Hopping trajectories due to long-range interactions determine surface accumulation of microalgae.

The accumulation of motile cells at solid interfaces increases the rate of surface encounters and the likelihood of surface attachment, leading to surface colonization and biofilm formation. The cell density distribution in the vicinity of a physical boundary is influenced by the interactions between the microswimmers and their physical environment, including hydrodynamic and steric interactions, as well as by stochastic effects. Disentangling the contributions of these effects remains an experimental challenge. Here, we use a custom-made four-camera view microscope to track a population of motile puller-type Chlamydomonas reinhardtii in a relatively unconstrained three-dimensional (3D) domain. Our experiments yield an extensive sample of 3D trajectories including cell-surface encounters with a planar wall, from which we extract a full description of the dynamics and the stochasticity of swimming cells. We use this large data sample and combine it with Monte Carlo simulations to determine the link between the dynamics at the single-cell level and the cell density. Our experiments demonstrate that the near-wall scattering is bimodal, corresponding to steric and hydrodynamic effects. We find, however, that this near-wall dynamics has little influence on the cell accumulation at the surface. On the other hand, we present evidence of a cell-induced surface-directed rotation leading to a vertical orbiting behavior and hopping trajectories, consistent with long-range hydrodynamic effects. We identify this long-range effect to be at the origin of the significant surface accumulation of cells.

The physical interactions of single-celled organisms with their surroundings have many ecological, medical, and technological implications (1–3). Interactions of planktonic algae or bacteria with solid substrates modulate the population distribution (4, 5), consequentially affecting the processes of surface attachment, colonization, and biofilm growth (6, 7). Elucidating the mechanisms governing these interactions is paramount to technological advances in biofouling prevention (8), to the development of cell manipulation strategies (9, 10), and to our understanding of microbial colonization and infections (11).

Physical mechanisms, suggested to influence cell-surface interactions and accumulation, include hydrodynamic interactions (12, 13), steric interactions and wall scattering (14–16), stochastic processes (17), and flow (18). The contributions of each of these mechanisms have been investigated in previous theoretical work. These studies have highlighted the role of hydrodynamic interactions in the scattering of both “pusher”- and “puller”-type motile cells at a no-slip solid surface (19–22). For pusher-type cells, steric interactions and stochastic processes have been shown theoretically to be sufficient to account for the surface accumulation of cells at the solid interface (14, 15).

Experimental studies of motile bacteria have linked hydrodynamic interactions with solid surfaces to wall trapping (13), to variations in the swimming speed (3), in the orientation relative to the surface (12), and in the orientation in the surface-parallel plane (23). Studies of puller organism Chlamydomonas reinhardtii have focused on the near-wall scattering dynamics. In shallow flow chambers, steric contact has been suggested to dominate surface interactions (10), while effects of hydrodynamic interactions have been evidenced in interactions with curved surfaces (24). Visualization and tracking constraints have often limited experimental studies of flagellates to two-dimensional (2D) flow cells (10, 24–26). Such confinement restricts the motion of the microorganisms in one dimension and alters the flow and the hydrodynamic effects compared with the dynamics of unconstrained swimmers interacting with an infinite planar wall predicted by theoretical studies (12, 19–21).

Here, we use a custom-made multiview microscope to track the unconstrained three-dimensional (3D) dynamics of C. reinhardtii interacting with planar surfaces. Previous 3D tracking experiments have been limited to single cells (27, 28) or small populations at very low density (29). Recent experiments using holography (30–33) have required low optical densities and relatively small measurement domains (34, 35). In addition, holographic measurements have a limited accuracy along the optical axis (36), which hinders the resolution of contact and noncontact cell wall interactions. Our experimental setup circumvents these limitations by using four cameras recording at four different angles to reconstruct the 3D dynamics and trajectories of a large population of cells, within a flow cell of dimension mm along the optical axis and within a position error smaller than the cell radius. Close to the wall, our measurement method resolves a bimodal distribution corresponding to contact and noncontact wall scattering interactions. Away from the wall, we report a long-range attraction effect leading to hopping trajectories and repeated surface approaches. The cell density distribution measured in the flow cell is compared with results from Monte Carlo simulations, which reveal that long-range interactions drive the surface population accumulation, not the detail of the contact/noncontact near-wall interactions.

