Chemotaxis of artificial microswimmers in active density waves
Alexander Geiseler, Peter Hänggi, Fabio Marchesoni, Colm Mulhern, Sergey Savel'ev
CChemotaxis of Artificial Microswimmers in Active Density Waves
Alexander Geiseler, ∗ Peter H¨anggi,
1, 2, 3
Fabio Marchesoni,
4, 5
Colm Mulhern, and Sergey Savel’ev Institut f¨ur Physik, University of Augsburg, D-86159, Germany Nanosystems Initiative Munich, Schellingstraße 4, D-80799 M¨unchen, Germany Department of Physics, National University of Singapore, 117551 Singapore, Republic of Singapore Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering,Tongji University, Shanghai 200092, People’s Republic of China Dipartimento di Fisica, Universit`a di Camerino, I-62032 Camerino, Italy Department of Physics, Loughborough University, Loughborough, LE11 3TU, United Kingdom (Dated: October 14, 2018)Living microorganisms are capable of a tactic response to external stimuli by swimming towardsor away from the stimulus source; they do so by adapting their tactic signal transduction pathwaysto the environment. Their self-motility thus allows them to swim against a traveling tactic wave,whereas a simple fore-rear asymmetry argument would suggest the opposite. Their biomimeticcounterpart, the artificial microswimmers, also propel themselves by harvesting kinetic energy froman active medium, but, in contrast, lack the adaptive capacity. Here we investigate the transport ofartificial swimmers subject to traveling active waves and show, by means of analytical and numericalmethods, that self-propelled particles can actually diffuse in either direction with respect to the wave,depending on its speed and waveform. Moreover, chiral swimmers, which move along spiralingtrajectories, may diffuse preferably in a direction perpendicular to the active wave. Such a varietyof tactic responses is explained by the modulation of the swimmer’s diffusion inside traveling activepulses.
I. INTRODUCTION
Taxis is the biased movement of bacteria, somatic cells,or multicellular organisms in response to external stim-uli such as light, electric currents, gravity and chemi-cals. Taxes are classified based on the type of activatingstimulus and on whether the organism’s resultant drift isoriented towards (positive) or away from (negative) thestimulus source [1]. Tactic cell migration plays a crucialrole in biological pattern formation and the organizationof complex biological organisms. For instance, bacteriafind food (e.g., glucose) or flee from poison (e.g., phe-nol) by swimming respectively up (positive chemotaxis)or down (negative chemotaxis) the concentration gradi-ent of the sensed chemical [2, 3].A biomimetic counterpart of cellular motility is theability of specially designed synthetic microparticles topropel themselves by harvesting kinetic energy froman active environment [4, 5]. Self-propulsion is fueledby stationary non-equilibrium processes, like directional“power-strokes” from catalytic chemical reactions or self-phoresis by short-scale (electric, thermal, or chemical)gradients, produced by the particle itself, in virtue ofsome built-in functional asymmetry [4–6]. Similarly tobacteria, artificial microswimmers are also known to dif-fuse up or down long-scale monotonic gradients of theactive medium [7–10]. However, bacteria regulate theirresponse to an external stimulus by adapting their (com-plex) tactic signal transduction pathways [11], which thusoperate like sensor-actuator loops. In contrast, due to ∗ Corresponding author:[email protected] their often sub-micron size and lack of an internal struc-ture, synthetic swimmers are unable to process the tacticsignal [12, 13], i.e. their response being strictly local.In most circumstances the tactic stimulus is modu-lated in space and time in the form of single or en-trained active pulses that sweep through the suspendedswimmer. Certain microorganisms are able to locate thesource of the pulses and move towards it, no matter whatthe sign of their response to a monotonic active gradi-ent. A study case is the chemotactic aggregation of theamoebae of the cellular slime mold (Dictyostelium dis-coideum) [14]. This process is directed by periodic se-quences (or waves) of symmetric concentration pulses ofa chemoattractant, irradiating from the aggregation cen-ter outwards. As amoebae exhibit positive chemotaxis,one would expect a cell movement towards the centerin the wave-front and away from it in the wave-back.As a consequence, amoebae would spend more time inthe back-wave than in the front-wave, thus undergoinga net drift in the direction of the active wave propaga-tion. Most remarkably, the conclusion of this argument(first proposed by Stokes in a more general fluidodynam-ics context [15, 16]) does not change by reversing thesign of amoeba chemotaxis: According to Stokes’ wavefore-rear asymmetry argument and contrary to experi-mental observations, swimmers with definite chemotaxis(positive or negative, alike) would always surf active den-sity waves (ADW), thus moving away from the the wavesource [17]. A proposed resolution of this apparent para-dox (termed “chemotactic wave paradox” in Ref. [18])for cellular microorganisms requires the standard modelfor cell chemotaxis to be modified so as to account fora finite adaptation time of the chemotactic pathways totemporally varying stimuli [18]. a r X i v : . [ c ond - m a t . s o f t ] J un FIG. 1. (a) Ideal experimental setup of a thermophoretic swimmer diffusing on a planar substrate irradiated by a laser beam[8, 25]. By pulling at constant speed a slit-screen sliding between the laser and the