Abstract
Spermatozoa of marine invertebrates are attracted to their conspecific female gamete by diffusive molecules, called chemoattractants, released from the egg investments in a process known as chemotaxis. The information from the egg chemoattractant concentration field is decoded into intracellular Ca^{2+} concentration ([Ca^{2+}]_{i}) changes that regulate the internal motors that shape the flagellum as it beats. By studying sea urchin speciesspecific differences in sperm chemoattractantreceptor characteristics we show that receptor density constrains the steepness of the chemoattractant concentration gradient detectable by spermatozoa. Through analyzing different chemoattractant gradient forms, we demonstrate for the first time that Strongylocentrotus purpuratus sperm are chemotactic and this response is consistent with frequency entrainment of two coupled physiological oscillators: i) the stimulus function and ii) the [Ca^{2+}]_{i} changes. We demonstrate that the slope of the chemoattractant gradients provides the coupling force between both oscillators, arising as a fundamental requirement for sperm chemotaxis.
Introduction
Broadcast spawning organisms, such as marine invertebrates, release their gametes into open water, where they are often subject to extensive dilution that reduces the probability of gamete encounter (Lotterhos et al., 2010). In many marine organisms, female gametes release diffusible molecules that attract homologous spermatozoa (Lillie, 1913; Miller, 1985; Suzuki, 1995), which detect and respond to chemoattractant concentration gradients by swimming toward the gradient source: the egg. Although it was in bracken ferns where sperm chemotaxis was first identified (Pfeffer, 1884), sea urchins are currently the bestcharacterized model system for studying sperm chemotaxis at a molecular level (Alvarez et al., 2012; Cook et al., 1994; Darszon et al., 2008; Strünker et al., 2015; Wood et al., 2015).
The sea urchin egg is surrounded by an extracellular matrix which contains spermactivating peptides (SAPs) that modulate sperm motility through altering intracellular Ca^{2+} concentration ([Ca^{2+}]_{i}) and other signaling intermediates (Darszon et al., 2008; Suzuki, 1995). The biochemical signals triggered by SAPs guide the sperm trajectory toward the egg.
The decapeptide speract is one of best characterized members of the SAP family due to its powerful stimulating effect on metabolism, permeability and motility in Strongylocentrotus purpuratus and Lytechinus pictus spermatozoa. The binding of speract to its receptor, located in the flagellar plasma membrane, triggers a train of [Ca^{2+}]_{i} increases in immobilized spermatozoa of both species (Wood et al., 2003). This calcium signal was proposed to regulate the activity of dynein motor proteins in the flagellum, and thus potentially modulate the trajectory of freeswimming spermatozoa (Brokaw, 1979; Mizuno et al., 2017).
A direct link between [Ca^{2+}]_{i} signaling and sperm motility was established through the use of optochemical techniques to rapidly, and nonturbulently, expose swimming sea urchin spermatozoa to their conspecific attractant in a wellcontrolled experimental regime (Böhmer et al., 2005; Wood et al., 2005). Currently, it is well established that the transient [Ca^{2+}]_{i} increases triggered by chemoattractants produce a sequence of turns and straight swimming episodes (the ‘turnandrun’ response), where each turning event results from a rapid increase in the [Ca^{2+}]_{i} (Alvarez et al., 2012; Böhmer et al., 2005; Shiba et al., 2008; Wood et al., 2005). The turnandrun response seems to be a general requirement for sperm chemotaxis in sea urchins, however it is not sufficient on its own to produce a chemotactic response (Guerrero et al., 2010a; Strünker et al., 2015; Wood et al., 2007; Wood et al., 2005).
In spite of 30 years of research since speract’s isolation from S. purpuratus oocytes (Hansbrough and Garbers, 1981; Suzuki, 1995), chemotaxis of S. purpuratus sperm in the presence of this peptide has not yet been demonstrated (Cook et al., 1994; Darszon et al., 2008; Guerrero et al., 2010a; Kaupp, 2012; Miller, 1985; Wood et al., 2015). A comparison between individual L. pictus and S. purpuratus sperm responses to a specific chemoattractant concentration gradient generated by photoactivating caged speract (CS) revealed that only L. pictus spermatozoa exhibit chemotaxis under these conditions (Guerrero et al., 2010a). In that study, L. pictus spermatozoa experience [Ca^{2+}]_{i} fluctuations and pronounced turns while swimming in descending speract gradients, that result in spermatozoa reorienting their swimming behavior along the positive chemoattractant concentration gradient. In contrast, S. purpuratus spermatozoa experience similar trains of [Ca^{2+}]_{i} fluctuations that in turn drive them to relocate, but with no preference toward the center of the chemoattractant gradient (Guerrero et al., 2010a).
In the present work, we investigate boundaries that limit sperm chemotaxis of marine invertebrates. Particularly, we examined whether the chemoattractant concentration gradient must have a minimum steepness to provoke an adequate, chemotactic sperm motility response. Previous studies of chemotactic amoebas crawling up a gradient of cAMP, have shown that the slope of the chemical concentration gradient works as a determinant factor in chemotaxis of this species, where the signaltonoise relationship of stimulus to the gradient detection mechanism imposes a limit for chemotaxis (Amselem et al., 2012). In addition, recent theoretical studies by Kromer and colleagues have shown that, in marine invertebrates, sperm chemotaxis operates efficiently within a boundary defined by the signaltonoise ratio of detecting ligands within a chemoattractant concentration gradient (Kromer et al., 2018).
If certain, this detection limit may have prevented the observation and characterization of chemotactic responses on S. purpuratus spermatozoa to date. In this study, we identify the boundaries for detecting chemotactic signals of S. purpuratus spermatozoa, and show that sperm chemotaxis arises only when sperm are exposed to much steeper speract concentration gradients than those previously employed by Guerrero et al. (2010a). Furthermore, we examined the coupling between the recruitment of speract molecules during sperm swimming (i.e. stimulus function) and the internal Ca^{2+} oscillator, and demonstrate that sperm chemotaxis arises through coupling of these physiological oscillators.
Results
Speciesspecific differences in chemoattractantreceptor binding rates: chemoattractant sensing is limited by receptor density in S. purpuratus spermatozoa
Spermatozoa measure the concentration and the changes in concentration of eggreleased chemoattractant during their journey. Cells detect chemoattractant molecules in the extracellular media by integrating chemoattractantreceptor binding events. A spermatozoon moving in a medium where the chemoattractant concentration is isotropic will collect stochastic chemoattractantreceptor binding events with a rate $J$, according to Equation (1) (Figure 1).
Where $D$ is the diffusion coefficient of the chemoattractant, $a$ is the radius of the cell, $\stackrel{}{c}$ is the mean chemoattractant concentration, N is the number of receptor molecules on the cell surface, s is the effective radius of the chemoattractant molecule, ${J}_{max}$ is the maximal flux that the cell can experience, and $\frac{N}{N+\pi a/s}$ is the probability that a molecule that has collided with the cell will find a receptor (Berg and Purcell, 1977). The quantity πa/s is the number of receptors that allows half maximal binding rate for any concentration of chemoattractant, which is hereafter denoted as N_{1/2} (see 1.1. On the estimate of maximal chemoattractant absorption in Appendix 1).
The expression above was used by Berg and Purcell (1977) to conclude that the chemoattractant binding and absorption rate saturate as a function of the density of receptors, becoming diffusion limited, that is when $N\gg {N}_{1/2}=\mathrm{\pi}a/s\mathrm{}$ the chemoattractant absorption flux becomes $J\cong {J}_{max}$ (see 1.1. On the estimate of maximal chemoattractant absorption in Appendix 1). If the density of the chemoattractant receptor is such that spermatozoa of the different species operate under this saturated or perfect absorber regime, then any postulated speciesspecific differences would have to be downstream.
In Supplementary file 1 we list the biophysical parameters considered for calculating the speciesspecific rate of binding as a function of the chemoattractant concentration. The different functions of the receptor density and the species receptor density are depicted in Figure 1—figure supplement 1. Our calculations (see Appendix 1, section 1.1. On the estimate of maximal chemoattractant absorption) indicate that only S. purpuratus spermatozoa operate in a regime for which the rate of chemoattractant uptake is limited by receptor density, therefore it cannot be considered as a perfect absorber. The actual number of speract receptors for this species is approximately 2×10^{4} per sperm cell which is fewer than the estimate of N_{1/2} ~ 3×10^{4} (Supplementary file 1). In contrast, L. pictus and A. punctulata spermatozoa seem to approximate toward operating as perfect absorbers (Figure 1—figure supplement 1 and Supplementary file 1). Both observations hold when considering the cylindrical geometry of the sperm flagellum. A low number of (noninteracting) receptors, sparsely covering the flagellum (i.e. with a large distance between receptors compared to receptor size) entails a nonsaturated diffusive flux that, hence, depends on the number of receptors. The cylindrical geometry of the flagellum strengthens the observation that the larger surface area of the cylinder gives a longer average distance between receptors and, hence, offsetting the saturation of the overall diffusive flux to higher receptor number (see section 1.1. On the estimate of maximal chemoattractant absorption in Appendix 1).
In conclusion, there are meaningful speciesspecific differences in chemoattractant receptor density which could by themselves explain differences in chemotactic behavior.
Receptor density constrains the chemoattractant concentration gradient detectable by spermatozoa
A functional chemotactic signaling system must remain unresponsive while the cell swims through an isotropic chemoattractant concentration field and must trigger a directional motility response if the cell moves across a concentration gradient (Figure 1a–c). This absolute prerequisite of the signaling system defines the minimal quantitative constraints for reliable detection of a gradient and therefore for chemotaxis.
A cell moving along a circular trajectory in an isotropic chemoattractant field (Figure 1a) will collect a random number of chemoattractantreceptor binding events during the half revolution time $\mathrm{\Delta}\mathrm{t}$, that has a Poisson distribution with mean J$\mathrm{\Delta}\mathrm{t}$ and standard deviation $\sqrt{J\mathrm{\Delta}\mathrm{t}}$. Because under these conditions there is no spatial positional information to guide the cell, the chemotactic signaling system must be unresponsive to the fluctuations in the number of binding events expected from the Poisson noise.
The chemotactic response should only be triggered when the cell moves into a concentration gradient (Figure 1b and c) sufficiently large to drive binding event fluctuations over the interval $\mathrm{\Delta}\mathrm{t}$ with an amplitude that supersedes that of the background noise. As derived in the Appendix 1, section 1.2. A condition for detecting a change in the chemoattractant concentration, the reliable detection of a chemoattractant gradient requires the following condition dependent on the maximal concentration difference experienced during half a revolution and on the mean chemoattractant concentration $\stackrel{}{c}$:
Noting that the lefthand side of the condition represents the chemotactic signal and the righthand side is a measurement of the background noise, Equation (2) can be rewritten in terms of signaltonoise ratio:
Where $v$ is the mean linear velocity ($\frac{\mathrm{\Delta}r}{\mathrm{\Delta}t}$), where $\mathrm{\Delta}r$ is the sampling distance or diameter of the swimming circle, and $\xi ={\stackrel{}{c}}^{1}\frac{\partial c}{\partial r}$ is the relative slope of the chemoattractant concentration gradient. The quantity $\xi $ measures the strength of the stimulus received when sampling a position r, relative to the mean concentration $\stackrel{}{c}$ (Figure 1c). As $\xi $ increases, the strength of the chemotactic signal increases.
Equation (3) means that the ability to reliably determine the source of the attractant depends critically on the relative slope of the chemoattractant concentration gradient $\xi $, which must be steep enough to be distinguishable from noise (Figure 1b and c, and Supplementary file 1; for further explanation see 1.2. A condition for detecting a change in the chemoattractant concentration in Appendix 1).
We modeled the SNR corresponding to different gradients, and within a range of mean concentrations of chemoattractant between 10^{−11} to 10^{−6} M for three sea urchin species: S. purpuratus, L. pictus and A. punctulata (Figure 1df). For all species studied, at high mean concentrations of chemoattractant (10^{−8} to 10^{−6} M), the change in chemoattractant receptor occupancy experienced at two given distinct positions allows reliable assessment of relatively shallow chemical gradients (ξ ~ [10^{−3},10^{−4}] µm^{−1}), with SNR > 1 for a wide range of $\xi $ (Figure 1df). However, at low concentrations of chemoattractant (below 10^{−8} M), keeping all other parameters equal, stochastic fluctuations begin to mask the signal. In this lowconcentration regime, the steepness of the chemoattractant concentration gradient is determinant for chemoattractant detection. Shallow gradients result in insufficient SNR, while steeper chemoattractant gradients (ξ > 10^{−3} µm^{−1}) are dependably detected by spermatozoa, that is SNR > 1 (Figure 1df).
