Abstract
We unveil orbital topologies of two nearby swimming microorganisms using an artificial microswimmer, called Quadroar. Depending on the initial conditions of the microswimmers, we find diverse families of attractors including dynamical equilibria, bound orbits, braids, and pursuit–evasion games. We also observe a hydrodynamic slingshot effect: a system of two hydrodynamically interacting swimmers moving along braids can advance in space faster than noninteracting swimmers that have the same actuation parameters and initial conditions as the interacting ones. Our findings suggest the existence of complex collective behaviors of microswimmers, from equilibrium to rapidly streaming states.
Similar content being viewed by others
Introduction
Microorganisms including all species of bacteria, protozoa, and also some of alga, are playing an important role in recycling nutrients in the Earth’s ecosystems^{1,2}. A recent report by National Science Foundation (NSF) estimates the presence of about one trillion species right now on the Earth, that only onethousandth of one percent of which have been studied^{3}. This fact has put the investigation of natural or artificial microswimmers under the spotlight^{4,5,6,7}.
Swimming microorganisms in nature nearly always come in groups, and understanding their collective behaviors in the presence of hydrodynamic interactions requires multiscale models^{8,9,10}. The main challenge in developing statistical and continuum models is how we deal with collisional and relaxation processes, which are basically determined by twobody interactions^{11}. This is exacerbated by the long range nature of hydrodynamic interactions at low Reynolds number conditions that makes the investigation of the swarm dynamics of microswimmers substantially different from other wellstudied swarms. Specifically, swimmers at small scales strongly affect (at distances several bodylength away) their fluidic environment and hence their nearby swimmers (compare this with, say, Quadcopters whose influence on nearby copters is limited to a fraction of their body length). An interesting example, showing the significance of hydrodynamic interactions, is that they may trigger the locomotion of otherwise nonswimming reciprocal swimmers^{12,13}.
Prior observations have revealed a glimpse of these complex behaviors in nature^{14,15,16,17,18}. For instance, the parallel motion of two flagelladriven bacteria have been shown to be unstable^{14}, two nematodes tied to a wall from one end eventually get entangled^{15}, and two Paramecia avoid each other solely due to hydrodynamic interactions^{16}. Previous studies have addressed twobody dynamics using simple minimal model swimmers^{19,20,21,22,23,24,25,26} and have been able to report few basic behaviors. For example, two pullertype squirmers experience a significant change in their orientations after an encounter^{19}, which together with the swimmers’ inertial effects, causes hydrodynamic attraction^{20}. For spherical swimmers with spatially confined circular trajectories, the only possible longtime cell–cell interaction is either an attraction or repulsion^{21}, whereas two rigid helices do not attract or repel each other while rotating inphase^{22}. Another example is a systems of two linkedsphere swimmers^{27} that may converge, diverge oscillate or stay parallel to each other^{23,24}.
Recent experiments, nevertheless, uncover more complex flow fields around flagellated microorganisms than what had been previously thought^{28}. Specifically, flagellated microorganisms such as Chlamydomonas reinhardtii induce an oscillatory flow field that alternates between the flow fields of basic puller and pushertype swimmers^{28,29}, complicating the nature of two or multibody interactions of such microorganisms. To gain insight into the two and threedimensional interactions of microorganisms, we simulate them using the Quadroar swimmer whose flow field^{30} is similar to that of C. reinhardtii^{28,29,31}. One of the major advantages of the Quadroar as an artificial microswimmer is that it consists of rotary disks and only one reciprocating actuator. This remarkably simplifies the realization of the Quadroar as linear actuators (in all scales) are hard to fabricate and assemble. In nanoscales, the science and engineering of making molecular rotary units have leapt forward and claimed the Nobel prize in Chemistry in 2016^{32}, and molecularscale linear actuators can, in principle, be made of certain proteins^{33}. The Quadroar is highly controllable and has full threedimensional maneuverability. It can therefore track any prescribed spatial path^{33}. This has been a challenge in the design of medical microbots^{34} that makes a submillimeterscale Quadroar also a suitable candidate for various biological applications^{35} such as drug delivery or autonomous surgery^{34}. In macro scales, the Quadroar can be hired as a robotic swimmer^{36,37} for inspection missions in highly viscous fluid reservoirs^{38}.
