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Dynamics of self-organized rotating spiral-coils in bacterial swarms† Szu-Ning Lin, Wei-Chang Lo and Chien-Jung Lo* Self-propelled particles (SPP) exhibit complex collective motions, mimicking autonomous behaviors that are often seen in the natural world, but essentially are generated by simple mutual interactions. Previous research on SPP systems focuses on collective behaviors of a uniform population. However, very little is known about the evolution of individual particles under the same global influence. Here we show selforganized rotating spiral coils in a two-dimensional (2D) active system. By using swarming bacteria Vibrio alginolyticus as an ideal experimental realization of a well-controlled 2D self-propelled system, we study the interaction between ultra-long cells and short background active cells. The self-propulsion of long cells and their interactions with neighboring short cells leads to a self-organized, stable spiral rotational state in 2D. We find four types of spiral coils with two main features: the rotating direction (clockwise or counter-clockwise) and the central structure (single or double spiral). The body length of the spiral coils falls between 32 and 296 mm and their rotational speed is within a range from 2.22 to 22.96 rad s1. The dynamics of these spiral coils involves folding and unfolding processes, which require local velocity
Received 7th August 2013 Accepted 6th November 2013
changes of the long bacterium. This phenomenon can be qualitatively replicated by a Brownian dynamics simulation using a simple rule of the propulsion thrust, imitating the reorientation of bacterial
DOI: 10.1039/c3sm52120f
flagella. Apart from the physical and biological interests in swarming cells, the formation of self-
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organized spiral coils could be useful for the next generation of microfabrication.
1. Introduction The emergent states of dense active particles show rich macroscopic patterns including swarms and vortices.1–4 The mutual interaction between these self-propelled particles (SPP) leads to dynamical patterns with uid-like features. These have been studied by discrete and continuous models regarding the SPP density,1–4 noises,1 and the energy transport.5 The experimental systems are limited due to technical difficulties in producing and controlling numerous autonomous agents. Biological samples at the cellular level are good candidates for reproducible experiments.5,6 In vitro experiments at the molecular level that use puried motor proteins and laments can also produce well-controlled experiments.7,8 Articial samples such as Janus particles driven by either catalytic processes6 or thermophoretic gradients,9 and vibration granules10 system are potential candidates for the experimental realization as well. In particular, bacterial swarm systems are exceptionally suitable for this purpose because bacteria can replicate themselves without much care and are easy to observe,11 and we have quite a good understanding of their chemotaxis and motility.12,13
Department of Physics and Graduate Institute of Biophysics, National Central University, Jhongli, Taiwan 32001, Republic of China. E-mail: [email protected] † Electronic supplementary 10.1039/c3sm52120f
information
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(ESI)
available.
See
DOI:
Furthermore, among these SPP systems, microswimmers have attracted a great deal of attention for their intriguing, diverse swimming mechanisms,14 and for their potential applications in devising articial swimmers. Micro-organisms such as bacteria have been extensively studied for decades.14–17 In an environment of low Reynolds numbers, the viscosity of uids dominates the inertia, and the hydrodynamic ow can be described by the Stokes equation. Most previous studies focused on the hydrodynamics of bulk liquids, free-standing lms, or uids near solid walls. On the other hand, our understanding of hydrodynamic interactions in bacterial swarms, typically existing in a quasi-two-dimensional liquid layer on an agar gel, is relatively low.18,19 Besides the research focus on the collective behavior of SPP system, passive particles embedded in SPP demonstrates enhanced diffusion.6 More interestingly, passive asymmetric micro-gear can be driven to rotate by active bacteria suspensions.20,21 Following research on passive micro-gears driven by SPP, we are interested in an active system that comprises autonomous SPP, which have the same propulsion mechanism but over different lengths. Vibrio alginolyticus swarm cells provide a good experimental system and simple geometric difference in length for observation. Very little is known of how polydisperse individual active particles evolve with each other. Here we demonstrate a self-organized stable structure in such an active system. We nd that, surprisingly, long swarming
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bacteria can form rotating spiral coils among the active short swarming bacteria in two dimensions. Our ndings suggest the existence of local, self-organized stable states in active systems. In such a system, the interactions between bacteria (e.g. the excluded volume interaction) and the orientation of agella are crucial to these self-organized stable states. The rich conformation of a long swarming bacterium and the self-organization of spiral coils are the consequence of interactions with other short background cells. We also demonstrate a Brownian dynamics simulation that is able to form a self-organized spiral coil by employing a simple propulsion rule and excluded volume interactions among constituent beads. The simulation results imply that the essential factor of forming a stable rotating spiral coil is the capability of reorientating the propulsion force generated by agella, which has been reported in the literature.22
2.
Experiments
When bacteria swarm on nutrient-rich agar surfaces, they elongate without division and express more agella.23,24 Vibrio alginolyticus swarm cells show a wide range of cell lengths, mainly 4 mm and up to 296 mm. This provides an ideal experimental realization to achieve our goal. Inspired by the observation of various conformations of long swarming cells in a Vibrio alginolyticus colony (Fig. 1a and b), we propose a new system of self-propelled particles, which consists of the same propulsion mechanism but over two very different aspect ratios in 2D. Vibrio alginolyticus has a dual-agella system that composes of a single Na+-driven agellum used for swimming in bulk liquid and lateral H+-driven agella for situations on surfaces or in highly viscous environments.25–27 Vibrio alginolyticus cells are excellent surface swarmers and can reach a speed of 29.0 12.9 mm s1. Bacterial strains and swarming assay Cells of Vibrio alginolyticus strain YM19, a polar agellumdefective mutant (Pof Laf+)28 were grown in VC medium (0.5% polypepton, 0.5% yeast extract, 0.4% K2HPO4, 3% NaCl, 0.2% glucose) overnight at 30 C. A 0.5 mL drop of saturated culture was inoculated on the central top of the VC agar (VC medium and 1% agar) and grown at 30 C for another 3 hours. The dish was then transferred to the Nikon Ti-U microscope with a 40 (NA 0.65) phase contrast objective. The images were taken by a CCD camera (AVT GE680) at 50 frames per second. Bacterial images analysis was performed using customized IDL programs and ImageJ. The cells are motile within the thin liquid layer between the agar and the air (Fig. 1a). The lateral agella can form bundles to propel cells in this environment.22 The spreading colony exhibits different cell densities and metabolic activities in the radial direction.29 The most metabolically active region is near the edge of the spreading colony with a few layers of cells. In the inner region of the swarm plate, the cell density reduces to a level at which we can observe individual cells in a single layer. We recorded the bacterial motions by an inverted optical
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microscope with the high speed camera. Shown in Fig. 1b are some typical images taken in the inner region of the colony, where a few long bacteria are interacting with high-density short cells (55% average coverage ratio) in a thin layer. On the swarm plate, Vibrio alginolyticus cells elongate and express lateral agella.25–27 The lateral agella only rotate in the counter-clockwise direction28 so that the cell can only change its moving direction while the agella are reorientated. Cells show persistent movement unless they are stopped by other cells. The agella density is 5 agella per mm.25 In the inner region, the length distribution has a very long tail with a mean of 4.3 1.2 mm, and the longest cell we have found is as long as 296 mm. In this paper, we focus on the dynamics of these long cells, which interact with a large number of short cells.