Previous reports show that A. punctulata spermatozoa are very sensitive to resact (presumably reacting to single molecules) due the high density of resact receptors (~3×10^{5} per cell), which allows them to sense this chemoattractant at low picomolar concentrations (Kashikar et al., 2012). In contrast, L. pictus and S. purpuratus spermatozoa bear lower densities of chemoattractants receptors, approximately 6.3×10^{4} and 2×10^{4} receptors/cell, respectively (Nishigaki et al., 2001; Nishigaki and Darszon, 2000). According to these speciesspecific differences in chemoattractant receptor densities, Figure 1d–f suggests that the spermatozoa of A. punctulata are likely more sensitive to resact, than those of either L. pictus or S. purpuratus species to the same mean concentration gradients of speract; with the spermatozoa of S. purpuratus being less sensitive than those of L. pictus species to equivalent speract gradients and mean concentrations. Moreover, the constraints on SNR imply that S. purpuratus spermatozoa should only respond to the chemoattractants at higher mean speract concentrations and at steeper gradients than those that elicit chemotaxis in L. pictus spermatozoa (compare Figure 1d and e).
To understand the differential sensitivity between the spermatozoa of S. purpuratus and L. pictus we analyzed the scenario in which the capacity to detect the gradient for both spermatozoa species were equal, that is they would have the same signaltonoise ratios, SNR_{purpuratus} = SNR_{pictus}. We compute the ratio of the slopes of the speract concentration gradient experienced by either S. purpuratus or L. pictus spermatozoa, which represents a scaling factor (SF) in the gradient slope, expressed as:
where $Z=\left(\frac{Na}{N+\text{}\pi a/s}\right)$ is the probability that a speract molecule that has collided with the cell will bind to a receptor (Berg and Purcell, 1977), multiplied by the radius $a$ of the cell.
The estimation of the scaling factor SF predicts that S. purpuratus spermatozoa should undergo chemotaxis in a speract gradient three times steeper than the gradient that elicits chemotaxis in L. pictus spermatozoa, with ${\xi}_{purpuratus}~\mathrm{}3{\xi}_{pictus}$.
In summary, the chemoreception model suggests that S. purpuratus spermatozoa detect chemoattractant gradients with lower sensitivity than those of L. pictus. It also predicts that S. purpuratus spermatozoa may detect chemoattractant gradients in the 10^{−9} M regime with sufficient certainty only if the slope of the chemoattractant concentration gradient is greater than 3×10^{−3} µm^{−1} (i.e. steep concentration gradients) (Figure 1d).
If the latter holds true, then S. purpuratus spermatozoa should be able to experience chemotaxis when exposed to steeper speract gradients than those tested experimentally so far. Given this prediction, we designed and implemented an experimental condition for which we expect S. purpuratus spermatozoa to experience chemotaxis. In general terms, this scaling rule for sensing chemoattractant gradients might also apply for other species of marine invertebrates.
S. purpuratus spermatozoa accumulate at steep speract concentration gradients
Our experimental setup is designed to generate determined speract concentration gradients by focusing a brief (200 ms) flash of UV light along an optical fiber, through the objective, and into a field of swimming S. purpuratus spermatozoa containing cagedsperact (CS) at 10 nM in artificial sea water (Guerrero et al., 2010a; Tatsu et al., 2002). To test experimentally whether S. purpuratus spermatozoa undergo chemotaxis, as predicted from the chemoreception model, we varied the slope of the speract concentration gradient by separately employing four optical fibers of distinct diameters, arranged into five different configurations (f1, f2, f3, f4, f5) (Figure 2c).
Each configuration produces a characteristic pattern of UV illumination within the imaging field (Figure 2c). The UV intensity, measured at the back focal plane of the objective for each fiber configuration, is shown in Supplementary file 2. The spatial derivative of the imaged UV light profile was computed as a proxy for the slope of the speract concentration gradient (Figure 2b). By examining these UV irradiation patterns, the highest concentration of speract released through photoliberation from CS is generated by the f5 fiber, followed by f4>f3>f2>f1 (Figure 2a). The steepest UV irradiation gradients are those generated by the f2, f3 and f5 fibers (Figure 2b).
Irrespective of the optical fiber used, the photoactivation of caged speract triggers the stereotypical Ca^{2+}dependent motility responses of S. purpuratus spermatozoa (Figure 2d, Video 1, Appendix 1—video 1 and Appendix 1—videos 4, 5 and 6). To determine whether these changes lead to sperm accumulation, we developed an algorithm, which automatically scores the number of spermatozoa at any of the four defined concentric regions (R1, R2, R3, and R4) relative to the center of the speract concentration gradient (Figure 3a and Figure 3—figure supplement 1).
As you can see in Table 1 Supplementary file 1, the photoliberation of speract through the different fibers used here triggered various response types (Figure 3b and c and Figure 3—figure supplement 2). Negative controls (Low [Ca^{2+}]_{i} or High extracellular K^{+} ([K^{+}]_{e}) for f2 gradient) did not show increased sperm numbers in any region (Figure 3b and Figure 3—figure supplement 2; Appendix 1—videos 2 and 3, respectively).
In summary, S. purpuratus spermatozoa accumulate significantly toward the center of the speract gradients generated by the f2 and f3fibers (Figure 3b), which provide UV light profiles with steeper slopes compared to the f1 and f4 fibers (Figure 2b). These observations agree with the chemoreception model, in that spermatozoa exposed to steeper gradients experience lower uncertainty (i.e. higher SNR) to determine the direction of the source of the chemoattractant.
Notably, the use of fibers f4 and f5 uncages higher concentrations of speract (by providing higher UV energies than other fibers) (Figure 2a and Supplementary file 2), yet they do not trigger the maximum accumulation of S. purpuratus spermatozoa at the center of the chemoattractant field.
S. purpuratus spermatozoa undergo chemotaxis upon exposure to steep speract gradients
The sperm accumulation responses observed in any of f2 and f3 conditions suggest that the slope of the chemoattractant concentration gradient might indeed function as a driving force for sperm chemotaxis. However, the accumulation of spermatozoa at the center of the field might also imply other factors, such as cell trapping, or cell death (Yoshida and Yoshida, 2011).
To more reliably scrutinize the trajectories described by S. purpuratus spermatozoa in response to speract gradients, chemotactic behavior was quantified using a chemotactic index (CI) that considers the sperm speed and direction both before and after the chemotactic stimulus (see Figure 4a and b). This CI takes values from −1 (negative chemotaxis) to 1 (positive chemotaxis), with 0 being no chemotaxis at all (Video 2). The temporal evolution of the CI, for each of f1, f2, f3, f4, f5 speract concentration fields, was computed (Figure 4c), and their distributions across time were analyzed by a binomial test (Figure 4d, and Appendix 1—video 7 (for further explanation, see Chemotactic index section in Materials and methods).
The speract fields created by fibers f2, f3 and f5 produce significantly positive CI values compared to other conditions (f1, f4 and negative controls), confirming that steeper speract concentration gradients trigger chemotactic responses in S. purpuratus spermatozoa. Again, the lack of chemotactic responses in S. purpuratus spermatozoa observed by Guerrero et al. (2010a), was reproduced through stimulation with f4, zero Ca^{2+}, or High K^{+} experimental regimes (a scrutiny of nonchemotactic cells is presented in Figure 4—figure supplement 1 and section 2.7. Sperm swimming behavior in different chemoattractant gradients in Appendix 1).
Chemotactic efficiency, which in our work is reported by CI, contains information regarding the capability of single cells to detect and undergo a direct response toward a chemotactic stimulus. It also provides information about the percent of responsive cells that, after detecting a stimulus, can experience chemotaxis. As sperm chemotaxis, and chemotaxis in general, has evolved to operate optimally in the presence of noise (Amselem et al., 2012; Kromer et al., 2018; Lazova et al., 2011), we examined the boundary of SNR where sperm chemotaxis operates efficiently for S. purpuratus spermatozoa (Figure 4e). Take into account that in the regime of SNR < 1, chemotactic efficiency scales monotonically; for SNR > 1, saturation or adaptation mechanisms might impinge on the chemotactic efficiency, as reported in other chemotactic signaling systems (Amselem et al., 2012; Kromer et al., 2018; Lazova et al., 2011). In agreement with these results, we found that the percentage of S. purpuratus spermatozoa experiencing relocation increases monotonically with the SNR (Figure 4f), within the noise limits of 0.1 < SNR < 0.8, which is also in agreement with the findings of sperm chemotaxis operating optimally in the presence of noise (Amselem et al., 2012; Kromer et al., 2018; Lazova et al., 2011).
The magnitude of slope of the gradient is a major determinant of sperm chemotaxis
The spatial derivative of the UV profiles shown in Figure 2b indicates that the steeper light gradients generated from UV irradiation are those of f2, f3 and f5, which are assumed to generate the steepest speract gradients of similar form. This assumption is strictly valid at the instant of UV exposure; subsequently the speract gradient dissipates over time with a diffusion coefficient of D ≈ 240 μm^{2}s^{−1}. However, the gradient steepness that each spermatozoon experiences during swimming is determined by the combination of UV flash duration, the speract diffusion time, and the sperm motility response by itself.
In nature, spermatozoa of external fertilizers tend to swim in spiral 3D trajectories. However, under the experimental conditions explored in this research, we analyzed sperm swimming in 2D circularlike trajectories confined at a few microns above the coverslip. The UV flash that sets the initial chemoattractant distribution was focused at the imaging plane (~1–4 µm above the coverslip) (Nosrati et al., 2015). Hence, the correct diffusion problem corresponds to that of a 2D diffusing regime. We sought to understand how the stimulus function, which S. purpuratus spermatozoa experience during the accumulation of bound speract throughout their trajectory, influences their motility response. For this purpose, we computed the spatiotemporal dynamics of the speract gradient for f1, f2, f3, f4 and f5 fibers (Figure 5a and b and Figure 5—figure supplement 1). and analyzed the trajectories of spermatozoa swimming in these five distinct speract gradient configurations (Figure 5c, Figure 5—figure supplement 2a and Figure 5—figure supplement 2c). Moreover, we examined the stimulus function of individual spermatozoa in response to each of the five speract gradient forms (Figure 5e, Figure 5—figure supplement 2b, Figure 5—figure supplement 2d and Video 3).
The model of chemoreception presented in the previous sections (see Equations (2) and (3)) predicts a scaling rule for chemotactic responses between S. purpuratus and L. pictus spermatozoa of SF ~ 3 (Equation (4)). The derivatives of the UVirradiation profiles shown in Figure 2b indicate that the f2, f3, and f5 fibers generate steeper speract gradients than the f1 and f4 fibers.
To determine the direction of the chemoattractant concentration gradient, the signal difference $\partial c$ between two sampled positions $\partial r$ must be greater than the noise (Figure 1a). To test the prediction of the chemoreception model, we computed the local relative slope of the chemoattractant concentration gradient $\xi $ detected by single spermatozoa exposed to a given speract concentration gradient, with $\xi ={\stackrel{}{c}}^{1}\frac{\partial c}{\partial r}$ (Figure 5e).
We found that, in agreement with the chemoreception model, the maximum relative slope of the chemoattractant concentration gradient ${\xi}_{max}=max({\xi}_{1},{\xi}_{2},{\xi}_{3},\dots ,{\xi}_{n})$ required by S. purpuratus spermatozoa to undergo chemotaxis is created when the f2 and f3 fibers are employed to generate speract gradients (Figure 5e). This relative slope of the chemoattractant concentration gradients is at least three times greater than that experienced when exposed to the f4generated speract gradient (Figure 6b). In addition, L. pictus spermatozoa undergo chemotaxis when exposed to the f4 speract gradient, which is 2–3 times smaller than that required by S. purpuratus (Figure 6b). These findings support the predicted scaling rule for the detection of the speract concentration gradient between L. pictus and S. purpuratus spermatozoa (Figure 6b and c).
The slope of the speract concentration gradient is the critical determinant for the strength of coupling between the stimulus function and the internal Ca^{2+} oscillator
Friedrich and Jülicher proposed a general theory that captures the essence of sperm navigation traversing periodic paths in a nonhomogeneous chemoattractant field, in which the sampling of a stimulus function S(t) is translated by intracellular signaling into the periodic modulation of the swimming path curvature k(t) (Friedrich and Jülicher, 2008; Friedrich and Jülicher, 2007; Riedel et al., 2005). As a result, the periodic swimming path drifts in a direction that depends on the internal dynamics of the signaling system. In this theory, the latency of the intracellular signaling (i.e. the [Ca^{2+}]_{i} signal), expressed as the phase shift between S(t) and k(t), is a crucial determinant of the directed looping of the swimming trajectory up the chemical concentration field (Friedrich and Jülicher, 2009; Friedrich and Jülicher, 2008).
