The stochastic dance of circling sperm cells: sperm chemotaxis in the plane

Biological systems such as single cells must function in the presence of fluctuations. It has been shown in a two-dimensional experimental setup that sea urchin sperm cells move toward a source of chemoattractant along planar trochoidal swimming paths, i.e. drifting circles. In these experiments, a pronounced variability of the swimming paths is observed. We present a theoretical description of sperm chemotaxis in two dimensions which takes fluctuations into account. We derive a coarse-grained theory of stochastic sperm swimming paths in a concentration field of chemoattractant. Fluctuations enter as multiplicative noise in the equations for the sperm swimming path. We discuss the stochastic properties of sperm swimming and predict a concentration-dependence of the effective diffusion constant of sperm swimming which could be tested in experiments.


Introduction
Sperm chemotaxis plays an important role for fertilization. Sperm cells steer up a gradient of chemoattractant, which is released by the egg, a process called chemotaxis. Sperm chemotaxis is well established in marine invertebrates with external fertilization (e.g. sea urchins [1]) and has been demonstrated in mammals (e.g. humans [2]).
Sperm cells possess a single eukaryotic flagellum of about 50 µm length. Dynein motors within this flagellum drive a regular beat of the flagellum which propels the sperm cell in a liquid [3]. Chemotaxis is mediated by a signaling system, which is located within the flagellum [1]. Specific receptors in the flagellar membrane are activated upon binding chemoattractant molecules and trigger a chain of signaling events which ultimately change the intra-flagellar calcium concentration [1,4,5]. The calcium concentration apparently regulates the activity of the flagellar dynein motors and results in a swimming response [6]. Interestingly, sperm cells can respond to chemotactic stimuli of subpicomolar concentration [4,5,7].
Here, we will focus on the chemotaxis of sea urchin sperm 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 cells swim in a shallow observation chamber under the microscope [1,4], [8]- [11]. In this situation, sperm cells become localized near the surfaces of the chamber where they swim along planar paths. In the absence of chemoattractant, the swimming paths are circular. The curvature of their swimming paths is a direct consequence of an asymmetry of the flagellar beat [4]. The rotational sense of circular swimming is mostly counterclockwise (when viewed in a direction parallel to the inward normal of the boundary surface).
In a concentration gradient of chemoattractant, the circular swimming paths drift toward regions of higher chemoattractant concentration; the swimming paths thus resemble trochoids [1,8,9]. Experiments indicate that both the intra-flagellar calcium concentration 3 and the curvature of the swimming path exhibit characteristic periodic modulations with the frequency of circular swimming when a sperm cell swims in a chemoattractant concentration gradient [1].
The mechanism of sperm chemotaxis can be formalized as follows [12]: while swimming along an approximately circular path in a concentration gradient of chemoattractant, a sperm cell traces a concentration stimulus from the concentration field, which is periodically modulated with the frequency of circular swimming. As a result, the chemotactic signaling system shows a periodic modulation of activity and causes a periodic modulation of the curvature of the swimming path. A periodic modulation of the curvature gives rise to a swimming path which is a drifting circle, or (prolate) trochoid. The relative direction of the drift with respect to the concentration gradient depends crucially on the phase shift between the stimulus and the curvature oscillations.
Experiments on sea urchin sperm chemotaxis reveal a pronounced variability of the observed sperm swimming paths. We argue that sperm chemotaxis is subject to fluctuations of various sources: the chemotactic stimulus, which the sperm cells receives exhibits fluctuations since the binding of chemoattractant molecules to their respective receptors is a discrete process. Additionally, intrinsic fluctuations of the chemotactic signaling system and fluctuations in the activity of the motor ensemble which drive the flagellar beat are present.
In this paper, we develop a framework to systematically account for fluctuations affecting sperm chemotaxis. Note that sperm swimming is an active process which is characterized by non-equilibrium fluctuations. Starting from a physical description of sperm chemotaxis, we derive an effective stochastic equation for the centerline of sperm swimming paths which follows stochastic trochoids. Different sources of fluctuations enter as multiplicative noise in this description. Our effective description characterizes sperm chemotaxis by a superposition of a chemotactic drift and effective diffusion. We show that the chemotaxis mechanism is robust in the presence of fluctuations and we discuss chemotactic success of sperm cells toward a single source of chemoattractant. We calculate a concentration-dependence of the diffusion constant of sperm swimming circles which could be tested in future experiments.

