrodynamic friction that is large enough to almost halt their swimming, it is not surprising that the measured flow field is dominated by the forces imposed on the fluid by the flagella and therefore a force-free 3-Stokeslet model is inappropriate. After all, the presence of non-hydrodynamic friction means that the cell exerts a net force on the fluid. It seems to me that this situation essentially resembles that of a tethered microorganism like Vorticella and I would have liked to see more discussion on the similarities between the two cases, both experimentally and from a modelling perspective. I do not think this is developed sufficiently in the manuscript.

We thank the reviewer for this relevant suggestion. We agree that the strongly confined cell resembles the case of a sessile filter feeder like Vorticella which is generally tethered to a substrate via its stalk. Below, we outline the similarities between these two cases from both experimental and theoretical perspectives.

The ciliary array of tethered filter feeders like Vorticella and Stentor, distributed along the periphery on the top of the bell-shaped body, beat at 30-50 Hz in metachronal coordination to generate a fluid flow towards themselves and feed from the passing fluid [Ryu et al., 2016]. Experimental measurements of the flow fields in the standard slide-coverslip setup with the tethered organism oriented parallelly and midway between the two surfaces show a dual-vortex flow structure [Nagai et al., 2009; Pepper et al., 2010; Ryu et al., 2016], similar to our strongly confined cell (Figure 3C). This vortical flow field generated by the ciliary carpet of Vorticella is much stronger than that of our strongly confined CR (2-cilia flow) with a maximum velocity of 360 μm/s drawing in fluid containing food particles about 450 μm from the body of the organism [Nagai et al., 2009; Ryu et al., 2016].

It is evident from these experimental flows that the tethered feeders exert a net force towards themselves which is modelled by a parallel Stokeslet between two rigid walls, directed towards the body of the organism, which explains the far-field flow features reasonably well [Pepper et al., 2010; Ryu et al., 2016]. This Stokeslet model can also explain our strongly confined CR’s flow-field, far from the organism, because the CR squeezed between the two walls is barely able to move and hence the flagella pull fluid from front to back resulting in a net force on the fluid towards the cell. However, we show that the near-field flow due to the two flagellar motion of the strongly confined CR is accurately described by two (like-signed, directed towards the cell body) Brinkman Stokeslets localized with a Gaussian spread on the approximate flagellar positions. Similarly, Pepper et al., considered an effective 2D Brinkman cylindrical model to account for finite-size effects, where the Vorticella is modelled as a cylinder (axis perpendicular to the walls) with a tangential velocity distribution on the surface to describe the multiciliary beating on the top of the filter feeder [Pepper et al., 2010]. This analytical model is slightly more involved than ours due to the coordinated beating of multiple cilia for filter feeders. To summarize, these Stokeslet and Brinkman flow models agree well with the experimentally observed flow vortices of Vorticella and strongly confined CR with appropriate consideration of the differences in their ciliary beating (multi-ciliated metachronal waves for Vorticella and two-ciliary flow for CR).

We acknowledge the reviewer’s suggestion by adding a short summary of the above in the discussion of our revised manuscript (page 21) in order to keep the storyline focussed.

Besides this, the range of gaps probed by the authors is rather coarse. The 10um gap is not much smaller than the smallest gaps that were probed in ref [12] for which -however- the flow field was qualitatively different. In my opinion, within the context of the effect of confinement on flow fields, it would have been interesting to explore this range in more detail, studying the transition from the hydrodynamic to the contact-friction case.

We thank the reviewer for raising this pertinent suggestion. It is a question that we ourselves have wondered about but can think of no simple experimental or theoretical approach to answer. We cannot produce double tape spacer with the necessary resolution. In principle, one can imagine designing an experiment where the cell and tracer suspension is injected into a microfluidic chamber made of glass walls and PDMS spacer, and the PDMS is pumped to gradually increase the chamber height while simultaneously observing the changes in the CR flow field. This can be perhaps explored in a future study. However, even in the 10 μm chamber, cells which are less confined (H10 Wobblers, D/H ≈ 1) have an irregular beat pattern due to frequent interactions of flagella with the rigid walls of the chamber during the spinning motion of the cells and therefore, it is difficult to assign a beat-averaged flow field that is universal for all H10 Wobblers. So, it is challenging to measure an ensemble averaged flow field with gradually decreasing confinement, especially when one attempts to observe the transition from contact friction case (our strongly confined cell, D/H ≈ 1.2, force-monopole flow, akin to a 2D source-dipolar flow pointing opposite to swimmer’s motion) to the hydrodynamic one (Jeanneret et al., 2019, D/H ≈ 0.7, force-free 2D-source-dipolar flow along the swimmer’s motion). It is quite clear that this transition is discontinuous, where individual cells in the chamber will be probing either of these two cases even when their sizes may be similar.