substrate, it is possible to modulate thelaser intensity, I , hitting the particle, thus realizing an effective ADW. (b),(c) Chemotactic shift induced by a Gaussian activepulse, v ( x ) = v exp( − x / σ ), traveling to the right with constant speed u (see inset of (c), where a simple sketch of themodel is depicted). The swimmer’s self-propulsion parameters, v = 53 µ m / s and D φ = 165 s − , were chosen to mimic theexperimental setup of Ref. [25], and the pulse width, σ = 1 µ m, was set three times the swimmer’s propulsion length l φ = v /D φ .Translational noise D and chiral torque Ω were set to zero to focus on the basic mechanism responsible for the emergence ofthe spatial shift, ∆( t ) = (cid:104) x ( t ) − x (0) (cid:105) . In (b) the particle displacement, ∆( t ), is plotted vs. t in units of the pulse crossing time t σ = σ/u for a slow and fast pulse. For the sake of comparison, we remind that the time the particle takes to diffuse a length σ in the bulk is τ σ = σ / D s . In (c) the final displacement ∆( ∞ ) is plotted as a function of the pulse speed. The stochasticintegration of the model Eqs. (1) (crosses), are compared with the numerical solution the corresponding Fokker-Planck Eq. (7)(solid curves). The contour plot in (d) illustrates the dependence of ∆( ∞ ) on both pulse parameters σ and u . The white curveseparates the regions with positive and negative ∆( ∞ ). This approach cannot be extended to the case of ar-tificial swimmers diffusing across a traveling ADW be-cause these respond to the instantaneous activation prop-erties of the surrounding medium with no temporal mem-ory [19]. However, controlling the transport of syn-thetic sub-micron particles, self-propelling in a spatio-temporally modulated active medium, is key to the suc-cess of nanorobotics in technological applications, like en-vironmental monitoring, intelligent drug delivery, or evenmore challenging biomedical tasks [6]. With this goal inmind, we numerically investigated the diffusive dynam-ics of artificial microswimmers at low Reynolds numbers,subjected to traveling symmetric active pulses of differ-ent waveforms. We observed that swimmers with posi-tive taxis in a monotonic active gradient, actually drifttowards or away from the pulse source, depending on thepulse sequence. Moreover, chiral swimmers, which due tosome configurational asymmetry tend to move in circles [20], may also drift orthogonally to the incoming activewave. The variability of swimmers’ tactic response isproven to result more from the spatio-temporal modula-tion of their active motion within a traveling pulse, thanthe pulse fore-rear gradient asymmetry.The paper is organized as follows. In Sec. II we intro-duce the model of the artificial microswimmer that is tobe considered. Already in this section, without resortingto the finer technical details (given later), a qualitativepresentation of the key results will be given. Here thetactic responses of the swimmer to single wave pulsesand also to active density waves are discussed. A morequantitative discussion of these results is given in Sec.III, where, in addition, related analytical results will bepresented. Finally, in Sec. IV, the details of the methodsused to obtain the results will be explained, including thenumerical schemes applied to integrate the Langevin andcorresponding Fokker-Planck equations.
II. RESULTSA. Model
Regardless of the details of the self-propulsion mecha-nism, at low Reynolds numbers the diffusion of an artifi-cial microswimmer on a surface is conveniently modeledby a set of simple Langevin equations (LE)˙ x = v ( x, t ) cos φ + (cid:112) D ξ x ( t ) , ˙ y = v ( x, t ) sin φ + (cid:112) D ξ y ( t ) , ˙ φ = Ω + (cid:112) D φ ξ φ ( t ) . (1)Three sources of fluctuations are explicitly incorporatedin the model: two translational of intensity D , and oneorientational of intensity D φ . All noises are Gaussian andstationary, with zero-mean and autocorrelation functions (cid:104) ξ i ( t ) ξ j (0) (cid:105) = 2 δ ij δ ( t ), with i, j = x, y, φ . As such noisesresult from a combination of thermal fluctuations in thesuspension fluid and randomness of the propulsion mech-anism, we treat the intensities D and D φ as indepen-dent parameters. Finally, depending on their geometry,2D swimmers often experience an additional torque, Ω,which makes them rotate counter-clockwise or clockwise.Positive and negative chiral effects impact the transportproperties of both biological and synthetic active swim-mers [20–23].When the swimmer floats in a homogeneous active sus-pension, its propulsion speed is a constant, v ( x ) = v . Asdetailed in Sec. IV, it then undergoes an active Brownianmotion with finite persistence time, τ φ = 1 /D φ , length, l φ = v τ φ , and bulk diffusion constant lim t →∞ (cid:104) [ x ( t ) − x (0)] (cid:105) = D + D s , with D s = v / D φ .To model the effects of an active pulse sweepingthrough the suspension fluid, we assume that the propul-sion speed, v ( x, t ), is a local function of the physio-chemical properties of the medium at the swimmer’s po-sition. For pulses propagating from left to right along the x axis, this amounts to inserting in Eqs. (1) an appro-priate function v ( x, t ) = v ( x − ut ), where u is the pulse’sspeed and the waveform v ( x ) is chosen so as to describesingle or entrained traveling pulses [4] with amplitude v .Like in Ref. [18], here we restrict our analysis to spatiallysymmetric waveforms, v ( x ) = v ( − x ), in order to avoidadditional ratchet effects [12, 17]. The exact mechanismunderlying the tactic stimulus modeled by the spatio-temporal function