Sperm require a sense of direction to locate the egg for fertilization. They follow gradients of chemical and physical cues provided by the egg or the oviduct. However, the principles underlying three-dimensional (3D) navigation in chemical landscapes are unknown. Here using holographic microscopy and optochemical techniques, we track sea urchin sperm navigating in 3D chemoattractant gradients. Sperm sense gradients on two timescales, which produces two different steering responses. A periodic component, resulting from the helical swimming, gradually aligns the helix towards the gradient. When incremental path corrections fail and sperm get off course, a sharp turning manoeuvre puts sperm back on track. Turning results from an ‘off’ Ca2+ response signifying a chemoattractant stimulation decrease and, thereby, a drop in cyclic GMP concentration and membrane voltage. These findings highlight the computational sophistication by which sperm sample gradients for deterministic klinotaxis. We provide a conceptual and technical framework for studying microswimmers in 3D chemical landscapes.
Many motile sperm rely on chemical and physical cues to locate the egg1,2,3,4. A beating hair-like filament, called the flagellum, serves both as an antenna that gathers sensory cues and as a motor that propels the cell. Receptors on the flagellar surface transduce these sensory cues into cellular signals. Ultimately, these signals modulate the wave-like beating of the flagellum that steers a sperm’s swimming path (for review see ref. 5).
Sperm from many species, in particular marine animals, are attracted to the egg by chemical factors—a process called chemotaxis. Previous chemotaxis studies tracked sperm only in two-dimensions (2D); at the glass/water interface of shallow observation chambers, sperm swim on a plane in circles6,7,8,9,10. While cruising on circular paths in a chemical gradient, sperm sample the chemoattractant concentration either continuously or intermittently and gradually adjust their swimming path—a mechanism called klinotaxis. The repetitive stimulation entrains a suite of Ca2+ bursts that modulate the waveform of the flagellar beat5,9,11,12. The ensuing alternate periods of symmetrical and asymmetrical beating give rise to a looping swimming pattern (‘drifting circles’) up a gradient. However, unrestricted sperm from species such as sea urchin swim on a 3D helical path13,14,15,16,17. The conceptual work by Crenshaw17,18 suggests that in a chemical gradient, sperm orient by helical klinotaxis, that is, by alignment of the helix towards the gradient. In addition, Crenshaw17,18 shows that such alignment could be theoretically achieved if the components of the cell’s rotational velocity are simple functions of the stimulus.
A generic quantitative theory of this chemotactic steering by Friedrich and Jülicher19 captures the essence of navigation along periodic paths: a cellular signalling system transforms the periodic stimulation s(t) (chemoattractant binding) into a periodic intracellular signal i(t) (Ca2+ bursts) that in turn modulates the swimming path curvature κp(t). The phase lag between s(t) and κp(t), that is, the latency of the Ca2+ signal, is a crucial determinant of the directed drift of circles up or down a 2D chemical gradient. This theory also predicts bending of the swimming helix in 3D. However, these theories have not been experimentally tested in a well-defined 3D chemical gradient; moreover, the mechanisms of 3D klinotaxis cannot be derived from 2D swimming at interfaces, not least because of the hydrodynamic interactions between the flagellum and the surface of the recording chamber20,21. In fact, 3D navigation mechanisms remain unknown for any swimming eukaryotic cell, because of the technical challenges to rapidly establish and quantitatively characterize a chemical gradient and to track rapidly moving cells as they traverse a complex chemical 3D landscape.
We chose the sea urchin Arbacia punctulata to study the search strategy of sperm in a 3D chemoattractant landscape. A. punctulata are broadcast spawners that release their gametes into the ocean, where sperm swim freely. For several reasons, A. punctulata sperm provide an unmatched model to address fundamental questions of cell navigation22: (1) chemotaxis has been well-established22,23; (2) the chemoattractant is known24, and the signalling pathway has been studied in depth25,26,27,28,29; (3) conditions for swimming in an aqueous medium can be readily emulated; finally, (4) Arbacia sperm, unlike mammalian sperm, represent a homogenous population, that is, most sperm are chemotactically active.
Here we study freely swimming sperm using digital inline high-speed holographic microscopy30. Moreover, 3D landscapes with defined spatio-temporal pattern are created by light using a caged chemoattractant31. In addition, we study by reverse optochemical engineering of signalling events, how steering responses are adjusted during navigation in a gradient. Finally, using hydrodynamic simulations and a minimal model of sperm signalling, we provide a theoretical framework accounting for sperm navigation in 3D.