In this study, we use the Quadroar swimmer and show that two model microswimmers in the Stokes regime–that generate flow fields resembling that of C. reinhardtii–have a rich twobody dynamics. Unlike other existing theoretical models that try to simulate the swimming mechanism of specific microorganisms^{39}, the Quadroar is designed to induce an oscillatory flow field with anterior, side and posterior vortices in its surrounding^{30}. Therefore, complex interactions that we find in the phasespace of two swimmers are generic characteristics of microorganisms generating anterior, side and posterior vortices.
Kinematics and Numerical Framework
The Quadroar consists of an Ishaped frame including an active chassis and two axles of length 2b (see Fig. 1). Each axle at its two ends is connected to two disks of radii a. The length of the chassis is variable and is equal to 2l + 2s(t) where s(t) is the contribution from the expansion/contraction of a linear actuator installed in the middle of the chassis. The angular position of each disk D_{ n } (n = 1, …, 4) with respect to the leg of its axle is denoted by − π ≤ ϑ_{ n } ≤ π. We define a bodyfixed Cartesian coordinate system (x_{1}, x_{2}, x_{3}) with its origin at the geometrical center of the frame. The (x_{1}, x_{2})plane lies in the plane of the swimmer and the x_{1}axis is along the chassis. We also define a global Cartesian coordinate system (X_{1}, X_{2}, X_{3}) as is shown in Fig. 1(b). The bodyfixed coordinates x_{ i } are related to the global coordinates X_{ i } (i = 1, 2, 3) through a transformation matrix R that depends on three orientation (Euler’s) angles of the swimmer. We assume that the influence of each disk on its surrounding environment can be modeled as a point force and a point torque^{40}.
For each of the two swimmers, j = A, B, and for each of their four disks n = 1, …, 4, the point forces (f_{ jn }) and torques (τ_{ jn }), expressed in the global coordinate frame are given by^{30}
where μ denotes dynamic viscosity of the surrounding fluid; v_{ jn } and ω_{ jn } are the linear and angular velocities of each disk with respect to the swimmer’s hydrodynamic center and bodyfixed coordinate frame, respectively; v_{j,c} is the absolute velocity of the hydrodynamic center and ω_{j,body} is the angular velocity of the jth swimmer expressed in terms of Euler’s angles; u_{ jn } and 2Ω_{ jn } are the velocity and vorticity fields of the fluid at the center of each disk; G is the isotropic rotation tensor and K_{ jn } is the translation tensor corresponding to disk n of swimmer j, defined as^{33}:
Since a selfpropelled swimmer in the Stokes regime is forcefree and torquefree, we must have \({\sum }_{n=1\,}^{4}{f}_{jn}\,=\,0\) and also \({\sum }_{n=1}^{4}[({{\bf{R}}}_{j}^{T}\cdot {{\boldsymbol{r}}}_{jn})\times {f}_{jn}+{{\boldsymbol{\tau }}}_{jn}]=0\) for j = A, B. These four sets of vectorial equations (two vectorial equations for each swimmer) require the values of velocities u_{ jn } and spins Ω_{ jn } to be complete and solvable for v_{ c } and ω_{body}. The linear nature of the Stokes equation allows us to invoke superposition and obtain^{30,40}:
where c_{0} = 8πμ, \({{\boldsymbol{X}}}_{kn,j}={{\bf{R}}}_{j}^{T}\cdot ({{\boldsymbol{r}}}_{jn}{{\boldsymbol{r}}}_{jk})\) and \({{\boldsymbol{X}}}_{kn,ij}=({{\boldsymbol{X}}}_{c,j}+{{\bf{R}}}_{j}^{T}\cdot {{\boldsymbol{r}}}_{jn})({X}_{c,i}+{{\bf{R}}}_{i}^{T}\cdot {{\boldsymbol{r}}}_{ik})\). The scalars z_{kn,j} and z_{kn,ij} are the magnitudes of the vectors X_{kn,j} and X_{kn,ij}, respectively, and r_{ jn } denotes the position vector of the nth disk in the swimmer’s local coordinate frame. In all expressions we have i,j = A, B with the condition i ≠ j in each expression.
