Swimming in a Crystal:

We study catalytic Janus swimmers and Escherichia coli bacteria swimming in a two-dimensional colloidal crystal. The Janus swimmers orbit individual colloids and hop between colloids stochastically, with a hopping rate that varies inversely with fuel (hydrogen peroxide) concentration. At high fuel concentration, these orbits are stable for 100s of revolutions, and the orbital speed oscillates periodically as a result of hydrodynamic, and possibly also phoretic, interactions between the swimmer and the six neighbouring colloids. Motile E.~coli bacteria behave very differently in the same colloidal crystal: their circular orbits on plain glass are rectified into long, straight runs, because the bacteria are unable to turn corners inside the crystal.

Understanding non-equilibrium systems is a grand challenge cutting across many areas of current physics. An exciting frontier concerns self-propelled swimmers, from motile bacteria to synthetic, catalytic 'Janus' particles dispersed in suitable fuel [1].
Swimmers in general generate dipolar or higher-order multipolar flow fields in the surrounding fluid and the symmetry of these flow fields is known to control generic aspects of their individual and collective behavior [2,3]. Dipolar swimmers may be extensile (pushers) or contractile (pullers), depending on whether fluid is ejected or pulled in along the swimming axis. These behave quite differently; e.g., a suspension of pushers is expected to have lower viscosity than the pure fluid, whereas pullers should increase the viscosity [2].
To develop the physics of self-propelled particles, it is important to obtain data from well-characterised, simple experimental systems for comparison with theories and simulations that deal with highly-idealised swimmers. While a number of ingenious synthetic self-propelled colloids exist [4], none is yet understood and well controlled enough to have emerged as a widely-accepted model. One popular class of swimmers uses local gradients generated by heterogeneous surface chemistry to drive phoretic motion, i.e. polystyrene (PS) particles partially coated with platinum (Pt) dispersed in aqueous hydrogen peroxide (H 2 O 2 ) [5]. A decade after such 'Janus' swimmers were proposed [6], their propulsion mechanism remains controversial [7][8][9], and their flow fields unmapped. This contrasts with a model natural swimmer, motile Escherichia coli bacteria, for which both swimming mechanism [10] and flow fields [11] are known.
We present an experimental study of these Pt-PS Janus swimmers in a 2D crystal of passive colloids. The swimmers are trapped in stable orbits around individual colloids, as previously seen for Pt-Au nano-rod swimmers [12]. By studying oscillations in their orbital speed, we show that these Janus swimmers are pushers. The far-field flow around these Janus swimmers therefore resembles that of E. coli, which are also pushers. However, a 2D colloidal crystal is found to have opposite effects on these two swimmers, due to their short-range interactions with the crystal. This serves as a salutary reminder that here, as is often the case in non-equilibrium systems, details matter! We also study the dependence of orbital trapping strength on the fuel concentration ([H 2 O 2 ]). We find an inverse dependence between [H 2 O 2 ], and the rate at which Janus swimmers hop to neighbouring colloids, leading to a qualitative behavioural transition, with the crystal becoming practically impenetrable to Janus swimmers at high [H 2 O 2 ]. Interestingly, the hopping rate is independent of swimming speed, indicating a nonhydrodynamic trapping mechanism. 2 We prepared Janus particles (5 nm Pt sputtered on 2 µm diameter fluorescent PS colloids (Invitrogen) [8]) and suspended them at volume fraction ≈ 10 −6 v/v in aqueous H 2 O 2 (Acros) solutions in chambers formed between glass slides (Menzel) and 22×22 mm 2 glass coverslips (Bettering) with 300 µm thick parafilm spacers. On the coverslips, 2D colloidal crystals were formed beforehand by depositing 10 µm diameter PS colloids (Thermo-Fisher) at 1% v/v in water and evaporating at 70 • C. The radii of the static colloids, R = 5.06±0.02 µm, and Janus swimmers, a = 0.96 ± 0.04 µm, were determined by repeated (25×) measurement of the interparticle distance in close-packed 2D crystals.
