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 vn, n˙ ϭ Ϫvt ϩ vb, b˙ ϭ Ϫvn. [6] r˙ ϭ vt, ˙t ϭ vn, n˙ ϭ Ϫvt, [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. If the curvature is constant, 0, the ϭ 2 ϩ 2 1/2 swimming is 0 v0 ( 0 0) . swimming path is a circle with radius r0 ϭ 1/0. The angular
Swimming along such a helical path in a chemoattractant SCIENCES frequency of swimming in circles is ϭ v if the speed v ϭ 0 0 0 concentration field leads again to a time-dependent stimulus s(t)
v is constant. APPLIED PHYSICAL 0 of the signaling system as in the two-dimensional case. In our We consider a swimming path in a concentration field c(x)of three-dimensional description, we assume that the output vari- chemoattractant, where x ϭ (x, y) denotes a position in two- able a(t) of this system modulates both the curvature (t) and the dimensional space. The swimming sperm perceives a temporal torsion (t) of the swimming path concentration stimulus s(t) ͑ ͒ ϭ ϩ ͑ ͑ ͒ Ϫ ͒ ͑ ͒ ϭ ͑ ͑ ͒͒ t 0 1 a t 1 s t c r t . [2] ͑ ͒ ϭ ϩ ͑ ͑ ͒ Ϫ ͒ [7] t 0 1 a t 1 . BIOPHYSICS This stimulus s(t) triggers a response of the chemotactic The system of Eqs. 2, 4, 6, and 7 uniquely determines the signaling network. In general, the signaling network is a dynamic swimming path r(t) for given concentration field and initial system that generates a time-dependent output that depends on conditions. Fig. 2a shows an example of a swimming path in a the history of the stimulus. We characterize this response by a radial concentration field, which leads toward the origin of the dimensionless output variable a(t) with a ϭ 1 in steady state. We concentration field. The swimming path r(t) is superhelical: It is assume that this output variable directly affects the curvature a perturbed helix that winds around a curved centerline R(t) (see of the swimming path by Fig. 2b). ͑ ͒ ϭ ϩ ͑ ͑ ͒ Ϫ ͒ The response of the torsion to weak stimuli is characterized by t 0 1 a t 1 . [3] a linear response coefficient ϭ exp(i), which is defined The variable a(t) plays a similar role as the internal calcium analogously to the linear response coefficient of the curvature concentration within the flagellum. The calcium concentration (Eq. 5). is controlled by the chemotactic signaling network and influ- ences the activity of dynein motors, thereby modulating the Properties of Swimming Paths flagellar beat pattern and the curvature of the swimming path. Motion in a Plane. Eqs. 1–4 for the swimming path r(t) can be ϭ ϩ For simplicity, we assume that the swimming speed is constant solved numerically. For a linear concentration field c(x) c0 ⅐ v ϭ v0 and not affected by chemotactic signaling. c1 x, the swimming paths are drifting circles. The overall motion We can capture the essential properties of the chemotactic of these circling paths is captured by the trajectory of the circle signaling network, namely its ability to adapt and its relaxation centers, which defines the centerline R(t). The centerline R(t)is ␣ dynamics by a simple dynamical system (18, 19) oriented approximately at constant angle with respect to the direction of the concentration gradient c1 (see Fig. 1a). In a a˙ ϭ ps Ϫ a radial concentration field with c(x) ϭ c /͉x͉, the centerline R(t) [4] 0 p˙ ϭ 1 Ϫ a. is a spiral that circles toward the origin at x ϭ (0, 0) (see Fig. 1b). This choice of the radial decay is is motivated by the steady-state Here, p(t) is an internal variable that governs adaptation, is a concentration field established in three dimensions by diffusion relaxation time, and has units of time per volume. For a from a source. Note that the spiral shape of the centerline does time-independent stimulus s(t) ϭ s0, the system