## Abstract

Many patterns in biological systems depend on the exchange of chemical signals between cells. We report a spatiotemporal pattern mediated by hydrodynamic interactions. At planar surfaces, spermatozoa self-organized into dynamic vortices resembling quantized rotating waves. These vortices formed an array with local hexagonal order. Introducing an order parameter that quantifies cooperativity, we found that the array appeared only above a critical sperm density. Using a model, we estimated the hydrodynamic interaction force between spermatozoa to be ∼0.03 piconewtons. Thus, large-scale coordination of cells can be regulated hydrodynamically, and chemical signals are not required.

Eukaryotic cilia and flagella are rodlike appendages that contain a conserved motile structure called the axoneme (*1*), an example of which is the tail of many animal spermatozoa. Oscillatory waves generated by the sperm tail propel spermatozoa through fluid, usually along helical paths. If spermatozoa approach planar surfaces, they become trapped at these surfaces and follow circular swimming paths with a strongly preferred handedness (*2*) (movie S1).

We found that the spermatozoa of sea urchins (*Strongylocentrotus droebachiensis* and *S. purpuratus*) self-organize at high surface densities into an array of vortices (Fig. 1, A and B, and movies S2 and S3) (*3*). At a density of 6000 cells/mm^{2}, each vortex contained 10 ± 2 spermatozoa (mean ± SD) circling clockwise (observed from inside the water phase) around a common center (Fig. 1, C to F). The circular paths had a radius of *R* = 13.2 ± 2.8 μm, the time for one revolution was *T* = 0.67 ± 0.09 s, and the swimming speed was *v* = 125 ± 21 μm/s. The beat frequency was *f* = 41.7 ± 3.7 Hz. Occasionally the hopping of spermatozoa between vortices and the fusion of two vortices were observed. The vortices were densely packed and their centers moved randomly with an apparent diffusion coefficient of *D* = 6.2 ± 0.9 μm^{2}/s. This apparent diffusion coefficient is much larger than the thermal diffusion coefficient *D* = 0.06 μm^{2}/s of a disk similar in size to a vortex [*D* = *kT*/γ; γ = (32/3) × η*R* = 0.07 μN·s/m, where radius *R* = 13 μm and friction in water η = 1 mPa·s] (*4*). This indicates that the array is out of thermal equilibrium because of the active propulsion of the spermatozoa (*5*, *6*), and hence the pattern is an example of self-organization (*7*–*9*). Slight changes of the microscopic parameters of such self-organized systems can lead to sudden changes in the overall pattern, making these systems amenable for regulation (*10*). We therefore analyzed the unexpected vortex array of spermatozoa to understand its underlying physical cause and to determine its possible relevance for related biological processes.

The vortex array reflected two levels of order: a clustering of spermatozoa into vortices and a packing of these vortices into an array. We assessed the packing order of the vortex array by measuring various correlation functions of the vortex centers. The pair-correlation function and the triplet-distribution function (*11*) revealed a local hexagonal order with an average vortex spacing of 49 ± 9 μm (Fig. 2). Furthermore, the bond-angular correlation function (*12*) showed an exponential decay indicating the absence of long-range order. Thus, the array is liquidlike rather than hexatic or crystalline (*12*).

We asked how the spermatozoa within a vortex influence each other (Fig. 3A and movie S4). Interactions could lead to changes in the circling radius, the swimming velocity, or the beat frequency. However, within experimental errors, we found no differences in these parameters whether spermatozoa were in a vortex or isolated. Instead, we did find a particular form of synchronization of the beating patterns of spermatozoa within a vortex: We described each spermatozoon by two variables: (i) the phase of the oscillation of the head during the beat of the spermatozoon, φ(*t*) [this oscillation is driven by and has the same frequency as the oscillation of the tail (Fig. 3B)] and (ii) the angular position of the head in its trajectory around the vortex, Φ(*t*) (Fig. 3C). No correlation in Φ(*t*) between any two spermatozoa in the same vortex was found. The same was true for φ(*t*). Hence, spermatozoa within a vortex swim at different speeds and beat at different frequencies. However, there is a strong correlation between the differences Δφ(*t*) and ΔΦ(*t*) between pairs of spermatozoa in the same vortex (Fig. 3, D to E). This implies, for example, that if one spermatozoon swims twice as fast as another then it also beats at twice the frequency. Thus, locally the tails are beating in synchrony and a trailing spermatozoon follows in the wake of the leading one. Because the spermatozoa swim in closed circular paths, there must be an integral number of wavelengths along the circumference of the vortex. The slope, Δφ/ΔΦ, was 4.2 ± 0.2 (Fig. 3E), consistent with a wave number of 4, which is determined by the geometry of the vortex: Dividing the circumference of the swimming path (2π*R, R* = 11.6 ± 3.0 μm for this particular vortex) by the beat wavelength on the sperm tail (λ = 17.6 ± 1.3 μm; along the curved centerline of the flagellar waveform, not along the arc length of the tail) gives 4.1 ± 1.4. Thus, hydrodynamic coupling of the sperm tails within a vortex leads to a quantized rotating wave with wave number 4 (Fig. 3F). This rotating wave is a generalization of the synchronization of the beats of spermatozoa swimming close to one another (*13*–*15*). Furthermore, it is related to the three-dimensional (3D) metachronal waves observed on the surfaces of ciliates and ciliated epithelia, which are important for swimming motility and the movement of mucus, where hydrodynamic interactions are also thought to play an important role (*16*, *17*).