Even though this conceptual framework provides insights into the mechanism governing sperm chemotaxis, it does not explore the scenario where chemoattractants trigger an autonomous [Ca^{2+}]_{i} oscillator (Aguilera et al., 2012; Espinal et al., 2011; Wood et al., 2003), which suggests that sperm chemotaxis might operate in a dynamical space where two autonomous oscillators, namely the stimulus function and the internal Ca^{2+} oscillator, reach frequency entrainment (Figure 6a).
To test the hypothesis that the slope of the speract concentration gradient regulates the coupling between the stimulus function and the internal Ca^{2+} oscillator triggered by speract, we made use of a generic model for coupled phase oscillators (Pikovsky et al., 2003). In its simplest form, the model describes two phase oscillators of intrinsic frequencies ω_{1} and ω_{2} coupled with a strength γ through the antisymmetric function of their phase difference ɸ = φ_{1}  φ_{2}. The time evolution of ɸ then follows an Adler equation dɸ/dt = Δω  2γ sin(ɸ), which is the leading order description for weaklycoupled nonlinear oscillators. In the present case, the two coupled oscillators are the internal Ca^{2+} oscillator and the oscillations in the stimulus function induced in spermatozoa swimming across a speract gradient (Figure 6a). The former occurs even for immotile cells, for which there are no stimulus oscillations under a spatially uniform speract field (Figure 6—figure supplement 1, and Appendix 1—video 8); while the latter exists under two tested negative controls: cells swimming in Low Ca^{2+} and in High K^{+} artificial sea water, both of which inhibit Ca^{2+} oscillations (see Figure 3c, Figure 3—figure supplement 2 and Appendix 1—videos 2 and 3, respectively).
Wood et al., showed that immobilized S. purpuratus spermatozoa might experience spontaneous Ca^{2+} transients (Wood et al., 2003) (see Figure 6—figure supplement 1). To provide insight into the mechanism of sperm chemotaxis we characterized and compared the spontaneous vs. the speractinduced [Ca^{2+}]_{i} oscillations, and conclude that they are of different oscillatory nature, hence the spontaneous oscillations do not have a role in sperm chemotaxis (see Figure 3—figure supplement 3 and section 2.8. Spontaneous vs. speractinduced [Ca^{2+}]_{i} oscillations in Appendix 1).
There are two immediate predictions from the Adler model: first, there is a minimum coupling strength necessary for the two oscillators to synchronize (γ_{min} = Δω/2). For weaker coupling (i.e. γ < γ_{min}), the two oscillators run with independent frequencies and, hence, the phase difference increases monotonically with time; second, and within the synchronous region (i.e. γ > γ_{min}), the phase difference between the oscillators is constant and does not take any arbitrary value, but rather follows a simple relation to the coupling strength (ɸ_{sync} = arcsin(Δω/2γ)). Figure 6d shows the two regions in the parameter space given by Δω and γ. The boundary between these two regions corresponds to the condition γ = γ_{min} and it delimits what is known as an Arnold’s tongue.
We measured the difference in intrinsic frequency by looking at the instantaneous frequency of the internal Ca^{2+} oscillator just before and after the speract gradient is established. The range of measured Δω is shown in Figure 6d as a band of accessible conditions in our experiments (mean of Δω, black line; mean ± standard deviation, green dashed lines). If the driving coupling force between the oscillators is the maximum slope of the speract concentration gradient, that is γ = ξ_{max}, we would expect to find a minimum slope ($\overline{\xee*max}$) below which no synchrony is observed. This is indeed the case as clearly shown in Figure 6b, e and f (magenta line). Moreover, and for cells for which synchronization occurs, the measured phase difference is constrained by the predicted functional form of ɸ_{sync} = ɸ_{sync}(Δω, γ) as can be verified from the collated data shown in Figure 6e and f within the theoretical estimates (see also Figure 6—figure supplement 2). Altogether, the excellent agreement of this simple model of coupled phase oscillators with our data, points to the slope of the speract concentration gradient as the driving force behind the observed synchronous oscillations and, as a result, for the chemotactic ability of sea urchin spermatozoa.
Discussion
What are the boundary conditions that limit a sperm’s capacity to determine the source of guiding molecules?
During their journey, spermatozoa must measure both the concentration and change on concentration of chemoattractants. Diffusing molecules bind to receptors as discrete packets arriving randomly over time with statistical fluctuations, imposing a limit on detection. By following the differences in the mean concentration of chemoattractants, sampled at a particular time, spermatozoa gather sufficient information to assess the source of the gradient. However, there is a lower detection limit to determine the direction of the chemical gradient, which depends on the swimming speed of the sperm, the sampling time, and as shown in this work, on the steepness of the slope of the chemoattractant concentration gradient.
For almost three decades, chemotaxis had not been observed for the widelystudied S. purpuratus species under diverse experimental conditions, raising doubts about their chemotactic capabilities in response to the speract concentration gradients (Cook et al., 1994; reviewed in Guerrero et al. (2010a); Guerrero et al. (2010b); Solzin et al. (2004). The observed lack of chemotactic responses by these spermatozoa has been recognized as an ‘anomaly’ in the field  if we aspire to generalize and interpret findings in sea urchin spermatozoa to chemotactic responses in other systems, then it is critical to accommodate and account for any apparent outliers, and not ignore them as inconveniently incongruent to the model.
To examine whether S. purpuratus spermatozoa are able to detect spatial information from specific chemoattractant concentration gradient, we use a model of chemoreception developed by Berg and Purcell (1977), which considers the minimal requirements needed for a single searcher (i.e. a sperm cell) to gather sufficient information to determine the orientation of a nonuniform concentration field. By considering the difference between L. pictus and S. purpuratus spermatozoa in terms of the number of chemoattractant receptors, receptor pocket effective size, cell size, sampling time, mean linear velocity, sampling distance, and the local mean and slope of the chemoattractant concentration gradient, our model predicts that S. purpuratus spermatozoa would need a speract gradient three times steeper than the gradient that drives chemotactic responses for L. pictus spermatozoa. We tested this experimentally by exposing S. purpuratus spermatozoa to various defined speract concentration gradients.
We showed that S. purpuratus spermatozoa can undergo chemotaxis, but only if the speract concentration gradients are sufficiently steep, as predicted by the chemoreception model (i.e. speract gradients that are at in the region of three times steeper than the speract concentration gradient that drives chemotaxis in L. pictus spermatozoa). This confirms and explains why the shallower speract gradients previously tested are unable to generate any chemotactic response in S. purpuratus spermatozoa (Guerrero et al., 2010a), despite inducing characteristic ‘turn and run’ motility responses.
These findings indicate that the guiding chemical gradient must have a minimum steepness to elicit sperm chemotaxis, where the signaltonoise relationship (SNR) of stimulus to the gradient detection mechanism imposes a limit for the chemotactic efficiency. Our results are in agreement with recent theoretical studies by Kromer and colleagues, indicating that sperm chemotaxis of marine invertebrates operates optimally within a boundary defined by the SNR of collecting ligands within a chemoattractant concentration gradient (Kromer et al., 2018). We showed that SNR can be tuned by the steepness of the chemical gradient, where higher SNR’s are reached at steeper gradients, hence increasing the probabilities of locating the source of the gradient.
The large majority of marine spermatozoa characterized to date, together with many motile microorganisms, explore their environment via helical swimming paths, whereupon encountering a surface these helices collapse to circular trajectories. The intrinsic periodicity of either swimming behavior commonly results in the periodic sampling of the cell chemical environment with direct implications for their ability to accurately perform chemotaxis.
The periodic sampling of chemoattractants by the sperm flagellum continuously feeds back to the signaling pathway governing the intracellular Ca^{2+} oscillator, hence providing a potential coupling mechanism for sperm chemotaxis. Indirect evidence for the existence of a feedback loop operating between the stimulus function and the Ca^{2+} oscillator triggered by chemoattractants has been found in L. pictus, A. punctulata and Ciona intestinalis (ascidian) species, whose spermatozoa show robust chemotactic responses toward their conspecific chemoattractants (Böhmer et al., 2005; Guerrero et al., 2010a; Jikeli et al., 2015; Shiba et al., 2008).
To investigate further the molecular mechanism involved in sperm chemotaxis, we measured both the stimulus function and the triggered [Ca^{2+}]_{i} oscillations for up to one thousand S. purpuratus spermatozoa exposed to five distinctlyshaped speract concentration gradients. We demonstrate that the steepness of the slope of the chemoattractant concentration gradient is a major determinant for sperm chemotaxis in S. purpuratus and might be an uncovered feature of sperm chemotaxis in general. A steep slope of the speract concentration gradient entrains the frequencies of the stimulus function and the internal Ca^{2+} oscillator triggered by the periodic sampling of a nonuniform speract concentration field. We assessed the transition boundary of the coupling term (the slope of the speract concentration gradient) for the two oscillators to synchronize and found it to be very close to the boundary where S. purpuratus starts to experience chemotaxis. The agreement of our data with a model of weaklycoupled phase oscillators suggests that the slope of the speract concentration gradient is the driving force behind the observed synchronous oscillations and, as a result, for the chemotactic ability of sea urchin spermatozoa.
It is not that surprising to find matching of frequencies when dealing with two oscillators coupled through a forcing term. Nonetheless, the boundaries of the ‘region of synchrony’ are by no means trivial. What is relevant to the former discussion is the existence of thresholds in the coupling strength, whose experimental calculations agree with our theoretical predictions based on the chemoreception model. In addition, such a minimal model for coupled oscillators is also able to predict computed functional dependencies that are well documented in the literature, that is the observed temporal and frequency lags between the stimulation and signaling responses of the chemoattractant signaling pathway (Alvarez et al., 2012; Böhmer et al., 2005; Guerrero et al., 2010a; Kaupp et al., 2003; Nishigaki et al., 2004; Pichlo et al., 2014; Shiba et al., 2008; Strünker et al., 2006; Wood et al., 2007; Wood et al., 2005).
Caution must be exercised with the interpretations of the agreement of our data with such a generic model for coupled phase oscillators, particularly when considering only a few steps of the oscillatory cycles. The latter is relevant for assessing frequency entrainment, which in some cases demands a certain delay before reaching the synchronized state, that is when the natural frequencies of the connected oscillators are very distinct. The chemotactic responses scored in the present study encompass a few steps (<10) of both the stimulus function and the internal Ca^{2+} oscillator triggered by speract (Figure 5—figure supplement 2, Figure 6—figure supplement 1 and Figure 6—figure supplement 2). Our data indicate that within the chemotactic regime, frequency entrainment of the stimulus function and the internal Ca^{2+} oscillator of S. purpuratus spermatozoa seems to occur almost instantaneously, within the first three oscillatory steps (Figure 6—figure supplement 2). Such interesting findings can be explained by the proximity of the natural frequencies of both oscillators (Figure 6d), which may relieve the need for a longer delay for reaching frequency entrainment. Whether the proximity of the frequencies of both oscillators is sculped by the ecological niche where sperm chemotaxis occurs is an open question, however, a nearinstantaneous entrainment would confer obvious evolutionary advantage under the reproductively competitive conditions of synchronized spawning as undertaken by sea urchins.
One can further hypothesize about the evolutionary origin of the described differences in sensitivity to chemoattractant concentration gradients between S. purpuratus and L. pictus spermatozoa if we consider their respective ecological reproductive niches. The turbulent environment where sea urchins reproduce directly impinges on the dispersion rates of small molecules such as speract, hence, imposing ecological limits that constrain permissive chemoattractant gradient topologies within different hydrodynamic regimes. For instance, the reproductive success of L. pictus, S. purpuratus and abalone species has been shown to peak at defined hydrodynamic shearing values (Hussain et al., 2017; Mead and Denny, 1995; Riffell and Zimmer, 2007; Zimmer and Riffell, 2011). What are the typical values of the chemoattractant gradients encountered by the different species in their natural habitat? The correct scale to consider when discussing the smallscale distribution of chemicals in the ocean is the Batchelor scale, l_{B} = (ηD^{2}/ζ)^{1/4}, where η is kinematic viscosity, D the diffusion coefficient and ζ is the turbulent dissipation rate (Aref et al., 2017; Batchelor et al., 1959). Turbulence stirs dissolved chemicals in the ocean, stretching and folding them into sheets and filaments at spatial dimensions down to the Batchelor scale: below l_{B} molecular diffusion dominates and chemical gradients are smoothened out.
