Present address: ]Pulmonary, Critical Care and Sleep Medicine, Department of
Internal Medicine, Yale School of Medicine, 300 Cedar Street, New Haven, CT
06519, USA Present address: ]Sensing Electromagnetic Plus Corp., 2450
Embarcadero Way, Palo Alto, CA-94303, USA

# Adaptive phototaxis of Chlamydomonas and the evolutionary
transition to multicellularity in Volvocine green algae

Kyriacos C Leptos<EMAIL_ADDRESS>Maurizio Chioccioli
<EMAIL_ADDRESS>[  Silvano Furlan<EMAIL_ADDRESS>[
Adriana I Pesci<EMAIL_ADDRESS>Raymond E Goldstein
<EMAIL_ADDRESS>Department of Applied Mathematics and
Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3
0WA, UK

###### Abstract

A fundamental issue in biology is the nature of evolutionary transitions from
unicellular to multicellular organisms. Volvocine algae are models for this
transition, as they span from the unicellular biflagellate Chlamydomonas to
multicellular species of Volvox with up to 50,000 Chlamydomonas-like cells on
the surface of a spherical extracellular matrix. The mechanism of phototaxis
in these species is of particular interest since they lack a nervous system
and intercellular connections; steering is a consequence of the response of
individual cells to light. Studies of Volvox and Gonium, a $16$-cell organism
with a plate-like structure, have shown that the flagellar response to
changing illumination of the cellular photosensor is adaptive, with a recovery
time tuned to the rotation period of the colony around its primary axis. Here,
combining high-resolution studies of the flagellar photoresponse with 3D
tracking of freely-swimming cells, we show that such tuning also underlies
phototaxis of Chlamydomonas. A mathematical model is developed based on the
rotations around an axis perpendicular to the flagellar beat plane that occur
through the adaptive response to oscillating light levels as the organism
spins. Exploiting a separation of time scales between the flagellar
photoresponse and phototurning, we develop an equation of motion that
accurately describes the observed photoalignment. In showing that the adaptive
time scale is tuned to the organisms’ rotational period across three orders of
magnitude in cell number, our results suggest a unified picture of phototaxis
in green algae in which the asymmetry in torques that produce phototurns arise
from the individual flagella of Chlamydomonas, the flagellated edges of Gonium
and the flagellated hemispheres of Volvox.

##  I Introduction

A vast number of motile unicellular and multicellular eukaryotic
microorganisms exhibits phototaxis, the ability to steer toward a light
source, without possessing an image-forming optical system. From
photosynthetic algae [1] that harvest light energy to support their metabolic
activities to larvae of marine zooplankton [2] whose upward phototactic motion
enhances their dispersal, the light sensor in such organisms is a single unit
akin to one pixel of a CCD sensor or one rod cell in a retina [3]. In
zooplanktonic larvae there is a single rhabdomeric photoreceptor cell [4]
while motile photosynthetic microorganisms such as green algae [5] have a
“light antenna” [6], which co-localizes with a cellular structure called the
eyespot, a carotenoid-rich orange stigma. For these simple organisms, the
process of vectorial phototaxis, motion in the direction of a source rather
than in response to a light gradient [7], relies on an interplay between the
detection of light by the photosensor and changes to the actuation of the
apparatus that confers motility, namely their one or more flagella. Evolved
independently many times [8], the common sensing/steering mechanism seen
across species involves two key features.

The first attribute is a photosensor that has directional sensitivity,
detecting only light incident from one side. It was hypothesized long ago [6]
that in green algae this asymmetry could arise if the layers of carotenoid
vesicles behind the actual photosensor act as an interference reflector. In
zooplankton this “shading” role is filled by a single pigment cell [4]. This
directionality hypothesis was verified in algae by experiments on mutants
without the eyespot, that lacked the carotenoid vesicles [9], so that light
could be detected whatever its direction. Whereas wild-type cells performed
positive phototaxis (moving toward a light source), the mutants might naively
have been expected to be incapable of phototaxis. Yet, they exhibited negative
phototaxis, a fact that was explained as a consequence of an additional effect
first proposed earlier [10]; the algal cell body functions as a convex lens
with refractive index greater than that of water. Thus, a greater intensity of
light falls on the photosensor when it was illuminated from behind than from
the front, and a cell facing away from the light erroneously continues
swimming in that direction, as if it were swimming toward the light.

The second common feature of phototactic microorganisms is a natural swimming
trajectory that is helical. Spiral swimming has been remarked upon since at
least the early 1900s, when Jennings [11] suggested that it served as a way of
producing trajectories that are straight on the large scale, while
compensating for inevitable asymmetries in the body shape or actuation of
cilia, and Wildman [12] presciently observed that chirality of swimming and
ciliary beating must ultimately be understood in terms of the genetic program
contained within chromosomes. While neither offered a functional purpose
related to phototaxis, Jennings did note earlier [13, 14] that when organisms
swim along regular helices they always present the same side of their body to
the outside. This implies that during regular motion the photosensor itself
also has a fixed relationship to the helix.

In Chlamydomonas, motility derives from the breaststroke beating of two
oppositely-oriented flagella emanating from near the anterior pole of the cell
body, as depicted in Fig. 1. The flagella, termed cis and trans for their
proximity to the eyespot, define a plane, the unit normal to which is the
vector $\hat{\bf e}_{1}$. Historical uncertainties around the precise three-
dimensional swimming motion of Chlamydomonas were resolved with the work of
Kamiya and Witman [15], the high-speed imaging study of Rüffer and Nultsch
[16] and later work by Schaller et al. [17], who together demonstrated three
features: (i) the eyespot is typically located on the equatorial plane of the
cell, midway between $\hat{\bf e}_{1}$ and the vector $\hat{\bf e}_{2}$ that
lies within the flagellar plane, pointing toward the cis flagellum, (ii) cells
rotate counterclockwise (when viewed from behind) the axis $\hat{\bf e}_{3}$
at frequency $f_{\mathrm{r}}\sim 1.5-2.5\,$Hz ($f_{r}=1.67\pm 0.35\,$Hz in a
recent direct measurement [18]), and (iii) positively phototactic cells swim
along helices such that the eyespot always faces outward. The rotation around
$\hat{\bf e}_{3}$ was conjectured to arise from a small non-planarity of the
beat, as has been recently verified [19], while helical swimming arises from
rotation around $\hat{\bf e}_{1}$ due to a slight asymmetry in the two
flagellar beats.

Figure 1: Phototaxis in Chlamydomonas. Cell geometry (in box) involves the
primary axis $\hat{\bf e}_{3}$ around which the cell spins at angular
frequency $\omega_{3}$, the axis $\hat{\bf e}_{2}$ in the flagellar beat
plane, and $\hat{\bf e}_{1}=\hat{\bf e}_{2}\times\hat{\bf e}_{3}$. As the cell
swims and spins around $\hat{\bf e}_{3}$, its eyespot (red) moves in an out of
the light shining in the direction of the blue arrows. This periodic light
stimulation leads to alternating cis and trans flagella dominance, producing
rotations around $\bf{\hat{e}_{1}}$ and hence a phototactic turn.

It follows from the above that the eyespot of a cell whose swimming is not
aligned to the light receives an oscillating signal at angular frequency
$\omega_{3}=2\pi f_{\mathrm{r}}$. Detailed investigation into the effect of
this periodic signal began with the work of Rüffer and Nultsch, who used cells
immobilized on micropipettes to enable high-speed cinematography of the
waveforms. Their studies [20, 21] of beating dynamics in a negatively-
phototactic strain showed the key result that the cis and trans flagella
responded differently to changing light levels by altering their waveforms in
response to the periodic steps-up and steps-down in signals that occur as the
cell rotates. This result led to a model for phototaxis [17] that divides
turning into two phases (Fig. 1): phase I, in which the eyespot moves from
shade to light, causing the trans flagellum to increase transiently its
amplitude relative to the cis flagellum, and phase II, in which the eyespot
moves from light to shade, leading to transient beating with the opposite
asymmetry. Both phases lead to rotations around $\hat{\bf e}_{1}$, and turns
toward the light. The need for an asymmetric flagellar response was shown in
studies of the mutant ptx1 [22, 23], which lacks calcium dependent flagellar
dominance [24] and can not do phototaxis.

These transient responses were studied further [25] through the photoreceptor
current (PRC) that can be measured in the surrounding fluid. Subjecting a
suspension of immotile cells (chosen to avoid movements) to rectified
sinusoidal light signals that mimic those received by a rotating cell, they
found that the PRC amplitude displays a maximum as a function of frequency,
with a peak close to the body rotation frequency $f_{r}$. This “tuning” of the
response curve was investigated in more detail—in a negatively-phototactic
strain—in the important work of Josef, et al. [26], who projected the image of
the cell onto a quadrant photodiode whose analog signal could be digitized at
up to $4000$ samples per second. While this device did not allow detailed
imaging of the entire waveform, it was able to capture changes in the forward
reach of the two flagella (termed the “front amplitude”) over significantly
longer time series than previous methods. Combined with later work that
analyzed the signals within the framework of linear systems analysis [27],
these studies showed how each of the two flagella exhibits a distinct, peaked
frequency response.

From the original measurements of transient PRCs induced by step changes in
light levels [25], it was evident that the response in time was biphasic and
adaptive —a rapid rise in signal accompanied by slower recovery phase back to
the resting state —and the presence of two timescales is implicit in the
existence of the peak in the frequency response. More recently, measurements
of the flagella-driven fluid flow around colonies of the multicellular alga
Volvox carteri [28] showed again this adaptive response, which could be
described quantitatively by a model previously to describe chemotaxis of both
bacteria [29] and spermatozoa [30]. In a suitably rescaled set of units, the
two variables $p$ and $h$ in this model respond to a signal $s(t)$ through the
coupled ODEs

$\displaystyle\tau_{r}\dot{p}$ $\displaystyle=s-h-p$ (1a)
$\displaystyle\tau_{a}\dot{h}$ $\displaystyle=s-h,$ (1b)

where $p$ governs some observable, $h$ represents hidden biochemistry
responsible for adaptation, $\tau_{r}$ is the rapid response time, and
$\tau_{a}$ is the slower adaption time. In bacteria, the adaptive response is
exhibited by the biochemical network governing rotation of flagella, while for
sperm curvature of the swimming path was altered linearly with $p$ in response
to a chemoattractant.

The model (1) was incorporated into a theory of Volvox phototaxis using a
coarse-grained description of flagella-driven flows akin to the squirmer model
[31], with a dynamic slip velocity ${\bf u}(\theta,\phi,t)$ as a function of
spherical coordinates on the colony surface. Without light stimulation, the
velocity is an axisymmetric function ${\bf u}_{0}(\theta)$ that varies with
the polar angle $\theta$, and is dominated by the first mode
$u_{1}\propto\sin\theta$ [32]. As (1) is meant to describe the fluid flow
associated with flagella of each of the somatic cells on the surface, it is
introduced into the slip-velocity model through response fields
$p(\theta,\phi,t)$ and $h(\theta,\phi,t)$ over the entire surface. Experiments
indicate that the photoresponse is accurately represented the form

${\bf u}(\theta,\phi,t)={\bf
u}_{0}(\theta)\left[1-\beta(\theta)p(\theta,\phi,t)\right],$ (2)

where the parameter $\beta$ encodes the latitude-dependent photoresponse of
the flagella (strong at the anterior of the colony, weak in its posterior).
The swimming trajectories were then obtained from integral relationships
between the slip velocity and the colony angular velocity [33].

Figure 2: Master plot of adaptive time scales in Volvocine green algae. For
each of Chlamydomonas (this paper), Gonium [34], and Volvox [28] the
dimensionless product of the rotation frequency $f_{r}$ around the primary
body-fixed axis and the flagellar adaptive time $\tau_{a}$ is plotted as a
function of the organism radius $R$ (bottom axis) and typical cell number $N$
(top axis).