reaches a steady not depend on the precise form of the radial decay. state with a ϭ 1 and p ϭ 1/s0. The system is adaptive because the To understand these results, we consider the limit of weak steady-state output is independent of the stimulus level s0. Eqs. gradients. We determine the swimming path in a linear concen- 1–4 determine a unique swimming path r(t) in a self-consistent tration field c(x, y) ϭ c0 ϩ c1 x by a perturbation calculation in 2 manner for given initial conditions and the concentration field. the small parameter ϭ r0 c1, which describes the strength of Examples of swimming paths for linear and radial concentration the perturbation of the swimming path by chemotactic signaling. fields are shown in Fig. 1. These paths can be described as circles Here we explain the logic of the calculation, a detailed deviation
Friedrich and Ju¨licher PNAS ͉ August 14, 2007 ͉ vol. 104 ͉ no. 33 ͉ 13257 Downloaded by guest on October 2, 2021 a b c R 30 r0 20
r(t) R(t) 10 ψ 0 1 1 1 2 6 3 ππ 2 π 3 π
Fig. 2. Three-dimensional swimming path of a sperm cell in a radial concentration field c(x) ϭ c0 /͉x͉.(a) The swimming path r(t) is superhelical and leads to the chemoattractant source (blue dot). (b) The centerline R(t) of the full swimming path. (c) Dynamics of the coarse-grained quantities R ϭ ͉R͉ and alignment angle between helix vector and gradient direction (red curve). Temporal information is provided by dot marks along the (R, )-trajectory, which are seperated by a time interval of 100 . For comparison, the (R, )-trajectory as obtained by our perturbation calculation for the small pertubation parameter (see Eq. 12)is shown in black. Although the value of the perturbation parameter rises from 0.1 to 1 along the trajectories, qualitative agreement between both trajectories persists. Numerical integration of Eqs. 2, 4, 6, and 7 was done for 1 ϭϪ0.1/r0, 0 ϭ 0.2/r0, 1 ϭ 1/r0, ϭ 10 c0 /, v0 ϭ r0 /, and initial conditions r (0) ϭ (15, 25,Ϫ10)r0, t(0)ϭ(1,0,0), n(0)ϭ(0,1,0), b(0) ϭ (0,0,1), a(0) ϭ 1, and p(0) ϭ 1/c(r(0)). The size of the box in (a) and (b) is approximately 25r0 ϫ 40r0 ϫ 40r0.
is provided in supporting information (SI) Text. When swimming This equation for the centerline can be illustrated for the case along a perturbed circular path, the sperm cell perceives a of a radial concentration field c(x) ϭ C(͉x͉). In this case, concentration stimulus that is periodically modulated. This according to Eq. 10, the centerline R(t) winds around the origin periodic stimulus elicits periodic modulations of the curvature of in a logarithmic spiral, R() ϭ R0 exp(Ϫcot(␣)), where is the the swimming path. The resulting swimming path is thus a polar angle and R ϭ ͉R͉ (see Fig. 3). It spirals inwards for Ϫ/2 Ͻ drifting circle with corresponding centerline ␣ Ͻ /2, which corresponds to a phase shift of the signaling system of Ͻ Ͻ 2. Otherwise, it spirals outwards. ͑ ͒ ϭ ͑ ␣ ␣͒ ϩ ᏻ͑2͒ R t vd cos , sin t [8] We have compared solutions to the dynamic equations (Eqs. 1–4) for the swimming path to solutions of Eq. 10 for the center ϭ ␦ ϩ ᏻ 2 ␦ ϭ 2 and drift velocity vd c1 ( ) where 1/2 v0r0. For line in our perturbation calculation and found good quantitative the angle with respect to the gradient, we obtain agreement for Շ 0.1. Furthermore, Eq. 10 could predict perfectly whether the path reached the egg. ␣ ϭ 3/2 Ϫ ϩ ᏻ͑͒. [9]