v ( x, t ) does not, in principle, need tobe specified; it can, for instance, be of chemo-, diffusio-,electro- or thermophoretic nature.A proof of concept of the tactic effects predicted inthis work is sketched in Fig. 1(a). In this ideal exper-iment a thermophoretic swimmer, floating on a planarsubstrate, is irradiated by a defocused laser beam [8].A traveling train of laser intensity pulses is created byinserting a slit screen between the light source and thesubstrate and sliding it at a constant speed. Since theself-propulsive speed of a thermophoretic swimmer is ap-proximately proportional to the laser intensity [24], one can thus tailor at will the spatio-temporal modulation ofthe velocity field, v ( x, t ).For the sake of concreteness, in the following we adoptthe term positive or negative chemotaxis to mean thedrift of an active swimmer parallel or anti-parallel to thedirection of a generic incoming active pulse or wave, re-gardless of its nature. Moreover, we remind the readerthat, like in the experimental setup of Fig. 1(a), thepropulsion speed of most artificial chemotactic swimmersgrows linearly with the concentration of the chemoacti-vants in the active suspension, whereas, as assumed inEqs. (1), their angular diffusion stays almost the same[6]. Such swimmers, when placed in a static velocityfield, that is for v ( x, t ) = v ( x ), are known to diffuse upthe velocity gradient [7, 26]. Other authors [8, 27] havedetected the dependence of the rotational diffusion onthe active density wave to be generally weaker than ofthe propulsion speed. Extending our analysis to an x -dependent D φ would not alter the general conclusions ofthe present work, except for some more laborious techni-cal details [28].Finally, we stress that in the ideal experiment of Fig.1(a) hydrodynamic effects can be largely suppressed, atleast in the absence of activant gradients. Indeed [3],(i) swimmers freely diffuse in the bulk away from thecontainer’s walls; (ii) their density can be lowered so asto avoid clustering [29]; and (iii) they can be fabricatedrounded in shape and so small in size (i.e., pointlike) tominimize hydrodynamic backflow effects. On the con-trary, the activant gradients considered in the presentwork surely cause additional hydrodynamic effects inthe form of a polarizing torque that tends to align theparticle’s velocity parallel or anti-parallel to the gradi-ent, depending on the swimmer’s surface properties [30].Our simulations show that chemotaxis is enhanced forswimmers aligned against the activant gradient and sup-pressed in the opposite case, as also reported in Ref. [8].However, we numerically checked that the chemotactic ef-fect persists even in the latter case, solely its magnitudeslightly diminishes. To keep our discussion as simple aspossible, polarization effects will be neglected here andfully investigated in a follow-up technical report.We now qualitatively discuss the numerical results ofFigs. 1-4, that were obtained by numerically integratingthe LEs (1) and the corresponding Fokker-Planck equa-tion (FPE) (see Sec. IV). A more technical analysis ofthese results will be presented in Sec. III. B. Chemotaxis by single active pulses
In Figs. 1(b)-(d) and Supplementary Movies 1 and 2[31] we illustrate the effects of an active Gaussian pulseof amplitude v traveling from left to right across a swim-mer at rest in the absence of translational fluctuations, D = 0. The swimmer undergoes a net longitudinal shift,∆( ∞ ) = lim t →∞ (cid:104) x ( t ) − x (0) (cid:105) : most notably, ∆( ∞ ) ismarkedly negative for slow pulses, u (cid:28) v , and positive . . . u / v L/l φ . . . v x / v (a) . . . u / v L/l φ . . v x / v (b) . . . u / v L/l φ . . v x / v (c) . . . u / v L/l φ . . . D x / ¯ D x (d) FIG. 2. Chemotaxis of an achiral swimmer with Ω = 0 in a sinusoidal traveling ADW. Unless differently stated, all modelparameters were chosen consistently with reported experimental values [25]: v = 53 µ m / s, D φ = 165 s − , and D = 2 . µ m / s.The ADW waveform, v ( x ) = w +( v − w ) sin ( πx/L ), has fixed maxima, v , and tunable minima w , with w < v . All resultsare from numerical integration of the model Eqs. (1) or the corresponding FPE (7) (see Sec. IV). Lengths and velocities are givenin units of l φ and v , respectively. (a)-(c) Contour plots of the longitudinal drift velocity, v x = lim t →∞ (cid:104) x ( t ) − x (0) (cid:105) /t in theplane ( L, u ); w = 0 and all other parameters are the same except for D = 0 in (b) and w = 0 . v x, max , and negative, v x, min , chemotaxis are marked by black crosses; the separatrix curves, u s ( L ), delimiting the regions ofpositive and negative chemotaxis are drawn in white; (d) Contour plot of the diffusion constant D x = lim t →∞ (cid:2) (cid:104) x (cid:105) − (cid:104) x (cid:105) (cid:3) / t for the model parameters of panel (a). The scaling factor, ¯ D x = D + v / D φ , is the swimmer’s diffusion constant in the averagevelocity field (cid:104) v ( x ) (cid:105) = v /
