Coherent Hydrodynamic Coupling for Stochastic Swimmers
aa r X i v : . [ phy s i c s . b i o - ph ] J u l epl draft Coherent Hydrodynamic Coupling for Stochastic Swimmers
A. Najafi and R. Golestanian Department of Physics, Zanjan University, Zanjan 313, Iran Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK
PACS – Locomotion
PACS – Micromechanical devices and systems
PACS – Chemical kinetics in biological systems
Abstract. - A recently developed theory of stochastic swimming is used to study the notion of co-herence in active systems that couple via hydrodynamic interactions. It is shown that correlationsbetween various modes of deformation in stochastic systems play the same role as the relativeinternal phase in deterministic systems. An example is presented where a simple swimmer canuse these correlations to hunt a non-swimmer by forming a hydrodynamic bound state of tunablevelocity and equilibrium separation. These results highlight the significance of coherence in thecollective behavior of nano-scale stochastic swimmers.
Introduction. –
Swimming strategies for microor-ganisms and microbots need to take into account the pe-culiarities that arise in low Reynolds number hydrody-namics [1–3]. When utilizing only a small number of de-grees of freedom, a careful non-reciprocal prescription ofcyclic deformations is needed to achieve swimming [4–15].While these ideas have been primarily developed to de-scribe the swimming of bacteria [16], sperms [17], andother micro-scale living systems [18], in recent years theyhave attracted additional interest with the advent of thefirst generation of artificial microswimmer prototypes [19].Swimmers of micro- or nanoscale need to face an addi-tional challenge, namely, the overwhelming fluctuationsthat would act against their targeted mechanical task. Inits most basic form, the effect of fluctuations on the mo-tion of swimmers that are not directionally constrainedor steered is to randomize their orientation via rotationaldiffusion [20–24]. The fluctuations can also interfere withthe propulsion mechanism and alter the swimming veloc-ity [25], for example via the density fluctuations in thecase of self-phoretic swimmers [26] or fluctuations in theconformational changes in deforming swimmers [27].Interacting swimmers [28] are known to have rathercomplex many-body behaviors [29], which can be under-stood in terms of instabilities in the context of continuumtheories (that are constructed based on symmetry consid-erations) [30]. Another fascinating consequence of long-range hydrodynamic interactions between active objectswith cyclic motions is the significance of internal phase as a key dynamical variable [12,31,32], and the possibility ofsynchronization [33]. However, most current theoreticalstudies of the collective behavior of swimmers (continuumtheories and simulations) ignore the possible effects of co-herence, and it is natural to wonder if this is justified.One could argue that the overwhelming fluctuationsthat are present at small scale may wash out any traceof coherence among swimmers. To examine the validityof this argument, we consider the following question: doesthe notion of relative internal phase apply to stochasticswimmers? We use a statistical description to model thedynamics of systems that undergo random conformationalchanges while interacting hydrodynamically. Using a spe-cific example of a three-sphere system coupled to a two-sphere system, we calculate the swimming velocities asfunctions of the statistical transition rates for the confor-mational changes. We show that coherence could be intro-duced in the system through the correlations between thedeformations, and that it can be used to create a stablebound state between the three-sphere swimmer and thetwo-sphere system (that cannot swim when isolated) witha tunable equilibrium distance and velocity.
