Permeation Dynamics of Active Swimmers Through Anisotropic Porous Walls

Natural habitats of most living microorganisms are distinguished by a complex structure often formed by a porous medium such as soil. The dynamics and transport properties of motile microorganisms are strongly affected by crowded and locally anisotropic environments. Using Chlamydomonas reinhardtii as a model system, we explore the permeation of active colloids through a structured wall of obstacles by tracking microswimmers' trajectories and analyzing their statistical properties. Employing micro‐labyrinths formed by cylindrical or elongated pillars, we demonstrate that the anisotropy of the pillar's form and orientation strongly affects the microswimmers' dynamics on different time scales. Furthermore, we discuss the kinetics of the microswimmer exchange between two compartments separated by an array of pillars.


Introduction
In experiments, microswimmers are often studied either in bulk [1][2][3][4] or in homogeneous quasi-2D geometries, such as thin films, [5] thin Hele-Shaw cells, [6] or at interfaces, [7] to facilitate the observation.However, the natural habitats of swimming microorganisms are very complex.Biological self-propelled particles like Chlamydomonas reinhardtii inhabiting soil or E. coli dwelling in mammalian guts encounter non-planar, rough surfaces and interact with suspended and sedimented passive particles.Such constraints profoundly affect the dynamics and collective behavior of microswimmers.
Understanding how microswimmers navigate complex environments is crucial in studying phenomena such as groundwater DOI: 10.1002/apxr.202300047contamination [8] and the spread of infections in animals or plants. [9,10]otential applications of selfpropelled particles, such as bacterial bioremediation [11] or tumour treatment, [12] would benefit from a deep understanding of how heterogeneous media affect microswimmers' motility.
Intriguing effects of geometrically complex and constrained environments, such as porous media, on the swimming behavior of active microorganisms were revealed in experiments with Escherichia coli. [13]These motile bacteria exhibit run-and-tumble behavior in bulk.But the tumbling rate has the opposite effect on the cell spreading in porous solidified agar than in a homogeneous aqueous buffer.Increasing the run time or, equivalently, decreasing the tumbling rate enhances the translational motion in a buffer.In contrast, in a porous environment, the cells spread more efficiently when they tumble more frequently to avoid trapping.Yet, a very high tumbling frequency reduces the diffusion constant even in the porous environment. [13]Trajectory analysis in a 3D porous medium showed that the environment may even induce trapping-hopping behavior rather than run-and-tumble motion. [14,15]nother surprising discovery was that strong confinement of E. coli in narrow channels can result in motility enhancement as bacteria switch to stable and fast 1D swimming in the center of the channels. [16]t was demonstrated that sperm cells can be directed through narrow channels with complex shapes using geometrical constraint, and that a preferred polarity of active motion can even be induced using a ratchet-like channel geometry. [17] recent study [18] showed that Janus particles switch from active Brownian motion with reorientation by rotational diffusion in the free-swimming state to island-hopping motion with obstacle-induced tumbling in the presence of colloidal clusters.The trajectories in the complex environment with colloidal islands exhibit a run-and-tumble-like character.
Extensive simulations of the behavior of active agents in porous media revealed rich dynamics, which are determined by the holes/obstacles' sizes, shapes, and flexibilities. [19,20]In order to understand how an active swimmer is affected by the geometric complexity of its surrounding, it is vital to investigate how swimmers interact with boundaries of different shapes.
[23][24] Scattering of active swimmers by obstacles differs significantly from specular reflection due to complex hydrodynamic and steric interactions, as well as self-propulsion-induced nonequilibrium phenomena.Experimental investigation of the motile puller-type algae interactions with a planar wall [21] revealed a constant outgoing angle determined by the swimmer geometry.If the obstacles are cylindrical, interactions are more complex and can be purely hydrodynamic. [22]In arrays of large axisymmetric pillars, preferred swimming directions arise in an ensemble of Chlamydomonas reinhardtii. [25]hin microchannels with a thickness of 20 μm, slightly larger than the cell diameter, restrict the motion of the swimming algae to a single plane.Such quasi-2D experimental geometry considerably simplifies the analysis of microswimmers' trajectories and allows exploring the confinement effects.In this paper, we explore the exchange of microswimmers Chlamydomonas reinhardtii between two compartments separated by a porous wall modelled by an array of pillars.Choosing rounded and elongated pillars (Figure 1) allows us to study the effect of the anisotropy on the exchange kinetics and the character of microswimmers' trajectories.We demonstrate that the pillar's form and orientation strongly affect the transmission and reflection coefficients of the swimming algae and the orientational anisotropy of their motion.

