Cooperation of Sperm in Two Dimensions: Synchronization, Attraction and Aggregation through Hydrodynamic Interactions
aa r X i v : . [ phy s i c s . b i o - ph ] O c t Cooperation of Sperm in Two Dimensions: Synchronization, Attraction and Aggregation throughHydrodynamic Interactions
Yingzi Yang, Jens Elgeti and Gerhard Gompper ∗ Theoretical Soft Matter and Biophysics Group, Institut f¨ur Festk¨orperforschung, Forschungszentrum J¨ulich, D52425 J¨ulich, Germany (Dated: July 12, 2018)Sperm swimming at low Reynolds number have strong hydrodynamic interactions when their concentrationis high in vivo or near substrates in vitro. The beating tails not only propel the sperm through a fluid, but alsocreate flow fields through which sperm interact with each other. We study the hydrodynamic interaction andcooperation of sperm embedded in a two-dimensional fluid by using a particle-based mesoscopic simulationmethod, multi-particle collision dynamics (MPC). We analyze the sperm behavior by investigating the relation-ship between the beating-phase di ff erence and the relative sperm position, as well as the energy consumption.Two e ff ects of hydrodynamic interaction are found, synchronization and attraction. With these hydrodynamice ff ects, a multi-sperm system shows swarm behavior with a power-law dependence of the average cluster sizeon the width of the distribution of beating frequencies. PACS numbers: 82.70.-y, 87.16.Qp, 87.17.Aa
I. INTRODUCTION
Sperm motility is important for the reproduction of animals.A healthy mature sperm of a higher animal species usually hasa flagellar tail, which beats in a roughly sinusoidal pattern andgenerates forces that drive fluid motion. At the same time,the dynamic shape of the elastic flagellum is influenced bythe fluid dynamics. The snake-like motion of the tail propelsthe sperm through a fluid medium very e ffi ciently. In the pastdecades, the e ff ort to quantitatively describe the fluid dynam-ics of sperm has been very successful [1, 2].However, despite considerable progress in modeling spermelementary structures and the behavior of a single sperm ina fluid medium [3, 4], relatively few studies have examinedthe fluid-dynamics coupling of sperm and other mesoscopic ormacroscopic objects, e.g., the synchrony of beating tails [5, 6],the tendency of accumulation near substrates [7, 8, 9, 10],etc. In nature, the local density of sperm is sometimes ex-tremely high. For example, in mammalian reproduction, theaverage number of sperm per ejaculate is tens to hundredsof millions, so that the average distance between sperm ison the scale of ten micrometers — comparable to the lengthof their flagellum. The sperm are so close that the interac-tion between them is not negligible. In recent years, exper-iments [11, 12, 13, 14, 15, 16] have revealed an interestingswarm behavior of sperm at high concentration, e.g. the dis-tinctive aggregations or ’trains’ of hundreds of wood-mousesperm [14, 15], or the vortex arrays of swimming sea urchinsperms on a substrate [16]. The mechanisms behind the abun-dant experimental phenomena are still unclear. In this paper,we focus on the hydrodynamic interaction between sperm andexplain its importance for the cooperative behavior.The higher animal sperm typically have tails with a lengthof several tens of micrometers. At this length scale, viscousforces dominate over inertial forces. Thus, the swimming mo-tion of a sperm corresponds to the regime of low Reynolds ∗ e-mail: [email protected] number [17]. Experimental observations of two parameciumcells swimming at low Reynolds number have shown thatthe changes in direction of motion between two cells are in-duced mainly by hydrodynamic forces [18]. Studies of modelmicro-machines indicate that hydrodynamic interaction is sig-nificant when the separation distance is comparable to theirtypical size [19]. The hydrodynamic interaction between tworotating helices, like bacterial flagella, has been investigatedboth experimentally [20] and theoretically [21, 22]. An arti-ficial microswimmer, which mimics the motion of a beatingsperm, has been constructed from a red blood cell as head anda flagellum-like tail composed of chemically linked param-agnetic beads; the propulsion is then induced by a magnet-ically driven undulation of the tail [23]. Simulations havebeen employed to study the motion of a single of these arti-ficial microswimmer [24], as well as the hydrodynamic inter-actions between two swimmers [25]. Even studies of a mini-mal swimming model of three linearly connected spheres [26]have shown a complicated cooperative behavior [27]. Thus,although there has been much progress on modeling and ob-serving a single swimmer, the understanding on the hydrody-namic coupling behavior of dense system of swimmers is stillpoor.In this paper, we focus on the cooperation behavior ofsperm in two dimensions. Although real swimming sperma-tozoa are certainly three-dimensional, the qualitatively similarphenomena, and the great saving of simulation time, makes itworthwhile to discuss the problem of cooperation in a viscousfluid in two dimensions. Furthermore, sperm are attracted tosubstrates in in-vitro experiments [7, 8, 9, 10] and are there-fore often swimming under quasi-two-dimensional condition(it has to be emphasized that hydrodynamic interactions in twodimensions and in three dimensions near a substrate are ofcourse di ff erent). Thus, we construct a coarse-grained spermmodel in two dimensions and describe the motion of the sur-rounding fluid by using a particle-based mesoscopic simula-tion method called multi-particle collision dynamics (MPC)[28, 29]. This simulation method has been shown to capturethe hydrodynamics and flow behavior of complex fluids overa wide range of Reynolds numbers very well [30, 31], and isthus very suitable for the simulation of swimming sperm.This paper is organized as follows. Section II gives a briefdescription of our sperm model and of the particle-based hy-drodynamics approach. In order to understand a complexmany-body system of micro-swimmers, a first-but-importantstep is to investigate the interaction between two swimmingsperm. Thus, in Sec. III, we look at the cooperative behaviorof two sperm. Two remarkable hydrodynamic e ff ects, syn-chronization and attraction, are found and discussed in detail.In Sec. IV, we analyze the clustering behavior of multi-spermsystems. In particular, we consider a sperm system with adistribution of beating frequencies, and determine the depen-dence of the cluster size on the variance of the frequency dis-tribution. II. SPERM MODEL AND MESOSCALEHYDRODYNAMICSA. Multi-Particle Collision Dynamics (MPC)
MPC is a particle-based mesoscopic simulation techniqueto describe the complex fluid behaviors for a wide range ofReynolds numbers [30, 31]. The fluid is modeled by N pointparticles, which are characterized by their mass m i , continu-ous space position r i and continuous velocity u i , where i = N . In MPC simulations, time t is discrete. During everytime step ∆ t , there are two simulation steps, streaming andcollision. In the streaming step, the particles do not interactwith each other, and move ballistically according to their ve-locities, r i ( t + ∆ t ) = r i ( t ) + u i ∆ t (1)In the collision step, the particles are sorted into collisionboxes of side length a according to their position, and inter-act with all other particles in same box through a multi-bodycollision. The collision step is defined by a rotation of all par-ticle velocities in a box in a co-moving frame with its centerof mass. Thus, the velocity of the i -th particle in the j -th boxafter collision is u i ( t + ∆ t ) = u cm , j ( t ) + R j ( α )[ u i − u cm , j ] (2)where u cm , j ( t ) = P j m i u i P j m i (3)is the center-of-mass velocity of j -th box, and R j ( α ) is a ro-tation matrix which rotates a vector by an angle ± α , with thesign chosen randomly. This implies that during the collisioneach particle changes the magnitude and direction of its veloc-ity, but the total momentum and kinetic energy are conservedwithin every collision box. In order to ensure Galilean invari-ance, a random shift of the collision grid has to be performed[32, 33].The total kinematic viscosity ν is the sum of two contri-butions, the kinetic viscosity ν kin and the collision viscosity FIG. 1: (Color online) Two-dimensional model of sperm. The modelconsists of three parts, the head (blue), the mid-piece (red) and thetail (cyan). Two sinusoidal waves are present on the beating tail. ν coll . In two dimension, approximate analytical expressionsare [34, 35], ν coll p k B T a / m = h (1 − cos α ) (cid:18) − ρ (cid:19) (4) ν kin p k B T a / m = h (cid:20) − cos α ρρ − − (cid:21) (5)where ρ is the average particle number in each box, m is themass of solvent particle and h = ∆ t p k B T / ma is the rescaledmean free path. In this paper, we use k B T = m = a = ∆ t = . α = π/ ρ =
10. This implies, in particular,that the simulation time unit ( ma / k B T ) / equals unity. Withthese parameters, the total kinematic viscosity of fluid is ν = ν coll + ν kin ≈ .
