\section*{Abstract}
Sperm perform extremely demanding tasks with minimal mechanosensory capabilities. The cells must quickly navigate in a noisy environment and 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 discover that the navigation strategy of sea urchin sperm is a well-established adaptive control technique known as extremum seeking. This bridge between mathematical control theory and the biology of taxis in microorganisms is expected to deepen our understanding of the process. For example, it suggests the characterization of the signaling pathway as an adaptive band pass filter. Moreover, it will guide engineers in developing bio-inspired miniaturized robots for source seeking with minimal sensors.

\section*{Introduction}

Source seeking is the problem of locating an object that emits a scalar signal (e.g. temperature, light intensity, or chemical concentration), typically without global positioning information \cite{krstic2008extremum}. Many living organisms are routinely faced with the source seeking problem. A well studied example is that of sperm chemotaxis in sea urchins. To locate an egg in open water, sea urchin sperm evolved to swim up the gradient of the concentration field established by the diffusion of a chemoattractant sperm-activating peptide (SAP) released by sea urchin eggs \cite{alvarez2014computational}. Notably, the navigation strategy of sea urchin sperm is deterministic; the cells employ the mean curvature of the flagellum controlled by intracellular calcium as a steering feedback mechanism to swim in circular paths that drift in the direction of the gradient in 2D, and in helical paths that align with the gradient in 3D \cite{crenshaw1993orientation,friedrich2007chemotaxis,jikeli2015sperm}. This feedback mechanism is mediated by a complex signaling pathway that regulates the influx and efflux of calcium in the cell \cite{kaupp2003signal,priego2020modular}. We bridge a gap between mathematical control theory and the taxis of microorganisms by framing the search for the egg as a source seeking problem and demonstrating that the navigation strategy of sea urchin sperm is in fact a natural implementation of a well established adaptive control paradigm known as extremum seeking \cite{krstic2008extremum,scheinker2014extremum}.

The hallmarks of an extremum seeking solution to the source seeking problem are: i) injection of periodic perturbation signals to sample the local signal strength, ii) a filter that measures the instantaneous signal strength and extracts the oscillations due to the perturbation signals, iii) demodulation of the local gradient information from the filter's output, and iv) an integrator that biases the motion in the direction of the gradient \cite{krstic2000stability}. We show that the kinematics of the swimming pattern naturally provide the roles of the periodic perturbation, demodulation, and integration components of the standard extremum seeking loop (Fig. \ref{fig:blk_diag_ct}). The characterization of sea urchin sperm chemotaxis as an extremum seeking problem suggests modeling the signaling pathway as an adaptive band pass filter. In this manner, the swimming kinematics of sea urchin sperm, together with the signaling pathway, naturally constitute an extremum seeking strategy for chemotaxis.