Equilibration Kinetics
In the microchannels, an array of pillars serving as a model system for a porous wall separates two obstacle-free compartments 1 and 2 (see Figure 2).Excursions of a microswimmer through the porous wall (compartment 3) may either result in the swimmer's permeation from one obstacle-free compartment (1 or 2) to the other ("transmission") or its reflection, i.e., the swimmer's return to the region without pillars it came from initially.A distribution mismatch in microswimmers can be induced by localized illumination.In thin microchannels (≈ 20 μm thickness), we could not observe any noticeable phototaxis; however, illumination by blue light (480 nm) induced adsorption of the microswimmers at the glass substrate.
Through adhesion, immotile algae were collected in the chamber exposed to the local illumination.We designate the algae in the adhered state as 'immotile' although they retain slow gliding  motility. [26,27]When the local illumination was terminated, the cells desorbed and transitioned into the motile state (Figure 3). Figure 4 shows the equalization of the microswimmer number density in two compartments of Sample E (parallel elongated pillars) after the illumination was removed.The numbers equalize on a time scale of about 200 s.Although there is a lightinduced disbalance in the total number density of cells n tot = n m + n ad, which slowly equilibrates (Figure 4a), the number density of swimming algae n m equilibrates between the two compartments nearly instantaneously (Figure 4b).This can be attributed to the low rate of the desorption process (Figure 3).During detachment, the number of adhered cells exhibits a nearly exponential decay to the number determined by the global illumination intensity and other conditions of the cells.The characteristic desorption time  dsp can be extracted from the single exponential fit giving  dsp = 135s.

Near Wall Accumulation
In contrast to the quick equilibration of the number of moving cells between compartments 1 and 2, there is a disbalance of the cell concentration across the width of the microchannel as the swimmers accumulate at the channel walls.
The tendency of active particles to stay close to boundaries has been observed in many theoretical [28,29] and experimental studies for systems as diverse as motile algae [30][31][32] and bacteria, [33]  sperm cells [34,35] and artificial microswimmers. [36,37]The underlying mechanisms range from hydrodynamic attraction [33,34] to steric reorientation and trapping. [28,29]he accumulation of Chlamydomonas reinhardtii at the walls of microchannels was recently reported by Williams et al. [32] Surprisingly, the swimmer accumulation at microchannel walls seems to be largely unaffected by the presence of pillars in microlabyrinths, even for Sample D, with elongated pillars oriented parallel to the channel walls.(see Figure 5).

Cylindrical Obstacles
The effect of geometrical constraints on the swimming behavior is manifested in the character of the cells' trajectories.Scattering ofalgae at the obstacles results in the reduction of swimming persistence.At the same time, regularly ordered obstacles affect the alignment of the microswimmers' trajectories achieved through sliding over the surface of the pillars.Typical trajectories for Sample A in compartment 3, containing the array of pillars, are shown in Figure 6.
Most trajectories exhibit small undulations.Chlamydomonas reinhardtii is known to swim along helical trajectories in bulk, [4] which can be attributed to the nonplanar flagellar beating and a small asymmetry in the flagellar driving forces. [39]The capillary confinement restricts the rotation of the algae and suppresses their helical motion.Instead, the undulated character of the trajectories occurs.A combination of hydrodynamic and steric interactions of the algae with the pillars results in their scattering, which has been studied by Contino et al. [22] In a periodic array of pillars, preferred swimming directions along lanes between the pillars arise. [25]he trajectories can start in one compartment and end in the other compartment, as exemplarily shown in Figure 6a.This case corresponds to the permeation through compartment 3 to get from 1 to 2 or from 2 to 1.Some algae can even turn around and,  [38] ).
after an excursion through compartment 3, return to the same compartment they came from (Figure 6b).This situation is designated as the reflection.
Although any alga can permeate or reflect, the reflection and permeation events strongly differ in their statistics (Figure 7).Reflecting algae have significantly smaller mean detention times in compartment 3 than permeating ones ( ref det ≈ 2.8 s,  perm det ≈ 8.5 s for Sample B).This can be explained by the fact that most algae scatter back at the first row of pillars.This can be well seen in Figure 7, where the distribution of the maximal penetration depth for reflecting trajectories is shown.The distribution function exhibits multiple maxima corresponding to the positions of the pillar planes.This suggests that the scattering effects are responsible for the reflection rather than single cells' tumbling or meandering motion (see also Figure S2, Supporting Information).
Preferred directions of motion in the lattice of pillars are revealed in orientational probability distribution functions (PDF) of displacements.In the case with high porosity (Sample B, see Figure 8a), the PDF shows isotropic distribution.Anisotropy arises as the inter-pillar-distance is decreased (Sample A, see Figure 8b).Peaks in the angular PDF correspond to the directions along which free straight paths exist (Figure 8c,d).Motion along the directions for which dips in the PDF occur would result in collisions with obstacles, leading to a reorientation of the algae.
Swimming anisotropy has previously been observed by Brun-Cosme-Bruny et al. [25] in a square lattice of large cylindrical pillars.In that case, a smaller number of distinguished directions emerged, apparently due to the difference in obsta- cle size [40] and lattice geometry.Following, [25] the swimming anisotropy can be analytically accounted for by introducing an anisotropic scattering rate.Without obstacles, Chlamydomonas reinhardtii are modelled as active Brownian particles exhibiting rotational diffusion characterised by the constant D r .In the presence of obstacles, the swimmers experience additional random direction changes due to collision-induced tumbling.The tumbling rate  depends on the direction of motion.In the case of a square lattice, described in ref. [25], () =  0 −  4 cos (4) and  0 =  4 such that the tumbling rate vanishes in directions 0, ±/2,  along which obstacle-free lanes exist.To adapt the model to the case of a hexagonal lattice of cylindrical obstacles, we describe the tumbling anisotropy via The angular distribution function, following the derivation described in ref. [25], is The experimental data can be fitted with Equation (3) assuming  6 =  0 (see Figure 8d), showing qualitative agreement with the description as an anisotropic scattering medium.

