Enhanced motility of a microswimmer in rigid and elastic confinement
aa r X i v : . [ c ond - m a t . s o f t ] S e p Enhanced motility of a microswimmer in rigid and elastic confinement
Rodrigo Ledesma-Aguilar ∗ and Julia M. Yeomans The Rudolf Peierls Centre for Theoretical Physics,University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom (Dated: November 10, 2018)We analyse the effect of confining rigid and elastic boundaries on the motility of a model dipolarmicroswimmer. Flexible boundaries are deformed by the velocity field of the swimmer in such away that the motility of both extensile and contractile swimmers is enhanced. The magnitude ofthe increase in swimming velocity is controlled by the ratio of the swimmer-advection and elastictimescales, and the dipole moment of the swimmer. We explain our results by considering swimmingbetween inclined rigid boundaries.
Introduction:–
Confinement in rigid and elastic envi-ronments is a key concept affecting fluid transport andlocomotion in microscopic systems, ranging from molec-ular motors to single and multicellular self-propelled or-ganisms [1]. The low-Reynolds number world of micro-swimmers is often crowded by passive and active, perme-able and impermeable, boundaries, such as viscoelasticgels, microtubules or cell walls. While these can act asbarriers or defence mechanisms against microorganisms,it has been suggested that they can be exploited by theswimmers to enhance their motility [1].Not surprisingly, the locomotion of pathogens close toflexible and rigid surfaces can play a key role in theirsuccess or failure as infectious agents. Motile pathogenicbacteria and parasites move through the extracellularmatrix of the host to invade the vascular system [2] and,in the case of neurodegenerative infections, even crossthe blood-brain barrier [3]. For example, pathogenicsphirochaetes responsible for syphilis [4], leptospirosis [5]and Lyme disease [6], have been reported to swim inmicrovasculature channels [2], renal tubules [5], and toinvade the intracellular junctions between endothelialcells [7]. Defensive mechanisms against infections also in-volve close interaction between complex elastic surfacesand pathogens. Flexible-surfaced leukocytes prey onmotile microorganisms [8], such as
E. coli and
P. aerugi-nosa , and the viscoelastic gel lining the stomach acts as abarrier against
H. pylori [9], a bacterium linked to chronicgastritis and stomach cancer. Understanding the wayin which microswimmers behave in confinement can alsolead to novel applications in microfluidics and biotechnol-ogy, ranging from bacterial rectification [10, 11] to con-trolled steering of artificial micro-robots [12, 13].Motivated by these observations, in this Letter we for-mulate the following questions: what is the effect of theactivity of a microscopic swimmer on a bounding elasticsurface and, what is the effect of the interaction with thesurface on the motility of the swimmer? While previousefforts have mainly focused on swimming strategies thatexploit rigid and elastic surfaces to overcome the scal-lop theorem [14–16], the effect of confinement by rigidand flexible boundaries on the motility of microswim- mers remains largely unexplored [17–19]. By providinga microscopic footing, single-particle interactions can beused to build up coherent long-wavelength hydrodynamicmodels of active matter subject to viscoelastic hetero-geneity [20, 21] and confinement [22–25] to model, e.g., peristaltic pumping and cytoplasmic streaming [26, 27].We begin our discussion by presenting 3D hydrody-namic simulations of a single microswimmer movingalong the centre line of an elastic tube [Figs. 1(a) and (b)].The surface of the tube is a rectangular mesh of points,coupled by stretching and bending elastic springs which,in equilibrium, form a cylinder. By choosing the meshsize and elastic coupling between points on the surfacewe can model both impermeable and permeable tubes, aswell as sets of uncoupled or cross-linked filaments whichmimic gel-like environments [see Supplementary Infor- -3 SpeedPowerEfficiency 0.911.11.21.310 -3 SpeedPowerEfficiency (c) (d)(a) (b)Extensile Contractile
