Sea urchin sperm exploit extremum seeking control to find the egg

Mahmoud Abdelgalil1,*, Yasser Aboelkassem2,3, Haithem Taha1

1 Mechanical and Aerospace Engineering Department, University of California at
Irvine, Irvine, CA 92697
2 College of Innovation and Technology, University of Michigan at Flint, Flint
MI 48502
3 Michigan Institute for Data Science, University of Michigan, Ann Arbor MI
48109

## 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.

## 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 [12]. 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 [2]. 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 [4, 6, 9]. This
feedback mechanism is mediated by a complex signaling pathway that regulates
the influx and efflux of calcium in the cell [10, 14]. 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 [12,
15].

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 [13]. 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. 2). 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.

### 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 [7]. The kinematics of a
rigid body is:

$\displaystyle\bm{\dot{p}}(t)$ $\displaystyle=\bm{R}(t)\bm{V},$
$\displaystyle\bm{\dot{R}}(t)$ $\displaystyle=\bm{R}(t)\bm{\widehat{\Omega}}$
(1)

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 [2]. A common model
of the effect of the chemotactic signaling pathway on the swimming kinematics
of sea urchin sperm is given by the relations:

$\displaystyle\bm{V}$
$\displaystyle=\left[\begin{array}[]{ccc}v&0&0\end{array}\right]^{\intercal},$
$\displaystyle\bm{\Omega}$
$\displaystyle=\left[\begin{array}[]{ccc}\omega_{\parallel}(t)&0&\omega_{\perp}(t)\end{array}\right]^{\intercal}$
(4)

where $v>0$ is constant, and the angular velocity components
$\omega_{\parallel}(t)$ and $\omega_{\perp}(t)$ are given by:

$\displaystyle\omega_{\parallel}\,(t)$
$\displaystyle=\omega_{\parallel,0}+\omega_{\parallel,1}\eta(t),$
$\displaystyle\omega_{\perp}(t)$
$\displaystyle=\omega_{\perp,0}+\omega_{\perp,1}\eta(t),$ (5)

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 [5, 6, 9].

### 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
$\bm{\Omega}_{0}$:

$\displaystyle\bm{\Omega_{0}}$
$\displaystyle=\left[\begin{array}[]{ccc}\omega_{\parallel,0}&0&\omega_{\perp,0}\end{array}\right]^{\intercal}$
(7)

and the average position and orientation as:

$\displaystyle\bm{\overline{p}}(t)$
$\displaystyle=\bm{p}(t)-\varepsilon\overline{\bm{R}}(t)\bm{\delta}(\omega
t),$ $\displaystyle\bm{\overline{R}}(t)$
$\displaystyle=\bm{R}(t)\,\text{exp}\big{(}\bm{\widehat{\Omega}}_{0}t\big{)}^{\intercal}$
(8)

whose dynamics can be exactly written as:

$\displaystyle\bm{\dot{\overline{p}}}(t)$
$\displaystyle=\bm{\overline{R}}(t)\bm{V}_{\eta}(\omega
t)\eta(t)\,\varepsilon+\bm{\overline{R}}(t)\bm{\overline{V}}$ (9)
$\displaystyle\bm{\dot{\overline{R}}}(t)$
$\displaystyle=\bm{\overline{R}}(t)\bm{\widehat{\Omega}}_{\eta}(\omega
t)\eta(t)$ (10)

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. 2 provides a block diagram description of the dynamical equations
[8]-[10] 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 [9] and [10]. 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.

Figure 1: 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.

Figure 2: 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,
$\zeta_{1}(0)=0.75c(\bm{p}(0)),\zeta_{1}(0)=1.25c(\bm{p}(0)),\rho(0)=1$. When
$l_{0}=200\,\mu\text{m}$, the velocities $v$, $\omega_{\perp,0}$ and
$\omega_{\parallel,0}$ agree with their experimentally observed values [5]. 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, 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.

### 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 [2] (relaxation). We model the
response of the signaling pathway by the difference between stimulation and
relaxation as in the following system of differential equations:

$\displaystyle\sigma_{1}\dot{\zeta}_{1}(t)=\zeta_{2}(t)-\zeta_{1}(t),\quad\quad\sigma_{2}\dot{\zeta}_{2}(t)=s(t)-\zeta_{2}(t)$
(11) $\displaystyle\zeta(t)=\zeta_{2}(t)-\zeta_{1}(t)$ (12)
$\displaystyle\mu\,\dot{\rho}(t)=\rho(t)\left(1-(\rho(t)\zeta(t))^{2}\right),\quad\eta(t)=\rho(t)\zeta(t)$
(13)

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 [11]-[12] represent a band pass filter. Equation
[13] 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 [2].
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 [8] and [9]):

$\displaystyle s(t)\approx s(t_{0})+\varepsilon\nabla
c(\bm{\overline{p}}(t_{0}))^{\intercal}\bm{\overline{R}}(t_{0})\left(\bm{\overline{V}}\omega\Delta
t+\bm{\delta}(\omega t)\right)$ (14)

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 [8].
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 [14] to the model of the signaling pathway given by
equations [11]-[13] leads to a response whose quasi-steady output $\eta(t)$
can be approximated by:

$\displaystyle\eta(t)$
$\displaystyle\approx\left(\rho_{1}\bm{\overline{V}}+\rho_{2}\bm{\delta}(\omega
t+\phi)\right)^{\intercal}\bm{\overline{R}}(t)\frac{\nabla
c(\bm{\overline{p}}(t))}{\lVert\nabla c(\bm{\overline{p}}(t))\lVert}$ (15)

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 [15] into the
average kinematics of sperm motion (i.e. equations [9] and [10]) 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. 2 shows a sample response of the system defined by
[1]-[5] under the feedback law defined by [11]-[13].

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
$\bm{\overline{p}}(t)$:

$\displaystyle\frac{d}{dt}\left(c(\bm{\overline{p}}(t))\right)$
$\displaystyle=-\gamma\,\omega_{\perp,0}\,\omega_{\perp,1}\sin(\phi)\lVert\nabla
c(\bm{\overline{p}}(t))\lVert,$ (16)

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 [16] is a
gradient ascent only if $\omega_{\perp,0}\,\omega_{\perp,1}\sin(\phi)<0$,
which is interestingly deduced in [6] 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 [9] and [10]. 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 [15, 3]; the current paper establishes a natural connection between
these algorithms and sperm chemotaxis.

## Discussion

Helical klinotaxis is a ubiquitous mode of taxis in microorganisms including
sperm cells [4]. 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 [14], and the binding of the SAP
molecules with receptors is a noisy process [11]. 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 [10, 1].
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. 2A), resulting in only slight
adjustment of the direction of average motion (the blue segments in Fig. 2B).
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. 2A), resulting in a vigorous
adjustment of the trajectory (the red segments in Fig. 2B). 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 [9, 11].

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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