Shape-Anisotropic Obstacles
Shape-anisotropic obstacles, in the form of elongated pillars, strongly affect the dynamics of microswimmers in compartment 3 exhibiting a different behavior than that observed in the case of cylindrical pillars.Pillars' shape and orientation also affect the exchange rates between the compartments 1 and 2. Exemplary trajectories of the motile algae in the porous environment of compartment 3 are shown in Figure 9 for elongated obstacles with their long axes perpendicular or parallel to the microchannel walls.
Similarly to the case of cylindrical pillars, reflections of the algae often occur at the outermost obstacle row (Figure 9a).This is especially well seen for orthogonal pillar orientation.Note that yet, trajectories of reflection events can exhibit more complexity (Figure 9b-e).The microswimmers can penetrate deeper into the obstacle array (Figure 9d,e).Algae that fully traverse the porous environment may find a straight path (Figure 9i,j) or meander through the labyrinth of pillars (Figure 9g,h).Hence, there is a wide spread of transmission path lengths and of the time spent in the obstacle array until the other compartment is reached.The trajectory in Figure 9h exhibits transient adhesion for less than 2 s.Owing to frequent encounters with the upper and lower microchannel wall in strong vertical confinement, this phenomenon, as well as periods of slow translation superimposed with a 'jiggle motion' are sometimes observed.The environment's anisotropy profoundly affects the orientational order of microswimmers' displacements inside the walls of pillars.In the case of the elongated pillars, the PDE exhibit distinctive maxima in the directions parallel to the pillars' long axes (Figure 10).This confirms the orientational ordering effect determined by the geometrical constraints.Shoulders in the orientational PDE correspond to the specific directions along which free straight paths exist, ±/4 and ±3/4, for Samples C and D (Figure 10b,c,e,f).Similarly, peaks in the PDF at approximately ±0.19, ±0.81 correspond to obstacle-free directions in Sample E (Figure 10h,i).The effective anisotropic scattering medium model proposed by Brun-Cosme-Bruny et al. [25] can be extended for application to Samples C and D. The direction dependence of the tumbling rate is modelled as where  0 ⩾ 0 and () ⩾ 0 ∀ ∈ [ − , ).The PDF has the form One should note that the PDF, other than the tumbling rate, contains a cos (6)-term.While the shape of the PDF is in good agreement with the experimental data (Figure 10b,e), the tum-bling rate is orders of magnitude larger than expected.There are several possible reasons for the deviation.First, Samples C and D have a high cell concentration, where reorientations may be caused by cell-cell-interactions.The model, however, is derived for a single cell.Second, orientation changes by collision-induced tumbling are modelled by choosing a random angle from the uniform distribution.Yet, elongated obstacles, similar to planar walls, [21] might cause systematic scattering.Indeed, interactions with the long sides of the elongated obstacles appear to be biased toward small outgoing angles (see Figure S5, Supporting Information).The existence of preferred scattering angles could also explain, why not all directions with obstacle-free lanes are equal, i.e., why directions of motion along the long axis of the obstacles appear as strong peaks in the angular PDF, whereas the free paths along ±/4 and ±3/4 result in less pronounced shoulders.Third, elongated obstacles may not only cause systematic scattering, but they can even suppress tumbling and promote swimmer alignment.
Despite the alignment induced by the anisotropy of the surrounding, typical trajectories retain their undulatory form (see e.g., Figure 9f).
