Sea urchin sperm exploit extremum seeking control to find the egg

Sperm cells perform extremely demanding tasks with minimal capabilities. The cells must quickly navigate in a noisy environment to find an egg within a short time window for successful fertilization without any global positioning information. Many research efforts have been dedicated to derive mathematical principles that explain their superb navigation strategy. Here we show that the navigation strategy of sea urchin sperm, also known as helical klinotaxis, is a natural implementation of a well-established adaptive control paradigm known as extremum seeking. This bridge between control theory and the biology of taxis in microorganisms is expected to deepen our understanding of the process. For example, the formulation leads to a coarse-grained model of the signaling pathway that offers new insights on the peculiar switching-like behavior between high- and low-gain steering modes observed in sea urchin sperm. Moreover, it may guide engineers in developing bioinspired miniaturized robots with minimal sensors.

Figure 1

A block diagram description of the simplest extremum seeking control scheme as represented by Eqs. (1).

Figure 2

The geometry of helical swimming. (a) The helical nature of the trajectory suggests decomposing the motion $(\mathbf{x},\mathbf{R})$ into an average part along the helical centerline $(\overline{\mathbf{x}},\overline{\mathbf{R}})$ and a periodic excursion $\overline{\mathbf{R}}\mathbf{d}\left(t\right)$ that is orthogonal to the centerline. (b) The direction of the gradient $\check{\nabla }c$ can be decomposed into two parts, one along the helical centerline $\overline{\mathbf{h}}$ (denoted as ${\check{\nabla }}_{\parallel }c$), and another orthogonal to it (denoted as ${\check{\nabla }}_{\perp }c$) which is contained in the same plane as the direction $\overline{\mathbf{q}}\left(t\right)$ of the periodic excursion $\overline{\mathbf{R}}\mathbf{d}\left(t\right)$.

Figure 3

A block diagram description of the chemotactic navigation of sperm represented by Eqs. (9) and (12).

Figure 4

The three cases of the behavior of the signaling pathway illustrated on a sample trajectory projected on the $xy$-plane, the response $\eta$, the average instantaneous concentration $c\left(\overline{\mathbf{x}}\right)$, and the angle $\psi ={cos}^{-1}\left({\overline{\mathbf{h}}}^{⊺}\check{\nabla }c\right)$ (in degrees) between the gradient and the average direction of motion $\overline{\mathbf{h}}$, in a radial concentration field $c\left(\mathbf{x}\right)=1/(1+0.5|\mathbf{x}{|}^2)$. The initial position is taken as ${\mathbf{x}}_0=(6,1,0)$, and the initial orientation is $\mathbf{R}\left(t_0\right)=\text{exp}(2\pi {\widehat{\mathbf{e}}}_2/5)$, where ${\widehat{\mathbf{e}}}_2=(0,1,0)$. The rest of the initial conditions and parameter values are in the table.