Initially, with the coverslip uppermost, swimmers migrated upwards due to gravitaxis [8,13], and remained swimming stably along the coverslip. The chamber was then inverted for epifluorescence observation under an inverted microscope (Ti Eclipse, Nikon, ×20 objective) with a CCD camera (Eosens, Mikrotron). This inversion left subsequent behavior qualitatively unchanged. Initially, most swimmers are outside the crystal. At 1% H 2 O 2 , swimmers colliding with a colloid at the edge of the crystal orbit that colloid in the wedge-like space between colloid and coverslip, Fig. 1a, before hopping out of the crystal or into orbit around another colloid. Thus, Janus swimmers rapidly spread through the crystal ( Fig. 1b and Fig. 1c, permits detailed study of their behaviour. Note first that the swimming speed inside the crystal u c > u p , the speed on plain glass, so that at 10% H 2 O 2 , u c /u p = 1.7 ± 0.3. Varying [H 2 O 2 ] between 0.1% and 10%, the hopping rate decreases steadily. In 10% H 2 O 2 , the swimmers remain trapped at the edge of the crystal for many minutes (Fig. 1d).
Such oscillations could only arise via a few sufficiently long-range interactions between the swimmer and the neighboring colloids lying ≥ 1 µm from the swimmer surface: electrostatics, spatial [H 2 O 2 ] fluctuations and HI. We can exclude the first two possibilities. Adding 100 µM NaNO 3 did not modify the observed oscillations within error, whereas the Debye length reduction from ∼100 nm (in 10% H 2 O 2 [8] (Fig. 1c) Taking these oscillations as due to HI, the measured positive retardation δ immediately implies that these swimmers are pushers. A pusher approaching a neighboring colloid pushes fluid against that colloid, retarding its own motion, whereas leaving the colloid, fluid ejected backwards accelerates it, consistent with the observed -90°  positive retardation (see Fig 3a). However, this symmetry argument strictly relies on the swimmer facing along the orbital tangent. Using the shadowing effect of the Pt [8,16], we measured the orientation, Fig. 3a, away from the tangent in the horizontal (φ 0 ) and, by looking along the plane of the coverslip, vertical (τ ) planes, to be φ 0 = 7 ± 2 • , and τ = 1 ± 2 • [14], small enough for this symmetry argument to remain valid.
To obtain quantitative results, we model the swimmers as spherical squirmers with a tangential slip velocity u s on their surface [17]: where θ is the polar angle (θ = 0 at the PS pole, and flow is in the θ direction) and P n are Legendre polynomials. We take n ≤ 3 here. The bulk swimming speed u 0 = 2B 1 /3 [17]. Defining b n = B n /B 1 , the Stokeslet dipole (the lowest order flow singularity present) is extensile (pusher) if b 2 < 0, and contractile (puller) if b 2 > 0, while b 3 determines higher order flow singularities.
To estimate the HI, we make several approximations. Interactions between swimmers and each neighboring col-loid are replaced by previously derived far-field interactions with planes [18] tangent to the colloid at the point on the colloid instantaneously nearest the swimmer. We ignore secondary interactions with the colloids and glass surface, and we assume u 0 ≈ u p , the average swimming speed on the coverslip.
We also measured the mean orbital radius ρ = 4.56 ± 0.02 µm, from which we estimate the gap between the swimmer and the central colloid, g c 200 nm [14]. We use g c = g p = 70 ± 10 nm, assuming equidistance of the swimmer from plane (g p ) and colloid. Finally, we checked for, and did not observe, competing oscillations in φ 0 and ρ, which are therefore taken as constant. For full calculation details see [14]. Fig. 3c shows the speed variations calculated with single nearest-neighbor colloids (at φ = 0) for various basic swimmers. These are pushers (b 2 = −10, b 3 = 0), pullers (b 2 = 10, b 3 = 0), neutral squirmers (b 2 = b 3 = 0) and quadrupolar squirmers (b 2 = 0, b 3 = 10). The slight deviation from antisymmetry (pusher, puller) or symmetry (neutral, quadrupolar) around φ = 0 is mostly due to the finite angles, φ 0 and τ . Summing over 6 nearestneighbors gives sinusoidal oscillations (Fig. 3d), and only the pusher has retardation (δ = 14 • ) consistent with experiment. However, the best fit for the measured retardation and amplitude is obtained by combining modes: Our central result, that these Janus swimmers are pushers, should be insensitve to the approximations made, since calculating higher order HI will not alter the symmetry of the flow field components. While the precise values of b 2 , b 3 are less secure, these results still have important implications.
