Taylor line swimming in microchannels and cubic lattices of obstacles

Microorganisms naturally move in microstructured fluids. Using the simulation method of multi-particle collision dynamics, we study an undulatory Taylor line swimming in a two-dimensional microchannel and in a cubic lattice of obstacles, which represent simple forms of a microstructured environment. In the microchannel the Taylor line swims at an acute angle along a channel wall with a clearly enhanced swimming speed due to hydrodynamic interactions with the bounding wall. While in a dilute obstacle lattice swimming speed is also enhanced, a dense obstacle lattice gives rise to geometric swimming. This new type of swimming is characterized by a drastically increased swimming speed. Since the Taylor line has to fit into the free space of the obstacle lattice, the swimming speed is close to the phase velocity of the bending wave traveling along the Taylor line. While adjusting its swimming motion within the lattice, the Taylor line chooses a specific swimming direction, which we classify by a lattice vector. When plotting the swimming velocity versus the magnitude of the lattice vector, all our data collapse on a single master curve. Finally, we also report more complex trajectories within the obstacle lattice.


I. INTRODUCTION
The motility of microorganisms in their liquid environment is important in various biological processes 1 . Microorganisms move in the low-Reynolds-number regime, where viscous forces dominate over inertia 2 . They have developed various swimming strategies to cope with the strong viscous forces 2 including beating flagellar appendages of sperm cells 3,4 , metachronal waves of collectively moving cilia on the cell surface of a paramecium 5 , rotating helical flagella in E.coli [6][7][8][9][10] , and periodic deformations of the whole cell body [11][12][13] . A first expression for the swimming speed of a simplified flagellar model was given by Taylor in 1951 14 . In this model a prescribed bending wave moves along a filament, which we call Taylor line in the following. A recent study with the Taylor line showed hydrodynamic phase locking of multiple flagellas 15 and Ref. 16 determined the optimal shape of a large amplitude wave. These insights into biological swimming mechanisms in Newtonian liquids inspired studies of artificial swimmers in unbounded 17,18 as well as bounded 19,20 fluids.
In 1979 L. Turner and H. C. Berg suggested that the geometric constraints of polymer networks in viscoelastic fluids can drastically enhance the swimming speed of microorganisms 47 . Based on experimental observations with helical bacteria they formulated the following picture. When rotating about their helical axis, bacteria with helical shape move through a polymeric liquid like through a quasi-rigid medium and similar to a corkscrew driven into cork. So, in the ideal case, after each full rotation the bacterium would proceed by one full pitch length. In this paper we will investigate another type of this geometrical swimming by studying the Taylor line in a cubic lattice of obstacles.
A typical example for obstacles in nature are erythro-cytes or red blood cells. The African trypanosome, the causative agent of the sleeping sickness, swims faster in the crowded environment of blood and thereby removes surface-bound antibodies with the help of hydrodynamic drag forces 45 . In this way, the parasite evades the immune response of its host. The motility of the African trypanosome in a Newtonian liquid was investigated in bulk fluid by computer modeling [63][64][65] and in Poiseuille flow 33 . Blood is a complex viscoelastic liquid containing a large amount of cellular components, which gives blood a non-Newtonian character. Its viscosity depends on the volume fraction of erythrocytes (hematocrit), shear rate, and temperature 66,67 . In order to understand the geometrical constraints of erythrocytes for the motility of the trypansome or how other obstacles influence the swimming of sporozoites or C. elegans, more controlled experiments were conducted. They use either suspended colloids 62,68 or fabricated lattices of posts 11,51,52,69,70 .
In lab-on-chip devices obstacle lattices are used to separate trypanosomes from erythrocytes with the idea to diagnose the sleeping sickness in an early stage 71 . Trypanosomes swimming in these lattices show a motility much more comparable to their in vivo motility due to interactions with the obstacles 11 . Similarly, Park et al. found that C. elegans swims up to ten times faster in an obstacle lattice compared to its swimming speed in bulk fluid 51 . The speed-up depended on the lattice spacing. A combined experimental and numerical study by Majmudar et al. with an undulatory swimmer such as C. elegans showed that most of the characteristics of this new type of swimming in an array of micro pillars can be explained by a mechanical model for the swimmer 52 . It does not need any biological sensing or behavior.
