During chemotaxis and phototaxis, sperm, algae, marine zooplankton, and other microswimmers move on helical paths or drifting circles by rhythmically bending cell protrusions called motile cilia or flagella. Sperm of marine invertebrates navigate in a chemoattractant gradient by adjusting the flagellar waveform and, thereby, the swimming path. The waveform is periodically modulated by Ca2+ oscillations. How Ca2+ signals elicit steering responses and shape the path is unknown. We unveil the signal transfer between the changes in intracellular Ca2+ concentration ([Ca2+]i) and path curvature (κ). We show that κ is modulated by the time derivative d[Ca2+]i/dt rather than the absolute [Ca2+]i. Furthermore, simulation of swimming paths using various Ca2+ waveforms reproduces the wealth of swimming paths observed for sperm of marine invertebrates. We propose a cellular mechanism for a chemical differentiator that computes a time derivative. The cytoskeleton of cilia, the axoneme, is highly conserved. Thus, motile ciliated cells in general might use a similar cellular computation to translate changes of [Ca2+]i into motion.

The flagellum of sperm serves both as a propeller and antenna that detects chemical cues released by the egg or associated structures (Kaupp et al., 2003, 2008). The binding of chemoattractant molecules to surface receptors initiates a series of signaling events that produce Ca2+ signals in the flagellum (Matsumoto et al., 2003; Wood et al., 2007; Kaupp et al., 2008; Shiba et al., 2008). Chemotactic signaling and behavior are most advanced in sperm of the sea urchin Arbacia punctulata. It involves binding of resact, the chemoattractant, to a receptor guanylyl cyclase, a rapid rise of the cellular messenger cyclic guanosine monophosphate (cGMP; Kaupp et al., 2003), a hyperpolarization as a result of the opening of K+-selective cyclic nucleotide-gated ion channels (Strünker et al., 2006; Galindo et al., 2007; Bönigk et al., 2009), and, finally, the opening of voltage-dependent Cav channels. The periodic stimulation of sperm during circular swimming in a chemoattractant gradient entrains periodic Ca2+ signals and alternating periods of high path curvature (turn) and low path curvature (run) that result in a looping swimming path toward the egg (Böhmer et al., 2005; Friedrich and Jülicher, 2007; Wood et al., 2007; Guerrero et al., 2010a,b).

The relationship between intracellular Ca2+ concentration ([Ca2+]i) and flagellar beat or path curvature has been primarily studied in sperm that had been demembranated by detergents and reactivated by addition of ATP and cAMP (Lindemann and Lesich, 2009). These studies show that the flagellar beat is more asymmetrical at high [Ca2+]i and more symmetrical at low [Ca2+]i (Brokaw, 1979; Lindemann and Goltz, 1988; Lindemann et al., 1991). The action of Ca2+ on the flagellar beat is mediated by CaM, is relatively slow (on a minute time scale), and is modulated by cAMP (Lindemann et al., 1991). Although these studies highlighted the importance of Ca2+ and cAMP in demembranated sperm, for several reasons, the significance for intact motile sperm is limited. First, sperm from both marine invertebrates and mammals respond to stimulation with a rapid Ca2+ signal and motor response on the subsecond to second time scale (Kaupp et al., 2003; Böhmer et al., 2005; Wood et al., 2005; Strünker et al., 2006, 2011; Kilic et al., 2009; Guerrero et al., 2010a). However, Ca2+ experiments in demembranated sperm lacked time resolution, and, consequently, rapid or transient changes in flagellar beat might have been missed. Second, the Ca2+ action critically depends on the extraction and reactivation protocol, giving rise to a wide range of Ca2+ sensitivities (Gibbons and Gibbons, 1972; Okuno and Brokaw, 1981). Third, in reactivated flagella, the concentration, dynamics, and location of molecular components important for flagellar bending (Goltz et al., 1988; Salathe, 2007) might have been severely altered. Finally, in intact sperm, high [Ca2+]i levels persist during low path curvature, i.e., straight swimming (Böhmer et al., 2005; Wood et al., 2005; Shiba et al., 2008; Guerrero et al., 2010a; Kambara et al., 2011), challenging the view that steady-state [Ca2+]i controls the flagellar beat directly. To overcome these limitations, time-resolved measurements of changes in [Ca2+]i and motor response in intact swimming sperm are required.