First, these results could help to explain why u c > u p (Fig. 1c). Simulations have shown that, for spherical squirmers, far-field approximations remain surprisingly accurate even for small surface-swimmer gaps [18]. For our swimmers, these approximations predict u c /u p = 1.2 ± 0.4 [14]. However, this prediction is not strong: the different modes make competing contributions to the speed within the crystal, so even the sign of this predicted change is uncertain, while other near-field effects, such as electroosmotic pumping [19], may also contribute to the observed speed-up within the crystal.
Second, the simplest model for Janus swimmer propulsion, self-diffusiophoresis with constant mobility, predicts B n = 0 for even n [18,20,21], inconsistent with a pusher. Instead, recently suggested self-electrophoretic propulsion mechanisms [8,9], imply a pusher, because, in this case, electric fields confined to the Pt surface propel the particle from the rear.
Third, the value of b 2 determines the far-field flow around the swimmer. The flow field around a pusher decays as a Stokeslet dipole u ∼ αr −2 , where α is the dipole strength and r the distance from the swimmer. here. Interestingly, this value is comparable to that measured for E. coli, α = 31.8 µm 3 s −1 [11].
Much of the current theory and simulation literature on active particles starts from swimmers specified on the level of their far-field flow, with the tacit assumption that this sufficiently determines their generic behavior. We explore the validity of this assumption here by repeating our experiments using motile GFP-labelled smoothswimming E. coli (strain AB1157) in motility buffer [22]. On glass, these bacteria circulate clockwise (viewed from above), Fig. 4a, due to their rotating flagella [23][24][25]. The crystal rectifies this circulation into straight trajectories, Fig. 4b.
The striking contrast with Janus swimmers can be seen by comparing their MSD, Fig. 4c. Like Janus swimmers, E. coli are initially ballistic on plain glass, with the MSD levelling off due to circular motion. Within the crystal, straight trajectories extend the ballistic regime. Hence, as indicated by the arrows in Fig. 4c, this porous environment has precisely opposite effects on E. coli and Janus swimmers, respectively increasing and decreasing their net diffusivity, despite both swimmers having similar far-field flow.
This contrasting behaviour stems from the swimmers' short-range interactions with the crystal. The behaviour of E. coli can be explained simply by steric hindrance. At ≈ 7 µm, their flagella are sufficiently long to hinder turning out of a straight channel of colloids, and their typical orbital radius (Fig. 4a) is much larger than the inter-colloid spacing, so preventing orbiting. Occasionally, bacteria do orbit individual colloids, but imaging E. coli with fluorescent flagella [14], shows that these cells typically have shorter, ≈ 3 µm flagella ( Fig. 4d and  SV2 [14]). They can therefore turn corners more easily, and should have a naturally tighter orbital radius than bacteria with longer flagella [24].
The Janus swimmers' behaviour is harder to understand. Several simulations and calculations have predicted purely hydrodynamic trapping of squirmers near surfaces [21,26,27], and in orbit round colloids [12]. One clear signature of such hydrodynamic trapping would be a direct speed dependence of the hopping rate between orbits Γ. Interestingly, the measured hopping rate of our Janus particles shows no direct speed dependence. At each [H 2 O 2 ], there is a wide variation in u c , presumably due to variation in the Pt coating. However, there is little systematic variation of Γ with u c at each single [H 2 O 2 ]: in Fig. 2d, the coloured lines, corresponding to exponential fits through the data for each [H 2 O 2 ], are all flatter than the exponential fit through the mean of each dataset (dashed, black line). Hence, trapping is strongly [H 2 O 2 ] dependent (Fig. 2c), but via some speedindependent mechanism.
This speed-independence does not entirely rule out purely hydrodynamic trapping, since trapping strength should depend on higher order modes, (B n>1 in Eq. 1) [21] which are possibly decoupled from u c . However, it indicates the need to also investigate nonhydrodynamic trapping mechanisms [28]. In particular, given the importance of electrostatics in these swimmers' propulsion [8,9], ionic currents and reaction-induced surface charging could be significant.