2. For animations of the swimmer’s diffusion under diverse dynamical conditions see SupplementaryMovies 3-6 [31]. for fast pulses, u ≥ v , see Figs. 1(c),(d). We explainthe existence of opposite chemotactic regimes by notic-ing that a swimmer with D = 0 diffuses only across thewidth of the incoming active pulse. Suspended inside aslow travelling pulse with u (cid:28) v , the particle quicklydiffuses either against the front or the rear of the pulse.Upon reaching either pulse’s edge, its diffusivity gets sup-pressed, that is its self-propulsive velocity, v ( x ), growssmaller than the pulse speed, u . The ensuing behaviourat the two sides of the pulse is different. After hitting ther.h.s., the particle is caught up again by the advancingpulse and resumes diffusing, whereas upon hitting thel.h.s., it is left behind and comes to rest. The two pulse’sedges behave, respectively, like travelling reflecting andadsorbing walls. As a result we expect, on average, a netshift of the particle to the left. In the opposite regime ofa fast travelling pulse, u (cid:29) v , the particle comes almostimmediately to rest when hitting the left pulse wall, whileit can travel a much longer distance to the right withoutbouncing against the right pulse wall. In the optimalcase, u (cid:39) v , such a distance is of the order of the per- sistence length, l φ . Accordingly, for a fixed pulse width,∆( ∞ ) attains a positive maximum around u (cid:39) v , andvanishes monotonically in the limit u/v → ∞ .Consistent with our interpretation, in Figs. 1(c)-(d)the transition from negative to positive chemotaxis oc-curs when the time the particle takes to diffuse a lengthof the order of l φ , grows longer than the correspondingpulse crossing time, namely for u/v (cid:39) /
2. Of course,under realistic experimental conditions, the apparent di-vergence of the shift, ∆( ∞ ) → −∞ , at vanishingly slowpulse speeds, Fig. 1(c), would be offset by the inevitabletranslational fluctuations with D >
0, neglected in thesimulations of Fig. 1. Moreover, for narrow pulses, σ (cid:46) l φ , the swimmer cannot diffuse much inside thepulse, but crosses it ballistically. Consequently, as illus-trated in Fig. 1(d), its positive drift is more pronouncedthan in the case σ (cid:29) l φ . Vice versa, for large pulse widthsthe particle’s dynamics is dominated by active diffusionand its displacement grows increasingly negative at small u . Of course, for σ → ∞ the activant gradient becomesnegligible and ∆( ∞ ) vanishes. Thus, in the regime ofnegative ∆( ∞ ), for any chosen u there exists an optimalpulse width where the negative particle displacement isma! ximum. C. Chemotaxis by active density waves
More generally, active pulses are generated in randomor periodic sequences. For the sake of simplicity, we con-sider here the case of periodic ADWs with waveform v ( x ) = w + ( v − w ) sin ( πx/L ) . Like in most experimental setups, we assume that the pa-rameters that regulate the swimmer’s dynamics, D , v , τ φ and l φ , are fixed, whereas the ADW parameters, u, L and w /v , can be tuned at the experimenter’s conve-nience. We checked that the swimmer’s chemotactic re-sponse is not appreciably modified by varying the se-quence or waveform of the active pulses.In the stationary regime, chemotaxis of achiral swim-mers with Ω = 0 is characterized in terms of the driftvelocity, v x = (cid:104) ˙ x (cid:105) . This is plotted in Fig. 2 for differ-ent values of the wave parameters. The regions of theplane ( L, u ) exhibiting positive or negative chemotaxisare delimited by separatrix curves, which depend also on D and w [Figs. 2(a)-(c) and 3(b)]. Like in the caseof single active pulses, negative chemotaxis is induced byslow ADWs only, with u < v (Supplementary Movies3-6 [31]). Indeed, for fast ADWs, the distance a swim-mer can travel without crossing a minimum of v ( x ) islonger to the right than to the left. Hence, one can ex-pect that v x >
0. Moreover, the distance the swimmercan surf a wave with u (cid:39) v is limited solely by its per-sistence time, τ φ , so that here positive chemotaxis is themost pronounced. Vice versa, for u (cid:28) v , the swimmercrosses the ADW troughs, with reduced self-propulsionspeed, both left and right. As the ADW propagates tothe right, the time the swimmer takes to cross a trough toits left is shorter than to its right, hence v x < D = 0, where—provided w = 0—the particlecan never cross a trough to the right, see Fig. 2(b) andSupplementary Movies 4 and 6 [31].The translational fluctuations, D >
0, help the swim-mer diffuse across the ADW troughs, suppressing the ve-locity rectification mechanism described above. Thus,with increasing D , the negative rectification effect be-comes smaller compared to the positive surfing effect,until eventually v x changes sign. As the same effect cannot only be achieved by raising the noise strength D ,but also by lowering the pulse periodicity L (translationalnoise can easily “kick” particles out a narrow pulse), theseparatrix bends downward, almost vertically, at a crit-ical value of L – see Fig. 2(a). The positive chemotaxisobserved in the bottom-left quadrant of Fig. 2(a) is an un-avoidable effect of the translational fluctuations D (cid:54) = 0.The dependence of the vertical branch of the separa-trix on the model parameters is illustrated by the curves of Figs. 3(b)-(c). We observe that its position along thehorizontal axis is (i) shifted to the right proportional to D ; (ii) shifted to the left proportional to w ; and (iii) in-dependent of l φ (not shown). This behavior points to theexistence of a critical value of D /Lv , ( D /Lv ) cr , abovewhich negative chemotaxis is suppressed. Moreover, Fig.3(c) clearly shows that ( D /Lv ) cr grows linearly with w , independently of l φ . This implies that the verticalbranch of the separatrix can be shifted to lower L eitherby lowering D or increasing w . However, these two op-tions for enlarging the negative chemotaxis region of the( L, u ) plane, have opposite impact on the modulus of v x – chemotaxis is enhanced by lowering D and suppressedby raising w at constant v [Fig. 3(c), inset].The horizontal branch of the separatrix also dependson w , but is insensitive to the noise intensities D and D φ . In the limit D →