Experimental Approach

We recorded the 3D dynamics of individual cells, in a dilute suspension of density , of the common model motile organism, the biflagellated green alga C. reinhardtii (37). The motility of the cell originates in the “breaststroke” motion of its two anterior flagella of length resulting in an average swimming velocity . In our experiments, we measured , , and cell body diameter, , in agreement with previous motility studies of C. reinhardtii (25, 37–40), see for details.

Suspensions of C. reinhardtii were loaded inside a flow chamber between two parallel glass coverslips, treated to prevent cell-surface adhesion (). The distance between the two planar glass surfaces corresponded to the smallest dimension of the flow cell . The geometry of the flow cell allows us to observe the interactions between motile cells and the planar surfaces of the glass coverslips in an unconstrained 3D fluid environment. The suspensions were illuminated using a light source (Luminus; PT-120-TE) at intensity . The wavelength of the illumination was , beyond the phototactic action spectrum of C. reinhardtii (37, 41–43), to avoid inducing a phototactic response. Experiments were performed at an ambient temperature of .

We used a custom-built multicamera microscopy setup to image a dilute suspension (Fig. 1). Videographic data were recorded simultaneously from four LaVision Imager sCMOS cameras at (exposure time ). The optical viewing path of each camera passed through a single large objective (Zeiss; , ), offset from the objective axis such that all cameras viewed a common measurement domain but from viewing directions separated by approximately (44, 45). The camera aperture was set such that the entire height of the 2-mm flow chamber was in focus ( has details).

Fig. 1.

(A) Schematic representation of the key elements of the experimental apparatus. A cylindrical fluidic domain containing a suspension of C. reinhardtii is illuminated at . Four Imager sCMOS cameras (LaVision) view the fluidic domain off center through a single objective lens (Zeiss). The minimum dimension of the measurement domain, from surface to surface, is . Coordinate systems and nomenclature used throughout the paper are given in the enlarged schematic view of the measurement domain. BSA, bovine serum albumin. (B) Example images from each camera of cells at suspension density cells per mL. (C) Sample of reconstructed 3D cell trajectories.

We first tracked the cells in the image plane of each camera view. These tracked objects were then matched between camera views and triangulated to find the 3D spatial coordinates using a custom-made tracking algorithm similar to ref. 46 (). The average skewness error (47) in this triangulation was (), which compares favorably with typical particle location uncertainty of cell diameters using holographic methods (36). The tangential vector, , is defined in relation to the velocity of each track as . The plane corresponds to the planar glass surface. We define as the angle between the cell velocity vector, , and the plane of the surface. Only tracks longer than 1 s, with median velocity , are considered here. Our experiment yields such tracks, the statistics of which are sufficient to reveal both near-surface scattering behavior and subtle effects in the far field.

Near-Surface Interactions and Scattering

First, we investigate the scattering dynamics of motile cells in the near-surface region, by only considering the trajectories approaching the surface within a distance of . We use our large data sample to characterize the scattering mechanisms from the kinematics of the cells and determine the importance of steric and hydrodynamic effects. For each of the trajectories in the “near-surface” region, we define as the distance traveled in the surface-parallel plane and report the cell trajectories near the surface. We call the time at which the cell is at a minimum distance from the surface. Fig. 2 represents the distribution of these trajectories in coordinates deduced from all trajectories recentered at the position of minimal approach to the surface . The residence time , for which a cell resides in the near-surface region , scales inversely with the median swimming speed of the cell (). The short residence time, , indicates that long-period trapping is minimal. This provides support for a deterministic departure mechanism, such as the combination of cell rotation and flagellar contact proposed by refs. 21 and 24, rather than the entrapment and noise-mediated release observed for other swimmer geometries (48).

Fig. 2.