Free swimming Free swimming
PSfrag replacements τ s /τ f τ s /τ f FIG. 1. (Colour online) (a) and (b) Simulations of dipolarextensile (pusher) and contractile (puller) swimmers passingthrough an elastic tube. The deformation of the channel, pro-truding on average towards pushers and away from pullers, isset by the competition between the swimmer velocity field(projected planes) and the elastic response of the boundary.(c) and (d) The increased friction due to the solid bound-aries leads to a change in the swimmer velocity, the powerconsumed and the swimming efficiency, relative to the free-swimming case. The amplitude of the response in motilitydepends on the ratio between swimmer and elastic timescales, τ s /τ f . mation for more details of the model]. The swimmer iscomposed of two beads of variable hydrodynamic frictioncoefficient coupled by an elastic spring whose rest lengthoscillates in time according to a prescribed swimmingstroke [28]. The resulting force-free swimmer acts on thefluid as a force dipole of strength p and has an advectiontimescale τ s that depends on the swimming stroke. Suchgeneric dipolar swimmers [1] can be ‘pushers’ or ‘pullers’,depending on their flow pattern [projections in Figs. 1(a)and (b)]. For pushers this is extensile (fluid is pushed outfrom the ends of the swimmer and drawn in to the sides)while for pullers it is contractile (fluid is pushed out fromthe sides and pulled in to the ends). The shape of theelastic boundary is set by the competition between theactivity of the swimmer, which tends to deform the tubeand is characterised by τ s and p , and the elastic responseof the surface, which resists such deformations and is con-trolled by the effective relaxation timescale τ f . Note thedifferent shape deformations for pushers and pullers.We characterise the effect on swimmer motility by mea-suring the ratio of the swimmer speed, v , to the freeswimming value, v s , resulting from the interaction be-tween the swimmer and the tube [Figs. 1(c) and (d)].Both kinds of swimmers tend to move faster through flex-ible channels–characterised by small ratios of the swim-ming and elastic time scales, τ s / τ f . Even though thepower consumed by the swimmer P increases with con-finement, the overall swimming efficiency ǫ ∼ v /P [29]increases for flexible boundaries ( τ s /τ f < shift and tilt deformations to the boundary.We show that shift deformations (towards the swimmer)always increase the swimmer speed by an amount that isproportional to the self-swimming speed and independentof the strength or direction of the velocity field created bythe swimmer. Conversely, tilt deformations, correspond-ing to a local inclination of the boundary, couple to theparticular swimming stroke and can result in enhancedor reduced motility depending on the swimming pattern.Using these results we argue that the hydrodynamic cou-pling between the swimmer and a flexible boundary al-ways leads to channel deformations which favour the pas-sage of the swimmer. Conversely, we find that the powerconsumed depends on the average distance to the walls,increasing with wall proximity, and not on their inclina-tion. As a result, the swimmer speed can increase dueto the local deformation of the walls with the power re-maining relatively constant, leading to a larger swimmingefficiency [Figs. 1(c) and (d)]. Model swimmer:–