02. The size of the simulation box is L x × L y ,where L x = L y = a , four times the length of the sperm tail,if not indicated otherwise. Periodic boundary conditions areemployed. B. Sperm Model in Two Dimensions
Although animal sperm di ff er from species to species, theirbasic structure is quite universal. Usually, a sperm consistsof three parts: a head containing the genetic information, abeating long tail, and a mid-piece to connect head and tail.Our two-dimensional sperm model, shown in Fig. 1, consistsof these three parts. The head is constructed of N head = l = . a with interaction potential V bond ( R ) = k ( | R | − l ) (6)into a circle of radius 2 a . Each of the head particles has amass m head =
20. The mid-piece consists of N mid = m mid =
10 connected by springs of length l = . a . The first particle of the mid-piece, which is fixedto the center of the head, is connected with every particle onthe head by a spring of length l head − mid = a , in order tomaintain the circular shape of the head, as well as to stabi-lize the connection between head and mid part. The tail has N tail =
100 particles of mass m tail =
10, linked together bysprings of length l . The spring constants are chosen to be k head − mid = , k head = , k mid = k tail = × , where k head − mid is the spring constant for the connection of the headparticles and the center, and k mid and k head are the spring con-stants for the tail and the mid-piece, respectively.A bending elasticity is necessary for the mid-piece and thetail to maintain a smooth shape in a fluctuating environment,and to implement the beating pattern. The bending energy is E bend = X i ∈ mid κ (cid:26) R i + − R i (cid:27) (7) + X i ∈ tail κ (cid:26) R i + − R ( l c s , tail ) R i (cid:27) (8)where κ denotes the bending rigidity, R ( l c s ) is a rotation ma-trix which rotates a vector anticlockwise by an angle l c s , and c s , tail is the local spontaneous curvature of the tail of the s -th sperm. We choose κ = , much larger than the thermalenergy k B T = c s , mid = c s , tail is a variable changingwith time t and the position x along the flagellum to create apropagating bending wave, c s , tail ( x , t ) = c , tail + A sin (cid:20) − π f s t + qx + ϕ s (cid:21) . (9)A detailed analysis of the beating pattern of bull sperm inRef. [36] shows that a single sine mode represents the beat toa very good approximation. The wave number q = π/ l N tail is chosen to mimic the tail shape of sea-urchin sperm [1], sothat the phase di ff erence between the first and the last parti-cles of the tail is 4 π , and two waves are present (see Fig. 1). f s is the beating frequency of the s -th sperm. The constant c , tail determines the average spontaneous curvature of the tail. ϕ s isthe initial phase of the first tail particle on the s -th sperm, and A is a constant related to the beating amplitude. We choose A = .
2, which induces a beating amplitude A tail = . a ofthe tail. As t increases, a wave propagates along the tail, push-ing the fluid backward at the same time propelling the spermforward. We keep A , k , T s , and ϕ s constant for each spermduring a simulation. Although the spontaneous local curva-ture is prescribed, the tail is elastic and its configuration isa ff ected by the viscous medium and the flow field generatedby the motion of neighboring sperm.In order to avoid intersections or overlaps of di ff erentsperm, we employ a shifted, truncated Leonard-Jones poten-tial V ( r ) = ǫ (cid:20)(cid:16) σ r (cid:17) − (cid:16) σ r (cid:17) (cid:21) + ǫ, r < / σ , r ≥ / σ (10)between particles belonging to di ff erent sperm, where r is thedistance between two particles. Parameters σ = ǫ = .