S. purpuratus is primarily found in the low intertidal zone. The purple sea urchin lives in a habitat with strong wave action and areas with shaking aerated water. These more energetic zones, including tidal channels and breaking waves, generate relatively high levels of turbulence (ζ ~10^{−4} m^{2}s^{−3}) which lead to somewhat small values of l_{B} and, hence, to steep gradients (i.e. 1/l_{B}). L. pictus, on the contrary, is mostly found at the edge of or inside kelp beds, well below the low tide mark where the levels of turbulence are much more moderate (ζ ~10^{−6} m^{2}s^{−3}) (Jimenez, 1997; Thorpe, 2007). This difference in the turbulent kinetic energy dissipation rate has a significant effect on the spatial dimensions of chemical gradients for sperm chemotaxis present in a particular habitat. The ratio of l_{B} for the different habitats scales as l_{Bpurpuratus}/l_{Bpictus} ~ (ζ_{pictus}/ζ_{purpuratus})^{1/4} ~ 3, which fits considerably well with the relative sensitivity to speract of the two species. Furthermore, we have shown that S. purpuratus spermatozoa experience chemotaxis toward steeper speract gradients than those that guide L. pictus spermatozoa, which is also compatible with the distinct chemoattractant gradients they might naturally encounter during their journey in search of an egg.
Materials and methods
Materials
Artificial seawater (ASW), and Low Ca^{2+} ASW were prepared as in Guerrero et al. (2010a), their detailed composition, together with an extended list of other materials is presented in the Appendix 1. Caged speract (CS), was prepared as described previously (Tatsu et al., 2002).
Loading of Ca^{2+}fluorescent indicator into spermatozoa and microscopy imaging
Request a detailed protocolS. purpuratus or L. pictus spermatozoa were labeled with fluo4AM (as described in section 2.2. Loading of Ca^{2+}fluorescent indicator into spermatozoa in Appendix 1), and their swimming behavior was studied at the waterglass interface on an epifluorescence microscope stage (Eclipse TE300; Nikon). The cover slips were covered with polyHEME to prevent the attachment of the cells to the glass. Images were collected with a Nikon Plan Fluor 40×/1.3NA oilimmersion objective. Temperature was controlled directly on the imaging chamber at a constant 15°C. Stroboscopic fluorescence excitation was provided by a Cyan LED synchronized to the exposure output signal of the iXon camera (2 ms illumination per individual exposure, observation field of 200×200 µm), the fluorescence cube was set up accordingly (see Appendix 1).
Image processing and quantification of global changes of spermatozoa number and [Ca^{2+}]_{i}
Request a detailed protocolTo study the dynamics of overall sperm motility and [Ca^{2+}]_{i} signals triggered by the distinct speract gradients, we developed an algorithm that provides an efficient approach to automatically detect the head of every spermatozoa in every frame of a given videomicroscopy file. A detailed description of the algorithm is provided in the Appendix 1.
Computing the dynamics of speract concentration gradients
Request a detailed protocolThe dynamics of the chemoattractant gradient was computed using Green’s function of the diffusion equation, considering diffusion in 2D:
Equation (5) for the concentration tells us that the profile has a Gaussian form, where D is the diffusion coefficient of the chemoattractant, ${c}_{b}$ is the basal concentration of the chemoattractant, t is the time interval, r is the distance to the center of the gradient and c_{0} is the initial concentration. The width of the Gaussian is $\sigma =\sqrt{4D(t+{t}_{0})}$, and hence it increases as the square root of time.
The speract concentration gradients were generated via the photolysis of 10 nM caged speract (CS) with a 200 ms UV pulse delivered through each of four different optical fibers with internal diameters of 0.2, 0.6, 2, and 4 mm (at two different positions). Light intensity was normalized dividing each point by the sum of all points of light intensity for each fiber and multiplying it by the fiber potency (measured at the back focal plane of the objective) in milliwatts (mW) (Supplementary file 2). Each spatial distribution of instantaneouslygenerated speract concentration gradient was computed by fitting their corresponding normalized spatial distribution of UV light (Residual standard error: 2.7 × 10^{−5} on 97 degrees of freedom), considering an uncaging efficiency of 5–10%, as reported (Tatsu et al., 2002).
The diffusion coefficient of speract has not been measured experimentally. However, the diffusion coefficient of a similar chemoattractant molecule, resact (with fourteen amino acids), has been reported, D_{resact} = 239 ± 7 µm^{2} s^{−1} (Kashikar et al., 2012). If we consider that speract is a decapeptide, the 1.4 fold difference in molecular weight between speract and resact would imply a (1.4)^{1/3} fold difference in their diffusion coefficients, which is close to the experimental error reported (Kashikar et al., 2012). For the sake of simplicity, the spatiotemporal dynamics of the distinct instantaneously generated speract gradients was modeled considering a speract diffusion coefficient of D_{speract} = 240 µm^{2} s^{−1}.
Computing [Ca^{2+}]_{i} dynamics and the stimulus function of single spermatozoa
Request a detailed protocolSpermatozoa were tracked semiautomatically by following the head centroid with the MtrackJ plugin (Meijering et al., 2012) of ImageJ 1.49u. Single cell [Ca^{2+}]_{i} signals were computed from the mean value of a 5x5 pixel region, centered at each sperm head along the time. The head position of each spermatozoa x was used to compute the mean concentration of speract at $r$ over each frame. The stimulus function of single spermatozoa $S=f\left(c\right)$ was computed by solving Equation (5) considering both their swimming trajectories, and the spatiotemporal evolution of a given speract concentration gradient. The profiles of UV light were used to compute the initial conditions at $c(r,{t}_{o})$.
The phase and temporalshifts between the time derivative of the stimulus function $dS/dt$ and the internal Ca^{2+} oscillator triggered by speract, were computed from their normalized crosscorrelation function.
Programs were written in R statistical software (R Development Core Team, 2016).
Chemotactic index (CI)
Request a detailed protocolEach sperm trajectory was smoothened using a moving average filter, with a window of 60 frames (two seconds approximately) (Figure 4b and Video 2). A linear model was then fitted to the smoothed trajectory; the corresponding line was forced to go through the mean point of the smoothed trajectory (orange point in Figure 4b and Video 2). The θ angle between red and black vectors was calculated in each frame from the second 4.5 to 10.
The chemotactic index is defined based on the progressive displacement of the sperm trajectory as $\mathrm{C}\mathrm{I}=\frac{\leftu\rightcos\theta \leftv\rightcos\phi}{\leftu\right+\leftv\right}$, being ϕ and θ the angles between gray and magenta, and red and black vectors, respectively; and v and u the magnitude of the sperm progressive speed before and after speract uncaging, respectively (Figure 4b and Video 2). The CI considers the sperm displacement before speract uncaging (i.e. unstimulated drift movement at 0–3 s), and then subtracts it from the speract induced effect (at 3–10 s). The CI takes values from −1 (negative chemotaxis) to 1 (positive chemotaxis), being 0 no chemotaxis at all.
Statistical analyses
Request a detailed protocolThe normality of the CI distributions, each obtained from f1 to f5 speract gradient stimuli, was first assessed using the ShapiroWilk test; none of them were normal (Gaussian), so each CI distribution was analyzed using nonparametric statistics (Figure 4d and Appendix 1—video 7). The curves obtained from medians of each CI distribution were smoothed using a moving average filter, with a window of 20 frames (0.6 s) (Figure 4c).
Data are presented for individual spermatozoa (n) collected from up to three sea urchins. All statistical tests were performed using R software (R Development Core Team, 2016). The significance level was set at 95% or 99%.
Appendix 1
1.Theory
1.1. On the estimate of maximal chemoattractant absorption
Berg and Purcell (1977) derived a simple expression for the mean chemoattractant binding and absorption flux by a cell in the steadystate, denoted $J$:
where $D$ is the diffusion coefficient of the chemoattractant; $a$ is the radius of the cell; $\stackrel{}{c}$ is the mean concentration of the chemoattractant;$N$ is the number of receptors on the membrane of the cell; $s$ is the effective radius of the receptor, assumed to be disklike on the cell surface and binding to chemoattractant molecules with high affinity; ${J}_{max}=4\pi Da\stackrel{}{c}$, which is the maximal flux of chemoattractant that a cell in the steadystate can experience; and the receptor term $\frac{N}{N+\pi a/s}$ is the probability that a molecule that has collided with the cell will find a receptor (Berg and Purcell, 1977).
The receptor term arises from the matching of two distinct limits: for a low number of receptors, the flux into independent patches leads to an overall diffusive flux into the sphere that is linear with the number of receptors. In the opposite limit of large surface coverage, the 'interactions between the effects of adjacent receptors' leads indeed to the saturation of chemoreception. The expression implies that for $N\gg \pi a/s={N}_{1/2}$ the flux of chemoattractant absorption becomes $J\cong {J}_{max}$, which means that coverage of only a small fraction of the cell surface by the receptors may lead to maximal flux. The flux becomes practically independent of the number of receptors, proportional to the concentration of the chemoattractant and limited by its diffusion, when receptor density is sufficiently large. For a given chemoattractant concentration the half maximal flux $(J=\frac{1}{2}{J}_{max})$ is reached when the number of receptors is ${N=N}_{1/2}=\pi a/s$.
It is worth getting a rough estimate of the number of receptors required for a maximal influx of chemoattractant, in the specific case of spermatozoa by calculating ${N}_{1/2}$, assuming a spherical cell with a surface area equivalent to that of the actual flagella. In the case of S. purpuratus sperm, the flagellar width h ≅ 0.2 μm, and length L ≅ 40 μm that would give us an approximate surface area A ∼ 25 μm^{2} or an equivalent spherical radius a_{e} ~ 1.4 µm and, hence, a ${J}_{max}\cong 0.1{J}_{max}^{sph}$, where ${J}_{max}^{sph}$ represents the maximal influx of chemoattractants for the spherical cell. The factor 0.1 relating the maximal flux ${J}_{max}$ in a cylindrical flagellum and ${J}_{max}^{sph}$ in a spherical cell, arises by recalling that the expression for ${J}_{max}$ in Equation (A1).
The later stems from an analogy with electrostatics such that the total current depends on the electrical capacitance C of the conducting material and, in particular, on the geometrical arrangement. The capacitance of a simple spherical conductor equals the radius a of the sphere but more generally we have ${J}_{max}=4\pi CD\stackrel{}{c}$ (Berg and Purcell, 1977).
Note that the spherical geometry is a first order approximation, which has been extremely useful and successful in the past in shedding light on many problems with more complex geometries. This includes the first estimate of diffusive fluxes in this same chemotaxis problem (as Berg and Purcell showed in 1977). Here, we have followed the same principle of ‘minimal modelling’ that captures the main physics but that, at the same time, allows for simple characterization of the relevant parameters (e.g. the dependence with the number of receptors).
A more accurate computation can be obtained by considering the nearly cylindrical shape of the flagellum. The capacitance of a finite cylinder can be obtained as a series expansion in the logarithm of the cylinder aspect ratio Λ = ln(L/h) (Maxwell, 1877) and, to a second order in 1/Λ, and is given by the following expression:
For the case of the slender sperm flagellum with h/L << 1, $\Lambda \cong 5$, this expression gives a maximal influx for the cylinder (${J}_{max}^{cyl})$ that is, again, approximately one tenth that of the equivalent sphere:
The above description is valid only in the limit of instantaneous adsorption at the receptors. For a finite rate of binding by the receptors we can simply modify the above expressions to include an effective size for the binding sites se = kon/D (Phillips et al., 2012). With kon = 24 µM^{1}s^{1} and 27 µM^{1}s^{1} being the corresponding affinity constants for speract and its receptor, calculated by Nishigaki in 2000 for L. pictus and in 2001 for S. purpuratus (Nishigaki and Darszon, 2000; Nishigaki et al., 2001), respectively. D = 240 µm^{2}s^{1}, se = 1.7 Å and 1.9 Å, respectively, which is indeed much smaller than the physical size of the receptors (Supplementary file 1). Note that the dimensions of the speract receptor radius are not known, however Pichlo et al. (2014) provided an estimation of the radius of the resact receptor (the extracellular domain of the GC) of 2.65 nm. The value of s ~ 0.19 nm used in this work is about one order of magnitude smaller than such estimation. This value arises not from estimates of either receptor or chemoattractant sizes, but rather from an estimate of the effective size of the binding site, based on experimental measurements of chemoattractant binding kinetics.