In this paper we present a detailed hydrodynamic study of an undulatory Taylor line swimming in a twodimensional microchannel and in a cubic lattice of obstacles. We use the method of multi-particle collision dynamics for simulating the hydrodynamic flow fields 72 . In the microchannel the Taylor line swims at an acute angle along a channel wall with a clearly enhanced swimming speed. In a dilute obstacle lattice swimming speed is also enhanced due to hydrodynamic interactions with the obstacles similar to a study by Leshansky 73 . Moving the obstacles closer together (dense obstacle lattice), the undulatory Taylor line has to fit into the free space of the obstacle lattice, where it performs geometric swimming. Here, the swimming speed is close to the wave velocity of the bending wave traveling along the Taylor line. In this regime, we classify the possible swimming directions by lattice vectors. When plotting the ratio of swimming and wave velocity versus the magnitude of the lattice vector (effective lattice constant), all our data collapse on a single master curve. This demonstrates the regime of geometric swimming. We also illustrate more complex trajectories.
The article is structured as follows. In Sec. II we introduce our computational methods including the method of multi-particle collision dynamics and the implementation of the Taylor line. In Sec. III we calibrate the parameters of the Taylor-line model by studying its swimming motion in the bulk fluid. In Secs. IV and IV we review the respective results for swimming in the microchannel and in the obstacle lattice. Sec. VI closes with a summary and conclusions.

A. Multi-particle collision dynamics
We employ the method of multi-particle collision dynamics (MPCD) to simulate the Taylor line in its fluid environment 74,75 . MPCD uses point particles of mass m 0 as coarse-grained fluid particles. Their dynamics consists of a ballistic streaming and a collision step, which locally conserves momentum. Therefore, the resulting flow field satisfies the Navier-Stokes equations but also inherently includes thermal fluctuations 72 .
In the streaming step the positions r i of all fluid particles are updated according to where v i is the particle velocity and ∆t c the MPCD time step between collisions 76 . After each streaming step the fluid particles are sorted into quadratic collision cells of linear dimension a 0 , so that on average each cell contains N particles with mass M = N m 0 . In each cell we redistribute the particles' velocities following a collision rule, for which we choose the Anderson thermostat with additional angular momentum conservation 76 . At first we calculate the total momentum, P cell = m 0 i∈cell v i , of each collision cell. Then, we assign to each velocity component of a particle relative to the mean velocity P cell /M a random component v i,rand from a Gaussian distribution with variance k B T /m 0 . Here, T is temperature and k B the Boltzman constant. Using the mean random momentum P rand = m 0 i∈cell v i,rand of each cell, we determine the new particle velocities after the collision: This collision rule conserves linear momentum but not angular momentum 72 . To keep the latter constant, we note that during the collision step the fluid particles have fixed distances. Therefore, one can apply a rigid body rotation, ∆ ω × r i , to replace the new velocities Here, the angular velocity is where Θ = m 0 i∈cell | r i | 2 is the moment of inertia of the particles in the cell. This rule restores angular momentum conservation keeping linear momentum constant. By definition, the collision rule based on the Anderson thermostat also keeps the temperature constant.
To restore Galilean invariance and the molecular chaos assumption, we always apply a random grid shift when defining the collision cells and take the shift from the interval [0, a 0 ] 77,78 . Transport coefficients of the MPCD fluid can be found in Ref. 79 .
In the following, we will measure quantities in typicial MPCD units. We will use the linear dimension of the collision cell a 0 as a unit for lengths, energies are measured in units of k B T , and mass in units of m 0 . Then the time unit becomes τ 0 = a 0 m 0 /k B T 80 . In this unit, our time step between collisions is always chosen as ∆t c = 0.01.