These results can be generalized to a nonlinear concentration Motion in Three-Dimensional Space. Our numerical solutions for field provided that the gradient is weak, ϽϽ 1, and the swimming paths in three dimensions shown in Fig. 2a demon- 2 strate that the modulation of curvature and torsion of the path .nonlinearities are small on the length-scale r0, ٌ͉ c͉ ϽϽ ٌ͉c͉/r0. Eq 8 for the centerline can then be generalized as by the signaling system lead to chemotactic behavior. In the absence of a gradient, the swimming path is a perfect helix with ϭ ͑␣ٌ͒ ٌ͉ ͉ ϩ ᏻ͑2͒ R˙ vdM c/ c [10] a straight centerline. If this trajectory encounters a linear concentration field, the helix bends until its axis is parallel or -where the operator M(␣) rotates the vector ٌc in the plane by antiparallel to the concentration gradient. In a radial concen the angle ␣ in a counterclockwise sense. Eq. 10 represents a tration field, swimming paths are deformed helices that wind differential equation for the centerline R(t) if the gradient ٌc of toward the origin of the concentration field. To understand these the concentration field is evaluated at any time at position R(t). numerical observations, we generalize the ideas developed in the previous section to three dimensions. R(t) r(t) R r(t) R h
θ ψ Δ c
Fig. 3. Schematic representation of a swimming path in two dimensions in a R radial concentration field of chemoattractant. The swimming path r(t) (black line) is a drifting circle whose center moves along the centerline R(t) (red line). In the limit of weak concentration gradients with the parameter small (see Fig. 4. A deformed helical path r(t) can be generated by the motion of a point text) this centerline is a logarithmic spiral whose tangent encloses a constant on an imagined solid disk that spins around the helix axis given by the vector angle of ␣ with the gradient direction at every point. The distance Rϭ͉R͉ of the h normal to the disk. The center of the disk moves along a curved centerline centerline to the source and the polar angle are used in the formula of the R(t) with tangent R˙ . The gradient direction ٌc and the helix vector h enclose logarithmic spiral (see text). an angle .
13258 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0703530104 Friedrich and Ju¨licher Downloaded by guest on October 2, 2021 path with phase shift and , respectively. These fast modu- lations result in slow variations of R˙ and h. Also, helix frequency ϭ 0 ϩ ᏻ() and helix pitch 2h ϭ 2h0 ϩ ᏻ() are shifted to first order in . In a linear concentration field, we obtain a dynamic equation for the alignment angle between h and the gradient vector ٌc (see Fig. 5) which reads
˙ ϭϪ sin ϩ ᏻ͑2͒, [11] where  ϭ ٌ͉c͉ ( Re Ϫ Re ) with ϭ 0 r0 h 0/2, ϭ 2 0 r0 /2. This equation has two fixed points at ϭ 0 and ϭ , which correspond to helical paths whose helix vectors are parallel and antiparallel to the gradient vector, respectively. The sign of  Fig. 5. In a linear concentration field c(x) ϭ c0ϩc1 ⅐ x with constant gradient determines which of the fixed points is stable. The rate at which vector (blue arrow), the swimming path is a deformed helix (black line), which the helix aligns with the gradient direction is ͉͉. Therefore, for bends and eventually aligns with the gradient: The angle between the the almost all initial conditions, the system exhibits chemotaxis gradient direction ٌc and the helix vector h decreases monotonically, see Eq. upwards the gradient if  Ͼ 0. This behavior is therefore robust 11. The path shown is a numerical solution to Eqs. 2, 4, 6, and 7 with and does not depend on fine-tuning of parameters. parameters ϭϪ0.5/r , ϭ 0.1/r , ϭ 0.5/r , ϭc /, v ϭ r /, and initial 1 0 0 0 1 0 0 0 0 Consider the case of a radial concentration field c(x) ϭ C(͉x͉). conditions r(0) ϭ (15,25,Ϫ10)r0, t(0) ϭ (1,0,0), n(0) ϭ (0,1,0), b(0) ϭ (0,0,1), a(0) ϭ 1, and p(0) ϭ 1/c(r(0)). The size of the box is approximately The five degrees of freedom represented by R and h can be ϭ ͉ ͉ 45r0 ϫ 20r0 ϫ 25r0. expressed by the distance to the origin R R , the alignment angle between helix vector and radial concentration gradient, as well as three Euler angles , , and (see SI Text). Because The net motion resulting from swimming along a deformed of rotational symmetry, the variables R and decouple from the helical path r(t) is captured by the trajectory of the centerline other three variables and obey the dynamic equations