Tracking sperm in 3D
Sperm were illuminated with coherent laser light. Light scattered by sperm interfered with the unscattered background, and this interference was recorded by a camera at 600 holograms per s (h.p.s.). Applying the Rayleigh–Sommerfeld back-propagator32, we determined numerically from each hologram the 3D coordinates of the sperm head33 (Supplementary Fig. 1). Far from boundaries, A. punctulata sperm swam on a regular helical path with a mean speed v=200±57 μm s−1 (Fig. 1a); mean helix parameters were: radius r0=8.4±3.1 μm, pitch p0=47.6±9.1 μm, helix period T=0.38±0.07 s, path curvature κp=0.065±0.013 μm−1 and path torsion τp=0.067±0.031 μm−1 (track duration 1 s; n=20 cells, mean±s.d.). For comparison, near a glass/water interface, sperm swim on circles of radius r0=23.5±0.9 μm with swimming speed v=160±29 μm s−1, and circle period T=0.9±0.2 s (n=6). Thus, 2D and 3D kinematic parameters are distinctively different, underscoring the notion that 2D and 3D navigation are fundamentally different20.
The sperm head wiggles around the average path with a frequency identical to that of the flagellar beat34; passive head wiggling counterbalances periodic forces generated by active flagellar bending35. For freely swimming sperm, head wiggling indicated a flagellar beat frequency of 43.5±3.5 Hz (Fig. 1b; n=20). Moreover, head wiggling was approximately planar, consistent with an approximately planar beat pattern. While sperm swim along a helix, the head-wiggling plane slowly rotates around the helix axis: the normal vector describes a circle on the unit sphere (Fig. 1c), characterized by a constant inclination to the helix axis (52.0°±11.5°; circular mean±circular s.d., n=20).
To gain insight into the mechanisms producing a helical path, we simulated swimming paths resulting from different flagellar waveforms using resistive-force theory35,36. The flagellar shape was characterized by a constant flagellar twist τf and a flagellar curvature κf(l,t)=K0+B cos(ω0t−λl) describing a travelling bending wave with beat amplitude B, wave speed ω0/λ and mean flagellar curvature K0 along the arc length l of the flagellum. We calculated the swimming path produced by three different flagellar waveforms: asymmetric/non-twisted (K0>0; τf =0), symmetric/twisted (K0=0; τf >0) and asymmetric/twisted (K0>0; τf >0). Without twist, the beat pattern and swimming path were planar (Fig. 2a), that is, sperm moved on circles of radius rp≈1.3/K0 that is set by the flagellar beat asymmetry. The symmetric/twisted beat produced a non-planar beat pattern, and sperm swam on twisted ribbons (Fig. 2b). A fraction of human sperm adopts this pattern37. Finally, an asymmetric/twisted beat produced a helical path, which is observed for some sperm species13,14, and head wiggling occurred in a plane that slowly rotated around the helix axis (Fig. 2c). Both features are borne out by our experiments (Fig. 1b). This beat pattern could result from a combination of periodic motor activity travelling down the flagellum, which drives planar bending waves and persistent motor activity, which twists the flagellum38.
3D steering is deterministic
Using a photosensitive caged form of the chemoattractant resact, we sculpted by light well-defined 3D profiles of chemoattractant. The intensity profile of the uncaging ultraviolet light was approximated by a Gaussian beam (Supplementary Fig. 2, Supplementary Movie 1). To mimic an egg that continuously releases resact, ultraviolet illumination was maintained during the measurement. The time-dependent concentration profile c(x,t) was reconstructed numerically from the light profile using the resact diffusion coefficient39 and the extinction coefficient and quantum yield of caged resact (Fig. 3a, Supplementary Movie 2). From the swimming path r(t) and the concentration profile c(x,t), we derived the chemoattractant stimulus encountered by sperm s(t)=c(r(t),t) (Fig. 3b, Supplementary Fig. 3, Supplementary Movies 3,4).
Sperm might employ either a deterministic or stochastic strategy of chemotactic steering. For deterministic steering, correcting path adjustments are directed towards the gradient. Alternatively, during stochastic steering, path adjustments are chosen at random; afterwards, the microswimmer decides whether this choice was favourable and acts accordingly. Bacteria follow a stochastic search strategy; they alternate between episodes of straight swimming (‘runs’) and stochastic tumbling, which randomizes cell orientation and, thereby, produces a random walk40. However, ‘runs’ up a gradient are longer, producing a biased drift towards the chemoattractant source. To distinguish between deterministic and stochastic search strategies, we analysed the distribution of helical adjustments relative to the local concentration gradient for sperm crossing a chemoattractant field. The bending of the helix axis (dh/dt) is biased towards the perpendicular component ∇⊥c of the local gradient, that is, the optimal direction for local alignment with the gradient (Fig. 3c). Thus, steering responses of sperm are deterministic rather than stochastic.
The sensori-motor transfer function
We identified two different steering modes while sperm navigate in a chemoattractant gradient. Steady bending of the helix aligned its axis with the gradient direction, thereby ramping up the mean stimulation level (‘on response’); occasionally bending was interrupted by a vigorous steering event that abruptly changed the swimming direction (‘off response’) (Fig. 3b). Using a Fourier filter, we decomposed the stimulus function s(t) into high and low frequency components. One component represents stimulus oscillations of about 2 Hz, resulting from the helical nature of the path; this 2 Hz component was superimposed on the steady increase or decrease of the mean stimulus level (Fig. 4a,d). Remarkably, oscillations in curvature and torsion of the swimming helix are highly correlated with the stimulus oscillations, suggesting that subtle changes in helix geometry are underlying its smooth alignment with the gradient, that is, the ‘on response’ (Fig. 4d,e).