B. No-slip boundary condition: Bounce-back rule and virtual particles
At bounding walls fluid flow obeys the no-slip boundary condition. To implement it within the MPCD method, we let the effective fluid particles interact with channel walls or obstacles using the bounce-back rule, see Fig. 1. When a fluid particle moves into an obstacle or a channel wall during the streaming step (position B), we invert the velocity v i, = − v i and let the particle stream to position C during half the collision time: Then, we move this particle to the closest spot on the obstacle surface or channel wall (position D) and let it stream with the reversed velocity during half the collision time to position E. In addition, the no-slip boundary condition is improved using virtual particles inside a channel wall or an obstacle, see Fig. 2. We uniformly distribute virtual particles (red dots in Fig. 2) in the areas of the collision cells,  where ri gives the bead position. The tangential vector ti = ri+1− ri connects two neighboring beads and is not normalized to one. The angles αi between the tangential vectors are used to define the sinusoidal bending wave running along the Taylor line. (b) Snapshot of the Taylor line, which swims along the unit vector e in a bulk fluid with superimposed thermal diffusion. The blue line represents the center-of-mass trajectory. The end-to-end distance of the Taylor line or its length along e is L = 2λ, where λ is the the wavelength of the bending wave along e and A its amplitude.
which extend into the channel wall or obstacles. The velocity components are chosen from a Gaussian distribution with variance k B T /m 0 . The virtual particles also take part in the collision step. So, close to bounding walls one has the same average number of particles in a collision cell as in the bulk. Both rules together implement the no-slip boundary condition at a bounding surface in good approximation 76,81 .

C. A discrete model of the Taylor line
The Taylor line propels itself by running a sinusoidal bending wave along its contour line. Figure 3(a) shows how we discretize the Taylor line by a bead-spring chain with N beads each of mass m = 10 m 0 . The beads at positions r i interact with each other by a spring and a bending potential. The spring potential implements Hooke's law between nearest neighbors 18 , Here l 0 = 1/2 a 0 is the equilibrium distance between the beads and | t i | = | r i+1 − r i | the actual distance, where t i denotes the tangent vectors. The contour length of the bead-spring chain, is approximately constant. We choose a large spring constant D = 10 6 to ensure that deviations from the equilibrium distance l 0 between the beads are smaller than 0.002l 0 . Finally, the spring force acting on bead i is The bending potential creates a sinusoidal bending wave that runs along the Taylor line. It was also used in two-dimensional studies of swimming sperm cells 42 and in simulations of the African typansome 63,64 . The bending potential has the form: where κ = pk b T is the bending rigidity and p the persistence length 82 . The rotation matrix R(α) rotates the tangential vector by an angle α about the normal of the plane, so the equilibrium shape of the Taylor line is not straight but bent. For the rotation angle at bead n we choose α n = l 0 c(n, t), where the equilibrium curvature, is a function of the position of bead n on the Taylor line (n ∈ {1, N }) and time t. It creates the sinusoidal bending wave running along the Taylor line with wavelength λ c (measured along the contour) and an amplitude A controlled by the parameter b. Unless stated otherwise, we choose the ratio of persistence to contour length as p/L c = 5 · 10 3 to ensure that bending forces are much stronger than thermal forces, in order to induce directed swimming 42 . This is investigated in more detail in Sec. III.