Dynamic equations of sperm motion
We study sperm chemotaxis in two dimensions for sperm cells swimming near a surface. We define the swimming path as the trajectory of the sperm head averaged over one cycle of the flagellar beat. This average eliminates rapid periodic movements of the head with the frequency of the flagellar beat. The planar swimming path is parameterized by time-dependent xand y-coordinates which we denote by r 1 (t) and r 2 (t), respectively. Throughout this paper, we will represent points in the plane (r 1 , r 2 ) as complex numbers r = r 1 + i r 2 ; an analogous convention applies for planar vectors. This notation will simplify the calculations.
The velocity of the swimming path r(t) is characterized by the speed v = |ṙ| and the tangent t =ṙ/v, where dots denote time derivatives. The normal vector is defined as n = i t. The Frenet-Serret equations in two dimensions can now be written as complex-valued differential equationṡ where κ =ṫ n * /v is the (signed) curvature of the swimming path and the star denotes the complex conjugate. The absolute value of the (signed) curvature satisfies the familiar relation |κ| = |ṫ|/v. From equation (2.1), we find for the time evolution of the argument ϕ of the tangent For constant curvature κ(t) = κ 0 > 0, the swimming path r(t) is a circle with radius r 0 = 1/κ 0 . The angular frequency of circular swimming is ω 0 = v 0 κ 0 if speed v(t) = v 0 is constant. In the following, we consider the case where sperm cells swim in a counterclockwise sense along their circular swimming paths r(t) such that κ 0 > 0. Our results also apply for swimming in a clockwise sense, however, r 0 and ω 0 attain negative values in this case. Note that a change of sign κ 0 → −κ 0 corresponds to using the complex-conjugate path r → r * .
We consider a sperm swimming path in a concentration field c(x) = c(x + iy) of chemoattractant. Due to the motion of the sperm cell, the local concentration c(r(t)) at the position r(t) of the sperm cell changes with time t. Chemoattractant molecules bind to specific receptors on the flagellar membrane with a total rate q(t) which is defined by an ensemble average and which we assume to be proportional to the local concentration We introduce the chemoattractant stimulus s(t) = δ(t − t j ) which counts binding events of single chemoattractant molecules to the receptors which occur at times t j . Since binding events are stochastic, the stimulus s(t) is a stochastic flux with expectation value This stimulus s(t) triggers a response of the chemotactic signaling network. In general the signaling network is a dynamic system which generates a time-dependent output which depends on the history of the stimulus. We characterize this response by a dimensionless output variable a(t) with a = 1 for a constant stimulus in the absence of fluctuations. This output variable a(t) affects the curvature κ(t) of the swimming path by modulating the activity of the flagellar motors. In our simple description [12], we write Here, we include a noise term ξ m with zero mean in (2.5) to account for fluctuations of the flagellar beat due to fluctuations in the activity of the motor ensemble. For simplicity, we assume that the swimming speed v(t) = v 0 is constant, unaffected by chemotactic signaling. We can capture the essential properties of the chemotactic signaling system, namely its ability to adapt and its relaxation dynamics, by a simple dynamical system [12]- [14] (2.6) Here, p(t) is an internal variable which governs adaptation and represents a dynamic sensitivity. This sensitivity p is reduced whenever the output a of the signaling system exceeds the value 1, and correspondingly is increased if a falls below this value. In the absence of any stimulus, s(t) = 0, the sensitivity p remains bounded with p = 1/s b . The parameter s b has units of a rate and corresponds to a detection limit for low stimuli which could for example result from a background activity of the chemoattractant receptors. The time constants σ and µ set the response timescale and the timescale of adaptation, respectively. The intrinsic fluctuations of the molecular network underlying chemotactic signaling are included in a generic fashion by the noise term ξ a with zero mean. For a time independent stimulus s(t) = s 0 , the stochastic system (2.6) reaches a stationary state with a ≈ 1 and p ≈ 1/(s b + s 0 ). The system is adaptive since the expectation value (and also the full statistics) of the steady-state output is independent of the stimulus level s 0 .
In the noise-free case, ξ a = 0, ξ m = 0, small periodic variations of the stimulus s(t) = s 0 + s 1 cos ω 0 t evoke a periodic response of the curvature κ(t) = κ 0 + ρ κ s 1 cos(ω 0 t + ϕ κ ) + O(s 2 1 ) with amplitude gain ρ κ and phase shift ϕ κ . The linear response coefficientχ κ (ω 0 ) = ρ κ e iϕ κ is obtained from the response functioñ of the signaling system described by equation (2.6), evaluated at the circling frequency ω 0 . Note that the phase-shift ϕ κ is independent of the stimulus strength s 0 . The amplitude gain ρ κ is a function of the stimulus level s 0 and can be written as with a constantρ κ that represents an amplification strength of the chemotactic signaling system.