Here, we study Ca2+ signals and steering responses of sperm while moving in a gradient of chemoattractant or after the release of the second messenger cGMP via photolysis of caged compounds. We identify the signal transfer function between [Ca2+]i and path curvature and analyze how the waveform of the Ca2+ signal controls the swimming path. Finally, we propose a chemical differentiator model by which cells translate the time derivative of Ca2+ signals to modulate the flagellar beat.

### Time derivative of [Ca2+]i controls the path curvature

To understand how changes in [Ca2+]i control the chemotactic steering response, we studied the dynamic relationship between [Ca2+]i and path curvature. Using caged compounds, Ca2+ oscillations were evoked by a step increase of either cGMP or the chemoattractant resact (Böhmer et al., 2005). First, we stimulated sperm by flash photolysis of caged cGMP and recorded the relative changes in fluorescence (Fr) of the Ca2+-sensitive dye Fluo-4. Binding of Ca2+ to and unbinding from BAPTA-derived fluorescent indicators occur within a few milliseconds (Naraghi, 1997; Faas et al., 2011), whereas Ca2+ signals occur on a subsecond to second time scale. Therefore, the kinetics of Ca2+ signals is not compromised by the kinetics of the dye. In addition, Fluo-4 fluorescence scales linearly with the [Ca2+]i for the regimen of concentrations found in sperm during chemotaxis (Fig. S1). Unstimulated sperm swam in circles with the net swimming speed v0 = 120 ± 13 µm/s (mean ± SD; n = 26) and constant path curvature κ0 = 39 ± 6 mm−1, corresponding to a radius of the swimming circle r0 = 26 ± 4 µm and a circle period T = 2π/(v0κ0) = 1.4 ± 0.2 s. Photorelease of cGMP elicited a train of sawtooth-shaped Ca2+ signals in the flagellum that were accompanied by brief spikelike increases of the path curvature κ, intermitted by longer periods of low κ values (Fig. 1 A). A characteristic feature was that κ peaked, whereas Fr was still rising, and steeply fell below resting values, whereas Fr was slowly declining (Fig. 1 A); thus, [Ca2+]i stayed elevated during the intermittent periods of straight swimming (Fig. 1 B). Both observations are inconsistent with a direct control of path curvature by absolute [Ca2+]i. In fact, analysis of the paths from 27 cells revealed no correlation between fluorescence intensity, i.e., [Ca2+]i and curvature (Fig. 1 F). However, the time derivative of the fluorescence signal (dFr/dt) and the curvature (κ) superimposed (Fig. 1 C). The good match also manifested in a high correlation coefficient between these two measures (Fig. 1 F). Moreover, dFr/dt, but not Fr, perfectly coincided with the turn episodes, i.e., the path segment with the highest curvature (Fig. 1 [B–E] and Video 1). Because after stimulation the fluorescence from the flagellum makes the largest contribution to the Ca2+ signal and because the correlation is highest for the flagellum compared with the head, we conclude that d[Ca2+]/dt in the flagellum is responsible for the changes in κ (Figs. S4 and S5).

The time course κ(t) after stimulation can be described by the linear relationship

$κ(t)=κ1+βdFrdt,$
(1)

where β is a proportionality factor, κ1 is a parameter that characterizes the basal curvature, and Fr = ΔF/F0 is the normalized relative change in fluorescence. Fit parameters were β = 0.04 ± 0.03 s/mm and κ1 = 12 ± 8 mm−1 (mean ± SD; n = 27). The goodness of the fit to Eq. 1 is given by the coefficient of determination, which is the square of the Pearson correlation coefficient R (Figs. 1 F and 2 D).