We used a simple model of phase-locked oscillations to fit the cross-correlation of stimulus and swimming response: Acos(Ω0Δt+φ)e−DΔt, where Ω0 is the helix frequency, φ is the phase shift between path curvature or torsion and the stimulus, and D accounts for phase diffusion (Fig. 4f). We find a phase shift between stimulus oscillations and oscillations of curvature and torsion of φκ=205.7° (184.8°, 233.2°) and φτ=−16.3° (−72.2°, 43.6°), respectively, where the numbers in parentheses indicate 95% confidence intervals by bootstrap (n=10 cells). For optimal alignment, theory predicts19 a φκ=180° and φτ=0°—in fair agreement with our experimental results (see Supplementary Note 1).
Strong steering responses were initiated when the slow stimulus component began to decline, that is, when sperm were about to lose track of the gradient, hence ‘off response’ (Fig. 4b,c). Nine out of 10 cells displayed such ‘off responses’, and during a total recording time of 77 s ‘off responses’ were observed on 31 occasions. Strong ‘off responses’ provide sperm with the means to evoke an emergency steering manoeuvre when steady helix alignment (‘on response’) is not sufficient.
In conclusion, sperm survey a chemical landscape on two different timescales simultaneously. While swimming up the gradient (mean stimulation level increases), the rapid periodic stimulus component smoothly and continuously bends the swimming helix towards the gradient. When swimming down the gradient (mean stimulation level decreases), sperm respond with a vigorous, yet almost optimally directed turn. While the smooth helix bending during the ‘on response’ involves only subtle adjustments of helix parameters, the strong ‘off response’ affords major transient distortions of helix geometry. After getting back on track, sperm resume regular helical swimming.
The behavioural ‘off response’ is generated by Ca2+
Sperm host a sophisticated pathway for Ca2+ signalling that controls the flagellar beat waveform (Fig. 5a). Briefly, chemoattractant stimulation produces a pulse of cyclic GMP (cGMP), the intracellular messenger31. The rise is caused by cGMP synthesis via the chemoreceptor, a guanylyl cyclase27 and the decay by cGMP hydrolysis via phosphodiesterase (PDE) activity. cGMP opens cyclic nucleotide-gated K+-selective channels28 (CNGK) and, thereby hyperpolarizes sperm25. During the decline from hyperpolarization, voltage-dependent Ca2+ channels (CatSper) open26. Ultimately, Ca2+ entering the sperm flagellum initiates a change in the flagellar waveform and a steering response7,9,11,31.
The behavioural ‘off response’ when swimming down the gradient indicates that not only an increase, but also a decrease of chemoattractant evokes a Ca2+ signal. To test this idea, we developed a technique for optochemical control of a cellular response. We imposed a temporal pattern of photo-stimulation on sperm loaded with membrane-permeable caged cGMP31,39. The cGMP dynamics inside sperm is controlled via the balance between extrinsic ‘synthesis’ by photolysis of caged cGMP and intrinsic hydrolysis by PDE. Thereby, we mimicked an increase or decrease of resact. We produced cGMP with 393-nm light and simultaneously followed the changes in Ca2+ concentration and membrane potential Vm by a Ca2+ indicator dye (Fluo-4) and a voltage probe (di-8-ANEPPS), respectively. As a control, we mimicked a brief puff of chemoattractant by a light pulse that is shorter than the latency of the Ca2+ response (<200 ms). The control stimulus, after a short latency, evoked a transient Ca2+ signal (Fig. 5b)31. Using a longer light pulse, the waveform of the Ca2+ signal changed in two respects: the amplitude of the initial Ca2+ signal became much smaller and, unexpectedly, a second Ca2+ signal was generated precisely when the photolyzing light was switched off. In keeping with our nomenclature, we refer to the first and second Ca2+ signal as ‘on’ and ‘off’ Ca2+ response, respectively.
To elucidate the cellular mechanism underlying the Ca2+ ‘off response’, we probed the cGMP-induced voltage response using the same light protocol. Light stimulation produced a hyperpolarization25 (Fig. 5b). For both brief and long light pulses, the waveform of the rising phase of the hyperpolarization was identical. However, for long light pulses (that is, with prolonged cGMP ‘synthesis’), the decline from the hyperpolarizing peak was slower and incomplete; after the light was switched off (that is, when cGMP production ceased), the recovery from hyperpolarization was rapidly completed. The rapid Vm drop coincided with the onset of the Ca2+ ‘off response’. The following model readily accounts for these observations22,26: Voltage-dependent CatSper channels open during recovery from hyperpolarization. While the photolyzing light is on, cGMP does not return to baseline levels as quickly and completely as compared with a brief light pulse. Consequently, a fraction of CNGK channels are kept open, and sperm stay partially hyperpolarized. When cGMP production by light ceases, cGMP levels rapidly drop due to PDE activity. The rapid return to resting Vm opens additional CatSper channels and, thereby, produces the Ca2+ ‘off response’. The behavioural ‘off response’ triggered by this Ca2+ signal results in a major correction of the swimming path. Along the new direction of swimming, sperm experience again an ascending chemoattractant concentration in time.