These equivalences were used to obtain the estimates in Supplementary file 1, which are discussed in the main text. From these estimates, we can compute N_{1/2} ~ 3×10^{4} as the total number of SAP receptors for the S. purpuratus sperm flagellum to act as a perfect absorber. As the actual number of SAP receptors for this species is lower than that figure, that is N < N_{1/2}, we cannot approximate the solution of Equation (A1) to that of a perfect absorber. More specifically, under these circumstances the absorption remains almost linearly dependent on the actual number of receptors on the flagellum (Figure 1—figure supplement 1).
1.2. A condition for detecting a change in the chemoattractant concentration
A cell uses the chemoattractant it samples from the medium as a proxy of the extracellular concentration of the chemoattractant at any given time. The flux of chemoattractant $J$, calculated in the previous section, measures the sampling rate. Because the number of chemoattractant molecules is finite and small, the actual number of molecules sampled by the cell in an interval of time $\mathrm{\Delta}\mathrm{t}$ is a random variable, denoted $n$, which is Poisson distributed with expected value $E\left[n\right]=J\mathrm{\Delta}\mathrm{t}$ and standard deviation $SD\left[n\right]=\sqrt{J\mathrm{\Delta}\mathrm{t}}$. The chemotaxis signaling system of the spermatozoon should remain unresponsive while the cell is swimming in an isotropic chemoattractant concentration field, as there are no spatial cues for guidance (Figure 1a), although motility responses may still be triggered. For example, the stereotypical turnandrun motility responses of S. purpuratus sperm in the presence of speract (isotropic fields or weak gradients) (Wood et al., 2007)., it has been previously reported that the turnandrun motility response is necessary, but not sufficient, for sea urchin sperm chemotaxis (Guerrero et al., 2010a). The Poisson fluctuations of sampled chemoattractant molecules, measured by $\sqrt{J\mathrm{\Delta}\mathrm{t}}$ can be understood as background noise and hence should not elicit a response. When the sperm is swimming confined to a plane in a chemoattractant gradient produced by the egg (Figure 1ab and c), the chemotactic responses should be triggered only when the amplitude of the sampling fluctuations are sufficiently large as to not be confused with the background noise, that is when the difference in concentration at the two extremes of the circular trajectory leads to fluctuations in chemoattractant sampling that are larger than the background noise. These considerations lead to a minimal condition for reliable detection of a chemotactic signal (Berg and Purcell, 1977; Vergassola et al., 2007), the corresponding condition can be stated as:
where $E\left[n\right]=J\mathrm{\Delta}\mathrm{t}=4\pi Da\stackrel{}{c}\frac{N}{N+\pi a/s}\mathrm{\Delta}t$; $\mathrm{\Delta}\mathrm{t}\mathrm{}$ is specifically the time the sperm takes to make half a revolution in its circular trajectory; $v$ is the mean linear velocity, defined as $v=\frac{\Delta r}{\Delta t}$, where $\Delta r$ is the diameter of the circumference in the 2D sperm swimming circle (Figure 1c and Supplemnetary file 1); and $\xi ={\stackrel{}{c}}^{1}\frac{\partial c}{\partial r}$ is the relative slope of the chemoattractant concentration gradient.
As described in the main text, by interpreting the lefthand side of the Equation (A4) as the minimal chemotactic signal; and the righthand side as a measurement of the background noise at a given mean concentration. Hence, one can obtain a minimal condition for the smallest signal to noise ratio (SNR) necessary to elicit a chemotactic response. Equation (A4) can be rewritten in terms of signaltonoise ratio:
Note that all previous Equations (A1A5) are only valid for small Peclet numbers (Pe ≤1) which is indeed the case for chemoattractant transport to the sperm. Pe estimates the relative importance of advection (directed motion) and diffusion (randomlike spreading) of ‘anything that moves’. We are studying the motion of chemoattractant molecules: they are transported (relative to the swimming sperm) by its swimming while jiggling around by Brownian motion at the molecular scale.
An evidencebased estimate of the Peclet number for chemoattractants can be provided by following the definition of the Peclet number Pe = UR/D, with the sperm swimming speed in the range U ~ [72–100 µm s^{−1}], diffusivity D ~ 240 µm^{2} s^{−1} for the chemoattractant. The critical length scale R for the diffusive problem can be estimated by either i) computing the influx transport problem in a cylindrical geometry with the fluid flow parallel to the flagellar long axis (i.e. the sperm swimming direction) for which R is the flagellar width ~0.2 µm; or ii) for the simplified spherical cell approximation for which R is simply the equivalent spherical radius a_{e} ~ [1.39–1.58 µm] (see section 1.1. On the estimate of maximal chemoattractant absorption). This renders Pe ~ [6e^{2}  6e^{1}] ≤ 1 for all experiments presented in this manuscript.
2.Extended materials and methods
2.1. Materials
Undiluted S. purpuratus or L. pictus spermatozoa (JAVIER GARCIA PAMANES, Ensenada, Mexico PPF/DGOPA224/18 Foil 2019, RNPyA 7400009200; and South Coast BioMarine San Pedro, CA 90731, USA respectively) were obtained by intracoelomic injection of 0.5 M KCl and stored on ice until used within a day. Artificial seawater (ASW) was 950 to 1050 mOsm and contained (in mM): 486 NaCl, 10 KCl, 10 CaCl_{2}, 26 MgCl_{2}, 30 MgSO_{4}, 2.5 NaHCO_{3}, 10 HEPES and 1 EDTA (pH 7.8). For experiments with L. pictus spermatozoa, slightly acidified ASW (pH 7.4) was used to reduce the number of spermatozoa experiencing spontaneous acrosome reaction. Low Ca^{2+} ASW was ASW at pH 7.0 and with 1 mM CaCl_{2}, and Ca^{2+}free ASW was ASW with no added CaCl_{2}. [Ser5; nitrobenzylGly6]speract, referred to throughout the text as caged speract (CS), was prepared as previously described (Tatsu et al., 2002). Fluo4AM and pluronic F127 were from Molecular Probes, Inc (Eugene, OR, USA). PolyHEME [poly(2hydroxyethylmethacrylate)] was from SigmaAldrich (Toluca, Edo de Mexico, Mexico).
2.2. Loading of Ca^{2+}fluorescent indicator into spermatozoa
This was done as in Beltrán et al. (2014), as follows: undiluted spermatozoa were suspended in 10 volumes of low Ca^{2+} ASW containing 0.2% pluronic F127 plus 20 µM of fluo4AM and incubated for 2.5 hr at 14°C. Spermatozoa were stored in the dark and on ice until use.
2.3. Imaging of fluorescent swimming spermatozoa
The cover slips were briefly immersed into a 0.1% wt/vol solution of polyHEME in ethanol, hotair blowdried to rapidly evaporate the solvent, wash with distilled water twice followed by ASW and mounted on reusable chambers fitting a TC202 Bipolar temperature controller (Medical Systems Corp.). The temperature plate was mounted on a microscope stage (Eclipse TE300; Nikon) and maintained at a constant 15°C. Aliquots of labeled sperm were diluted in ASW and transferred to an imaging chamber (final concentration ~2×10^{5} cells ml^{−1}). Epifluorescence images were collected with a Nikon Plan Fluor 40×/1.3NA oilimmersion objective using the Chroma filter set (ex HQ470/40x; DC 505DCXRU; em HQ510LP) and recorded on a DV887 iXon EMCCD Andor camera (Andor Bioimaging, NC). Stroboscopic fluorescence illumination was supplied by a Cyan LED no. LXHLLE5C (Lumileds Lighting LLC, San Jose, USA) synchronized to the exposure output signal of the iXon camera (2 ms illumination per individual exposure). Images were collected with Andor iQ 1.8 software (Andor Bioimaging, NC) at 30.80 fps in fullchip mode (observation field of ~200×200 µm).
2.4. Image processing
The background fluorescence was removed by generating an average pixel intensity timeprojection image from the first 94 frames (3 s) before uncaging, which was then subtracted from each frame of the image stack by using the Image calculator tool of ImageJ 1.49 u (Schneider et al., 2017). For Figure 2d, the maximum pixel intensity time projections were created every 3 s from backgroundsubtracted images before and after the UV flash.
2.5. Quantitation of global changes of spermatozoa number and [Ca^{2+}]_{i}
To study the dynamics of overall sperm motility and [Ca^{2+}]_{i} signals triggered by the distinct speract gradients we developed a segmentation algorithm that efficiently and automatically detects the head of every spermatozoa in every frame of a given videomicroscopy archive (C/C++, OpenCV 2.4, Qtcreator 2.4.2). Fluorescence microscopy images generated as described previously were used. The following steps summarize the workflow of the algorithm (Figure 3—figure supplement 1):
Segment regions of interest from background: This step consists of thresholding each image (frame) of the video to segment the zones of interest (remove noise and atypical values). Our strategy includes performing an automatic selection of a threshold value for each Gaussian blurred image (I_{G}) (σ = 3.5 µm) considering the mean value (M_{I}) and the standard deviation (SD_{I}) of the image I_{G}. The threshold value is defined by: T_{I} = M_{I} + 6SD_{I}.
Compute the connected components: The connected components labeling is used to detect connected regions in the image (a digital continuous path exists between all pairs of points in the same component  the sperm heads). This heuristic consists of visiting each pixel of the image and creating exterior boundaries using pixel neighbors, accordingly to a specific type of connectivity.
Measure sperm head fluorescence. For each region of interest, identify the centroid in the fluorescence channel (sperm head) and measure the mean value.
Compute the relative positions of the sperm heads within the imaging field, and assign them to either R1, R2, R3 or R4 concentric regions around the centroid of the UV flash intensity distribution. The radii of R1, R2, R3 or R4, were 25, 50, 75 and 100 µm, respectively.
Repeat steps 1 to 4 in a framewise basis.
Step 1 of the algorithm filters out shot noise and atypical values; step two divides the images into N connected components for the position of the sperm heads; step three quantitates sperm head fluorescence, and finally step four computes the relative sperm position on the imaging field. A similar approach has been recently used to identify replication centers of adenoviruses in fluorescence microscopy images (Garcés et al., 2016).
We automatically analyzed 267 videos of S. purpuratus spermatozoa, each containing tens of swimming cells, exposed to five distinct speract concentration gradients.
2.6. Analysis of speract induced Ca^{2+} transients with immobilized spermatozoa
Imaging chambers were prepared by coating cover slips with 50 µg/ml polyDlysine, shaking off excess, and allowing to airdry. Coated cover slips were then assembled into imaging chambers. Fluo4 labeled spermatozoa were diluted 1:40 in ASW, immediately placed into the chambers, and left for 2 min, after which unattached sperm were removed by washing with ASW. The chambers were then filled with 0.5 ml of ASW containing 500 nM of caged speract and mounted in a TC202 Bipolar temperature controller (Medical Systems Corp.). Images were collected with Andor iQ 1.7 software (Andor Bioimaging, NC) at 90 fps in fullchip mode, binning 4×4 (observation field of 200 µm x 200 µm). The imaging setup was the same as that used for swimming spermatozoa. The caged speract was photoreleased with a 200 ms UV pulse delivered through an optical fiber (4 mm internal diameter) coupled to a Xenon UV lamp (UVICO, Rapp Opto Electronic). The optical fiber was mounted on a ‘defocused’ configuration to minimize the generation of UV light heterogeneities.
Images were processed offline using ImageJ 1.45 s. Overlapping spermatozoa and any incompletely adhered cells, which moved during the experiment, were ignored. Fluorescence measurements in individual sperm were made by manually drawing a region of interest around the flagella with the polygon selections tool of ImageJ.
2.7. Sperm swimming behavior in different chemoattractant gradients
The sperm swimming behavior in response to a chemoattractant concentration gradient can be classified accordingly to their orientation angle (θ), which is formed between their reference and velocity vectors (Figure 4b). For the sake of simplicity, chemotactic drifts (toward the source of the chemoattractant gradient) were considered to fall within the category of (θ < 60°). The drift of swimming sperm in a direction perpendicular to the gradient results from orientation angles falling within the range 60° ≤ θ ≤ 120°. The instances of negative chemotactic drifts (opposite to the source of the chemoattractant gradient) were classified as those having higher orientation angles θ > 120° (Figure 4—figure supplement 1).