\subsection*{Modeling the sperm motion} 
Swimming in the low Reynolds number fluid regime is dominated by viscous forces, which enables the use of kinematic models as good approximations to the motion of micro-swimmers, including sperm cells \cite{friedrich2010high}. The kinematics of a rigid body is:
\begin{align}
    \label{eq:orig_kin} \bm{\dot{p}}(t) &= \bm{R}(t)\bm{V}, & \bm{\dot{R}}(t) &= \bm{R}(t)\bm{\widehat{\Omega}}
\end{align}
where the vectors $\bm{V}$ and $\bm{\Omega}$ are the linear and angular velocity vectors as represented in the body frame, $\bm{\widehat{\Omega}}$ denotes the skew-symmetric matrix corresponding to the angular velocity vector $\bm{\Omega}$, $\bm{p}(t)$ is the instantaneous position of the body with respect to the origin of a fixed frame of reference, and $\bm{R}(t)$ is the instantaneous rotation matrix that relates the body frame to the fixed frame. In the case of sea urchin sperm, the mean curvature and torsion of the flagellar beating pattern, which are regulated by the chemotactic signaling pathway, control the angular velocities in the body frame \cite{alvarez2014computational}. A common model of the effect of the chemotactic signaling pathway on the swimming kinematics of sea urchin sperm is given by the relations:
\begin{align}
    \bm{V} &= \left[\begin{array}{ccc} v & 0 & 0 \end{array}\right]^\intercal, &\bm{\Omega} &= \left[\begin{array}{ccc} \omega_\parallel(t) & 0 & \omega_\perp(t) \end{array}\right]^\intercal
\end{align}
where $v>0$ is constant, and the angular velocity components $\omega_\parallel(t)$ and $\omega_\perp(t)$ are given by:
\begin{align}\label{eq:omega_def}
    \omega_\parallel\,(t) &= \omega_{\parallel,0} + \omega_{\parallel,1}\eta(t) , & \omega_\perp(t) &= \omega_{\perp,0} + \omega_{\perp,1} \eta(t) ,
\end{align}
with $\omega_{\perp,0},\omega_{\perp,1},\omega_{\parallel,0},\omega_{\parallel,1}$ as constant coefficients, and $\eta(\cdot)$ as a dynamic feedback term regulated by the signaling pathway \cite{crenshaw1996new,friedrich2007chemotaxis,jikeli2015sperm}. 
\subsection*{An extremum seeking loop} The constant angular velocity components $\omega_{\parallel,0}$ and $\omega_{\perp,0}$ lead to a periodic swimming pattern which injects periodic perturbations with zero average into the position and orientation of the cell. To study the average motion of the cell, we remove these periodic variations from the motion by defining the constant angular velocity vector $\bm{\Omega}_0$:
\begin{align}
    \bm{\Omega_0} &= \left[\begin{array}{ccc} \omega_{\parallel,0} & 0 & \omega_{\perp,0} \end{array}\right]^\intercal
\end{align}
and the average position and orientation as: 
\begin{align}
    \label{eq:avg_mot_1} \bm{\overline{p}}(t)&= \bm{p}(t)-\varepsilon \overline{\bm{R}}(t)\bm{\delta}(\omega t), & \bm{\overline{R}}(t)&= \bm{R}(t)\,\text{exp}\big(\bm{\widehat{\Omega}}_0 t\big)^\intercal
\end{align}
whose dynamics can be exactly written as:
\begin{align}
\label{eq:pos_kin}
    \bm{\dot{\overline{p}}}(t)&=\bm{\overline{R}}(t)\bm{V}_\eta(\omega t)\eta(t)\,\varepsilon +\bm{\overline{R}}(t)\bm{\overline{V}} \\
\label{eq:rot_kin}
    \bm{\dot{\overline{R}}}(t) &=\bm{\overline{R}}(t)\bm{\widehat{\Omega}}_\eta(\omega t)\eta(t)
\end{align} 
where $\omega=(\omega_{\perp,0}^2+\omega_{\parallel,0}^2)^{\frac{1}{2}}$ is the temporal frequency of the periodic swimming pattern, $\varepsilon = 1/\omega$, the vector $\bm{\overline{V}}$ is the average velocity vector represented in body coordinates without any feedback from the signaling pathway, the vector $\bm{\delta}(\omega t)$ is the periodic perturbation in the position injected by the swimming pattern, and the vectors $\bm{V}_\eta(\omega t)$ and $\bm{\Omega}_\eta(\omega t)$ are the feedback coefficients of the average kinematics (see the SI appendix for exact expressions). The periodic perturbation $\bm{\delta}(\omega t)$ has zero average, whereas the coefficients $\bm{V}_\eta(\omega t)$ and $\bm{\Omega}_\eta(\omega t)$ are periodic with non zero average. 