Dynamical conformations The bacterial cell wall is a rigid structure to maintain the cell shape. In aqueous environments, long cells are almost straight. However, while interacting with the background short cells, the long cells show very dynamical changes to their conformations (Fig. 1b). To characterize the length-dependent dynamical properties of cell conformations, we measured their end-point positions,~ r 1 and~ r 2, and the end-to-end vector, d~ r ¼~ r 1 ~ r 2. The mean square displacement (MSD, denoted as gd) of ~ r 1, ~ r 2, and the mean square variation (gv) of the end-to-end vector d~ r show the dynamical transition over different cell lengths (L) (Fig. 1c–e). In 2D, gd,v ¼ 4Dta, where a ¼ 2 indicates ballistic motion, a ¼ 1 diffusive motion (random walk), and a ¼ 0 caged motion (xed distance). Fig. 1c shows a short cell of 3.2 mm long whose two ends move ballistically, and the end-to-end distance remains almost a constant. For the longer cells (Fig. 1d), the cell body can be bent by the external cluster of cells. The two ends show ballistic motion, but the end-to-end distance shows superdiffusive motion. For the ultra-long cells (Fig. 1e), the two ends are still ballistic while the tted a for the end-to-end vector becomes 2 as well. This implies that the motions of the two ends are not synchronized. The tted a from gv of the observed cells with different L are plotted in Fig. 1f (triangle), illustrating the transition from short hard-rods to long semi-exible chains. In this inner region of the colony, we can nd long cells that are folded into spiral coils and persistently rotate. Once the spiral coil formed, it can stably rotate for at least ve minutes. Fig. 2a displays a typical eight-round double spiral coil. Fig. 2b shows its cumulative angle, and the rotational speed of the spiral coil is 12 rad s1. The neighboring short-cell clusters interacted with the spiral coil in two different ways. For single or small numbers of cells colliding with the spiral coil, they just owed through the edge like streaming interactions. On the contrary, large clusters could push the coil to move it. Fig. 2c shows the spiral coil displacement with time. It shows uctuating displacements and occasionally some vast displacements when a large cluster interacted with it. Fig. 2d illustrates the bacterial velocity eld calculated by Particle Image Velocimetry (PIV) at the time of the largest displacement shown in Fig. 2c. Nonetheless, the MSD of the spiral center suggests a caged
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Fig. 1 The bacterial swarm system. (a) A schematic diagram of the swarm plate and a Vibrio alginolyticus swarm cell with lateral flagella. Bacteria are swarming in the liquid layer on top of the agar plate with a glass support below. (b) Typical images of the swarm plate with different lengths of cells. Lower row: original time sequence images. Upper row: un-selected cells removed to highlight the configuration of long, medium and short cells. Long cells show more dynamical changes of configurations while short cells show straighter configurations. Scale bar, 10 mm. (c–e) Mean square displacement analyses of two end points (triangle and open circle, gd ¼ h[~ r i(t) ~ r i(t0)]2i) and mean square variation of end-to-end vectors (square, gv ¼ h[d~ r(t) d~ r(t0)]2i) in a short (3.2 mm, c), medium (18.7 mm, d) and long (154.3 mm, e) cell, respectively. All curves are fitted by MSD ¼ Ata over the short time region (t ¼ 20–100 ms). The fitted a of (gd1,gd2,gv) are (1.85, 1.85, 0.38) for (c), (1.85, 1.82, 1.64) for (d), and (1.87, 1.90, 1.85) for (e) respectively. (f) The fitting exponent a of the gv versus the total cell length of experimental data (open triangle) and simulation data (circle).
behavior. Compared to the MSD of unfolded long cells, the displacement hints at very different dynamics in these cases.
Rotating spiral coils We nd that there are four kinds of rotating spiral coils with two different features in the swarm plate: the rotational direction and the central structure. As shown in Fig. 3, there are single clockwise (SCW), single counter-clockwise (SCCW), double clockwise (DCW), and double counter-clockwise (DCCW) spiral coils. The single spiral coil can be described as an Archimedes'
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spiral with the relation R ¼ bq/2p, where R is the spiral radius, q the spiral angle and b the cell width for the close packing. The double spiral can be viewed as two mirror single spirals connected smoothly at the center with R ¼ bq/p for each arm. There is a wide range of spiral coil lengths, which falls between 32 and 296 mm. The shortest coil has two rounds. Table 1 shows the statistics of the spiral coils. There are about the same number of single and double spirals, but more CW spirals than their CCW counterparts. The lateral agellar motors of Vibrio alginolyticus can only rotate in the CCW direction.28 The cell body will rotate in the CW direction relative
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Table 1
Statistics of four types of spiral coilsa
Coil types
Rotation speed (rad s1)
Total length
% of all types
SCCW SCW DCCW DCW
12.52 13.95 6.50 9.70
85.24 99.57 138.56 127.21
14.70 35.29 4.90 45.09
a
Total 102 spiral coils.
Folding and unfolding
Self-organized rotating bacterial spiral coil. (a) Image of a stable rotating bacterial spiral coil with surrounding short cells. The radius of the spiral coil is 8 mm. Scale bar, 10 mm. (b) The cumulative rotational angle (q) and rotational frequency (u) of the spiral coil in (a). (c) Displacement of the spiral coil. The largest displacement occurs at 17.2 s. (d) The PIV calculation of the time at which the spiral coil shows the largest displacement. The white circle indicates the location of the spiral coil. Fig. 2
Fig. 3 Four types of self-organized rotating bacterial spiral coils. Time sequence images of rotating spiral coils and the mathematical model plots. (a) Single clockwise spiral coil; (b) single counter-clockwise spiral coil; (c) double clockwise spiral coil; (d) double counter-clockwise spiral coil. Arrows indicate the rotational directions. Scale bar, 10 mm.
to its axis to counter the angular momentum such that the cell body experiences a force, due to the surface-shear, to move the cell toward its starboard side.30 This is likely the explanation of the number ratio between the CW and CCW spiral coils in our observations.