For parallel elongated obstacles (Sample D), like in the case of cylindrical pillars, reflecting algae have significantly smaller mean detention times  ref det ≈ 1.3 s in compartment 3 then permeating ones ( perm det ≈ 3.5 s) (Figure 11a,b).The reflections occur at the tips of the pillars that correspond to the maxima in the penetration length distribution function (Figure 11c).In the case of elongated pillars oriented perpendicular to the capillary axis (Sample C), the wall detention time for permeating algae,  perm det ≈ 7.2 s, becomes larger than in the case of parallel aligned pillars (Figure 12).This can be attributed to the trapping through the alignment of the trajectories in the orthogonal direction (see Figure 9e,g).Indeed, the angular displacement distribution is strongly anisotropic, with maxima aligned orthogonally to the capillary axis (Figure 10a-c).
It should be noted that trajectories whose distance to the lateral microchannel walls is smaller than the steric interaction distance are excluded from the data set.In the case of Sample C, this results in a small set of only 30 transmission trajectories, which makes claims about the permeation detention time disputable.Yet, when letting go of the constraint and taking into account trajectories that may include steric wall interactions, the wall detention time for permeation,  perm det ≈ 6.2 s (from 283 trajectories, see also Figure S6, Supporting Information), is still outstandingly high.
The different behavior of the motile algae, depending on the geometry of their surrounding, is reflected by the distributions of the path length of permeating trajectories.Two aspects must be considered when comparing path lengths: The availability of short paths and the probability of swimmers finding short paths.The former aspect is purely geometric, whereas the latter involves swimmer-obstacle interactions.If we are interested in both aspects, we contemplate the mean transmission path length Ltrans measured in units of the width w of compartment 3 with obstacles.Alternatively, we consider the mean transmission path lengths normalized by the length of the shortest available path L min trans that reflects to what extent the geometry guides swimmers along short paths.The quantity Ltrans ∕L min trans serves as a measure for the enhancement or hindrance of permeation by a microlabyrinth and emphasizes the effects of swimmer-obstacle interactions, i.e., alignment or randomization of swimmer motion.The shortest available paths are straight lines following obstaclefree lanes from compartment 1 to 2. In some cases, algae might be able to find slightly smaller paths if they exploit the finite lane width to travel under a smaller angle to the horizontal.Practically, this deviation seems negligible.
The shortest paths can be found for samples D and E with parallel pillar orientation, where a straight path parallel to the microchannel walls exists.These are the samples with the smallest mean transmission path lengths in units of w (see Table 1).The geometry in Sample D also favors the shortest path, as shown by the smallest value of L trans ∕L min trans across all samples.In contrast, L trans ∕L min trans in Sample E is larger than for the Samples A and B with cylindrical obstacles.This implies that the parallel obstacles enhance permeation efficiency for the case with a larger porosity (Sample D) and reduce it for the smaller inter-obstacle distance in Sample E. Swimmers are more likely to change lanes in E than in D (see Figure S4, Supporting Information) which increases the path length.
Not surprisingly, Sample C with perpendicularly oriented elongated obstacles has the longest mean transmission path length.One should mention that, due to the strong inhibition of transmission, the values for Ltrans in Table 1 are based on only 30 data points.Letting go of the restriction that only trajectories that stay far from the lateral microchannel walls are taken into account, one obtains values L trans ∕w = 2.36, L trans ∕L min trans = 1.67, showing a similar trend, from a much larger data set of 283 transmission events.The mean reflection path lengths of most samples are comparable, with the exception of Sample C, where a large fraction of the channel cross-section is occupied by obstacles, resulting in frequent reflection at the first row of obstacles and consequently low Lrefl .