Statistical analysis of many Volvox colonies shows that there is tuning of the
response in that the product $f_{r}\tau_{a}\approx 1$ ($f_{r}\tau_{a}=1.20\pm
0.44$) [28], as indicated in Fig. 2. The significance of the product
$f_{r}\tau_{a}$ being of order unity can be understood as follows: when a
region of somatic cells rotates to face a light source, the fluid flow it
produces will decrease as $p$ rapidly increases on a time scale $\tau_{r}$,
and if the time $\tau_{a}$ it takes for $p$ to recover is comparable to the
colony rotation period $1/f_{r}$ then the fluid flow along the dark side will
be stronger than than on the light side, and the colony turns to the light.

A similar tuning phenomenon is found with Gonium [34], a member of the
Volvocales typically composed of $16$ cells arranged in a flat sheet as in
Fig. 2. The flagella of the four central cells beat in a Chlamydomonas-like
breaststroke waveform that propels the colony in the direction of the body-
fixed axis $\hat{\bf e}_{3}$ perpendicular to the sheet. The flagella of the
outer $12$ cells beat at an angle with respect to the plane; their dominant
in-plane component rotates the colony at frequency $f_{r}$ about $\hat{\bf
e}_{3}$, while the out-of-plane component adds to the propulsive force of the
central cells. Experiments show that the peripheral cells display the same
kind of biphasic, adaptive response as do Volvox colonies. This light-induced
“drop-and-recover response” produces an axial force component $f_{\parallel}$
from the peripheral flagella of the form

$f_{\parallel}(\theta,t)=f_{\parallel}^{(0)}\left[1-p(\theta,t)\right],$ (3)

where $f_{\parallel}^{(0)}$ is the uniform component in the absence of
photostimulation. Again, the directionality of the eyespot sensitivity leads
to a photoresponse $p$ that is greatest (and $f_{\parallel}$ that is smallest)
for those cells facing the light, and this nonuniformity in $f_{\parallel}$
leads to a net torque about an in-plane axis which, balanced by rotational
drag, leads to phototactic turning toward the light. The data for Gonium also
supports tuning, with the product $f_{r}\tau_{a}=0.95\pm 0.50$, as shown in
Fig. 2.

In the present work we complete a triptych of studies in Volvocine algae by
examining Chlamydomonas, the unicellular ancestor of all others [35]. Our
purpose is to construct, in a manner that parallels that for Volvox and
Gonium, a theory that links the photoresponse of flagella to the trajectories
of cells turning to the light. We base the description on the kinematics of
rigid bodies, where the central quantities are the angular velocities around
body-fixed axes. This model bears some similarity to an earlier study of
phototaxis [36], in which the asymmetric beating of flagella—modelled as
spheres moving along orbits under the action of prescribed internal forces
responding to light on the eyespot—was related to rotations about body-fixed
axes, but the response to light was taken to be instantaneous and non-
adaptive.

Results reported here on Chlamydomonas show that $f_{r}\tau_{a}$ is close to
unity ($f_{r}\tau_{a}=1.27\pm 0.41$), from which we infer that tuning is an
evolutionarily conserved feature spanning three orders of magnitude in cell
number and nearly two orders of magnitude in organism radius (Fig. 2). We
conclude that, in evolutionary transitions to multicellularity in the
Volvocine algae, the ancestral photoresponse found in Chlamydomonas required
little modification in order to work in vastly larger multicellular spheroids.
The most significant change is basal body rotation [37] in the multicellulars
in order that the two flagella on each somatic cell beat in parallel, rather
than opposed as in Chlamydomonas. In Gonium, this arrangement in the
peripheral cells leads to colony rotation, while for the somatic cells of
Volvox the flagellar beat plane is tilted with respect to meridional lines,
yielding rotation around the primary colony axis.

The presentation below proceeds from small scales to large, following a
description in Sec. II of experimental methods used in our studies of the
flagellar photoresponse of immobilized cells at high spatio-temporal
resolution, and of methods for tracking phototactic cells. In Sec. III we
arrive at an estimate of rotations about the body-fixed axis $\hat{\bf e}_{1}$
arising from transient flagellar asymmetries induced by light falling on the
eyespot, and thus a protocol to convert measured flagella dynamics to angular
velocities within the adaptive model. Section IV incorporates those results
into a theory of phototactic turning. Exploiting a separation of time scales
between individual flagella beats, cell rotation, and phototactic turning, we
show how the continuous-time dynamics can be approximated by an iterated map,
and allow direct comparison to three-dimensional trajectories of phototactic
cells. By incorporating an adaptive dynamics at the microscale, one can
examine the speed and stability of phototaxis as a function of the tuning
parameter $f_{r}\tau_{a}$ and deduce its optimum value. These results explain
the many experimental results summarized above, and put on a mathematical
basis phenomenological arguments [17] about the stability of phototactic
alignment in Chlamydomonas.

Figure 3: Experimental methods. Setups to measure (a) the flagellar
photoresponse of cells immobilized on a micropipette, and (b) swimming
trajectories of phototactic cells in a sample chamber immersed in an outer
water tank to minimize thermal convection, as described in text.

##  II Experimental methods

Culture conditions. Wild type Chlamydomonas reinhardtii cells (strain CC125
[38]) were grown axenically under photoautotrophic conditions in minimal media
[39], at 23∘C under a 100 µE$\cdot$s${}^{-1}\cdot$m-2 illumination in a
diurnal growth chamber with a $14:10\,$h light-dark cycle.

Flagellar photoresponse of immobilized cells. The flagellar photoresponse of
C. reinhardtii was captured at high spatio-temporal resolution using the
experimental setup shown in Fig. 3(a), which builds on previous studies [40,
28, 41]. Cells were prepared as described previously [41] – centrifuged,
washed and gently pipetted into a bespoke observation chamber made of
polydimethylsiloxane (PDMS). Chambers were mounted on a Nikon TE2000-U
inverted microscope with a $\times 63$ Plan-Apochromat water-immersion long-
working-distance (LWD) objective lens (441470-9900; Carl Zeiss AG, Germany).
Cells were immobilized via aspiration using a micropipette (B100-75-15;
Sutter, USA) that was pulled to a ⌀5-µm tip, and the flagellar beat plane was
aligned with the focal plane of the objective lens via a rotation stage. Video
microscopy of immobilized cells was performed using a high speed camera
(Phantom v341; Vision Research, USA) by acquiring $15\,$s movies at $2,000$
fps.

The light used for photostimulation of cells was provided by a $470$ nm Light
Emitting Diode (LED) (M470L3; Thorlabs, USA) that was controlled via an LED
driver (LEDD1B; Thorlabs, USA), coupled to a ⌀$50\,$µm-core optical fiber
(FG050LGA; Thorlabs, USA). This fiber is much smaller than that used in
previous versions of this setup in order to accommodate the smaller size of a
Chlamydomonas cell relative to a Volvox spheroid. The LED driver and the high-
speed camera were triggered through a data-acquisition card (NI PCIe-6343;
National Instruments, USA) using in-house programs written in LabVIEW 2013
(National Instruments, USA), for both step- and frequency-response
experiments. Calibration of the optical fiber was performed as follows: A
photodiode (DET110; Thorlabs, USA) was used to measure the total radiant power
$W$ emerging from the end of the optical fiber for a range of voltage output
values (0-5 V) of the LED driver. The two quantities were plotted and fitted
to a power-law model which was close to linear.

Cells were stimulated at frame $2896$ ($\approx 1.45\,$s into the recording).
A light intensity of $\approx$ 1 µE$\cdot$s${}^{-1}\cdot$m-2 (at $470\,$nm)
was found empirically to give the best results in terms of reproducibility,
sign, i.e. positive phototaxis, and quality of response; we conjecture that
the cells could recover in time for the next round of stimulation. For the
step-response experiments, biological replicates were $n_{\mathrm{cells}}=3$
with corresponding technical replicates $n_{\mathrm{tech}}=\\{4,2,2\\}$. For
the frequency-response experiments, biological replicates were
$n_{\mathrm{cells}}=3$ with each cell stimulated to the following amplitude-
varying frequencies: 0.5 Hz, 1 Hz, 2 Hz, 4 Hz and 8 Hz. Only the cells that
showed a positive sign of response for all 5 frequencies are presented here.
This was hence the most challenging aspect of the experimental procedure.

To summarize, the total number of high-speed movies acquired was
$n_{\mathrm{mov[hs]}}=24$. All downstream analysis of the movies was carried
out in MATLAB. Image processing and flagella tracking was based on previous
work [41], and new code was written for force/torque calculations and
flagellar photoresponse analysis.

Phototaxis experiments on free-swimming cells. Three-dimensional tracking of
phototactic cells was performed using the method described previously [42]
with the modified apparatus shown in Fig. 3(b). The experimental setup
comprised of a sample chamber suspended in an outer water tank to eliminate
thermal convection. The modified sample chamber was composed of two acrylic
flanges (machined in-house) that were clamped in a watertight manner onto an
open-ended square borosilicate glass tube ($2\,\text{cm}\times
2\,\text{cm}\times 2.5\,\text{cm}$; Vetrospec Ltd, UK). This design allowed a
more accurate and easy calibration of the field of view and a simpler and
better loading system of the sample via two barbed fittings. This new design
also minimized sample contamination during experiments. Two $6$ megapixel
charge-coupled device (CCD) cameras (Prosilica GT2750; Allied Vision
Technologies, Germany), coupled to two InfiniProbeTM TS-160s (Infinity, USA)
with Micro HM objectives were used to achieve a larger working distance than
in earlier work ($48\,$mm vs. $38\,$mm) at a higher total magnification of
$\times 16$. The source of phototactic stimulus was a $470\,$nm blue-light LED
(M470F1; Thorlabs, USA) coupled to a solarization-resistant optical fiber
(M22L01; Thorlabs, USA) attached to an in-house assembled fiber collimator
that included a ⌀12.7 mm plano-convex lens (LA1074-A; Thorlabs, USA).
Calibration of the collimated optical fiber was performed similarly to the
experiments with immobilized cells. The calibration took account of the
thickness of the walls of the outer water tank and the inner sample chamber,
as well as the water in between.

The two CCD cameras and the blue-light LED used for the stimulus light were
controlled using LabVIEW 2013 (National Instruments, USA) including the image
acquisition driver NI-IMAQ (National Instruments, USA). The cameras were
triggered and synchronized at a frame rate of $10\,$Hz via a data-acquisition
device (NI USB 6212-BNC; National Instruments, USA). For every tracking
experiment ($n_{\mathrm{mov[3d]}}=6$), two $300$-frame movies were acquired
(side and top) with the phototactic light triggered at frame $50$ ($5\,$s into
the recording). The intensity of the blue-light stimulus was chosen to be
either $5$ or 10 µE$\cdot$s${}^{-1}\cdot$m-2. To track the cells we used in-
house tracking computer programs written in MATLAB as described in [42].
Briefly, for every pair of movies cells were tracked in the side and top
movies corresponding to the $xz$-plane and in the $xy$-plane respectively. The
two tracks were aligned based on their $x$-component to reconstruct the three-
dimensional trajectories. The angle between the cell’s directional vector and
the light was then calculated for every time point.

##  III Flagellar dynamics

Figure 4: Flagellar photoresponse of an immobilized cell after a step-up in
light. Light is from the left (blue arrow), toward the eyespot. The panels
show overlaid flagellar waveforms of a single beat (a) in the dark and (b)
starting at $50\,$ms ($52.5\,$ms) for the cis (trans) flagellum after the step
increase in light. Scale bar is $5\,$µm.