Fig. \ref{fig:blk_diag_ct} provides a block diagram description of the dynamical equations [\ref{eq:avg_mot_1}]-[\ref{eq:rot_kin}] representing the navigation strategy of sea urchin sperm, where $c(\bm{p})$ is the SAP concentration at the point $\bm{p}$ and the special integration symbol $\fint$ denotes the nonholonomic kinematic integrator defined by equations [\ref{eq:pos_kin}] and [\ref{eq:rot_kin}]. The block diagram reveals a vivid connection between sperm chemotaxis and extremum seeking.  In this diagram, most components (i.e. perturbation, demodulation and integration) of extremum seeking are clearly present. An immediate outcome of the proposed connection between sperm chemotaxis and extremum seeking is the realization that the signaling pathway must be playing the role of filtration. 
\begin{figure*}
    \centering
    \begin{minipage}[t]{0.475\linewidth}
    \centering
        \includegraphics[width=\columnwidth]{figs/Chemotaxis_Block_Diagram.pdf}
        \caption{\textbf{Block diagram description of sperm chemotaxis}. The swimming pattern injects periodic perturbations into the instantaneous position of the cell which leads to oscillations in the stimulus. The signaling pathway relays these periodic perturbations to the angular velocities through flagellar deformation. Then, the periodic feedback coefficients (i.e. $\bm{V}_\eta$ and $\bm{\Omega}_\eta$) of the dynamics of average motion demodulate the gradient information carried by the feedback signal $\eta(t)$ through signal multiplication. Finally, the kinematic integrator biases the motion in the direction of the gradient. Most hallmarks (i.e. periodic perturbation, demodulation, and integration) of extremum seeking  are apparent in this feedback loop. As such, the extremum seeking formulation suggests that the filtration role is played by the signaling pathway.} \label{fig:blk_diag_ct}
    \end{minipage}\hfill
    \begin{minipage}[t]{0.475\linewidth}
    \centering
    \includegraphics[width=\columnwidth]{figs/combined_off_res.pdf}
    \caption{\textbf{Sample response and trajectory in a radial concentration field.}  The simulation parameters are: $v=l_0\,\text{sec}^{-1},~\omega_{\perp,0}=-7\,\text{sec}^{-1},~\omega_{\parallel,0}=-5\,\text{sec}^{-1},~\sigma_1=2\omega^{-1},~\sigma_2=\omega^{-1},~\mu=\omega^{-1}/3,~\omega_{\perp,1}=-1,~\omega_{\parallel,1}=-5$, $l_0$ is a characteristic length, and the concentration field is $c(\bm{p})= l_0 \lVert\bm{p}\lVert^{-1}$. 
    The initial conditions are: $\bm{p}(0)=-(l_0,l_0,3.5l_0)$, $\bm{R}(0)$ is the identity matrix, $\zeta_1(0)=0.75 c(\bm{p}(0)),\zeta_1(0)=1.25 c(\bm{p}(0)), \rho(0)=1$. When $l_0=200\,\mu\text{m}$, the velocities $v$, $\omega_{\perp,0}$ and $\omega_{\parallel,0}$ agree with their experimentally observed values \cite{crenshaw1996new}. \textbf{A:} the stimulus signal $s(t)=c(\bm{p}(t))$ and the feedback signal $\eta(t)$ where black arrows indicate the intervals when the stimulus is not increasing and high amplitude steering response, \textbf{B:} the spatial trajectory of the cell $\bm{p}(t)$ where the black arrows indicate the segments of the trajectory where a strong deformation takes place.}
    \label{fig:sample_traj}
    \end{minipage}
\end{figure*}
\subsection*{Chemotactic sensing}
 Having established the connection between the chemotactic navigation strategy of sea urchin sperm and extremum seeking, which suggests the characterization of the dynamics of the signaling pathway as a band pass filter, it is interesting to connect such dynamics to the intracellular processes involved in sea urchin sperm chemotaxis. Experiments show that a step increase in the SAP concentration initiates rapid synthesis of cGMP in the cell which triggers a cascade of cellular events leading to the influx of calcium by opening calcium permeable channels (stimulation). Hydrolysis of cGMP and the efflux of calcium relax the response to resting levels \cite{alvarez2014computational} (relaxation). We model the response of the signaling pathway by the difference between stimulation and relaxation as in the following system of differential equations:
\begin{gather}\label{eq:pthway_1}
    \sigma_1\dot{\zeta}_1(t) = \zeta_2(t)-\zeta_1(t), \quad\quad
    \sigma_2\dot{\zeta}_2(t) = s(t)-\zeta_2(t) \\
    \label{eq:pthway_2}
    \zeta(t) = \zeta_2(t)-\zeta_1(t)\\
    \label{eq:pthway_3}
    \mu\,\dot{\rho}(t) = \rho(t)\left(1-(\rho(t)\zeta(t))^2\right),\quad 
    \eta(t) = \rho(t)\zeta(t)
\end{gather} 
where $\mu,\sigma_1,\sigma_2$ are positive constants, and $\sigma_2\leq\sigma_1$. We note that the signals $\zeta_2(t)$ and $\zeta_1(t)$ model the stimulation and relaxation processes, respectively. It is clear that equations [\ref{eq:pthway_1}]-[\ref{eq:pthway_2}] represent a band pass filter. Equation [\ref{eq:pthway_3}] models the adaptation of the signaling pathway. During the course of motion, the cell is exposed to a time varying stimulus $s(t)=c(\bm{p}(t))$ through the binding of SAP molecules with receptors along the flagellum \cite{alvarez2014computational}. This stimulus may be approximated by its Taylor series over an $\mathcal{O}(\varepsilon)$ time duration $t_0\leq t\leq t_0+\varepsilon$ (relying on the relations [\ref{eq:avg_mot_1}] and [\ref{eq:pos_kin}]):
\begin{equation}\label{eq:stim}
    \begin{aligned}
        s(t) \approx s(t_0)+\varepsilon \nabla c(\bm{\overline{p}}(t_0))^\intercal\bm{\overline{R}}(t_0)\left(\bm{\overline{V}}\omega \Delta t+ \bm{\delta}(\omega t)\right)
    \end{aligned}
\end{equation}
where $\Delta t=t-t_0$. The gradient information is modulated in the stimulus on two components: a ramp component and a harmonic component. One can simply extract the gradient information from the stimulus by time differentiation. However, a simple differentiator is notoriously known for high frequency noise amplification leading to deteriorated performance in the case of noise-corrupted signals which is the typical case in chemotaxis \cite{friedrich2008stochastic}. Hence, a band pass filter whose quasi steady state carries the gradient information is more robust in the presence of noise. Applying the stimulus $s(t)$ as in equation [\ref{eq:stim}] to the model of the signaling pathway given by equations [\ref{eq:pthway_1}]-[\ref{eq:pthway_3}] leads to a response whose quasi-steady output $\eta(t)$ can be approximated by:
\begin{align}\label{eq:eta_approx}
    \eta(t) &\approx  \left( \rho_1 \bm{\overline{V}}+  \rho_2 \bm{\delta}(\omega t+\phi)\right)^\intercal \bm{\overline{R}}(t)\frac{\nabla c(\bm{\overline{p}}(t))}{\lVert\nabla c(\bm{\overline{p}}(t))\lVert}
\end{align}
where $\rho_1$ and $\rho_2$ are non negative gain functions that are independent from the magnitude of the gradient, and $\phi$ is the phase lag introduced by the dynamics of the signaling pathway (see the SI appendix for exact expressions). Closing the feedback loop with equation [\ref{eq:eta_approx}] into the average kinematics of sperm motion (i.e. equations [\ref{eq:pos_kin}] and [\ref{eq:rot_kin}]) modifies the net motion of the cell via two mechanisms: i) the change in the orientation due to $\bm{\Omega}_\eta(\omega t)\eta(t)$, and ii) the bias in the direction of motion due to $\bm{V}_\eta(\omega t)\eta(t)$. A standard averaging analysis in the fast time variable $\sigma=\omega  t$ may be used to study the average motion of the cell under this feedback (see the SI appendix for more details). Fig. \ref{fig:sample_traj} shows a sample response of the system defined by [\ref{eq:orig_kin}]-[\ref{eq:omega_def}] under the feedback law defined by [\ref{eq:pthway_1}]-[\ref{eq:pthway_3}].