How is the vortex array formed? Because we did not observe vortex arrays at low sperm surface densities, we suspected that density might play a role in the self-organization process. To quantify the order at the different densities, we defined an order parameter χ as follows. The binary images of each movie showing only sperm heads were summed such that each pixel value in the resulting image was proportional to the number of different spermatozoa that swam over that pixel (Fig. 4A) (*3*). If the swimming paths of different spermatozoa were uncorrelated, then these pixel values would be binomially distributed. However, if spermatozoa accumulated in a vortex they would trail each other and the distribution would differ from a binomial one because low and high pixel values (corresponding to centers of the vortices and swimming trails, respectively) would be overrepresented (Fig. 4B). In this case, the variance of the measured distribution () will be larger than that of the binomial distribution (). This motivated our definition of the order parameter , which had the expected properties: zero for a random configuration, and greater than zero if spermatozoa shared similar swimming paths. The value of χ depended on the average number of spermatozoa per vortex and how well the centers of their circular swimming paths colocalized. χ was a robust measure for the correlation among the objects and was related to the pair-correlation function [supporting online material (SOM) text]. Furthermore, χ required no labor-intensive object tracking, and hence it might be useful for quantifying order in other spatiotemporal patterns involving tracks of multiple particles or signals such as intracellular organelle transport (*18*) or ant trails (*19*).

We measured the order parameter χ for various sperm surface densities (Fig. 4C) and found a rapid change in the slope of the curve at ∼2500 cells/mm^{2} (fitting a Hill equation revealed a cooperativity factor of 5). This suggested a bifurcation separating a disordered and an ordered regime: one where the swimming paths of the spermatozoa were random and one where the correlation among the swimming paths increased, reflecting an increasingly pronounced vortex array.

To support this interpretation and to gain insight into the physical mechanisms underlying the pattern formation, we propose a simplified model. Each spermatozoon is represented by a point particle located at the center of its circular swimming path. These particles move randomly with an apparent diffusion coefficient of *D* = 9.0 ± 2.0 μm^{2}/s, measured for isolated spermatozoa. A short-range pairwise attraction, arising from the hydrodynamic forces leading to the observed synchronization (*20*), and a longer range repulsion, which could be of steric or hydrodynamic origin (*21*), are assumed (Fig. 4D). Although one cannot describe circular flow by a potential (*22*), the important features of the observed pattern are captured by our model.

Stochastic simulations of this model (SOM text) also revealed two regimes: a random distribution of particles at low densities with a transition toward a hexagonal array of clusters at a critical particle density (Fig. 4E). Assigning to each particle a spermatozoon circling around that position, we generated simulated movies (*3*) mimicking the experimental observation (Fig. 4F versus Fig. 1B). Moreover, the order parameter χ computed for different simulated sperm densities agreed with the experimentally observed dependency (Fig. 4C). Our numerical results were further supported by a 1D mean-field analysis (SOM text), which indicated the existence of a supercritical pitch-fork bifurcation at a critical sperm density (*23*). This critical density was proportional to the interaction strength and inversely proportional to the diffusion coefficient, the latter being associated with the noise in the system. This analysis demonstrates how the activity of biological processes can be regulated by critical points or bifurcations. For example, ciliary metachronal waves (*16*, *24*) might be switched on and off by small physiologically controlled changes of the activity of the individual cilia, thereby tuning the critical density for the onset of the metachronal wave.

The only free parameter in our model was the ratio of the maximum interaction potential to the drag coefficient, *V*_{0}/γ = 5 μm^{2}/s, which was chosen to match the critical density (Fig. 4C). This allowed us to estimate the interaction force between two spermatozoa *F*_{int} = |grad(*V*)| = (*V*_{0}/γ) × γ/*R* ∼0.03 pN (using *R* = 13 μm and γ = 0.07 μN·s/m from above). This force is about 1% of the forward propulsion force of spermatozoa *F*_{for} ∼5 pN (*25*). Although this hydrodynamic interaction force is smaller than typical adhesion forces involved in sperm cooperation (*26*), it is evidently large enough to coordinate the cells and to regulate large-scale pattern formation in the absence of chemical signals (*27*).

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/309/5732/300/DC1

Materials and Methods

SOM Text

Figs. S1 to S6

References

Movies S1 to S5