Hydrodynamic Model. –
Consider the a three-sphere system that is located collinearly at a distancefrom a two-sphere system, as shown schematically in Fig.1. The two systems undergo conformational changes byopening and closing of the three arms, which could onlylead to net swimming for the three-sphere system (andnot the two-sphere system) when isolated, due to scallopp-1. Najafi and R. Golestaniantheorem [3]. When at a finite distance D , the two sys-tems interact hydrodynamically, and their dynamics willbe coupled to each other. The system on the left (Fig. 1)is made up of three spheres of radius R that are connectedby arms of lengths L + u L ( t ) and L + u L ( t ), while the sys-tem on the right consists of two similar spheres connectedby an arm of length L + u R ( t ). For simplicity, we assumethat the linker do not interact with the fluid. To Ana-lyze the dynamics of the system, we use the linearity ofthe Stokes equation—the equation for hydrodynamics inzero Reynolds number—and express the velocity of eachsphere v i as a linear combination of the force f j acting ona different sphere j : v i = e X j = a M ij f j , (1)where the details of the hydrodynamic interactions areentailed in the coefficients M ij . Using Oseen’s approx-imation, we can write simple closed form expressions forthe coefficients when the spheres are considerably far fromeach other. Denoting the positions of the spheres by x i ,we have: M ij = πηR , i = j, πη | x i − x j | , i = j, (2)where η is the viscosity of the fluidic medium. Equation(1) thus gives us five equation for the ten unknowns v i and f i ( i = a, · · · , e ). Maintaining force-free conditionson the two systems, namely, f a + f b + f c = 0 and f d + f e = 0, provides two additional equations. The final threeequations are obtained by the kinematic constraints v b − v a = ˙ u L , v c − v b = ˙ u L , and v e − v d = ˙ u R , where the dotdenotes d/dt .Considering the case where the two systems are far fromeach other and the deformations are small compared to theaverage length of the arms L , such that R ≪ u ≪ L ≪ D , we can set up a perturbative scheme to investigatethe effect of the hydrodynamic interactions [14]. Solvingthe above linear system of ten equations, we can find allthe velocities and the forces, from which we can calculatethe average swimming velocity of the three-bead system V L = h v a + v b + v c i and that of the two-bead system V R = h v d + v e i . To the leading order in perturbationtheory, we find V L = 712 RL (cid:10) ˙ u L u L (cid:11) − RLD (cid:2)(cid:10) u L ˙ u R (cid:11) − (cid:10) u L ˙ u R (cid:11)(cid:3) , (3) V R = RLD (cid:20) − (cid:10) ˙ u L u L (cid:11) + 32 (cid:10) u L ˙ u R (cid:11) + 32 (cid:10) u L ˙ u R (cid:11)(cid:21) . (4)We can also extract the average forces acting on the beads.To the leading order in deformations, this yields: h f a i = 54 πη R L (cid:10) ˙ u L u L (cid:11) − πη R LD (cid:2)(cid:10) u L ˙ u R (cid:11) + 2 (cid:10) u L ˙ u R (cid:11)(cid:3) , h f c i = 54 πη R L (cid:10) ˙ u L u L (cid:11) + 3 πη R LD (cid:2) (cid:10) u L ˙ u R (cid:11) + (cid:10) u L ˙ u R (cid:11)(cid:3) , (11): (21): (31): (41): (12): (22): (32): (42): a L uL + d b c e L uL + R uL + D R u L u L u R (11) (41) (31) (12) (42) (32) (22) (21) δ δ δ x Fig. 1: Schematic view of a three-sphere swimmer interactingwith a two-sphere system (top) and the three dimensional con-figuration space (middle) representing the eight possible dis-tinct conformational states of the combined system (bottom).We denote by ( iα ) the state where the three-bead system is instate i and the two-bead system is in state α . h f d i = 9 πη R LD (cid:2)(cid:10) u L ˙ u R (cid:11) + (cid:10) u L ˙ u R (cid:11)(cid:3) . Note that h f b i = −h f a i − h f c i and h f e i = −h f d i .The above expressions for the velocities and forces aregiven in terms of the three average quantities (cid:10) ˙ u L u L (cid:11) , (cid:10) u L ˙ u R (cid:11) , and (cid:10) u L ˙ u R (cid:11) , which correspond to the averagerates of sweeping enclosed areas in the three perpendicularsections of the three dimensional ( u L , u L , u R ) configura-tion space of the system, respectively. For deterministic conformational changes of the form u L = d cos(Ω t − ϕ L ) ,u L = d cos(Ω t − ϕ L ) ,u R = d cos(Ω t − ϕ R ) , we can calculate them using time averaging over a period.This