75 are chosen.During the MPC streaming step, the equation of motionof the sperm particles is integrated by a velocity-Verlet algo-rithm, with a molecular-dynamics time step ∆ t s = × − ,which is 1 /
50 of the MPC time step ∆ t . The sperm only inter-acts with the fluid during the MPC collision step. This is doneby sorting the sperm particles together with the fluid particlesinto the collision cells and rotating their velocities relative tothe center-of-mass velocity of each cell.Since energy is injected into the system by the actively beat-ing tails, we employ a thermostat to keep the fluid temperature constant by rescaling all fluid-particle velocities in a collisionbox relative to its center-of-mass velocity after each collisionstep. This procedure has the advantage that the energy con-sumption per unit time of the sperm can be easily extractedthrough the rescaling of the particle velocities.We start with a single-sperm system with c , tail = f = / u single = . ± . ff usion coe ffi cient of a sperm due to the thermal fluctu-ations of the MPC fluid is very small, on the order of 10 − [37]. This implies that the time the sperm needs to cover adistance of half the length of its flagellum by passive di ff u-sion is more than a factor 10 larger than the time to travelthe same distance by active swimming. Therefore, di ff usionplays a negligible role in our simulations. The energy con-sumption per unit time P single = . ± . P single ≃ k B T f . Thus, we estimate a Reynolds num-ber Re = A tail u single /ν ≃ .
03 for our sperm model, where A tail = . a is the beating amplitude of the tail. III. TWO-SPERM SIMULATIONSA. Symmetric Sperm
Two sperm, S1 and S2, are placed inside the fluid, initiallywith straight and parallel tails at a distance d = i.e. withtouching heads). They start to beat at t = ff erentphases ϕ and ϕ . The initial positions of sperm do not mattertoo much, because two freely swimming sperm always havethe chance to come close to each other after a su ffi ciently longsimulation time. We consider two sperm with the same beatfrequency f = / c , tail = ff ects can be distinguished, a short time“synchronization” and a longer time “attraction” process. Ifthe initial phase di ff erence ∆ ϕ = ϕ − ϕ at time t = ff ect denoted “synchronization”takes place, which is accomplished within a few beats. Thisprocess is illustrated in Fig. 2a-e by snapshots at di ff erent sim-ulation times. The synchronization time depends on the phasedi ff erence, and varies from about two beats for ∆ ϕ = . π (see Fig. 2) to about five beats for ∆ ϕ = π . A di ff erencein swimming velocities adjusts the relative positions of thesperm. After a rapid transition, the velocities of two cells be-come identical once their flagella beat in phase. Because theinitial distance between tails d = A tail = . a , the sperm tails can touch when theystart to beat for 0 . π < ∆ ϕ < . π . This geometrical e ff ectis reduced by the hydrodynamic interaction, which a ff ects thebeating amplitude. In case contact occurs, it accelerates thesynchronization. In order to avoid this direct interaction due tovolume exclusion, we have also performed simulations of twosperm with initial distance d =
10, and find the synchronizedstate achieved within several beats, as in the simulations with d =
5. Thus, the synchronization e ff ect is of purely hydrody- (a)(b)(c)(d)(e)(f)(g) FIG. 2: (Color online) Snapshots of two sperm with phases ϕ (up-per), ϕ (lower), and phase di ff erence ∆ ϕ = ϕ − ϕ = . π . (a) t f = / t f = /
3; (c) t f = /
6; (d) t f = / t f = /
6; (f) t f = ; (g) t f = . From (a) to (e), thesynchronization process takes place. The tails are already beating inphase in (e). From (e) to (g), two synchronized sperm form a tightcluster due to hydrodynamic attraction. namic origin. Since the beating phase at time t is determinedby f and ϕ s , which are kept constant in our simulations, ourmodel sperm can only achieve synchronization by adjustingthe relative position.Our results are in good agreement with the prediction ofTaylor [5], based on an analytical analysis of two-dimensionalhydrodynamics, that the viscous stress between sinusoidallybeating tails tends to force the two waves into phase. Thesame phenomenon has also been observed by Fauci and Mc-Donald [6] in their simulations of sperm in the presence ofboundaries, and has been called “phase-locking” e ff ect. Asimilar e ff ect of undulating filaments immersed in a two- dimensional fluid at low Reynolds