Physical Sensing of Surface Properties by Microswimmers – Directing Bacterial Motion via Wall Slip

Bacteria such as Escherichia coli swim along circular trajectories adjacent to surfaces. Thereby, the orientation (clockwise, counterclockwise) and the curvature depend on the surface properties. We employ mesoscale hydrodynamic simulations of a mechano-elastic model of E. coli, with a spherocylindrical body propelled by a bundle of rotating helical flagella, to study quantitatively the curvature of the appearing circular trajectories. We demonstrate that the cell is sensitive to nanoscale changes in the surface slip length. The results are employed to propose a novel approach to directing bacterial motion on striped surfaces with different slip lengths, which implies a transformation of the circular motion into a snaking motion along the stripe boundaries. The feasibility of this approach is demonstrated by a simulation of active Brownian rods, which also reveals a dependence of directional motion on the stripe width.

, Fp and h eff shown in Table S1. We find that the force for a no-slip surface F 0 x = F max s + Fp > 0 and the force for a perfect-slip surface F ∞ x = Fp < 0. The effective width h eff from the linear approximation of the fluid velocity profile in the gap is comparable, but not equal to the gap width h. The comparison here confirms that Eq. (1) describes the hydrodynamic interaction of rotating spherical and ellipsoidal bodies with surfaces over a wide range of slip lengths very well. The slip length b0, at which the force Fx vanishes, is also shown in Table S1. For a sphere of diameter d = 0.9 µm, one obtains b0 ≈ 65 nm at h = 0.02265 d = 20 nm.
Mesoscale hydrodynamics simulations.-We briefly describe the hybrid simulation method here; for details, we refer to Refs. 3,4. The model E. coli has a spherocylindrical body and four left-handed helical filaments and is constructed by particles of mass M . We choose the body length b = 2 − 4 µm, body diameter d = 0.9 µm, flagellar helix radius 0.2 µm, pitch 2.2 µm and angle 30 • from experiments. 5,6 The elastic bending and twist moduli of the filaments are chosen according to the experimental range from about 10 −24 to 10 −21 N m 2 . [7][8][9] The details of the model will be published elsewhere.
The solvent is modelled by a collection of point-like particles of mass m. Their dynamics comprises of alternating streaming and collision steps. In the streaming step, the solvent particles move ballistically and the position rs of particle s with velocity vs is updated according to rs(t + ∆t) = rs(t) + vs(t) ∆t with ∆t the time interval between collisions, while the dynamics of the body and flagellar particles is described by the Newton's equations of motion. In the collision step, all particles are sorted into cubic cells of length a and the velocity vi of particle i in cell c is renewed via the collision rule 10,11 v new where vc and rc are the velocity and position of the center of mass of all particles in the cell c, mj the mass of particle j in c, v ran i a random velocity sampled from the Maxwell-Boltzmann distribution, and I the moment-of-inertia tensor of all particles in c. The collision rule (i) conserves both linear and angular momentum locally, i.e. in each collision cell, (ii) includes thermal fluctuations of the solvent, and (iii) maintains a constant temperature. To satisfy Galilean invariance, a random grid shift of the cells is performed before each collision step. 12 Length and time scales.-By mapping the cell-body width d = 9 a to 0.9 µm for swimming E. coli 6 , we obtain the length scale a = 100 nm, which sets the resolution of hydrodynamics in our simulation method. Comparison of the flagellar rotation rate ω f = 0.0356 kBT /ma 2 in our model to the experimental value of 2π × 120 Hz 6,13 leads to the time scale ∆t = 2.4 µs.
Simulation setup.-Our simulations are performed in cubic boxes of length L = 120 a with periodic boundaries in the xand y-directions and two planar surfaces implemented at z = 0 and z = L. Additional simulations with L = 150 a are run and the obtained average curvature is consistent with that from L = 120 a within numerical errors, indicating the periodic boundaries do not affect our simulation results. We choose the collision time step ∆t = 0.05 ma 2 /kBT and the solvent density ρ = 10 m/a 3 , leading to the solvent viscosity η = 7.15 √ mkBT /a 2 and the Schmidt number Sc = 20, for which momentum transport dominates over mass transport. Newton's equations of motion for the particles of the E. coli model are integrated with a time step δt = ∆t/25 using the velocity-Verlet algorithm.
Slip length of the surface.-We obtain partial-slip surfaces with different slip lengths by randomly mixing no-slip and perfect-slip boundary conditions. Figure S2 shows the slip length b as a function of the mixing ratio p, defined as the probability of applying no-slip boundary condition at each collision step (1 − p for perfect-slip). b is measured from the velocity gradient of the fluid under shear.  Table S1. Insets in (a)-(c) are close-up of the first four points. See Fig. 3(b) for the notations.  Table S1: Properties of a particle rotating parallel to a nearby surface, as obtained from the fits in Fig. S1. F max s and F p are rescaled by 3πηd 2 ω/2. Movie S2: Simulation animation of E. coli swimming near a no-slip surface. The circular trajectory with clockwise motion is viewed from above the surface, compare also Fig. 2(a). The geometry of the bacterium is the same as in Fig. 1(a) and Movie S1.
Movie S3: Simulation animation of active Brownian rod swimming near a striped surface. The trailing path in color (red to blue) represents the swimming trajectory for the past 40 seconds. R − and R + are the radii of curvature for the clockwise and counterclockwise trajectories on the alternating stripes with width L, respectively. The geometry, swimming velocity and diffusion coefficients of the active rod are in agreement with experimental values of E. coli.