Characterization of noise terms
2.2.1. Stochastic chemoattractant stimulus. Sperm cells 'measure' the local chemoattractant concentration c = c(r(t)) by 'reading out' the number of their activated chemoattractant receptors. A chemoattractant receptor becomes activated upon binding a chemoattractant molecule and deactivates again on a timescale of 300 ms [7]. Activation and deactivation of receptors is subsumed together with downstream signaling in the generic signaling system (2.6) whose input s(t) is the stochastic flux of binding events of individual chemoattractant molecules. Let us discuss first a homogeneous concentration field of chemoattractant c(x) = c 0 . We consider the case where the binding events of individual chemoattractant molecules to the receptors are independent and the series of binding events is therefore a Poissonian spike train s(t) = j δ(t − t j ) with expectation value equal to the rate q 0 = λc 0 of binding events. Here, t j denotes the time of the jth binding event.
In an inhomogeneous concentration field, the local chemoattractant concentration c(r(t)) changes with time; thus q(t) changes as well. We describe the statistics of the event times t j by a non-homogeneous Poisson process, i.e. by a renewal process with conditional probability density This choice gives s(t) = q(t) and a Poissonian statistics of binding times for slowly varying q(t).
The generic signaling module (2.6) serves as a low-pass filter which averages the incoming stimulus over the relaxation time σ of the signaling module (2.6). Thus the relevant timescale to which the rate q of binding events should be compared is the relaxation time σ . When the rate q is large compared to σ −1 and q(t) changes on a timescale slow compared to the mean 6 inter-event-interval 1/q, then we can replace s(t) by a coarse-grained version known as the diffusion limit . In this limit, the parameter η = (qσ ) −1/2 which characterizes the relative noise strength of s(t) is small. Here, the relative noise strength of s(t) averaged over a time τ is defined as with δs = s − s , where · τ denotes an average over a time interval τ . Hence, η 2 equals the relative noise strength of s for an averaging time τ = σ .

Intrinsic noise of chemotactic signaling.
Chemotactic signaling of sperm cells refers to the chain of events within the sperm's flagellum relating the activation of chemoattractant receptors to a motor response which changes the swimming path of the sperm cell. Early steps of the signaling cascade are likely to involve only low numbers of the messenger molecules cGMP [11]. Thus we expect, that the chemotactic signaling system is a source of fluctuations as well. We describe these fluctuations effectively by a noise term ξ a with power spectrumS a (ω).

Fluctuations of flagellar propulsion.
Sperm cells are propelled in a liquid by the periodic beat of their flagellum, which is driven by an ensemble of dynein motors. We expect fluctuations in the propulsion generated by the flagellar beat due to noise in the activity of the motor ensemble. This noise causes fluctuations ξ m , in the curvature of the swimming path r(t), which we characterize by the power spectrumS m (ω) of ξ m . In general, fluctuations in the activity of the motor ensemble will cause also fluctuations of the swimming speed v(t). For simplicity, we neglect speed fluctuations here. Including speed fluctuations in our description is straightforward and does not change results significantly.

Hydrodynamic flows.
In the open sea, complex hydrodynamic flows convect the sperm cell and perturb the concentration field of chemoattractant. Here, we do not take these effects into account. Equations (2.1)-(2.6) describe stochastic sperm swimming paths in a concentration field of chemoattractant. The equations can be solved numerically; figure 1(B) shows an example of a stochastic swimming path r(t) in a linear concentration field of chemoattractant.

Statistical properties of sperm swimming paths
We first discuss in section 3.1 statistical properties of sperm swimming paths for circular swimming with fluctuating curvature. We relate the effective diffusion coefficient of sperm swimming circles to the fluctuation spectrum of the curvature. We then discuss diffusion and drift of swimming paths in a weak concentration gradient. We focus on two limit cases: (i) low chemoattractant concentration (section 3.2), and (ii) high chemoattractant concentration (section 3.3). We find analytical expressions for the effective diffusion coefficient for Integration time was t end = 100 s. The initial conditions were a(0) = 1, p(0) = 1/(s b + λc 0 ), r(0) = −r 0 ie iϕ 0 , and ϕ 0 ∈ [0, 2π) drawn randomly from a uniform distribution. For the noise term ξ a , Gaussian white noise with ξ a (t 1 )ξ a (t 2 ) =S a δ(t 1 − t 2 ) was chosen. (C) Endpoints R(t end ) of centerlines of swimming paths in a linear concentration field for n = 10 4 different realizations (black dots). All centerlines (not shown) start at the origin. The same parameters as for panel (B) were used. Also shown is the contour plot of a histogram of the centerline endpoints (red contour lines). The contour lines enclose regions which contain 10, 50 and 90% of the endpoints, respectively. The histogram was computed from n = 10 6 realizations. The expectation value R(t end ) is indicated by a red dot. (D) Contour plot of the spatial distribution function of the endpointsR(t end ) of the coarse-grained centerlineR(t) determined by the effective equation of motion (4.1) (black contour lines). As in panel (C), the contour lines enclose regions, which contain 10, 50 and 90% of the endpoints, respectively. The distribution function ofR(t end ) was obtained by numerically integrating the Fokker-Planck equation associated with (4.1). The expectation value R (t end ) is indicated by a black dot. For comparison, the contour plot from panel (C) is shown again (red contour lines).