From the bending potential we derive a bending force acting on bead j: where R T (α j ) means transposed matrix. Then, the total force F i = F H i + F B i determines the dynamics of the Taylor line. In our simulations we update the positions of the beads during the streaming step using the velocity Verlet algorithm with time step δt = 0.01∆t c 64 . In addition, the beads with mass m = 10m 0 participate in the collision step and the components of their random velocities v i,rand are chosen from a Gaussian distribution with k B T /10m 0 . The beads thereby interact with the fluid particles which ultimately couples the Taylor line to the fluid environment. Note, since the beads of the Taylor line have a different mass than the fluid particles, in all the formulas of Sec. II A one has to replace m 0 i∈cell . . . by i∈cell m i . . ., where m i is the mass of either the fluid particles or the Taylor line beads. The latter also interact with channel walls or obstacles by the bounce-forward rule, which is very similar to the bounce-back rule used for the fluid particles. Upon streaming into an obstacle or wall, we place the particle onto position D, see Fig. 1. However, in contrast to the bounce-back rule, only the velocity component of the bead orthogonal to the surface is inverted. This ensures that the Taylor line can slip along a surface.
We introduce the normalized end-to-end vector of the Taylor line, to quantify the mean swimming direction and denote the end-to-end distance by L. Unless mentioned otherwise, we always fit two complete bending wave trains onto the Taylor line, meaning L = 2λ, where λ is the wavelength measured along e || [see Fig. 3 (b)]. Note that λ is different from the wavelength λ c along the contour introduced in Eq. (10). In the following, we will vary the amplitude A of the bending wave keeping the end-to-end distance with L = 2λ fixed. Therefore, we always have to adjust the contour length of the Taylor line by adding or removing some beads. Typically, we use Taylor lines with L = 42a 0 and the number of beads ranges from N = 88 to 125.

III. TAYLOR LINE IN THE BULK FLUID
In the following we discuss the swimming velocity of the Taylor line as a function of the dimensionless persistence length p/L c . Thermal fluctuations noticeably bend an elastic line on lengths comparable to the persistence length. So, in our case the Taylor line should have the form of a sine wave when p is much larger than its contour length L c . In addition, the Taylor  (blue line), reminiscent of a Brownian particle. In case (b) with p/L c = 10 the bending wave is clearly visible, although still distorted by thermal fluctuations, and the Taylor line exhibits persistent motion. The Taylor line has a fully undistorted, sinusoidal contour in case (c) at p/L c = 500. The trajectory of the center of mass shows directed swimming superimposed by Brownian motion. The total displacement over a complete simulation run is larger compared to (b) and the Taylor line has reached its maximum propulsion speed.
To discuss directed swimming more quantitatively, we introduce the swimming velocity v || = d r · e || /∆t, where we project the center-of-mass displacement d r during time ∆t onto the mean direction of the Taylor line defined in Eq. (12) and indicated in Fig. 3(b). We then define the stroke efficiency It compares the mean swimming speed, averaged over the whole swimming trajectory, with the phase velocity c, at which the bending wave travels along the Taylor line. Then, S = 1 indicates optimal swimming of the Taylor line. In three dimensions this situation is similar to a corkscrew screwed into the cork. It moves at a speed that equals the phase velocity of the helical wave traveling along the rotating corkscrew.
In Fig. 5 we plot the stroke efficiency S versus persistence length p/L c . For p = L c the stroke efficiency is approximately zero as already observed from the trajectory (a) in Fig. 4. The efficiency S increases nearly linearly in log(p/L c ) until at ca. p/L c = 10 2 it reaches a plateau value. A linear fit gives the plateau value S 0 = 0.098 typical for low Reynolds number swimmers. For example, for C. Elegans studied in Ref. 51 we estimate S = 0.12. In the following we always use the persistence length p/L c = 5 · 10 3 to be on the safe side.
Within resistive force theory, one derives for the swimming speed of the Taylor line in the limit of A λ: with the wave number k = 2π/λ and angular frequency ω = 2πν. The parameters ξ ⊥ and ξ are the respective local friction coefficients per unit length for motion perpendicular and parallel to the local tangent 1 . Originally, Taylor used ξ ⊥ = 2ξ valid for an infinitely long filament. We are able to reproduce the linear relationship between swimming speed v || and ω in our simulations (see inset of Fig. 5). Whereas A/λ = 0.1 (blue) and 0.14 (red) confirm the expected scaling with kA 2 , the straight line for A = 0.04λ = 0.9a 0 deviates from it, possibly because the amplitude is too small to be correctly resolved in the MPCD simulations. Note, for large ω (data not shown) we observe deviations since the MPCD fluid becomes compressible 80 .