As for the theoretical approach to studying the transition between the two limits, we would need to build a model that accounts for binding and unbinding of the molecular entities on the cell surface with the walls. Such a model will of course involve several phenomenological parameters that would act as additional degrees of freedom while comparing experimental data to the theoretical predictions. Furthermore, such an effort would also demand a more controlled experimental system wherein the surface chemistry can be tuned to vary the frictional interactions.

2) Regarding the modelling, I have two main comments. Firstly, I appreciate the idea of a diffused Stokeslet. However, it does look very close to the idea of a regularised Stokeslet (Cortez 2005), which is not even mentioned in the manuscript. I think this is quite surprising. I would expect the manuscript to comment on differences between the two approaches.

We thank the reviewer for appreciating our idea of a diffused Stokeslet and seeking comparison of our approach with that of Cortez et al., 2005. Cortez have introduced the idea of a regularized Stokeslet by including a radially symmetric smooth function ϕϵ (e.g., Gaussian, Lorentzian etc.) in the forcing term to stabilize the singularities in the Stokeslet expression for practical computation [Cortez, 2001]. This is done by introducing a small cut-off parameter ϵ, in the function ϕϵ, which controls the spreading. Cortez et al., have provided a methodology to compute the closed-form analytical solutions for any such functional regularization ϕϵ to the Stokeslet [Cortez, 2001; Cortez et al., 2005]. Our approach of including a Gaussian form of regularization to the Brinkman Stokeslet is in a similar spirit. We have convolved our Brinkman Stokeslet to a 2D Gaussian with standard deviation σ (which is similar to ϵ of Cortez et al.,) to numerically obtain the flow-field. That is, our approach is only different from Cortez et al., in its implementation. We have given numerical solution to the regularized Stokeslet so that it explains our experimental data while Cortez et al., have provided analytical expressions for a Lorentzian regularization in an unbounded domain [Cortez, 2001; Cortez et al., 2005]. To summarize, our convolution approach gives numerical solution to any form of regularization to the Stokeslet whereas Cortez et al., approach gives analytical closed-form solutions to the same.

We understand the spirit of the concern raised by the reviewer and acknowledge it by elaborating the following sentence while introducing the Gaussian regularization in our revised manuscript (page 15):

“We, therefore, associate a 2D Gaussian source of standard deviation σ, to Equation 1 instead of the point-source δ(r), in a manner similar to the regularized Stokeslet approach [cite Cortez et al., 2005]”

Secondly, I do not understand the FT approach to finding the 2D Stokeslet for the Brinkman equation, when this is actually known analytically. It has been published in

Pushkin, D. O. and Bees, M. A. Bugs on a slippery plane: Understanding the motility of microbial pathogens with mathematical modelling. Adv. Exp. Med. Biol. 915, 193-205 (2016).

which is cited in ref [12] of the manuscript (Jeanneret et al., PRL 2019). This paper also shows that a model based on the point forces from Liron and Mochon does not perform well (see Suppl Mat), while one made of Brinkman Stokeslets with a 2D source dipole does. Therefore I do not think that it is surprising to see that the same approach works well for the current manuscript. In fact, given the similarity between the current setup and that in [12], I would have expected a comparison between the current approach and the full model from [12] at least in the Supplementary Material.

We appreciate the reviewer for seeking comparison of our theoretical model with that of Jeanneret et al., 2019 [12] which uses the Pushkin-Bees (PB) solution [Pushkin and Bees, 2016] in constructing their 2D model.