###  III.1 Forces and torques

We begin by examining the response of the two flagella of an immobilized
Chlamydomonas cell to a change in the light level illuminating the eyespot.
Figure 4 and Supplementary Video 1 [43] shows a comparison between the
unstimulated beating of the flagella and the response to a simple step up from
zero illumination. These are presented as overlaid flagellar waveforms during
a the single beat in the dark and one that started $50\,$ms after the step. In
agreement with previous work cited in Sec. I [21, 26], we see that the
transient response involves the trans flagellum reaching further forward
toward the anterior of the cell, while the cis waveform contracts
dramatically. The photoresponse is adaptive; the marked asymmetry between the
cis and trans waveforms decays away over $\sim 1-2\,$s, restoring the
approximate symmetry between the two. This adaptive timescale is much longer
than the period $T_{\mathrm{b}}\sim 20\,$ms of individual flagellar beats.

Table 1: Geometry of flagellar beats. Data are from the present study except for the flagellum radius. Quantity | Symbol |  Mean ${\bm{\pm}}$ SD
---|---|---
flagellum length | $L$ |  $13.5\,\pm\,0.8\,$µm
flagellum radius [46] | $a$ |  $0.125\,$µm
cell body radius | $R$ |  $4.4\,\pm\,0.3\,$µm
beat frequency | $\hat{f}_{\mathrm{b}}$ |  $43.2\,\pm\,8.0\,$Hz
anchor angle | $\varphi_{a}$ | $(0.05\pm 0.01)\pi$
initial angle | $\hat{\varphi}_{0}$ | $(0.26\,\pm\,0.05)\pi$
sweep angle | $\hat{\varphi}_{\mathrm{b}}$ | $(0.33\,\pm\,0.06)\pi$
Figure 5: Flagellar beat cycles. (a) Angles $\Theta$ on each flagellum (red
for cis, blue for trans) relative to symmetry axis $\hat{\bf e}_{3}$ (green)
of the cell are used to define the cycle. Scale bar is $5\,$µm. (b) Typical
time series of the two angles.

We wish to relate transient flagellar asymmetries observed with immobilized
cells, subject to time-dependent light stimulation, to cell rotations that
would occur for freely-swimming cells. We begin by examining the beating of
unstimulated cells to provide benchmark observations. We analyze high-speed
videos to obtain the waveforms of flagella of length $L$, radius $a$, in the
form of moving curves ${\bf r}(\lambda,t)$ parameterized by arclength
$\lambda\in[0,L]$ and time. Within Resistive Force Theory (RFT) [44, 45], and
specializing to planar curves, the hydrodynamic force density on the filament
is

${\bf f}(\lambda,t)=-\left(\zeta_{\perp}\hat{\bf n}\hat{\bf
n}+\zeta_{\parallel}\hat{\bf t}\hat{\bf t}\right)\cdot{\bf r}_{t}(\lambda,t),$
(4)

where $\hat{\bf t}={\bf r}_{s}$ and $\hat{\bf n}$ (with $\hat{\bf
n}_{i}=\epsilon_{ji}\hat{\bf t}_{j}$) are the unit normal and tangent at
$\lambda$, and $\zeta_{\perp}$ and $\zeta_{\parallel}$ are drag coefficients
for motion perpendicular and parallel to the filament. We assume the
asymptotic results $\zeta_{\perp}=4\pi\mu/c_{\perp}$ and
$\zeta_{\parallel}=2\pi\mu/c_{\parallel}$, where $c_{\perp}=\ln({\cal
L}\sqrt{e})$ and $c_{\parallel}=\ln({\cal L}/\sqrt{e})$, with ${\cal L}=L/a$
the aspect ratio. Table 1 gives typical values of the cell parameters; with
${\cal L}\approx 108$, we have $c_{\perp}\approx 5.2$ and
$c_{\parallel}\approx 4.2$. To complete the analysis, we adopt the convention
shown in Fig. 5(a) to define the start of a beat, in which chords drawn from
the base to a point of fixed length on each flagellum define angles
$\Theta_{\mathrm{cis,trans}}$ with respect to $\hat{\bf e}_{3}$, Local minima
in $\Theta_{\mathrm{cis,trans}}(t)$ [Fig. 5(b)] define the beat endpoints.

Figure 6: Flagellar forces and torques of unstimulated cells. Propulsive force
(a) and torque about the cell center (c) of cis (red) and trans (blue)
flagella during beat cycle of a representative cell, with cycle averages
indicated by dashed lines. (b) and (d) show boxplots of peak values (∗) and
beat-average quantities ($\langle\rangle$) computed from $n=48$ flagella in 24
movies.

Using a hat ($\,\hat{}\,$) to denote quantities measured without
photostimulation, Fig. 6 shows the results of this analysis for the propulsive
component of the total force,

$\hat{\cal F}(t)=\hat{\bf e}_{3}\cdot\int_{0}^{L}\\!d\lambda\,{\bf
f}(\lambda,t)$ (5)

and the torque component around $\hat{\bf e}_{1}$,

$\hat{\mathcal{T}}(t)=\hat{\bf e}_{1}\cdot\int_{0}^{L}\\!d\lambda\,{\bf
r}\times{\bf f}(\lambda,t),$ (6)

where ${\bf r}$ is measured from the cell center. The smoothness of the data
arises from the large number of beats over which the data are averaged. The
force $\hat{\mathcal{F}}$ varies sinusoidally in time, offset from zero due to
the dominance of the power stroke over the recovery stroke, with a peak value
$\hat{\mathcal{F}}^{*}=16.0\pm 5.2\,$pN and mean over a beat period of
$\langle\hat{\mathcal{F}}\rangle=3.6\pm 1.1\,$pN per flagellum. These findings
are in general agreement with previous studies of Chlamydomonas cells held by
an optical trap [47], micropipette-held cells [48], measurements on swimming
cells confined in thin fluid films [49] and in bulk [50], and more recent work
using using micropipette force sensors [51].

Aggregating all data obtained on the unstimulated torques exhibited by cis and
trans flagella, we find a peak magnitude $\hat{\mathcal{T}}^{*}=187\pm
48\,$pN$\cdot$µm and cycle-average mean value
$\langle\hat{\mathcal{T}}\rangle=82\pm 17\,$pN$\cdot$µm. As a consistency
check we note that the ratio torque/force should be an interpretable length,
and we find $\hat{\mathcal{T}}^{*}/\hat{\mathcal{F}}^{*}\sim 12\,$µm, a value
that is very close to the mean flagellar length $L=13.5\,$µm. Across a sample
size of $n=24$ we find the sum

$\langle\hat{\mathcal{T}}\rangle=\langle\hat{\mathcal{T}}_{\rm
cis}\rangle+\langle\hat{\mathcal{T}}_{\rm trans}\rangle,$ (7)

is $\langle\hat{\mathcal{T}}\rangle=-6.2\pm 15.9\,$ pN$\cdot$µm, and thus is
consistent with symmetry of the two flagella, but the data clusters into two
clear groups; a cis-dominant subpopulation ($n=9$), with
$\langle\hat{\mathcal{T}}\rangle^{\rm cis}=11.2\pm 7.0\,$pN$\cdot\upmu$m and a
trans-dominant subpopulation ($n=15$) with
$\langle\hat{\mathcal{T}}\rangle^{\rm trans}=-16.6\pm 8.6\,$pN$\cdot$µm. As
discussed below, such differences would generally distinguish between
positively and negatively phototactic cells, and the presence of both in our
cellular population likely reflects the detailed growth and acclimatization
conditions. For consistency, we focus here only on positively phototactic
cells. For them, the residual torque provides information on the pitch and
amplitude of helical trajectories of unstimulated cells.

Figure 7: The pivoting-rod model of the power stroke.

###  III.2 Heuristic model of flagellar beating

Below we extend the quantification of flagellar beating to a transient
photoresponse like that in Fig. 4, with the goal of inferring the angular
velocity $\omega_{1}$ around $\hat{\bf e}_{1}$ that a freely-swimming cell
would experience, and which leads to a phototurn. The constant of
proportionality between the torque and the angular velocity is an effective
rotational drag coefficient that can be viewed as one of a small number of
parameters of the overall phototaxis problem. Our immediate goal is to develop
an estimate of this constant to provide a consistency check on the final
theory in its comparison to experiment.

A simple model of the power stroke shown in Fig. 7 can be used to understand
the peak values $\hat{\mathcal{F}}^{*}$ and $\hat{\mathcal{T}}^{*}$: a
straight flagellum attached at angle $\varphi_{a}$ to a spherical body of
radius $R$, whose beat angle $\varphi(t)$ sweeps from $\hat{\varphi}_{0}$ to
$\hat{\varphi}_{0}+\hat{\varphi}_{b}$. Table 1 and Fig. 8 summarize data on
these geometric quantities. Relative to the center $\mathrm{C}$ a point at
arclength $\lambda\in[0,L]$ is at position

${\bf r}(\lambda,\varphi)={\bf R}_{a}+\lambda\hat{\bf t}(\varphi),$ (8)

where ${\bf R}_{a}=R\left[\sin{\varphi_{a}}\hat{\bf
e}_{x}+\cos{\varphi_{a}}\hat{\bf e}_{y}\right]$ is the vector $\mathrm{CO}$
and $\hat{\bf t}(\varphi)=\sin{\varphi}\,\hat{\bf
e}_{x}+\cos{\varphi}\,\hat{\bf e}_{y}$ is the unit tangent. The velocity of a
point on the filament is ${\bf
r}_{t}(\lambda,\varphi)=-\lambda\dot{\varphi}\hat{\bf n}$.

Figure 8: Distributions of geometric quantities for flagellar beats, computed
from $n=24$ movies: (a) instantaneous flagellar beat frequency
$\hat{f}_{\mathrm{b}}$ (b) flagellar length $L$, (c) cell-body radius $R$,
(d-f) per-beat flagella-line initial angle $\hat{\varphi}_{0}$, sweep angle
$\hat{\varphi}_{b}$ and anchor angle $\varphi_{a}$.

The integrated force of the pivoting rod in Fig. 5 is

$\bm{{\cal F}}=\frac{1}{2}\zeta_{\perp}L^{2}\dot{\varphi}\hat{\bf n},$ (9)

Considering the sinusoidal variation of the quantities in Figs. 6(a,c), we
estimate $\varphi$ by the lowest mode that has vanishing speed at beginning
and end of the power stroke,

$\varphi(t)=\hat{\varphi}_{0}+\frac{1}{2}\hat{\varphi}_{b}\left[1-\cos\left(\pi
t/\eta T_{b}\right)\right],$ (10)

for $t\in[0,\eta T_{b}]$, where $T_{b}$ is the full beat period and
$\eta\simeq 0.7$ is the fraction of the period occupied by the power stroke.
Using the data in Table 1, and the fact that the maximum projected force
occurs very close to the time when $\varphi=\pi/2$, we obtain the estimate

$\hat{\cal F}^{*}\sim\frac{\pi^{2}\mu L^{2}f_{b}\hat{\varphi}_{b}}{\eta
c_{\perp}}\sim 22\,{\rm pN},$ (11)

which is $\sim\\!1$ s.d. above the experimental mean. While it is not
surprising that the pivoting rod model overestimates the propulsive force
relative to the actual undulating flagellum, the fact that this overestimate
is small indicates that the essential physics is contained in (11).