0, the separatrix is a smoothfunction of L , u s ( L ), with limits u s (0) and u s ( ∞ ) of thesame order of magnitude, both limits being functions of w and smaller than v . In view of these results we con-clude that negative chemotaxis is a robust property ofthe system, since it sets in under the most affordable ex-perimental conditions of ADWs traveling with low speed, u (cid:28) v , and long wavelength, L (cid:29) l φ . D. Transverse chemotaxis of chiral swimmers
An intrinsic rotational torque of the swimmer canbe either the accidental result of fabrication defects orthe desired effect obtained, e.g., by bending an activenanorod [32]. In any case, chirality strongly impactsswimmers’ chemotaxis. In Figs. 4(b)-(c) the longitudinaland transversal drift velocities, v x = (cid:104) ˙ x (cid:105) and v y = (cid:104) ˙ y (cid:105) ,are plotted against the torque frequency, Ω, for condi-tions of, respectively, the largest positive and negativechemotaxis at Ω = 0 [marked by crosses in the contourplot of Fig. 2(a)]. A few remarkable properties are im-mediately apparent (see also Supplementary Movies 7-10[31]): (i) Chirality induces a transverse chemotactic drift, v y (cid:54) = 0. Such an effect is the strongest in the regime ofpositive longitudinal chemotaxis of achiral particles; (ii) v x (Ω) and v y (Ω) are respectively even and odd functionsof Ω, consistent with the symmetry of the model Eqs. (1)under the transformation φ → − φ ; (iii) Chirality tendsto suppress longitudinal chemotaxis. This effect is bestnoticeable in Fig. 4(b), where for Ω τ φ ∼ π the longitudi-nal drift velocity, v x , drops to zero, while v y develops apeak of height comparable with v x (0). Under these con-ditions, the chemotactic effect of the incoming ADW isfully transverse and with the same sign as Ω. At evenhigher frequencies, v x changes sign from positive to neg-ative; (iv) In the regime of negative chemotaxis of achiralparticles and zero translational noise [see Fig. 2(b)], thecurves v x (Ω) and v y (Ω) have the same sign and no zerosfor Ω >
0. On raising D , without changing the signof v x , transverse chemotaxis is suppressed and its signvaries with increasing Ω.The mechanism responsible for transverse chemotaxisis illustrated in Fig. 4(a). An active particle subject toa positive torque, Ω >
0, bounces against the ADWtroughs, thus tracing spiraling circles of radius r d ∼ v / Ω, which go up (down) the right (left) hand side ofthe trough. For fast traveling ADWs with u > v , bounc-ing trajectories only take place on the r.h.s. of the wavetroughs, so that v y >
0. Of course, transverse chemotaxisis most pronounced when the bouncing process is syn-chronized with the wave modulation, namely, when thebouncing time, π/ Ω, is of the order of the wave period,
L/u . Moreover, such a mechanism grows more effectivefor swimmer persistence times larger than the bouncingtimes, that is, for Ω τ φ (cid:38) π . Both conditions are satisfiedin Fig. 4(b), so that, in an appropriate Ω τ φ range, trans-verse chemotaxis supersedes longitudinal chemotaxis.For slow ADWs, spiraling trajectories develop on bothsides of the wave minima. As noticed above, trough cross-ings from right to left take a shorter time than from left toright; accordingly, upward bouncing trajectory arcs haveshorter span than the downward ones. This observationexplains why, in Fig. 4(c) for u (cid:28) v and D = 0, trans-verse chemotaxis and chirality have opposite signs. Also,similarly to achiral chemotaxis, translational noise easesADW trough crossings in both directions, thus suppress-ing transverse chemotaxis and eventually reversing itsorientation. It is also suppressed by reducing the ADWamplitude. III. DISCUSSION
We next present a more quantitative analysis of ourresults based on the approximation schemes detailed inSec. IV.
A. Ballistic regime
The curves of the drift velocity, v x , versus the ADWspeed, u , in Fig. 3(a) exhibit a characteristic resonant be-havior for both positive and negative chemotaxis. In theballistic regime, i.e. for L (cid:28) l φ , their decay is satisfac-torily described in the two-state approximation, where,for sufficiently long persistence times (compared to thepulse crossing time), the swimmer’s dynamics is modeledas the superposition of a non-fluctuating, self-propellingdrift to the right and to the left. In the absence of trans-lational noise, the predicted average drift velocity, Eq.(5), decays like: (i) v x /v ∝ v /u for u (cid:29) v (to holdfor any value of L/l φ ); (ii) v x /v ∝ − u/v for u (cid:28) v and w >
0; and (iii) v x /v ∝ − (cid:112) u/v for u (cid:28) v and w = 0. These predictions closely agree with thenumerical fits of Fig. 3(a). In addition, Eq. (5) sug-gests that the positive chemotaxis maxima, v x, max , oc-cur for u (cid:39) v , as expected, and the negative minima, v x, max , for u (cid:46) w . The two-state model also provides − . .
05 0 .
01 1 100 v x / v u/v L/l φ = 1 . L/l φ = 6 . (cid:5) : ∝ u? : ∝ √ u ∝ /u ?? (cid:5) (cid:5)(cid:5) (a) .
11 0 . u / v L/l φ (0 | | . | . | . | . | (cid:16) w /v (cid:12)(cid:12)(cid:12) D /D (0)0 (cid:17) = (b) . . . ( D / L v ) c r w /v numericsanalytics − .
050 0 0 . v x / v − w /v v x, max v x, min × ∝ (1 − w /v ) ∝ (1 − w /v ) (c) FIG. 3. Chemotaxis of an achiral swimmer in a sinusoidaltraveling ADW. Unless differently stated, all parameters werechosen as in Fig. 2(a). Lengths and velocities are given inunits of l φ and v , respectively. (a) v x vs. u in the ballisticregime, L/l φ = 1 .
6, and diffusive regime,
L/l φ = 6 .
