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Chemotaxis of Sperm Cells

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Chemotaxis of Sperm Cells Chemotaxis of sperm cells Benjamin M. Friedrich* and Frank Ju¨licher* Max Planck Institute for the Physics of Complex Systems, No¨thnitzer Strasse 38, 01187 Dresden, Germany Edited by Charles S. Peskin, New York University, New York, NY, and approved June 20, 2007 (received for review April 17, 2007) We develop a theoretical description of sperm chemotaxis. Sperm In this case, sperm swim on helical paths. In the presence of a cells of many species are guided to the egg by chemoattractants, chemoattractant concentration gradient, the helices bend, even- a process called chemotaxis. Motor proteins in the flagellum of the tually leading to alignment of the helix axis with the gradient (8). sperm generate a regular beat of the flagellum, which propels the Chemotaxis is mediated by a signaling system that is located sperm in a fluid. In the absence of a chemoattractant, sperm swim in the sperm flagellum (10). Specific receptors in the flagellar in circles in two dimensions and along helical paths in three membrane are activated upon binding of chemoattractant mol- dimensions. Chemoattractants stimulate a signaling system in the ecules and start the production of cyclic guanine monophosphate flagellum, which regulates the motors to control sperm swimming. (cGMP). A rise in cGMP gates the opening of potassium Our theoretical description of sperm chemotaxis in two and three channels and causes a hyperpolarization of the flagellar mem- dimensions is based on a generic signaling module that regulates brane. This hyperpolarization triggers the opening of voltage- the curvature and torsion of the swimming path. In the presence gated calcium channels and the membrane depolarizes again. of a chemoattractant, swimming paths are drifting circles in two The overall effect of this signaling cascade is the generation of dimensions and deformed helices in three dimensions. The swim- a transient increase of the internal calcium concentration along ming paths can be described by a dynamical system that exhibits the flagellar length (9, 10, 13). This calcium signal is thought to different dynamic regimes, which correspond to different chemo- regulate the activity of dynein motor proteins in the flagellum, tactic behaviours. We conclude that sampling a concentration field thus affecting the geometry of the swimming path (14). Inter- of chemoattractant along circular and helical swimming paths is a estingly, this signaling system operates over a vast range of robust strategy for chemotaxis that works reliably for a vast range chemoattractant concentrations ranging from picomolar to of parameters. micromolar concentrations (13). This observation suggests that adaptation plays an important role in sperm chemotactic ͉ ͉ ͉ helical klinotaxis sperm swimming flagellar propulsion signaling. ͉ signaling systems adaptation In this article, we develop a theoretical description of sperm chemotaxis. We characterize a simple and general mechanism perm chemotaxis plays an important role for fertilization. It for chemotaxis in two dimensions that is motivated by recent Simplies that a swimming sperm cell steers upwards a gradient experiments (9–11, 13). Sperm swimming on circular paths in a of a chemoattractant that is released by the egg. Sperm chemo- concentration gradient of a chemoattractant sample a periodic taxis is well established in marine invertebrates with external concentration stimulus. The signaling system transfers this stim- fertilization (e.g., sea urchins) (1) and has been demonstrated in ulus into a periodic modulation of the curvature of the swimming mammals (e.g., humans) (2). path. As a result, the circular path drifts in a direction that In bacteria, chemotaxis is well understood both experimen- depends on the internal dynamics of the signaling system. We tally and theoretically. It has been shown that bacteria undergo show that this principle is more general and also works for helical a biased random walk toward the source of a chemoattractant. trajectories in three dimensions. We discuss conditions under This is achieved by stochastic switching between running and which a swimming sperm reaches the egg, in both two and three tumbling modes of motion which is controlled by a chemotactic dimensions. Our work characterizes a principle for the chemo- signaling system in a way that depends on the history of the taxis of microorganisms that is based on the idea of helical chemotactic stimulus (3–5). This chemotactic system provides a klinotaxis introduced by the earlier work of Crenshaw (15–17). general mechanism for chemotaxis of microorganisms (3, 5). The swimming of sperm is very different from bacterial Theoretical Description of Sperm Chemotaxis Ϸ swimming. Sperm possess a single eukaryotic flagellum of 50 Chemotaxis in a Plane. Chemotaxis can be studied in two dimen- ␮ m length. Dynein motors in the microtubule based axonem of sions for sperm swimming near a surface along a path r(t). The this flagellum generate a bending wave with a frequency of Ϸ50 Ϸ ␮ swimming path is defined as the position of the head averaged Hz that propels the sperm forward at a speed of 200 m/s (6, over one cycle of the flagellar beat as a function of time. This 7). Sperm of many species do not swim on straight trajectories average eliminates rapid periodic movements of the head at the but instead move in circles or helical paths (1, 6, 8–10). The frequency of the flagellar beat. The swimming velocity is char- differences of the swimming motion of sperm cells as compared acterized by the speed v ϭ ͉r˙͉ and the tangent vector t ϭ r˙/v of to bacteria suggest that fundamentally different principles of the swimming path. Dots denote time derivatives. The geometry chemotaxis are at work in sperm. of the swimming path is described by the Frenet–Serret equa- Here, we will focus on the chemotaxis of sea urchin sperm tions in two dimensions because of the large amount of experimental data available for this system. Chemotaxis of sea urchin sperm is usually observed under experimental conditions where sperm swim in a small Author contributions: B.M.F. and F.J. designed research, performed research, and wrote the observation chamber under the microscope. In this situation, paper. sperm become localized near the surfaces of the chamber where The authors declare no conflict of interest. they swim on circular paths (1, 9–12). The curvature of their This article is a PNAS Direct Submission. swimming path is a consequence of the asymmetry of the *To whom correspondence may be addressed. E-mail: [email protected] or julicher@ flagellar beat (9). In a concentration gradient of a chemoattrac- pks.mpg.de. tant, the circular swimming path drifts toward positions with This article contains supporting information online at www.pnas.org/cgi/content/full/ higher concentration (1, 10). It has also been possible to track 0703530104/DC1. sperm swimming far from surfaces in three-dimensional space. © 2007 by The National Academy of Sciences of the USA 13256–13261 ͉ PNAS ͉ August 14, 2007 ͉ vol. 104 ͉ no. 33 www.pnas.org͞cgi͞doi͞10.1073͞pnas.0703530104 Downloaded by guest on October 2, 2021 a b whose centers drift along well defined trajectories. Small periodic variations of the stimulus s(t) ϭ s0 ϩ s1 cos ␻0t r(t) ␬ ϭ ␬ ϩ Δ r(t) evoke a periodic response of the curvature (t) 0 c ␳ ␻ ϩ ␾ ϩ ᏻ 2 ␳ ␬s1 cos ( 0t ␬) (s1) with amplitude gain ␬ and phase R(t) shift ␾␬. For the signaling system (Eq. 4), the linear response ␹ ϭ ␳ ␾ R(t) coefficient ␬ ␬exp(i ␬) reads ␬ Ϫ i␻ ␹ ϭ 1 0 ␬ ␴␻2 Ϫ ␮ Ϫ ␻ . [5] Fig. 1. Swimming paths r(t) in two dimensions for two different chemoat- s0 0 s0 / i 0 tractant concentration fields. (a) In a linear concentration field c(x) ϭ c0 ϩ c1 ⅐ x with constant gradient vector (blue arrow), the swimming path (black line) This linear response will play a key role to characterize swim- is a drifting circle. This drift can be described by the motion of the center of the ming paths. circle R(t) (red line). This centerline encloses an angle of ␣ with the gradient direction. (b) In a radial concentration field c(x) ϭ c0/͉x͉, the swimming path is Chemotaxis in Three-Dimensional Space. Far from any surface, a circle that drifts along a spiral to the center of the radial distribution (blue sperm cells swim along helical paths if no chemoattractant is dot). The paths shown are numerical solutions to Eqs. 1–4 with parameters in present (8). The geometry of the swimming path r(t) is char- ͉ ͉ ϭ ␬ ϭ ␮ϭ ␴ ϭ ␴ ␬ ϭ ␮ ϭ (a) c1 0.1 c0/r0, 1 1/r0, 5 c0/ , v0 0.5 r0/ , and in (b) 1 1/r0, 0.5 ϭ ϭ ˙ ͉˙͉ ␴ ϭ ␴ ␴ acterized by the tangent t r˙/v, the normal n t/ t and the c0/ , v0 0.5 r0/ , where c0,r0, set the concentration, length and time scales ϭ ϫ of the problem, respectively. In both cases, initial conditions were r(0) ϭ (10, binormal b t n. The time evolution of these vectors is given by the Frenet–Serret equations in three dimensions Ϫ1) r0, a(0) ϭ 1, and p(0) ϭ 1/c(r(0)). r˙ ϭ vt, ˙t ϭ v␬n, n˙ ϭ Ϫv␬t ϩ v␶b, b˙ ϭ Ϫv␶n. [6] r˙ ϭ vt, ˙t ϭ v␬n, n˙ ϭ Ϫv␬t, [1] For constant curvature ␬(t) ϭ ␬0 and torsion ␶(t) ϭ ␶0, the ϭ ␬ ␬2 ϩ ␶2 ␬ swimming path is a perfect helix with radius r0 0/( 0 0) and where the normal n is orthogonal to t, and is the local curvature ␲ ϭ ␲␶ ␬2 ϩ ␶2 ␬ ϭ ␬ pitch 2 h0 2 0/( 0 0). The angular frequency of helical of the swimming path.
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