We assume that disks D_{ n } (n = 1, …, 4), of each swimmer j = A, B are spinning with angular velocities \({\dot{\vartheta }}_{j1}={\dot{\vartheta }}_{j2}={c}_{0}{\omega }_{s}\) and \({\dot{\vartheta }}_{j3}={\dot{\vartheta }}_{j4}={c}_{0}{\omega }_{s}+\delta \omega \) where δω is a detuning parameter, and the length of the linear actuator at the middle of the chassis varies according to s(t) = s_{0}[1 − cos(ω_{ s }t)]/2. Throughout our simulations we set a = 1, b/a = 4, l/a = 4, s_{0}/l = 1/2, and ω_{ s } = 1. The characteristic time scale of the twobody system is T_{ s } = 2π/ω_{ s }. The parameter c_{0} affects both the swimmer’s dynamics and flow field around it. For c_{0} ≈ 0.5, it has been shown^{30} that the flow field induced by the Quadroar closely resembles that of C. reinhardtii alga^{29}. Our numerical experiments show that the resemblance holds for almost any c_{0} ≥ 1. This is similar to the recent experimental observations^{41} that swimming speed or beat frequency do not have a considerable effect on bioconvection behavior of C. reinhardtii cells. The similarity in behaviors for this broad range of c_{0}, which even includes the singlefrequency case (i.e., c_{0} = 1), adequately addresses the concern about whether any of emerging dynamical regimes is affected by the presence of two different frequencies. This further highlights the significance of having an oscillatory flow field. To speed up numerical simulations, we set c_{0} = 50. For individual swimmers, nonzero values of δω significantly increase the number of orbital families, and in some cases lead to densely interwoven quasiperiodic rosetteshaped trajectories capable of inducing chaotic mixing in the surrounding environment^{30}. Here, to focus on the basics of mutual interactions, we consider δω = 0. The results of this study are still valid for small δω, but start to deviate and become more involved as δω increases. Note that a = 1 μm leads to a Quadroar of ∼8–12 μm, which is similar to the size of a C. reinhardtii cell. Moreover, setting c_{0} = 50 results in the frequency of 50 Hz for the disks, reminiscent of the flagella beat frequency for a C. reinhardtii cell^{29}. For a more detailed description of the flow field around an isolated Quadroar the reader is referred to earlier publications on the Quadroar dynamics^{30}.
Results
Two of our swimmers, depending on their relative initial locations (dX_{1}, dX_{3}) portray a range of various trajectories as a result of their mutual hydrodynamic interactions. These trajectories range from converging, diverging, and oscillatory motions (which are also seen in other artificial microswimmers^{23}), to forming braids [Fig. 2(b)], and even dynamical equilibria [Figs 2(e)and 3(b)] which, to the best of our knowledge, have never been observed in lowReynoldsnumber swimming. We also report capture into bound orbits [Fig. 3(c)] for two interacting microorganisms swimming in an infinite unbounded fluid. Interestingly, a similar behavior is observed in the lab for two Volvox colonies attracted by the chamber ceiling^{42}.