4, for finite D and w = 0 (dashed curves), D = 0 and w = 0 . D = 0 and w = 0 (solid curves). All curvesdecay asymptotically with power laws, as indicated; (b) sepa-ratrix, u s vs. L for different values of D and w , where D (0)0 denotes the standard value of 2 . µ m /s used in the previousfigures. The vertical branch of u s shifts to lower L propor-tional to the noise level, D , and inversely proportional to theADW baseline, w ; (c) ( D /Lv ) cr vs. w from the numer-ical integration of the exact Eq. (7) (blue crosses), and theapproximated FPE in the ballistic regime (red crosses), seeEq. (11). In the inset the maxima and minima of v x , v x, max and v x, min , are shown to increase faster than linearly withthe amplitude of the ADW, v − w . Note, however, thatthe modulus of v x, min increases with w for w (cid:28) v , as alsoshown in (c). a close estimate of the horizontal branch of the separa-trix, u s ( L ). The limit L/l φ → D = 0, u s (0),can be approximated analytically by setting v x = 0 inthe second line of Eq. (5) and solving for u . Hence u s (0) /v = [( w + v ) / v ][1 + (cid:112) v w / ( v + w ) ],in quantitative agreement with the numerical data of Fig.3(b). The full curves u s ( L ) plotted there were obtainedby locating the zeros of the average current, Eq. (9), asa function of u and L .As remarked in Sec. II, translational fluctuations tendto suppress chemotaxis at large, and negative chemo-taxis in particular. In the two-state model notation, thisis a consequence of the noise-induced “creeping effect”mentioned in Sec. IV, an effect more conveniently ad-dressed in the FPE formalism. For instance, upon ex-panding the ballistic approximation of the average cur-rent, Eq. (11), in powers of u/v , the chemotactic speedis proven to grow proportionally to u (and not √ u ) evenat w = 0, with a sign that turns from negative to posi-tive on increasing D [Fig. 3(a)]. More interestingly, Eq.(11) allowed us to locate the vertical separatrix branchof the contour plots of Figs. 2(a)-(c), namely the quan-tity ( D /Lv ) cr plotted in Fig. 3(c). For an analyticalestimate of the same quantity, we notice that transla-tional noise provides an additional diffusion mechanismthat competes with self-propulsion. The self-propulsionmechanism eventually prevails when the time the parti-cle takes to diffuse a half ADW wavelength due to trans-lational noise, ( L/ / D , is shorter than the averagetime to cross the same distance with an average speed( v + w ) / L/ ( v + w ). This occurs for D /Lv (cid:38) ( D /Lv ) cr ,with ( D /Lv ) cr ∝ w /v , in agreement with the nu-merical data in Fig. 3(c). We remarked in Sec. II that( D /Lv ) cr is independent of τ φ ; therefore, this analyti-cal estimate holds in the ballistic and diffusive regimes,alike.Contrary to the Stokes drift [16], v x at small u growswith exponent clearly smaller than 2, irrespective of w and D (and, therefore, of its own sign): this means thatthe chemotactic effect studied here is not governed by thefore-rear ADW gradients so much as by the swimmer’sdiffusion across the traveling ADW troughs. B. Diffusive regime
In the diffusive regime, i.e. for L (cid:29) l φ , the persistencetime τ φ can be taken as vanishingly small. Accordingly,the swimmer’s diffusion in the wave direction is closelydescribed by the multiplicative LE in Eq. (6). StandardStratonovitch calculus [33, 34] yields the finite drift term v x = [( v − w ) / D φ ] (cid:104) ( d/dx ) sin [ π ( x − ut ) /L ] (cid:105) , (2)where (cid:104) . . . (cid:105) denotes a stationary average over thestochastic trajectories of x ( t ). This average was com-puted explicitly by solving the corresponding FPE, Eq. (12): For u < v chemotaxis indeed turns out to be nega-tive with v x ∝ − u for D > u (cid:28) v , and v x ∝ −√ u for D = 0 and u (cid:28) v , as in the fits of Fig. 3(a).On the other hand, upon increasing u larger thanthe modulus of the multiplicative term in Eq. (6), u > |(cid:104) v ( x ) cos φ (cid:105)| (cid:39) ( w + v ) / √
2, the dynamical effect ofthe angular fluctuations is suppressed with respect to thedragging action of the ADW, so that chemotaxis changessign from negative to positive. The r.h.s. of the above in-equality approximates the horizontal asymptote of u s ( L )in the limit L/l φ → ∞ . As displayed in Fig. 3(b), u s ( ∞ )is a function of w , only.The role of the separatrix is further illustrated by thecontour plot of Fig. 2(d), where we plotted the diffusionconstant D x = lim t →∞ (cid:2) (cid:104) x (cid:105) − (cid:104) x (cid:105) (cid:3) / t for the modelparameters of Fig. 2(a). The computed values of D x have been compared with the swimmer’s average diffu-sion constant, ¯ D x , defined in the figure caption. One seesimmediately that D x / ¯ D x (cid:39) L, u ) plane, except across the (hot) horizontal and (cold)vertical branches of the separatrix, where D x / ¯ D x > D x / ¯ D x <
1, respectively. Indeed, the chemotaxissign inversion observed upon increasing the ADW speed, u , signals a locked-running transition in the two-statemodel Eqs. (3), a mechanism known to produce excessdiffusion [35–37]. On the other hand, the chemotaxisinversion obtained by increasing the ADW wavelengthis governed by translational noise (it never occurs for D = 0), which acts there as a sort of lubricant, thussuppressing the net swimmer diffusivity. With a viewto experimental demonstration, we remark that for ar-tificial swimmers, chemotaxis by traveling ADWs is nomore dispersive than regular transport in the bulk. C. Chiral chemotaxis
The FPE formalism also allows a better characteriza-tion of chiral chemotaxis. To shed light on the underly-ing irreducible 2D mechanism we considered the modelEqs. (1) in the opposite adiabatic limits Ω τ φ (cid:28) τ φ (cid:29)