Model of chemotactic steering
Previous theoretical models did not address how the flagellar beat steers a cell in a gradient15,19. Moreover, they predicted steady adjustment of the swimming path by a constant feedback mechanism. We developed further the theory of chemotactic steering in two respects: a dynamic regulation of flagellar waveform and an adaptive steering feedback. For steering, sperm must alter the waveform of their flagellum. Therefore, we first computed helix parameters for constant waveforms (characterized by mean flagellar curvature K0 and constant flagellar twist); this allowed the identification of a unique parameter set that reproduces the helix geometry in the absence of stimulation (Fig. 6a,b). Furthermore, 2D experiments suggest that a chemotactic signalling pathway regulates the mean flagellar curvature9,12. Therefore, we simulated the 3D swimming path resulting from variations of K0. A small-amplitude, periodic modulation of K0 causes steady helix bending (Fig. 6c), whereas a single large-amplitude modulation elicits a sharp turn (Fig. 6d). Helix bending and sharp turn are reminiscent of the ‘on’ and ‘off response’, respectively. Second, in an extended model of chemotactic steering, K0 is dynamically adjusted to the stimulus concentration sampled by the swimming cell (see Supplementary Note 1). For up the gradient swimming, when only small course corrections are needed to align the helix, the signalling feedback strength is small; consequently, the amplitude of K0 oscillations is small too, resulting in steady alignment (‘on responses’). If, however, sperm swim down a gradient, an upregulated feedback strength allows for sharp turns (‘off responses’). Simulations using this generic model account for the chemotactic ‘on’ and ‘off’ steering responses and show robust chemotaxis towards the summit of a resact concentration profile (Fig. 6e,f).
We identify the principal features of sperm navigation in a 3D chemical landscape. Sperm probe the chemoattractant concentration along helical paths and, thereby, a spatial gradient is translated into a temporal stimulus pattern that oscillates with the 2 Hz periodicity of helical movement. Thus, the swimming path organizes the temporal stimulus pattern perceived by sperm, a principle known as information self-structuring41. The periodic component is superimposed on a mean stimulus level (baseline) that either increases or decreases slowly when swimming up or down a continuous gradient, respectively. The rapid stimulus oscillations provide a sense of direction, whereas the baseline slope controls the response strength. A positive slope signifying a chemoattractant increase produces weak ‘on responses’; a negative slope when losing track evokes strong ‘off responses’. This regulation of response strength allows sperm to tune klinotaxis behaviour ranging from subtle helix bending to abrupt emergency turns.
Response regulation can be achieved by dynamic adjustment of feedback strength, which is superior to a previous model with only constant feedback19. What might be the cellular signalling mechanisms underlying dynamic feedback strength? The cGMP dynamics, resulting from cGMP degradation by PDE activity and inactivation of the receptor guanylyl cyclase27, controls the recovery from hyperpolarization and, thereby, the amplitude and timing of Ca2+ responses. Based on our macroscopic Ca2+ measurements, we propose a model in which slowly increasing stimulation during helix alignment keeps cGMP levels elevated and the recovery from hyperpolarization is, therefore, slower and incomplete; consequently, Ca2+ responses turn out small. When stimulation is waning while drifting down the gradient, cGMP hydrolysis outcompetes cGMP synthesis. As a result, CNGK channels close swiftly and Vm rapidly drops to resting values, thus triggering a large ‘off’ Ca2+ signal that drives an ‘off response’. We propose that the regulated feedback strength resides in the dynamic balance between cGMP synthesis and hydrolysis. Future work needs to combine holographic microscopy with fluorescence imaging techniques to monitor the signalling events in single cells during 3D chemotaxis.
While sperm from many marine invertebrates reach the egg by free 3D swimming, sperm from some fish swim on the 2D surface of the large egg during their search for a small fertilization site, known as the micropyle42. Moreover, mammalian sperm swim along the narrow oviduct and intermittently interact with the convoluted epithelium that lines the oviduct43, thus mammalian sperm probably switch between 2D and 3D navigation. How similar are 2D and 3D navigation mechanisms? For sea urchin sperm, the drifting-circle pattern observed during 2D chemotaxis on a surface is equivalent to the helix-bending mode of 3D klinotaxis. Because the ‘off response’ is inherently rooted in the design of cellular signalling, we expect that the control logic is similar for 2D and 3D scenarios. Thus, why have ‘off responses’ escaped detection in studies of 2D navigation? For 2D swimming at interfaces, movements are constrained and subject to hydrodynamical interactions with the surface. Consequently, the ballistic component of movement along the helical path is missing because movement across the surface is not possible. The degree of freedom for 3D swimming is manifold larger and thus is the likelihood that sperm eventually will swim away from the chemical source. Therefore, the ‘off response’ is more important and prominent for 3D than for 2D swimming. Future work needs to address the behavioural ‘off response’ under 2D conditions, for example, by applying the approach of reversed optochemical engineering at the single cell level.