Transmissivity and Reflectivity
Excursions of microswimmers through the compartment 3 with the obstacle array can be classified as transmission or reflection events to define reflection and transmission coefficients R and T as the ratio of the number of reflection/transmission events and the total number of observed excursions through compartment 3, i.e.,

R =
N refl N refl + N trans (6)   T = N trans N refl + N trans (7)   where N refl, trans is the number of observed reflection or transmission events, respectively.The boundaries of compartment 3 are located 17 μm from the outermost obstacle row (see also Figure 6), which represents the distance below which steric swimmer-obstacle interactions can occur (assuming a swimmer body radius of 5 μm and a flagellar length of 12 μm [38] ).Transmission and reflection events occurring close to the lateral channel walls are discarded to exclude the effects of steric swimmer-wall interactions that may occlude the influence of obstacle geometry and arrangement.
Reflection and transmission coefficients are listed in Table 2 for channels with different geometries and porosities of the pillar lattice.The porosity was defined as where A is the total area of a unit cell, and A obstacles is the area of the unit cell occupied by pillars.
The transmission was strongly suppressed when elongated obstacles were oriented perpendicularly to the channel's main axis, and nearly all algae were reflected (Sample C, Table 2).Transmission coefficients for parallel orientation of elongated obstacles and cylindrical pillars were significantly larger than for perpendicularly oriented elongated obstacles.
Two effects contribute to the suppression of transmission for perpendicularly oriented stadium-shaped obstacles: First, this geometry has a low aperture, i.e., the obstacles occupy a large part of the channel cross-section.Second, the preferred swimming direction of Chlamydomonas reinhardtii inside the porous wall is unfavourable for transmission (Figure 10a-c).
Since perpendicular obstacles direct algae toward the lateral microchannel walls, one might be concerned that the transmission/reflection statistics get distorted by the aforementioned ex-clusion of trajectories approaching the channel walls.While the absolute number of transmission events increases substantially if the trajectories in the vicinity of the channel walls are considered, the transmission coefficient for perpendicular obstacle orientation takes the value T = 4 %, thus remaining remarkably small.
A higher porosity leads to a larger transmission probability for channels with the same obstacle shape and orientation (compare Samples A and B or D and E, Table 2).

Mean Squared Displacement and Persistence Time
The mean squared displacement (MSD) of the active swimmers is a measure of the transport efficiency.Under the assumption of exponential decorrelation of velocities (9)   the mean squared displacement of a self-propelled particle restricted to a 2D plane follows the equation where  is the persistence time, D is the (passive) translational diffusion constant and v is the self-propulsion speed.Note that Equation (10) slightly differs from the commonly used Howse Equation, [2] which is valid for the projection of 3D motion on a plane rather than for the motion constrained to two dimensions.Equation ( 10) can be fitted to the MSD calculated, taking all trajectories and all starting times t 0 in each of the different compartments (see Figure S7a-e, Supporting Information).Due to the very limited compartment size, especially for compartment 3, MSD calculations are only reliable for small time lags Δt.For longer Δt, the limited size of the region of interest (ROI) leads to artifacts that falsely suggest subdiffusion.(see Figure S7f, Supporting Information).Swimming speeds and persistence times can be determined from the MSD fits.The translational diffusion constant was fixed at D = 4.29 • 10 −2 μm 2 s −1 , based on the Einstein relation for Stokesian friction on a 5 μm-sphere in water at room temperature.It is to expect that the presence of pillars affects the persistence time of active swimmers.Indeed, a reduction of the persistence by the presence of obstacles was observed consistently (see Table 3).Similar behavior was reported for E. coli in the presence of pillars, where a reduction of the pore size decreased the persistence length, thereby hindering swimmer transport. [41]The simulations demonstrated the impact of the run-length scale on the spread of active agents. [20]dapting a simplified approach that ignores possible alignment effects of obstacles, the persistence time  depends on the intercollision time  collision as where  is a constant and  free is the persistence time in the absence of obstacles, i.e., when  collision → ∞.A similar model was successfully applied to E. coli. [41]An equation of this form was also used for Chlamydomonas reinhardtii in ref. [42], where the density-dependence of persistence caused by swimmer-swimmer-interactions was investigated in the absence of obstacles.The swimmer density varies indeed between the different samples used here.However, the density and velocitydependence of swimmer-swimmer-intercollision times is contained in  free .As discussed above, density gradients equilibrate very quickly in the microchannels such that there is no substantial swimmer concentration difference between different compartments in the same channel.Consequently, for compartment 3,  free , the persistence time if pillars were removed, can be estimated from the persistence time in obstacle-free compartments 1 and 2. In approximation, the mean value is taken The swimmer-obstacle-intercollision time  collision is approximated roughly as where v is the swimming speed obtained from MSD-fits and  min is the minimal interobstacle distance for a given lattice.To check the validity of the model (Equation ( 10)), the persistence times in the porous environment of compartment 3  center are determined from MSD-fits and  free and  collision are calculated using Equations ( 12) or (13), respectively.The fit shown in Figure 13 shows that (equivalent to Equation (10) for the central compartment 3) is a surprisingly good approximation, keeping in mind that the lattice geometry was represented in the model only via the minimal separation between pillars  min .