In order to systematically study different possibilities of twoswimmer wiggling, induced by hydrodynamic interactions, we simultaneously consider the effects of swimming direction and relative initial locations, which also cover phase shift effects. Since there is no explicit time dependency in the Stokes equation, swimmers with an arbitrary phase shift between them (as a result of being launched at different times) can be considered as two swimmers with initial locations described at the moment that the second swimmer is turned on. For simplicity, we focus on the planar phase space. Nevertheless, our findings can be inherently generalized to 3D space. Our study has also been conventionally arranged in two general categories: (i) the two swimmers are released in the same direction such that their initial x_{3}axes are parallel and both aligned with the positive X_{3}axis (cf. Fig. 1), and (ii) the two swimmers are initially facing opposite directions such that at t = 0 the following conditions hold: \({\hat{{\boldsymbol{x}}}}_{3A}\) ⋅ \({\hat{{\boldsymbol{X}}}}_{3}=1\), and \({\hat{{\boldsymbol{x}}}}_{3B}\) ⋅ \({\hat{{\boldsymbol{X}}}}_{3}=1\), where the hat sign denotes unit vector. The resulting parameter space for each of these general cases is still valid for small perturbations. For larger perturbations, however, the parameter space starts to deviate from the presented plot and gradually tends to that of the other general case. For example, by changing the relative angle between the swimmers’ initial x_{3}axes from zero to π, the corresponding parameter space diagram will gradually transform from Fig. 2a (swimming in the same direction) to Fig. 3a (swimming in opposite direction).
The parameter space for the trajectories of two swimmers released parallel and in the same direction [case (i)] is displayed in Fig. 2(a) with sample trajectories demonstrated in Fig. 2(b–e). In these figures, the swimmers would follow dashed lines in the absence of hydrodynamic interactions. If the two swimmers are released close to each other, and depending on their relative locations, they form a variety of braids with different shapes [Fig. 2(b)]. Interestingly, we find that forward translational motion along a braid is faster, sometimes by a factor of two, than the motion of individual swimmers in the absence of hydrodynamic interactions. This phenomenon, which we refer to as hydrodynamic slingshot effect, can be easily deduced from Fig. 2(b): two hydrodynamically interacting swimmers advance along braids, and therefore in space (motion along colored lines), faster than noninteracting swimmers (moving along dashed lines) whose actuation parameters and initial conditions exactly match those of the interacting ones. The slingshot effect is caused by a synergistic process: each swimmer induces an advection field that sums with the relative velocity of its companion swimmer with respect to the background fluid, boosting the absolute velocity of the companion swimmer. Figure 4 demonstrates a snapshot of the flow field and corresponding streamlines induced by the system of two swimmers advancing along a braidlike trajectory. It shows how each swimmer induces an advection field at the geometric center of the other swimmer, boosting its absolute velocity. The net flow field could also be described as a constructive interference of the two swimmers’ flow fields (see Fig. 4). The resulting net flow field is similar to the one induced by a single Quadroar swimmer^{33} but with a more powerful propellers (higher values of c_{0}).
Other families of trajectories that we observe for interacting swimmers belong to a general family of nonorbiting paths including diverging and converging trajectories [Fig. 2(c)]. Nonorbiting paths may occur as pursuitevasion games when one of the swimmers chases the other one [Fig. 2(d)]. The most interesting nonorbiting path that we have found happens when the swimmers get in a reverse motion [Fig. 2(e), colored dark blue in Fig. 2(a)], eventually reaching to a dynamical equilibrium. In dynamical equilibrium states, the swimmers’ propellers are working continuously and their flow fields form a saddle structure (Fig. 5). The net flow of the saddle structure is zero, so follows the equilibrium state. In the space between the swimmers, fluid is pumped out in a direction almost parallel to the chassis of both swimmers, and is sucked back normal to the chassis. Four prominent vortices are formed around the propellers of the swimmers. These vortices are enclosed by a largescale hyperbolic structure. Our longterm simulations show that dynamical equilibria are stable to small perturbations. This is a counterintuitive property because the existence of hyperbolic structures usually implies local instability. The existence of dynamical equilibria for N > 2 swimmers is an unsolved problem, whose solution can sharpen our understanding of bacterial clustering and motile cell accumulations^{43,44}.