In conclusion, we have studied the behavior of PS-Pt Janus swimmers in 2D colloidal crystals. From observed oscillations in the speed with which they orbit individual colloids in the 2D crystal we conclude that these swimmers are pushers. This is a methodological contribution. The flow fields around biological microswimmers have previously been determined by particle-imaging velocimetry [11,29] and from the pair scattering of two swimmers [30]. We have shown that flow-field information can also be obtained from interactions with wellcharacterised porous media.
Our measurements show that the far-field flow around these Janus swimmers is similar to that around motile E. coli bacteria. Yet we find that they respond to a 2D colloidal crystal in diametrically opposite ways. This emphasises that in highly non-equilibrium systems such as active colloids, details matter, so that claims of generic behavior require thorough experimental testing.

GEOMETRICAL MEASUREMENTS
In this section, we give details of how we estimate the gap sizes and inclination angles between the surface of the swimmer, and the static colloid and glass surfaces.
As the swimmer orbits a single colloid, we wish to measure the radius ρ of its orbit, the azimuthal angle of the swimmer around its orbit φ, and the inclination φ 0 and τ of the swimmer's orientation away from the tangent to that orbit (see Fig. S1c). However, since the Janus particle has non-uniform fluorescence intensity, we cannot straightforwardly determine the centre of the particle. We instead measure equivalent parameters (ρ , φ , φ 0 ) for an ellipse fitted to a thresholded image of the swimmer at each frame, whose centre will be offset from the true centre of the swimmer by some small distance ∆c along the swimmer's orientation vector.
The expected shape of the image of the swimmer is not clear, since the Pt coating appears to only partially block out the underlying fluorescence (see supplementary video SV2). We estimate ∆c from the aspect ratio of the fitted ellipse by performing idential ellipse fitting in MATLAB on two models of the changing thresholded shape of the swimmer which have the lower half of the image either a half-ellipse or a truncated semicircle (Fig. S1a).
We plot the relationship between the difference ∆L in the fitted major and minor axis lengths, and the offset of the centroid ∆c in these two models, and use the average of these two curves to estimate the experimental value of ∆c. The radius of the Janus swimmers is a = 0.96 ± 0.04 µm, and, averaging over 17 videos, we find ∆L = 360 ± 20 nm, giving ∆c = 135 ± 30 nm, where the difference between the two curves in Fig. S1a has been taken into account in the uncertainty. To lowest order in ∆c, the corrections to ρ, φ and φ 0 are then given by: The corrections are approximately 20 nm, and 2 • respectively, and these have already been applied here and in the main text, to give ρ = 4.56 ± 0.02 µm and φ 0 = 7 ± 2 • . From the average value of the orbital radius ρ , we calculate the size of the gaps between the swimmer surface and the static colloid g c or the plane glass surface g p . The geometric construction in Fig. S1b gives the following expression for g c and g p where the radius of the static colloids, R = 5.06 ± 0.02 µm, and the averages . . . has been dropped for convenience. This single equation cannot be used to solve for both g c and g p . However, since both gap sizes must be positive, we can obtain upper bounds on each:  S1. a) Results of estimating the offset of a fitted ellipse ∆c/a from the difference in fitted axis lengths ∆L/a based on two models shown here and described in the text. b) Side view of a swimmer orbiting a colloid (not to scale) with geometrical construction to determine the size of the gaps gc and gp between the swimmer and the colloid or plane. c) Plan and side views of a Janus swimmer defining ρ, φ, φ0 and τ . φ = 0 is defined as the angle to one of the neighbouring colloids. d-e) Diagrams of the sample cell for observation along the plane of the coverslip. Observation is from below coverslip B, through the crystal and along the plane of coverslip A.
where we have ignored small terms quadratic in g p , g c . Calculating these limits gives g c < 200 nm and g p < 300 nm, taking the largest value within experimental error. If instead, we assume that g c = g p = g, Eq. S2 gives: or g = 70 ± 10 nm.