number was seen by Fauciin Ref. [38].Synchronization is a fast process, which is achieved in atmost ten beats in our simulations. Another hydrodynamic ef-fect, which we denote “attraction”, takes much longer time.Two synchronized and separated sperm gradually approacheach other when they are swimming together, as if there wassome e ff ective attractive interaction between them. The onlyway in which the sperm can attract each other in our simula-tions is through the hydrodynamics of the solvent. This ef-fect takes several ten beats to overcome the initial distance of d = ff ect between sperm. Hence in somecases, they could only see a synchronization e ff ect, and nei-ther a clear towards-wall tendency nor a distinguishable at-traction e ff ect. The hydrodynamic attraction was masked bythe presence of the walls.To analyze the cooperating sperm pair in more detail, wechoose the head-head distance d h to characterize the attrac-tion and synchronization, because it is easy experimentally totrack the head position. The dependence of d h on the phasedi ff erence is symmetric with respect to ∆ ϕ = ∆ ϕ >
0. There is a plateau at about d h = a for ∆ ϕ < . π , which corresponds to the sperm heads touch-ing each other. For ∆ ϕ > . π , d h increases linearly with ∆ ϕ .Finally, for ∆ ϕ > . π , the phase di ff erence is so large thatthe attraction is not strong enough to overcome the thermalfluctuations and pull the sperm close together. Although syn-chronization still occurs at the beginning, the two sperm leaveeach other soon after.Riedel et al. [16] also see such a linear relation in their ex-periments of sea-urchin sperm vortices. They define the beat-ing phase of a sperm by its head oscillation, and an angularposition of the sperm head within the vortex. In this way, thebeating phase di ff erence of the sperm in the same vortex wasfound to have a linear relation with the angular position dif-ference, which corresponds to the head-head distance in oursimulations.So far, we have considered sperm with a single beat fre-quency. In nature, sperm of the same species always have awide distribution of beat frequencies. For example, the beatfrequency of sea-urchin sperm ranges from 30Hz to 80Hz[1], and the frequency of bull sperm ranges from 20Hz to30Hz [36]. Thus, we assign di ff erent beat frequencies totwo sperm, f = /
120 and f = / .
4, corresponding to ∆ f / f ≈ . ϕ s =
0. Thisimplies that the phase di ff erence of the beats between the two tf d h / a FIG. 3: (Color online) Head-head distance d h of two cooperatingsperm. Simulation data are shown for fixed phase di ff erence (red, (cid:3) ), with error bars denoting the standard deviation. The interpolating(red) line is a linear fit for 0 . π < ∆ ϕ < . π . The distance d h is alsoshown as a function of time t (top axis) in a simulation with a 0 . ff erence in the beat frequencies of the two sperm (solid line).FIG. 4: Snapshots of two synchronized human sperm in experimentat di ff erent times [39, 40]. (Left) Two sperm with initially well syn-chronized tails and very small phase di ff erence; (Middle) the spermare still swimming together and are well synchronized after 4 sec-onds; a phase di ff erence has developed; (Right) the sperm begin todepart after 7 seconds. The scale bar corresponds to a length of 25 µ m. sperm increases linearly in time, ∆ ϕ = π ( f − f ) t . (11)Fig. 3 shows the head-head distance versus time. It agreesvery well with the data for fixed phase di ff erences. At t f =
150 where ∆ ϕ ≃ . π , the sperm trajectories begin to depart.Fig. 4 shows two cooperating human sperm swimming inan in-vitro experiment near a glass substrate [39, 40]. Thetwo sperm swim together for more than 6 seconds at a beatfrequency of approximately 8Hz. Their tails remain synchro-nized during this time, while the head-head distance and phasedi ff erence increases with time (see Fig. 4). After a while, thesperm leave each other because the phase di ff erence becomestoo large. There is no indication of a direct adhesive interac-tion between the sperm.An interesting question is whether the cooperation of asperm pair reduces the energy consumption. Fig. 5 displaysthe energy consumption of two sperm with the same beat fre- tf P FIG. 5: (Color online) Energy consumption per unit time, P , of twocooperating sperm. Symbols show simulation data for fixed phasedi ff erence (red, (cid:3) ), where error bars denote the standard deviation. P versus time t in a simulation with a 0 .