yields (cid:10) ˙ u L u L (cid:11) = 12 d Ω sin( ϕ L − ϕ L ) , (5) (cid:10) u L ˙ u R (cid:11) = 12 d Ω sin( ϕ R − ϕ L ) , (6)p-2oherent Hydrodynamic Coupling for Stochastic Swimmers (11) (41) (31) (12) (42) (32) (22) I (21) I I I I I j j j j j Fig. 2: Two dimensional projection of the 3D configurationspace of Fig. 1 showing the different probability current loops. (cid:10) u L ˙ u R (cid:11) = 12 d Ω sin( ϕ R − ϕ L ) . (7)The above equations manifestly show that the relativeimportance of these three conformational space area-sweeping rates is determined by the relative phases ofthe deformations. We now aim to address the questionof whether such a concept can exist at small scales wherethe conformational changes are stochastic. Stochastic Systems. –
To construct a statistical the-ory for the deformations of the two systems we assumethat they have distinct conformational states and the de-formations can be modeled as stochastic jumps betweenthese states that occur at given rates [27, 34]. The three-sphere system can be described with four states and thetwo-sphere system with two states, which make a total ofeight distinct conformational states in the three dimen-sional configuration space, as shown in Fig. 1. Morespecifically the states of the three-sphere swimmer arelabeled by the index i as follows: i = 1 the two armsare closed ( u L = 0 , u L = 0), i = 2 the right arm isopen ( u L = δ, u L = 0), i = 3 the two arms are open( u L = δ, u L = δ ), and i = 4 the left arm is open( u L = 0 , u L = δ ). For the two-sphere system, we onlyhave two possibilities: α = 1 the arm is closed ( u R = 0)and α = 2 it is open ( u R = δ ). To describe the instan-taneous state of the system we denote the probability offinding the left swimmer at state i ( i = 1 , · · · ,
4) and theright two-bead system at state α ( α = 1 ,
2) by P iα . Theseprobabilities are normalized as X i,α P iα = 1 . The kinetics of the conformational transitions of the twocoupled systems is given by introducing the correspondingtransition rates. We assume that the conformational changes happen oneat a time, which means that transitions are only allowedbetween states that are nearest neighbors in the cubic con-figuration space shown in Fig. 1. We denote the tran-sition rate for the jump from state i to state j for theleft swimmer when the two-bead system is in state α by k Lji ( α ). Similarly, the transition rate for the two-bead sys-tem jumping from state α to state β when the three-spheresystem is in state i is denoted as k Rβα ( i ). Note that therates for conformational changes within each system inprinciple depend on the state of the other system. The cu-bic configuration space has six current loops correspondingto six faces, as shown in Fig. 2. These currents, however,are subject to an overall conservation law, which impliesthat only five independent currents exist in the system.We define the following currents j = P k L (1) − P k L (1) ,j = P k L (1) − P k L (1) ,j = P k L (1) − P k L (1) ,j = P k R (1) − P k R (1) ,j = P k L (2) − P k L (2) , in terms of the probabilities and the rates, and can usethem to calculate the currents in the loops as follows (seeFig. 2): I = 16 ( − j + j + j + j + j ) ,I = 16 (3 j − j + j + j + j ) ,I = 16 (3 j + j − j + j + j ) ,I = 16 ( − j + j + j − j + j ) ,I = 16 (3 j + j + j + j + j ) ,I = − I − I − I − I − I . We can now solve the steady state master equation for thesystem and calculate the probabilities and the currents.Using the currents in the loops, we can write down ex-pressions for the average rates of sweeping areas in thethree perpendicular sections of the configuration space, asshown in Fig. 2. The results, which are the statisticalanalogs of eqs. (5), (6), and (7), read (cid:10) ˙ u L u L (cid:11) = δ ( I − I ) , (8) (cid:10) u L ˙ u R (cid:11) = δ ( I − I ) , (9) (cid:10) u L ˙ u R (cid:11) = δ ( I − I ) . (10)Due to the sign convention used in the definition of thecurrents the current running through opposite faces of thecube in Fig. 2 have opposite signs. Therefore, the totalrate of sweeping a certain projected area in the configu-ration space is the difference between the currents of thecorresponding opposite faces in the cube, as eqs. (8), (9),p-3. Najafi and R. Golestanian - - - - Ε Ε Stable
Unstable