How common are navigation strategies of sperm across phyla? For mammalian sperm, three different mechanisms of guidance have been proposed: chemotaxis44,45, thermotaxis46,47 and rheotaxis4,48. Human sperm, when rapidly stimulated with progesterone, a putative chemoattractant, released from caged progesterone49 undergo hyperactive episodes that change the swimming direction45. Similar events are observed in a thermotaxis assay after a temperature shift to lower values47. Superficially, these hyperactive episodes observed in human sperm are reminiscent of the ‘off responses’. However, unlike ‘off responses’, they initiate a stochastic rather than deterministic reorientation and are more akin to stochastic tumbling episodes of bacterial chemotaxis.
Klinotaxis is a basic mechanism of navigation displayed by many motile cells and organisms50, including the nematode Caenorhabditis elegans, larva from Drosophila, Platynereis and zebrafish primordial germ cells, protists and even some bacteria51,52,53,54,55. Sensory modalities as diverse as phototaxis, thermosensation, taste, olfaction and electroreception employ klinotaxis. In many animals, neuronal circuits are dedicated to analyse the temporal pattern of sensory stimuli that instruct the klinotactic response. Moreover, neuronal circuits are composed of ‘on’ and ‘off cells’ that register the ups and downs of sensory stimuli. Temporal sampling during klinotaxis is either accomplished along a periodic path or by undulatory movements of the body. It is quite remarkable that a signalling pathway can encompass these computational features in a single cell.
On a final note, chemoattractant landscapes around an egg in a natural habitat are not known and are expected to be rather complex. Local chemoattractant concentrations might be rapidly changing and distorted by turbulent fluid flow and drifting of eggs, giving rise to ‘plumes’ rather than Gaussian-shaped continuous gradients56. To emulate these native conditions and to study how sperm, and any other microswimmers, cruise in such complex chemical landscapes is the next frontier of enquiry. The optochemical and microscopy techniques presented here combined with computational models provide the means to generate and quantify complex landscapes and to reveal the search strategy during microswimmer navigation in general.
Sperm spawning was evoked by injecting 0.5 ml of 0.5 M KCl into the body cavity of sea urchins from the species A. punctulata. Sperm were collected using a Pasteur pipette and stored on ice. Artificial seawater (ASW) contained (in mM) 423 NaCl, 9.27 CaCl2, 9 KCl, 22.94 MgCl2, 25.5 MgSO4, 0.1 EDTA and 10 HEPES. The pH was adjusted to pH 7.8 with NaOH.
Dry sperm were diluted 6.6 × 104 fold with ASW supplemented with 0.5% Pluronic F127 (Sigma-Aldrich) and studied in a custom-made observation chamber (11.5 × 11.5 × 1 mm3). The chamber featured two openings for sperm injection. To minimize convective flow, the laboratory and the microscope incubator were set to 18 °C. The temperature of 18 °C was chosen because it roughly corresponds to the average temperature of the northern Atlantic coast during the fertility season of A. punctulata57.
Freely swimming sperm were tracked using an inverted microscope (IX71; Objective × 20, 0.75 numerical aperture, UPLSAPO; Olympus). Coherent illumination was achieved with a laser light source (LDH-D-C-510, PicoQuant GmbH) and the corresponding controller (Sepia II Multichannel Processor, PicoQuant GmbH). Laser light was coupled into a multimode fibre. A custom-made adapter was used to position the fibre parallel to the optical axis of the objective. The illumination intensity was adjusted to use the dynamical range of the camera (12-bit; PCO Dimax HD). Movies were collected at 600 frames per second; each frame represents a holographic image containing the complete 3D information of the sperm cell.
Caged compounds were photolyzed using a 365-nm LED (M365L2-C; Thorlabs). The ultraviolet light was coupled into a liquid guide (77566; Newport) followed by two Plano-convex lenses (LA 1951-A f=25.4 mm; LA 1509-A f=100 mm; Thorlabs) and coupled to the imaging optical path with a dichroic filter (ff 495-Di03; Semrock). The irradiation power (0.8 mW) was measured with a power meter (detector PowerMax and head model PS19Q; Coherent). The light spectrum was recorded with a spectrometer (51024 DW; Ocean Optics). Photolysis and data acquisition were synchronized using a wave generator (33500B; Agilent).