Conclusion
In this paper, we demonstrated that the diffusion (permeation) of the microswimmers Chlamydomonas reinhardtii through a porous wall, determined by the short time-scale dynamics is strongly affected by the shape anisotropy of the obstacles comprising the porous medium.Describing the transport using reflection and transmission coefficients, we show that the geometrical constraints and density of the obstacles determine the permeation through the wall and the reflection of the microswimmers.Reduction in the porosity of the wall results in the obstruction of the permeation of the microswimmers and an enhancement of the reflection.
Through the steric and hydrodynamic interactions between the microswimmers and the pillars, the shape anisotropy of the pillars results in the alignment of the microswimmers' trajectories, introducing bias in the diffusive process.Scattering at the pillar arrays appears to contribute the most to the reflection events.
Depending on the pillars' orientation and mediated by the interactions with the microchannel's walls, such alignment may even lead to trapping, as was observed in the case of orthogonally oriented pillars in Sample C.
Our research has shown that controlling the persistence length of microswimmers in a microfluidic device can locally manipulate their dynamics by taking into account the anisotropy of the environment constrained by obstacles.Controlling the obstacle orientation (by magnetic field or light) will allow the permeabilities in microchannels to be further tuned.Furthermore, the effective anisotropic scattering medium model in ref. [25] can be successfully applied for more complicated scattering geometries by accounting for lower symmetries in the direction-dependent tumbling rate.