The proportion of spermatozoa orientated with low θ angles, i.e. toward the source of the chemoattractant concentration gradient is enriched in those gradients that give chemotactic responses: f2 (pvalue<0.001) and f5 (pvalue=0.003); compare with f1 (pvalue=0.2) and f4 (pvalue=0.51). Statistical comparisons were performed with the Pearson's Chisquared test considering a probability of success of 1/3 for each type of response (nonresponding cells were not considered).
The two tested negative controls for chemotaxis (Low Ca^{2+} or High extracellular K^{+} ([K^{+}]_{e}) for f2 gradient) showed a complete distinct distribution that the corresponding f2 gradient (f2.0Ca pvalue<0.001, f2.K pvalue=0.01, Fisher’s exact test), i.e. as expected the proportion of cells experiencing chemotactic drift was significantly reduced on the negative controls.
Interestingly, the f3 gradient provides the major stimulation of cell motility (the frequency of nonresponsive cells drops down to ~2%), however in this experimental condition the proportion of cells responding toward the source of the chemoattractant gradient was not significantly distinct from the other two types of responses (pvalue=0.12, Pearson's Chisquared test).
In any tested gradient, the distributions of orientation angles have the same proportions between perpendicular and opposite to the source responses: f1 (pvalue=0.63), f2 (pvalue=1), f3 (pvalue=0.4), f4 (pvalue=0.84) and f5 (pvalue=0.15). Statistical comparisons were performed with the exact binomial test considering a hypothesized probability of success of 0.5.
2.8. Spontaneous vs. speractinduced [Ca^{2+}]_{i} oscillations
We characterized and compared the spontaneous vs. the speractinduced Ca^{2+} oscillations (Figure 3—figure supplement 3) and conclude that they are completely different phenomena. Spontaneous Ca^{2+} oscillations are only observed in about 10% of the analyzed population of spermatozoa (see Statistical analysis section in Materials and methods on the manuscript). Most of the time only one spontaneous oscillation is observed, and in the cases where more than one spontaneous oscillation is present (which accounts for ~20% of the spontaneous oscillations, i.e. only 2% of the total cells analyzed), they are significantly different in nature to the speractinduced Ca^{2+} oscillations, judged as follows: they display a larger period and amplitude (~one order of magnitude) when compared to the speract induced oscillations (Figure 3—figure supplement 3c and Figure 3—figure supplement 3d). When these Ca^{2+} spontaneous oscillations occur, if not very large, the cell will change direction randomly. If the oscillation is large enough, and is beyond a certain [Ca^{2+}]_{i} threshold, the cell stops swimming altogether (see for example: Wood et al., 2005; Guerrero et al., 2013). After detection, we discarded cells undergoing spontaneous oscillations in the present work.
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Decision letter

Naama BarkaiSenior Editor; Weizmann Institute of Science, Israel

Raymond E GoldsteinReviewing Editor; University of Cambridge, United Kingdom

Jörn DunkelReviewer; Massachusetts Institute of Technology, United States

Robert AustinReviewer; Princeton University, United States
In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.
Acceptance summary:
The work of RamirezGomez et al., is an important contribution to our understanding of sperm chemotaxis in sea urchins, a historically important class of organisms in the unravelling of this phenomenon. It was in these that sperm activating peptides such as speract were first identified; these play a role in triggering calcium increases that regulate dynein motor activity and thereby control motility. Through studies of various species of sea urchins and theoretical analysis of the limits of gradient detection the authors identify the boundaries for detecting chemotactic signals of S. purpuratus spermatozoa, and show that sperm chemotaxis arises only when sperm are exposed to sufficiently steep speract concentration gradients They show further that sperm chemotaxis arises through coupling between recruitment of speract molecules during sperm swimming and the internal Ca^{2+} oscillator.
Decision letter after peer review:
Thank you for sending your article entitled "Sperm chemotaxis is driven by the slope of the chemoattractant concentration field" for peer review at eLife. Your article is being evaluated by Naama Barkai as the Senior Editor, a Reviewing Editor, and three reviewers.
Given the list of essential revisions, the editors and reviewers invite you to respond within the next two weeks with an action plan and timetable for the completion of the additional work. We plan to share your responses with the reviewers and then issue a binding recommendation.
The reviewers had mixed opinions on this work, but reviewer #3 has raised a number of technical issues that need a clear response from you in order that we can reach a formal decision. Please pay particular attention to those items.
Reviewer #2:
There is nothing wrong with this paper. It gives a very thorough review of the wellresearched field of chemotaxis including some solid modeling. The problem is that it isn't new or surprising (to me). The same manuscript was put online in June 2017 in bioRxiv with little note. The statement that "For almost three decades, chemotaxis had not been observed for the widelystudied S. purpuratus species under diverse experimental conditions, raising doubts about their chemotactic capabilities in response to the speract concentration gradients" is made without citation in both versions of the paper, but it doesn't seem to have been much of a mystery. The receptor density of low on S. purpuratis, which then requires a steeper gradient to overcome noise, as the authors have shown.
While there is nothing wrong here, it seems very academic and has previously attracted little attention, so I question why it should be published in eLife.
Reviewer #3:
I started the manuscript with excitement but it did not take long to recognise that the theoretical work has been not been implemented to sufficient standards of diligence; it appears unchecked for errors, both minor and fundamental, with examples of the latter including modelling assumption, equation solution and dimensions.
The limits of detection and the limits of when oscillators couple (e.g. that pendula on a wall are sufficiently coupled to synchronise) is interesting and is the concept in the context of chemotaxis explored here. However, thresholds by their nature are sensitive – the number of theoretical errors means that discussing and examining thresholds does not appear to be sound (as opposed to using controlled and justified approximations).
Hence, I am afraid I cannot recommend the manuscript for publication, with further details are below. I should note I have less confidence in the experimental aspects and leave this to other reviewers.
Equation (1):
 A list of assumptions should be provided with such equations. The authors have assumed a large Peclet number and it is not clear the Peclet number is large (it is, if the flagellum radius is used as the length scale, but the interaction of the fluid flow with the concentration field means such an assumption is not obviously valid). Appeal to Berg's paper is insufficient as the Peclet number is larger for bacteria, as the smaller the length scale the greater the effect of diffusion, and diffusion is dominant for an isolated bacterium. The assumption that Pe >> 1 is therefore a substantial one and should be justified – it is not clear, either way, whether it is true or false.
 The authors assume a spherical geometry for the flagellum. This is flawed. It is commented on in the SI and an alternative is given, but not used. I did not understand why an inappropriate and inaccurate approximation is used in the main text when the authors know this is an issue; no explanation seems to be present.
 The calculation of the receptor term, N/[N + πa/s], is for a spherical flagellum only and arises from interactions between the effects of adjacent receptors. For a fixed volume, as assumed, the sphere has minimal surface area and thus receptor interaction is highest given they are assumed placed at random. Thus, using the correct geometry with a fixed volume will reduce this receptor interaction effect and yet it is fundamental to the paper. Hence the authors are overestimating the influence of one of the primary features they are testing for – this seems fundamental and one of the reasons I am recommending reject.
 s is the receptor effective radius not the chemoattractant radius (subsection “Speciesspecific differences in chemoattractantreceptor binding rates”).
Equation (5):
 This is also flawed. It is dimensionally incorrect as stated.
 To within a scaling to fix dimensions, it is the onedimensional solution. Either the twodimensional or threedimensional solution is required here (depending on the gap between the cover slips, which does not appear to be given – my one comment on the experiments – the geometry should be clearly stated). The use of the incorrect spatial dimension changes the gradient term in the SDR expression and thus has major downstream impact, explicitly affecting what the authors are testing. Again, this seems fundamental in testing the chemoreception model and thus is a further reason I am recommending reject.
 Equation (5) does not respect the fact any solution without initial conditions imposed will satisfy invariance to shifts t > t+q for any q and so cannot be correct (it needs t+t_{0} rather than t in the square root or t_{0} must be zero).
Equation (S4). Appendix 1 subsection “1.2. A condition for detecting a change in the chemoattractant concentration”. The chemoreception model uses the circumference of the sperm's circular path (v∆t) rather than its diameter, yet the diameter governs the range of concentrations the cell experiences. Given the ratio of circumference to diameter is π this constitutes a factor of π in the definition of SNR and feeds through to the rest of the paper. For studying thresholds, a factor of three can be very important and this contributes to my overall decision.
Appendix 1 subsection “1.1. On the estimate of maximal chemoattractant absorption” There are dimensional errors in the expression for the effective size of the binding site.
There are numerous points of presentation, only the more general are below as opposed to detailed minor points (e.g. no equation punctuation, use of ∂ for increments… which I have not documented).
 "Caution needs to be taken with the interpretations of the agreement of our data with such a generic model for coupled phase oscillators" – such a generic model does not pin down mechanism and an oscillator inheriting the frequency of its forcing does not seem unexpected, so I am also hesitant about what can be learnt from the interpretations. Similarly, for (over)statements in the manuscript e.g. "that spermatozoa exposed to steeper gradients experience lower uncertainty (i.e. higher SNR) to determine the direction of the source of the chemoattractant". Any theory with sensible monotonic relationships will show this trend so I am also not clear what is learnt from such observations. This is probably a point of presentation only, but I struggled on such points.
 It is unclear sometimes what is model prediction as statements that are derived from the models are often not stated as such making it harder to follow.
[Editors’ note: The editors accepted the authors’ plan for revisions asking for further expansion on certain points.]
3.2) Given the conviction of the authors, this should just be a simple case of providing an evidencebased estimate of the Peclet number for sperm (not bacteria – these are much smaller and thus unreliable for inferring the correct physical scales) to demonstrate transport is diffusively dominated. Such a demonstration is required.
3.3) Please evidence that is a legitimate approximation or use the expression for a cylinder. There is no demonstration that the difference between the two geometries does not change the presented results. Such evidence is required.
3.4) Please provide explicit evidence for your claims such that the use of this expression – based on a spherical flagellum – does not impact on the theoretical predictions and subsequent experimental comparisons, predictions, conclusions etc.
https://doi.org/10.7554/eLife.50532.sa1Author response
Reviewer #2:
2.1) There is nothing wrong with this paper. It gives a very thorough review of the wellresearched field of chemotaxis including some solid modeling. The problem is that it isn't new or surprising (to me). The same manuscript was put online in June 2017 in bioRxiv with little note.
Our work was uploaded in bioRxiv as part of the reviewing process in eLife. During these two years, and as a result of the positive interaction with the reviewers we have made significant improvements to our work. Since then, it raised the interest of several researchers of the community who provided personal feedback, enriching the process of revising the manuscript. The Abstract has been read 1,820 times, the HTML text downloaded 102 times, and the pdf 728 times. We therefore argue that these data are sufficient to demonstrate a wide interest in our findings. Furthermore, till this work, chemotaxis in S. purpuratus sperm has not been demonstrated and more importantly, an explanation of why it has not been observed under previously tested conditions was lacking (see detailed referenced answer below).
2.2) The statement that "For almost three decades, chemotaxis had not been observed for the widelystudied S. purpuratus species under diverse experimental conditions, raising doubts about their chemotactic capabilities in response to the speract concentration gradients" is made without citation in both versions of the paper, but it doesn't seem to have been much of a mystery. The receptor density of low on S. Purpuratis, which then requires a steeper gradient to overcome noise, as the authors have shown.
The reviewer seems to suggest that demonstrating sperm chemotaxis in S. purpuratus was a matter of time and is therefore trivial. In this regard, we counter by asking why has nobody found it before? To date, it has been more than three decades since the isolation of the founder member of the family of Sperm Activating Peptides (SAP) that regulate sperm motility, speract. Following the advice of this reviewer we will include references that state the observations made by some colleagues on the field of sperm motility:
Cook et al., 1994 cite: “Of the many identified echinoderm egg peptides (Suzuki and Yoshino, 1992), only resact is known to produce swimming responses (Ward et al., 1985), and a detailed understanding of signal transduction exists only for speract (Cook and Babcock, 1993a, 1993b)”.
In that article, the authors propose a model to explain the minimal molecular and cellular mechanism required for sperm chemotaxis, taking the current knowledge about speract signaling to extrapolate to the regulation of sperm motility during chemotaxis, while recognizing that speract was not demonstrated to be a chemoattractant itself. In the same manuscript the authors state:
“The changes in flagellar waveform and swimming paths produced by speract strongly indicate that a physiological role of this egg peptide is to modify sperm swimming behavior. Although chemotaxis to speract remains to be established, the similarities between the Ca^{2+} responses to speract and to the known chemoattractant peptide resact suggest that these represent part of a common mechanism for sperm chemotaxis”.