To obtain the inclination τ w.r.t. the glass plane (Fig. S1c), 10 Janus swimmers orbiting colloids in the crystal were observed along the plane of the coverslip using a custom-built sample chamber, shown in Fig. S1d-e. A colloidal crystal was formed at the edge of a 22×22 mm 2 coverslip (A), as in the main text. Coverslip A was attached with ∼ 600 µm parafilm to a glass slide previously cut down to 50 mm, so that the edge of coverslip A was flush with the long edge of the slide, with the crystal facing inwards. The slide was then glued onto a 22×50 mm 2 coverslip (B), the crystal lying next to coverslip B. Janus swimmers in 10% H 2 O 2 solution were added as usual, and viewed through coverslip B using a 100× oil immersion objective. Swimmers were recorded orbiting single colloids at the lower edge of the crystal, and images were captured with a CoolSNAP (Photometrics) camera using MicroManager [1] (see SV4). The inclination τ = 1 ± 2 • of the swimmers w.r.t. coverslip A was determined by fitting ellipses to thresholded images of the swimmers, as above.

SPEED OSCILLATIONS
We present in this section details of the approximate calculation of the hydrodynamic interaction between a swimmer following a circular orbit, and a ring of static colloids outside that orbit. We approximate the hydrodynamic interactions between the swimmer and static colloids by replacing the swimmers with point-like particles having low-order flow singularities, and replacing each colloid with a plane tangent to the colloid at the instantaneous point of closest approach between colloid and swimmer. We then make use of approximate analytical solutions for the interaction between point-like swimmers and planes [2]. This approach can be justified by noting that the typical distance between the swimmer and the nearest colloid is of order the swimmer diameter. It is therefore large enough that the far-field approximations will be at least qualitatively correct, yet much smaller than the size of the colloid, so that the planar approximation is also reasonable. We also ignore the effect of the glass slide and the central colloid, and of interactions between the flow fields arising from different neighbouring colloids, so that the interaction with multiple colloids is taken to be the sum over interactions with individual colloids.
We first specify the spatial relationship of the swimmer and the neighbouring colloids. The swimmer is located at r, with a neighbouring colloid fixed at X. The glass surface and the central colloid are not included in the calculation, since the interactions between the swimmer and these surfaces should remain constant, though we look at the effects of these surfaces later. At each moment, the swimmer's orientation is specified by unit vectorv, whilê p is the unit vector tangent to the swimmer's trajectory. In general,v =p. We define l = r − X as the displacement vector of the swimmer from the static colloid, with centre-to-centre distance l = |l|. The radius of the large colloid is R, and h = l − R is the distance between the centre of the swimmer and the surface of the neighbouring colloid.
Next we must decompose the swimmer's orientation into components perpendicular and parallel to the static colloid's surface, in order to use the expressions for interactions given in [2]. The unit vectorl = l/l points along l, while the unit vectork = (l × (v ×l))/(|v ×l|), is parallel to the colloid surface, and is in thev,l plane. The orientationv of the swimmer can then be decomposed as:v =k cos σ +l sin σ , where σ is the inclination of the swimmer away from the tangent plane to the colloid's surface. sin σ =v ·l, and we define two other angles likewise: sin ψ =p ·l, and cos χ =p ·v.
The hydrodynamic interactions between a free swimmer, moving originally at speed u 0 along directionv, and the tangent plane, would in general result in an additional swimmer velocity ∆u, which can be decomposed alongl andk: In the case, however, of a particle in an orbit, the particle velocity is constrained to lie on the tangent to the orbit, p, so the observed variation in swimmer speed will be u =p · ∆u, or: For the velocity components u l and u k , we use the farfield interaction formulae given in [2]. Translating into our coordinate system: where α, β and γ give the coefficient of the Stokeslet dipole, source dipole and Stokeslet quadrupole components of the flow field, and higher order components, decaying as h −4 and faster have been excluded. We then express these flow field components as functions of the surface flow on the squirmer. We consider a spherical squirmer of radius a with a tangential slip velocity profile given by [3]: where θ is the polar angle (θ = 0 at the PS pole, and flow is in the θ direction) and P n are Legendre polynomials. For simplicity, we consider only n ≤ 3. The bulk self propulsion speed is u 0 = 2B 1 /3 [3]. B 2 determines whether the Stokeslet dipole (the lowest order flow singularity present) is extensile (pusher, B 2 < 0) or contractile (puller, B 2 > 0), while B 3 and B 1 specify higher order flow fields. Following [4], we define b n = B n /B 1 . In [4], the connection is made between the far-field flow components, and the slip velocity components: γ = (−5b 3 /16)a 3 u 0 , and combining Eq. S7, S8, S10 then gives the predicted fractional speed variationũ 0 = u /u 0 .