If the initial distance of the swimmers is large enough, their hydrodynamic interaction will be very small, drifting the swimmers slightly off their straight trajectories. We have also observed a switch between different trajectories as the twobody system evolves: a converging motion may end up in an equilibrium state, or pursuitevasion game may bifurcate to either of converging or diverging paths. In the parameter space, we have marked these cases with two or more colors [Fig. 2(a)].
Two swimmers starting their motions in opposite directions [case (ii)] exhibit different orbital topologies from what we observed for codirectional ones. Their twobody dynamics depends on the impact parameter dX_{1} [Fig. 3(a)]. When the impact parameter is relatively small (\(d{X}_{1}/a\lesssim 5\)) and the swimmers initially move towards each other, dX_{3} > 0, we always obtain an equilibrium state [Fig. 3(b)]. Similar to the equilibria of case (i), the actuators of the swimmers are operational at the equilibrium state and energy is consumed only for flow generation (and not translation). For large impact factors, \(d{X}_{1}/a\gtrsim 15\), trajectories are deflected similar to the lensing/refraction of light rays [Fig. 3(d)]. The deflecting trajectories have also been observed in linkedsphere swimmers^{24,27}. Our results are in agreement with the anglepreserving confrontation of two Tdual swimmers^{25}. For intermediate impact factors, we observe a capture phenomenon as the microswimmers begin to orbit each other after a translational phase [Fig. 3(c)]. It is noted that capture into a quasiperiodic orbit is a transitional state between dynamical equilibria and deflecting trajectories. Such transitional states fill a complex fractalshaped region of the parameter space, showing high degree of sensitivity to initial conditions [see the zoomedin box in Fig. 3(a)] with the dominant lengthscale of a disk radius. This result suggests the existence of highly chaotic Nbody systems of swimming microorganisms. Although details of trajectories in an orbiting motion can be complex, the bounded nature of the overall twobody motion in an infinite fluid domain is a unique physical process, for which many applications can be sought. Examples include mixing by microswimmers and trapping microorganisms by artificial microswimmers.
Conclusions and Discussion
Here we have shown that two microswimmers in Stokes regime can stop each other by forming a dynamical equilibrium in an infinite fluid domain. Furthermore, depending on where the two swimmers are released, they may also get trapped into bounded orbits and revolve about each other indefinitely. We have systematically studied the entire phase space of a hydrodynamically interacting twoswimmer system, and identified the basins of dynamical equilibria and periodic orbits in the parameter space. We also found other diverse sets of orbits including closely winding braids, and pursuit–evasion dynamics. Sensitivity to initial conditions, slingshot effect for motions along braids, dynamical equilibria, and capture into bound orbits, as demonstrated in this study, can have unexpected implications to motion of microorganisms. Nonlocal models of passive and active stresses due to hydrodynamic and steric interactions^{9} will then need modifications as diffusion in the phase space cannot be modeled only as a function of macroscopic streaming velocity.
Methods
Numerical Techniques
At each time step, we first substitute 1 and 2 into the force and torque balance equations. Then, together with equations presented in 4, the system is solved for the 20 vectorial (60 scalar) unknowns: v_{j,c}, ω_{j,body}, u_{ jn } and Ω_{ jn } (j = A, B; n = 1, …, 4). We then find the position and orientation of each swimmer by integrating v_{j,c} and ω_{j,body} in time. The angular velocity of each swimmer, denoted by ω_{j,body}, is related to the yawpitchroll sequence of Euler’s angles α = (ϕ, θ, ψ) through
Since there is a coordinatetype singularity in T, when θ = ±π/2, all computations have been carried out in the space of unit quaternions q, and then outputs are mapped back onto the space of Euler’s angles α^{33}:
Threedimensional Beads Model Simulation
In order to validate our models of the point forces and torques of the disks, we develop a full threedimensional beads realization of the disks^{45,46}. We first briefly explain the concept of beads model using Fig. 6a, then compare the results of our numerical method with those of beads model simulation.