On photolysis, the cGMP increase occurs almost instantaneously and triggers intrinsic Ca2+ oscillations (Böhmer et al., 2005). For swimming in a gradient of resact, however, the cGMP increase is slower owing to the synthesis by the guanylyl cyclase, and Ca2+ oscillations are entrained to the periodic stimulation of sperm with the chemoattractant (Böhmer et al., 2005). Therefore, we tested whether the relation between dFr/dt and κ also holds true during chemotactic navigation. We recorded Ca2+ signals of sperm swimming in a chemical gradient. The gradient was produced by the release of resact from caged resact using a radial (Gaussian) profile of UV light (Böhmer et al., 2005; Friedrich and Jülicher, 2007). In the resact gradient, sperm moved along drifting circles toward the top of the concentration profile (Fig. 2 A). Again, the curvature correlated with dFr/dt and not with Fr (Fig. 2, B–D). In some cases, the peaks of d[Ca2+]i/dt and κ do not perfectly align (Fig. 1 C). A shift might result from data acquisition and processing involving finite frame rates (30 Hz) and spatiotemporal smoothing that slightly limits temporal resolution. We conclude that the relation between d[Ca2+]i/dt and κ is preserved, regardless of whether Ca2+ oscillations are evoked by a step increase of cGMP or by navigation in a gradient of chemoattractant.

### Waveform of Ca2+ signals determines the swimming path

The waveform and frequency of Ca2+ signals evoked by cGMP or resact greatly vary between cells depending on the stimulus strength or steepness of the gradient. We examined which features of the Ca2+ waveform shape the swimming path. Photorelease of cGMP evoked an initial increase of [Ca2+]i followed by oscillations of smaller amplitude that are superimposed on the elevated Ca2+ level (top graphs in Fig. 3, A–C). Quite generally, steep and long rising phases of [Ca2+]i produced sharp turns, whereas the characteristics of the Ca2+ decline determined the duration and curvature of the run periods (bottom illustrations in Fig. 3, A–C). Despite the large variation of Ca2+ waveforms, path curvature and dFr/dt always superimposed (middle graphs in Fig. 3, A–C).

To gain insight into how exactly the Ca2+ waveform shapes the path, we numerically reconstructed the swimming path generated by model Ca2+ signals using Eq. 1. The slope of the Ca2+ rise determines the sharpness of turns and, thereby, the angle between sperm orientation before and after a turn (Fig. 4 A). In addition, the orientation angle is controlled by the duration of the rising phase (Fig. 4 B).

The slope of the Ca2+ decline determines the curvilinearity of the path during the run period (Fig. 4 C). A fast Ca2+ decline can even give rise to negative values of the curvature (i.e., the path temporarily curves in clockwise rather than counterclockwise direction; Fig. 4 C, right). In fact, this prediction from the simulation is borne out by experiment. We took advantage of the observation that a second cGMP increase delivered at the peak of a Ca2+ signal produced a large and rapid Ca2+ drop, probably owing to the closure of Cav channels and enhanced Na+/Ca2+ exchange activity (Fig. 5 A; Nishigaki et al., 2004; Kashikar, 2009). We recorded the path before and after the second stimulus. During the rapid Ca2+ drop, whereas Fr was still elevated, dFr/dt was minimal, and the curvature adopted negative values (Fig. 5, A–C). As a consequence, the swimming path curved for a short period in the clockwise direction before it returned to the counterclockwise mode (Fig. 5, D–F).

Periodic Ca2+ signals with various shares of positive and negatives slopes gave rise to a great variety of swimming paths ranging from slow drifting circles to saltatoric paths characterized by narrow turns and wide arcs of run periods (Fig. 4 D). Finally, using Eq. 1 and considering the swimming speed of the cell, the overall path of sperm swimming in a chemoattractant gradient was also predicted with reasonable precision from the experimentally determined Ca2+ signal (Fig. 6).