IV. TAYLOR LINE IN A MICROCHANNEL
In the following we present our simulation data of the Taylor line swimming in a microchannel and discuss it in detail.  acute tilt angle along the channel wall. Earlier simulations of swimming sperm cells have attributed the attraction to the wall to a pusher-like flow field, which drags fluid in at the sides of the swimmer 37 . Thereby, the sperm cells are hydrodynamically attracted by the wall. Additional flow at the free end of the flagellum pushes the tail of the sperm cell up. In Fig. 6(c) we confirm this picture. Below the wave crests fluid is strongly pulled towards the Taylor line, while fluid flow towards the wall below the wave troughs is much weaker. Hence, the Taylor line is attracted to the wall. In addition, fluid flow towards the wave crest at the front is stronger compared to the second wave crest, which obviously tilts the Taylor line as Fig. 6 (b) demonstrates. In order to investigate the tilt angle φ at the channel walls in more detail, in Fig. 7 we plot φ versus channel width for several amplitude-to-wavelength ratios A/λ. Each curve except for the smallest amplitude A starts with a small region of the channel width d/A ∈ [2, 3], where the tilt angle is ca. 0.01π and hardly depends on d/A. Then, at the width d/A ≈ 3 the tilt angle increases and ultimately reaches a plateau value at d/A ≈ 8 meaning that the Taylor line does not interact with the other channel wall at widths d/A 8. The inset plots the plateau or maximum tilt angle φ max versus A 2 /λ 2 . It is determined as the average of all tilt angles for d/A 8. The maximum tilt angle φ max needs to be an even function in A since −A only introduces a phase shift of π in the bending wave, which does not change the steady state of the Taylor line. Indeed, we can fit our data by where φ 2 = 1.944 and φ 0 = 0.046 are fit parameters.

B. Speed enhancement at the channel wall
The swimming speed v W of the Taylor line along the channel wall is enhanced compared to the bulk value v and strongly depends on the channel width. To discuss this effect thoroughly, we define a speed enhancement factor In Fig. 8 we plot it versus the channel width d/A. Starting from d/A ∈ [1,2], where the Taylor line squeezes into the channel, γ increases and goes through a maximum at d/A ≈ 3. Interestingly, the maximum value of γ is approximately the same, only for the smallest amplitude the maximum is larger and shifted towards d/A ≈ 4. As before, at d/A 8 the factor γ reaches a plateau value γ ∞ . Obviously, this happens when the other channel wall does no longer influence the swimming Taylor line by hydrodynamic interactions. So the presence of both channel walls helps to speed up the Taylor line with an optimal channel width at d/A ≈ 3. The inset shows how γ ∞ decreases with increasing wave amplitude A and reaches nearly one at A/λ = 0.24. This suggest the following interpretation. The Taylor line uses the no-slip condition of the fluid at the channel wall to push itself forward. This is more effective the closer the Taylor line swims at the wall, i.e., for small A. In contrast, with increasing A also the mean distance of the Taylor line from the wall increases and one expects to reach the bulk value of the swimming speed (γ ∞ = 1) at large A. The dashed line in the inset is an exponential fit to γ ∞ −γ 0 = γ 1 exp(−γ 2 A/λ). We find that γ 0 = 1.08 deviates from the ideal large-amplitude value of one. This is due to a numerical artifact since for large A the MPCD fluid is no longer incompressible 80 .

V. TAYLOR LINE IN A CUBIC OBSTACLE LATTICE
We now study the Taylor line swimming in a cubic lattice of obstacles with lattice constant d. Fig. 9 shows the cubic unit cell. The obstacles have a diameter 2R/λ, which we always refer to the wavelength λ = 21a 0 of the Taylor line. By varying d and R, the Taylor line enters different swimming regimes, which we will discuss in detail in what follows.