Experimental
Cultivation of Chlamydomonas reinhardtii: Chlamydomonas reinhardtii of the strain SAG 11-32a were purchased from the culture collection of algae at Göttingen University.The cells were grown under illumination by fluorescent lamps (Osram Fluora L15W/77) on a 14 h:10 h day-nightcycle in Tris-acetate-phosphate-medium (TAP-medium).Air bubbling promoted gas exchange in the liquid cultures and, by inducing flow, mitigated biofilm formation.
The strain was maintained by weekly subculturing.On the 7 th day after the inoculation of a culture, its cell density was determined using a hemocytometer.Then, a defined volume of the liquid culture was transferred to a new flask with TAP in a clean bench to inoculate a fresh culture with an initial cell density of 10 5 cells per ml.Experiments were performed with vegetative cells taken from the cultures during the daytime on the 6 th to 8 th day after inoculation.
Microfluidic Chip Fabrication: Microfluidic labyrinths were fabricated by replicating the master wafer in the transparent, flexible polymer Polydimethylsiloxane (PDMS).The structures were designed and sketched using AutoCAD (Autodesk) software.Next, soft lithography mask based on Soda Lime (c) glass, Al and Cu was fabricated at Compugraphics Jena GmbH according to the prepared sketch.
The master mould fabrication was implemented using a soft lithography process on a 4″ silicon wafer with softened process parameters.Mainly, the temperature of the soft bake and post-exposure bake was reduced compared to a standard to ensure the ablation of the photoresist in circular openings with a radius of 10 μm.Thus, the silicon wafer was dehydrated at 200°C over 5 min, and the photoresist SU8-25 (MicroChem GmbH) was spin-coated (LabSpin 6, SüSS MicroTec GmbH) on it at 750 rpm for 6 s and subsequently at 2000 rpm for 30 s.The soft bake step was done at 65°C over 3 min, followed by 85°C over 10 min.After the samples were completely dry, they were exposed through the prefabricated shadow mask with an exposure dose of 160mJcm -2 and wavelength of 365nm.Post-exposure bake (PEB) was done at 65°C over 2 min, followed by 85°C over 10 min.After PEB, the samples were slowly cooled down over 2 h.Finally, wafers were washed for 17 min with constant stirring in the mr-Dev600 developer to remove the non-exposed photoresist.After development, the master molds were immersed in Isopropanol for 2 min and dried with a nitrogen gun.
The microstructures of the master mold (Figure S1a, Supporting Information) were pre-sputtered with 25 nm gold and analyzed using a profilometer (FRT MicroProf) and a scanning electron microscope (SEM, Zeiss EVO50).
To obtain PDMS microfluidic channels, silicone-based polymer was mixed with 10:1 base to a curing agent, respectively, and poured on the prepared wafer.Curing took place at room temperature for over 48 h.Once the PDMS was cured, it was removed from the mold and cut.Next, the 170 μm thick glass slides and PDMS structures (Figure S1b, Supporting Information) were treated with oxygen plasma for 1 min, brought into contact, and heated for over 10 min at 80°C.
Once the PDMS was bonded to the glass slide, the microfluidic chips were filled with DI water immediately to exploit the temporary hydrophilicity of the PDMS and, in this way, avoid air bubbles in the microlabyrinth.
Preparation of Samples: Chlamydomonas reinhardtii were extracted from culturing flasks under sterile conditions.The suspensions of algae were centrifuged (1000 g for 10min) and the supernatant was removed.Subsequently, the cells were re-dispersed in fresh TAP-medium.
The sample was then filled into the PDMS-channels using a pipette.The channel inlets were sealed with grease and the microfluidic chips were kept in water under the same lamp that was used for cell culturing for several hours.It appeared, that a few hours after filling the channels, cells were slow and sparse in the channel.If channels were incubated overnight before usage, more cells had spread from the inlets into the channel and the swimmers tended to be faster.
In some cases, the light-switchable adhesion of Chlamydomonas reinhardtii [43] was exploited to prepare concentration gradients.Blue light was applied locally using a halogen lamp (Zeiss HAL 100) with a filter ( ≈ 480 nm, I ≈ 750 μW cm −2 ).The cells were exposed to ambient red LED illumination ( ≈ 628 nm, I ≈ 110 μW cm −2 ) to keep cells motile in the whole microchannel.Swimming cells that reached the blue spot often switched into the adhered state.The channels were kept under these conditions for about one hour to accumulate adhered cells in the blue spot.Then, the red ambient illumination was switched off and global blue illumination was applied for about 10 min using a cold light source (Schott KL 2500 LCD) and a blue bandpass filter ( max ≈ 513 nm, I ≈ 335 μW cm −2 ).In this way, cell adhesion was triggered globally.Total adhesion was not achieved, but a significant gradient of the adhered cell concentration could be prepared in this way.The cells switched back into the swimming state after the illumination conditions were changed to red ambient light and red observation illumination.
Observation: An inverted microscope (Zeiss Axio Observer.D1) equipped with a digital camera (Canon EOS M6 Mark II) was used to record videos of Chlamydomonas reinhardtii motion in the patterned microchannels (see Figure 1a) at the resolution of 1920x1080 px 2 and the frame rate of 50s −1 .
The microscope was used in transmission in phase contrast mode with a 10x-objective (Zeiss A-Plan 10x/0,25 Ph1).The halogen lamp (Zeiss HAL 100) serving as a light source for observation was operated through a red glass filter ( ⩾ 630 nm) to enhance the cell motility. [43,44]Additionally, red ambient light was applied with LEDs ( ≈ 628 nm, I ≈ 110 μWcm −2 ).
Cell Tracking: Videos of Chlamydomonas reinhardtii in patterned microchannels were cropped to a size of 920x920 px 2 and rotated to achieve horizontal orientation of the microchannel borders.To facilitate the detection of moving algae, the background was subtracted using a mixture of Gaussian algorithms (BackgroundSubtractorMOG2 from OpenCV in Python) or minimum projection (in Matlab).The background image was refreshed as the video progressed (typically in intervals of 1000 frames) to account for dynamic changes due to the adhesion and detachment of algae at the glass bottom of the channel.The detection and tracking was performed using Trackmate plugin in Fiji (ImageJ) [45,46] employing a Laplacian of Gaussians filter for edge detection and a Kalman tracker to extract the trajectories.

Figure 1 .
Figure 1.a) Sketch of the setup for observation of Chlamydomonas reinhardtii in patterned microchannels.b) Microchannel samples A -E consisting of two obstacle-free chambers separated by porous walls (see also Figure 2).c) Illustration of the geometry of pillar lattices in the microchannels and differentiation of the samples.Samples A, B contain cylindrical pillars and differ in the center-to-center-distance between adjacent pillars d.The pillars in samples C, D, E are stadium-shaped pillars, and the lattices vary with respect to the pillar orientation and the transversal inter-pillar-distance  t .