### Pharmacological interference with chemotactic signaling does not alter the relation between d[Ca2+]i/dt and κ

Next, we pharmacologically altered the Ca2+ waveform to test whether Ca2+ kinetics affects chemotactic efficacy. Niflumic acid alters Ca2+ oscillations in sperm either by inhibition of Ca2+-activated Cl channels (Wood et al., 2003, 2007) or by targeting other signaling components such as hyperpolarization-activated cyclic nucleotide-gated channels (Gauss et al., 1998; Cheng and Sanguinetti, 2009). Whatever the underlying mechanism might be, Niflumic acid offers a means to study how the Ca2+ waveform affects the swimming path and, thereby, chemotaxis efficacy. Niflumic acid distorted the cGMP-evoked Ca2+ signals in a characteristic way (Fig. 7): the initial increase of [Ca2+]i was followed by Ca2+ oscillations that appeared as small ripples on an elevated Ca2+ level (Fig. 7 A). Accordingly, the curvature pattern consisted of closely spaced spikes and was lacking longer intermittent periods of low κ values. Despite these profound changes in Ca2+ waveform, the relation between dFr/dt and κ remained unaltered, and both quantities superimposed (Fig. 7 B). The altered Ca2+ dynamics entailed characteristic changes in the swimming path. Control sperm usually make sharp turns and wide arcs of running. With Niflumic acid, turns and runs were both shorter. Furthermore, the curvature changed sign, and the runs initially curved in clockwise rather than counterclockwise direction, producing a trefoil-like pattern of the path (Fig. 7 C).

How do the changes in swimming path affect the ability of sperm to accumulate in a chemical gradient? We followed the redistribution of sperm in a resact gradient produced by UV irradiation of caged resact using a Gaussian intensity profile. In the presence of Niflumic acid, the ability of sperm to accumulate was impaired although not completely abolished (Fig. 7 D). Chemotactic behavior requires a characteristic latency between the stimulus and the steering response (Friedrich and Jülicher, 2007). We presume that Niflumic acid detuned this time delay and, thereby, impaired chemotactic accumulation.

### The chemical differentiator model

Despite the simple relationship between d[Ca2+]i/dt and path curvature, it is not apparent how cells implement such a differentiator using a reaction network to control flagellar bending waves. In flagella and cilia, the shape and frequency of the beat are modulated by several Ca2+-binding proteins (B), notably CaM. These proteins control molecular components of the axoneme, in particular motor proteins (DiPetrillo and Smith, 2009; King, 2010). Here, we propose a simple biochemical mechanism that could generate a cellular differentiator. The second-order kinetics for Ca2+ binding to a protein B reads as follows:

$Ca2++B↔koffkonCaB.$

In the terminology of signal processing, the concentration [CaB] of Ca2+ complexes serves as a low-pass filter that faithfully tracks relatively slow variations of the input [Ca2+] but effectively filters any fast changes. The second-order reaction can be linearized using two simplifying assumptions. First, in the limit where kon[Ca2+] is much less than koff, [B] is approximately constant, and the kinetics of the reaction becomes pseudo–first order with a characteristic time constant of low-pass filtering τ1 = 1/koff. Second, in the limit of small variations of the total Ca2+ concentration around its steady-state value, the characteristic time constant τ2 = 1/(kon[Ca2+] + koff + kon[B]).

Our simple model consists of two distinct proteins, B1 and B2, coupled to a motor protein. We assume that binding of Ca2+ to B1 and B2 enhances and diminishes flagellar asymmetry, respectively, for example by enhancing either principal or reverse flagellar bending on opposite microtubule doublets. For sake of simplicity, we assume the following direct relationship between the concentration difference [CaB1] − [CaB2] and flagellar asymmetry C:

$C=α×([CaB1]−[CaB2]).$
(2)

The difference of occupancy of B1 and B2 by Ca2+ can act as a chemical differentiator, provided the kinetics of binding of CaB1,2 to the motor protein is not identical, B1 and B2 are of similar concentration, and the Kd values are approximately equal. Of note, an equivalent mechanism can be established with a single Ca2+-binding protein hosting two different Ca2+-binding sites.

To study the performance of the chemical differentiator model, we chose the following set of parameters: the rate constants and concentrations were derived from the Ca2+-binding protein light chain 4 (LC4). LC4 forms part of the outer dynein arm of Chlamydomonas reinhardtii. This protein complex is required for changes in flagellar symmetry associated with the photophobic response. LC4 possesses two EF-hand motifs and binds Ca2+ with a dissociation constant Kd = 3 × 10−5 M. LC4 forms a complex with the dynein heavy chain–γ and undergoes Ca2+-dependent conformational changes (Sakato et al., 2007). Assuming a 1:1 stoichiometry between LC4 and outer dynein arm (24-nm axial periodicity), a flagellar length of 50 µm, and a flagellar volume of 1.6 femtoliters, we obtain [LC4]total = 20 µM. We assume [B1] = [B2] = [LC4]total and a maximal free [Ca2+] of 500 nM (Cook et al., 1994). The reaction rates used were k1on = 5 × 106 M−1s−1, k1off = 150 s−1, k2on = k1on/2, and k2off = k1off/2, compatible with the Kd value reported for LC4. We studied the input–output relation of the chemical differentiator by using sinusoidal [Ca2+] test stimuli with varying frequency f as follows:

$[Ca2+]=A0+A1cos(2πf×t).$

We find that the output signals also display sinusoidal waveforms that can be described by

$C=α×([CaB1]−[CaB2])=ρA1cos(2πf×t+φ),$

with amplitude gain ρ and phase shift ϕ depending on the input frequency f (Fig. 8, A and B).

Strict mathematical differentiation of the test stimuli leads to the following expression:

$d[Ca2+]dt=−2πfA1sin(2πf×t)=2πfA1cos(2πf×t+π2).$

Thus, differentiation requires both an output signal with a phase shift ϕ = π/2 radians or 90° and a gain that increases linearly with the input frequency. Indeed, we find that the chemical differentiator model fulfils these two requirements within the differentiation band (f ≤ 6 Hz; Fig. 8, A and B).

An inherent property of a strict mathematical differentiator is that the output gain approaches zero for slow input frequencies. As a consequence, at steady state, the differentiator is insensitive to the basal input. Within the differentiation band, the chemical differentiator indeed displays this property. Therefore, the chemical differentiator exhibits perfect adaptation: the steady-state output C is zero if the input [Ca2+] does not change with time. In other words, the output returns to baseline levels even in the presence of an elevated steady input level.

On the contrary, for high input frequencies (f > 6 Hz), the chemical differentiator deviates from the strict mathematical equivalent. This unexpected property results in a more robust mechanism because input signals with high frequencies (usually associated with stochastic noise) are filtered out. Of note, the frequency of Ca2+ oscillations in sperm is typically ∼1 Hz. Indeed, when using prototypical test Ca2+ signals, the output C follows the time derivative d[Ca2+]i/dt of the Ca2+ signal (Fig. 8, C and D).

In terms of systems biology, our model represents an incoherent biparallel motif that also behaves like a band-pass filter (Milo et al., 2002). However, it lacks robustness with respect to parameter variations. Such robustness could require feedback mechanisms not considered here (Barkai and Leibler, 1997). In particular, direct mechanical feedback arising from the flagellar beat could provide such robustness (Howard, 2009). Our system is intended to illustrate a simple mechanism of chemical differentiation by cellular messengers that produce opposite actions on different time scales in response to an input signal.

In a general model of sperm chemotaxis, a signaling pathway translates a periodic stimulus into a periodic modulation of the path curvature (Friedrich and Jülicher, 2007). Here, we identified the signal transfer function between the input (Ca2+ signal) and the output (path curvature). Although Ca2+ spikes trigger the steering response, previous studies on sperm of marine invertebrates noted that the path curvature is not directly correlated with [Ca2+]i and that sperm swim on straighter paths at elevated [Ca2+]i (Böhmer et al., 2005; Wood et al., 2007; Shiba et al., 2008; Guerrero et al., 2010a), an observation at odds with studies on demembranated sperm (Brokaw, 1979; Lindemann and Goltz, 1988; Lindemann et al., 1991). Moreover, it was noted that κ peaked long before [Ca2+]i has risen maximally (see Fig. 3 B from Guerrero et al., 2010a). Our findings provide an explanation for these observations: the curvature κ is highly correlated with d[Ca2+]i/dt rather than [Ca2+]i, and, consequently, [Ca2+]i is still high when κ is low. By inference, our data suggest that the flagellar asymmetry is controlled by d[Ca2+]i/dt as well. Moreover, numerical simulations using this signal transfer function reproduce the richness of swimming modes—from looping paths to smoothly drifting circles—observed for sperm of many marine invertebrates (Miller, 1985).