Circular swimming with fluctuating curvature
As an introductory example, we study planar sperm swimming paths r(t) in the absence of a concentration gradient of chemoattractant: sperm cells then swim along circles of radius r 0 with a curvature κ(t) that fluctuates around its mean κ 0 = 1/r 0 κ(t) = κ 0 + ξ κ (t).
(3.1) Here, ξ κ is a stationary stochastic process with mean zero and power spectrumS κ (ω) which represents the curvature fluctuations.
These curvature fluctuations result in an effective diffusion of the center R(t) of the swimming circle. In the following, we characterize the diffusion of R(t) in the limit of weak curvature fluctuations with the limit parameter ε small, where ε 2 = r 2 0 ξ 2 κ . We require the higher moments of ξ κ to be of the order of at least ε 3 , i.e. r n 0 ξ n κ O(ε 3 ), n 3. We introduce the moving center R(t) of the swimming path r(t), see figure 1(A) R = r + r 0 n = r + r 0 i t.
(3.2) In the noise-free case, ε = 0, this definition yields the centerpoint R of the perfect circle r(t). By differentiation, we obtain the following stochastic differential equations for R(t) and ϕ(t) Note that the second equation contains multiplicative noise. These equations also hold for the case where ξ κ is Gaussian white noise provided the Stratonovich interpretation is used. We assume that the argument ϕ 0 of the tangent t(0) = e iϕ 0 at time t = 0 is uniformly distributed in [0, 2π). Without loss of generality, we choose R(0) = 0. Then the rotational symmetry of the problem implies The expectation value · averages over an ensemble of stochastic paths with R(0) = 0 and random ϕ 0 (uniformly distributed). In the following, we compute the variance matrix C kl = R k R l where k, l = 1, 2 and R = R 1 + iR 2 . The trace of the variance matrix C kk = |R| 2 is computed as a double integral Here, the autocorrelation function S κ (t 1 − t 2 ) = ξ κ (t 1 )ξ κ (t 2 ) is the Fourier transform ofS κ (ω). For times t much larger than the correlation time of ξ κ , we can approximate |R| 2 as Note that the variance |R| 2 grows linearly in time as is typical for a diffusion process. The complex number R 2 characterizes an anisotropy of the diffusion process To leading order in ε, for a uniform distribution of ϕ 0 . Indeed, the rotational symmetry of the problem already implies R(t) 2 = 0. We conclude, that curvature fluctuations result in effective isotropic diffusion of the center R(t) of the swimming circles with an effective diffusion constant D that is proportional to the power spectrum of the curvature fluctuations evaluated at the circle frequency The approximation (3.9) is valid up to terms of order O(ε 3 ). Note that the diffusion coefficient is proportional to the power spectrumS κ of the curvature fluctuations evaluated at the frequency ω 0 , see [15] for a related result in a different context. For the particular model given in (2.1)-(2.6), the power spectrumS κ (ω) has contributions from (i) the stochastic stimulus, (ii) the intrinsic fluctuations of the chemotactic signaling system and (iii) from the fluctuations in the activity of the motor ensemble. Below we give an approximation forS κ in the limit of weak noise, see equation (3.25). In general,S κ and D are functions of the concentration c 0 of a homogeneous concentration field of chemoattractant. Figure 2 displays the concentration dependence of D; a pronounced maximum of D can be observed close to c * = s b /λ. Using the same reasoning as above, one can derive an exact expression for the diffusion constant D Here,C κ (ω) is the Fourier transform of the characteristic function C κ (t) = exp iv 0 κ (t) of κ (t) = t 0 dτ ξ κ (τ ). Expanding (3.10) in powers of ε reproduces (3.9) as leading order contribution. The special case where the curvature fluctuations ξ κ are described by Gaussian white noise with ξ κ (t 1 ) ξ κ (t 2 ) = 2D κ δ(t 1 − t 2 ), has been discussed in [16] and gives C κ (t) = exp(−v 2 0 D κ |t|) and 4D = 2D κ v 4 0 /(ω 2 0 + D 2 κ v 4 0 ).

Chemoattractant gradient with low concentration
There are strong indications, that sperm cells can detect single chemoattractant molecules, when chemoattractant concentration is subpicomolar [4,7]. To characterize sperm chemotaxis at very low chemoattractant concentrations, we study the set of equations (2.1)-(2.6) in the limit of a low chemoattractant concentration with a chemoattractant molecule binding rate q small compared with σ −1 , where σ is the relaxation timescale of the signaling system in (2.6). For simplicity, we neglect the intrinsic noise of chemotactic signaling and the fluctuations in the activity of the motor ensemble, i.e. ξ a = 0 and ξ m = 0.

Statistics of sperm swimming paths.
We study the stochastic equations of motion (2.1)-(2.6) of sperm swimming paths for a linear concentration field of chemoattractant, where we assume for simplicity that the gradient is parallel to the x-axis. We consider the limit of (i) a shallow concentration gradient with ν = c 1 r 0 /c 0 1, and (ii) a binding rate q(t) of chemoattractant molecules low compared with σ −1 . Between two binding events, t j t < t j+1 , the swimming path is circular once the jth swimming response relaxes r(t) = R(t) − ir 0 e iω 0 t+iϕ 0 . This results in a periodic modulation of the binding rate q(t) = λ c(r(t)) = λ c(R(t)) + λ c 1 r 0 sin(ω 0 t + ϕ 0 ).