1, both in the ballistic and diffusive regimes intro-duced above. A systematic perturbation approach (notreported here) led us to conclude that v x = c ( v − w )and v y = s ( v − w ), where c and s are, respectively,even and odd functions of Ω. In particular, for Ω τ φ (cid:28) c ∝ Ω and s ∝ Ω for any D ; for Ω τ φ (cid:29) c decays like Ω − at D = 0 and Ω − at D (cid:54) = 0, whereas s ∝ Ω − , independent of the translational noise strength D .We conclude by underscoring that chemotaxis of artifi-cial microswimmers is a robust phenomenon that lends it-self to accessible laboratory demonstrations and promis-ing applications to nanotechnology and medical sciences.Our numerical results and interpretation go beyond theearlier “chemotactic wave paradox” debate, insofar as theswimmer’s tactic response results from its ability of dif-fusing within a traveling active pulse and not just the y / L x/L v Ω u (a) − . . −
10 0 10 v x , y / v Ω τ φ v x v y ∝ Ω ∝ Ω − ∝ Ω − ∝ Ω − ∝ Ω − (b) − . . − v x , y / v Ω τ φ v x v y × ∝ Ω ∝ Ω − ∝ Ω − (c) FIG. 4. Chemotaxis of a chiral swimmer with Ω (cid:54) = 0 in a sinusoidal traveling ADW. Unless differently stated, all modelparameters were chosen as in Fig. 2(a). All results are from numerical integration of the model Eqs. (1) or the correspondingFPE (7). The drift velocities and chiral frequency are given in units of v and D φ = τ − φ , respectively. For animations seeSupplementary Movies 7-10 [31]. In (a) is a sketch of the upward spiraling trajectory of a swimmer bouncing against theleft trough of an ADW, where the dotted black line indicates the half-width. The longitudinal and transverse drift velocities, v x = lim t →∞ (cid:104) x ( t ) − x (0) (cid:105) /t and v y = lim t →∞ (cid:104) y ( t ) − y (0) (cid:105) /t are plotted vs. Ω τ φ for (b) L/l φ = 2, u/v = 1; and (c) L/l φ = 7, u/v = 0 . D = 0 are also drawn for the sake of comparison (dashed curves).The exponents of fitted power-laws coincide with those predicted in Sec. III. asymmetry of the fore-rear pulse gradients. Moreover,the direction and magnitude of the tactic response are ex-tremely sensitive to the self-propulsion mechanism, whichsuggests the design of tactic devices to control the pro-duction and transport of artificial swimmers. IV. METHODS
The model Eqs. (1) describe the spatial diffusion of anactive over-damped Brownian particle subject to a trav-eling field of force v ( x, t ). Contrary to the previous litera-ture on Brownian Stokes’ drift [16, 17], here the particle’smotion is characterized by a finite persistence time, τ φ ,and related length l φ = v τ φ . A simple calculation [38]shows that the angular factors, cos φ and sin φ , decayexponentially, (cid:104) cos φ ( t ) cos φ (0) (cid:105) = (cid:104) sin φ ( t ) sin φ (0) (cid:105) =(1 /
2) exp( − D φ | t | ), hence τ φ = 1 /D φ . Persistence (ormemory effects) become appreciable when the Brownianparticle is confined to geometries with characteristic sizesmaller than l φ . In our model, the standard distinctionbetween a ballistic regime with L (cid:28) l φ , and a diffusiveregime with L (cid:29) l φ , must be revised to account for thefinite wave period L/u , as discussed below.
A. The Langevin equations
The stochastic differential Eqs. (1) were numerically in-tegrated by means of a standard Euler-Maruyama scheme[39]. The stochastic averages were taken over an ensem-ble of trajectories with random initial swimmer orienta-tion, φ (0) ∈ [0 , π ]. The two-state model.
In the ballistic regime, the ana-lytic treatment of Eqs. (1) with Ω = 0 is greatly simplifiedby assuming that an achiral swimmer moves in the wave’s direction with equal probability to the right, φ = 0, orto the left, φ = π , i.e. with velocity ± v ( x, t ), thus to-tally ignoring its transverse motion along y . The ensuingswimmer dynamics is modeled through two independentLEs˙ x = ± w ± ( v − w ) sin [ π ( x − ut ) /L ] + (cid:112) D ξ x ( t ) , (3)obtained by setting cos φ = ± x (cid:48) = x − ut , Eqs. (3) can rewrittenin the form of two LEs for for a tilted potential withdifferent tilting, ˙ x (cid:48) = − u ± [( v + w ) − ( v − w ) cos(2 πx (cid:48) /L )] / √ D ξ x ( t ) . (4) The time a noiseless swimmer takes to cross a wave-length L can be calculated analytically from Eqs. (4)with D = 0. In the steady-state with φ = 0 (ori-ented to the right), we obtained ˙ x (cid:48) > ≤ u ≤ w ,˙ x (cid:48) = 0 for w ≤ u ≤ v , and ˙ x (cid:48) < u ≥ v ; in thesteady-state with φ = π (oriented to the left), ˙ x (cid:48) < u . The coordinate x (cid:48) is thus locked for φ = 0 and w ≤ u ≤ v , and running under all remaining condi-tions, with | v ( ± ) x (cid:48) | = L/t ± for φ = 0 and φ = π , respec-tively, where t ± = L/ (cid:112) ( v ∓ u )( w ∓ u ). Accordingly,on transforming back to the coordinate x , we obtain twodistinct solutions for (cid:104) ˙ x (cid:105) , v (+) x and v ( − ) x . Finally, aver-aging over the two φ states corresponds to taking thearithmetic mean v x = [ v (+) x + v ( − ) x ] /
2, that is v x = u + (cid:104)(cid:112) ( v − u )( w − u ) − (cid:112) ( v + u )( w + u ) (cid:105) ≤ u ≤ w u − (cid:112) ( v + u )( w + u ) w ≤ u ≤ v u − (cid:104)(cid:112) ( u − v )( u − w ) + (cid:112) ( u + v )( u + w ) (cid:105) u ≥ v (5) As shown in Sec. III, despite the rather rough assump-tions detailed above, this result proves to be an effectiveinterpretation tool.Such a ballistic scheme holds in a strict sense underthe condition that τ φ > max { t + , t − } , namely for u/v > [ (cid:112) (2 L/l φ ) + (1 − w /v ) +(1+ w /v )] /
2. This impliesthat Eq. (5) applies for u > v when L/l φ (cid:28)
1, and for u/v > L/l φ when L/l φ (cid:29)
1. Moreover, since in thestate φ = 0 with w < u < v the particle is locked, i.e. t + = ∞ , the solution for v x holds approximately underthe weaker condition τ φ > t − . This is why in Sec. III weextended our two-state model interpretation to u (cid:28) v and L (cid:28) l φ .To solve the two-state model of Eqs. (3) in the presenceof translational noise we had recourse to the FPE formal-ism, as discussed below. We anticipate that for D > v (+) x (cid:48) > w ≤ u ≤ v [34]:translational noise smooths the locked-running transi-tion, thus acting as a dynamical lubricant (creeping ef-fect).