How general is this mechanism of using d[Ca2+]i/dt to control the swimming path of sperm? We analyzed the changes in [Ca2+]i and swimming path for several datasets of two other sea urchin species and a single dataset of Ciona intestinalis. For all three species, we obtained similar results, suggesting that this mechanism is not restricted to A. punctulata sperm (Figs. S2 and S3). It will be interesting to study this relationship also in mammalian sperm or, quite general, in motile cilia whose beat also depends on [Ca2+]i.

However, regulation of κ might be more complex. For example, before stimulation, sperm swim with a basal curvature (κ0) larger than after stimulation (κ1). Moreover, the mean curvature after stimulation stays low for some time and then slowly returns to resting values within ≤60 s (Böhmer et al., 2005). Accordingly, when sperm have returned to swimming again in regular circles, their diameter is larger (i.e., the κ1 value is lower; Kaupp et al., 2003; Böhmer et al., 2005; Guerrero et al., 2010a). When [Ca2+]i eventually returns to baseline level, the circles constrict, and the curvature returns to the basic value κ0 (Kaupp et al., 2003; Böhmer et al., 2005). Thus, in intact sperm, the relationship between flagellar asymmetry and steady-state [Ca2+]i is inverse to that observed for demembranated sperm. Probably, the modulation of flagellar bending by [Ca2+]i is altered in demembranated sperm and might not mirror the control in intact motile sperm. In conclusion, [Ca2+]i possibly controls the flagellar beat on two different time regimes: on a rapid time scale, the curvature follows on the heels of d[Ca2+]i/dt, whereas at quasi–steady state, κ is inversely related to [Ca2+]i (i.e., low κ corresponds to higher [Ca2+]i). On a final note, the ciliary and flagellar beat is also affected by cAMP and pH (Salathe, 2007). It needs to be addressed by future studies whether cAMP and pH are involved in the slow modulation of κ.

Our studies have been restricted to sperm swimming at the glass–water interface in shallow chambers. When swimming in a plane (2D), the path curvature is controlled by d[Ca2+]i/dt. In 3D, however, sperm swim on helical paths (Crenshaw, 1993; Corkidi et al., 2008). Chemotaxis in 2D only requires modulation of the flagellar beat in a plane, yet, for freely swimming sperm, more complex 3D waveforms of the beat might be required. Notwithstanding, the reorientation of the axis of the path helix toward the chemoattractant gradient might also be governed by d[Ca2+]i/dt rather than [Ca2+]i itself.

For most cellular systems, the response to a continuous stimulus is transient, a property termed adaptation. In sperm, nothing was known about adaptation along the cellular signaling pathway. The dependence of curvature on d[Ca2+]i/dt constitutes a novel chemomechanical mechanism of perfect adaptation. The mechanical transducer only responds to changes in [Ca2+]i (i.e., at steady state, the output is constant). Such an adaptive system enables sperm to maintain their responsiveness at elevated levels of both chemoattractant and [Ca2+]i during their sojourn to the egg.

The mechanism how sperm compute a time derivative of [Ca2+]i and how this time derivative controls the flagellar beat could be based on two Ca2+-binding reactions with different time constants and opposite action on flagellar bending. In the axoneme of Chlamydomonas flagella, ∼27 proteins have been identified that carry an EF-hand, a common structural motif for Ca2+-binding proteins (Pazour et al., 2005). CaM, a key Ca2+ sensor in cilia, is associated with the radial spoke stalk and the central pair, and several CaM-binding proteins have been identified (DiPetrillo and Smith, 2009, 2010; King, 2010). Moreover, Ca2+ binds more rapidly to CaM than to other Ca2+-binding proteins (Faas et al., 2011). Furthermore, two Ca2+ ions cooperatively bind to both the N- and C-lobe of CaM. The binding kinetics and affinity of Ca2+ to the two lobes are orders-of-magnitude different (Faas et al., 2011). These features provide a structural basis for the chemical differentiator model. Finally, the N- and C-lobe of CaM might sense different Ca2+ pools (i.e., global vs. local Ca2+ in nanodomains), as suggested for Cav channels (Tadross et al., 2008). Our minimalist model illustrates a general concept by which a cell could compute the time derivative of an input signal.