R(t). For given centerline, the helical path can be described by SCIENCES tracing the trajectory of a point on the circumference of an R˙ ϭ Ϫh cos Ϫ ␥ sin2 and APPLIED PHYSICAL imagined solid disk with radius r0, rotating in its plane with a rotation rate ⍀ and with its center moving along R(t). The 1 [12] 3 ˙ ϭ Ϫsin ͩ Ϫ ͑h Ϫ ␥ cos ͒ͪ , orientation of the disk is characterized by the unit vector h R normal to the disk, which we call the helix vector (see Fig. 4). For ϭ ٌ͉ ͉ ␦ Ϫ ␦ ␦ ϭ ␥  a comparison of the full swimming path r(t) and its centerline where is as above and c ( Im Im ) with 0 r0 2 ϩ 2 ␦ ϭ 2 R(t), see Fig. 2 a and b. (2r 0 3h0)/4, 0 r0 h0 /4 (see SI Text). As in the case of a linear
As in the planar case, the chemotactic feedback loop for three concentration field, we can analyze under what conditions swim- BIOPHYSICS space dimensions (Eqs. 2, 4, 6, and 7) can be studied in the limit ming paths find an egg of radius Regg at the origin by discussing of weak gradients by a perturbation calculation in the small phase space trajectories of this dynamical system. We restrict ͉␥͉ Ͻ parameter ϭ (/0ϩ/0) r0 ٌ͉c͉, see SI Text for details. In the ourselves to the case 0 h0 relevant to our perturbation limit of a weak gradient, the swimming path is a perturbed helix. calculation and assume for simplicity constant values of  and ␥,as In a concentration field, this helical swimming path samples a well as ϭ 0, h ϭ h0. These assumptions correspond to a signaling time-dependent chemotactic stimulus that has a periodic com- network that has constant linear response coefficients , , ponent that is governed by the projection ٌЌc ϭ (1 Ϫ hh⅐) ٌc independent of the average stimulus level, and perfectly adapts to of the three-dimensional gradient on the plane of the disk. This ramp stimuli. However, the following analysis will not change periodic component of the stimulus generates an output signal qualitatively if these assumptions are not imposed, provided that  that periodically modulates both curvature and torsion of the and ␥ do not switch sign. Depending on the two parameters  and
a b c R R R h0 h0 h0 20 20 20
0 0 0 π 0 ψ π π 0 ψ π π 0 ψ π
Fig. 6. Flow field and typical trajectories in the (R, )-phase space for the dynamic system given by Eq. 12, which describes chemotaxis in three dimensions in a radial concentration field. Here R is the distance of the centerline to the chemoattractant source, and is the angle between the helix vector and the gradient direction. Distance R is shown relative to h0 where 2h0 is the pitch of the unperturbed helix. (a), (b), and (c) are examples for the three regimes A, B, and C, respectively (see text for details). Typical (R, )-trajectories (black lines) as well as the gradient vector field (gray arrows) are shown. (a) In regime A, the system possesses two repulsive fixed points (black dots). The (R, )-trajectories are spirals which spiral away from the fixed points. (b) In regime B, the fixed points are attractive and (R, )-trajectories spiral toward the fixed points. (c) In regime C, no fixed points exist, the (R, )-trajectories are U-shaped and eventually move away from the chemoattractant source. The parameter values used were  ϭ 0.1 0, ␥ ϭ 0.1 0 h0 (a);  ϭ 0.2 0, ␥ ϭϪ0.05 0 h0 (b); and  ϭϪ0.1 0, ␥ ϭϪ0.1 0 h0 (c).