Locomotion at low Reynolds num-ber relies on swimming strokes that are non-reciprocalin time. Many microorganisms achieve this by deform- ing their shape in such a manner that the local dragcoefficient varies along their bodies. We coarse grainsuch a feature by considering a model swimmer madeup of anterior and posterior spheres, a and p respec-tively, joined by a link of prescribed but variable length l [28] [see Fig. 2(a)]. The local drag on each spherefollows from the Stokes Law, F i = ξ i v i , where v i isthe velocity of the i -th sphere and ξ i its friction coef-ficient. The swimmer is subject to the force-free con-dition, F p + F a = 0, and to the kinematic constraintdictated by the swimming stroke, v a − v p = ˙ l , where thedot implies differentiation with respect to time, t . Asa consequence, the instantaneous speed of the swimmerobeys v ≡ ( v p + v a ) / l ( ξ p − ξ a ) / ξ p + ξ a ) . For free, unbounded, swimmers locomotion can onlybe achieved if the friction coefficients vary in time, intro-ducing a non-reciprocal deformation of the body of theswimmer. This can be readily verified by considering theStokes drag coefficient ξ i = 6 πηa i , where η is the vis-cosity of the fluid and a i is the instantaneous radius ofthe i -th sphere. Without loss of generality, we can set l = l + δl sin( ωt ), a p = a + δa sin( ωt + ∆ ψ + δψ ), and a a = a + δa sin( ωt +∆ ψ − δψ ) to obtain the average speedof the swimmer over one stroke, v s ≡ R π/ω dtv/ (2 π/ω ) = δlωδa sin δψ cos ∆ ψ/ a + O ( δa/a ) [30].The instantaneous forces exerted by the swimmer onthe fluid can be averaged over a stroke to obtain thenet flow induced by the swimmer. Due to the force-free condition the far-field velocity field is dipolar, h u i = p (3 cos φ − ˆr / πηr , where r ≡ r ˆr is the displacementvector from the position of the centre of mass of the swim-mer (also stroke-averaged) [31] to a point in the fluid,and φ measures the angle from the axis of the swim-mer to r in the counter-clockwise direction. The dipolestrength p = 3 πηδlωδal cos δψ sin ∆ ψ/ ψ and δψ , determines whether theflow pattern is extensile ( p >
0) or contractile ( p <
Rigid boundaries:–
We first focus on the interactionbetween the model swimmer and a bounding solid sur-face. We consider the situation where the swimmer andthe surface are aligned along a single axis of revolu-tion, as shown in Fig. 2(a). For such configurationsthe drag coefficient increases, and can be written as ξ i = 6 πηa i / (1 + ζ [ a i /h i ]) , where ζ ( x ) < ζ is a decreasing function of a i /h i , andits functional dependence is specific to the geometry ofthe surface. The interaction with the solid will lead to anet contribution to the swimming speed, v w . For planarwalls ( h i = h ) and prescribed deformations, the swim-mer ‘grips’ the surface, exploiting the higher resistanceoffered by the walls. Expanding v w for small swimmerdeformations we find v w = − v s (cid:18) a h (cid:19) ζ ′ ζ , (1)where ζ ≡ ζ [ a /h ], etc. The scaling of Eq. (1) indi-cates that the swimmer always moves faster in parallelconfinement ( ζ ′ < h i donot vary along the swimmer trajectory. This, however,changes if the walls are inclined relative to the swim-mer path. As illustration, consider small wall inclina-tions such that h i = h + δ i . This leads to an additionalcontribution to the speed of the swimmer. Expandingin powers of the δ i this contribution reads v i = v s (cid:18) a h (cid:19) (cid:18) ∆ + ∆ tan ∆ ψ tan δψ (cid:19) Z + O (∆ i ∆ j ) . (2)The first term, which reflects the net displacement of theposition of the surface from the reference value, h , iscontrolled by the shift , ∆ ≡ ( δ p + δ a ) / h . The sec-ond term results from the local inclination of the surface,measured by the tilt , ∆ ≡ ( δ p − δ a ) / h . Both termsscale with the function Z ≡ Z[ a /h ] <
0, which en-codes the strength of the resistance provided by the wallthrough Z ≡ (cid:2) ζ ′ − ( a /h ) (cid:0) ζ ′ / (1 + ζ ) − ζ ′′ (cid:1)(cid:3) / (1 + ζ ) . While the first term in Eq. (2) can be absorbed as asimple offset of h in Eq. (1), the ∆ -term introduces aninterplay between the local structure of the wall and thedetails of the swimming stroke, as indicated by the phasedependence. Both kinds of swimmers achieve propulsionby pushing on their posterior end during the expansionof the link ( a p > a a ), and by pulling on their anteriorend during contraction ( a a > a p ). For pushers the ‘powerstep’ corresponds to the extension step, where the sphereshave relatively large radii [Fig. 2(b)]. Diverging constric-tions, ∆ <