(3.16)
For q σ 1, we can treat individual binding events independently and find for the center R of the circular swimming path r for t > 0 (3.17) Here, the t j denotes the times of the individual binding events and ϕ j = ϕ(t j ). From the periodic modulation of the binding rate (3.16), we determine the probability density P(ϕ) of the angles ϕ j 2π P(ϕ) = 1 + ν sin ϕ + O(ν 2 ). |δR(t)| 2 = |u| 2 λc 0 t + O(ν 2 ). with δR = R − R . Thus, the circle center R is both (i) drifting with drift speed v d = ρ κ λc 1 r 2 0 v 0 /2 in a direction which encloses an angle α = 3π/2 − ϕ κ with the concentration gradient and (ii) diffusing in both space dimensions with an effective diffusion coefficient D = |δR| 2 /(4t) = (r 0 v 0 ) 2 ρ 2 κ λc 0 /4 + O(ν 2 ). The average drift of the circle center given by (3.19) is the same as the one obtained for a deterministic description of sperm chemotaxis [12]. Thus the deterministic approach is providing a mean-field description. Diffusion is essentially isotropic; the quantity δR 2 = δ R 2 1 − δ R 2 2 + 2i δ R 1 δ R 2 , which characterizes anisotropy of diffusion vanishes to linear order in ν, δR 2 = 0 + O(ν 2 ). The relative strength of the chemotactic drift compared to the undirected diffusive motion of the center of the swimming circle is characterized by a dimensionless Peclet number This expression for the Peclet number only holds if intrinsic fluctuations of chemotactic signaling and noise in the activity of the flagellar motor ensemble can be neglected. In the next section, we derive an expression for the effective diffusion coefficient D where these noise sources have been included, see equations (3.25) and (3.26). With these noise sources, we find that the Peclet number is proportional to the relative concentration gradient |∇c|/c for high concentrations c.

Chemoattractant gradient with high concentration
To gain further insight into the stochastic description of sperm chemotaxis, we study the set of equations (2.1)-(2.6) in the weak noise limit for a linear concentration field of chemoattractant c(x + iy) = c 0 + c 1 x. More precisely, we consider the limit of (i) a shallow concentration gradient with ν = c 1 r 0 /c 0 1, and (ii) a high chemoattractant concentration c 0 with a binding rate q(t) = λc(r(t)) of chemoattractant molecules much higher than σ −1 , where σ sets the 12 relaxation timescale of the chemotactic signaling system in (2.6). In this limit, the parameter η = (qσ ) −1/2 is small; η characterizes the relative noise strength of s(t). For simplicity, we first neglect the intrinsic noise of chemotactic signaling and the fluctuations in the activity of the motor ensemble, i.e. ξ a = 0 and ξ m = 0. The general case will be studied in equation (3.25).
We develop a systematic perturbation theory for the statistics of centerlines R(t) of stochastic swimming paths for the small limit parameters ν and η. In a systematic expansion of R , |δR| 2 and δR 2 in powers of ν and η, many terms vanish due to the symmetries of the problem; to leading order we find . The expansion of R in powers of ν can only contain odd powers of ν since reversing the direction of the gradient changes the sign of both ν and R . Likewise the expansions of both |R| 2 and R 2 contain only even powers of ν. Terms proportional to η yield zero in the noiseaverage. Since δR 2 measures anisotropy, this expression must vanish in the case ν = 0 when the rotational symmetry is not broken. For the leading order coefficients of R and |R| 2 we recover essentially the results obtained for the case of low concentration So far, we have neglected the intrinsic noise of chemotactic signaling and the fluctuations in the activity of the motor ensemble. Now we include these noise terms and consider the limit of weak noise with η 1, ξ 2 a η 2 and r 2 0 ξ 2 m η 2 . We also assume that the higher order moments of ξ a and ξ m are of the order of at least O(η 3 ). In this limit,S κ (ω) is a sum of contributions from the different noise sources where ρ κ =ρ κ /(s b + λc 0 ) is the amplitude gain of the chemotactic signaling system (2.6). The second summand of the right-hand side of equation (3.25) describes a contribution tõ S κ stemming from the intrinsic noise of chemotactic signaling; whereas the third summand describes a contribution due to fluctuations in the activity of the motor ensemble. We define the concentration independent part ofS κ (ω 0 ) asS κ (ω 0 ) (0) =ρ 2 κS a (ω 0 ) +S m (ω 0 ). We can thus writẽ S κ (ω 0 ) as the sum of a concentration independent partS κ (ω 0 ) (0) and a concentration-dependent partS κ (ω 0 ) (c) = ρ 2 κ λc 0 . From ρ κ =ρ κ /(s b + λc 0 ), we see thatS κ (ω 0 ) (c) scales like c 0 /(s b + λc 0 ) 2 . As a consequence,S κ (ω 0 ) is a non-monotonic function of concentration c 0 with a maximum at c * = s b /λ. Hence, also the effective diffusion constant