Solzin et al. (2004) compare the distinct signaling responses triggered by speract and resact on their conspecific spermatozoa, where similarities and differences between species are scrutinized, the corresponding section of the manuscript states: “speract, the peptide of S. purpuratus, does not display chemotactic activity (Cook et al., 1994). Despite this fact, the current model of sperm chemotaxis was readily generalized (Cook et al., 1994).
Moreover, in the same manuscript the lack of chemotactic responses of S. purpuratus to speract is stressed, citing the corresponding observations: “In a capillary assay, we also found no evidence for chemotactic activity of speract (unpublished data)”.
We note that the authors suggest that speract may act as a chemoattractant, but also leave open the possibility that it may serve functions other than chemotaxis. We did not and do not intend to provide here an exhaustive compendium of positive and negative results concerning assays tailored to explore the motility alterations driven by speract on S. purpuratus spermatozoa. Instead, we want to stress that the observed lack of chemotactic responses on these sperm cells has been recognized as a “mystery” in the field – if we aspire to generalize and interpret findings in sea urchin spermatozoa to chemotactic responses in other systems, then it is critical to accommodate and account for any apparent outliers, and not ignore them as inconveniently incongruent to the model.
Discussions about the molecular differences between species, that could explain the discrepancies between the motility responses triggered by either speract or resact, are present in the literature and remained unresolved until now. Quantitative dissimilarities on the number of receptors, swimming velocities, size of the flagellum, ecological reproductive niche, among others, have been noted and discussed in previous publications (reviewed in Guerrero et al., 2010). The present investigation is unique in the sense that it collects such information and, taking the seminal work provided by Berg about the physics of chemoreception as starting point (Berg and Purcell, 1977), and formulates a succinct but sufficiently complete model, which predicts the chemoattractant receptor density as an important quantity. Our findings indicate that the number of receptors determine the sensitivity that sperm have to reliably sample the egg positional information on a noisy background of signals, through measuring the slope of the chemoattractant concentration gradient. Our findings go much further, and are more generally relevant, than the singular observation of chemotaxis on S. purpuratus. They contribute to the general understanding of how information is transferred between the source and the searcher, when the channel of information is of chemical nature. The later becomes relevant when considering the ecological context where the chemical gradients are established, which for the case of marine species are shaped by the sliding of water bodies.
In summary, we present a rationale which establishes links that go from the molecular (number of receptors and coupling stimulation and signaling oscillators) to cellular (regulation of sperm motility) and ecological regimes (by understanding the scaling of the chemoattractant gradients as result of the hydrodynamical regime that shape them), where reproduction of marine invertebrates takes place.
2.3) While there is nothing wrong here, it seems very academic and has previously attracted little attention, so I question why it should be published in eLife.
This answer is a continuation of 2.1 and 2.2, which develop further explanations about the relevance of the present investigation. To complement this point we would like to recall the comment from reviewer #1: “In my opinion, the information content and quality of the figures seems now appropriate for eLife. Thus, overall, I think this joint experimental and theoretical work can be a valuable original contribution to the literature in the field of sperm chemotaxis”.
Reviewer #3:
3.1) The limits of detection and the limits of when oscillators couple (e.g. that pendula on a wall are sufficiently coupled to synchronise) is interesting and is the concept in the context of chemotaxis explored here. However, thresholds by their nature are sensitive – the number of theoretical errors means that discussing and examining thresholds does not appear to be sound (as opposed to using controlled and justified approximations).Hence, I am afraid I cannot recommend the manuscript for publication, with further details are below. I should note I have less confidence in the experimental aspects and leave this to other reviewers.
We would like to start our reply with a general comment on the points raised by reviewer #3, summarizing the major points raised before discussing them one by one in dedicated answers.
Of major concern to the reviewer are several points that stem from the physical and mathematical grounds that sustain distinct aspects of the theoretical approaches used in our manuscript and, hence, of the validity of the statements derived from them. In what follows, we provide a discussion about the physical concepts used in our work, which are mostly based on the elaboration of minimal models whose approximations need to be stated and justified clearly. Unfortunately, these might have not been done to the required level of detail, leading to misunderstandings and erroneous interpretations. In particular, we emphasize that several of this referee’s comments appear to interpret our conclusions and findings as having the opposite meaning to that which we intended (i.e. 3.2, 3.3 and 3.9), hence, raising the doubts expressed in their comments. We are confident that the answers provided below will satisfy such inquiries.
Specifically, the major concerns were:
 Whether the geometric considerations that support the proposed chemoreception model justify our interpretations. In particular, the number density of speract receptors and their impact on the flux of chemoattractant (being or not within the linear regime).
 Clarify whether advective terms have to be taken into account for the elaboration of the chemoreception model.
 Whether the effective receptor size is computed correctly, and if not, whether it biases the observations gathered from the modeling approach.
 The use of an oversimplified diffusion model (1D) which might introduce errors in our estimate of the gradients explored by the sperm cells during chemotaxis. This, in turn, might impinge on the evaluated SNR, which is fundamental when testing the chemoreception model.
We show that the criticisms raised by the reviewer derive mostly from a misunderstanding of how we present concepts and approximations, and we hope that the current explanations lead the reviewer to revisit his/her appreciation about this manuscript, which currently departs from the appreciation of the other reviewers. For instance, reviewer #1 states that “the theoretical analysis […] and arguments presented by the authors in the revised version [is] convincing.” And reviewer #2 mention that this paper “gives a very thorough review of the wellresearched field of chemotaxis including some solid modeling”. We hope that, in light of the provided answers, both the editor and the reviewers.
3.2) Equation (1):
– A list of assumptions should be provided with such equations. The authors have assumed a large Peclet number and it is not clear the Peclet number is large (it is, if the flagellum radius is used as the length scale, but the interaction of the fluid flow with the concentration field means such an assumption is not obviously valid). Appeal to Berg's paper is insufficient as the Peclet number is larger for bacteria, as the smaller the length scale the greater the effect of diffusion, and diffusion is dominant for an isolated bacterium. The assumption that Pe >> 1 is therefore a substantial one and should be justified – it is not clear, either way, whether it is true or false.
The assumption behind Equation (1) is, in fact, quite the opposite to what the referee seems to imply here i.e. we are dealing with a purely diffusive flux into the spherical absorber. This is indeed the condition we should expect both for the individual bacterium discussed in Berg’s original work, and equally for the individual sperm cells discussed in the present work. The corresponding Peclet numbers remain small, i.e. diffusion dominates over advective transport, whether we use the flagellum radius or any other characteristic length scale for the swimming cell. This is a common scenario in the microbial world. To find high Pe (transport dominated by advection) at these micro scales we have to look, for instance, into collective behavior. That is something some of us have studied extensively in other contexts (e.g. Tuval et al., 2005) and which can, in practice, induce collective fluid flows at scales much larger than the individuals.
3.3) The authors assume a spherical geometry for the flagellum. This is flawed. It is commented on in the SI and an alternative is given, but not used. I did not understand why an inappropriate and inaccurate approximation is used in the main text when the authors know this is an issue; no explanation seems to be present.
We consider that using a spherical geometry per se is valid. It is a first order approximation that has been extremely useful and successful in the past in shedding light on a large number of problems with more complex geometries. This includes the first estimate of diffusive fluxes in this same chemotaxis problem (as Berg and Purcell showed in 1977). We have followed the same principle of "minimal modelling" that captures the main physics but that, at the same time, allows for simple characterization of the relevant parameters (e.g. the dependence with the number of receptors).
However, and for the sake of completeness, we have also included in the Supplementary information of our manuscript the more accurate computation of the absolute diffusive flux on a cylinder. This is a significantly more complicated expression than the compact solution for the spherical case. Nonetheless, and to a first order in an expansion in the slenderness of the flagellum, it is in fact approximately proportional (i.e. equal modulo a pre factor/proportionality constant) to the flux on a sphere. Hence, we strongly consider that the spherical approximation mostly used throughout the main manuscript to be neither "inappropriate" nor "inaccurate" as for comparing relative fluxes for which the aforementioned prefactor/proportionality constant cancels out and, hence, becomes irrelevant. In the new version of the manuscript, we have now expanded the explanation for the used approximations and their validity as requested by the referee. We thank the reviewer for raising this concern.
3.4) The calculation of the receptor term, N/[N + πa/s], is for a spherical flagellum only and arises from interactions between the effects of adjacent receptors. For a fixed volume, as assumed, the sphere has minimal surface area and thus receptor interaction is highest given they are assumed placed at random. Thus, using the correct geometry with a fixed volume will reduce this receptor interaction effect and yet it is fundamental to the paper. Hence the authors are overestimating the influence of one of the primary features they are testing for – this seems fundamental and one of the reasons I am recommending reject.
The receptor term, as originally deduced by Berg and Purcell, (1977), arises from the matching of two distinct limits: for a low number of receptors, the flux into independent patches leads to an overall diffusive flux into the sphere that is linear with the number of receptors. In the opposite limit of large surface coverage, the "interactions between the effects of adjacent receptors" leads indeed to the saturation of chemoreception. However, we believe "the primary features [we] are testing for" are the exact opposite to what the referee states: a low number of (noninteracting) receptors, sparsely covering the flagellum (i.e. with a large distance between receptors compared to receptor size) entails a nonsaturated diffusive flux that, hence, depends on the number of receptors.
The cylindrical geometry, if anything, strengthens our assumption: the larger surface area of the cylinder gives a larger average distance between receptors and, hence, offsetting the saturation of the overall diffusive flux to higher receptor number. As a result, the flux strongly depends on the number of receptors, which is indeed one of the main points of our manuscript: it is incorrect to assume that the number of receptors is in fact large enough for all species for the perfect absorber approximation to be a valid one. Thus, by using the approximate spherical case, we are actually underestimating this effect, instead of overestimating as suggested by the reviewer. Such explanation has been now included in the manuscript.
3.5) s is the receptor effective radius not the chemoattractant radius (subsection “Speciesspecific differences in chemoattractantreceptor binding rates”).
Continuing from 3.4, our model derives from Berg’s original model of a sphere covered with N perfect absorber patches of radius s, the effective receptor radius. The dimensions of the speract receptor radius is not known, however Pichlo el al., (2014), provided an estimation of the radius of the resact receptor (the extracellular domain of the GC) of 2.65 nm. The value of s ~ 0.19 nm used in this work is about one order of magnitude smaller than such estimation. This value arises not from estimates of either receptor or chemoattractant sizes, but rather from an estimate of the effective size of the binding site, based on experimental measurements of chemoattractant binding kinetics (seeSupplementary file 1, and 3.9 below).
3.6) Equation (5):
– This is also flawed. It is dimensionally incorrect as stated.
– To within a scaling to fix dimensions, it is the onedimensional solution. Either the twodimensional or threedimensional solution is required here (depending on the gap between the cover slips, which does not appear to be given – my one comment on the experiments – the geometry should be clearly stated). The use of the incorrect spatial dimension changes the gradient term in the SDR expression and thus has major downstream impact, explicitly affecting what the authors are testing. Again, this seems fundamental in testing the chemoreception model and thus is a further reason I am recommending reject.
We thank the reviewer for raising these observations, and we agree with both comments to a certain extent. Equation (5) is indeed dimensionally incorrect as stated, in the sense that there is a normalization constant/prefactor that has not been explicitly included. It is, however, implicitly accounted for through the fitting procedure to the different UV profiles and, hence, this typo does not propagate any errors into the evaluated variables. We have now made this choice more explicit when first defining c_{0}.
Equation (5) is indeed (modulo the aforementioned prefactor) the solution to the 1D diffusion problem. Although this was done for the sake of simplicity, we agree with the reviewer in that using different spatial dimensions would have an effect on the estimated timedependent gradients.
In nature, external fertilizers sperm cells tend to swim in spiral 3D trajectories. However, under the experimental conditions explored in this research, we are analyzing only the confined sperm swimming trajectories within the imaging chamber, swimming in 2D circularlike trajectories confined at a few microns above the coverslip. The UV flash that sets the initial chemoattractant distribution was focused at the imaging plane (~14 𝜇m above the coverslip). Hence, the correct diffusion problem indeed corresponds to that of a 2D diffusing regime.