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For the single spiral coils, the folding process requires an initialization of turning directions. The cell wall is stiff enough to prevent the self-turning. The turning events happen when one end encounters other short cells. Once the end is turning towards the long cell itself, this leads to self-trapping and initialization of the formation of spiral coils, see Fig. 4a. We also nd that about half of the spiral coils are double spiral coils with an S-shaped center. The folding process requires a hair-pin structure to be formed in the rst place. It occurs as the middle part of the long cell is buckling due to other groups of cells. Hair-pin structures are common among the long cells. Aerward it requires another step in which the hair-pin becomes a new leading ‘head’ and turning into a spiral, as shown in Fig. 4c. During these processes, the agella on the front-half may change orientations and therefore the hair-pin becomes a new leading end. On the other hand, for the unfolded long cells, there is no head-tail preference; both ends are moving seemingly independently. We never nd inversed double spiral coils whose two ends are closed in the center. It appears to be difficult for the two ends to move together and be trapped at the spiral center. Moreover, we also observed unfolding events with the stable rotating spiral coils unfolded to long strings. The unfolding processes require two steps. First the rotational speed of the coil is interrupted, for example, by short cells intruding to the coil through the gap between rounds. When this happens, there is possibility that the agella change orientations for the open ends to vary their moving directions. Once the ends change directions, the whole structure is no longer stable and will unfold immediately, see Fig. 4b. The typical time scales of folding and unfolding are merely a few seconds. The most important feature of the folding and unfolding events is that they are a consequence of a simple mechanism, that the thrustgenerating agella can change orientation in low speed conditions.22
3.
Brownian dynamics simulation
Since the bacterial swarm is conned in a liquid layer on top of an agar gel, as mentioned in the Introduction, the hydrodynamic ow is expected to be involved in the dynamic behavior of these bacteria. Despite the lack of detailed microscopic theory for such a condition, generally speaking one should consider both near-eld and far-eld ows because of the high density Soft Matter, 2014, 10, 760–766 | 763
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interaction, to see whether these ingredients (the steric effect and agellar reorientation) are sufficient to generate stably rotating spiral coils. The Brownian dynamics simulation is employed via a C program with subroutines from Numerical Recipes in C. The cells are semi-exible chains of connected self-propelled, steric beads (diameter s ¼ 1 mm) whose motion is conned on a plane, see Fig. 5a. There are two types of cells; background short cells and one long cell in an area of 60 60 mm2 with periodic boundary conditions. Each bead obeys the overdamped Langevin equation as follows: x
n o ~n ðtÞ ðnÞ ðnÞ dR ~n þ f~p ðtÞ þ f~ ðtÞ; ¼ VU R dt
(1)
where x is the friction constant, U({~ Rn}) the interaction potential ~(n) the random between beads, ~ f (n) p the propulsion force and f
Fig. 4 Folding, unfolding and the model of spiral coils. (a) The folding process of a single counter-clockwise spiral coil. Left: time sequence images of the folding process. The selected cell is artificially colored. Right: the end-to-end distance and the radius of gyration of the cell. (b) The unfolding of a double clockwise spiral coil. Left: time sequence images of the unfolding process. The selected cell is artificially colored. Right: the end-to-end distance and the radius of gyration of the cell. (c) Schematic diagrams of the folding processes of spiral coils. Scale bar, 10 mm.
and density uctuations in the bacterial swarm.31 Other than the hydrodynamic interactions between the bacteria, the ow in the liquid layer can penetrate into the porous media below (the agar gel), which introduces a different boundary condition from that of a rigid wall.18 However, based on the experimental observation we have seen, the MSD of short bacteria suggests a ballistic regime for the time scale on the order of 102 to 101 second. This implies that the interaction between the bacteria could be mainly of the excluded volume interaction. Furthermore, we are most interested in the formation of spiral coils and their stability, which seem to be involved with the reorientation of agella. As a consequence, we attempt to employ a simple, coarse-grained Brownian dynamics simulation, excluding the hydrodynamic
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Fig. 5 The Brownian dynamics simulations of self-propelled semiflexible chains. (a) A schematic diagram of the chain model of the long cell and short ones. (b) A typical simulation snapshot of a stiff long chain exhibits buckling caused by the neighboring short chains. (c) Time sequence images of the simulated folding process of a single counter-clockwise spiral coil. (d) The end-to-end distance and the radius of gyration of the cell in (c). (e) The cumulative rotational angle and rotational frequency of the simulated spiral coil after folding.
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force acting on the bead n. Although a microswimmer should be treated as a force-free swimmer (i.e. with a force dipole rather than a force monopole used here),15 in our model the liquid is merely a passive viscous medium to dissipate energy. Therefore, one shall expect that the result may quantitatively change if the force-free condition and the hydrodynamic ow are properly implemented into the model. The interaction potential contains the Weeks–Chandler–Andersen (WCA) potential for the excluded volume interaction, the nitely extensible nonlinear elastic (FENE) potential for the bonding, and a typical elastic bending energy. More details are in the ESI.† In order to model the change of propelling directions of agella on the bacteria, we propose a simple way to imitate the propulsion force generated by agella. First we assume that each bead is propelled by a constant propulsion force | ~ f (n) p | ¼ 0.2 pN. Despite the xed magnitude, the direction of the propulsion can be changed according to the current speed of the bead. The angle deviation from the tangential direction of the long cell is Dqp, Dqp ¼
2p
1 1 X ; cX ˛R : \X \ ; 2 2 v 1 þ Co ~
(2)
where X is a uniformly distributed random variable. The coefcient Co controls how large the angle deviation can be due to the current speed |~ v|. From our observation, when a bacterium is blocked by other bacteria, it can reverse and escape from the obstacles without changing the motor rotating direction.22 Therefore, the agella must reorientate in order to reverse the bacterial moving direction. Fig. 5b shows a typical snapshot of a long cell with short cells. The tted a from gv of the simulated cells with different L are plotted in Fig. 2f (circles), showing characteristic dynamics that are consistent with the experiments. We also nd that the long cell self-organizes and folds into a spiral coil, in Fig. 5c and d, in our simulation, and rotates stably, as shown in Fig. 5e.