Effective equation for the centerline
We derive an effective equation of motion for a coarse-grained centerlineR(t) in a linear concentration field (3.14). We restrict ourselves to the limit case of weak noise with η 1 as specified in the previous section 3. 3. Additionally, we assume that the concentration gradient is shallow with ν = c 1 r 0 /c 0 1. On short timescales t < T = 2π/ω 0 , the centerline R(t) is a stochastic trajectory that depends on the particular shape of the spectraS a andS m and therefore has non-generic properties. For longer time intervals t = nT , however, we find simple, general expressions for the drift R( t) − R(0) and for the variance |R( t)| 2 , see equations (3.22)-(3.23).
In order to derive a genuine coarse-grained description, we introduce a discrete centerlinê R(t k ), which is defined at the discrete times t k = k t, k = 0, 1, 2, . . ., and which reproduces (to leading order) the second-order statistics of R(t) at the discrete times t k , see appendix A for details. We then perform a continuum limit to find a continuous-time coarse-grained centerlinê R(t) which obeys d dtR (t) = v d e iα ∇c/|∇c| + ξ . Here ξ = ξ 1 + i ξ 2 is a complex random variable whose components ξ j are uncorrelated Gaussian random variables with ξ k (t 1 )ξ l (t 2 ) = 2D δ(t 1 − t 2 ) δ kl for k, l = 1, 2. The concentration gradient ∇c is the complex number ∇c(x + iy) = ∂ x c(x + iy) + i ∂ y c(x + iy). The time-evolution of the coarse-grained centerlineR described by equation (4.1) is the superposition of (i) a deterministic drift with drift velocity v d in a direction that encloses an angle of α = 3 2 π − ϕ κ with the concentration gradient, and (ii) pure diffusion with an effective diffusion coefficient of D. Since the diffusion coefficient D depends on concentration, the noise term in equation (4.1) is effectively multiplicative. Equation (4.1) is defined by using the Itō interpretation; see [17,18] for a discussion of Itō and Stratonovich stochastic differential equations. Note that the Itō and the Stratonovich version of (4.1) differ by a noise-induced drift term which is of the order of ∼ νη 2 . We can generalize (4.1) for a nonlinear concentration field c(x), provided the nonlinearities of the concentration field are small on the length scale r 0 , i.e. |∇ 2 c| |∇c|/r 0 . Figure 1(D) compares (i) the distribution of endpoints R(t end ) of centerlines R(t) of stochastic swimming paths r(t) obtained from the full dynamic equations of stochastic sperm motion (2.1)-(2.6), and (ii) the distribution of endpointsR(t end ) of coarse-grained centerlineŝ R(t) obtained from the effective equation of motion (4.1). We find good agreement between both endpoint distributions. In particular, both endpoint distributions are anisotropic. This is remarkable, since the distribution of endpointsR(t end ) of coarse-grained centerlines is obtained from simulating equation (4.1), which has an isotropic noise term. The apparent anisotropy is a consequence of the position dependence of the drift speed v d (R). Note that in the special case α = 0, the stochastic equation of motion (4.1) of the coarse-grained centerlineR describes the trajectory of a particle diffusing in an effective potential E = ∇ ln(s b + λc) which satisfies −∇ E(R) = v d ∇c/|∇c|. Here =ρ κ r 2 0 v 0 /2 is a constant.