A single spherical bead, moving with velocity v_{0}, in Stokes regime induces a wellknown velocity field in the surrounding fluid. For an arbitrary point in cylindrical coordinate system, this velocity field in radial and tangential directions is given by:
where R_{0} is the radius of the bead, and re_{ r } is the position vector of the arbitrary point with respect to the bead’s center. Figure 6a demonstrates two beads B_{1} and B_{2}, which are moving with absolute velocities v_{1} and v_{2} in the stationary frame. With respect to the background fluid, B_{1} (B_{2}) has a hydrodynamic velocity v_{1H} (v_{2H}), and thus induces a velocity field v_{1H,2} (v_{2H,1}) at the position of B_{2} (B_{1}). So, the hydrodynamic velocity of each bead is given by the following implicit formula:
Generalization of this simple idea, in order to formulate hydrodynamics of a system composed of N beads, results in the following system of linear algebraic equations:
The velocity v_{jH,i}, which is induced at the position of B_{ i } due to the motion of B_{ j }, follows from equation (6) as:
where r_{ ji } = r_{ i } − r_{ j }, r_{ i } and r_{ j } are position vectors of beads B_{1} and B_{2}, and θ_{ ji } is defined by \(cos{\theta }_{ji}=\frac{{{\boldsymbol{r}}}_{ji}\cdot {{\boldsymbol{v}}}_{jH}}{{{\boldsymbol{r}}}_{ji}{{\boldsymbol{v}}}_{jH}}\). To put the system of equations (8) into the standard format of \({\mathscr{A}}{\mathscr{X}}={\mathscr{B}}\), the general hydrodynamic relations between beads is defined here as:
\({S}_{3\times 1}^{(k)}\) for each k ∈ {1, 2, 3} is a column vector given by:
where r_{ mn } = r_{ n } − r_{ m } = (x, y, z). Applying this representation to the general formulation of the system of N beads presented in equation (8), leads to the following implicit system of linear equations:
This system of equations can then be solved using standard linear algebra methods. The inputs of the system are absolute velocities, v_{ i }, which are assigned to individual beads that assemble a rigid body, and outputs are hydrodynamic velocities.
We now validate our numerical method of modeling hydrodynamic interactions using a 3D beads model realization of two nearby rotating disks, where each disk is composed of a large number of beads (see Fig. 6b). The optimum number of beads required to model each disk is determined through the convergence of results. For the presented case in Fig. 6b, as an example, the optimum number of beads is 331, which corresponds to R_{0}/a = 1/21 ≈ 0.05. It should be noted that the thickness (2R_{0}) of each disk can be neglected compared to its diameter (2a), as expected by the swimmer model. Then, the general set of equations (14) must be solved for the entire system of the beads. Drag element exerted on each bead is then determined by multiplying the translational drag coefficient, 6πμR_{0}, to the bead’s consequent hydrodynamic velocity. At the end, the final results of this threedimensional simulation is compared to our numerical results of the point force and torque models.
Figure 6b demonstrates the schematics of the problem setup, where two disks of radius a and located at a distance of d are rotating with angular velocities ω_{1} and ω_{2}. The interaction of these two disks is modeled using: (i) our point force and torque models, and (ii) the full threedimensional beads simulation. Our results displayed in Fig. 7 show a good agreement between the two models, even for the smallest possible distance, d = 2a, between the disks. Figure 7 represents the magnitudes of the total force and torque exerted on disks as a function of their distance. The disks are counterrotating with ω_{1} = ω_{2} = 0.5ω_{ s }, and the total exerted force and torque (on each) are computed instantaneously for different distances between them.
References
Christner, B. C., Morris, C. E., Foreman, C. M., Cai, R. & Sands, D. C. Ubiquity of biological ice nucleators in snowfall. Science 319, 1214–1214 (2008).