5% di ff erence in the frequenciesof two sperm (solid black line). quency f = /
120 as a function of the phase di ff erence. Theenergy consumption P is nearly constant at small phase dif-ference. It increases for ∆ ϕ ≥ . π roughly linearly until itreaches another plateau for ∆ ϕ ≥ . π . The second plateaucorresponds to two sperm swim separately, so that energy con-sumption is twice the value of a single sperm. Our results arein agreement with the conclusion of Taylor [5] that less energyis dissipated in the fluid if the tails are synchronized.Fig. 5 also shows the energy consumption of two spermwith f = / f = / . ϕ = ϕ = t . In this simulation, we start with two sperm which areparallel and at a distance d =
5. For t f <
25, the energy con-sumption decreases as the sperm are approaching each other.The data agrees quantitatively very well with results for con-stant ∆ ϕ , and reaches a plateau when the cooperating spermpair departs.The synchronization and attraction also exists in our sim-ulation of swimming flagella without heads. In this case, thetime-reversal symmetry of Stokes flow implies that no syn-chronization nor attraction is possible at zero Reynolds num-ber. In our simulations, the thermal fluctuations and a finiteReynolds number break the time-reversal symmetry. B. Asymmetric Sperm
In nature, sperm have an abundance of di ff erent shapes. Inparticular, these shapes are typically not perfectly symmet-ric. The asymmetric shape can cause a curvature of the spermtrajectory [7, 41]. For example, sea-urchin sperm uses thespontaneous curvature of the tail to actively regulate the spermtrajectory for chemotaxis [42, 43]. In our simulations, we im-pose an asymmetry of the tail by employing a non-zero spon-taneous curvature c , tail .We consider curved sperm tails, with c , tail = . / a , which -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.05101520 = -0.7 = 0.7 d h / a FIG. 6: (Color online) Head-head distance d h of two cooperatingsperm with spontaneous curvature c , tail = . / a as a function of thephase di ff erence ∆ ϕ . The error bars represent standard deviations.Lines are linear fits to the data in the range − . π < ∆ ϕ < − . π and 0 . π < ∆ ϕ < . π , respectively. The inset shows two typicalconformations with positive and negative phase di ff erence. -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.03540455055 P FIG. 7: (Color online) Energy consumption per unit time, P , versusphase di ff erence ∆ ϕ of two sperm with spontaneous curvature c , tail = . / a of their tails. The error bars represent standard deviations. results in a mean curvature of the trajectory of a single spermof c t a = . ± . d h ( ∆ ϕ ) is not symmetric about ∆ ϕ = ∆ ϕ is defined as the phase of thesperm on the inner circle minus the phase of the sperm on theouter circle. The steric repulsion of the heads causes a plateauof the head-head distance at d h = ff er-ences ∆ ϕ , as in Fig. 3 for symmetric sperm. For ∆ ϕ < − π/ ∆ ϕ > π/
4, the head distance increases linearly with in-creasing phase di ff erence, with a substantial di ff erence of theslopes for ∆ ϕ < ∆ ϕ >
0, see Fig. 6. The two spermdepart when ∆ ϕ > . π . For ∆ ϕ . − . π , the sperm pairbriefly looses synchronicity, but then rejoins with a new phase di ff erence ∆ ϕ ′ = ∆ ϕ + π .The energy consumption P for sperm with spontaneous cur-vature (see Fig. 7) also increases sharply at ∆ ϕ = . π andstays at the plateau with P = . ± . ∆ ϕ , as thesperm are swimming separately. However, for ∆ ϕ < − π/ P increases rather smoothly until the cooperation is lost for largephase di ff erences.We conclude that the strong curvature of the tail breaks thesymmetry of the head-head distance d h and the energy con-sumption P in ∆ ϕ , but the e ff ect of synchronization and at-traction are still present and play an important role in the co-operation of sperm pairs. IV. MULTI-SPERM SYSTEMS
When two sperm with the same beating period happen toget close and parallel, they interact strongly through hydro-dynamics and swim together. With this knowledge of hydro-dynamic interaction between two sperm, we now study a sys-tem of 50 sperm in a simulation box of 200 ×