Example: radial concentration field
We discuss the case of a radial concentration field c(x) = C(|x|), which arises if chemoattractant diffuses away from a single source such as an egg. The chemotactic drift speed v d and the effective diffusion coefficient D of sperm swimming circles depend on the concentration c, see equations (3.24) and (3.26). In a radial concentration field, these quantities become functions v d = v d (R) and D = D(R) of the distance R = |R| from the source. We introduce polar coordinates (R, ψ) such that R = R e iψ . From (4.1) and the rules of stochastic calculus [17], we find a stochastic differential equation for R and ψ in the Itō interpretation, see appendix Ḃ where ξ 1 and ξ 2 are uncorrelated Gaussian white noise terms with zero mean and ξ k (t 1 ) ξ l (t 2 ) = 2D δ(t 1 − t 2 ) δ kl . The expression forṘ in equation (5.1) contains a noise-induced drift term D/R, which results from the nonlinearity of the coordinate transformation R = R e iψ → (R, ψ). We introduce the probability density S(R, t) such that S dA is the probability to find a sperm cell in the area element dA at time t. The total number of sperm cells in the system equals dA S(R, t). It is useful to introduce also the radial probability density P(R, t) = R 2π 0 dψ S(Re iψ , t). Note that P dR is the probability to find a sperm cell in a thin annulus of radius R and width dR. In our effective equation of motion (5.1), the dynamics of R decouples from ψ reflecting the symmetries of the problem. Therefore, the radial probability P(R, t) obeys a Fokker-Planck equation which reads [19] ∂ ∂t P = L P (5.2) with a Fokker-Planck operator L given by For completeness, we mention that the equation for the probability density S(R, t) = S(Re iψ , t) has the form R ∂ ∂t S = L(R S) + ∂ ∂ψ −v d cos ϕ κ RS + ∂ ∂ψ DS . We are interested in the rate at which sperm cells reach the egg and study in the following an idealized scenario, where (i) we neglect the concentration dependence of the effective diffusion coefficient D(R) = D 0 , (ii) the sensitivity threshold of chemotactic signaling vanishes, s b = 0, and (iii) the radial chemoattractant concentration field decays as a power-law c(x) ∼ 1/|x| β with exponent β. Note that the steady-state concentration field established by diffusion in three-dimensional space from a spherical chemoattractant source with constant release rate of chemoattractant corresponds to β = 1. For adaptive chemotactic signaling (without sensitivity threshold), the relevant input quantity is the relative concentration gradient |∇c|/c, which for our choice of concentration field scales inversely with distance as |∇c|/c = β/R. Hence, the drift speed in the radial direction reads v d sin ϕ κ = − 0 /R where 0 = −βρ κ r 2 0 v 0 sin ϕ κ 15 is a constant. Likewise, the drift speed in the azimuthal direction is given by v d cos ϕ κ = − 0 cot ϕ κ /R. To mimic the egg, we consider a disk-shaped target of radius R target located at the origin of the concentration field which absorbs any sperm cell hitting its boundary. The absorbing target gives rise to a boundary condition P(R target ) = 0. We study the steady-state solution of equation (5.2) subject to this boundary condition and the condition P 0 (R) ≈ 2π R S ∞ for large R where S ∞ is the sperm concentration far from the target. For positive chemotaxis with 0 > 0, the stationary probability density reads S 0 (R) = P 0 (R)/(2π R) In steady-state, sperm cells arrive at the target with a constant rate k 0 = 2π S ∞ 0 . The stationary state is characterized by a net flux in the azimuthal direction J ψ = v d cos ϕ κ S 0 (R), i.e. k ψ = J ψ dR is the rate by which sperm cells pass through a section of length dR on a ray emanating from the origin. Thus there is a net circular current whirling around the origin. Such an azimuthal flux has been already discussed previously [20]. Note that if chemotaxis is absent or negative with 0 0, then there exists no steady-state probability distribution S 0 (R) for fixed concentration S ∞ at infinity. If we start in this case with a homogenous distribution of sperm cells S(R, 0) = S ∞ at time t = 0, a growing region around the target becomes depleted of sperm cells and the time-dependent rate k(t) of sperm cells arriving at the target vanishes in the limit of long times. We thus find that the system undergoes a dynamic second-order phase transition as a function of chemotactic strength 0 where k 0 = lim t→∞ k(t) plays the role of an order parameter.