Further heuristic insight into the flagellar forces produced can be gained by
estimating the resultant motion of the cell body, assumed to be a sphere of
radius $R$. This requires incorporating the drag of the body and that due to
the flagella themselves. A full treatment of this problem requires going
beyond RFT to account for the effect of flows due to the moving body on the
flagellar and vice versa. In the spirit of the rod model, considerable insight
can be gained in the limit of very long flagella, where the fluid flow is just
a uniform translational velocity ${\bf u}(t)$ and the velocity of a point on
the rod is

${\bf r}_{t}(\lambda,\varphi)=-\lambda\dot{\varphi}\hat{\bf n}+{\bf u}.$ (12)

Symmetry dictates that the net force from the downward sweeps of two mirror-
image flagella is along $\hat{\bf e}_{3}$, as is the translational velocity
${\bf u}=u\hat{\bf e}_{y}$ of the cell body. Adding mirror-image copies of the
force (9) and the drag force on the body $-\zeta u\hat{\bf e}_{y}$, where
$\zeta=6\pi\mu R$ is the Stokes drag coefficient for a sphere of radius $R$,
the condition that the total force vanish yields

$u(t)=v\frac{\sin\varphi\sin\left(\pi t/\eta
T_{b}\right)}{1+d_{\perp}\sin^{2}\varphi+d_{\parallel}\cos^{2}\varphi},$ (13)

where $d_{\perp}=2\zeta_{\perp}L/\zeta=4L/3Rc_{\perp}$ and
$d_{\parallel}=2L/3Rc_{\parallel}$. The speed $v$ is given by the maximum
force (11) as

$v=\frac{2\hat{\mathcal{F}}^{*}}{\zeta}=\frac{\pi f_{b}\phi_{b}L^{2}}{3\eta
c_{\perp}R},$ (14)

and is independent of the viscosity $\upmu$, as it arises from a balance
between the two drag-induced forces of flagellar propulsion and drag on the
spherical body, the latter ignoring the contribution in the denominator of
(13) from flagellar drag. For typical parameters, $d_{\perp}\approx 0.8$, and
the denominator is $\approx\\!1.8$ when $u$ is maximized (at $\varphi=\pi/2$),
while $v\sim 540\,$µm/s. Thus, the peak swimming speed during the power stroke
would be $u^{*}\sim v/1.8\sim 300\,$µm/s, consistent with measurements [49],
which also show that over a complete cycle, including the recovery stroke, the
mean speed $\bar{u}\sim u^{*}/4$. We infer that $\bar{u}\sim 75\,\upmu$m/s,
consistent with observations [49, 52].

We now use the rod model to estimate the maximum torque
$\hat{\mathcal{T}}^{*}$ produced on an immobilized cell, to compare with the
RFT calculation from the experimental waveforms. As in (12), the force density
on the moving filament is ${\bf f}=\zeta_{\perp}\lambda\dot{\varphi}\hat{\bf
n}$, the torque density is $\left({\bf R}+\lambda\hat{\bf t}\right)\times{\bf
f}$, and the integrated torque component along $\hat{\bf e}_{1}$ is

$\hat{\mathcal{T}}=\frac{1}{2}\zeta_{\perp}\dot{\varphi}R^{3}\left(\ell^{2}\cos\left(\varphi-\varphi_{a}\right)+\frac{2}{3}\ell^{3}\right).$
(15)

where $\ell=L/R$ and $\varphi$ is again given by (10). The two terms in (15),
scaling as $RL^{2}$ and $L^{3}$, arise from the distance offset from the cell
body and the integration along the flagellar length, respectively.

Examining this function numerically we find that its peak occurs approximately
midway through the power stroke, where $\varphi-\varphi_{a}\simeq\pi/3$,
leading to the estimate

$\hat{\mathcal{T}}^{*}=\frac{\pi^{2}\mu f_{b}\hat{\varphi}_{b}R^{3}}{\eta
c_{\perp}}\left(\frac{1}{2}\ell^{2}+\frac{2}{3}\ell^{3}\right)\sim 250\,{\rm
pN}\cdot\upmu{\rm m}.$ (16)

and, for average torque,

$\langle\hat{\mathcal{T}}\rangle\simeq\frac{2}{\pi}{\mathcal{T}}^{*}\sim
160\,{\rm pN}\cdot\upmu{\rm m}.$ (17)

Here again these estimates are slightly more than $1$ s.d. above the
experimental value, giving further evidence that the rod model is a useful
device to understand the scale of forces and torques of beating flagella.

The essential feature of (13) is an effective translational drag coefficient
$\tilde{\zeta}$ that is larger than that of the sphere due to the presence of
the very beating flagella that cause the motion. For flagella oriented at
$\varphi=\pi/2$, $\tilde{\zeta}=\zeta+2\zeta_{\perp}L$, a form that reflects
the extra contribution from transverse drag on the two flagella. We now
consider the analogous rotational problem and estimate an effective rotational
drag coefficient $\tilde{\zeta}_{r}$ in terms of the bare rotational drag
$\zeta_{r}=8\pi\mu R^{3}$ for a sphere. If we set in rotational motion at
angular speed ${\bm{\Omega}}$ a sphere with two flagella attached at angles
$\pm\varphi_{a}$, the velocity of a rod segment at $\lambda$ is
${\bm{\Omega}}\times{\bf r}$ and the calculation of the hydrodynamic force and
torque proceeds as before, yielding

$\displaystyle\frac{\tilde{\zeta_{r}}}{\zeta_{r}}=1$
$\displaystyle+\frac{\ell\cos^{2}(\varphi-\varphi_{a})+\ell^{2}\cos(\varphi-\varphi_{a})+\ell^{3}/3}{c_{\perp}}$
$\displaystyle+\frac{\ell\sin^{2}(\varphi-\varphi_{a})}{2c_{\parallel}}.$ (18)

For typical parameters, $\zeta_{r}\sim 2.1\,$pN$\cdot\upmu$m$\cdot$s, and the
added drag of the flagella is significant; the ratio in (18) varies from
$4.8-3$ as $\varphi$ varies from $\hat{\varphi}_{0}$ to
$\hat{\varphi}_{0}+\hat{\varphi}_{b}$. At the approximate peak of the power
stroke we find $\tilde{\zeta}_{r}\simeq 9\,$pN$\cdot\upmu$m$\cdot$s, a value
we use in further estimates below.

The effective rotational drag coefficient can be used to estimate the
unstimulated angular speed due to the small torque imbalance noted below (7)
for the cis-dominant subpopulation,

$\hat{\omega}_{1}\sim\frac{\langle\hat{\mathcal{T}}\rangle^{\rm
cis}}{\tilde{\zeta}_{r}}\simeq 1\,{\rm s}^{-1},$ (19)

which can be compared to the angular speed $|\omega_{3}|\simeq 10\,$s-1 of
spinning around the primary axis. In Sec. IV we show that the small ratio
$\hat{\omega}_{1}/|\omega_{3}|\sim 0.1$ implies that the helices are nearly
straight.

###  III.3 Adaptive dynamics

The results of the previous section constitute a quantitative understanding of
the phototactic torques produced within a given flagellar beat, which
typically lasts $20-25\,$ms. As mentioned previously, the timescale for the
full photoresponse associated with a change in light levels falling on the
eyespot is considerably longer, on the order of $0.5\,$s. This separation of
timescales is illustrated in Fig. 9, where we have schematically shown the
time-resolved, oscillating phototactic torque of each of the two flagella, the
signed sum, and its running average. It is precisely because of the separation
of timescales between the rapid beating and both the slow response and the
slow phototurns that a theory developed in terms of the beat-averaged torques
is justified.

Figure 9: Schematic of flagellar photoresponse. A step-up in light at $t=0$
(green) leads to a biphasic decrease in the mean value and oscillation
amplitude of the cis phototorque (red), and a biphasic increase in the
magnitude of the mean value of the trans phototorque (blue). The net torque
(purple), the signed sum of the two contributions, has a biphasic response in
both the oscillation amplitude and its running mean value. Figure 10: Dynamics
of the flagellar photoresponse. (a) The mean (dark blue) and standard
deviation (light-blue) of the proxy torque during a step-up stimulus for one
cell ($n_{\mathrm{tech}}$ = 4) fitted to (25a) (red line). (b) Inset showing
fitted ($\tau_{r}$, $\tau_{a}$) pairs for $n_{\mathrm{cells}}$ = 4 upon step-
up stimulation. (c) As in (b) but for the delay time $\tau_{d}$ and time of
maximum photoresponse amplitude $t^{*}$.

In the following, we measure phototorques relative to the unstimulated state
of the cell, and define the two (signed) beat-averaged quantities

$\delta{\mathcal{T}}_{\rm cis}=\langle{\mathcal{T}}_{\rm
cis}\rangle-\langle\hat{\mathcal{T}}_{\rm cis}\rangle,\ \ \ \
\delta{\mathcal{T}}_{\rm trans}=\langle{\mathcal{T}}_{\rm
trans}\rangle-\langle\hat{\mathcal{T}}_{\rm trans}\rangle,$ (20)

and their sum, the net beat-averaged phototactic torque

${\mathcal{T}}_{\rm p}=\delta{\mathcal{T}}_{\rm cis}+\delta{\mathcal{T}}_{\rm
trans}.$ (21)

${\mathcal{T}}_{\rm p}>0$ when the cis flagellum beats more strongly and
${\mathcal{T}}_{\rm p}<0$ when the trans flagellum does. Our strategy is to
determine ${\mathcal{T}}_{\rm p}$ from experiment on pipette-held cells and to
estimate the resulting angular speed $\omega_{1}^{\rm p}$ using
$\tilde{\zeta}_{r}$.

The scale of net torques expected during a transient photoresponse can be
estimated from the pivoting-rod model. From step-up experiments such as that
shown in Fig. 4, we observe that there are two sweep angles $\varphi_{b}^{\rm
cis,trans}$ whose difference $\Delta\varphi_{b}^{*}=\max\\{\varphi_{b}^{\rm
cis}-\varphi_{b}^{\rm trans}\\}$ can be used in (17) to obtain
${\mathcal{T}}_{\rm p}^{*}$, the maximum value of the beat-averaged sum
(corresponding to the most negative value of the purple running mean in Fig.
9). Averaging over $8$ photoresponse videos, we find
$\Delta\varphi_{b}^{*}\sim-\pi/14$, which yields the estimate

${\mathcal{T}}_{\rm p}^{*}\sim-34\,{\rm pN}\cdot\upmu{\rm m}.$ (22)

From the effective drag coefficient the corresponding peak angular speed in
such a photoresponse is

$\omega_{1}^{*}\sim\frac{{\cal T}_{\rm p}^{*}}{\tilde{\zeta}_{r}}\sim-4\,{\rm
s}^{-1}.$ (23)

To put this in perspective, consider the photoalignment of an alga swimming
initially perpendicular to a light source. If sustained continuously, complete
alignment would occur in a time $(\pi/2)/\omega_{1}^{*}\sim 0.4\,$s, whereas
our observations suggest a longer timescale of $\sim 2\,$s. This will be shown
to follow from the variability of $\omega_{1}$ during the trajectory in accord
with an adaptive dynamics.

While the estimate in (22) gives a guide to scale of the torques responsible
for phototurns, we may calculate them directly within RFT from flagellar
beating asymmetries in the same manner as in the unstimulated case. Figure 10
shows the response of a single cell to a step-up in illumination (of which
Fig. 4 is a snapshot), in which the results are presented both in terms of
${\mathcal{T}}_{\rm p}$ and the estimated $\omega_{1}^{\rm p}$. To obtain
these data, the oscillating time series of cis and trans torques were
processed to obtain beat-averaged values whose sum yields the running average,
as in Fig. 9. The overall response is $<0$, indicating that the trans
flagellum dominates, and the peak value averaged over multiple cells of
$-37\pm 12\,$pN$\cdot$nm is consistent with the estimate in (22). The biphasic
response, with a rapid increase followed by a slow return to zero, is the same
form observed in Volvox [28] and Gonium [34].