Fig. 3: Phase diagram of possible states. Steady state solutionsin the form of bound states with constant separation ( ˙ D = 0)are highlighted, and the Stable and Unstable regions are shown. and (10) show. The area of each projection is equal to δ . The above expressions can be used in eqs. (3) and (4)to calculate the average swimming velocities of the twosystems.In the most general case with arbitrary rates the explicitform of the resulting probabilities and velocities is cum-bersome and therefore not shown here. To illustrate thegeneric features of the solution, we focus on a simplifiedexample with the following choices for the transition rates.For transitions in the three-sphere system we choose k Lij ( α ) = (1 + ǫ ) ω, i = 1 , j = 2 , α = 1 , (1 + ǫ ) ω, i = 1 , j = 2 , α = 2 ,ω, other states (11)For the two-sphere system, we choose k Rβα ( i ) = ω, for all states . The above choices allow us to only focus on the effect ofthe correlation between the two devices, as having differentvalues for ǫ and ǫ means that the rate of the three-spheresystem going from state 1 to state 2, which means openingits right arm, depends on whether the two-sphere systemis in the closed or the open state. If ǫ = ǫ = 0, detailedbalance holds and neither of the two components has a netmotion. With the above choices, the average velocities ofthe two systems can be found as V L = Rω (cid:18) δL (cid:19) " y − (cid:18) LD (cid:19) y , (12) V R = Rω (cid:18) δL (cid:19) (cid:18) LD (cid:19) (cid:20) − y + 32 y (cid:21) , (13)where y = 12( ǫ + ǫ ) + 5 ǫ ǫ ǫ + ǫ ) + 15 ǫ ǫ + 192 , (14) y = 6( ǫ − ǫ )56( ǫ + ǫ ) + 15 ǫ ǫ + 192 . (15) The term proportional to y in eq. (12) is the spontaneousswimming velocity of the three-sphere system, while the y contribution in eq. (13) is the passive velocity at the loca-tion of the two-sphere system caused by the swimming ofthe three-sphere system. The contributions proportionalto y in eqs. (12) and (13) are active contributions origi-nating from a coherence between u L and u R in the formof (cid:10) u L ˙ u R (cid:11) = 0. Note that the coherence will disappearwhen ǫ = ǫ .An interesting consequence of the coherent coupling isthat the two systems can form a moving hydrodynamicbound state at a fixed separation. Noting that dd t D ( t ) = V R − V L ord D ( t )d t = Rω (cid:18) δL (cid:19) "(cid:18) LD ( t ) (cid:19) (2 y − y ) − y , (16)we can find the conditions at which stable and unstablebound states are possible in the effective dynamical equa-tion for D ( t ), as shown in Fig. 3. The equilibrium dis-tance between the two systems in the stable hydrodynamicbound states is given as D eq = L (cid:18) (cid:19) / (cid:20) − ǫ (24 + 5 ǫ )12( ǫ + ǫ ) + 5 ǫ ǫ (cid:21) / , (17)which can be controlled by changing the transition rates.We note that the hydrodynamic-induced formation ofbound states of a pair of microorganisms has been recentlyobserved experimentally [35]. Discussion. –
Our analysis shows that the conceptof relative internal phase and coherence between a num-ber of systems that undergo stochastic deformations in ahydrodynamic medium at low Reynolds number is welldefined. Stochastic coherence could result from averagecorrelations that can be induced between various modes ofthe conformational transitions, and need not exist instan-taneously to lead to average correlated behavior. Com-parison between eqs. (5), (6), and (7) that are defined fordeterministic systems and eqs. (8), (9), and (10) that aredefined for stochastic systems shows how average relativephase between various modes of the conformational tran-sitions can be defined and probed in terms of the currentsin the configuration space of the system.In the present study, coherence between the two sub-systems is introduced via the rates defined in eq. (11).When ǫ = ǫ , the rate of opening of the right arm of thethree-sphere swimmer is chosen to depend on whether thetwo-sphere system is in the closed or open state; it is ex-actly this difference that leads to the correlation term pro-portional to y in eqs. (12) and (13), as can be seen fromthe explicit dependence of y ∝ ( ǫ − ǫ ) in eq. (15). Thismeans that while it is possible to have coherence betweendifferent parts of the system when undergoing stochasticfluctuations, this coherence still needs to be imposed viathe different rates in the kinetic equations. Physically,what this means is that the coherence needs to introducedp-4oherent