Thermodynamic Limits of Sperm Swimming Precision

Sperm swimming is crucial to fertilize the egg, in nature and in assisted reproductive technologies. Modeling the sperm dynamics involves elasticity, hydrodynamics, internal active forces, and out-of-equilibrium noise. Here we give experimental evidence in favor of the relevance of energy dissipation for sperm beating fluctuations. For each motile cell, we reconstruct the time evolution of the two main tail's spatial modes, which together trace a noisy limit cycle characterized by a maximum level of precision $p_{\text{max}}$. Our results indicate $p_{\text{max}}\sim {10}^2{\text{s}}^{-1}$, remarkably close to the estimated precision of a dynein molecular motor actuating the flagellum, which is bounded by its energy dissipation rate according to the thermodynamic uncertainty relation. Further experiments under oxygen deprivation show that $p_{\text{max}}$ decays with energy consumption, as it occurs for a single molecular motor. Both observations are explained by conjecturing a high level of coordination among the conformational changes of dynein motors. This conjecture is supported by a theoretical model for the beating of an ideal flagellum actuated by a collection of motors, including a motor-motor nearest-neighbor coupling of strength $K$: When $K$ is small the precision of a large flagellum is much higher than the single motor one. On the contrary, when $K$ is large the two become comparable. Based upon our strong-motor-coupling conjecture, old and new data coming from different kinds of flagella can be collapsed together on a simple master curve.

Figure 1

Precision of stochastic clocks. Examples are shown, in different systems, of the coarse-grained coordinate $\theta \left(t\right)$ useful to define the precision $p$ of Brownian clocks. (a) Sketch of the supposed chemical cycle of the dynein ATP-ase, counterclockwise starting from the top: (i) rest (apo) state, (ii) ATP-binding, (iii) detachment of the stalk, (iv) ATP $\to$ $\text{ADP}+\text{P}$ reaction with rejoining of the stalk to the upper substrate, in a forward position, and (v) stroke of the linker (red spring) with consequent dragging of the upper substrate. (b) Sketch of the elastomechanical cycle of a sperm cell. The tail shape is approximated by a curve close to $A\left(t\right)cos\left(kx\right)+B\left(t\right)sin\left(kx\right)$. (c) Mean-square displacement from a trajectory generated by the numerical integration of the equation $\dot{\theta }=J+\sqrt{2D}\eta \left(t\right)$, with $\eta \left(t\right)$ white noise with unitary amplitude, $J=1$, and $D=0.1$. The insets show the trajectory $\theta \left(t\right)$ with a different time range, i.e., (i) $t\in [0,0.1]$, (ii) $t\in [0,1]$, and (iii) $t\in [0,10]$. It is evident how the mean current $\overline{\dot{\theta }}=J$ can be appreciated only at times much larger than $1/p=D/J^2$. (d) Successive snapshots [taken at 0.04-s intervals with an optical microscope (see Appendix pp1)] of the tail dynamical sequence in a caged sperm experiment. The purple box represents the region of interest and the red dot and red arrow identify the parameters $a$ and $b$ ($y$ position and slope of the tail close to the origin, respectively) which approximate $A$ and $B$ in Eq. (3), here corresponding to $(A,B)\sim (0,1)\to (-1,0)\to (0,-1)\to (1,0)$. The white arrows indicate the $\widehat{x}$ and $\widehat{y}$ directions (the $\widehat{z}$ direction is perpendicular to both). The white scale bar represents $10\mu \text{m}$.

Figure 2

Extracting precision from tail tracking. (a) Histogram of the positions in the plane $a\left(t\right)\text{−}b\left(t\right)$ for the 12 sperms with precision $p>20{\text{s}}^{-1}$. The light blue curve traces the $a\left(t\right)\text{−}b\left(t\right)$ path for a particular sperm in 0.2 s (10 frames). (b) Integrated phase-space current $\theta \left(t\right)$ for a few observed sperms. The inset shows the probability density function (PDF) (over all observed sperms) of the average current $J$. (c) PDF (over a 20-s acquisition), for a few sperms, of $\delta \theta =\theta (t+dt)-\theta \left(t\right)$ (shifted by the mean and scaled by the standard deviation), where $dt=0.02$ s. The inset shows a few autocorrelations $C\left(t\right)=\langle \widetilde{\delta \theta }\left(t\right)\widetilde{\delta \theta }\left(0\right)\rangle$, where $\widetilde{\delta \theta }=\delta \theta -\langle \delta \theta \rangle$. (d) Mean-square displacement of $\theta \left(t\right)$ for a given sperm and its fit according to the model $\text{MSD}\left(t\right)=2Dt+{\left(Jt\right)}^2$. The inset shows the PDF of diffusivity $D$ over all observed sperms. (e) Diffusivity $D$ versus average current $J$ together with decaying exponential fit $D\sim {10}^2{e}^{-J/25}{\text{s}}^{-1}$. (f) Precision $p$ versus the noise-to-signal ratio (NSR) computed from the signals $a\left(t\right)$ and $b\left(t\right)$. The inset shows $p$ versus the current $J$. In (e) and (f) error bars for $J$ and $D$ denote 3 times the standard deviation of $J$ and $D$ estimated when fitting the MSD by ${\chi }^2$ optimization. Data come from the observation of 54 different sperm cells, if not specified otherwise.