32 3 2 ha u ¼ 0:77 1015 W. Previous experi3 ments using passive asymmetric micro-gears in an active bacterial suspension20,21 have shown that the gear can rotate a few revolutions per second with signicant uctuations and a translated power 1015 W. The main contribution driving objects to rotate in the study of micro-gear is from stochastic collisions, but in our case the rotation is driven by the motor output of the rotating spiral coil itself. As for how the spiral coil is able to stably rotate, we consider a simple argument of force balance to explain it. Suppose that there is a rotating spiral coil with a constant angular velocity. On each point of the spiral coil, as modeled by an Archimedes' spiral, there are three forces in balance: the propulsion force, the drag, and the elastic restoring force due to bending.33 Since the degree of bending is decreasing while one follows the contour from the center toward the outer rim, the elastic restoring force is reducing accordingly, as shown in Fig. 6 (more details are in the ESI†). Meanwhile, the local drag is increasing because of the increasing translational velocity. As a result, the propulsion force has to counteract the net force of these two to achieve force balance of the spiral coil, as shown in the inset of Fig. 6. Nevertheless, before the rigid long cell forms a spiral coil, the deformation requires extra forces, contributed by short cells. The short active cells can stiffen or bend the long cells depending on the relative moving directions.34 As a result, active long cells display much richer dynamical congurations compared to passive long polymers in 2D.35 The dynamical properties also originate from the exclusively CCW rotating agella, which can change the orientation in swarm conditions.22 Even with the lateral agellar motor unable to switch, we can observe the cells reversing directions when they collide with long cells in the experiments. This indicates that the agellar orientation can be changed without switching. For the folding and unfolding of spiral coils, these processes require changes in the direction of local speeds that are results of changing agella orientations. In our Brownian dynamics estimated as P ¼
4. Discussion E. coli's agella can generate an average bundle thrust of 0.28 pN in aqueous medium32 with 4–8 agella. Therefore, each agellum can generate about 0.05 pN thrust. A single bacterial agellar motor can output a torque of 2.0 1018 N m1.12,13 The output torque for the spiral coil in Fig. 1 is estimated to be 32 T ¼ ha3 u ¼ 6:5 1017 N m1 , where h is the viscosity of the 3 medium, a (8 mm) the spiral radius, and u (11.86 rad s1) the rotational speed. Assuming only the agella in the spiral’s outermost round contributes to the rotation, and the torque is balanced with the rotational viscous drag. The torque generated 0 by the agella is T ¼ NF a ¼ T, where N is the number of agella, thus the cell needs about 160 agella to make it happen. In the outermost round there are 250 agella, estimated from average the Vibrio alginolyticus swarm cells agella density, 5 agella per mm.25 Despite this estimation being not far from the known characteristics, it is as yet unclear how these thin agella function in a crowded space. The output power is This journal is © The Royal Society of Chemistry 2014
Elastic restoring force per unit arc length of a spiral coil versus the spiral angle q. The restoring force due to bending is reducing along the contour from the center toward the outer rim, as shown in the figure. As a result, for the spiral coil to stably rotate at a fixed position, the forces exerted on each part of the spiral have to balance with each other, as shown in the inset. As the restoring force and the drag vary along the contour, the magnitude and direction of the propulsion force must change accordingly.
Fig. 6
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simulation, the novel ingredient, different from previous numerical research, is that the propulsion force of selfpropelled particles can change direction according to certain conditions. By incorporating this rule, we can reproduce the self-organized processes of spiral coils.
5.
Conclusion
So far we have shown that the formation of spiral coils in 2D requires the following conditions: (i) a self-propelled long cell interacting with other active particles, and (ii) local changes of propulsion direction that depends on translational speed. We have demonstrated that the routes of folding and unfolding of single and double spiral coils involve the change of local speeds of the long cells. In our simulation, we use a coarse-grained model with merely the excluded volume interaction, cell stiffness, and a propulsion force, which is able to reorientate its direction, to examine the fundamental requirements of forming the spiral coil. With the 2D constraints, long self-propelled strings can form stable rotating spiral coils. Similarly, one may expect such a system to construct more complicated structures in three dimensions, such as knots, that are locally stable as well. Moreover, the inclusion of hydrodynamic interactions may quantitatively change the characteristics of the system, for instance, the density uctuation and the mean size of a cluster. Further studies are needed in order to fully understand an active system similar to the bacterial swarm investigated in this work. Our ndings in this work suggest that richer dynamics can be seen in polydisperse or heterogeneous active system of selfpropelled particles. The presented work and the understanding of these systems may help develop future applications, and further investigations are necessary for integrating the active systems into new technologies.
Acknowledgements This work is supported by the National Science Council of the Republic of China under contract no. NSC98-2112-M-008-010MY3 and no. NSC101-2112-M-008-008. We thank Michio Homma and Seiji Kojima (Nagoya University, Japan) for providing the bacteria Vibrio alginolyticus strains. We also thank Ikuro Kawagishi, Yoshiyuki Sowa and Hsuan-Yi Chen for helpful discussion. WC Lo acknowledges the support of a postdoctoral fellowship from HY Chen.
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5 T. Ishikawa, N. Yoshida, H. Ueno, M. Wiedeman, Y. Imai and T. Yamaguchi, Phys. Rev. Lett., 2011, 107, 028102. 6 G. Mino, T. E. Mallouk, T. Darnige, M. Hoyos, J. Dauchet, J. Dunstan, R. Soto, T. Wang, A. Rousselet and E. Clement, Phys. Rev. Lett., 2011, 106, 048102. 7 V. Schaller, C. A. Weber, B. Hammerich, E. Frey and A. R. Bausch, Proc. Natl. Acad. Sci. U. S. A., 2011, 108, 19183. 8 Y. Sumino, K. H. Nagai, Y. Shitaka, D. Tanaka, K. Yoshikawa, H. Chate and K. Oiwa, Nature, 2012, 483, 448. 9 H. R. Jiang, N. Yoshinaga and M. Sano, Phys. Rev. Lett., 2010, 105, 268302. 10 A. Kudrolli, G. Lumay, D. Volfson and L. S. Tsimring, Phys. Rev. Lett., 2008, 100, 058001. 11 H. P. Zhang, A. Beer, E.-L. Florin and H. L. Swinney, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 13626. 12 M. Baker and R. Berry, Contemp. Phys., 2009, 50, 617. 13 H. C. Berg, Annu. Rev. Biochem., 2003, 72, 19. 14 E. Lauga and T. R. Powers, Rep. Prog. Phys., 2009, 72, 096601. 15 E. M. Purcell, Am. J. Phys., 1977, 45, 3. 16 S. Mishra, A. Baskaran and M. C. Marchetti, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2010, 81, 061916. 17 K. Drescher, R. E. Goldstein, N. Michel, M. Polin and I. Tuval, Phys. Rev. Lett., 2010, 105, 168101. 18 J. Lega and T. Passot, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2003, 67, 031906. 19 J. Elgeti and G. Gompper, Europhys. Lett., 2009, 85, 38002. 20 A. Sokolov, M. Apodaca, B. Grzybowski and I. Aranson, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 969. 21 R. Leonardo, L. Angelani, D. Dell'Arciprete, G. Ruocco, V. Iebba, S. Schippa, M. P. Conte, F. Mecarini, F. De Angelis and E. Di Fabrizio, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 9541. 22 L. Turner, R. Zhang, N. C. Darnton and H. C. Berg, J. Bacteriol., 2010, 192, 3259. 23 M. F. Copeland and D. B. Weibel, So Matter, 2009, 5, 1174. 24 C. J. Gode-Potratz, R. J. Kustusch, P. J. Breheny, D. S. Weiss and L. L. McCarter, Mol. Microbiol., 2011, 79, 240. 25 W. E. De Boer, C. Golten and W. A. Scheffers, Neth. J. Sea Res., 1975, 9, 197. 26 L. McCarter, J. Mol. Microbiol. Biotechnol., 2004, 7, 18. 27 N. Li, S. Kojima and M. Homma, Genes Cells, 2011, 16, 985. 28 K. Kojima, R. Kudo, T. Yakushi, M. Homma and I. Kawagishi, Mol. Microbiol., 2007, 64, 57. 29 N. C. Darnton, L. Turner, S. Rojevsky and H. C. Berg, Biophys. J., 2010, 98, 2082. 30 E. Lauga, W. R. DiLuzio, G. M. Whiteesides and H. A. Stone, Biophys. J., 2006, 90, 400. 31 T. Ishikawa, J. Locsei and T. Pedley, J. Fluid Mech., 2008, 615, 401. 32 N. C. Darnton, L. Turner, S. Rojevsky and H. C. Berg, J. Bacteriol., 2007, 189, 1756. 33 M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, 1986, ch. 8.8. 34 N. Kikuchi, A. Ehrlicher, D. Koch, J. A. Kas, S. ramaswamy and M. Rao, Proc. Natl. Acad. Sci. U. S. A., 2009, 106, 19776. 35 B. Maier and J. O. Radler, Phys. Rev. Lett., 1999, 82, 1911.