(A) Distribution of cell trajectories in coordinates within of a close surface encounter. Green markers indicate the mean trajectory, with motion from the red marker toward blue. The vertical blue dashed line indicates the moment of closest approach, . (B and C) Evolution of PDFs of pitching rate, , and tangential acceleration, , through all close surface approaches. The gray dashed lines in B and C correspond to the times at which the peak values are obtained, which we use to construct the joint PDF in D. (D) Joint PDF of pitching rate, , at and tangential acceleration, , at . The 2D Gaussian distributions fitted to the two peaks of the observed bimodal distribution are indicated by black contours. The dashed black curve delineates , which we use as a criterion for classifying individual trajectories into C-mode or NC-mode cell-surface interactions. (E) Distributions of closest approach distance for contact and noncontact interactions. (F and I) Similar to A for contact and noncontact interactions, respectively. (G and J) PDF of surface departure angle, , given approach angle, , for (G) contact and (J) noncontact events. The blue dashed line indicates . The mean departure angle conditional on is indicated in each case by the solid black line. (H and K) For contact and noncontact events, respectively, joint PDFs of trajectory locations projected into the surface-parallel plane and aligned into the coordinate system, , relative to the cell orientation at .

Steric interactions with the wall have been evidenced to play an important role in the scattering dynamics of ciliates (10). It is reasonable to assume that such direct contact interactions may be characterized by stronger fluctuations in the velocity and the swimming direction, as the cilia come in sudden contact with the wall, leading to changes in the cell velocity and orientation. To identify such interactions and quantify their prevalence in our experiments, we consider the rate of change of the pitch angle relative to the surface, as well as the rate of change of the swimming velocity . Fig. 2 represents the distribution for the pitching rate and the tangential acceleration , both given as a function of time. Fig. 2 shows a short positive increase of around the time of closest approach , corresponding to a brief cell rotation away from the surface and consistent with surface scattering. Fig. 2 highlights another signature of wall interaction in the fluctuations of , with a reduction in swimming velocity before , followed by reacceleration to the original velocity after .

Close inspection of the distribution of reveals, however, that many trajectories are not characterized by an abrupt deceleration and acceleration at the surface. Rather, for many trajectories, scattering occurs at a constant velocity (Fig. 2). These differences in scattering dynamics can be identified on the joint probability density function for the peak values of and (Fig. 2). This joint probability density function (PDF) reveals a bimodal distribution in the kinematics of the scattering trajectories. This indicates two distinct modes of cell-surface interactions, exhibiting qualitatively different scattering behavior. The first mode is characterized by large fluctuations in the kinematics around the point of nearest approach, with and (contact mode [C mode] in Fig. 2). The second mode includes trajectories with near-constant velocity scattering peak at and small pitching rate (noncontact mode [NC mode] in Fig. 2).

The impulsive kinematics of trajectories within the C mode bear the signature of close contact interaction with the surface. Hence, we proceed by considering the distribution of minimal approach distance for the C and NC modes separately. For this, we fit a 2D Gaussian distribution to each peak (Fig. 2) to distinguish the trajectories within each scattering mode. For the C mode, the distribution of is maximum for and rapidly decreases to zero at larger distances from the surface (Fig. 2). For the NC mode, on the other hand, the distribution of is small in the vicinity of the solid surface and rapidly increases for . Given the average cell diameter, , and length of the anterior flagella of C. reinhardtii, , we confirm that for the trajectories within the impulsive C mode, the cells have come in direct contact with the wall. From the distribution of closest approach distances , we can conclude that within the continuous NC mode, cells have scattered from the wall without direct contact with the solid interface and suggest a scattering due to hydrodynamic interactions.

Fig. 2 represents the distribution and average of all trajectories in the C and NC modes, respectively. The scattering trajectories in both modes are strikingly different. Within the C mode, trajectories are asymmetric with cells approaching the wall over a range of predominantly steep angles and departing predominantly at shallow angles (Fig. 2). Within the NC mode, the trajectories are symmetric, and both the approach and departure angles are much smaller than in the C mode (Fig. 2). These results are in agreement with previous theoretical work by Spagnolie and Lauga (19). We define, for each trajectory, the approach angle and the departure angle (Fig. 1) and represent the joint probability distribution of for the C and NC modes in Fig. 2 , respectively. Contact interactions occur for a range of steep and depart at a shallower angle. The angle is independent of (black line in Fig. 2), therefore defining a critical departure angle of , consistent with previous experimental work (10). By contrast, noncontact interactions occur for shallow approach angles, which are smaller than the critical departure angle . In addition, the scattering trajectories are symmetric, such that the distribution follows the line (Fig. 2). Our experimental observations are in qualitative agreement with theoretical work by Spagnolie and Lauga (19). For sufficiently steep values of , hydrodynamic effects are insufficient to prevent surface contact (19), and cells leave the surface only after their orientation exceeds a critical angle, . Only for do hydrodynamic forces prevent contact. The signatures of the trajectories projected in the plane of the solid surface also differ; contact interactions are characterized by the trajectories rapidly turning to the right (Fig. 2), while for noncontact interactions, they remain mostly straight (Fig. 2).