We plan to shortly revisit our analyses within this 2D diffusive framework but, in the meantime, we provide here a first estimate for the induced errors. In Author response image 1, we compare the solution for the diffusive spreading between the 1D approximation and the 2D case. While the time evolution of the chemoattractant concentration profiles significantly differs (see Figure 1A and 1C), the relative gradients (i.e. locally normalized derivatives) are reasonably similar (Figure 1B and 1D). Remember that our proposed chemoreception model considers relative concentration gradients, instead of absolute ones. For a typical chemotactic sperm sampling chemoattractant concentration gradients over the course of a few seconds, the average error committed by assuming a 1D diffusive spreading (which would increase as a function of time as seen in Figure 1F) would be of the order of 25%. This error is significant and must be corrected for, which is the reason we plan to perform a recalculation prior to resubmission. However, we would like to stress that a 25% error in the estimated relative gradients will not modify substantially the main conclusions of our work, as these are based on an over 300% differences in experienced gradients between the different experimental conditions (i.e. different light profiles and different species sensitivities) as we have measured and presented in the manuscript.
3.7) Equation (5) does not respect the fact any solution without initial conditions imposed will satisfy invariance to shifts t> t+q for any q and so cannot be correct (it needs t+t_{0} rather than t in the square root or t_{0} must be zero).
The referee is correct. This was actually a typo in Equation (5). Instead of “t”, one should have written “t+t_{0}” as is present at a couple of lines below in the same paragraph (subsection “Computing the dynamics of speract concentration gradients”) when computing σ asσ=4D(t+t0). The value of t_{0} was adjusted together with c_{0}such that Equation (5) at t=t_{0} accurately describes the shape of the UV light flash.
3.8) Equation (S4). Appendix 1 subsection “1.2. A condition for detecting a change in the chemoattractant concentration”. The chemoreception model uses the circumference of the sperm's circular path (v∆t) rather than its diameter, yet the diameter governs the range of concentrations the cell experiences. Given the ratio of circumference to diameter is π this constitutes a factor of π in the definition of SNR and feeds through to the rest of the paper. For studying thresholds, a factor of three can be very important and this contributes to my overall decision.
We fully agree with this comment, and believe the criticism stands from a simple misunderstanding: it is indeed the diameter of the swimming circle which "governs the range of concentrations the cell experiences". And this is the scale we have actually used to define our chemoreceptor estimates. This is specified, for instance, in Supplementary file 1 where we wrote “v, the mean linear speed of the spermatozoa, i.e. Δr/Δt, where Δr is the sampling distance (the diameter of the swimming circle).” We have now also made this choice of scale more explicit in the main text as well.
3.9) Appendix 1 subsection “1.1. On the estimate of maximal chemoattractant absorption” There are dimensional errors in the expression for the effective size of the binding site.
We understand the source of the confusion here and we apologize for not clarifying this estimate in more detail in the text. There are no dimensional errors in this expression. Following Phillips et al., "Physical Biology of the Cell", it is presented in a somewhat unusual manner: the estimated effective size is based on the affinity constant, s_{e} = k_{on}/D where k_{on}[s^{1}M^{1}s^{1}] and D[m^{2}s^{1}], which requires first converting concentrations [M] into inverse volume [m^{3}]. This leads to the correct dimensions for s_{e}[m] as discussed in the aforementioned textbook, and which is explicitly cited alongside this estimate in the Appendix.
3.10) There are numerous points of presentation, only the more general are below as opposed to detailed minor points (e.g. no equation punctuation, use of ∂ for increments… which I have not documented).
Thank you for raising this concern, the manuscript has been reviewed in search of punctuation errors and misuse of mathematical notation.
3.11) "Caution needs to be taken with the interpretations of the agreement of our data with such a generic model for coupled phase oscillators" – such a generic model does not pin down mechanism and an oscillator inheriting the frequency of its forcing does not seem unexpected, so I am also hesitant about what can be learnt from the interpretations.
We fully agree with the reviewer in that it is not that surprising to find matching of frequencies when dealing with two oscillators coupled through a forcing term. But we want to stress that the boundaries of the “region of synchrony” are by no means trivial. We recall that the experimental proof of the fact that the slope of the gradient is the driving force responsible for the oscillator coupling is a significant contribution of the present work. But what is most relevant to the former discussion is the existence of thresholds in the coupling strength, whose experimental calculations agree with our theoretical predictions based on the chemoreception model. In addition, such minimal model for coupled oscillators is also able to predict computed functional dependencies well documented in the literature, i.e. the observed temporal and frequency lags between the stimulation and signaling responses of the chemoattractant signaling pathway (Kaupp, 2003; Nishigaki et al., 2004; Bohemer et al., 2005; Wood et al., 2005, 2007; Strunker et al., 2006; Shiba et al., 2008; Guerrero et al., 2010; Alvarez et al., 2012, Pichlo et al., 2014).
3.12) Similarly, for (over)statements in the manuscript e.g. "that spermatozoa exposed to steeper gradients experience lower uncertainty (i.e. higher SNR) to determine the direction of the source of the chemoattractant". Any theory with sensible monotonic relationships will show this trend so I am also not clear what is learnt from such observations. This is probably a point of presentation only, but I struggled on such points.
Reviewer #3 is correct in the sense that we developed several arguments starting from a simple theory showing monotonic relationships, i.e. as the slope of the chemoattractant concentration gradient increases so does the SNR. However, the present research goes further than a theoretical proposal, as it develops alongside an experimental study that corroborates its predictions, i.e. sensing the presence of a chemoattractant gradient requires overcoming a boundary of detection at which the shaping of the decay length of the gradient, ‘the slope’, has a major role for sperm chemotaxis (for further discussion see responses to 2.3, 3.1 and 3.11).
3.13 It is unclear sometimes what is model prediction as statements that are derived from the models are often not stated as such making it harder to follow.
We revisited the whole manuscript to ensure the proper explanation of the cases where the statements originate either from the predictions of the model, or from the experimental observations.
[Editors’ note: The editors accepted the authors’ plan for revisions asking for further expansion on certain points. What follows is the authors’ additional responses to these points.]
3.2) Given the conviction of the authors, this should just be a simple case of providing an evidencebased estimate of the Peclet number for sperm (not bacteria – these are much smaller and thus unreliable for inferring the correct physical scales) to demonstrate transport is diffusively dominated. Such a demonstration is required.
Pe estimates the relative importance of advection (directed motion) and diffusion (randomlike spreading) of "anything that moves". We are studying the motion of chemoattractant molecules: they are transported (relative to the swimming sperm cell) by its swimming while jiggling around by Brownian motion at the molecular scale. In this regard, there are distinct ways to conceptualize Pe, one for the chemoattractant molecules and one for the sperm cells themselves, which might explain discrepancies in Pe estimates, and hence interpretations. The Pe in our problem is small (<1), as we develop below. In contrast, Friedrich and Jülicher (2008) get to a more complex scenario where the Pe depends on the concentration gradient, taking values between 1 to 100. These two Pe are not the same, in the Friedrich and Jülicher approach, they are not looking at the thermal diffusion of chemoattractant molecules, but rather to an "effective" diffusivity used to characterize nondirected motion of the sperm cells themselves.
An evidencebased estimate of the Peclet number for chemoattractants can be provided by following the definition of the Peclet number Pe = UR/D, with the sperm swimming speed in the range U ~ [72100 µm/s], diffusivity D ~ 240 µm^{2}/s for the chemoattractant. The critical length scale R for the diffusive problem can be estimated by either i) computing the influx transport problem in a cylindrical geometry with the fluid flow parallel to the flagellar long axis (i.e. the sperm swimming direction) for which R is the flagellar width ~ 0.2 µm; or ii) for the simplified spherical cell approximation for which R is simply the equivalent spherical radius ae ~ [1.391.58 µm]. This renders Pe ~ [6e2 – 6e1] ≤ 1 for all experiments, which we have now included in Supplementary file 1 and which demonstrate that transport is diffusively dominated for an isolated single sperm cell (Acrivos and Taylor, 1962).
3.3) Please evidence that is a legitimate approximation or use the expression for a cylinder. There is no demonstration that the difference between the two geometries does not change the presented results. Such evidence is required.
As already mentioned in our previous reply, the use of the approximate spherical geometry is accurate when comparing relative fluxes (J/J_{max}, see Figure 1—figure supplement 1 & Equation (A1)). The reason being Equations (A2) and (A3) in our Appendix 1 that show how the correct influx toward a cylinder, computed as a series expansion in slenderness, is approximately proportional (to a first order in the flagella slenderness) to the diffusive influx to an equivalent sphere; i.e. they are identical modulo a prefactor. This prefactor (~ 0.1 in our case) cancels out when computing relative fluxes (J/J_{max}, see Figure 1—figure supplement 1 & Equation (A1)). Moreover, truncating out the second order term in the expansion (Equation (A2)) gives a < 5% error in these estimates. Hence, we stand by the simplest approximation when computing all our results.
3.4) Please provide explicit evidence for your claims such that the use of this expression – based on a spherical flagellum – does not impact on the theoretical predictions and subsequent experimental comparisons, predictions, conclusions etc.
The receptor term, as originally deduced by Berg and Purcell (1977), arises from the matching of two distinct limits: for a low number of receptors, the flux into independent patches leads to an overall diffusive flux into the sphere that is linear with the number of receptors. In the opposite limit of large surface coverage, the "interactions between the effects of adjacent receptors" leads indeed to the saturation of chemoreception (Figure 1—figure supplement 1). However, we believe that primary features we are testing for are the exact opposite to that which the referee states: a low number of (noninteracting) receptors, sparsely covering the flagellum (i.e. with a large distance between receptors compared to receptor size) entails a nonsaturated diffusive flux that, hence, depends on the number of receptors (Figure 1—figure supplement 1). For a fixed volume, the cylindrical geometry strengthens our assumption: the larger surface area of the cylinder gives a larger average distance between receptors and, hence, offsetting the saturation of the overall diffusive flux to higher receptor number. As a result, the flux strongly depends on the number of receptors, which is indeed one of the main points of our manuscript: it is incorrect to assume that the number of receptors is in fact large enough for all species for the perfect absorber approximation to be a valid one (see Figure 1—figure supplement 1). The sphere is the geometric 3D figure of smaller area for a given volume. Thus, by using the approximate spherical case, we are actually underestimating this effect, instead of overestimating as suggested by the reviewer. Such explanation has been now included in the manuscript.
References:
Acrivos A, Taylor TD. 1962. Heat and mass transfer from single spheres in stokes flow. Phys Fluids 5:387–394. doi:10.1063/1.1706630
https://doi.org/10.7554/eLife.50532.sa2Article and author information
Author details
Funding
Consejo Nacional de Ciencia y Tecnología (Fronteras 71)
 Alberto Darszon
 Adán Guerrero
Consejo Nacional de Ciencia y Tecnología (Ciencia basica 252213 y 255914)
 Alberto Darszon
 Adán Guerrero
Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (IA202417)
 Adán Guerrero
Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (IN205516)
 Alberto Darszon
Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (IN206016)
 Carmen Beltran
Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (IN215519)
 Carmen Beltran
Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (IN112514)
 Alberto Darszon
Ministerio de Economía y Competitividad (FIS201348444C21P)
 Idan Tuval
Ministerio de Economía y Competitividad (FIS201677692C21 P)
 Idan Tuval
Japan Society for the Promotion of Science (JSPS/236, ID no. S16172)
 Adán Guerrero
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
The authors thank Dr. Tatsu Yoshiro for providing the caged speract, and Drs. Hermes Gadêlha, David Smith and Nina Pastor for stimulating discussions and a critical reading of the manuscript. AG thanks Dr. Manabu Yoshida and Dr. Kaoru Yoshida for feedback regarding sperm chemotaxis in marine invertebrates, and to the Japan Society for the Promotion of Science (JSPS invitation fellowship for research in Japan to A.G., short term JSPS/236, ID no. S16172). AD performed part of this work while carrying out a Sabbatical at the Instituto Gulbenkian de Ciencia (IGC) supported by UNAM/DGAPA and IGC.
Ethics
Animal experimentation: All of the animals were handled according to approved institutional animal care and use committee protocols (# 44, 142, 188, 193, 285) of the Instituto de Biotecnología of the Universidad Nacional Autónoma de México.
Senior Editor
 Naama Barkai, Weizmann Institute of Science, Israel
Reviewing Editor
 Raymond E Goldstein, University of Cambridge, United Kingdom
Reviewers
 Jörn Dunkel, Massachusetts Institute of Technology, United States
 Robert Austin, Princeton University, United States
Publication history
 Received: July 25, 2019
 Accepted: March 6, 2020
 Accepted Manuscript published: March 9, 2020 (version 1)
 Version of Record published: March 24, 2020 (version 2)
Copyright
© 2020, RamírezGómez et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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