Discussion
In most experimental studies of sperm swimming in a chemoattractant field, sperm cells become localized near a boundary surface and swim in a plane [1], [4]- [11]. Observed swimming paths exhibit a large variability. Here, we presented a description of sperm chemotaxis in a plane which takes into account sources of fluctuations such as (i) a finite precision of the cell's measurement of the local chemoattractant concentration, (ii) intrinsic noise of chemotactic signaling, and (iii) fluctuations in the activity of the propulsion mechanism. In the absence of such fluctuations, sperm swimming paths are trochoids, or drifting circles [12]. In the presence of fluctuations, the paths can be described as noisy trochoids, resulting from the superposition of a chemotactic drift and unbiased random motion. We find expressions for the chemotactic drift velocity and the effective diffusion coefficient in the limit of a weak concentration gradient. Note that the assumption of a weak concentration gradient is satisfied in most experimental conditions, except perhaps in the close vicinity of a chemoattractant source such as an egg or a micropipette releasing chemoattractant. We can estimate parameters relevant for our stochastic description of sperm chemotaxis. In experiments on sea urchin sperm, typical values observed for the swimming speed and radius of swimming circles are v 0 ≈ 100-200 µm s −1 , r 0 ≈ 20-40 µm, respectively [1,21]. For comparison, the radius of the egg is R egg ≈ 100 µm [22]. From the drift speed v d ≈ 25-50 µm s −1 of swimming circles in a concentration gradient with |∇c|/c ∼ 1/100 µm reported in [1], we can estimate the amplification strength of the chemotactic signaling system as ρ κ ≈ 2v d /(r 2 0 v 0 |∇c|/c) ≈ 0.05 µm −1 . The drift angle α was found to peak around α ≈ 0 [1,10].
From stopped-flow experiments, it is known that chemotactic signaling operates on timescales 0.1-0.5 s [5], we thus expect the intrinsic relaxation times of the chemotactic signaling system (2.6) σ , µ to fall in that range. Sperm cells probably respond to single chemoattractant molecules [4,7], we therefore expect s b 1/σ . Chemoattractant molecules bind to receptors on the sperm flagellum at a rate q(t), proportional to concentration q(t) = λc(r(t)). where λ ≈ 20 pM −1 s has been estimated [1]. In an experiment with sea urchin sperm in a shallow observation chamber in the absence of chemoattractant, sperm cells were observed to swim along circular paths r(t) with radius r 0 = 40 µm and a swimming speed v 0 = 100 µm s −1 .
For long time, the center R(t) of the swimming circle diffused with an effective diffusion constant D = 9 ± 2 µm 2 s −1 [23]. Using equation (3.26), we find that the swimming noise stemming from the intrinsic fluctuations of chemotactic signaling and the fluctuations in the activity of the flagellar motor ensemble is characterized byS (0) In a homogeneous concentration field of chemoattractant, our theory predicts a diffusion coefficient of the center of the swimming circle which depends on concentration in a nonmonotonic way with a maximum at a finite concentration c * = s b /λ, see figure 2. This concentration dependence of the diffusion constant results from the fact that fluctuations of the chemotactic stimulus elicit a stochastic chemotactic response and contribute to the effective diffusion constant. For low concentrations c 0 c * , the chemotactic stimulus and its fluctuations are small, producing only a negligible contribution to the effective diffusion constant. For high concentrations c 0 c * , the chemotactic signaling system adapts its sensitivity to an averaged stimulus and thus generates an output with small fluctuations. The concentration dependence of the effective diffusion constant of sperm swimming paths could be tested in future experiments. With the parameter estimates given above, we find a value for c * in the subpicomolar range, c * = 0.1-1 pM. The motion of sperm cells in a concentration gradient is characterized by a Peclet number which compares drift speed and the effective diffusion coefficient on the scale sperm swimming circles. For fixed relative strength |∇c|/c of the concentration gradient, this Peclet number increases with concentration c and saturates for high concentrations c c * . For a concentration gradient with |∇c|/c = 1/100 µm as employed in [1], we estimate Pe ≈ 1 at c ≈ c * and Pe ≈ 10-100 for c c * .
In the absence of fluctuations, chemotaxis of sea urchin sperm cells relies on a robust mechanism which depends only on a few generic properties of the signaling system and does not require fine-tuning of parameters [12]: the swimming circles of sperm cells drift toward regions of higher chemoattractant concentration whenever −π < ϕ κ < 0, where ϕ κ is the phaseshift between a periodic stimulus and the output generated by the chemotactic signaling system. The equation (2.6) used in this work to describe the signaling system differs slightly from the one employed in [12]. In particular, the timescale of adaptation is independent of the stimulus level for the signaling system used here. It has been shown in stopped-flow experiments that the timescales of the chemotactic response change only marginally when stimulus strength is varied from picomolar to micromolar concentrations [1]. Additionally, the phase-shift ϕ κ does not depend on the stimulus level for the modified signaling system (2.6). We expect that for the chemotactic signaling system of sea urchin sperm this phase-shift is largely independent of chemoattractant concentration to ensure successful chemotaxis at different concentrations.
The chemotactic strategy of sea urchin sperm cells differs fundamentally from the mechanism employed by bacteria such as Escherichia coli [24,25]. These bacteria perform a biased random walk switching stochastically between a running mode with net motion and a tumbling mode during which the cell is randomly reoriented. Chemotaxis in a concentration gradient of chemoattractant is achieved by adjusting the rate of tumbling events in a way that depends on the history of the chemotactic stimulus. Thus bacterial chemotaxis is intrinsically stochastic and cannot be understood without considering noise. The mechanism of sperm chemotaxis, on the other hand, does not require noise. As we show here, this mechanism nevertheless also shows a high degree of robustness in the presence of fluctuations.
So far, we have discussed planar swimming paths of sperm cells close to a surface. Far from boundary surfaces, sea urchin sperm cells swim along helical paths [26,27] as a consequence of the chirality of their flagellar beat [7,28,29]. When swimming in a concentration gradient of chemoattractant, the helical swimming path tends to align with the gradient direction [26,30]. Theory predicts that the rate of alignment depends on the strength of the concentration gradient. The speed of net displacement along the helix centerline, however, is almost unaffected by the concentration gradient [12]. Hence, a sperm cell swimming along a helical path will directly head for a target releasing chemoattractant once the helix axis has aligned with the concentration gradient. This behavior is quite different from the case of planar chemotaxis along a circular path where the drift speed of sperm swimming circles toward a target is always proportional to the strength of the gradient. Thus sperm chemotaxis in three-dimensional space along helical paths is more efficient than along circular paths in a plane close to a boundary surface. In a forthcoming publication, we will extend the stochastic description of sperm chemotaxis presented here for planar swimming paths to the case of helical swimming paths in threedimensional space.
In summary, we conclude that the circular and helical swimming paths of sea urchin sperm cells provide an effective means of sampling the local chemoattractant concentration field and detecting a concentration gradient. Whether noise can enhance sperm-egg encounter rates remains an interesting but nontrivial question [31,32].