Figure 2 .
Figure 2. Exemplary illustration of the division of microchannels into three compartments, shown for Sample B. Trajectories are colored in green/red/blue in compartment 1/2/3.Compartment 3 with pillars acts as a porous environment separating the obstacle-free compartments 1 and 2.

Figure 3 .
Figure 3. Desorption kinetics: Time dependence of the number density of adhered (non-swimming) cells in compartment 1, which had been illuminated by localised blue light before the measurement.The red line is the exponential fit.

Figure 4 .
Figure 4. Equilibration of the microswimmers between two compartments after termination of the local illumination in compartment 1. a) Total number density of motile and immotile (adhered) swimmers, b) number density of motile swimmers only.

Figure 5 .
Figure 5. Distribution of the time-averaged cell number density n m over the distance from the upper microchannel wall d for Sample D. Compartment 3 (blue curve) contains pillars parallel to the channel walls.The inset illustrates the definition of d and the division of the channel into three compartments.

Figure 6 .
Figure 6.Selected trajectories in compartment 3 of Sample D containing cylindrical pillars.a) Trajectories resulting in the microswimmers' permeation from one compartment to another.b) Trajectories of microswimmers that start and end in the same compartments (reflections).The starting and ending points for two selected trajectories are designated by the pairs I-I′ and II-II′.The boundaries of compartment 3 (dashed lines) are located 17 μm from the outermost obstacle row, which represents the distance below which steric swimmer-obstacle interactions can occur (assuming a swimmer body radius of 5 μm and a flagellar length of 12 μm[38] ).

Figure 7 .
Figure 7. Distribution of wall detention times for a) permeating and b) reflecting trajectories for Sample B with cylindrical pillars.c) Corresponding distribution of the maximal penetration depth.

Figure 8 .
Figure 8. Angular probability distribution functions of displacements over Δt = 2s in a) Sample B and b) Sample A in different compartments.Distinguished directions in the periodic pillar array are shown in c).d) shows the PDF for compartment 3 in Sample A with the fit according to Equation (3) (black line).

Figure 9 .
Figure 9. Exemplary trajectories of Chlamydomonas reinhardtii in the presence of elongated obstacles.The background image is a phase contrast micrograph of the microchannel containing the motile algae.The long obstacle axes were either perpendicular (vertical) or parallel (horizontal) to the microchannel walls.Scalebars have a length of 20 μm.

Figure 10 .
Figure 10.Angular PDF of the swimmer displacement over time periods of Δt = 2 s.Each row of figures corresponds to one of the Samples C (a-c), D (d-f) and E (g-i).The first column (a,d,g) shows the PDF in different compartments.Figures in the second column (b,e,h) show only the PDF in compartment 3 containing pillars.Vertical lines mark distinguished directions.For Samples C (b) and D (e) black curves show the fit obtained by applying the effective anisotropic scattering medium model introduced in ref. [25] to the direction-dependent tumbling rate Equation (4).The third column of figures (c,f,i) illustrates the distinguished directions in the different geometries.

Figure 11 .
Figure 11.Distribution of wall detention times for a) permeating and b) reflecting trajectories for Sample D with elongated pillars.c) Corresponding distribution of the maximal penetration depth.

Figure 12 .
Figure 12.Distribution of wall detention times for a) permeating and b) reflecting trajectories for Sample C with elongated pillars.c) Corresponding distribution of the maximal penetration depth.

Figure 13 .
Figure 13.Terms of persistence and intercollision times are plotted to test the validity of Equation (14).The red line represents a linear fit without offset with r 2 = 0.958 and  = 0.403 ± 0.042.

Table 1 .
Mean path lengths of trajectories in compartment 3 with obstacles for reflected and transmitted algae.Mean path lengths are normalized by the width of the region width of the compartment w or by the shortest possible path length for transmission L min trans .

Table 2 .
Transmission and reflection coefficients T and R for Chlamydomonas reinhardtii at obstacle arrays of different porosity P. Obstacles were either cylindrical (○) or elongated, in which case the orientation of their long axis was either parallel (∥) or perpendicular (⊥) to the main axis of the channel.See Figure1for details of the channel geometry for different samples.Porosities were determined from images.

Table 3 .
Persistence times  in the obstacle-free compartments  free and in the compartment three with obstacles  center , swimming speed (all determined from MSD-fits), minimum distance between obstacles  min and swimmer-obstacle intercollision time  collision estimated according to Equation(13).