Long-Range Interactions and Vertical Hopping Behavior

We now characterize the swimming dynamics far from the surface and consider all recorded cell trajectories within the height of the flow chamber. We focus on wall interactions and report the dynamics along the wall normal direction. The motion of a cell along the axis is determined by the pitching rate for a given pitch angle and position in the flow chamber. By considering the pitch dynamics of the cells throughout the flow chamber, we identify regions of wall repulsion and attraction.

Fig. 3 presents our results for . The black dashed vertical line delineates downward () and upward () swimming cells.

Fig. 3.

(A) The mean value of cell pitching rate conditional to the pitch angle and surface-normal coordinate across the span of the experimental chamber. The vertical black dashed line indicates surface-parallel cell orientations. B and C provide expanded views of this state space in the near-surface region. A horizontal black dashed line delineates , within which cell-surface contact is possible. By integrating through this state space, the expected mean deterministic trajectory of a cell near a surface can be extracted. The example in B (black curve) begins at with constant velocity magnitude of . A pseudoperiodic motion is observed, growing in amplitude before reaching a limit cycle of amplitude . An example of a measured trajectory displaying this orbiting behavior in coordinates is given in C (black curve). A sample of measured trajectories shown in D demonstrates the oscillations in the cell-surface separation distance, , associated with this orbiting and leading to hopping trajectories in (, z) coordinates.

We first consider the lower surface of the flow chamber at . Close to the surface, within , is positive for all values of , corresponding to cells turning away from the surface and therefore, to surface repulsion. This is consistent with the surface scattering near the wall discussed previously (Fig. 2). It is noteworthy that this repulsion region extends up to m and therefore, beyond the contact region (Fig. 3 ). This is further evidence for the existence of noncontact scattering. For , the sign of the pitching rate changes, with for a wide range of values of (Fig. 3 ). In this region, the swimming cells reorient toward the surface. This reorientation is strongest for cells swimming toward the surface and weaker for cells swimming away from the surface with a small positive pitch angle . For large positive pitch angles, the pitching rate is negative, indicative of surface repulsion. At distances greater than m, we find such that the cells are neither attracted to nor repelled by the surface. The same pitching dynamic is observed on the top surface of the flow chamber at m, with similar magnitude and opposite signs for and (Fig. 3). The hydrodynamic interaction of a dipole with a surface elicits a rotational response toward or away from the surface, depending on dipole orientation (12). Our experimental data indicate a corresponding behavior in C. reinhardtii. The key difference is a bias toward surface-directed rotation over rotation away from the surface. Cells oriented toward the surface exhibit rotation toward the surface, while the rotational response of cells oriented away from the surface is closer to neutral. We hypothesize that this difference is due to the front to back asymmetry of swimming C. reinhardtii.

The repulsion region for m and the attraction region for m suggest the existence of trajectories, moving periodically closer and away from the surface. We can verify the existence of these pseudoperiodic trajectories by integrating the first-order equation using the empirical pitching map in Fig. 3. Such a trajectory is represented in black on Fig. 3. The amplitude of the motion grows until it reaches a limit cycle of amplitude m and period , where the cell periodically comes within a close distance to the wall. The robustness of our tracking method supports continuous tracking for long time periods, and we can therefore verify the existence of such hopping trajectories. Fig. 3 presents one such track in agreement with the attraction–repulsion cycles inferred by the pitching dynamics in coordinates. Several of such recorded trajectories are represented in the plane (Fig. 3) and display the trajectories characterized by successive surface approaches.