200 collisionboxes, which corresponds to a density of about three spermper squared sperm length. The initial position and orienta-tion for each sperm are chosen randomly. Considering that inreal biological systems the beat frequency is not necessary thesame for all sperm, we perform simulations with Gaussian-distributed beating frequencies. The initial phases of all spermare ϕ s = ff erent width δ f = h ( ∆ f ) i / / h f i of the Gaussian frequency distribution. Here, h ( ∆ f ) i is the mean square deviation of the frequency distri-bution, and h f i = /
120 is the average frequency.For δ f =
0, once a cluster has formed, it does not disinte-grate without a strong external force. A possible way of break-up is by bumping head-on into another cluster. For δ f > ffi ciently longtime, since the phase di ff erence to other cells in the cluster in-creases in time due to the di ff erent beat frequencies (compareSec. III A). At the same time, the cluster size can grow by col-lecting nearby free sperm or by merging with other clusters.Thus, there is a balance between cluster formation and break-up, as shown in the accompanying movie [44]. Obviously, theaverage cluster size is smaller for large δ f than for small δ f (see Fig. 8).To analyze the multi-sperm systems, we define a cluster asfollows. If the angle between vectors from the last to the firstbead of the tails of two sperm is smaller than π/
6, and at thesame time the nearest distance between the tails is smallerthan 4 a , which is approximately 1 /
10 of the length of the tail,then we consider these two sperm to be in the same cluster.By this definition, we find the evolution of the average clustersize < n c > shown in Fig. 9. Here, < n c > is the averagenumber of sperm in a cluster, < n c > = X n c n c Π ( n c ) , (12)where Π ( n c ) is the (normalized) cluster-size distribution. For (a)(b)(c) FIG. 8: (Color online) Snapshots from simulations of 50 symmetricsperm with di ff erent widths δ f of a Gaussian distribution of beatingfrequencies. (a) δ f =
0; (b) δ f = . δ f = . δ f =
0, the average cluster size continues to increase withtime. Both systems in Fig. 9 with δ f > δ f of the frequencydistribution. We find a decay with a power law, < n c > ∼ δ − γ f (13)with γ = . ± .
01. The error for γ is estimated from a fit of t < f > < n c > f =0 f =0.9% f =4.5% FIG. 9: (Color online) Time dependence of the average cluster size, < n c > , in a system of 50 symmetric sperm with various widths δ f ofthe frequency distribution, as indicated. N =50 N =25 < n c > f ( % ) FIG. 10: (Color online) Dependence of the average stationary clustersize, < n c > , on the width of the frequency distribution δ f . Data areshown for a 50-sperm system ( (cid:4) ) and a 25-sperm system ( ◦ ). Thelines indicate the power-law decays < n c > = . δ − . f (upper) and < n c > = . δ − . f (lower). the data for both 50-sperm and 25-sperm systems. The neg-ative power law indicates that the cluster size diverges when δ f →
0. This tendency is also implied by the continuouslyincreasing cluster size for δ f = ff ects. Similarly, in a system ofself-propelled rods with volume exclusion, a crossover from ( n c ) n c ( n c ) n c FIG. 11: (Color online) Cluster size distribution, Π ( n c ). Data areshown for 25 sperm in a 200 × a box ( • ), 100 sperm in a 400 × a box ( (cid:3) ) [note that both systems have the same sperm density],and 50 sperm in a 200 × a box ( △ ). The lines correspond to anexponential distribution. The inset shows the same data in a double-logarithmic representation. The line indicates a power law n − . c . power-law behavior at small cluster-sizes to a more rapid de-cay for large cluster sizes has also been found [47]. A power-law decay of the cluster-size distribution is indeed consistentwith our results for smaller cluster sizes, as shown in the insetof Fig. 11. The rather similar value of the exponent with thatof Ref. [46] is probably fortuitous. We attribute the exponen-tial decay of the cluster-size distribution for larger cluster size,which is apparent in Fig. 11, to finite-size e ff ects. Simulationsof larger system sizes are required to confirm this conclusion.To analyze the energy consumption of sperm clusters, weconsider a special case where sperm of the same frequency areprearranged to pack tightly and to be synchronized, as shownin Fig. 12a. A simple linear relationship between the energyconsumption of the sperm cluster and the cluster size is shownin Fig. 12a. From the linear fit of the data, we obtain an energyconsumption per sperm for an infinitely large cluster, P / n c = .