A. Dilute obstacle lattice
To define the dilute obstacle lattice, we introduce the width of the gap between two neighboring obstacles, For d surf > 2A the Taylor line with amplitude A can freely swim through the gap, whereas for d surf < 2A it has to squeeze through the gap and therefore adjusts its swimming direction. This leads to what we call geometrical swimming, which we will discuss in the following section.
We illustrate the first case, d surf > 2A, in Fig. 9, which shows the probability density P ( r) for all the beads of the Taylor line to visit a position r in the cubic unit cell. The probability density with the blue thin stripes shows that the Taylor line never leaves its lane. This is also true  for other values of d/λ as long as the Taylor line cannot freely rotate in the space between the lattices. A closer inspection also shows a thin white region (1) around the obstacles, which the Taylor line never enters. Nevertheless, the probability of the beads for being in region (2) in the narrow gap between the obstacles is much higher than for being in region (3) between the four obstacles. We understand this as follows. The beads move up and down while moving with the Taylor line. In region (2) the beads reach their largest displacement equal to A and slow down to invert their velocity. So, they spend more time in region (2), which explains the high residence probability not only in (2) but also in region (4).
In Fig. 10 we plot the stroke efficiency as a function of d surf /A for different 2R/λ. For d surf /A > 2 the stroke efficiency ultimately is proportional to 1/d surf as the inset demonstrates. In addition, at constant d surf the efficiency S is roughly the same, stronger deviations only occur at the smallest 2R/λ = 0.29. This means S is mainly determined by the gap width, through which the Taylor line has to move when A is kept constant. For d surf < 2A the Taylor line has to squeeze through the obstacle lattice. In the main plot of Fig. 10 one realizes a transition in all the curves, where S increases sharply. As we discuss in Sec. V B, this is where the swimming Taylor line fits perfectly along one of the lattice directions and geometric swimming takes place.

B. Geometric swimming in a dense obstacle lattice
In dense obstacle lattices (d surf < 2A) a new swimming regime occurs when the lattice constant d is appropriately tuned. Starting to swim in horizontal direction (see movie M1 in the supplemental material), the Taylor line adjusts its swimming direction along a lattice direction with lattice vector g = d(m e x + n e y ), which defines the swimming mode (m, n). We call this regime geometrical swimming. Figure 11 shows a few examples each with three snaphots of the Taylor line in green, red, and blue, where the time difference between the snapshots is between T and 2T . Perfect geometrical swimming occurs when one wave train fits perfectly into the lattice meaning where we have introduced the magnitude of the relevant lattice vector d eff = | g|. The (2,1) mode in the movie M1 is a good example for geometric swimming. Depending on radius R and amplitude A, the Taylor line also pushes against the obstacles. Obviously, for perfect geometrical swimming the swimming velocity v and the phase velocity c have to be identical: v = c. The Taylor line swims with an efficiency S = 1. It behaves like a corkscrew, which is twisted into a cork; after a full rotation the corkscrew has advanced by exactly one pitch. Differently speaking, the Taylor line converts the bending wave optimally into a net motion without any slip between Taylor line and viscous fluid. However, geometrical swimming also occurs when the perfect swimming condition is only approximately fullfilled, λ ≈ d √ m 2 + n 2 . In this case, the Taylor line pushes against the obstacles and the swimming velocity deviates from c but can even achieve values larger than c. We discuss this in the following. Note that several of these swimming modes, in particular the (1,1) mode, have been observed in experiments for C. elegans in an obstacle lattice 51,52 .