Cell Distribution and Surface Accumulation

Following the characterization of cell-surface interactions, the outstanding questions are whether and how these surface interactions modulate the surface accumulation and the cell distribution (12, 14, 15). To address this question, we characterize the distribution of cells in our experimental chamber. Our experiments are performed at low cell concentration, such that we average the density distribution over the entire recording. Fig. 4 , blue circles represents this time-averaged cell population density normalized by the bulk density. We find a statistical accumulation of C. reinhardtii at the surface and characterize the cell distribution with a penetration length .

Fig. 4.

(A) Cell density distribution along the direction. Blue markers indicate the measured distribution, calculated in bins. Red and green markers indicate the population distribution predicted via MCMC simulation. Both measured and random near-surface scattering dynamics are tested, as well as the effect of the long-range surface-directed rotation. The black line gives the expected distribution of run-and-tumble dynamics at a Péclet number, with . (A, Inset) Density distributions in linear coordinates. B–F present the probability distributions sampled in constructing the MCMC simulations. (B and C) Probability of Lagrangian rotation rate components and at each simulation time step, conditional on the previous value, for . (D) The distribution over of . (E and F) The probability distributions describing the scattering of cells from surfaces, including residence time and departure angle . (G) The MSD of measured and simulated cell trajectories far from the surface (). Each shows ballistic () behavior for , with diffusive behavior () thereafter. For clarity, only a selection of points is plotted. (H) Example of measured and simulated trajectories. (I) Near-surface hopping behavior of an example simulated trajectory.

The extensive dataset gathered provides a full description of the different aspects of the swimming dynamics, including its stochastic nature. We use this data sample to implement Markov chain Monte Carlo (MCMC) simulations to reproduce the dynamics of the algae in the flow chamber. We take advantage of the MCMC simulations to disentangle the contributions of the different features of cell-surface interactions identified in our study and directly probe their effect on the cell population distribution. In particular, we first investigate the influence of the rotational diffusion, before including the effects of the near-wall scattering dynamics and later, the long-range vertical wall attraction behavior. We find that, in our experiments, the long-range behavior dictates the accumulation of cells at the interface.

Our approach integrates the stochastic dynamics for the cell position through time, assuming a constant swimming velocity of for simplicity (). The tangent vector , which characterizes the swimming direction, is evolved stochastically according to the empirical data collected in our experiments. We first consider the diffusive rotational behavior of the swimming cells, which we extract from the empirical data. For each trajectory recorded far from the surfaces, we define the tangent , normal , and binormal local unit vector in the Frenet–Serret frame (Fig. 1). In this frame of reference, we determine from the empirical data the conditional probability distributions of the rotation rates and (Fig. 4 , respectively). In this framework, about the normal vector is by definition zero. Rotation rates were drawn at each time step by sampling the corresponding inverse cumulative distributions (49). The resulting stochastic dynamics of the cells were integrated in time with a time step equal to the experimental time step, .

Fig. 4 illustrates a representative trajectory obtained from these simulations, which is qualitatively similar to the experimentally measured trajectories. In addition, the time evolution of the mean square displacement (MSD) (Fig. 4) of the experimentally measured cell position indicates a diffusive behavior, from which we can estimate the diffusion coefficient as and the characteristic diffusive timescale . These measurements are in agreement with previously reported values (50). We obtain the identical diffusive behavior by computing the MSD from the solutions to the stochastic simulations, which validates our approach.

Near-surface accumulation has been shown to arise from a purely diffusive motion interacting with a planar no-slip surface (17). To investigate this, we simulated cells placed initially at random locations and orientations. The trajectories were integrated until convergence to a final steady-state population distribution (). Interactions with the surface were modeled by assuming evenly distributed random scattering angles and an exponentially distributed residence time distribution with characteristic timescale, , equal to the measured diffusive kinematics. Fig. 4 reports the resulting cell distribution (red crosses). The extent of surface accumulation agrees with that reported for the run-and-tumble simulations from ref. 17 (Fig. 4). The penetration length scale of this cell distribution, , is however much shorter than the one extracted in our experiments, and therefore, the diffusive nature of the cell kinematics is not at the origin of the significant surface accumulation reported in our experiments.

Next, we investigate whether the near-wall scattering dynamics can account for this discrepancy. We do this by repeating the stochastic simulations but including the near-wall scattering dynamics observed in the experiments. For each encounter of a cell with a surface, the near-surface scattering parameters, and , are stochastically sampled from the empirically determined distributions and (Fig. 4 ). We find that the near-wall scattering dynamics does not significantly affect the cell distribution and does not account for the cell accumulation measured in the experiments (Fig. 4).

At last, we study the effect of the long-range interactions described in Fig. 3. We include in our stochastic simulations the long-range effect on the pitching dynamics in the evolution of the swimming direction . At each time step, the surface-directed rotation rate, , is selected via inverse transform sampling from the probability distribution, , conditional on the instantaneous position of the cell in the state space presented in Fig. 3. This stochastically chosen rotation is added to the rotations, and , about the Lagrangian body frame. By including these long-range effects, our simulations yield trajectories characterized by successive surface approaches (Fig. 4), which are similar to the trajectories recorded experimentally (Fig. 3). The distribution of cell density for the simulations including the surface-directed rotation, , reveals a significant increase in cell accumulation at the wall, in agreement with experiments (Fig. 4). The agreement between simulations and experiments is excellent. In particular, the penetration length of the computed distribution compares well with our measurements. This applies generally regardless of the near-surface scattering applied (compare Fig. 4 , green vs. red squares), indicating that long-range cell-surface interactions dominate over both diffusion and the near-surface scattering dynamics in determining the degree of population accumulation near a surface.

Discussion

We have investigated the interactions between C. reinhardtii and flat no-slip surfaces. In the near-wall region, we elucidate the characteristics of close cell-surface encounters and distinguish between two scattering modes, corresponding to cells that come in direct contact with the surface and cells that do not. For contact interactions, our experiments in a 3D fluidic domain confirm the surface departure angle to be in the range , as previously observed in quasi-2D experiments. The departure angle is found to be independent of the approach angle, , even for very shallow angles and supports the argument that this critical departure angle is determined by geometric considerations of the organism during contact (10, 24). While we find contact scattering to occur more often than noncontact scattering, we still observe frequent noncontact scattering with planar walls, accounting for a significant fraction of close (m) surface encounters. These noncontact interactions are characterized by the symmetry of the trajectories, which reflects the time reversibility of the Stokes equations, in qualitative agreement with theoretical studies of the hydrodynamics of cell-surface interactions (19). The near-wall scattering dynamics observed in our experiments differs from those observed in flow chambers confining the flow field generated by the microswimmers between two narrowly separated plates (10). This indicates that near-wall scattering is affected by the confinement of the flow. We also observe that C. reinhardtii circle near surfaces. Our analysis indicates that, unlike the circling exhibited by pusher-type bacterial cells (23), the circular motion of C. reinhardtii on surfaces is nonhydrodynamic in nature, with turning in the surface plane observed only when contact is made between cell and surface.

Away from the wall, for , C. reinhardtii cells swimming toward the wall, for , rotate toward the surface, which is consistent with long-range hydrodynamic interactions for puller-type microswimmers (12, 19, 21). For cells moving away from the surface, , far-field hydrodynamic theory indicates a repulsive reorientation. In contrast, we observed that although the trend of in the state space matches that predicted by this theory, there is a strong bias toward surface-directed rotation. This bias induces cells leaving the surface to reorient toward the surface and explains our observations of hopping trajectories, characterized by periodic surface interactions. Qualitatively similar oscillatory attraction–repulsion limit cycles have previously been predicted to result from cell-surface hydrodynamic interactions using various theoretical swimmer models (e.g., refs. 51–53).

Using MCMC simulations, we establish the link between the dynamics of the motility near interfaces on the micrometer scale and the distribution of cells throughout the flow chamber on the millimeter scale. We conclude that the near-wall scattering dynamics, and in particular the distinction between steric contact (10) and close-range repulsive hydrodynamic forces (19), weakly affect the cell density at the solid interface and are not responsible for the large accumulation of cells observed in our experiments of C. reinhardtii. Taking into account the long-range surface-directed rotation described in Fig. 3 increases the magnitude and length scale of the surface accumulation. The length scale of the resulting periodic motion toward and away from the surface sets the penetration length scale of surface accumulation of cells. Any surface topography design aimed at modulating near-surface population density in order to mediate cell-surface adhesion and subsequent biofilm growth must therefore account not only for scattering dynamics but more importantly, for long-range interactions.