7. Thus, a freely swimming sperm can reduce its energyconsumption by almost a factor 2 by joining a cluster.The swimming speed of a sperm cluster decreases slowlywith increasing sperm number, as shown in Fig. 12b. Whenflagella are very close, with distances smaller than the size of aMPC collision box, hydrodynamic interactions are no longerproperly resolved. Instead, the collision procedure yields asliding friction for the relative motion of neighboring flagella.Thus, the energy of the beat is not only used for propulsion,but also to overcome the sliding friction. The energy con-sumption of tail-tail friction is proportional to the number ofneighbor pairs, and the hydrodynamic resistance of movingthe whole cluster is proportional to the cluster size and speed.Thus, the total energy consumption can be written as P = Cn c v + p f ( n c − , (14)where p f is the energy consumption due to tail-tail friction,and C is a constant. With the relation P = . n c + . (b) n c v cluster FIG. 12: (Color online) (a) Energy consumption per unit time, P ,of sperm clusters as a function of cluster size n c . Symbols indicatesimulations results. The fit line (red) is given by P = . n c + . n c . The fit line (red) is given by v = . √ . + . / n c . obtained above, the data for the cluster speed can be fittedto Eq. (14), which yields p f = .
28 and C = . × .Thus, the cluster speed reaches a non-zero asymptotic value[(13 . − p f ) / C ] / ≃ . V. SUMMARY AND DISCUSSION
We have simulated the hydrodynamic interaction betweensperm in two dimensions by the multi-particle collision dy-namics (MPC) method. Two e ff ects of the hydrodynamicinteraction were found in our simulations. First, when twosperm are close in space and swimming parallel, they syn-chronize their tail beats by adjusting their relative position.This process can be accomplished in a very short time, lessthan 10 beats. Second, two synchronized sperm have a ten-dency to get close and form a tight pair. This process takesmuch longer time then synchronization. It usually takes about100 beats to overcome a distance of 1 /
10 tail length betweensperm in our simulations.These hydrodynamic e ff ects favor the cooperation of spermin motile clusters. For a multi-sperm system, the average clus-ter size diverges if all sperm have the same beating frequency.A distribution of frequencies leads to a stationary cluster-sizedistribution with a finite average cluster size, which decreaseswith a power law of the variance of the frequency distribution.Furthermore, the average cluster size increases with increas-ing sperm density. The probability to find a cluster decreaseswith a power law for small cluster sizes; an exponential decayfor large cluster sizes is attributed to finite-size e ff ects.In sperm experiments, large bundles have been found insome species, like fish flies [11, 13] and wood mouse [12, 15].For fish-fly sperm, this has been attributed to some agglu-tination of the sperm heads to keep the size and structureof the bundles. Wood mouse sperm were released into anin-vitro laboratory medium, initially in single cell suspen-sion [12]. Within 10 minutes, large bundles containing hun-dreds or thousands of sperm were formed as motile ’trains’of sperm. Motile bundles of 50-200 sperm were also foundin the after-mating female’s body, as well as many non-motilesingle sperm. The hook structure on the head of wood mouse sperm is believed to favor the formation of such huge clusterin in-vitro experiments.In our simulations, sperm clusters are always seen, e.g.as marked in Fig. 8, after the system has reached a dy-namically balanced state of cluster sizes. Thus, we predictthat hydrodynamic synchronization and attraction play animportant role in the cluster formation of healthy and motilesperm, such as the bundles and trains observed for fish-flyand wood-mouse sperm at high concentrations, respectively.Furthermore, since the cluster size decreases with increasingwidth δ f of the distribution of beat frequencies, our resultsare consistent with the experimental observation that if thesperm are hyperactivated [12], which is an abnormal beatmode, or if some sperm are dead, the clusters fall apart. Acknowledgments:
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