When the motion is confined to the plane (i.e. when $\omega_{\parallel,0}=\omega_{\parallel,1}=0$), the averaging analysis yields the following differential equation for the average position $\bm{\overline{p}}(t)$:
\begin{align}\label{eq:avg_conc}
    	\frac{d}{dt}\left(c(\bm{\overline{p}}(t))\right) &=- \gamma\,\omega_{\perp,0}\,\omega_{\perp,1} \sin(\phi)\lVert\nabla c(\bm{\overline{p}}(t))\lVert,
\end{align}
where $\gamma$ is a positive constant, which is a gradient based strategy. That is, the periodic swimming pattern of the sperm, along with the tuned band-pass filter dynamics of the signaling pathway, manage to guide the average motion of the sperm in the direction of the gradient without direct access to any gradient information, which is the typical task of any extremum seeking algorithm. It should be noted that the dynamics in equation [\ref{eq:avg_conc}] is a gradient ascent only if $\omega_{\perp,0}\,\omega_{\perp,1} \sin(\phi)<0$, which is interestingly deduced in \cite{friedrich2007chemotaxis} from a different perspective. The details for the 3D case are more intricate due to the nonholonomic nature of the kinematic integrator defined by equations [\ref{eq:pos_kin}] and [\ref{eq:rot_kin}]. Thus, we include the averaging analysis and its results for the 3D case in the SI appendix. Remarkably, similar versions of the navigation strategy presented here have been independently proposed as extremum seeking algorithms for nonholonomic vehicles \cite{scheinker2014extremum,cochran20093}; the current paper establishes a natural connection between these algorithms and sperm chemotaxis.
\section*{Discussion}
Helical klinotaxis is a ubiquitous mode of taxis in microorganisms including sperm cells \cite{crenshaw1993orientation}. In this manuscript, we established a connection between helical klinotaxis and the adaptive control paradigm known as extremum seeking. This connection sheds light on the role played by the chemotactic signaling pathway in sea urchin sperm and suggests the characterization of its dynamics as an adaptive band pass filter. The signaling pathway involved in the chemotaxis of sea urchin sperm is complex \cite{priego2020modular}, and the binding of the SAP molecules with receptors is a noisy process \cite{kromer2018decision}. However, it is remarkable that the observed response of the signaling pathway in sea urchin sperm exhibits the essential features of the response of a band pass filter \cite{kaupp2003signal,alvarez2012rate}. Moreover, our analysis and simulations show that when the average direction of motion is aligned with the gradient and the average stimulus is increasing, the periodic component in the response $\eta(t)$ of the signaling pathway model is attenuated (the blue intervals in Fig. \ref{fig:sample_traj}A), resulting in only slight adjustment of the direction of average motion (the blue segments in Fig. \ref{fig:sample_traj}B). On the other hand, when the average direction of motion is mostly orthogonal to the gradient and the average stimulus is non increasing, the periodic component is amplified  (the red intervals in Fig. \ref{fig:sample_traj}A), resulting in a vigorous adjustment of the trajectory (the red segments in Fig. \ref{fig:sample_traj}B). As such, the suggested characterization of the signaling pathway as an adaptive band pass filter naturally reproduces the experimentally observed switching behavior between low gain and high gain steering modes, also known as the ``on'' and ``off'' responses, in sea urchin sperm 3D chemotaxis \cite{jikeli2015sperm,kromer2018decision}.

It is intriguing that all these observations of the behavior of the signaling pathway corroborate its characterization as an adaptive band-pass filter --- an immediate implication of the proposed extremum seeking formulation of the navigation strategy of sea urchin sperm.

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