The rate of change in Ca(2+) concentration controls sperm chemotaxis

Abstract

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 Ca(2+) oscillations. How Ca(2+) signals elicit steering responses and shape the path is unknown. We unveil the signal transfer between the changes in intracellular Ca(2+) concentration ([Ca(2+)](i)) and path curvature (κ). We show that κ is modulated by the time derivative d[Ca(2+)](i)/dt rather than the absolute [Ca(2+)](i). Furthermore, simulation of swimming paths using various Ca(2+) 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 [Ca(2+)](i) into motion.

Figure 1.

Time derivative of [Ca²⁺]_i controls curvature of swimming path. (A) Changes in [Ca²⁺]_i and path curvature κ. (B) Color-coded F_r along the swimming path of a cell. The blue arrow indicates the swimming direction. After photorelease of cGMP (black arrow), the cell displays Ca²⁺ oscillations and abandons swimming in closed circles. Of note, [Ca²⁺]_i was still rising even after the curvature peaked. (C) Time course of the time derivative dF_r/dt and κ superimposes. (A and C) Photolysis of caged cGMP was at t = 0 (dashed lines). (D) Color-coded dF_r/dt along the path of the same cell as in B. (E) Detail of the swimming path. F_r is maximal while the cell swims straight, whereas dF_r/dt is maximal when the cell turns. (F) Pearson correlation between path curvature and dF_r/dt or F_r. Individual data points are displayed as black circles (n = 27), mean correlation is in red, and SD is in blue (box length is twice the SD).

Figure 2.

Trajectory of a sperm cell in a radial gradient of resact is determined by the time derivative of [Ca²⁺]_i. (A) Swimming path of a sperm cell in a radial gradient of resact. The UV profile used for uncaging resact is indicated by shades of gray. The cell swims on circles drifting up the gradient while [Ca²⁺]_i oscillates. Arrows indicate the swimming direction. (B) Changes in F_r and path curvature κ oscillate with identical frequency, but the oscillations do not superimpose. (C) Changes in dF_r/dt and path curvature κ superimpose and are strongly correlated. (D) Pearson correlation between path curvature and dF_r/dt or F_r. Individual data points are displayed as black circles (n = 27), mean correlation is in red, and SD is in blue (box length is twice the SD).

Figure 3.

Waveform of Ca²⁺ signals determines path shape. (A–C, top graphs) Ca²⁺ signals and changes in path curvature κ evoked by release of cGMP at t = 0 (dashed lines) in three cells. Cyan dots on the Ca²⁺ signals correspond to local maxima of dF_r/dt. (middle graphs) dF_r/dt and curvature κ superimpose on each other. (bottom illustrations) Swimming paths of the same cells before (green) and after (red) release of cGMP (black arrows and yellow bars). Green and red arrows indicate the swimming direction of the cell before and after release of resact, respectively. From left A to right C, cells vary from fast and symmetric Ca²⁺ oscillations to slower and skewed oscillations. Correspondingly, path curvature varies from fast oscillations (on a high [Ca²⁺]_i) to slow and asymmetric oscillations with brief episodes of steep peaks (turns) and long episodes of low curvature (runs). Swimming paths for the respective cells vary from short turn and run episodes with almost symmetrical but opposite curving (A) to sharp turns and long runs (C). Cyan dots indicate local maxima of dF_r/dt.

Figure 4.

Numerical reconstruction of swimming paths. Model Ca²⁺ signals and corresponding reconstructed paths using the fit parameters κ₁ and β of Eq. 1. The swimming speed was considered constant for simplicity (v = 200 µm/s). (A) The slope of the rising phase (Ca²⁺ influx) determines the sharpness of the chemotactic turn and the orientation of the cell after the turn. (B) The length of the rising phase determines the duration of the turn and, therefore, codetermines the orientation of the cell before the run. (C) The slope of the Ca²⁺ decline determines the length, the straightness, and sign of the curvature during the run period, depending on the parameters κ₁ and β. (D) The kinetics of the rising and declining phases and the amplitude of the Ca²⁺ signal determine the drifting speed and direction of circles.

Figure 5.

Rapid Ca²⁺ decay reverses sign of path curvature. (A) Mean Ca²⁺ signal elicited by release of caged cGMP (n = 7) by one (red) or two (blue) identical flashes 800 ms apart. (B) Time course of the mean path curvature for cells stimulated with a single flash (red, n = 11) or two flashes (blue, n = 16). (C) Mean dF_r/dt calculated from data in A for one (red) or two (blue) identical flashes. (A–C) The UV flashes are indicated by black dashed lines. (D) Superposition of single-cell trajectories (light blue, n = 17) and mean trajectory (dark blue) around the position where the second stimulus was delivered (yellow boxes). (E) Superposition of single-cell trajectories (light red, n = 11) and mean trajectory (dark red) around the same time point position as in D for those cells receiving only one stimulus. (F) Comparison of mean swimming paths for cells stimulated with a single flash (red, n = 11) or two flashes (blue, n = 17). The error at each point is represented by an ellipse with minor and major axes showing the SEM in the direction of the abscissa and the ordinate.

Figure 6.

Reconstruction of sperm swimming paths in a chemical gradient. (A) Trajectory of the cell shown in Fig. 2 before and after the release of resact. (B) Numerical reconstruction of the swimming path using the time derivative dF_r/dt and the speed. The reconstructed path reproduces many of the essential features of the original path. (C) Experimental Ca²⁺ signal and speed of the sperm cell used for the reconstruction. The release of resact from the caged derivative occurs at t = 0 (dashed line).

Figure 7.

Interfering with the signaling pathway. (A) Changes in [Ca²⁺]_i and path curvature evoked by release of cGMP at t = 0 (dashed line) in the presence of 10 µM Niflumic acid. The cell displays small, rapid Ca²⁺ oscillations that entail similarly fast curvature oscillations (Pearson correlation, −0.17; mean Pearson correlation, −0.01 ± 0.28; n = 6). (B) Path curvature and dF_r/dt superimpose (Pearson correlation, 0.50; mean Pearson correlation, 0.48 ± 0.13; n = 6). (C) Swimming path of a sperm cell before (green) and after (red) cGMP release by a UV flash (black arrow and yellow bar) in the presence of 10 µM Niflumic acid. (D) Sperm accumulation in a Gaussian gradient of chemoattractant with and without Niflumic acid. Relative changes in the standard distance D_r of sperm around the center of the flash after release of resact (t = 0) without (blue) or with (red) 10 µM Niflumic acid are shown. Error bars denote SEM (12 experiments).

Figure 8.

Chemical differentiator model. (A) Input–output relation of the chemical differentiator. Phase shift between input and output signals is shown. Within the differentiator band, the phase shift ϕ approximately equals 90°, with a tolerance of 30°. (B) Amplitude gain as function of the input frequency. For input frequencies 0 < f < 6 Hz (the differentiation band is in gray), the amplitude gain ρ increases linearly. (C) Model Ca²⁺ signal used as input for the chemical differentiator (green) and the output parameter C (purple) resulting from the model. (D) Time derivative of the input Ca²⁺ signal and the output C perfectly superimposes. a.u., arbitrary unit.