Hydrodynamic Coupling for Stochastic Swimmersin the system via correlations between different confor-mational states of the system and the rates of transitionsbetween them. The correlations are reminiscent of the al-losteric interactions between proteins [36], and could inprinciple be engineered for artificial systems using similarstrategies. These strategies could involve physical interac-tions arising from electrostatic forces etc , hydrodynamicinteractions, and other effects that could modify the tran-sition rates by introducing additional mechanical energycontributions (costs) in the deformation process and henceaffecting the transition rates via the exponential (Arrhe-nius) dependence on energy change. Alternatively, thesecorrelations could be induced via external means such aslaser pulses that would affect transition rates only in cer-tain conformational states. However they are enforced, thepresent study asserts that such correlations could lead toa sustainable notion of coherence between stochasticallyfluctuating nano-scale devices in water.We have considered the transition rates for the confor-mational changes of the two small systems to be time in-dependent. If this assumption is not valid for any reason,the time dependence in the rates could weaken the degreeof coherence in the system and ultimately fully eliminateit if it is sufficiently strong. This is equivalent to intro-ducing time dependence in the phases in the deterministiccase [described by eqns. (5), (6), and (7)], which could de-stroy the phase coherence. Such a time dependence couldoccur due to temperature fluctuations [37], which in localthermodynamics approximation could affect the transitionrate through a dependence of the form ω ∼ exp (cid:20) − f δk B T (cid:21) , (18)where f represents a typical force involved in the confor-mational change. We can estimate the magnitude of f using the typical drag force experienced by a sphere of ra-dius R and moving velocity v = δω , namely f ≈ πηRδ ω .In the present work we have neglected temperature fluc-tuations. To examine the validity of this assumption, wecan use the complementarity relation ∆ E ∆(1 /T ) ≈ − k B ,which relates the strength of energy and temperature fluc-tuations, to estimate the magnitude of temperature fluc-tuations as ∆ T = k B T /C , where C is the heat capacity[37]. Using this simplified picture, we can estimate theeffect of temperature fluctuations on the transition ratesvia ∆ ωω = f δk B T × ∆ TT ≈ πηRδ ωk B T r k B C . (19)In order to have ∆ ω/ω ≪
1, which would guarantee thatthe above assumption is valid (i.e. temperature fluctua-tions can be ignored) the following condition must hold: ω ≪ k B T πηRδ r Ck B . (20)Putting R = δ = 1 nm at room temperature for water, we find the following condition: ω ≪ (cid:0) s − (cid:1) × r Ck B . (21)While there is a debate in the literature about the correctchoice for C , namely whether it is the heat capacity of theentire system [38] or that of the subsystem only [39], eqns.(20) and (21) show that in our case neglecting temperaturefluctuations is justified either way. Note also that ouranalysis has ignored quantum fluctuations, which meansthat the following criterion should also hold: ω ≪ k B T ¯ h .Finally, we note that we have made the specific choiceof a three-sphere system and a two-sphere system, becauseits configuration space is 3D and can be easily visualized(Figs. 1 and 2). The same analysis can be easily gener-alized to the case of two three-sphere swimmers, in whichcase the configuration space will be 4D and the bookkeep-ing of the projected areas in the graph where the proba-bility currents are flowing is slightly more delicate.In conclusion, we have shown that stochastic swimmerscould actively couple to each other using hydrodynamicinteractions. As an example, we demonstrated that thecoupling could be tailored such that an active swimmercould hunt a non-swimming system into a stable movingbound state, by “tuning” into the right correlated tran-sition rates. Our results show that the notion of internalphase and coherence could be important even for fluctu-ating systems that are coupled via hydrodynamic interac-tions at the nano-scale. ∗ ∗ ∗ We acknowledge financial support from MPIPKS(A.N.), CNRS (R.G.), and EPSRC (A.N. and R.G.).
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