We now argue that the adaptive model (1) used for those cases can be recast as
an evolution equation for the angular speed itself, setting $p=\omega_{1}$ and
$s=gI$, with $I$ the light intensity and $g$ a proportionality constant,

$\displaystyle\tau_{r}\dot{\omega_{1}}$ $\displaystyle=s-h-\omega_{1}$ (24a)
$\displaystyle\tau_{a}\dot{h}$ $\displaystyle=s-h.$ (24b)

For constant $s$ the system (24) has the fixed point
$(\omega_{1}^{*},h^{*})=(0,s)$. If $\omega_{1}=h=s=0$ for $t\leq 0$, followed
by $s=s_{0}$ for $t>0$, then

$\displaystyle\omega_{1}(t)$
$\displaystyle=\frac{s_{0}}{1-\rho}\left(e^{-t/\tau_{a}}-e^{-t/\tau_{r}}\right)$
(25a) $\displaystyle h(t)$
$\displaystyle=s_{0}\left(1-e^{-t/\tau_{a}}\right),$ (25b)

where $\rho=\tau_{r}/\tau_{a}$. The result (25a), illustrated in Fig. 11(a)
for the case $\rho=0.1$ and a square pulse of duration long compared to
$\tau_{a}$, shows clearly the biphasic response of the data in Fig. 10. This
behavior is like two coupled capacitors charging and discharging one another,
particularly in the limit $\rho\ll 1$. At early times, $h$ remains small and
$\omega_{1}$ relaxes toward $s_{0}$ with the rapid timescale $\tau_{r}$.
Later, when $t\sim\tau_{a}$, $h$ relaxes toward $s_{0}$, and $\omega_{1}$
relaxes instead toward zero, completing the pulse.

Figure 11: Dynamics of the adaptive model. (a) Response of the variables $p$
and $h$ to a square pulse of stimulus (green), for $\rho=0.1$. (b) Response to
rectified sinusoids.

After a step up, $h$ has relaxed to $s_{0}$, and if $s$ is then stepped down
to zero, $\omega_{1}$ rapidly tends toward $-s_{0}$, then later reverses its
negative growth and returns to zero. If, as in Fig. 11, the pulse width is
much larger than $\tau_{a}$, the step down response is simply the negative of
the step-up response. For smaller step duration, the step-down response is
still negative, but is not a mirror image of the step-up dynamics. Taking
$s_{0}$ to be positive, this antisymmetric response implies that as the
eyespot rotates into the light there is a step-up response with
$\omega_{1}>0$, corresponding to transient cis flagellar dominance, and when
the eyespot rotates out of the light then $\omega_{1}<0$, associated with
trans flagellum dominance. This is precisely the dynamics shown in Fig. 1 that
allows monotonic turning toward the light as the cell body rotates.

Note that the adaptive dynamics coupling $\omega_{1}$, $s$, and $h$ is left
unchanged by the simultaneous change of signs $\omega_{1}\to-\omega_{1}$,
$h\to-h$, and $s\to-s$. This symmetry allows us to address positive and
negative phototaxis in a single model, for if a step-up in light activates a
transient dominant trans flagellum response in the cell orientation of Fig. 2,
with $\omega_{1}<0$, we need only take $s<0$.

Since the model (24) is constructed so that $\omega_{1}$ is forced by the
signal $s$, the opposite-sign response to step-up and step-down signals is not
an obvious feature. Yet, in the standard manner of coupled first-order ODEs,
the hidden variable $h$ can be eliminated, yielding a single, second-order
equation for $\omega_{1}$. It can be cast in the simple form

$\left(\tau_{r}\frac{d}{dt}+1\right)\left(\tau_{a}\frac{d}{dt}+1\right)\omega_{1}=\tau_{a}\dot{s},$
(26)

which is explicitly forced by the derivative of the signal, thus driven
oppositely during and and down steps.

Previous studies [27] found a measurable time delay $\tau_{d}$ between the
signal and the response that, in the language of the adaptive model, is
additive with the intrinsic offset determined by the time scales $\tau_{r}$
and $\tau_{a}$. This can be captured by expressing the signal in (24) as
$s(t-\tau_{d})$. The maximum amplitude of $\omega_{1}(t)$ then occurs at the
time

$t^{*}=\frac{\tau_{r}}{1-\rho}\ln(1/\rho)+\tau_{d},$ (27)

at which point the amplitude is $s_{0}A(\rho)$, where

$A(\rho)=\frac{\rho^{\rho/1-\rho}-\rho^{1/1-\rho}}{1-\rho}.$ (28)

A fit to the step-response data yields $\tau_{d}=28\pm 11\,$ms and
$t^{*}=97\pm 18\,$ms. This delay between stimulus and maximum response has a
geometric interpretation. The angle through which the eyespot rotates in the
time $t^{*}$ is $|\omega_{3}|t^{*}=(0.28\pm 0.05)\pi$, which is very nearly
the angular shift $\kappa=\pi/4$ of the eyespot location from the ($\hat{\bf
e}_{2}$-$\hat{\bf e}_{3}$) flagellar beat plane (Fig. 1). Since
$\omega_{3}<0$, the eyespot leads the flagellar plane and thus $t^{*}$ is the
time needed for the beat plane to align with the light. In this configuration,
rotations around $\hat{\bf e}_{1}$ are most effective [17, 27].

Since the function $A(\rho)$ decreases monotonically from unity at $\rho=0$,
we identify the maximum angular speed $\omega_{1}^{*}$ attainable for a given
a stimulus as $s_{0}$. With $s_{0}=gI$, we can remove $g$ from the problem by
instead viewing $\omega_{1}^{*}$ as the fundamental parameter, setting

$s(t)=\omega_{1}^{*}(I).$ (29)

As indicated in (29), there is surely a dependence of $\omega_{1}^{*}$ on the
light intensity, not least because the rotational speed will have a clear
upper bound associated with the limit in which the subdominant flagellum
ceases beating completely during the transient photoresponse.

Turning now to the oscillating light signals experienced by freely-rotating
cells, the directionality of the eyespot implies that the signal will be a
half-wave rectified sinusoid (HWRS). Figure 11(b) shows the response of
$\omega_{1}$ to two single half-period signals of this type. Compared to the
square pulses of equal duration and maximum [Fig. 11(a)] the maximum response
amplitude is reduced due to the lower mean value and slower rise of the
signal. The frequency response of the adaptive model is most easily deduced
from (26), and if $s(t)=\tilde{s}e^{i\omega t}$, then there will be a
proportionate amplitude $\tilde{\omega}_{1}$. We define the response
$\mathscr{R}(\omega)=\tilde{\omega}_{1}/\tilde{s}$, gain
$G(\omega)=|\mathscr{R}(\omega)|$ and phase shift $\chi=\tan^{-1}[{\rm
Im}(\mathscr{R})/{\rm Re}(\mathscr{R})]$. These are

$\displaystyle G(\omega)$
$\displaystyle=\frac{\alpha}{\sqrt{\left(1+\alpha^{2}\right)\left(1+\rho^{2}\alpha^{2}\right)}}$
(30a) $\displaystyle\chi$ $\displaystyle=\pi+\chi_{0}-\delta,\ \ \ \
\chi_{0}=\tan^{-1}\left(\frac{1-\rho\alpha^{2}}{\left(1+\rho\right)\alpha}\right),$
(30b)

where $\alpha=\omega\tau_{a}$, $\delta=\omega\tau_{d}$ and the additive term
of $\pi$ in the phase represents the sign of the overall response. Figures
12(a,b) show these quantities as a function of the stimulus frequency $\omega$
for various values of $\rho$. The peak frequency is at
$\alpha^{*}=1/\sqrt{\rho}$, or $\omega^{*}=1/\sqrt{\tau_{r}\tau_{a}}$, at
which $G(\omega^{*})=1/(1+\rho)$ and $\chi=\pi-\omega^{*}\tau_{d}$. Fig. 12(a)
shows that the peak is sharp for large $\rho$ and becomes much broader as
$\rho\to 0$. The peak amplitude decays in a manner similar to (28) for a step
response [Fig. 12(c)].

Figure 12: Dynamics of the adaptive model. (a) Gain (30a) for various values
of $\rho=\tau_{r}/\tau_{a}$. (b) As in (a), but for the phase shift $\chi$
(30b), with $\tau_{d}=0$. (b) Comparison of peak amplitude for step-up and
oscillatory forcing as a function of $\rho$.  Figure 13: Frequency response of
immobilized cells. (a) Measured beat-average phototactic torque determined
from RFT (for positive phototaxis) at five stimulus frequencies (0.5, 1, 2, 4
and 8 Hz) for $n_{\mathrm{cells}}=3$ (blue) fitted to (30a) (red line). (b) As
in (a), but for the phase $\chi$ of the response, fitted to (30b). The
photoresponse shown (in blue) for three different stimulus frequencies (in
green): (c) $1$ Hz, (d) $2$ Hz, and (e) $4$ Hz.

The peaked response function amplitude (30a) and phase shift (30b) are
qualitatively similar to those obtained experimentally by Josef, et al. [27],
who analyzed separately the cis and trans responses and found distinct peak
frequencies for the two, and investigated the applicability of more complex
frequency-dependent response functions than those in (30). In the spirit of
the analysis presented here we do not pursue such detailed descriptions of the
flagellar responses, but it would be straightforward to incorporate them as we
discuss in Sec. IV.2.

Using the same protocol as for the step function response in Fig. 10, we
measured the frequency dependent photoresponse by subjecting cells to an
oscillating light intensity at five distinct frequencies, analyzing the
transient waveforms using RFT and determining the beat-average torque
magnitude. The results of this study (Fig. 13), were fit to the form (30a),
from which we obtained the time constants $\tau_{r}=0.009\pm 0.002\,$s and
$\tau_{a}=0.52\pm 0.10\,$s, and thus $\rho=0.02$. This strong separation
between response and adaptation time scales is consistent with that seen under
the step response (Fig. 10(a,b)) and leads to the broad peak of the frequency
response curve. The peak frequency ($\simeq 2\,$ Hz) is in very close
agreement with recent direct measurements of rotation frequency about the cell
body axis on free-swimming cells [18]. The phase data shown in Fig. 13(b) are
well-described by the adaptive model with the parameters determined from the
fit to the amplitude, with a time delay of $\tau_{d}=38\pm 5\,$ms, a value
that is consistent with that obtained from the step response. Note that for
frequencies $\omega$ near $\omega^{*}$ and for $\rho\ll 1$, the phase has the
simple form

$\chi\simeq\pi-2\tau_{r}\left(\omega-\omega^{*}\right)-\omega^{*}\tau_{d}+\cdots.$
(31)

This result shows that while negative detuning from $\omega^{*}$ by itself
increases the phase above $\pi$, the time delay can be a more significant
contribution, leading to $\chi<\pi$. Such is indeed the case in Fig. 13, where
the peak frequency is $\simeq 2\,$Hz, but $|\omega_{3}|\tau_{d}\simeq 0.13\pi$
and $\chi(\omega^{*})<\pi$.

##  IV Dynamics of phototactic turns

###  IV.1 Helices, flagellar dominance, and eyespot shading

We now consider the larger length scales associated with the swimming
trajectory of cells, and note the convention that rotation around an axis
${\bf e}_{i}$ is taken to have a positive angular velocity $\omega_{i}$ if the
rotation is clockwise when viewed along the direction that ${\bf e}_{i}$
points. Chlamydomonas spins about ${\bf e}_{3}$ with an angular velocity
$\omega_{3}<0$, and we define the positive frequency $f_{r}=-\omega_{3}/2\pi$.
Its helical trajectories arise from an additional angular velocity
$\hat{\omega}_{1}{\bf\hat{e}}_{1}$, and we assume that $\omega_{1}$,
$\omega_{3}$, and the translational speed $u$ along ${\bf\hat{e}}_{3}$ are
sufficient to define the trajectories, without invoking an angular velocity
$\omega_{2}$.

