The steering gaits of sperm

Abstract

Sperm are highly specialized cells, which have been subject to substantial evolutionary pressure. Whereas some sperm features are highly conserved, others have undergone major modifications. Some of these variations are driven by adaptation to mating behaviours or fitness at the organismic level. Others represent alternative solutions to the same task. Sperm must find the egg for fertilization. During this task, sperm rely on long slender appendages termed flagella that serve as sensory antennas, propellers and steering rudders. The beat of the flagellum is periodic. The resulting travelling wave generates the necessary thrust for propulsion in the fluid. Recent studies reveal that, for steering, different species rely on different fundamental features of the beat wave. Here, we discuss some examples of unity and diversity across sperm from different species with a particular emphasis on the steering mechanisms. This article is part of the Theo Murphy meeting issue 'Unity and diversity of cilia in locomotion and transport'.

Keywords: chirality; cilia; cytoskeleton; navigation; symmetry-breaking.

Figure 1.

Unity and diversity in sperm morphology and ultrastructure. Exemplary variations in sperm morphology (a–d) and axonemal structures (e–h). (a) Sperm from the sea urchin Arbacia punctulata characterized by the stereotypical ‘primitive’ sperm morphology. (b) Human sperm featuring a thicker flagellum with outer dense fibres surrounding the 9 + 2 structure. (c) Sperm cell from the fruit fly, Drosophila melanogaster, with a 2 mm-long flagellum. (d) Multiflagellated sperm from the plant Cycas revoluta. Reprinted with permission from [11]. (e) Schematic cross-section of the highly conserved 9 + 2 axonemal structure present in most motile cilia and flagella. Nine microtubule doublets (with tubules A and B) are arranged cylindrically around an additional pair of microtubule singlets located at the centre. Two dynein arms (green), referred to as inner and outer dynein arms, contain different subsets of dynein molecular motors. Arms are projected from the A tubule of one microtubule doublet towards the B tubule of the adjacent doublet in a clockwise fashion. (f) 9 + 2 axonemal structure of sperm from sea urchin. (g) 9 + 0 axonemal structure from sperm of the eel Anguilla anguilla. Only the inner dynein arms are visible (red arrow). Instead of outer dynein arms, a small electron-dense structure can be occasionally seen (black arrow). Adapted from [12]. (h) The 3 + 0 axoneme from the sperm of the parasite Diplauxis hatti is the simplest axoneme known. Adapted from [13]. Panels (a,b) indicate the two tubules of the microtubule doublets. M indicates the plasma membrane. Axonemal structures (e–h) are viewed from the head towards the flagellar tip. Dynein motors in (f) and (h) have not been resolved.

Figure 2.

Chiral flagellar beat patterns result in net cell rotation. (a–c) Prototypical flagellar beat patterns corresponding to (a) a symmetric travelling wave with amplitude C₁, (b) a superposition of a symmetric travelling wave and a circular arc with constant curvature C₀, and (c) a superposition of a symmetric travelling wave and a second harmonic component with amplitude C₂. The two half periods of the second harmonic component, otherwise overlapping, have been shifted vertically for better visualization. The superposition of the average curvature C₀ and the travelling wave (b) results in an asymmetric beat in space that can be used as a rudder to make a turn. The beat pattern in (c) is characterized by a symmetric average shape (red line), but nonetheless breaks symmetry as it is reflected by the asymmetric envelope of the resultant beat pattern. This beat pattern can also be used to make a turn. (d,e) Computed instantaneous rotation velocity Ω_i normalized by the angular beat frequency during three consecutive beat cycles (d) and swimming paths (e) for sperm featuring a simple travelling wave (olive), or a travelling wave plus an average curvature (blue) or a second harmonic (red). Numerical computations used resistive-force theory and headless sperm models for simplicity. Dashed lines in (d) represent the time-average. Parameters: flagellar length L = 40 µm, C₀ = −1.5/L, C₁ = 0.75k, C₂ = C₁/2, ϕ = π/2, k = 2π/λ and λ = L.

Figure 3.

Parameter space of flagellar beat patterns relevant for changing direction. (a) Swimming path for the prototypical flagellar waveform of equation (3.4b) for a time-dependent mean curvature C₀(t). C₀ is varied in time as indicated by the colour code, whereas C₁ = 0.05 µm⁻¹ is kept constant. (b) Swimming paths for the flagellar waveform of equation (3.4c) with time-dependent amplitude C₂(t) of the second harmonic, constant C₁ = 0.1 µm⁻¹ and second harmonic phase shift ϕ = π/2. (c) Analogous to panel (b), but for constant C₂ and a time-varying phase ϕ(t). Black arrows indicate swimming direction. (d–f) Net rotation velocity for different (static) parameter values of prototypical flagellar beat patterns, as in equation (3.4b) (d) or equation (3.4c) (e,f). Solid lines represent numerical computations using the resistive-force theory for a headless sperm cell. Dashed lines represent the approximate analytical solutions calculated from the small-curvature approximation (equations (4.1) or (4.2)). Parameters: ϕ = π/2 in (e) and C₁ = 0.05 µm⁻¹ in (f).

Figure 4.

The average flagellar curvature and the second harmonic of the beat are used in nature to change swimming direction. (a) Schematic of the experimental conditions used to record the flagellar beat of human and sea urchin sperm. Sea urchin sperm were freely moving, whereas human sperm were tethered at the head to prevent cell rolling and facilitate flagellar tracking. Tethering resulted in a rotation around the head. (b) Average power spectrum of the flagellar curvature for human sperm (blue; average from n = 6 cells) and sea urchin sperm (pink; average from n = 4 cells). (c) Exemplary time course of the normalized rotational velocity (blue) of human sperm, as well as the time course of the average flagellar curvature (green) and the effective strength of the second harmonic (red). (d) The same as (c), but for sea urchin sperm. (e,f) Normalized rotational velocity versus the average curvature for human (e) and sea urchin sperm (f). (g,h) Normalized rotational velocity versus the effective strength of the second harmonic for human (g) and sea urchin sperm (h). Different colours in e–h represent different cells. Black dashed line corresponds to a global linear fit. The phase ϕ_eff is chosen such as to maximize the correlation coefficient between the rotational velocity and the second harmonic strength.