0, amplify the strength of this step, leadingto an additional speed-up. For converging constrictions,however, the power step is weakened by an amount con-trolled by ∆ , which can lead to a net slow-down. Simi-larly, contractile swimmers generally swim faster in con-verging constrictions [Fig. 2(c)], relative to the planarwall reference configuration.The interplay between wall proximity and wall incli-nation can lead to both speed-ups and slow-downs forthe swimmer [see contour plots in Figs. 2(d) and 2(e)].Note that v i can be of the same order of magnitudeas v s . To obtain these plots we have set Z accord-ing to Fax´en’s correction to Stokes Law [32], whereby ζ ( x ) = Ax + Bx + Cx , with A = − . B = 0 . C = − . . This particular friction law is valid (b) (c)(d) (e) -- - - - -- - - - (a) p a PSfrag replacements l h v p v p v a v a v s v s h a h p ∆ ∆ ∆ ∆ FIG. 2. (Colour online) Swimming in confinement. (a) Dipo-lar swimmer composed of two spheres, a and p, in the middleof planar, inclined walls. (b) Extensile swimmers push onthe posterior end of their bodies (black arrows), and experi-ence a speed-up when in diverging channels, light red arrow.(c) Conversely, contractile swimmers pull on their anteriorend during the power step of their stroke (black arrows), andtheir speed is increased in converging channels (light greenarrow). (d) and (e): Contour plots of the normalised con-tribution to the speed of pushers and pullers due to inclinedboundaries, v i /v s , as a function of the average displacement,∆ , and net amplitude, ∆ , of the channel. The dotted pathscorrespond to the variation of ∆ and ∆ along the top curvesin Figs. 3(c) and 3(d). for a pair parallel plates, relevant for swimmers confinedin microfluidic chambers, and is expected to hold for ourtreatment in the limit of gently, locally, inclined surfaces.The additional drag induced by the walls causes anet increase in the average rate of energy consumedby the swimmer over a cycle, P = R π/ω dt ( F a v a + F p v p ) / (2 π/ω ). For parallel boundaries we find P w ≈ P s / (1 + ζ ), larger than the free-swimming value P s by afactor increasing for stronger confinement. However, fortapered channels the leading order contributions to thetotal power consumption depend only on the boundaryshift, ∆ , i.e. , P = P w (cid:18) (cid:18) a h (cid:19) ζ ′ ζ (cid:19) + O (∆ i ∆ j ) , (3)and are unaffected by the inclination of the walls. Asa consequence, the speed of the swimmer can vary dueto the kinematic coupling between the swimming strokeand the orientation of the boundaries at constant powerconsumption. Elastic boundaries:–
We now test the applicability ofthe shift-tilt theory to understand the enhanced motil-ity of swimmers observed in our simulations of flexi-ble channels. Qualitatively, pushers deform the bound-ary to create a locally diverging channel, while contrac-tile swimmers induce a local converging geometry [seeFigs. 1(a) and (b)]. These configurations lead to the ex- -0.0200.020.04 -(cid:7) -5 -4 -3 -0.0200.020.04 (cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15) (cid:16)(cid:17) -5 -4 -3 (a) (b) -2-1012-3 -2 -1 0 1 2 3 -2-1012-3 -2 -1 0 1 2 3 ( c(cid:18) (d) PSfrag replacements v f /v s x/l v f / v s v f / v s y/l τ s /τ f τ s /τ f δ /h = 1 . δ /h = 1 . δ /h = 0 . δ /h = 0 . δ /h = 0 . δ /h = 0 . y / h y / h x/h x/h δ p /h δ a /h λ/h v s v s FIG. 3. (Colour online) Typical deformations of an elasticboundary caused by (a) pushers and (b) pullers. Due to theflow-field pattern, shown as light coloured arrows, the inter-face is deformed to a locally diverging or converging shapefor pushers and pullers, respectively. The local deformationof the filaments is quantified by interpolating δ p and δ a atthe x − coordinate of the respective beads. (c) and (d) Cor-responding change in speed, v f = v − v s , relative to the free-swimming speed. The amplitude of the response is controlledby the shape of the boundary, which is set by the ratio ofswimmer and filament timescales, τ s /τ f , and by the swimmerdipolar strength, p ( δ ∼ p ) . pected speed-up, according to our arguments. Two maineffects set the deformation of the filaments, and in