The control of the flagellar beat by Ca2+ is not restricted to sperm. The intrinsic beat of most if not all motile cilia is modulated by [Ca2+]i (Salathe, 2007). Motile cilia serve diverse functions. They transport liquid and, thereby, generate gradients of morphogens during embryonic development (Okada et al., 2005), clear mucus in airway epithelium (Marshall and Nonaka, 2006), and control the phototactic or mechanosensitive steering response of Chlamydomonas (Witman, 1993), Volvox (Drescher et al., 2010), Paramecium (Naito and Kaneko, 1972), and zooplankton (Jékely et al., 2008). Because the motile cytoskeletal element of cilia, the axoneme, is highly conserved, it will be interesting to know whether cilia in general use a similar signal transfer function to modulate the beat pattern.

### Optical setup

The swimming behavior of sperm was studied in a shallow observation chamber (depth of 100 µm). Sperm swim in a plane close to the glass–water interface. The restriction to 2D motion facilitated both recording and analysis of swimming paths. Motility was recorded with an inverted microscope (IX71; Olympus) equipped with a 20× magnification objective (0.75 NA; UPLSAPO; Olympus). Photolysis of caged compounds was achieved using a 100-W mercury lamp (U-RFL-T; Olympus). The irradiation time (30 ms) was controlled by a mechanical shutter (Uniblitz VS25; Vincent Associates), and the intensity of the UV flash was adjusted using a set of neutral-density filters (Qioptiq Photonics). The 488-nm line from an argon/krypton laser (Innova 70C; Coherent, Inc.) was used for excitation of Fluo-4. To produce sharp images of swimming sperm, laser-stroboscopic illumination (2-ms pulse) was generated using an acoustooptical tunable filter (AA Opto-Electronic Company). The fluorescence was filtered by a 500-nm long-pass filter (Omega Optical, Inc.). Images were collected at 30 frames per second using a back-illuminated electron-multiplying charge-coupled device camera (DU-897D; Andor Technology). To reconstruct a cell’s trajectory, a motorized stage (SCAN IM; Märzhäuser Wetzlar GmbH & Co.) was used, and the position of the stage at each frame was recorded. The setup was synchronized using an acquisition program written in LabVIEW and data acquisition hardware PCI-6040E (National Instruments).

### Quantitative motility analysis

We used custom-made postacquisition programs written in MATLAB (MathWorks) to track swimming sperm and to measure time-resolved changes in fluorescence. The relative changes in fluorescence Fr were computed using the formula Fr(t) = 100−[F(t) − F(0)]/F(0), where F(t) is the total fluorescence of the cell at time t. Our analysis has to account for the complex fine structure of sperm swimming paths: for a swimming sperm cell, the sperm head wiggles around an averaged path with the same frequency as the flagellar beat as a result of balancing forces generated by the flagellar beat (Gray, 1955; Friedrich et al., 2010).

We used a second-order Savitzky–Golay filter to extract the averaged path from tracking of the sperm head. Sperm swim preferentially in an anticlockwise direction on the upper surface of the recording chamber (Gray, 1955). This preferred direction is reversed on the opposite surface. To compare cells from both glass/water interfaces, we flipped planar swimming paths to assure anticlockwise curvature before stimulation. Signed curvature was then calculated from the averaged path using the following formula:

$κ=(x.y..-x..y.)/v3.$

To reconstruct the swimming path from the speed and the Ca2+ signals, we numerically calculated and rescaled the time derivative of the fluorescence signal using the parameters κ1 and β resulting from a least mean squares fit of Eq. 1 to the data. Finally, the path was reconstructed from the curvature κ and the displacement (s = v × Δt) by numerical integration of the 2D Frenet–Serret equations.