1D reduced model.
Brownian motion can be treatedas purely diffusive when its persistence length is muchshorter than all other length scales of the system. Forthe swimmer of Eq. (1) this condition amounts to re-quiring that the persistence time, τ φ , is shorter thanboth crossing times t + and t − , namely for u/v ≤ [ (cid:112) (2 L/l φ ) − (1 − w /v ) − (1 + w /v )] /
2. In the limitof vanishingly small τ φ , the random orientation factorin Eq. (1), cos φ ( t ), can be replaced by an effectivenoise source with zero mean and autocorrelation func-tion (cid:104) cos φ ( t ) cos φ (0) (cid:105) (cid:39) δ ( t ) / (2 D φ ) [38], so that thedynamics of x (cid:48) = x − ut is well described by the effective1D LE,˙ x (cid:48) = − u + v ( x (cid:48) ) ξ φ ( t ) /D φ √ (cid:112) D ξ x ( t ) , (6)where the multiplicative stochastic term on the r.h.s.must be interpreted in Stratonovitch sense [33, 40]. B. The Fokker-Planck formalism
For a more detailed analysis of the swimmer’s stochas-tic dynamics we turn to the Fokker-Planck equation(FPE) associated with the model Eqs. (1) [33, 34], ∂ t P ( r , φ, t ) = −∇ i J i ( r , φ, t ) , (7) J ( r , φ, t ) = − D ∂ x + v ( x ) cos φ − u − D ∂ y + v ( x ) sin φ − D φ ∂ φ + Ω P ( r , φ, t ) , and ∇ = ( ∂ x , ∂ y , ∂ φ ). Here, r = ( x, y ) denotes the par-ticle’s spatial coordinates in the ADW moving frame, x − ut → x and y → y (the prime sign used in Eq. (4) hasbeen dropped for simplicity). The 3D functions P and J denote, respectively, the probability density and currentof the particle in the state ( r , φ ) at time t . The FPE (7)was numerically integrated by combining the method oflines [41] with a second-order backward-difference scheme[42]. Periodic boundary conditions were assumed for all three variables x , y and φ , with relevant periods L , L and2 π . This amounts to the one-zone reduced formulationof probability density [43]ˆ P ( r , φ, t ) = ∞ (cid:88) k,n,m = −∞ P ( x + kL, y + nL, φ + 2 πm, t ) . (8)The particle’s drift velocities in the ADW frame are thencomputed as the stationary currents in the x and y di-rection, v x,y = L (cid:90) d x L (cid:90) d y π (cid:90) d φ lim t →∞ J x,y ( r , φ, t ) . (9)The final result for v x is obtained by transforming it backto the laboratory frame, v x → v x + u . For achiral par-ticles, Ω = 0, the transverse coordinate y is dispensable,so that, upon integration over y , the FPE (7) is immedi-ately reduced to a partial differential equation for the 2Dprobability density P ( x, φ, t ). In the following we shortlyaddress two limiting cases of such a reduced FPE, cor-responding, respectively, to the two-state model and thereduced 1D model approximations of the LEs (1) withΩ = 0, as introduced above. Ballistic approximation.
In the ballistic regime, theparticle’s orientation is almost constant during its spatialrelaxation. Accordingly, the FPE can be further reducedto a 1D partial differential equation in x for a fixed, butarbitrary φ . The stationary reduced probability currenthas a manageable expression as a function of φ [43], i.e.ˆ J ( φ ) = D L (cid:26) − exp (cid:20) − Lv D (cid:18) v + w v cos φ − uv (cid:19)(cid:21)(cid:27) × (cid:90) d ξ (cid:90) d ζ exp (cid:20) − Lv D (cid:18)(cid:18) v + w v cos φ − uv (cid:19) ζ − v − w πv cos φ (cid:16) sin(2 π ( ζ + ξ )) − sin(2 πξ ) (cid:17)(cid:19)(cid:21) − , (10)which, upon averaging over a uniform φ distribution with φ ∈ [0 , π ] and transforming back to the laboratoryframe, yields the most accurate estimates of the driftvelocity in the ballistic regime, v x = L π π (cid:90) d φ ˆ J ( φ ) + u. (11) Diffusive approximation.
In the opposite limit of fastangular relaxation, the dependence on the coordinate φ can be projected out by means of a mapping procedure[44] to be detailed in an upcoming technical report. Inleading order of the perturbation parameter l φ /L , we ob-tained a 1D partial differential equation ∂ t P ( x, t ) = ∂ x a ( x ) P ( x, t ) − ∂ x b ( x ) P ( x, t ) , (12)0with a ( x ) = v ( x ) / D φ + D and b ( x ) =( d/dx ) v ( x ) / D φ − u . This is the FPE correspondingto the reduced multiplicative process derived in Eq. (6).Upon computing the stationary probability current,ˆ J = − exp − L (cid:90) d x b ( x ) a ( x ) × L (cid:90) d x L (cid:90) d y a ( x ) exp y + x (cid:90) x d z b ( z ) a ( z ) − , (13) the drift velocity is finally expressed as v x = L ˆ J + u . ACKNOWLEDGMENTS
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