First, we compare our Brinkman equation with that of Pushkin and Bees, 2016 for the 2D fluid velocity v(x,y). Eq 12.2 of [Pushkin and Bees, 2016] for the 2D fluid velocity, averaged over the film thickness of height H in the z-direction, is given by the following,

p(x,y)+η2v(x,y)ηλ2v(x,y)=0

Pushkin and Bees considered the permeability length to be λ=H for a channel that is formed by 2 parallel solid plates with no-slip boundary conditions i.e., a Hele-Shaw cell [Pushkin and Bees, 2016]. Jeanneret et al., corrected this factor to be λ=H12 for the z-averaged mean 2D Poiseuille flow in a Hele-Shaw cell, relevant for the comparison with their experimental flow field which is averaged over the depth of focus of 10 μm (20X objective; numerical aperture, NA, 0.4) [Jeanneret et al., 2019]. On the other hand, we write the quasi-2D Brinkman equation at the z=0 plane (Equation 1 and associated text in our manuscript) for appropriate comparison with our experimental data which is acquired at or nearby the mid-plane (z=0 between the solid walls at z=±H2) with a 40X objective that has very low depth of focus (1 μm). The permeability length of our Equation 1, on comparing with the above one, is λ=Hπ. Therefore, we use the exact analytical solution given by Pushkin and Bees in Equation 12.6-12.8 [Pushkin and Bees, 2016] with λ=Hπ for comparison with our Brinkman Stokeslet’s numerical solution using the Fourier Transform (FT) method. We find that the flow field obtained for 2-Stokeslets using the analytical solution (PB) is identical to that of our numerical one (revised Figure 4A in our manuscript) computed on the same grid size and spacing. Hence, our FT methodology for solving the Brinkman equation from first principles is an alternative approach to using the complete analytical solution of Pushkin and Bees. We appreciate the reviewer’s concern and acknowledge it by including the following sentence in our revised manuscript below Equation 2:

“This solution is identical to the analytical closed-form expression of Pushkin and Bees, 2016.”

We also agree with the reviewer that Jeanneret et al., 2019 [12] have shown that their theoretical model including a force-free combination of Pushkin-Bees (Brinkman) Stokeslets along with a 2D source dipole performs well over the spinning force-free combination of Liron and Mochon (LM) Stokeslets. However, our approach is not exactly the same. We show that force-free Brinkman Stokeslets (Figure 3 —figure supplement 1A) does not explain our experimentally observed flow in H=10 μm (Figure 3C) even qualitatively. On the other hand, we show that a quasi-2D Brinkman Stokeslet/force-monopole with a Gaussian regularization (spatially distributed between 2 flagellar positions; revised Figure 4C) matches our experimental flow field very well. Jeanneret et al., 2019 added the extra 2D source dipole because they observed that the spinning force-free combination of LM Stokeslets lacks the dipolar symmetry due to finite-sized body effects when compared with the experiments (Figure S6 of [Jeanneret et al., 2019]). However, when we subtract the theoretical 2-Stokeslet LM flow (not force-free; see Appendix1 – figure 1A) from our H10 experimental one (Figure 3C), we observe that we need spatially-averaged or regularized forces instead of point forces to explain our experiment.

The reason why our theoretical approach is not the same as Jeanneret et al., is because there are two major differences in our experimental observations:

a. In our case, the strongly confined CR exerts a net force on the fluid due to thepresence of strong non-hydrodynamic contact friction from the walls, unlike that of [Jeanneret et al., 2019]. This leads to a 96% reduction of the swimming speed of the CR cells when strongly confined in 10 μm chamber (D/H ~ 1.2) as compared to 30 μm chamber (D/H ~ 0.3) in our experiments. This coupling between motility and confinement is not observed by Jeanneret et al., likely due to the slightly weak confinement (D/H ~ 0.7) produced by their experimental methodology (Table S1 of [Jeanneret et al., 2019]), where the stresses present in the system are mostly hydrodynamic. It is therefore appropriate for them to use the conventional 3-Stokeslet theoretical model for CR which is force-free (apart from the source dipole contribution) whereas in our case, the nearly absent hydrodynamic drag experienced by the cell body leads to a monopolar flow with only 2 Stokeslets (like-signed) localized with a Gaussian spread around the approximate flagellar positions.

b. The CR cells in our experiment do not spin around their body axis in the strong confinement of H=10 μm, contrary to the experimental observation of Jeanneret et al. They added the extra 2D source dipole in their theoretical model to the force-free Pushkin-Bees Stokeslets to account for both finite-sized effects of the cell body and spinning motion of the cells (explained in Figure 1(c) of [Jeanneret et al., 2019]).

To further illustrate our point and appreciating the suggestion of the reviewer, we provide a detailed comparison of our theoretical model with that of Jeanneret et al., 2019 [12] in a new section in the Appendix of our revised manuscript – ‘5. Comparison of our theoretical model of strongly confined flow with that of Jeanneret et al.’.