In the geometric swimming regime, the swimming efficiency S = v /c can be rewritten in pure geometric quantities. Using v = d eff ν and c = λν, we immediately arrive at In Fig. 12 we plot this relation as dashed line together with the gray shaded region to indicate the geometricswimming regime. The figure plots the stroke efficiency of a Taylor line swimming predominantly along the diagonal direction in the lattice as a function of d diag , which is the diagonal distance of the obstacles. The curve parameter is the obstacle radius R/λ. The sharp increase of S in the orange curve (2R/λ = 0.62) at d diag = 0.9 indicates a transition from a swimming mode, where the Taylor line has to squeeze through the obstacle lattice, to the geometric-swimming regime. Then, a sharp decrease in S follows and ultimately S decreases slowly. Increasing d diag at constant R makes the gaps between the obstacles wider and at the sharp decrease the Taylor line enters the regime of dilute obstacle lattices discussed in the previous section.
The regime of geometric swimming extends over a finite interval in d diag . One recognizes that geometric swimming can also be implemented when d diag = λ is not exactly fulfilled. Even swimming velocities larger than the wave velocity c (S > 1) are realized. Figure  13 illustrates the mechanism for d diag > λ. It shows the probability density P ( r) summed over all beads to occupy a position between the obstacles. P ( r) reveals two sliding tracks of the Taylor line. A closer inspection shows that the head (nl 0 ∈ [0, 0.2L c ]) and middle (nl 0 ∈ [0.2L c , 0.7L c ]) sections move on the "pushing" track. When the bending wave passes along the Taylor line, the Taylor line pushes against the obstacles (indicated by the red arrows), which helps it to swim faster than in the ideal case. This is nicely illustrated in movie M1 in the supplemental material for the (1,1) mode. The other track is mainly occupied by the tail section (nl 0 ∈ [0.7L c , L c ]) which does not contribute to the increased propulsion. In between the tracks there is a blurry area indicating that the part of the Taylor line between the middle and tail section has to transit from the pushing to the other track.
At larger obstacle diameters in Fig. 12 (red, blue, and purple line) the sharp decrease in S after the geometric swimming indicates a different transition. The Taylor line changes direction and swims along the (1,0) direction since then the wavelength λ fits better to the spatial period, λ ≈ d. The local maximum in the red curve develops into a shoulder, which for the purple curve belongs to the (1, 0) mode of geometrical swimming. Finally, for the black line (2R/λ = 0.86) geometric swimming along the (1,0) direction is more developed. In Fig. 14 we show the positional probability density of all beads of the Taylor line exactly at the local maximum of the red curve in Fig. 12. With d/λ = 0.87 the Taylor line is not in the geometric swimming regime. Even though the distribution is much more blurred than before, there is still a clear sinusoidal track visible. The Taylor line pushes against the obstacles, which helps it to move through the narrow gap. Finally, the red curve in Fig. 12 becomes flat when the Taylor line enters the dilute-lattice regime.
For lattice constants d well below λ and smaller obstacle diameters 2R, one also observes the higher modes (2, 0), (2, 1), and (3, 1) visualized in Fig. 11. In Fig. 15 we summarize all our results by plotting S for the dif- cork.

C. More complex trajectories
In Fig. 16 we show examples of trajectories that do not show geometric swimming along a defined direction as discussed in Sec. V B but exhibit more complex shapes. They are also nicely illustrated in movie M2 of the supplemental material. Depending on the specific values for lattice constant d/λ and obstacle diameter 2R/λ, we can identify trajectories of different types. They either define new swimming modes [ Fig. 16 (a), (c) and (d)] or combine two geometric-swimming modes [ Fig. 16 (b)]. In Fig. 16 (a) the obstacle lattice is so dense that the Taylor  for four or two lattice constants, respectively, which results in a trajectory of rectangular shape. Figure 16 (b) shows the Taylor line while it switches its running mode between the (1,1) and (3,1) swimming direction (see also movie M2). A new trajectory type occurs when both the obstacle diameter 2R/λ and the lattice constant d/λ roughly agree with the wavelength (see also movie M2). In this case, after some transient regime the Taylor line is trapped and swims around a square of the same four obstacles [trapped circle mode in Fig. 16 c)] or around a single obstacle [trapped circle mode in Fig. 16 d)].