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Cite this: Soft Matter, 2014, 10, 760
Dynamics of self-organized rotating spiral-coils in bacterial swarms† Szu-Ning Lin, Wei-Chang Lo and Chien-Jung Lo* Self-propelled particles (SPP) exhibit complex collective motions, mimicking autonomous behaviors that are often seen in the natural world, but essentially are generated by simple mutual interactions. Previous research on SPP systems focuses on collective behaviors of a uniform population. However, very little is known about the evolution of individual particles under the same global influence. Here we show selforganized rotating spiral coils in a two-dimensional (2D) active system. By using swarming bacteria Vibrio alginolyticus as an ideal experimental realization of a well-controlled 2D self-propelled system, we study the interaction between ultra-long cells and short background active cells. The self-propulsion of long cells and their interactions with neighboring short cells leads to a self-organized, stable spiral rotational state in 2D. We find four types of spiral coils with two main features: the rotating direction (clockwise or counter-clockwise) and the central structure (single or double spiral). The body length of the spiral coils falls between 32 and 296 mm and their rotational speed is within a range from 2.22 to 22.96 rad s1. The dynamics of these spiral coils involves folding and unfolding processes, which require local velocity
Received 7th August 2013 Accepted 6th November 2013
changes of the long bacterium. This phenomenon can be qualitatively replicated by a Brownian dynamics simulation using a simple rule of the propulsion thrust, imitating the reorientation of bacterial
DOI: 10.1039/c3sm52120f
flagella. Apart from the physical and biological interests in swarming cells, the formation of self-
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organized spiral coils could be useful for the next generation of microfabrication.
1. Introduction The emergent states of dense active particles show rich macroscopic patterns including swarms and vortices.1–4 The mutual interaction between these self-propelled particles (SPP) leads to dynamical patterns with uid-like features. These have been studied by discrete and continuous models regarding the SPP density,1–4 noises,1 and the energy transport.5 The experimental systems are limited due to technical difficulties in producing and controlling numerous autonomous agents. Biological samples at the cellular level are good candidates for reproducible experiments.5,6 In vitro experiments at the molecular level that use puried motor proteins and laments can also produce well-controlled experiments.7,8 Articial samples such as Janus particles driven by either catalytic processes6 or thermophoretic gradients,9 and vibration granules10 system are potential candidates for the experimental realization as well. In particular, bacterial swarm systems are exceptionally suitable for this purpose because bacteria can replicate themselves without much care and are easy to observe,11 and we have quite a good understanding of their chemotaxis and motility.12,13
Department of Physics and Graduate Institute of Biophysics, National Central University, Jhongli, Taiwan 32001, Republic of China. E-mail: [email protected] † Electronic supplementary 10.1039/c3sm52120f
information
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(ESI)
available.
See
DOI:
Furthermore, among these SPP systems, microswimmers have attracted a great deal of attention for their intriguing, diverse swimming mechanisms,14 and for their potential applications in devising articial swimmers. Micro-organisms such as bacteria have been extensively studied for decades.14–17 In an environment of low Reynolds numbers, the viscosity of uids dominates the inertia, and the hydrodynamic ow can be described by the Stokes equation. Most previous studies focused on the hydrodynamics of bulk liquids, free-standing lms, or uids near solid walls. On the other hand, our understanding of hydrodynamic interactions in bacterial swarms, typically existing in a quasi-two-dimensional liquid layer on an agar gel, is relatively low.18,19 Besides the research focus on the collective behavior of SPP system, passive particles embedded in SPP demonstrates enhanced diffusion.6 More interestingly, passive asymmetric micro-gear can be driven to rotate by active bacteria suspensions.20,21 Following research on passive micro-gears driven by SPP, we are interested in an active system that comprises autonomous SPP, which have the same propulsion mechanism but over different lengths. Vibrio alginolyticus swarm cells provide a good experimental system and simple geometric difference in length for observation. Very little is known of how polydisperse individual active particles evolve with each other. Here we demonstrate a self-organized stable structure in such an active system. We nd that, surprisingly, long swarming
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bacteria can form rotating spiral coils among the active short swarming bacteria in two dimensions. Our ndings suggest the existence of local, self-organized stable states in active systems. In such a system, the interactions between bacteria (e.g. the excluded volume interaction) and the orientation of agella are crucial to these self-organized stable states. The rich conformation of a long swarming bacterium and the self-organization of spiral coils are the consequence of interactions with other short background cells. We also demonstrate a Brownian dynamics simulation that is able to form a self-organized spiral coil by employing a simple propulsion rule and excluded volume interactions among constituent beads. The simulation results imply that the essential factor of forming a stable rotating spiral coil is the capability of reorientating the propulsion force generated by agella, which has been reported in the literature.22
2.