The natural description of swimming trajectories is through the Euler angles
($\phi,\theta,\psi$) that define its orientation. In the standard convention
[53], their time evolution is given by angular velocities
($\omega_{1},\omega_{2},\omega_{3}$) as follows,

$\displaystyle\omega_{1}$
$\displaystyle=\dot{\phi}\sin\theta\sin\psi+\dot{\theta}\cos\psi$ (32a)
$\displaystyle\omega_{2}$
$\displaystyle=\dot{\phi}\sin\theta\cos\psi-\dot{\theta}\sin\psi$ (32b)
$\displaystyle\omega_{3}$ $\displaystyle=\dot{\phi}\cos\theta+\dot{\psi}.$
(32c)

The transformation from the body frame ${\bf x}$ to the laboratory frame ${\bf
x}^{\prime}$ is ${\bf x}^{\prime}={\bf A}\cdot{\bf x}$ and the reverse
transformation is via ${\bf x}=\tilde{\bf A}\cdot{\bf x}^{\prime}$, where
$\tilde{\bf A}={\bf A}^{-1}={\bf A}^{T}$, with

$\tilde{\bf A}=\begin{pmatrix}{\rm c}\psi\,{\rm c}\phi-{\rm c}\theta\,{\rm
s}\phi\,{\rm s}\psi&-{\rm s}\psi\,{\rm c}\phi-{\rm c}\theta\,{\rm s}\phi\,{\rm
c}\psi&{\rm s}\theta\,{\rm s}\phi\\\ {\rm c}\psi\,{\rm s}\phi+{\rm
c}\theta\,{\rm c}\phi\,{\rm s}\psi&-{\rm s}\psi\,{\rm s}\phi+{\rm
c}\theta\,{\rm c}\phi\,{\rm c}\psi&-{\rm s}\theta\,{\rm c}\phi\\\ {\rm
s}\theta\,{\rm s}\psi&{\rm s}\theta\,{\rm c}\psi&{\rm c}\theta\end{pmatrix},$
(33)

and we have adopted the shorthand $c\psi=\cos\psi$, etc.

The connection between helical swimming trajectories and the angular
velocities $\omega_{i}$ has been made by Crenshaw [54, 55, 56] by first
postulating helical motion and then finding consistent angular velocities. We
use a more direct approach, starting from the Euler angle dynamics (32). If
there is motion along a helix, and $\hat{\omega}_{1}$ and $\omega_{3}$ are
nonzero and constant, then apart from the degenerate case of orientation
purely along $\hat{\bf e}_{z}$, where “gimbal locking” occurs, we must have
$\dot{\phi}={\rm constant}$, $\dot{\theta}=0$, and $\dot{\psi}=0$. If we thus
set $\dot{\phi}=-\gamma$ (the sign choice taken for later convenience),
$\theta=\theta_{0}$ (with $-\pi/2\leq\theta_{0}\leq\pi/2)$, then a solution
requires $\psi=\pi/2$ and the primary body axis is

${\bf\hat{e}}_{3}=-\sin\theta_{0}\left[\sin\gamma
t\,{\bf\hat{e}}_{x}+\cos\gamma
t\,{\bf\hat{e}}_{y}\right]+\cos\theta_{0}\,{\bf\hat{e}}_{z}.$ (34)

If the organism swims along the positive ${\bf\hat{e}}_{3}$ direction at speed
$u$, then $\hat{\bf e}_{3}$ is the tangent vector $\hat{\bf t}$ to its
trajectory and we can integrate (34) using ${\bf\hat{t}}=(1/u){\bf r}_{t}$ to
obtain

${\bf r}(t)=\frac{u\sin\theta_{0}}{\gamma}\left[\cos\gamma
t\,{\bf\hat{e}}_{x}-\sin\gamma
t\,{\bf\hat{e}}_{y}\right]+ut\cos\theta_{0}\,{\bf\hat{e}}_{z},$ (35)

which is a helix of radius $R_{h}$ and pitch $P_{h}$ given by

$R_{h}=\frac{u\left|\sin\theta_{0}\right|}{\left|\gamma\right|},\ \ \ \
P_{h}=\frac{2\pi u\cos\theta_{0}}{|\gamma|}.$ (36)

With the parameters $(\gamma,\theta_{0})$ taking either positive or negative
values, there are four sign choices: $(+,+)$, $(+,-)$, $(-,+)$, and $(-,-)$.
Since the $z$ coordinate in the helices (35) increases independent of those
signs, we see that when $\gamma>0$ the $x-y$ components of the helices are
traversed in a clockwise (CW) manner and the helices are left-handed (LH),
while when $\gamma<0$ the in-plane motion is CCW, and the helices are right-
handed.

We are now in a position to describe quantitatively the helical trajectories
of swimming Chlamydomonas in the absence of photostimulation. From the
estimated angular speed $\hat{\omega}_{1}\sim 1\,$s-1 in (19), the typical
value $|\omega_{3}|\sim 10\,$s-1 and swimming speed $u\sim 100\,\upmu$m/s, we
find $R_{h}\sim 1\,\upmu$m and $P_{h}\sim 60\,\upmu$m, both of which agree
well with the classic study of swimming trajectories (Fig. 6 of [16]), which
show the helical radius is a small fraction of the body diameter and the pitch
is $\sim 6$ diameters.

Next we describe in detail the helical trajectories adopted by cells in
steady-state swimming, either toward the light during positive phototaxis, or
away from it during negative phototaxis. For such motions, the relevant
angular rotations are $\omega_{3}$ and the intrinsic speed $\hat{\omega}_{1}$.
As noted earlier [17], there are four possible configurations to be considered
on the basis of the sense of rotation around ${\bf\hat{e}}_{1}$ as determined
by cis dominance ($\hat{\omega}_{1}>0$) or trans dominance
($\hat{\omega}_{1}<0$). In both cases the relationship between the helix
parameters $\gamma$ and $\theta_{0}$ is

$\displaystyle\hat{\omega}_{1}$ $\displaystyle=-\gamma\sin\theta_{0}$ (37a)
$\displaystyle\omega_{3}$ $\displaystyle=-\gamma\cos\theta_{0},$ (37b)

with $\omega_{3}<0$ in both cases. In trans dominance, $\hat{\omega}_{1}<0$
and a solution of (37) has $0\leq\theta_{0}\leq\pi/2$, whereas for cis
dominance, $\hat{\omega}_{1}>0$ and $-\pi/2\leq\theta_{0}\leq 0$. Thus,

$\theta_{0}=\tan^{-1}(\hat{\omega}_{1}/\omega_{3}),\ \ \
\gamma=\sqrt{\hat{\omega}_{1}^{2}+\omega_{3}^{2}}.$ (38)

Setting $\tilde{t}=\gamma t$, we obtain the helical trajectories

$\displaystyle{\bf r}(t)$ $\displaystyle=\pm
R_{h}\left[\cos\tilde{t}\,{\bf\hat{e}}_{x}-\sin\tilde{t}\,{\bf\hat{e}}_{y}\right]+\frac{P_{h}\tilde{t}}{2\pi}{\bf\hat{e}}_{z},$
(39a) $\displaystyle{\bf\hat{e}}_{1}$
$\displaystyle=\cos\theta_{0}\left[\sin\tilde{t}\,{\bf\hat{e}}_{x}+\cos\tilde{t}\,{\bf\hat{e}}_{y}\right]\pm|\sin\theta_{0}|\,{\bf\hat{e}}_{z},$
(39b) $\displaystyle{\bf\hat{e}}_{2}$
$\displaystyle=-\cos\tilde{t}\,{\bf\hat{e}}_{x}+\sin\tilde{t}\,{\bf\hat{e}}_{y}.$
(39c)

for trans (+) and cis ($-$) dominance.

We can now express quantitatively features regarding the eyespot orientation
with respect to the helical trajectory that have been remarked on
qualitatively [17]. While there is some variability in the eyespot location,
it is typically in the equatorial plane defined by $\hat{\bf e}_{1}$ and
$\hat{\bf e}_{2}$, approximately midway between the two. We take it to lie at
an angle $\kappa\in[0,\pi/2]$ with respect to $\hat{\bf e}_{2}$, such that the
outward normal ${\bf\hat{o}}$ to the eyespot is

${\bf\hat{o}}=\sin\kappa\,{\bf\hat{e}}_{1}+\cos\kappa\,{\bf\hat{e}}_{2}.$ (40)

The outward normal vectors to the helix cylinder are $\hat{\bf
n}=\pm\left(\cos\tilde{t}\,\hat{\bf e}_{x}-\sin\tilde{t}\,\hat{\bf
e}_{y}\right)$, so the projection of the eyespot normal on $\hat{\bf n}$ have
the time-independent values

$\hat{\bf n}\cdot\hat{\bf o}=\begin{cases}-\cos\kappa<0;&{\it trans}\,{\rm
dominance},\\\ +\cos\kappa>0;&{\it cis}\,{\rm dominance}.\end{cases}$ (41)

Thus, for any $\kappa\in[0,\pi/2]$, the eyespot points to the inside (outside)
of the helix for trans (cis) dominance. This confirms the general rule that
any given body-fixed spot on a rigid body executing helical motion due to
constant rotations about its axes has a time-independent orientation with
respect to the helix. When $\kappa=0$, $\hat{\bf o}=\hat{\bf e}_{2}$, which
points to the cis flagellum, and we see that the dominant flagellum is always
on the outside of the helix.

If light shines long some direction $\hat{\bm{\ell}}$, its projection on the
eyespot is $-\hat{\bm{\ell}}\cdot\hat{\bf o}$; for light shining down the
helical axis, $\hat{\bm{\ell}}=-\hat{\bf e}_{z}$, then the projections in the
two cases are

$-\hat{\bm{\ell}}\cdot\hat{\bf
o}=\begin{cases}+\sin\kappa\,\sin\theta_{0}>0;&{\it trans}\,{\rm
dominance},\\\ -\sin\kappa\,|\sin\theta_{0}|<0;&{\it cis}\,{\rm
dominance}.\end{cases}$ (42)

Table 2: Left-handed helical swimming of Chlamydomonas. dominant |  | swimming relative | eyespot | eyespot
---|---|---|---|---
flagellum | $\hat{\omega}_{1}$ | to light source | orientation | status
cis | $+$ | toward | outside | shade
cis | $+$ | away | outside | light
trans | $-$ | toward | inside | light
trans | $-$ | away | inside | shade
Figure 14: Helical trajectory of a cis-dominant cell which is swimming in a
direction aligned with the light source located at the bottom. Principal body
rotation axes are depicted as cyan, magenta and black arrows for
$\bf{{\hat{e}}}_{1}$, $\bf{{\hat{e}}}_{2}$ and $\bf{{\hat{e}}}_{3}$ axes
respectively. Eyespot is always located in the shading hemisphere of the cell
body and is pointing outwards from the helix, as is the cis flagellum (not
shown).

These various situations are summarized in Table 2 and shown in Fig. 14. As
remarked many years ago, trans dominance holds in the case of negative
phototaxis, and cis dominance in positive phototaxis. The conclusion from
Table 2 is that when negatively phototactic cells swim away from the light or
positively phototactic cells swim toward the light their eyespots are shaded,
and thus the “stable” state is one minimal signal. so that when the first of
these conditions holds during negative phototaxis, whereas the second occurs
for positive phototaxis. Note that in the degenerate case $\theta_{0}=0$, when
the helix reduces to a straight line, the projections (42) vanish, so a cell
moving precisely opposite its desired direction would receive no signal to
turn. A small helical component to the motion eliminates this singular case.