We now calculate the interaction between our swimmer and a single neighbouring colloid. The plane glass surface sits on the x − y plane, with z pointing into the sample. The origin is at the point of contact between the central colloid and the plane. We take the swimmer to be a small distance g p above the glass plane, and orbiting at horizontal distance ρ from the z-axis through the centre of the central colloid (x = y = 0), and define its position r in terms of the azimuthal angle φ: r = ρ cos φ, ρ sin φ, a + g p . (S11) The neighbouring colloid is fixed at: while the tangent to the circular orbit of the swimmer is: and the orientation of the swimmer is: where φ 0 is the fixed angle between the tangent to the orbit and the orientation of the swimmer in the x-y plane, and τ is the fixed inclination of the swimmer away from the horizontal plane (Fig. S1c). This gives cos χ = cos φ 0 cos τ .
In Fig. S2a-b we plot the speed oscillation contribution from each of the flow singularities (α, β, γ), for a swimmer with a single neighbouring colloid. We take the experimental parameters: a = 0.96 µm, R = 5.06 µm, ρ = 4.56 µm and g p = 0.07 nm (so that the swimmer is equidistant from colloid and plane). In a), we set φ 0 = τ = 0, whereas in b), we use the experimental values: φ 0 = 7 • , τ = 1 • . This allows one to see the symmetry of each contribution. In a), the pusher-puller mode (α) is the only mode which is odd about the origin, and therefore the only mode which could result in the close to 15 • retardation seen experimentally. Taking into account the non-zero inclination of the swimmer to the tangent (Fig. S2b), these modes lose their symmetry slightly.
The total squirmer flow field always contains a fixed B 1 contribution, and variable contributions from the two other modes, B 2 and B 3 according to Eq. S10. In Fig. S2c, we plot the fractional speed variation produced by four basic swimmers. These are pushers (b 2 = −10, b 3 = 0), pullers (b 2 = 10, b 3 = 0), neutral squirmers (b 2 = b 3 = 0) and quadrupolar squirmers (b 2 = 0, b 3 = 10). The pusher accelerates as it passes closest to the neighbouring colloid, at φ = 0, whereas the puller slows down. As discussed in the main text, this is as expected, considering the extensile and contractile flow fields around the two swimmers. The neutral and quadrupolar swimmers, on the other hand, show almost symmetrical behaviour around φ = 0, in line with the symmetry of the source dipole and Stokeslet quadrupole (β, γ) flow components, which are the only components contributing to the flow fields of these swimmers.
To calculate the interaction between the swimmer and multiple spheres, we add the contributions from single spheres located at 60 • intervals around the central axis. In Fig. S2d, we plot the speed oscillations produced by six neighbouring colloids for the same set of swimmers. All of the oscillations are well described by sinusoids with period 60 • , but the retardation and amplitude of the oscillation depends on the type of swimmer: the pusher matches the experimentally observed retardation of ∼ 15 • , whereas the puller has a retardation of ≈ −15 • , and the other swimmers retardation close to 0 • . Fig. S2ef show the retardation and fractional amplitude variation of this 60 • period oscillation as a function of b 2 and b 3 .
The fractional amplitude calculated here isũ 0 = u /u 0 , whereas experimentally, we measureũ = u /u g where u g is the average swimming speed in the crystal. Assuming u 0 = u p , the swimming speed on plain glass,we requireũ 0 = 13 ± 2% to match experimental values (using u g /u p = 1.7 ± 0.3 andũ = 7.7 ± 0.5%). The best fit for the experimentally observed amplitude and retardation is then b 2 = −15 ± 7, b 3 = −8 ± 8, where the uncertainty is estimated from the experimental uncertainty in each input parameter.