Experiments
When bacteria swarm on nutrient-rich agar surfaces, they elongate without division and express more agella.23,24 Vibrio alginolyticus swarm cells show a wide range of cell lengths, mainly 4 mm and up to 296 mm. This provides an ideal experimental realization to achieve our goal. Inspired by the observation of various conformations of long swarming cells in a Vibrio alginolyticus colony (Fig. 1a and b), we propose a new system of self-propelled particles, which consists of the same propulsion mechanism but over two very different aspect ratios in 2D. Vibrio alginolyticus has a dual-agella system that composes of a single Na+-driven agellum used for swimming in bulk liquid and lateral H+-driven agella for situations on surfaces or in highly viscous environments.25–27 Vibrio alginolyticus cells are excellent surface swarmers and can reach a speed of 29.0 12.9 mm s1. Bacterial strains and swarming assay Cells of Vibrio alginolyticus strain YM19, a polar agellumdefective mutant (Pof Laf+)28 were grown in VC medium (0.5% polypepton, 0.5% yeast extract, 0.4% K2HPO4, 3% NaCl, 0.2% glucose) overnight at 30 C. A 0.5 mL drop of saturated culture was inoculated on the central top of the VC agar (VC medium and 1% agar) and grown at 30 C for another 3 hours. The dish was then transferred to the Nikon Ti-U microscope with a 40 (NA 0.65) phase contrast objective. The images were taken by a CCD camera (AVT GE680) at 50 frames per second. Bacterial images analysis was performed using customized IDL programs and ImageJ. The cells are motile within the thin liquid layer between the agar and the air (Fig. 1a). The lateral agella can form bundles to propel cells in this environment.22 The spreading colony exhibits different cell densities and metabolic activities in the radial direction.29 The most metabolically active region is near the edge of the spreading colony with a few layers of cells. In the inner region of the swarm plate, the cell density reduces to a level at which we can observe individual cells in a single layer. We recorded the bacterial motions by an inverted optical
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microscope with the high speed camera. Shown in Fig. 1b are some typical images taken in the inner region of the colony, where a few long bacteria are interacting with high-density short cells (55% average coverage ratio) in a thin layer. On the swarm plate, Vibrio alginolyticus cells elongate and express lateral agella.25–27 The lateral agella only rotate in the counter-clockwise direction28 so that the cell can only change its moving direction while the agella are reorientated. Cells show persistent movement unless they are stopped by other cells. The agella density is 5 agella per mm.25 In the inner region, the length distribution has a very long tail with a mean of 4.3 1.2 mm, and the longest cell we have found is as long as 296 mm. In this paper, we focus on the dynamics of these long cells, which interact with a large number of short cells.
Dynamical conformations The bacterial cell wall is a rigid structure to maintain the cell shape. In aqueous environments, long cells are almost straight. However, while interacting with the background short cells, the long cells show very dynamical changes to their conformations (Fig. 1b). To characterize the length-dependent dynamical properties of cell conformations, we measured their end-point positions,~ r 1 and~ r 2, and the end-to-end vector, d~ r ¼~ r 1 ~ r 2. The mean square displacement (MSD, denoted as gd) of ~ r 1, ~ r 2, and the mean square variation (gv) of the end-to-end vector d~ r show the dynamical transition over different cell lengths (L) (Fig. 1c–e). In 2D, gd,v ¼ 4Dta, where a ¼ 2 indicates ballistic motion, a ¼ 1 diffusive motion (random walk), and a ¼ 0 caged motion (xed distance). Fig. 1c shows a short cell of 3.2 mm long whose two ends move ballistically, and the end-to-end distance remains almost a constant. For the longer cells (Fig. 1d), the cell body can be bent by the external cluster of cells. The two ends show ballistic motion, but the end-to-end distance shows superdiffusive motion. For the ultra-long cells (Fig. 1e), the two ends are still ballistic while the tted a for the end-to-end vector becomes 2 as well. This implies that the motions of the two ends are not synchronized. The tted a from gv of the observed cells with different L are plotted in Fig. 1f (triangle), illustrating the transition from short hard-rods to long semi-exible chains. In this inner region of the colony, we can nd long cells that are folded into spiral coils and persistently rotate. Once the spiral coil formed, it can stably rotate for at least ve minutes. Fig. 2a displays a typical eight-round double spiral coil. Fig. 2b shows its cumulative angle, and the rotational speed of the spiral coil is 12 rad s1. The neighboring short-cell clusters interacted with the spiral coil in two different ways. For single or small numbers of cells colliding with the spiral coil, they just owed through the edge like streaming interactions. On the contrary, large clusters could push the coil to move it. Fig. 2c shows the spiral coil displacement with time. It shows uctuating displacements and occasionally some vast displacements when a large cluster interacted with it. Fig. 2d illustrates the bacterial velocity eld calculated by Particle Image Velocimetry (PIV) at the time of the largest displacement shown in Fig. 2c. Nonetheless, the MSD of the spiral center suggests a caged
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Fig. 1 The bacterial swarm system. (a) A schematic diagram of the swarm plate and a Vibrio alginolyticus swarm cell with lateral flagella. Bacteria are swarming in the liquid layer on top of the agar plate with a glass support below. (b) Typical images of the swarm plate with different lengths of cells. Lower row: original time sequence images. Upper row: un-selected cells removed to highlight the configuration of long, medium and short cells. Long cells show more dynamical changes of configurations while short cells show straighter configurations. Scale bar, 10 mm. (c–e) Mean square displacement analyses of two end points (triangle and open circle, gd ¼ h[~ r i(t) ~ r i(t0)]2i) and mean square variation of end-to-end vectors (square, gv ¼ h[d~ r(t) d~ r(t0)]2i) in a short (3.2 mm, c), medium (18.7 mm, d) and long (154.3 mm, e) cell, respectively. All curves are fitted by MSD ¼ Ata over the short time region (t ¼ 20–100 ms). The fitted a of (gd1,gd2,gv) are (1.85, 1.85, 0.38) for (c), (1.85, 1.82, 1.64) for (d), and (1.87, 1.90, 1.85) for (e) respectively. (f) The fitting exponent a of the gv versus the total cell length of experimental data (open triangle) and simulation data (circle).
behavior. Compared to the MSD of unfolded long cells, the displacement hints at very different dynamics in these cases.