turnthe back flow v f ≡ v − v s . On the one hand, the filamentis deformed by the velocity field of the passing swim-mer, of typical magnitude v ∼ p/ηh , over the swimmerself-propagation timescale τ s = l /v s . On the other, theelastic resistance of the filament can be quantified bythe bending relaxation timescale τ f ∼ ηa f λ /G, where λ ∼ h is the wavelength of the perturbation, G ≈ k b r eq is the bending modulus and r eq is the rest length betweenpoints in the filament. The growth rate of perturbationsto the boundary thus obeys, ˙ δ ≈ v − δ/τ f , which sug-gests a scaling δ ∼ δ [1 − exp( − τ s /τ f )] / ( τ s /τ f ) , where δ ≡ v τ s is the typical amplitude for freely-deformablechains. To explore the interplay between these effects indetail, we ran simulations considering a pair of initiallyparallel filaments [Figs. 3(a) and (b)] and explored a widerange in τ s /τ f . This simple configuration only reduces themagnitude of the back flow with respect to the many-filament simulations, and does not alter results qualita-tively. We focus on the dependence of v f on τ s /τ f forpushers and pullers of identical free-swimming speed v s but with different dipole strength, p [Figs. 3(c) and (d)] .As suggested by the simple scaling argument, curves tendto reach the rigid-boundary limit at τ s /τ f ≈
1, when therigidity of the filaments suppresses deformations. Addi-tionally, the amplitude of the curves scales with δ , as ex-pected for swimmers with stronger dipole moments thatcan induce larger deformations to the boundary.The shape of the curves shown in Fig. 3 is set by the subtle interplay between the local inclination and shiftinduced by the swimmer on the elastic boundary. Toillustrate this point, we measured the local amplitudes δ p and δ a [see Fig. 3(a)] for the top curve in Figs. 3(c)and (d). We then calculated the corresponding ∆ and∆ values, which we depict as paths in (∆ , ∆ ) − spacein Figs. 2(d) and 2(e). Extensile swimmers tend to pullon the boundary, shifting it to closer positions and tohigher inclinations simultaneously. Both effects decreasewith increasing filament rigidity, leading to a trajectoryin (∆ , ∆ ) − space that runs monotonically from negativevalues of both parameters to the origin, and is reflected inthe smooth decrease of v f with τ s /τ f shown in Fig. 3(c).Contractile swimmers, on the contrary, strongly deformthe boundary, creating high local inclinations, but onlyshift its position weakly. For intermediate rigidity of thefilament, while the inclinations die out, the boundary isshifted to a closer position to the swimmer. This con-tributes to speeding it up. However, for higher rigid-ity the shift decays and this effect vanishes. This com-petition leads to a turning trajectory in (∆ , ∆ )-spacewhich shows a good correlation with the non-monotonicbehaviour observed for v f − the maximum of the top curveof Fig. 3(d) corresponds to the right-most point of thepath in (∆ , ∆ )-space (red dot) in Fig. 2(e).We have described the hydrodynamic coupling betweenthe self-propulsion of microscopic swimmers and the re-sistance offered by elastic confining surfaces. Both push-ers and pullers deform the surfaces in such a way that en-hances their motility. A similar microscopic mechanismis likely to underlie the enhanced motility observed in vis-coelastic fluids [1, 33]. We have also shown that locallyinclined constrictions can enhance or hinder the motilityof swimmers depending on their swimming stroke. Thisasymmetry has potential as a means of separation or se-lective permeation of microorganisms. We hope that ourresearch will motivate experiments to explore these pos-sibilities.We thank Henry Shum and Mitya Pushkin for fruitfuldiscussions. R.L.-A. acknowledges support from MarieCurie Actions (FP7- PEOPLE-IEF-2010 no. 273406),and JMY from the ERC Advanced Grant (MiCE). ∗ [email protected][1] E. Lauga and T.R. Powers. Rep. Prog. Phys. , 72:096601,2009.[2] T.J. Moriarty, M.U. Norman, P. Colarusso, T. Bankhead,P. Kubes, and G. Chaconas.
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