Reconstruction of the 3D swimming path
The 3D swimming path was reconstructed in several steps. First, we created a ‘background’ image from the average intensity value of the hologram sequence (usually 24,000 frames). Because moving objects in the hologram sequence are averaged out, the background image contains only the non-scattered light and the fringes resulting from interference of the source with non-moving particles. Dividing each hologram by the background image results in distinct interference patterns of moving sperm32. To curb the data volume, single sperm cells were tracked within the plane of the image using a custom-made tracking programme written in Java (Java 1.6 24; ImageJ v.1.47 m). The determined position was used to define a moving region-of-interest (ROI) of typically 300 × 300 pixels around the sperm cell for each frame; this ROI was used for further analysis.
The Rayleigh–Sommerfeld back-propagation scheme was used to numerically refocus each background-free hologram. Numerical refocusing resulted in a focus stack of computed images. The z-position of the cell corresponded to the point of contrast inversion (Gouy phase anomaly58; Supplementary Fig. 1). Contrast inversion was identified by an image-processing filter that highlights axial gradients in the image stack, such as the Sobel filter33,59,60.
Caged compounds and flash photolysis
N-Fmoc-S-(2-Nitroveratryl)-L-cysteine was obtained as described61 and used for solid-phase peptide synthesis (Biosyntan GmbH, Berlin, Germany) to obtain caged Cys8-S-DMNB-resact31. The extinction coefficient of caged resact (20 μM) in ASW was measured by an absorption spectrophotometer (Varian Carry 5000). To determine the photochemical quantum yield (φchem), a 50 μM solution of the caged peptide in ASW was irradiated with a 365±5 nm ultraviolet-LED providing an power flux of 90 mW cm−2 (NCSU033A(T), Nichia, Japan). At several time points aliquots were taken and analysed via HPLC (n=3). From the initial slope of linear release (0–15% conversion) and the rate of photon absorption, determined by ferrioxalate actinometry62, we determined a photochemical quantum yield of φchem=0.4±0.1%. DEACM-caged cGMP was synthesized as described63.
Characterization of the 3D gradient
The ultraviolet-light profile was characterized by measuring optical sections along the z-axis. A stack of seven glass cover-slides (150-μm thickness each) was placed at the location of the observation chamber. The ultraviolet profile was measured between slides by adding fluorescein (10 μl; 1 mM) and imaging the fluorescence emission (emission filter ET 510 LP; Chroma) that resulted from excitation of fluorescein using the LED ultraviolet source. Because the geometry of the spatial light profile is fixed with respect to the objective lens, focusing on the individual planes was achieved by moving the camera to the corresponding conjugate plane. The resulting image was rescaled to compensate for the change in magnification. The scaling factor was measured by imaging a grid (DV2 calibration grid; Photometrics) at each camera position. Individual fluorescence pictures, corresponding to the ultraviolet projection on the different planes, were fitted to a Gaussian function (R2>0.977). The change in width along the different optical sections was well-described by a Gaussian beam64 (Supplementary Fig. 2).
To calculate the resact concentration in time and space, we numerically solved Fick’s second law in 3D using the Euler method for both resact and caged resact. Space was discretized on cubes of 5-μm side, and the integration time step chosen was 1.67 ms. The diffusion of resact39 (Dresact=239 μm2 s−1) and caged resact were assumed to be identical. The release of resact from its caged analogue (100 nM caged resact) was calculated on each iteration based on the power and spectrum of the ultraviolet source, the extinction coefficient (ɛ=4,100 M−1 cm−1) and the photochemical quantum yield of DMNB-caged resact. Programmes were written in MATLAB (Mathworks).
Quantification of the helical swimming path
From the tracked sperm head trajectory, we define an averaged path r(t) by averaging out wiggling of the sperm head at the frequency of the flagellar beat, using a least-square fit that simultaneously minimizes the average disance of r(t) to the head trajectory and the third time derivative of track coordinates. The implicit prior on the third spatial derivative ensures smoothness of the averaged path without introducing a bias on the path curvature. We then fitted perfect helices to short r(t) segments using sliding windows of 0.75 s. This defines a helical path centre line R(t), and the helix vector h=dR/dt/|dR/dt|.
We define s(t)=c(r(t),t) as a proxy for the concentration stimulus sampled by the cell in a concentration field c(x,t) and compute the local gradient direction ∇c(R,t). This gradient can be decomposed into components parallel (∇||c=(∇c·h)h) and perpendicular (∇⊥c=∇c −∇||c) to the helix vector (Fig. 3c). If the helix were perfectly aligned with the gradient direction, ∇⊥c would be zero.
We characterize helix bending (dh/dt) with respect to the local gradient direction ∇c as
where g1=∇⊥c/|∇⊥c|, g2=h × g1 and h form a local coordinate system that moves along the helix centre line. Thus, aligment occurs when the helix bends towards g1 (γ1>0). Any bending in the direction g2 does not contribute to chemotaxis. For the histogram showing the direction of helix bending relative to the gradient direction (Fig. 3c), we analysed n=10 cells along their full tracks every 0.75 s, corresponding to 152 data points.