Materials and Methods

Cell Culture Conditions.

C. reinhardtii of the wild-type CC-125 strain obtained from the Chlamydomonas Resource Center are maintained on solid agar slants (Alfa Aesar; H26724.22) under broad spectrum low-intensity () constant illumination. From this maintenance culture, of Tris-minimal growth medium (37) was inoculated and then, exposed to a 14-h/10-h light/dark cycle () at with aeration for 4 d, by which time the culture density is cells per mL and the growth is in the exponential phase. For each culture, the cells were counted using a hemocytometer (Neubauer; Marienfeld) and diluted using Tris minimal to either or . This is the same protocol followed in refs. 39 and 40.

Surface Preparation.

The -thick glass coverslips (VWR; 631-1572), which formed the no-slip boundary with which the cells were interacting, were precleaned and treated to minimize cell-surface adhesion. They were immersed in a sonic bath in surfactant solution (Sigma-Aldrich; Z805939-1EA) for 30 min, again for 30 min in de-ionized water only, and then surface treated by immersion for min in a 1% bovine serum albumin (Sigma-Aldrich; A7906-50G) protein solution in phosphate buffer (Sigma-Aldrich; P4417-50TAB) in order to prevent cell adhesion. The coverslips were affixed to the acrylic chambers using Norland Optical Adhesive (Thorlabs; NOA81).

Cell Imaging.

Excess scattered light was minimized by placing an iris between the light panel and the fluidic chamber, such that only light scattered from the cells is observed by the camera sensors. The cameras do not collect direct transmitted light but only the forward scattered light at a scattering angle, resulting in dark-field imaging. Swimming cells appear as individual bright spots of diameter, against a dark background.

Tracking Algorithm.

Cell images were first tracked in two dimensions in each camera plane using the nearest-neighbor method, constrained to within a maximum displacement limit of 7 pixels. Corresponding tracks in other camera views were then found using an computationally efficient linear projective geometry framework through matching epipolar lines of each track projected through the other camera image planes and constrained integer assignment (46). A least squares minimization triangulated each track in physical 3D space. Track noise was eliminated via a third-order, 25-point Savitsky–Golay filter allowing higher-order kinematic derivatives, such as curvature and torsion, to be accessed analytically.

The relationship between each camera and physical space was defined using a pinhole camera model, constructed by traversing a grid pattern with regular spacing of through the measurement volume in increments. The calibration was performed in air, and the foreshortening of the coordinate due to refraction at the fluidic chamber boundaries was corrected post hoc.

Cells immobilized on the no-slip surface were identified by applying a maximum displacement threshold over an entire temporal record. A planar estimate of the no-slip surface location was fitted to their mean position, with an offset of to account for the typical cell radius. The SD of the stationary cell locations gives the uncertainty of the estimation of the surface location as approximately ().

MCMC Simulations.

Locations and orientations of simulated cells were initialized according to a uniform distribution on and . The position of each cell was computed by the time integration of the first-order dynamic , with a time step equal to the time resolution of the experiment and assuming . The swimming direction is stochastically updated at each time step using our experimental data. This is done by rotating the Frenet–Serret coordinate frame, , of each cell at a rotation rate sampled via the inverse transform method at each time step from the experimentally measured rotation rate distributions. The bulk kinematics is represented by the rotation rates about the body coordinates and , which were drawn from the distributions and , constructed only from data recorded in the far-field region, at . We verified that the distributions of these kinematic variables were statistically independent in our experiments, such that they could be sampled independently. For this, we calculated the correlation coefficients between the body-frame rotation rates both near () and far () from the surface, which were found to be close to 0: and . An additional rotation, about the global coordinates, was defined by sampling to model the long-range pitching dynamics.

Simulated cells were considered in contact with the surface at , whereupon each was assigned a residence time and scattering angle sampled from the measured distributions, and . We verified that the residence time and the scattering angle were statistically independent by computing the correlation coefficient (), such that the distributions could be sampled independently. The comparison “random” scattering behavior was defined with uniformly distributed on and distributed exponentially with equal to the diffusion timescale, , at . The population density distribution, , was computed as the time average of the instantaneous cell locations. The residual on the population density distribution displayed first-order convergence and was of when the simulations were stopped after time steps ().