Mooshammer, M. et al. Adjustment of microbial nitrogen use efficiency to carbon: nitrogen imbalances regulates soil nitrogen cycling. Nature Communications 5 (2014).
Locey, K. J. & Lennon, J. T. Scaling laws predict global microbial diversity. Proceedings of the National Academy of Sciences 201521291 (2016).
Elgeti, J., Winkler, R. G. & Gompper, G. Physics of microswimmers–single particle motion and collective behavior: a review. Reports on Progress in Physics 78, 056601 (2015).
Simmchen, J. et al. Topographical pathways guide chemical microswimmers. Nature Communications 7 (2016).
Jeanneret, R., Pushkin, D. O., Kantsler, V. & Polin, M. Entrainment dominates the interaction of microalgae with micronsized objects. Nature Communications 7, 12518 (2016).
Qiu, T. et al. Swimming by reciprocal motion at low reynolds number. Nature Communications 5 (2014).
Wensink, H. H. et al. Mesoscale turbulence in living fluids. Proceedings of the National Academy of Sciences 109, 14308–14313 (2012).
Dunkel, J. et al. Fluid dynamics of bacterial turbulence. Physical Review Letters 110, 228102 (2013).
Bricard, A. et al. Emergent vortices in populations of colloidal rollers. Nature Communications 6 (2015).
Marconi, U. M. B. & Maggi, C. Towards a statistical mechanical theory of active fluids. Soft Matter 11, 8768–8781 (2015).
Alexander, G. P. & Yeomans, J. Dumbbell swimmers. Europhysics Letters 83, 34006 (2008).
Lauga, E. & Bartolo, D. No manyscallop theorem: Collective locomotion of reciprocal swimmers. Physical Review E 78, 030901 (2008).
Ishikawa, T., Sekiya, G., Imai, Y. & Yamaguchi, T. Hydrodynamic interactions between two swimming bacteria. Biophysical Journal 93, 2217–2225 (2007).
Backholm, M., Schulman, R. D., Ryu, W. S. & DalnokiVeress, K. Tangling of tethered swimmers: Interactions between two nematodes. Physical Review Letters 113, 138101 (2014).
Ishikawa, T. & Hota, M. Interaction of two swimming paramecia. Journal of Experimental Biology 209, 4452–4463 (2006).
Ariel, G. et al. Swarming bacteria migrate by lévy walk. Nature Communications 6 (2015).
Ishikawa, T. Suspension biomechanics of swimming microbes. Journal of The Royal Society Interface 20090223 (2009).
Ishikawa, T., Simmonds, M. P. & Pedley, T. J. Hydrodynamic interaction of two swimming model microorganisms. Journal of Fluid Mechanics 568, 119–160 (2006).
Li, G., Ostace, A. & Ardekani, A. M. Hydrodynamic interaction of swimming organisms in an inertial regime. Physical Review E 94, 053104 (2016).
Michelin, S. & Lauga, E. The longtime dynamics of two hydrodynamicallycoupled swimming cells. Bulletin of mathematical biology 72, 973–1005 (2010).
Kim, M. & Powers, T. R. Hydrodynamic interactions between rotating helices. Physical Review E 69, 061910 (2004).
Pooley, C., Alexander, G. & Yeomans, J. Hydrodynamic interaction between two swimmers at low reynolds number. Physical Review Letters 99, 228103 (2007).
Farzin, M., Ronasi, K. & Najafi, A. General aspects of hydrodynamic interactions between threesphere lowreynoldsnumber swimmers. Physical Review E 85, 061914 (2012).
Alexander, G. P., Pooley, C. & Yeomans, J. M. Scattering of lowreynoldsnumber swimmers. Physical Review E 78, 045302 (2008).
Gilbert, A. D., Ogrin, F. Y., Petrov, P. G. & Winlove, C. P. Motion and mixing for multiple ferromagnetic microswimmers. The European Physical Journal E 34, 1–9 (2011).