To compute path curvature κp(t) and torsion τp(t) of r(t), we fitted perfect helices to sliding path segments of duration 0.25 s. To stabilize the nonlinear helix fitting algorithm, we included a prior to the cost function that penalizes strong deviations from the helix parameters obtained for fitting a helix to a longer path segment of 0.75-s duration. The path curvature and path torsion were obtained from the radius and pitch of the fitted helical segments. We define the oscillatory part of s(t), κp(t) and τp(t) by subtracting a baseline defined by a Fourier filter with cut-off frequency at 1 Hz. To determine the phase-shifts (φκ and φτ) between stimulus and curvature/torsion oscillations, we fitted a model Acos(Ω0Δt+φ)e−DΔt of phase-locked oscillations to their normalized cross-correlation, where Ω0 is the helix frequency and D accounts for phase diffusion.
To characterize the fraction of cells displaying ‘off responses’, we selected those cells for which at least 3,000 frames (equivalent to 5 s) had been successfully tracked using our 3D tracking algorithm. ‘Off responses’ were attributed to alignment events where the angle between the helical axis and the gradient abruptly changed from values exceeding 90°.
We determined the plane of head wiggling by fitting planes to short segments of the head trajectory of 150-ms duration, using singular-value-decomposition on centred coordinate data.
Numerical simulations of swimming sperm
We numerically reconstructed the swimming path that would result from prototypical flagellar waveforms using resistive-force theory35,36. Specifically, we computed the anisotropic hydrodynamic friction forces associated with shape changes of the slender flagellum. This allowed us to self-consistently solve for the instantaneous motion of the cell’s material frame under conditions of force balance. Instantaneous translations and rotations were integrated using an explicit Euler scheme and the formalism of rigid body transformations. The time-dependent shape of the flagellum was characterized by the position vector rf(l,t) of the flagellar centre line as a function of arc length l along the flagellum and time t, as well as a Cosserat frame consisting of three orthonormal vectors e1(l,t), e2(l,t) and e3(l,t), such that e3 is tangential to the flagellar centre line rf, and e1 and e2 span a normal cross-section of the flagellum. Bending and twisting of the flagellum are characterized by rotations of the Cosserat frame:
where κf(l,t) denotes flagellar curvature and τf denotes flagellar twist. We assume a constant flagellar twist τf and a flagellar curvature κf given by a travelling bending wave,
where K0 is mean curvature, B the amplitude, ω0 the angular frequency and 2π/λ the wavelength of the flagellar bending waves.
We assumed parallel and normal flagellar friction coefficients ξ||=0.99η, and ξ⊥=1.81ξ||, respectively, where η denotes the dynamic viscosity of the surrounding liquid35. The flagellar length (41 μm), flagellar wavelength (2π/λ=29.6 μm) and amplitude of the flagellar curvature wave (B=0.160 μm−1) were estimated from flagellar tracking of sperm swimming close to a water/glass interface. The beat period (2π/ω0=23 ms) was estimated from the frequency of head wiggling for sperm swimming in 3D. Mean flagellar curvature (K0=0.0351 μm−1) and flagellar twist (τf=0.00477 μm−1) were obtained by a nonlinear fit to reproduce the experimentally measured helix parameters, see Fig. 5a,b. The sperm head was approximated by an ellipsoid with axes 2.5, 1.25 and 1.25 μm, as estimated from electron micrographs. Finally, the proximal end of the flagellum (neck) is assumed collinear with the long axis of the sperm head.
Numerical simulation of sperm chemotaxis
Using our hydrodynamic simulation scheme, we computed sperm swimming paths r(t) in a concentration landscape c(x,t) of chemoattractant. We assume a dynamic regulation of the shape of the flagellar beat in response to the concentration stimulus s(t) sampled by the cell along its path:
This minimal description embodies key characteristics of the sperm chemotactic signalling system as observed in experiments, namely an adaptation of sensitivity according to the baseline stimulus level39, as well as relaxation to the rest state (a=1) for any stimulus that does not change in time9. Further characteristics of the signalling module (equation (5)) are provided in the Supplementary Information text. The output variable a(t) regulates the mean flagellar curvature K0 of the flagellar beat according to:
Here Kb denotes the mean flagellar curvature of unstimulated sperm and χ denotes a feedback strength. For simplicity, all other parameters of the flagellar beat are assumed constant. A dynamic regulation of mean flagellar curvature on chemotactic stimulation has been demonstrated in previous 2D experiments12.
To conceptualize our novel experimental finding of ‘off responses’, we employ a dynamic feedback strength χ(t) that can alternate between a low (χon) and a high value (χoff) for steady swimming up or down the gradient, respectively. We introduce a signalling variable q(t), which tracks changes of the stimulus baseline, and that obeys the following low-pass filter dynamics:
where η denotes a relaxation time-scale. In our minimal description, the dynamic feedback strength χ(t) takes either of the two values, depending on whether q(t) exceeds a threshold θ:
For the simulations, we employed the concentration landscape c(x,t) corresponding to the experiment shown in Fig. 3b. Parameters: μ=150 ms, η=500 ms, θ=0.95, χon=Kb, χoff/χon=8 and Kb=0.0351 μm−1.