Friedrich and Ju¨licher PNAS ͉ August 14, 2007 ͉ vol. 104 ͉ no. 33 ͉ 13259 Downloaded by guest on October 2, 2021 a b c R(0) R(0) R(0)
h0 h0 h0 20 20 20
0 0 0 π 0 π π 0 π π 0 π ψ (0) ψ (0) ψ (0)
Fig. 7. Chemotactic success as a function of initial conditions of swimming paths according to the dynamic system given by Eq. 12. The initial condition of a path is characterized by the initial distance R(0) of the centerline to the chemoattractant source and the initial angle (0) between the helix axis and the gradient direction. Distance R is shown relative to h0 where 2h0 is the pitch of the unperturbed helix. For a given initial condition, chemotaxis is unsuccessful if the correspondent swimming path has a distance R(t) to the chemoattractant source which is always greater than an egg radius Regg, R(t) Ͼ Regg. Initial conditions with unsuccessful chemotaxis are shown as blue dots, blue lines, and blue hatched regions. The radius of the egg Regg is indicated by a dashed line. (a), (b), and (c) are examples for the three regimes A, B, and C, respectively (see text for details). (a) In regime A, where repulsive fixed points exist (see Fig. 6a), chemotaxis is robust and fails only at the fixed points and the marginal case (0) ϭϮ.(b) In regime B, fixed points are attractive (see Fig. 6b). Chemotaxis is successful in a large range of initial conditions and fails only in a neighborhood of the fixed points as well as in the marginal case (0) ϭϮ.(c) In regime C, no fixed points exist and chemotaxis is repulsive with respect to the chemoattractant source (see Fig. 6c). Chemotaxis is unsuccessful except for those initial conditions where the initial distance to the source is already small and the helix axis is nearly aligned with the gradient direction. The parameter values used were Regg ϭ h0 and  ϭ 0.1 0, ␥ ϭ 0.1 0h0 (a);  ϭ 0.2 0, ␥ ϭϪ0.05 0 h0 (b); and  ϭϪ0.1 0, ␥ ϭϪ0.1 0h0 (c).
␥, the flow of the dynamical system can exhibit three different circles in two dimensions and helices that are bent and tilted in regimes: Regime A for  Ͼ 0, ␥ Ͼ 0, B for  Ͼ 0, ␥ Ͻ 0, and C for three dimensions. We discuss the geometry of these paths and  Ͻ 0 (see Fig. 6). In regime A and B with  Ͼ 0, two symmetric determine the conditions under which the system moves to fixed points (R0, 0) exist, where R0 ϭ 0 h0 / ϩ ᏻ(), 0 ϭϮ/2 ϩ regions of high chemoattractant concentration. We find that key ᏻ(), which correspond to swimming paths for which the centerline parameters for chemotactic success are the phase shifts and circles around the origin and R is constant. Linear stability analysis between stimulus and modulations of curvature and torsion, reveals that for regime A, these fixed points are repulsive while for respectively. In both two and three dimensions, there exist large regime B they are attractive (see Fig. 6 a and b). In both regimes, ranges of these parameters for which chemotaxis is reliable. (R, )-trajectories of the dynamical system are spirals near the fixed There is an extensive overlap of those ranges where chemotaxis points which correspond to swimming paths that repeatedly (i) align works for the same parameters in two and three dimensions. with the gradient vector, (ii) approach the origin, (iii) loose align- Therefore, chemotaxis is a robust property of the system that ment when they come near the origin, and (iv) move away again. As does not require fine-tuning of parameters if the signaling system a consequence, R subsequently decreases and increases, with an is adaptive. Our work is related to earlier work by Crenshaw, who increasing amplitude of R changes in regime A and a decreasing suggested that chemotaxis could be realized for helical paths by amplitude in regime B. For regime C with  Ͻ 0, no fixed points periodically modulating rates of translation and rotation of exist, the trajectories are U-shaped and move toward large R at long swimming sperm (15–17). Several works have studied chemo- times (see Fig. 6c). taxis for helical paths using computer simulations (20) or by In the presence of an egg with radius Regg, the centerline of experiments with robots (21). swimming paths reaches the egg for almost all initial conditions Our results are consistent with experimental observations in case A (see Fig. 7a). For case B, (R, )-trajectories starting in both in two and three dimensions. When sperm swims close to a finite neighborhood of the fixed points (R0, 0) correspond to a surface, observed swimming paths in a concentration gra- swimming paths that do not reach the egg. Further away from dient resembled drifting circles (1, 10). Recent experiments this neighborhood, trajectories can reach the egg before they have shown that, upon periodic stimulation, the signaling spiral to the fixed point (see Fig. 7b). In case C, swimming paths network generates calcium spikes which are