Rotating spiral coils We nd that there are four kinds of rotating spiral coils with two different features in the swarm plate: the rotational direction and the central structure. As shown in Fig. 3, there are single clockwise (SCW), single counter-clockwise (SCCW), double clockwise (DCW), and double counter-clockwise (DCCW) spiral coils. The single spiral coil can be described as an Archimedes'
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spiral with the relation R ¼ bq/2p, where R is the spiral radius, q the spiral angle and b the cell width for the close packing. The double spiral can be viewed as two mirror single spirals connected smoothly at the center with R ¼ bq/p for each arm. There is a wide range of spiral coil lengths, which falls between 32 and 296 mm. The shortest coil has two rounds. Table 1 shows the statistics of the spiral coils. There are about the same number of single and double spirals, but more CW spirals than their CCW counterparts. The lateral agellar motors of Vibrio alginolyticus can only rotate in the CCW direction.28 The cell body will rotate in the CW direction relative
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Table 1
Statistics of four types of spiral coilsa
Coil types
Rotation speed (rad s1)
Total length
% of all types
SCCW SCW DCCW DCW
12.52 13.95 6.50 9.70
85.24 99.57 138.56 127.21
14.70 35.29 4.90 45.09
a
Total 102 spiral coils.
Folding and unfolding
Self-organized rotating bacterial spiral coil. (a) Image of a stable rotating bacterial spiral coil with surrounding short cells. The radius of the spiral coil is 8 mm. Scale bar, 10 mm. (b) The cumulative rotational angle (q) and rotational frequency (u) of the spiral coil in (a). (c) Displacement of the spiral coil. The largest displacement occurs at 17.2 s. (d) The PIV calculation of the time at which the spiral coil shows the largest displacement. The white circle indicates the location of the spiral coil. Fig. 2
Fig. 3 Four types of self-organized rotating bacterial spiral coils. Time sequence images of rotating spiral coils and the mathematical model plots. (a) Single clockwise spiral coil; (b) single counter-clockwise spiral coil; (c) double clockwise spiral coil; (d) double counter-clockwise spiral coil. Arrows indicate the rotational directions. Scale bar, 10 mm.
to its axis to counter the angular momentum such that the cell body experiences a force, due to the surface-shear, to move the cell toward its starboard side.30 This is likely the explanation of the number ratio between the CW and CCW spiral coils in our observations.
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For the single spiral coils, the folding process requires an initialization of turning directions. The cell wall is stiff enough to prevent the self-turning. The turning events happen when one end encounters other short cells. Once the end is turning towards the long cell itself, this leads to self-trapping and initialization of the formation of spiral coils, see Fig. 4a. We also nd that about half of the spiral coils are double spiral coils with an S-shaped center. The folding process requires a hair-pin structure to be formed in the rst place. It occurs as the middle part of the long cell is buckling due to other groups of cells. Hair-pin structures are common among the long cells. Aerward it requires another step in which the hair-pin becomes a new leading ‘head’ and turning into a spiral, as shown in Fig. 4c. During these processes, the agella on the front-half may change orientations and therefore the hair-pin becomes a new leading end. On the other hand, for the unfolded long cells, there is no head-tail preference; both ends are moving seemingly independently. We never nd inversed double spiral coils whose two ends are closed in the center. It appears to be difficult for the two ends to move together and be trapped at the spiral center. Moreover, we also observed unfolding events with the stable rotating spiral coils unfolded to long strings. The unfolding processes require two steps. First the rotational speed of the coil is interrupted, for example, by short cells intruding to the coil through the gap between rounds. When this happens, there is possibility that the agella change orientations for the open ends to vary their moving directions. Once the ends change directions, the whole structure is no longer stable and will unfold immediately, see Fig. 4b. The typical time scales of folding and unfolding are merely a few seconds. The most important feature of the folding and unfolding events is that they are a consequence of a simple mechanism, that the thrustgenerating agella can change orientation in low speed conditions.22
3.
Brownian dynamics simulation
Since the bacterial swarm is conned in a liquid layer on top of an agar gel, as mentioned in the Introduction, the hydrodynamic ow is expected to be involved in the dynamic behavior of these bacteria. Despite the lack of detailed microscopic theory for such a condition, generally speaking one should consider both near-eld and far-eld ows because of the high density Soft Matter, 2014, 10, 760–766 | 763
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interaction, to see whether these ingredients (the steric effect and agellar reorientation) are sufficient to generate stably rotating spiral coils. The Brownian dynamics simulation is employed via a C program with subroutines from Numerical Recipes in C. The cells are semi-exible chains of connected self-propelled, steric beads (diameter s ¼ 1 mm) whose motion is conned on a plane, see Fig. 5a. There are two types of cells; background short cells and one long cell in an area of 60 60 mm2 with periodic boundary conditions. Each bead obeys the overdamped Langevin equation as follows: x
n o ~n ðtÞ ðnÞ ðnÞ dR ~n þ f~p ðtÞ þ f~ ðtÞ; ¼ VU R dt
(1)
where x is the friction constant, U({~ Rn}) the interaction potential ~(n) the random between beads, ~ f (n) p the propulsion force and f
Fig. 4 Folding, unfolding and the model of spiral coils. (a) The folding process of a single counter-clockwise spiral coil. Left: time sequence images of the folding process. The selected cell is artificially colored. Right: the end-to-end distance and the radius of gyration of the cell. (b) The unfolding of a double clockwise spiral coil. Left: time sequence images of the unfolding process. The selected cell is artificially colored. Right: the end-to-end distance and the radius of gyration of the cell. (c) Schematic diagrams of the folding processes of spiral coils. Scale bar, 10 mm.
and density uctuations in the bacterial swarm.31 Other than the hydrodynamic interactions between the bacteria, the ow in the liquid layer can penetrate into the porous media below (the agar gel), which introduces a different boundary condition from that of a rigid wall.18 However, based on the experimental observation we have seen, the MSD of short bacteria suggests a ballistic regime for the time scale on the order of 102 to 101 second. This implies that the interaction between the bacteria could be mainly of the excluded volume interaction. Furthermore, we are most interested in the formation of spiral coils and their stability, which seem to be involved with the reorientation of agella. As a consequence, we attempt to employ a simple, coarse-grained Brownian dynamics simulation, excluding the hydrodynamic
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Fig. 5 The Brownian dynamics simulations of self-propelled semiflexible chains. (a) A schematic diagram of the chain model of the long cell and short ones. (b) A typical simulation snapshot of a stiff long chain exhibits buckling caused by the neighboring short chains. (c) Time sequence images of the simulated folding process of a single counter-clockwise spiral coil. (d) The end-to-end distance and the radius of gyration of the cell in (c). (e) The cumulative rotational angle and rotational frequency of the simulated spiral coil after folding.