Measurement of changes in [Ca2+]i and membrane voltage Vm
We measured changes in [Ca2+]i and Vm using a rapid-mixing device (SFM-400; Bio-logic) in the stopped-flow mode. Changes in [Ca2+]i and Vm were measured with the Ca2+ indicator Fluo-4 AM and the voltage-sensitive indicator di-8-ANEPPS (Molecular Probes), respectively. Dry sperm were suspended 1:6 (vol/vol) in loading buffer containing ASW and the indicator, in the absence (di-8-ANEPPS) or presence (Fluo-4 AM) of 0.5% Pluronic F127 (Molecular Probes). After incubation for at least 45 min with Fluo-4 AM or 5 min for di-8-ANEPPS at 17 °C, the sample was diluted 1:20 with ASW. Sperm were allowed to equilibrate in the new medium for 5 min. In the stopped-flow device, the sperm suspension was rapidly mixed 1:1 (vol/vol) with resact in ASW or ASW alone. Concentrations of resact are given as final concentrations after mixing. Fluorescence was excited by a SpectraX Light Engine (Lumencor). Emission was recorded by photomultiplier modules (H9656-20; Hamamatsu Photonics). The signal was amplified and filtered through a voltage amplifier (DLPVA-100-B-S; Femto Messtechnik). Data acquisition was performed with a data acquisition pad (PCI-6221; National Instruments) and Bio-Kine software v. 4.49 (Bio-Logic). For Ca2+ and Vm recordings, the excitation light was passed through a BrightLine 475/28 nm filter (Semrock) (SpectraX Light Engine). For Ca2+ measurements, the emitted light was passed through a BrightLine 536/40-nm filter (Semrock). Ca2+ signals represent the average of at least two recordings and are depicted as the per cent change in fluorescence (ΔF) with respect to the mean of the first 5–10 data points before the onset of the signal (F0). The control (ASW) ΔF/F0 signal was subtracted from the resact- or cGMP-induced signals. The Vm signals were recorded in dual-emission mode. The filters in front of the two photomultipliers were BrightLine 536/40 nm and BrightLine 628/40 nm (Semrock). The Bio-Logic software was used to record the fluorescence in the ratiometric dual-emission mode. The Vm signals are the ratio of F536/628 or R. The control (ASW) R signal was subtracted from the resact- or cGMP-induced signals. The mean R of the first 5–10 data points before the onset of the changes in fluorescence was set to 0, yielding ΔR. The Vm signals represent the average of at least three recordings and were digitally smoothed with five-point average smoothing. The changes in di-8-ANEPPS fluorescence were calibrated into Vm (mV) by stimulating sperm with 2.5 nM resact concentrations in ASW (9 mM K+) and ASW of 30 mM and 100 mM extracellular K+. Plotting the resact-evoked ΔR against [K+]o allowed the interpolation of Vrest and ΔR/mV as previously described25. Calibration of ΔR to mV was performed within each set of experiments. The data obtained from ensemble measurements were analysed using OriginPro 9.0 (OriginLab Corporation).
All data are given as mean±s.d. unless otherwise stated.
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This work was supported by the German Research Foundation (DFG; SPP microswimmers 1726) and the Rowland Institute at Harvard. We thank Drs Seifert and Wachten for discussions and critically reading the manuscript.
The authors declare no competing financial interests.
Supplementary Figures 1-4, Supplementary Note 1 and Supplementary References (PDF 1277 kb)
Gaussian UV light beam used for photolysis. The vertical grey line indicates the centre of the photolyzing beam along the optical axis. (MOV 1129 kb)
Reconstructed chemoattractant landscape. The chemoattractant concentration (in nM) is color-coded. The vertical grey line indicates the centre of the photolyzing beam along the optical axis. (MOV 951 kb)
Reconstructed swimming path of a sperm cell in a gradient. The vertical grey line indicates the centre of the photolyzing beam along the optical axis. Red arrowheads indicate abrupt chemotactic turns. The swimming path unfolds at half the real speed. (MOV 2841 kb)
3D Anaglyph of the reconstructed swimming path of a sperm cell in a gradient. The vertical grey line indicates the centre of the photolyzing beam along the optical axis. Red arrowheads indicate abrupt chemotactic turns. The swimming path unfolds at half the real speed. The movie should be watched with 3D red cyan glasses. (MOV 11520 kb)
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Jikeli, J., Alvarez, L., Friedrich, B. et al. Sperm navigation along helical paths in 3D chemoattractant landscapes. Nat Commun 6, 7985 (2015). https://doi.org/10.1038/ncomms8985
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