Finally, I think that comparing the magnitude of the flow fields as in Figure 5B,D is insufficient. One should instead show that both magnitude and direction of the flow fields are well captured by the model. This cannot really be grasped by comparing by eye Figure 3C and Figure 5A, C.

We thank the reviewer for raising this concern. We have already given the root mean square deviation (RMSD) between the experimental and theoretical flows in vx, vy and |v|, calculated at all grid points, for a quantitative comparison for the direction as well as the magnitude of flow fields. To further address the reviewer’s concern, we now show the comparison of the x and y components of the velocity field (vx and vy), whose magnitudes are responsible for determining the direction of the flow. We add another figure in the revised manuscript (revised Figure4 —figure supplement 2) which shows the azimuthal variation of vx and vy for representative radial distances of the H10 experimental flow fields and the corresponding theoretical models. We also add the corresponding comparison between H30 experiment and its theory in subfigures C and D of the revised Figure4 —figure supplement 3.

We do not add the comparison in the velocity components between H10 experiment and Liron-Mochon theory as this model is not appropriate for describing the experiment, as expected, and described in the manuscript. This deviation is already captured while comparing their flow magnitudes (Appendix1-Figure1) and hence we do not expect their directions to match either.

3) As for the question of nutrient transport, I find the claim on the enhancement for the 10um-cells not completely convincing. It is clear that the fluid fluxes for the 30um-cells and the 10um-cells are organised in an opposite way. However, Figure 4D shows that the positive part of the flux is of a very similar magnitude in both cases, in particular for the average fluxes. This is true also when one compares the negative fluxes. Given the similarities, I do not find a compelling reason why the cell should have a much higher nutrient uptake in one of the two cases. This might be the case, but there is not enough evidence in the paper to support this statement.

We appreciate the point raised by the reviewer. We understand that comparison of the average fluid fluxes may not be completely convincing to our claim of increased nutrient availability to the organism when strongly confined. We now support our claim of enhanced mixing and thereby nutrient transport through the fluid with an additional analysis which directly shows that diffusivity of passive tracers increases due to the strongly confined (yet slow-swimming) CR’s flow field. We measure the displacement and the mean-squared displacement (MSD) of the passive tracer particles (200 nm microspheres) when a single swimmer passes through the field of view (179 μm × 143 μm) in our experiments [Kurtudulu et al., 2011, Leptos et al., 2009]. Please refer to revised Figure 5 and revised Figure 5 — figure supplement 1, where we plot these quantities.

Nutrient uptake by any organism will depend on how much nutrient molecules the flow is bringing to the organism and the spatial scale/structure of the flows (apart from the absorption by the cell). Below, we summarize the results from our previous as well as new analyses which corroborate our statement of enhanced fluid transport/mixing for the 10 μm cells which have a positive influence on the nutrient uptake of these cells as they cannot move far enough to outrun diffusion of nutrient molecules because of slow swimming speed.

i. The slower decay in the fluid velocity correlation in the strongly confined case (revised Figure 5A) shows that the characteristic structure and spatial scale of the H10 flow is larger than the H30 one. 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 and correlation length λ) of the fluid particles on time scales λ/Vrms, in strong confinement.

ii. Figure 5, B and C showing colour coded tracer trajectories (according to their maximum displacement within a fixed Δt=0.2s) due to the motion and flow field of a representative H30 and H10 cell through the field of view also supports the above point as follows. The tracer trajectories close to the swimming path of the H30 swimmer (black dashed arrow) are mostly advected by the flow whereas those far away from the cell involve mostly Brownian components (revised Figure 5-A). However, most of tracers in the full field of view are perturbed/advected due to the H10 flow, those in the close vicinity being mostly affected (revised Figure 5-B). Their advective displacements are larger than that of the tracers due to H30 flow. Together both these representative images (revised Figure 5) show that the spatial range to which a swimmer motion advects the tracers – radius of influence, Rad – is higher in the case of H10 flow (Rad35 μm) when compared to the H30 one (Rad15 μm).

iii. Revised Figure 5 — figure supplement 1 shows the MSD of approximately 500 tracers in the whole field of view for each video where a single cell is swimming through the field of view and then ensemble average over 6 such videos. These plots with a scaling Δr2(Δt)Δtα show a higher MSD exponent in H10 (α1.55) than H30 (α1.25) indicating enhanced anomalous diffusion in strong confinement. Therefore, MSD calculation of the tracers quantify the relative increment in the advective transport of the H10 flow with respect to the H30 one.