D. Variation of the length of the Taylor line
In Fig. 17 we plot the stroke efficiency S versus diagonal obstacle distance d diag for different lengths L/λ of the Taylor line. We keep wavelength and obstacle radius constant. For L/λ = 0.5 the Taylor line hardly swims persistently, neither when it is strongly confined by the obstacles (d diag /λ < 1.3) nor when it does not touch the obstacles at all (d diag /λ > 1.3). This is nicely illustrated by movie M3. For L/λ = 2 and 3 the Taylor lines first are clearly in the geometric-swimming regime along the (1, 1) direction. The strong decrease of S at around d diag /λ = 1.2 indicates the transition to swimming along the (1, 0) direction. Right at the deep mimimum of the red curve (L/λ = 2) the Taylor line gets more or less stuck before it enters the (1, 0) swimming direction. At ca. d diag /λ > 1.4 the obstacles are sufficiently apart from each other and the Taylor line does not push against them anymore.
At length L/λ = 1 and ca. d diag /λ = 1.1 a new feature occurs. The Taylor line switches between geometric swimming along (1, 1) and (1, 0) direction. This is illustrated by the two branches of the green curve in Fig. 17 and in movie M4 for d diag /λ = 1.13. In the following broad mimimum of the green curve (1.17 < d diag /λ < 1.25), the Taylor line exhibits some stick-slip motion. It first pushes frequently against one obstacle and then swims more or less continuously for one lattice constant (see movie M4 for d diag /λ = 1.2). Again, at d diag /λ > 1.4 the Taylor line does not push anymore against the obstacles while swimming.

VI. SUMMARY AND CONCLUSIONS
We have implemented an undulatory Taylor line in a Newtonian fluid using the method of multi-particle collision dynamics and a sinusoidal bending wave running along the Taylor line. We have calibrated the parameters such that its peristence length is much larger than the contour length in order to observe regular undulatory shape changes and directed swimming.
In microchannels the Taylor line swims to one channel wall. Swimming speed is enhanced due to hydrodynamic interactions and the Taylor line is oriented with an acute tilt angle at the wall similar to simulations of sperm cells 37 . The acute angle can be understood by monitoring the initated flow fields. In wide channels the tilt angle increases quadratically with the amplitude A of the bending wave, while the speed enhancement decreases exponentially with increasing A since the Taylor line swims, on average, further away from the wall. In narrow channels the swimming speed has a maximum at rougly d/A ≈ 3. The Taylor line uses the no-slip condition of the fluid at the walls to effectively push itself forward.
In a dilute obstacle lattice swimming speed is also enhanced due to hydrodynamic interactions with the obstacles. In the dense obstacle lattice we could reproduce the geometrical swimming observed in the case of C. elegans 51,52 even though we did not consider any finite extension of the Taylor line. In addition, we found more complex swimming modes, which occur due to the strong confinement between the obstacles. In the geometrical swimming regime the Taylor line strongly interacts with the obstacles and swims with a speed close to the phase velocity of the bending wave, thus much more efficiently than in a pure bulk fluid. Geometrical swimming oc-curs when the wavelength of the Taylor line fits into the lattice along one specific direction. Thus, the swimming efficiencies of various geometrical swimming modes, plotted versus the ratio d eff /λ of effective obstacle distance and undulation wavelength, all collapse on the same master curve. Increasing d eff /λ beyond one, even swimming speeds larger than the phase velocity of the bending wave occur but ultimately the Taylor line enters a different swimming mode. Thus, one can control the swimming direction of undulatory microorganisms by tuning the lattice constant of an obstacle lattice. This might be used for a microfluidic sorting device.
The concept of geometrical swimming goes back to Berg and Turner in order to explain the enhanced swimming of helical bacteria in polymer networks of viscoelastic fluids 47 . Further studies on the undulatoryTaylor line should investigate the enhanced swimming speed in more disordered obstacle suspensions and when the obstacles are allowed to move, which models more realistic environments such as blood.