Figure 3

Theoretical model. (a) Sketch of the model. (b) Precision $p$ versus number of motors $N$ for different choices of the coupling parameter $K$, showing the $O\left(N\right)$ scaling for uncoupled motors ($K=0$) and the $O\left(1\right)$ scaling for large $K$. In all the simulations we have used $\alpha =\eta =0.5, k/\xi \mathrm{\Omega }=10$, and $\alpha N{U}_0/\mathrm{\Omega }{\ell }^2\xi =0.6$. The error bars are obtained to error propagation based upon the error in the measurement of $D$. Such error is the estimated standard deviations of $D$ in the nonlinear least-squares fit of the exponential decay of the phase correlation (see Appendix pp3).

Figure 4

Sperm precision and thermodynamics. Observations were made for caged sperms in experiments within a sealed chamber. (a) Decay of beating frequency [as measured from the spectrum peak of $a\left(t\right)$ and $b\left(t\right)$] with time. The average values of the frequencies at each hour of the experiment are also marked as white circles, together with a sigmoidlike fit $f\left(t\right)=c_1{e}^{-c_{2}t}s\left(t\right)+c_3{e}^{-c_{4}t}[1-s\left(t\right)]$ with sigmoid $s\left(t\right)={[1+e^{(t-t_0)/\tau }]}^{-1}$ and best-fit values $t_0=143, \tau =10.2, c_1=7.1, c_2=1.1\times {10}^{-3}, c_3=3.5$, and $c_4=2\times {10}^{-3}$. (b) Reduction of average current and diffusivity along time, shown as violin plots, each from an hour bin (e.g., 1 means all observation is done in the first hour, etc.). (c) Decay of precision with time, again in the form of a violin plot. The inset shows the decay of $p_{\text{max}}$ (estimated as the average of the top $25\%$ population) together with the normalized (by $N={10}^5$) power consumption extracted by the Taylor formula [Eq. (4)] using the frequency decay fit shown in Fig. 3, with amplitude fixed at the average observed value $5.2\mu \text{m}$. Error bars represent standard deviations normalized by the square root of the number of data in the aforementioned percentile. The sizes of samples in (b) and (c) are 41 sperms in the first hour, 24 in the second hour, 34 in the third hour, 25 in the fourth hour, and 16 in the fifth hour.

Figure 5

Collapse of correlation lengths in several experiments. The data from different previous experiments with sperms and flagella of C. reinhardtii algae are summarized here by plotting the correlation length ${\mathcal{L}}_{\text{TUR}}=\frac{\dot{W}}{p{k}_{B}T}$, discussed in Appendix pp2, versus the length of the flagella $N$ (both quantities are given as numbers of molecular motors). The data are also reported in Table 1. The light colored points allow one to distinguish the different experiments, while the circles identify the averages at each given flagellar length. Bars represent statistical errors. The dashed line marks the scaling law ${\mathcal{L}}_{\text{TUR}}\sim N$, which is expected under the hypothesis of strong coordination among adjacent motors in the axoneme.

Figure 6

Details about the tail-tracking procedure. (a) Examples of the signals $a\left(t\right)$ and $b\left(t\right)$ from trail tracking. One can see that $a\left(t\right)$ anticipates $b\left(t\right)$ of an angle between approximately $\pi /2$ and approximately $\pi$. (b) Spectra (modulus of the fast Fourier transform) of the signals $a\left(t\right)$ and $b\left(t\right)$. (c) Spectra after signal filtering. The orange and light blue dashed lines indicate the signal level and the noise level, respectively (the noise-to-signal ratio is defined as the ratio between the latter and the former).

Figure 7

Theoretical model. (a) and (d) Evolution of the system in the force-coordinate phase space: (a) $K=0$ and (d) $K=10$. The angle $\theta \left(t\right)$ measures the phase of this limit cycle after suitable rotation and normalization of the axis. (b) and (e) Evolution of $cos\left[\theta \right(t\left)\right]$ that allows us to evaluate the stability of the periods in the two cases: (b) $K=0$ and (e) $K=10$. (c) and (f) Real part of the autocorrelation of $e^{i\theta \left(t\right)}$ and exponential fit of approximately $e^{-Dt}$ of its envelope: (c) $K=0$ and (f) $K=10$. In all the simulations we have used $\alpha =\eta =0.5, k/\xi \mathrm{\Omega }=10$, and $\alpha N{U}_0/\mathrm{\Omega }{\ell }^2\xi =0.6$.