We understand the spirit of the concern raised by the reviewer and acknowledge it by removing the volume flux calculation and subfigures from our revised manuscript. Instead, we rephrase and modify the concerned section: “Enhancement of fluid mixing in strong confinement” with these new analyses (please see revised Figure 5 and Figure 5—figure supplement 1) that directly address the queries of the reviewer.

4) Finally, regarding the "paddling" state, I'm afraid this is the normal flagellar shock response in Chlamydomonas and not some sort of previously unknown state of the cells. It is known that the shock response can be elicited both by intense light stimuli or by mechanical stimuli. It is very likely that this is just a mechanosensitive shock, which comes from the mechanical interactions between the flagella and the upper/lower walls. In turn, these are possible due to the fact that cells of smaller size than the gap can spin around their body enough to touch the walls with their flagella. I think that the authors would need to investigate further the existing literature on flagellar shock response in Chlamydomonas and put appropriately in context the "paddling" behaviour they observe.

We thank the reviewer for raising this important concern. Mechanosensitive ion channels in CR, activated by mechanical agitation and/or compressive stresses, induce an influx of calcium ions which are responsible for changes in flagellar waveform (from breastroke to undulatory) leading to transient backward motion of the cell [Yoshimura et al., 1997; Fujiu et al., 2011] and/or increased beat frequency [Wakabayashi et al., 2009]. They also affect the inter-flagellar coordination through calcium-sensitive basal-body associated fibrous structures [Ruffer and Nultsch, 1998]. We agree with the reviewer that these calcium-mediated mechanosensitive shock responses to the flagellar beating in CR are very likely the reason for the ‘paddling’ behaviour observed in our experiments due to frequent flagella-wall interactions during the spinning motion of the cells when they are slightly smaller than the gap between these walls. We acknowledge the reviewer’s suggestion by adding/modifying texts in our revised manuscript (pg. 7-8) related to the paddler flagellar beat.

References:

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https://doi.org/10.7554/eLife.67663.sa2

Article and author information

Author details

  1. Debasmita Mondal

    Department of Physics, Indian Institute of Science, Bangalore, India
    Contribution
    Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing - original draft, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-8265-6876
  2. Ameya G Prabhune

    Department of Physics, Indian Institute of Science, Bangalore, India
    Present address
    Department of Physics, University of Colorado, Boulder, United States
    Contribution
    Initial preliminary theoretical calculations
    Competing interests
    No competing interests declared
  3. Sriram Ramaswamy

    Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore, India
    Contribution
    Conceptualization, Methodology, Project administration, Resources, Supervision, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-7726-8556
  4. Prerna Sharma

    Department of Physics, Indian Institute of Science, Bangalore, India
    Contribution
    Conceptualization, Funding acquisition, Methodology, Project administration, Resources, Supervision, Writing - original draft, Writing - review and editing
    For correspondence
    prerna@iisc.ac.in
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-4988-9560

Funding

The Wellcome Trust DBT India Alliance (IA/I/16/1/502356)

  • Prerna Sharma

Science and Engineering Research Board (J C Bose Fellowship)

  • Sriram Ramaswamy

The funders had no role in study design, data collection, and interpretation, or the decision to submit the work for publication.

Acknowledgements

We acknowledge Aparna Baskaran, Ramin Golestanian, Ayantika Khanra, Swapnil J Kole, Malay Pal, Balachandra Suri, and Ronojoy Adhikari for useful discussions. This work is supported by the DBT/Wellcome Trust India Alliance Fellowship (grant number IA/I/16/1/502356) awarded to PS. SR acknowledges support from a J C Bose Fellowship of the SERB (India) and from the Tata Education and Development Trust.

Senior Editor

  1. Anna Akhmanova, Utrecht University, Netherlands

Reviewing Editor

  1. Raymond E Goldstein, University of Cambridge, United Kingdom

Publication history

  1. Received: February 18, 2021
  2. Accepted: November 16, 2021
  3. Accepted Manuscript published: November 22, 2021 (version 1)
  4. Version of Record published: January 13, 2022 (version 2)
  5. Version of Record updated: January 17, 2022 (version 3)

Copyright

© 2021, Mondal et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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  1. Debasmita Mondal
  2. Ameya G Prabhune
  3. Sriram Ramaswamy
  4. Prerna Sharma
(2021)
Strong confinement of active microalgae leads to inversion of vortex flow and enhanced mixing
eLife 10:e67663.
https://doi.org/10.7554/eLife.67663

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