The efficiency of sperm chemotaxis was determined by an accumulation assay: a sperm suspension (5 × 105 cells/ml) was imaged under the microscope using dark-field microscopy while performing chemotaxis in a Gaussian gradient. The gradient was established by photolysis of 1 µM caged resact using five consecutive UV flashes to maintain the gradient for a longer period of time (flash duration of 250 ms and time between flashes of 1 s). The flashes displayed a Gaussian intensity profile with a characteristic width of σ = 115 µm. The distribution of cells around the center of the flash was quantified using relative changes of the weighted standard distance as

$D(t)=∑i,jIij(t)×[(ri−rx)2+(rj−ry)2]∑i,jIij(t),$

where (rx, ry) are the coordinates of the center of the flash, (ri, rj) are the coordinates of the pixels in each frame, and Iij represents the intensity of the pixel (ri, rj). Assuming that the brightness of each pixel is proportional to the density of cells, the quantity D represents the dispersion of cells. D depends on the size of the field of view. To account for this, the sum over the indices i and j is taken for pixels located at a distance ≤3σ of the center of the UV flash. Finally, the relative changes of D were calculated using the formula Dr = 100 ΔD/D0.

### Double-flash experiments

To calculate the mean swimming path after the second flash, individual trajectories were aligned at a time point shortly before the second flash (100 ms). Such an alignment consisted of a translation to the origin of coordinates and a rotation that made coincident the tangent of each trajectory at this time point with an angle of 45° with respect to the abscissa.

### Chemical differentiator

Based on the second-order chemical equations proposed in our model, the kinetics of each binding reaction was resolved numerically by direct integration of the mass balance equations. Changes in the free concentration of B1,2 and CaB1,2 were calculated for every iteration using the expressions

$[Bi]t+Δt=[Bi]t+Δt(koff[CaBi]t−kon[Ca2+]t[Bi]t) and$
$[CaBi]t+Δt=[CaBi]t+Δt(kon[Ca2+]t[Bi]t−koff[CaBi]t),$

where [Ca2+] is the input of the model, the integration time step Δt was 1 ns, and the initial conditions were [B1]0 = [B2]0 = [LC4]total and [CaB1]0 = [CaB2]0 = 0. Finally, the parameter C was calculated at each time point from its definition given in Eq. 2.

### Online supplemental material

Fig. S1 presents data of Fluo-4 fluorescence versus free [Ca2+] for sperm treated with the Ca2+ ionophore A23187. The fluorescent signal scales linearly with Ca2+ within the regimen of [Ca2+]i observed in sperm during chemotaxis. Fig. S2 shows evidence for a common mechanism of flagellar control by d[Ca2+]/dt in other sea urchins. Fig. S3 presents evidence for a common mechanism in ascidian sperm. Fig. S4 shows the contributions to Fr of the different parts of the cell (head and flagellum) and the correlation between Fr or dFr/dt versus the path curvature for each part. Fig. S5 displays a comparison of Fr or dFr/dt and path curvature for a particular sperm cell. Video 1 illustrates and recapitulates our findings; a sperm cell, loaded with the calcium indicator Fluo-4 AM and caged cGMP, is shown before and after the release of cGMP with a UV flash.

We thank H. Krause for preparing the manuscript, B. Faßbender, P. Perez-Prat, and A. Takemi for assistance with the analysis, and Drs. T. Strünker, R. Seifert, and D. Wachten for critical reading of the manuscript. We thank Drs. M. Yoshida and K. Shiba for kindly providing us with their data on ascidian sperm. The caged compounds were kindly provided by Dr. V. Hagen.

L. Alvarez and U. Benjamin Kaupp conceived the project and designed experiments. L. Alvarez, L. Dai, B.M. Friedrich, N.D. Kashikar, I. Gregor, and R. Pascal performed experiments and analyzed the data. L. Alvarez, L. Dai, and B.M. Friedrich developed the tools for analysis. L. Alvarez, L. Dai, and I. Gregor established the experimental setup. L. Alvarez, B.M. Friedrich, and U. Benjamin Kaupp developed the theoretical model. L. Alvarez, U. Benjamin Kaupp, and B.M. Friedrich wrote the manuscript. All authors revised and edited the manuscript.

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Abbreviations used in this paper:

• AM

acetoxymethyl ester

•
• ASW

artificial sea water

•
• cGMP

cyclic guanosine monophosphate

## Author notes

N.D. Kashikar’s present address is Neurobiology Division, Medical Research Council Laboratory of Molecular Biology, Cambridge CB2 0QH, England, UK.

B.M. Friedrich’s present address is Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany.