Self-Sustained Density Oscillations of Swimming Bacteria Confined in Microchambers
DDOI: 10.1103/PhysRevLett.115.188303
Self-Sustained Density Oscillations of Swimming Bacteria Confined in Microchambers
M. Paoluzzi , ∗ R. Di Leonardo , , and L. Angelani , Dipartimento di Fisica Universit`a Sapienza, P.le A Moro 2, 00185 Rome, Italy NANOTEC-CNR, Institute of Nanotechnology,Soft and Living Matter Laboratory, Piazzale A. Moro 2, I-00185, Roma, Italy Istituto dei Sistemi Complessi (ISC-CNR), UOS Sapienza, P.le A Moro 2, 00185 Rome, Italy
We numerically study the dynamics of run-and-tumble particles confined in two chambers con-nected by thin channels. Two dominant dynamical behaviors emerge: ( i ) an oscillatory pumpingstate, in which particles periodically fill the two vessels and ( ii ) a circulating flow state, dynamicallymaintaining a near constant population level in the containers when connected by two channels.We demonstrate that the oscillatory behaviour arises from the combination of a narrow channel,preventing bacteria reorientation, and a density dependent motility inside the chambers. Introduction. — Self-sustained oscillators are ubiqui-tous in physics and biology [1]. From the Van der Pol os-cillator to the heartbeat these systems are characterisedby a periodic motion that is sustained against dissipationby some form of energy source. Active systems constantlyconsume an internal energy source to sustain persistentmotions in a highly dissipative environment [2, 3]. Thisresults in a strongly out of equilibrium dynamics that cangive rise to unidirectional actuation of micro-objects [4–7], spontaneous accumulation of passive colloids over tar-get sites [8], long lived density fluctuations [9–11], large-scale vortex lattice [12], frozen steady states [13], activeliquid crystals [14] or macroscopic directed motion [15].It has been shown that the large scale collective patternsthat spontaneously break and reform in many active sys-tems can be stabilized by confining microstructures [16–18]. However spatial confinement always results in longlived unidirectional flows that only rarely and randomlycan switch direction [18].In this Letter we investigate the possibility of using ge-ometric confinement to obtain self-sustained oscillationsin active matter systems. In particular, by using nu-merical simulations, we show that active particles canalternately fill and empty two micro chambers connectedby thin channels. We find that the number of particlesinside each chamber fluctuates with a distribution thatbecomes increasingly bivariate for large particle densities.The narrowness of the channels ensures single file dynam-ics [19–24] and inhibits cell reorientation. This, togetherwith a density dependent motility of swimmers inside thechambers, gives rise to the observed oscillatory behavior.In addition, when the chambers are connected by twochannels we observe also a circulating flow maintaininga near constant population level in the containers.
Results and discussion. — We perform Molecular Dynam-ics simulations of N run-and-tumble swimmers [25–28] oflength (cid:96) and thickness a , with a/(cid:96) = 1 /
2, in two dimen-sions [4, 29, 30]. The interaction between the swimmersis purely repulsive. The system is confined in two cham- (a)(b)
FIG. 1: Snapshots of the simulations of active particles con-fined in two chambers connected by thin channels. Panel (a):alternating pumping, in which particles alternately fill andempty the chambers in the case of a single channel. Panel(b): alternating pumping and circular flow in the case of twochannels connecting the chambers. bers connected by one or two channels. Details of themodel can be found in [31].We first consider two circular chambers of radius R = 7 (cid:96) connected by a thin channel of length L = 50 (cid:96) and trans-verse size σ = (cid:96)/ N was varied from 160 to 464 cor-responding to area fractions φ going from 0.21 to 0.62[34]. We found that the presence of persistent currentsin thin channels gives rise to large fluctuations in particledistributions between the two chambers. The symmet-ric state, where the two chambers are equally populated,becomes unstable due to the spontaneous formation oflarge currents with a characteristic lifetime that increaseswith the total number of swimmers. As a consequence,one of the two chambers is progressively filled with par-ticles up to a point where it triggers the formation of a a r X i v : . [ c ond - m a t . s o f t ] N ov reversed current. This mechanism gives rise to an oscil-latory pumping between the two containers as evidencedin the snapshots of the numerical simulations (panel (a),Fig. (1) and the movie in [31]). In Fig. (2a) the timeevolution of the fraction of particles in the two reservoirs n , are shown for a sample run with φ = 0 .
47. Fig.(2b) shows the corresponding channel current defined asthe sum of the velocities of particles inside the channel, j = (cid:80) (cid:48) k v k ( t ) /L . A clear oscillation is observed with aperiod of about 2000. In the following we will considerthe probability distribution for coarse grained variables P ( x ) and P ( x, y ), defined as P ( x ) ≡ (cid:104) δ ( x − x ( t )) (cid:105) and P ( x, y ) ≡ (cid:104) δ ( x − x ( t )) δ ( y − y ( t )) (cid:105) where · · · indicates thetime average and the angular brackets (cid:104) . . . (cid:105) averages over120 independent runs. Calling δn = n − n the asym-metry parameter, we compute the joint probability den-sity P ( δn, j ) and the power spectrum S ( ω ) = | δ ˆ n ( ω ) | ,with δ ˆ n ( ω ) the Fourier transform of δn ( t ). In the up-per panels of Fig. (3) the quantity P ( δn, j ) is shown for φ = 0 .
30 and φ = 0 .
47. Contrary to the case of passivethermal particles, where a stable equilibrium point existsat ( δn = 0, j = 0), here we observe a stationary limit cy-cle corresponding to fluxes that alternately empty and fillthe containers (a video of a typical trajectory followed bythe system in the ( δn, j ) plane is shown in Supplemen-tal Material [31]. This oscillating behaviour appears asa peak in the power spectrum S ( ω ) (Fig. (3)d). Thehigh frequency behavior of S ( ω ) is well approximated bya ∼ ω − tail (the solid black lines in Fig. (3)d), indicat-ing that the dynamics of δn ( t ) is uncorrelated on smalltime-scales. When the number of particles is increased, ittakes a longer time to fill or empty the chambers result-ing in a decrease of the peak frequency. For high particledensities δn displays fluctuations that are large and longlived as shown by the pronounced peaks at δn ∼ ± P ( δn, j ) (Fig. 3b). The two chambers alternately filland empty almost completely although transitions be-tween these two states lose periodicity and the peak in S ( ω ) disappears (Fig. 3d).In order to analyze in more details the role playedby the swimmers density we investigate the behavior of P ( δn ) at different particles number. As one can see inpanel (c) of Fig. (2), the distribution P ( δn ) becomesbimodal by increasing φ . The bimodality of P ( δn ) in-dicates that density oscillations take place between twohigh density and long lived states, as a result the twochambers are alternatively almost filled and δn → ± δn max , defined as the value of | δn | wherethe distribution reaches its maximum value, continuouslygrows with φ , up to the threshold value φ = 0 .
47. Forhigher densities one of the two chambers fills up com-pletely and the corresponding internal motility vanishes[31] giving rise to long lived jammed states (the grey areain Fig. 3c). Varying the channel length L from 50 (cid:96) to25 (cid:96) , at least in the analyzed range, seems to have littleeffects on the results [31]. Performing simulations at dif- n -1 0 1 0 2000 4000 6000 8000 j t[r.u.] n
0 2000 4000 -1 0 1 j t[r.u.] (d) FIG. 2: Time evolution of the fraction of swimmers n , ( t )inside the vessels and the channel current j ( t ) for a typicalsingle run. Left panels refer to the case of one channel ( φ =0 . φ = 0 . j · j > ferent channel size σ/(cid:96) = 1 / , , / , L/(cid:96) = 25 , φ = 0 .
47, i.e. where P ( δn ) is stronglybimodal) we observe that δn max decreases toward 0 byincreasing σ , indicating that the channel thickness playsa crucial role in the observed alternate pumping phe-nomenon [31].At the microscopic level the mechanism responsible forthe observed phenomenology relies on the fact that parti-cles in a thin channel are unable to reverse their directionof motion, thus forming long files of pushing bacteria. Asmall difference in the number of left and right orientedbacteria determines the sign of the net flux, and counter-current swimming bacteria are then pushed back to theiroriginal chamber, with a consequent increase of the netflux in the channel. As a result small fluctuations are am-plified, giving rise to long lived flows of particles in thesystem. For non interacting particles, the rate of bacteriapassing from one chamber to the channel will be simplyproportional to the number of bacteria in the chamber N i with i = 1 , N , the motility will be reducedwith a consequent reduction in the entrance rate [31]. Asimple form for the rate that is consistent with this be-haviour is α = βN i (1 − N i /N ). The rate of entrance inthe channel is also affected by the sign of j that makesit harder for bacteria to enter a channel when there’s acounter-flowing current. To account for this last obser-vation we suppose that β can assume two different values β + , β − according to the sign of the current j . We foundthat the above expression for α fits very well our simula-tion data Fig. (4) panel (a). The origin of the oscillatingbehaviour becomes now clear if one follows the time evo-lution of N , over the curves in Fig. (4,a). Let’s startfrom the case where the total number of particles N isless than N and bacteria are equally distributed in thetwo chambers. Due to the narrowness of the channel,bacteria will conserve their direction from the momentthey enter the channel until they exit. A current rever-sal, for example from positive to negative, can then betriggered only if the rate of left going bacteria enteringfrom the right chamber is higher than the number of rightgoing bacteria entering from the left chamber. If we as-sume an initial positive current than the left chamberwill loose bacteria and N will move on the upper curvein Fig. (4,b) towards left. Since the total number of par-ticles has to be conserved N will move symmetricallyon the lower curve towards right. This flow of bacteriafrom the left chamber to the right chamber will continueuntil there’s a higher chance for bacteria to enter thechannel from the right and trigger a current reversal. Atthis point N will move on the lower curve, N on theupper curve and a reversed cycle will occur. When thetotal number of particle is equal to N , N and N willstart from the middle of the graph (Fig. (4) panel c) andthe two rates α will only become equal when one struc-ture is full and the other is empty and both values ofalpha vanish. At this point a current reversal can onlyoccur by an unbiased random fluctuation and the processis reversed with no defined period. This simple picturequantitatively describes the observed increase of oscilla-tion amplitudes with increasing φ (see inset in Fig.3c).We now consider the case in which two channels con-nect the chambers. Performing numerical simulations atdifferent particles area fractions φ , from 0.20 to 0.58 [34],we observe two dominant behaviors – see snapshots inpanel (b) of Fig. (1) and the movies in [31]. The firstone is the oscillatory pumping of swimmers between thetwo chambers – top of panel (b) in Fig. (1) – in anal-ogy with what observed in the presence of one channel.Again, differently from the equilibrium case, the fractionof particles in the containers is affected by strong fluctu-ations that favour flows that alternately empty and fillthe chambers. The presence of the second channel hasthe effect of reinforce the flows, resulting in higher peakfrequencies (shorter oscillation times) with respect to theone-channel case. The second new behavior, forbidden byconstruction when only one channel connects the cham-bers, is a circular flow of swimmers in the system. Insuch a case the currents j and j in the two channelshave the same orientation and the number of particles n and n inside the chambers turn out to be nearly con-stant. This circulating flow persists until a fluctuation inthe number of right and left oriented particles in a chan-nel gives rise to an inversion of the flux, thus resultingin two channel flows pointing towards the same cham-ber falling back into the previous situation of oscillatorypumping behavior. This kind of intermittent behaviorbetween the two phases is made clear by looking at the -1 0n - n -1 0 1 j -1 0 1n - n (a) (b)(c) P n - n (d) -6 -4 -2 -3 -2 -1 S [ a . u .] (cid:1) [r.u.] (cid:1) -2 (cid:98) n m a x (cid:113) FIG. 3: Single channel. Top panels: the join probabil-ity P ( n − n , j ) at two different densities, φ = 0 .