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force acting on the bead n. Although a microswimmer should be treated as a force-free swimmer (i.e. with a force dipole rather than a force monopole used here),15 in our model the liquid is merely a passive viscous medium to dissipate energy. Therefore, one shall expect that the result may quantitatively change if the force-free condition and the hydrodynamic ow are properly implemented into the model. The interaction potential contains the Weeks–Chandler–Andersen (WCA) potential for the excluded volume interaction, the nitely extensible nonlinear elastic (FENE) potential for the bonding, and a typical elastic bending energy. More details are in the ESI.† In order to model the change of propelling directions of agella on the bacteria, we propose a simple way to imitate the propulsion force generated by agella. First we assume that each bead is propelled by a constant propulsion force | ~ f (n) p | ¼ 0.2 pN. Despite the xed magnitude, the direction of the propulsion can be changed according to the current speed of the bead. The angle deviation from the tangential direction of the long cell is Dqp, Dqp ¼
2p
1 1 X ; cX ˛R : \X \ ; 2 2 v 1 þ Co ~
(2)
where X is a uniformly distributed random variable. The coefcient Co controls how large the angle deviation can be due to the current speed |~ v|. From our observation, when a bacterium is blocked by other bacteria, it can reverse and escape from the obstacles without changing the motor rotating direction.22 Therefore, the agella must reorientate in order to reverse the bacterial moving direction. Fig. 5b shows a typical snapshot of a long cell with short cells. The tted a from gv of the simulated cells with different L are plotted in Fig. 2f (circles), showing characteristic dynamics that are consistent with the experiments. We also nd that the long cell self-organizes and folds into a spiral coil, in Fig. 5c and d, in our simulation, and rotates stably, as shown in Fig. 5e.
32 3 2 ha u ¼ 0:77 1015 W. Previous experi3 ments using passive asymmetric micro-gears in an active bacterial suspension20,21 have shown that the gear can rotate a few revolutions per second with signicant uctuations and a translated power 1015 W. The main contribution driving objects to rotate in the study of micro-gear is from stochastic collisions, but in our case the rotation is driven by the motor output of the rotating spiral coil itself. As for how the spiral coil is able to stably rotate, we consider a simple argument of force balance to explain it. Suppose that there is a rotating spiral coil with a constant angular velocity. On each point of the spiral coil, as modeled by an Archimedes' spiral, there are three forces in balance: the propulsion force, the drag, and the elastic restoring force due to bending.33 Since the degree of bending is decreasing while one follows the contour from the center toward the outer rim, the elastic restoring force is reducing accordingly, as shown in Fig. 6 (more details are in the ESI†). Meanwhile, the local drag is increasing because of the increasing translational velocity. As a result, the propulsion force has to counteract the net force of these two to achieve force balance of the spiral coil, as shown in the inset of Fig. 6. Nevertheless, before the rigid long cell forms a spiral coil, the deformation requires extra forces, contributed by short cells. The short active cells can stiffen or bend the long cells depending on the relative moving directions.34 As a result, active long cells display much richer dynamical congurations compared to passive long polymers in 2D.35 The dynamical properties also originate from the exclusively CCW rotating agella, which can change the orientation in swarm conditions.22 Even with the lateral agellar motor unable to switch, we can observe the cells reversing directions when they collide with long cells in the experiments. This indicates that the agellar orientation can be changed without switching. For the folding and unfolding of spiral coils, these processes require changes in the direction of local speeds that are results of changing agella orientations. In our Brownian dynamics estimated as P ¼
4. Discussion E. coli's agella can generate an average bundle thrust of 0.28 pN in aqueous medium32 with 4–8 agella. Therefore, each agellum can generate about 0.05 pN thrust. A single bacterial agellar motor can output a torque of 2.0 1018 N m1.12,13 The output torque for the spiral coil in Fig. 1 is estimated to be 32 T ¼ ha3 u ¼ 6:5 1017 N m1 , where h is the viscosity of the 3 medium, a (8 mm) the spiral radius, and u (11.86 rad s1) the rotational speed. Assuming only the agella in the spiral’s outermost round contributes to the rotation, and the torque is balanced with the rotational viscous drag. The torque generated 0 by the agella is T ¼ NF a ¼ T, where N is the number of agella, thus the cell needs about 160 agella to make it happen. In the outermost round there are 250 agella, estimated from average the Vibrio alginolyticus swarm cells agella density, 5 agella per mm.25 Despite this estimation being not far from the known characteristics, it is as yet unclear how these thin agella function in a crowded space. The output power is This journal is © The Royal Society of Chemistry 2014
Elastic restoring force per unit arc length of a spiral coil versus the spiral angle q. The restoring force due to bending is reducing along the contour from the center toward the outer rim, as shown in the figure. As a result, for the spiral coil to stably rotate at a fixed position, the forces exerted on each part of the spiral have to balance with each other, as shown in the inset. As the restoring force and the drag vary along the contour, the magnitude and direction of the propulsion force must change accordingly.
Fig. 6
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simulation, the novel ingredient, different from previous numerical research, is that the propulsion force of selfpropelled particles can change direction according to certain conditions. By incorporating this rule, we can reproduce the self-organized processes of spiral coils.
5.
Conclusion
So far we have shown that the formation of spiral coils in 2D requires the following conditions: (i) a self-propelled long cell interacting with other active particles, and (ii) local changes of propulsion direction that depends on translational speed. We have demonstrated that the routes of folding and unfolding of single and double spiral coils involve the change of local speeds of the long cells. In our simulation, we use a coarse-grained model with merely the excluded volume interaction, cell stiffness, and a propulsion force, which is able to reorientate its direction, to examine the fundamental requirements of forming the spiral coil. With the 2D constraints, long self-propelled strings can form stable rotating spiral coils. Similarly, one may expect such a system to construct more complicated structures in three dimensions, such as knots, that are locally stable as well. Moreover, the inclusion of hydrodynamic interactions may quantitatively change the characteristics of the system, for instance, the density uctuation and the mean size of a cluster. Further studies are needed in order to fully understand an active system similar to the bacterial swarm investigated in this work. Our ndings in this work suggest that richer dynamics can be seen in polydisperse or heterogeneous active system of selfpropelled particles. The presented work and the understanding of these systems may help develop future applications, and further investigations are necessary for integrating the active systems into new technologies.
Acknowledgements This work is supported by the National Science Council of the Republic of China under contract no. NSC98-2112-M-008-010MY3 and no. NSC101-2112-M-008-008. We thank Michio Homma and Seiji Kojima (Nagoya University, Japan) for providing the bacteria Vibrio alginolyticus strains. We also thank Ikuro Kawagishi, Yoshiyuki Sowa and Hsuan-Yi Chen for helpful discussion. WC Lo acknowledges the support of a postdoctoral fellowship from HY Chen.
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