###  IV.2 Phototactic steering with adaptive dynamics

Now we merge the adaptive photoresponse dynamics with the kinematics of rigid
body motion. The dynamics for the evolution of the Euler angles in the limit
$\omega_{2}=0$ is obtained from (32), yielding

$\displaystyle\dot{\phi}$
$\displaystyle=\omega_{1}\frac{\sin\psi}{\sin\theta},$ (43a)
$\displaystyle\dot{\theta}$ $\displaystyle=\omega_{1}\cos\psi,$ (43b)
$\displaystyle\dot{\psi}$
$\displaystyle=\omega_{3}-\omega_{1}\frac{\sin\psi\cos\theta}{\sin\theta},$
(43c)

Given the assumption $\omega_{2}=0$, these are exact. As we take $\omega_{3}$
to be a constant associated with a given species of Chlamydomonas, it remains
only to incorporate the dynamics of $\omega_{1}$ and the forward swimming
speed to have a complete description of trajectories. The angular speed
$\omega_{1}$ is the sum of intrinsic and phototactic contributions,

$\omega_{1}=\hat{\omega}_{1}+\omega_{1}^{p},$ (44)

where $\omega_{1}^{p}$ is described by the adaptive model.

It is natural to adopt rescalings based on the fundamental “clock” provided by
the spinning of Chlamydomonas about $\hat{\bf e}_{3}$. Recalling that
$\omega_{3}<0$, these are

$\displaystyle T=$ $\displaystyle|\omega_{3}|t,\ \ \ \
\hat{P}=\frac{\hat{\omega}_{1}}{|\omega_{3}|},\ \ \ \
P=\frac{\omega_{1}^{p}}{|\omega_{3}|},\ \ \ \ \alpha=|\omega_{3}|\tau_{a},$
$\displaystyle\beta$ $\displaystyle=|\omega_{3}|\tau_{r},\ \ \ \
H=\frac{h}{|\omega_{3}|},\ \ \ \ S=\frac{s}{|\omega_{3}|}.$ (45)

To incorporate the photoresponse, the light signal at the eyespot must be
expressed in terms of the Euler angles. Henceforth we specialize to the case
in which the a light source shines in the $x-y$ plane along the negative
$x$-axis, so $\hat{\bm{\ell}}=-\hat{\bf e}_{x}$, and the normalized projected
light intensity $J=-\hat{\bm{\ell}}\cdot\hat{\bf o}$ on the eyespot can be
written as

$J=\sin\left(\kappa-\psi\right)\cos\phi-\cos\left(\kappa-\psi\right)\cos\theta\sin\phi.$
(46)

We assume for simplicity that eyespot shading is perfect, so that the signal
sensed by the eyespot is

$S=P^{*}J{\cal H}(J),$ (47)

where $P^{*}=\omega_{1}^{*}/|\omega_{3}|$ is negative (positive) for positive
(negative) phototaxis, and ${\cal H}$ is the Heaviside function.

Figure 15: Positive phototaxis from model for non-helical swimming. (a-c) Time
evolution of adaptive variables, Euler angles, and eyespot signal during a
phototurn, i.e. from swimming orthogonal to the light to moving directly
toward it. (d) Trajectory of the turn showing initial (S) and final (T)
orientations of the cell (also in magnified insets). (e-h) Analogous to panels
above but with dynamics starting from an orientation facing nearly away from
the light. Parameters are $\hat{P}=0$, $P^{*}=-0.4$, $\alpha=7$, $\beta=0.14$,
and $U=2$, with the eyespot along $\hat{\bf e}_{2}$ (i.e. $\kappa=0$ but shown
in canonical position), and $\tau_{d}=0$. Color scheme of cell’s principal
axes is same as in Fig. 14. Figure 16: Positive phototaxis with helical
swimming. As in Fig. 15 but with $\hat{P}=0.3$.

With these rescalings, the dynamics reduces to

$\displaystyle\phi_{T}$
$\displaystyle=\left(\hat{P}+P\right)\frac{\sin\psi}{\sin\theta},$ (48a)
$\displaystyle\theta_{T}$ $\displaystyle=\left(\hat{P}+P\right)\cos\psi,$
(48b) $\displaystyle\psi_{T}$
$\displaystyle=-1-\left(\hat{P}+P\right)\frac{\sin\psi\cos\theta}{\sin\theta},$
(48c) $\displaystyle\beta P_{T}$ $\displaystyle=S-H-P,$ (48d)
$\displaystyle\alpha H_{T}$ $\displaystyle=S-H.$ (48e)

These five ODEs, supplemented with the signal definition in (46) and (47),
constitute a closed system. To obtain the swimming trajectory, we use the cell
body radius $R$ to define the scaled position vector ${\bf R}={\bf r}/R$, so
that the dynamics ${\bf r}_{t}=u\hat{\bf e}_{3}$ becomes

${\bf
R}_{T}=U\left\\{\sin\theta\left[\sin\phi\,{\bf\hat{e}}_{x}-\cos\phi\,{\bf\hat{e}}_{y}\right]+\cos\theta\,{\bf\hat{e}}_{z}\right\\},$
(49)

where $U=u/(R|\omega_{3}|)$ is the scaled swimming speed. For typical
parameter values ($u=100\,$µm/s, $R\sim 5\,$µm, and $f_{r}=1.6\,$Hz), we find
$U\sim 2$. Given $(\theta(T),\phi(T))$, we integrate (49) forward from some
origin ${\bf R}(0)$ to obtain ${\bf R}(T)$, and use the triplet
$(\theta(T),\phi(T),\psi(T))$ and the matrix ${\bf A}$, the inverse of
$\tilde{\bf A}$ in (33), to obtain $\hat{\bf e}_{1}(T)$ and $\hat{\bf
e}_{2}(T)$.

An important structural feature of the dynamics is its partitioning into sub-
dynamics for the Euler angles and the flagellar response. The connection
between the two is provided by the response variable $P(T)$, via the signal
$S$, such that any other model for the response (for example, one
incorporating distinct dynamics for the cis and trans flagella), or the signal
(including only partial eyespot directionality, or cell body lensing) can be
substituted for the adaptive dynamics with perfect shading.

The model (48) has $4$ main parameters: $\hat{P}$ determines the unstimulated
swimming helix, $P^{*}$ sets the maximum photoresponse turn rate, and $\alpha$
and $\beta$ describe the adaptive dynamics. Additional parameters are the
eyespot angle $\kappa$ (40) and time delay $\tau_{d}$. To gain insight, we
first adopt the simplification that the eyespot vector $\hat{\bf o}$ is along
$\hat{\bf e}_{2}$ ($\kappa=0$), set $\tau_{d}=0$, and solve the initial value
problem in which a cell starts swimming in the $x-y$ plane ($\theta(0)=\pi/2$)
along the direction $-\hat{\bf e}_{y}$ ($\phi(0)=\pi/2$) with its eyespot
orthogonal to the light ($\psi(0)=0$; $\hat{\bf e}_{1}=\hat{\bf e}_{x}$ and
$\hat{\bf e}_{2}=\hat{\bf e}_{z}$) and about to rotate into the light. Figure
15 shows the results of numerical solution of the model for the non-helical
case $\hat{P}=0$, with $P^{*}=-0.4$ for positive phototaxis, $\alpha=7$,
$\beta=0.15$ and $U=2$. We see in Fig. 15(a) how the initially large
photoresponse when the cell is orthogonal to the light decreases with each
subsequent half turn as the angle $\phi$ evolves toward $\pi/2$ [Fig. 15(b)].
The signal at the eyespot [Fig. 15(c)], is a half-wave rectified sinusoid with
an exponentially decreasing amplitude. Note that for this non-helical case the
Euler angle $\theta$ remains very close to $\pi/2$ during the entire
phototurn, indicating that the swimmer remains nearly in the $x-y$ plane
throughout the trajectory.

If the initial condition is nearly opposite to the light direction, the model
shows that the cell can execute a complete phototurn (Fig. 15(e). In the
language of Schaller, et al. [17], since the eyespot is “raked backward” when
the cell swims toward the light, and the eyespot is shaded, when the cell
swims away from the light it picks up a signal and can execute a full $\pi$
turn to reach the light.

Next we include helicity in the base trajectory, setting $\hat{P}=0.3$. In the
absence of phototactic stimulation this value leads to helical motion with a
ratio of helix radius to pitch of $R_{h}/P_{h}=\hat{P}/2\pi\simeq 1/20$, a
value considerably larger than to that seen experimentally [16], but useful
for the purposes of illustration. The phototurn dynamics shown in Fig. 16
exhibits the same qualitative features seen without helicity, albeit with much
more pronounced oscillations in the evolution of the Euler angles,
particularly of $\phi$ and $\theta$. Averaged over the helical path the
overall trajectory is similar to that without helicity, and does not deviate
significantly from the $x-y$ plane.

To make analytical progress in quantifying a phototurn, we use the
simplifications that are seen in Fig. 15 for the non-helical case. First, we
neglect the small deviations of $\theta$ from $\pi/2$, and simply set
$\theta=\pi/2$. Second, we note that the time evolution of $\psi$ is dominated
by rotations around $\hat{\bf e}_{3}$, and thus we assume $\psi=-T$. This
yields a simplified model in which the remaining Euler angle $\phi$ is driven
by the cell spinning, subject to the adaptive dynamics.

$\displaystyle\phi_{T}$ $\displaystyle=-P\sin T,$ (50a) $\displaystyle\beta
P_{T}$ $\displaystyle=S-H-P,$ (50b) $\displaystyle\alpha H_{T}$
$\displaystyle=S-H.$ (50c)

where we allow for a general eyespot location, using
$J=\cos\phi\sin(T+\kappa)$, and thus

$S=P^{*}\cos\phi\sin\left(T+\kappa\right)\,{\cal
H}\left(\sin\left(T+\kappa\right)\right).$ (51)

As it takes a number of half-periods of body rotation to execute a turn, we
can consider the angle $\phi$ to be approximately constant during each half
turn $n$ ($n=1,2,\ldots$) at the value we label $\phi_{n}$. For any fixed
$\phi_{n}$, the signal $S$ is simply a HWRS of amplitude $P^{*}\cos\phi_{n}$.
We explore two approaches to finding the evolution of $\phi_{n}$: (i) a quasi-
equilibrium one in which the steady-state response of the adaptive system to
an oscillating signal is used to estimate $P$, and (ii) a non-equilibrium one
in which the response is the solution to an initial-value problem.

In the first approximation, we decompose the HWRS eyespot signal (51) into a
Fourier series,

$\displaystyle S=P^{*}\cos\phi_{n}\,\Bigl{[}$
$\displaystyle\frac{1}{\pi}+\frac{1}{2}\sin\left(T+\kappa\right)$
$\displaystyle-\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{\cos(2n\left(T+\kappa\right))}{(2n)^{2}-1}\Bigr{]}.$
(52)

From the linearity of the adaptive model it follows that each term in this
series produces an independent response with magnitude $G$ and phase shift
$\chi$ appropriate to its frequency $n\omega_{3}$, for $n=0,1,2,\ldots$. Since
the magnitude $G$ in (30a) vanishes at zero frequency, the contributing terms
in the photoresponse are

$\displaystyle P=P^{*}$
$\displaystyle\cos\phi_{n}\Biggl{\\{}\frac{1}{2}G(\omega_{3})\sin\left[T+\kappa+\chi\left(\omega_{3}\right)\right]$
(53)
$\displaystyle-\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{G(2n\omega_{3})}{(2n)^{2}-1}\cos\left[2n\left(T+\kappa\right)+\chi\left(2n\omega_{3}\right)\right]\Biggr{\\}}$

The first term dominates, as it is at the same frequency as the r.h.s. of the
equation of motion $\phi_{T}=-P\sin T$. Keeping only this term, we integrate

$\phi_{T}\simeq-\frac{1}{2}P^{*}G(\omega_{3})\cos\phi_{n}\sin
T\sin(T+\kappa+\chi)$ (54)

over one half period ($T=\pi$) and obtain the iterated map

$\phi_{n+1}=\phi_{n}+\xi\cos{\phi_{n}},$ (55)

where

$\xi=-\frac{\pi}{4}P^{*}G(\omega_{3})\cos\left[\kappa+\chi(\omega_{3})\right],$
(56)

with $\xi>0$ for $P^{*}<0$ in positive phototaxis and $\xi<0$ for negative
phototaxis. An alternative approach involves the direct integration of the
equations of motion over each half-turn. The lengthy algebra for this is given
in Appendix A, where one finds a map analogous to (55), but with an
$n$-dependent factor $\xi_{n}$ that converges for large $n$ to that in (56).
Supplementary Video 2 [43] illustrates the cell reorientation dynamics under
this map.