phase-locked to are repelled from the egg and chemotaxis acts down the gradient the stimulus with a phase shift of about Ϫ0.4 ϫ 2 (11). This (see Fig. 7c). Therefore, we find again that chemotaxis is a robust value is in the range where chemotaxis is successful in two property that does not require fine-tuning of parameters. It is dimensions. Note, however, that between the calcium spike most reliably in case A for which a sufficient condition is given and the curvature modulation, there could be an additional by Ͻ Ͻ 3/2, 0 Ͻ Ͻ /2. phase shift stemming from the dynamic dependence of the curvature on the flagellar beat pattern. In three dimensions, Discussion tracking experiments showed that sperm swim along helices We have presented a theoretical description of sperm swimming (8) as had been suggested earlier (6, 22). Furthermore, it was paths, taking into account chemotactic signaling. Our main observed that in a concentration gradient the helix axis aligned assumptions are (i) that the curvature and torsion of the with the gradient vector (8). swimming path are modulated by the signaling system, and (ii) From experiments, we can estimate parameter values which that the signaling system receives a temporal chemoattractant are relevant to our description. Typical values for the swimming Ϫ1 concentration stimulus implying that concentration differences speed, average curvature and torsion are v0 Ϸ 200 ms , 0 Ϸ Ϫ1 Ϫ1 along the length of the flagellum are irrelevant. We study 0.025Ϫ0.05 m , and 0 ϷϪ0.025 m , respectively (7, 8, 10). swimming paths both in two and three dimensions and for linear The radius of the egg is Regg Ϸ 100 m (23). From the drift speed Ϫ1 and radial concentration fields. In all cases, periodic components vd Ϸ 25–50 ms of swimming circles reported in ref. 11, we ϭ 2 ␦ ϭ Ϸ occur in the stimulus that elicit periodic variations of curvature estimate the perturbation parameter vdr0 / 2vd/v0 and torsion of the path. Using both numerical and analytical 0.25Ϫ0.5. In an early experiment with bracken fern spermato- methods, we show that the resulting swimming paths are drifting zoids, Brokaw measured the bending rate of the helix as a
13260 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0703530104 Friedrich and Ju¨licher Downloaded by guest on October 2, 2021 function of the concentration gradient. He suggested Eq. 11 as Helical swimming paths are ubiquitous in nature and are a a phenomenological description of his observations and could direct result of an asymmetric propulsion mechanism. In addi- Ϫ -estimate c/ٌ͉c͉ Ϸ 350 ms 1 (22). Furthermore, he observed tion to sperm, it has been observed for ascidian larvae, eukary that bending of the helix occurred in the plane spanned by otic flagellates such as Chlamydomonas, flagellated plant sper- Јϭ the helix vector and the gradient, which suggests that matozoids, and even some bacteria such as Thiovulum majus .(c͉(ImϪ Im ) Ϸ 0 (see Eq. S14 in SI Textٌ͉ (24–27). In the case of ascidian larvae, both positive and negative We used a simplified description of the signaling network which captures essential properties such as adaptation and a phototaxis have been observed in different developmental simple relaxation dynamics that generates a phase shift between stages. Such a change in phototactic behavior can be accounted stimulus and output. Experimental studies of the signaling for by our theory by a simple change of the phase shift between system revealed that the flagellar membrane potential exhibits the stimulus and the response (24). spikes on a 100-ms time scale with an amplitude that increases In summary, we have shown that temporal sampling of a from 2 to 45 mV when stimulus concentration is varied over 5 concentration field along a helical path provides a robust orders of magnitude from 250 fM to 25 nM. The spike amplitude strategy for chemotaxis if curvature and torsion are modulated Ϸ saturates for stimulus concentrations as low as 25 pM (13). This in response to the stimulus. This mechanism can work reliably nonlinear response suggests a more complex dynamics than the over a large concentration range, does not require fine-tuning of simple system described by Eq. 4. If the signaling network parameters, and depends only on a few generic properties of the generated spikes of constant amplitude that are phase-locked to a periodic stimulus, the perturbation parameter would be signaling network. independent of concentration. We speculate that an adaptation mechanism ensures that does not change significantly over a We thank U. B. Kaupp for stimulating discussions and a critical reading large range of concentrations. of the manuscript.
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Friedrich and Ju¨licher PNAS ͉ August 14, 2007 ͉ vol. 104 ͉ no. 33 ͉ 13261 Downloaded by guest on October 2, 2021