30 (panel(a)) and φ = 0 .
47 (panel (b)). Panel (c): probability distri-bution function P ( n − n ) increasing the number of swim-mers for L = 50 (cid:96) and σ = (cid:96)/
2. The full lines are fits with f ( δn ) = exp( a + bδn + cδn ). Inset: δn max (blue symbols) as afunction of φ , the red line is the amplitude of the oscillationspredicted by the model. Densities in the grey area correspondto jammed states where all bacteria fill one of the two cham-bers almost completely. Panel (d): power spectrum S ( ω ) atfive different densities. For clarity the individual curves havebeen vertically shifted multiplying for 10 n with n = 0 , ..., φ are (from bottom to top)0 .
21 (blue symbols), 0 .
26 (red), 0 .
30 (green), 0 .
40 (grey) and0 .
47 (purple). time evolution of n i and j i – see as example the reportedcase in Fig. 2, panel (c) and (d). When j and j havethe same sign, j · j >
0, a circular flow sets in (grey areain the figure), with a nearly constant population level inthe two containers. In the case j · j < P ( j + j , δn )and P ( j − j , δn ). Circulating flows correspond to thetwo spots of P ( j + j , δn ) at δn ∼ j + j (cid:54) = 0 (Fig.(5)a) and to the single spot of P ( j − j , δn ) at the origin(Fig. (5)b). The oscillatory behavior produces, instead,accumulation of particles into the chambers, producingthe spots at δn ∼ ± φ -dependence of the probability density P ( j + j ).We observe that P ( j + j ) changes its shape by increasing φ , developing a three peaks structure at high densities, φ > .
24 – Fig. (5), panel c. The two peaks at j + j (cid:54) = 0 are due to the circulating flows. The peak inzero corresponds to the limiting situations of maximumdensity unbalance produced by the alternating pumping ↵ N/ ↵ N/ c hanne l en t r an c e r a t e number of bacteria in the chamberdownstreamupstream ↵ oscillatorystochastic (a) (b)(c) FIG. 4: Left panel: variation of the number of swimmers inthe channel as a function of N i for downstream (green sym-bols) and upstream (red symbols). The full lines are the fitswith α . Right panels: the sketch of the mechanism responsi-ble of the alternating pumping for the oscillatory regime (top)and in the case of the stochastic regime (bottom), i. e., whenthe current is reversed with no defined period. state (the spots at δn ∼ ± j + j ∼ P ( j + j ) as a function of φ (see the insetin Fig. (5)c). Changing the chamber size at fixed densitydoes not have relevant effects on the reported behavior[31]. Conclusions. — We have shown that confining activeparticles in micro-chambers connected by thin channelsgives rise to self-sustained density oscillations, alternatelyfilling the two containers. When adding a second chan-nel, this oscillatory behaviour is still present togetherwith circulating flows. The basic ingredients of such aneffect are a density dependent motility combined withthe narrowness of the channels, which only permits sin-gle file dynamics and inhibits cell reorientation. Thereported numerical findings suggest a possible route togenerate self-sustained oscillations in active matter. Theproposed mechanism could be investigated experimen-tally since bacteria such as
E. coli and
B. subtilis remainmotile even when constrained in thin channels with a sizethat is comparable to their diameter [35, 36].We acknowledge support from MIUR-FIRB projectRBFR08WDBE. RDL acknowledges funding from theEuropean Research Council under the European UnionsSeventh Framework Programme (FP7/2007-2013)/ERCgrant agreement No. 307940. -1 0 1n - n -1 0 1 j + j -1 0 1n - n j - j (a) (b) (c) P j + j V a r (cid:113) FIG. 5: Double channel. Top panels: the join probabilities P ( j + j , δn ) (panel (a)) and P ( j − j , δn ) (panel (b)) for φ =0 .
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