Najafi, A. & Golestanian, R. Simple swimmer at low reynolds number: Three linked spheres. Physical Review E 69, 062901 (2004).
Klindt, G. S. & Friedrich, B. M. Flagellar swimmers oscillate between pusherand pullertype swimming. Physical Review E 92, 063019 (2015).
Guasto, J. S., Johnson, K. A. & Gollub, J. P. Oscillatory flows induced by microorganisms swimming in two dimensions. Physical Review Letters 105, 168102 (2010).
Jalali, M. A., Khoshnood, A. & Alam, M.R. Microswimmerinduced chaotic mixing. Journal of Fluid Mechanics 779, 669–683 (2015).
Ratcliff, W. C. et al. Experimental evolution of an alternating uniand multicellular life cycle in chlamydomonas reinhardtii. Nature Communications 4 (2013).
Koumura, N. et al. Lightdriven monodirectional molecular rotor. Nature 401, 152–155 (1999).
Jalali, M. A., Alam, M.R. & Mousavi, S. Versatile lowreynoldsnumber swimmer with threedimensional maneuverability. Physical Review E 90, 053006 (2014).
MedinaSánchez, M., & Schmidt, O. G. Medical microbots need better imaging and control. Nature 545, 406–408 (25 May 2017).
Nelson, B. J., Kaliakatsos, I. K. & Abbott, J. J. Microrobots for minimally invasive medicine. Annual review of biomedical engineering 12, 55–85 (2010).
Or, Y. & Murray, R. M. Dynamics and stability of a class of low Reynolds number swimmers near a wall. Physical Review E 79, 045302 (2009).
Zhang, S., Or, Y. & Murray, R. M. Experimental demonstration of the dynamics and stability of a low Reynolds number swimmer near a plane wall. American Control Conference 4205–4210 (2010).
Mavroidis, C. & Ferreira, A. Nanorobotics: past, present, and future. Nanorobotics, 3–27 (2013).
Friedrich, B. M. & Jülicher, F. Flagellar synchronization independent of hydrodynamic interactions. Physical Review Letters 109, 138102 (2012).
Lopez, D. & Lauga, E. Dynamics of swimming bacteria at complex interfaces. Physics of Fluids 26, 400–412 (2014).
Kage, A. & Mogami, Y. Individual Flagellar Waveform Affects Collective Behavior of Chlamydomonas reinhardtii. Zoological science 32, 396–404 (2015).
Drescher, K. et al. Dancing volvox: hydrodynamic bound states of swimming algae. Physical Review Letters 102, 168101 (2009).
BenJacob, E., Cohen, I. & Levine, H. Cooperative selforganization of microorganisms. Advances in Physics 49, 395–554 (2000).
Peruani, F. et al. Collective motion and nonequilibrium cluster formation in colonies of gliding bacteria. Physical Review Letters 108, 098102 (2012).
Chandran, P. L. & Mofrad, M. R. Averaged implicit hydrodynamic model of semiflexible filaments. Physical Review E 81(3), 031920 (2010).
Ota, S. et al. Brownian motion of tethered nanowires. Physical Review E 89(5), 053010 (2014).
Acknowledgements
Authors acknowledge the support by the National Science Foundation grant CMMI1562871.
Author information
Authors and Affiliations
Contributions
M.M. performed research including the modeling and simulations; M.A.J. and M.R.A. supervised the project. All authors contributed to the writing of the paper.
Corresponding author
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Mirzakhanloo, M., Jalali, M.A. & Alam, MR. Hydrodynamic Choreographies of Microswimmers. Sci Rep 8, 3670 (2018). https://doi.org/10.1038/s4159801821832w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4159801821832w
This article is cited by

Controlled swarm motion of selfpropelled microswimmers for energy saving
Journal of MicroBio Robotics (2021)

Independent control of multiple magnetic microrobots: design, dynamic modelling, and control
Journal of MicroBio Robotics (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.