Finally, though the far-field approximations made here cannot generally be applied to swimmers very close to surfaces, it has been found in simulations [2] that for spherical squirmers, the parallel speeds calculated from far-field approximations remain surprisingly accurate for gap sizes down to ∼ 10% of the colloid radius. In this case, we can estimate the overall speed of the swimmer within the crystal u c and on plain glass u p : where these speeds have been projected along the direction of travel, either in orbit around the central colloid situated at X = 0, 0, R , or parallel to the glass surface, and where u c and u p are the parallel (u k ) components of ∆u calculated from Eq. S8: u c = u 0 cos φ 0 cos τ 128 a a + g c 2 −72b 2 sin σ + a a + g c 4 7 − 27 sin 2 σ + 3b 3 1 + 9 sin 2 σ , u p = u 0 cos τ 128 a a + g p 2 −72b 2 sin τ + a a + g p 4 7 − 27 sin 2 τ + 3b 3 1 + 9 sin 2 τ , where sin σ = ρ sin φ 0 cos τ − (R − a − g p ) sin τ R + a + g c , (S17) and where u n is the constant speed variation arising from the 6 neighbouring colloids (the speed offsets in Fig. S2d).
Qualitatively, the speed increase inside the crystal comes from the extensile flow (b 2 < 0) pushing away from the colloid surface, and since φ 0 > 0, driving the swimmer forwards. However, the negative b 3 slightly retards the motion, as does the interaction with the neighbouring colloids (u n ≈ −0.3u 0 ).

FLAGELLA STAINED E. COLI
Construction of the smooth swimming E. coli strain AB1157 cheY has been described previously [5]. For the current work, the strain was further modified by replacement of the chromosomal copy of the fliC gene with a modified copy encoding a mutant FliC protein in which the serine amino acid at position 353 is replaced with a cysteine amino acid. Strain HCB1668 is a Tn5 fliC null derivative of AW405 in which FliC S353C is expressed from the plasmid pBAD33 [6]. This plasmid was used as a template to amplify 803 bp of fliC by PCR. This encompassed the AGT to TGC mutation which was flanked on each side by 400 bp of the fliC gene. The primers used for amplification were GCAACTCGAGCAATTGAGGGTGTTTATACTGA and GCAAGTCGACCCTGGTTAGCTTTTGCCAACA. Restriction sites for XhoI and SalI were included. The PCR product was purified, digested with XhoI and SalI and ligated into the plasmid pTOF24, which had been digested with the same enzymes. The resultant recombinant plasmid pTOF24 fliC was transformed into AB1157 cheY and used to replace the wild type fliC allele with the fliC mutation by plasmid mediated gene replacement using the method of [7]. Correct insertion of the mutation was verified by sequencing.
The resultant strain AB1157 cheY pHC60 FliC S353C was grown from a single colony in 10 ml Luria-Bertani broth containing 30 µgml −1 kanamycin and 5 µgml −1 tetracycline overnight at 30 C and 200 rpm. Bacteria were diluted 1:100 into 35 ml tryptone broth containing antibiotics as above and grown for further 4 h. Next, three washes were performed using phosphate motility buffer (6.2 mM K 2 HPO 4 , 3.8 mM KH 2 PO 4 , 67 mM NaCl, 0.1 mM EDTA at pH 7.0) and cells concentrated to a total volume of ∼3 ml. To perform flagella labelling the protocol of [6] was followed. Briefly, 10 µl of Alexa Flour 633 C5 maleimide (1 mgml −1 in dimethyl sulfoxide, Molecular Probes) was added to 1 ml of washed bacteria and incubated at room temperature and 100 rpm for 60 min. Three washes were performed as described above and final density was adjusted to optical density 0.3 at 600 nm in motility buffer containing 0.002 wt% TWEEN 20.
The flagella-labelled E. coli were observed in the sample chambers described in the main text, using confocal microscopy with 488 nm and 633 nm laser excitation on a Zeiss Confocal Microscope at 4 fps. We added 0.2 wt% TWEEN 20 to minimise adhesion of bacteria to the glass.