Figure 17: Iterated map of the reorientation model. (a) Cobwebbing of
iterations starting from $\phi$ near $-\pi/2$ for positive phototaxis (upper
branch in yellow) and near $+\pi/2$ for negative phototaxis (lower branch in
purple), as indicated by triangles, for light shining toward $-\hat{\bf
e}_{x}$. Values of $\xi=\mp 0.29$ are used which are calculated based on (56)
using values from Table 3. (b) Response factor $1/Q$ in (57) for number of
half-turns needed for alignment as a function of tuning parameter, for various
values of $\rho$ as indicated.

The iterated map (55) has fixed points at $\phi_{\pm}=\pm\pi/2$. Linearizing
about those values by setting $\phi_{n}=\pm\pi/2+\delta\phi_{n}$, we obtain
$\delta\phi_{n+1}=(1\mp\xi)\delta\phi_{n}$ and thus
$\delta\phi_{n}\propto(1\mp\xi)^{n}\delta\phi_{1}$. Hence: (i) the angle
$+\pi/2$ is stable for positive phototaxis when $0\leq\xi\leq 2$ and becomes
unstable for $\xi>2$, while it is unstable for negative phototaxis ($\xi<0$).
(ii) the angle $-\pi/2$ is unstable for positive phototaxis for any $\xi>0$,
while it is stable for negative phototaxis in the range $-2\leq\xi\leq 0$ and
unstable for $\xi<-2$. Thus, positively phototactic cells orient toward
$+\pi/2$ and negatively phototactic cells orient toward $-\pi/2$, except for
values of $|\xi|>2$. These exceptional cases correspond to peak angular speeds
$\omega_{1}^{*}>4|\omega_{3}|/3\sim 13\,$s-1.

Figure 17(a) shows the iterated map (55) for both positive and negative
phototaxis. In the usual manner of interpreting such maps, the “cobwebbing” of
successive iterations shows clearly how the orientation $+\pi/2$ is the global
attractor for positive phototaxis, and $\phi=-\pi/2$ is that for negative
phototaxis. When $|\xi|$ is small, the approach to the stable fixed point is
exponential, $\delta\phi\sim\exp(-n/N)$, where $N=N_{0}/Q(\omega_{3})$, with

$Q(\omega_{3})=G(\omega_{3})\cos\left(\kappa+\chi_{0}-\delta\right),\ \ \ \
N_{0}=\frac{4}{\pi|P^{*}|},$ (57)

is the characteristic number half-turns needed for alignment. The number
$N_{0}$ reflects the bare scaling with the maximum rotation rate around
$\hat{\bf e}_{1}$. For the typical value $P^{*}\simeq 0.1$ we have $N_{0}\sim
6$. The presence of the gain $G$ in denominator in (57) embodies the effect of
tuning between the adaptation timescale and the rotation rate around $\hat{\bf
e}_{3}$, while the term $\cos\left(\kappa+\chi_{0}-\gamma\right)$ captures the
feature discussed in frequency response studies in Sec. III.3, namely that the
flagellar asymmetries have maximum effect (and thus $Q$ is maximized) when the
negative phase shift $\chi_{0}-\delta$ offsets the eyespot location). Figure
17(b) shows the dependence of $N/N_{0}$ on the scaled relaxation time
$f_{r}\tau_{a}$ for various values of $\rho$. For the experimentally observed
range $\rho\simeq 0.02-0.1$ there is a wide minimum of $N/N_{0}$ around
$f_{r}\tau_{a}\sim 1$. This relationship confirms the role of tuning in the
dynamics of phototaxis, but also shows the robustness of the processes
involved.

Returning to the evolution equation (54) for $\phi$, we can also average the
term $\sin T\sin(T+\kappa+\chi)$ over one complete cycle to obtain the
approximate evolution equation

$\phi_{T}=\frac{1}{4}|P^{*}|Q(\omega_{3})\cos\phi.$ (58)

For positive phototaxis and with $\phi(0)=0$, the solution to this ODE can be
expressed in unrescaled units as

$\Phi(t)=2\tan^{-1}\left(e^{t/\tau_{x}}\right),$ (59)

where $\Phi=\pi/2+\phi$, with $\Phi(0)=\pi/2,\Phi(\infty)=\pi$, and the
characteristic time in physical units is

$\displaystyle\tau_{x}$
$\displaystyle=\frac{4}{|\omega_{1}^{*}|Q(\omega_{3})}$
$\displaystyle=\frac{4\tilde{\zeta}_{r}/|{\mathcal{T}}_{\mathrm{p}}^{*}|}{G(\omega_{3})\cos\left(\kappa+\chi_{0}-\delta\right)}.$
(60)

This is a central result of our analysis, in that it relates the time scale
for reorientation during a phototurn to the magnitude and dynamics of the
transient flagellar asymmetries during the photoresponse. As discussed above,
the function $Q(\omega_{3})$ embodies the optimality of the response—in terms
of the tuning between the rotational frequency and the adaptation time, and
the phase delay and eyespot position—but also captures the robustness of the
response through the broad minimum in $Q$ as a function of both frequency and
eyespot position.

Figure 18: Phototactic swimmers tracked in three-dimensions. (a) A U-turn:
trajectory in black is prior to light stimulation, that in green is
afterwords. Blue arrows indicate direction of light. The cropped trajectory
used for fitting the reorientation dynamics is bounded by the points S and T.
(b) Dynamics of the reorientation angle $\Phi$ (blue) for the cropped
trajectory fitted using (59). (c) Box plot of distribution of fitted
$\tau_{x}$ with estimates, steady-state (green) and nonequilibrium (purple),
derived from micropipette experiments.

Using the 3D tracking system described in Sec. II, we analyzed $6$ pairs of
movies, within which we tracked $283$ trajectories with duration greater than
$10\,$s and which included the trigger frame. From those, $44$ showed both
positive phototaxis and included a full turn to $\phi=\pi/2$ as shown in Fig.
18(a) and Supplementary Video 3 [43]. These trajectories were cropped to
include any points for which $-\pi/2\leqslant\phi\leqslant\pi/2$ and which
could then be fitted to (59) to determine the experimental time constant
$\tau_{x}$. The boxplot in Fig. 18 shows the experimental mean value
$\tau_{x}=0.96\pm 0.36\,$s (also in Table 3). This can be compared to the
estimates obtained within the steady-state approximation (60) and the
transient analysis (Appendices A and B), both of which are based on the mean
and standard error value of the peak flagellar phototorque of $-35.0\pm
8.8\,$pN$\cdot\upmu$m, the mean values of the adaptive flagellar response
time-scales ($\tau_{r}=0.018\,$s and $\tau_{a}=0.764\,$s in Table 3) and the
effective drag coefficient $\tilde{\zeta}_{r}$. The steady-state estimate of
$\tau_{x}$ is $1.04\pm 0.26\,$s (with $Q/4=0.243$), while the transient
estimate is $0.96\pm 0.24\,$s (with the nonequilibrium counterpart
$\bar{\mathcal{Q}}=0.265$ in (76)). This agreement provides strong validation
of the model of adaptive phototaxis developed here.

Table 3: Time scales that define the reorientation dynamics. Experimental data are from the present study except for the cell body rotation frequency. Quantity | Symbol |  Mean ${\bm{\pm}}$ SD
---|---|---
adaptation time | $\tau_{\mathrm{a}}$ |  $0.764\,\pm\,0.190\,$s
response time | $\tau_{\mathrm{r}}$ |  $0.018\,\pm\,0.009\,$s
time delay | $\tau_{\mathrm{d}}$ |  $0.028\,\pm\,0.011\,$s
cell body rotation frequency [18] | $f_{\mathrm{r}}$ |  $1.67\,\pm\,0.35\,$Hz
reorientation time scale | $\tau_{\mathrm{x}}$ |  $0.96\,\pm\,0.36\,$s

##  V Discussion

This study has achieved three goals: the development of methods to capture
flagellar photoresponses at high spatio-temporal resolution, the estimate of
torques generated during these responses and the measuremnt of relevant
biochemical time scales that underlie phototaxis, and the integration of this
information into a mathematical model to describe accurately the phototactic
turning of Chlamydomonas. In developing a theory for phototurns, our work also
puts on a more systematic mathematical foundation qualitative arguments [17]
for the stability of phototactic trajectories based on eyespot orientation in
both positive and negative phototaxis.

We have emphasized that rather than seek to develop a maximally detailed model
of the dynamics of individual flagellar responses involved in phototaxis, we
aimed to provide, in the context of one simple microscopic model, a multiscale
analysis of the connection between such responses and the phototactic
trajectories in a manner than can be easily generalized. Thus we obtain from
experiment the values for microscopic and macroscopic time scales, as shown in
Table 3, and derive relation between them, culminating in (60) (and (76)).

This analysis highlights the dual issues of optimatility and robustness. As
noted in the introduction, the former was first addressed using a paralyzed-
flagella mutant strain (pf14) and an electrophysiological approach on a bulk
sample by [25]. In those experiments, a suspension of immotile cells was
exposed to an oscillating light stimulus (wavelength $500\,$nm) and the
resulting photoreceptor current was measured in a cuvette attached to two
platinum electrodes. The experiment using relatively high light intensities
observed a frequency response peak of $1.6\,$Hz when stimulated with
$\approx$160 µE${\mkern 1.0mu\cdot\mkern 1.0mu}$s${}^{-1}{\mkern
1.0mu\cdot\mkern 1.0mu}$m-2 and a frequency response peak of $3.1\,$Hz when
stimulated with $\approx$ 40 µE${\mkern 1.0mu\cdot\mkern
1.0mu}$s${}^{-1}{\mkern 1.0mu\cdot\mkern 1.0mu}$m-2. The former observation is
in very good agreement with our results in Fig. 13 (peak response at $\simeq
2\,$Hz), even though we used light stimulus intensities of $\approx$1
µE${\mkern 1.0mu\cdot\mkern 1.0mu}$s${}^{-1}{\mkern 1.0mu\cdot\mkern
1.0mu}$m-2. We have not seen any evidence of cells having flagellar
photoresponse dynamics that would corroborate the latter result of $3.1\,$Hz
and this is a matter open to further study.

In addition, this study has addressed issues relating to past observations.
With respect to the lag time $\tau_{d}$ of the photoresponse, we have measured
by detailed study of the flagellar waveforms a value of $28\,\pm\,11\,$ms that
is very similar to the value $30-40\,$ms observed earlier [21]. In addition,
we have shown through the adaptive dynamics that the peak flagellar response
is at a larger total delay time $t^{*}$ given by (27) that corresponds
accurately to the time between the eyespot receiving a light signal and the
alignment of the flagellar beat plane with the light. Analysis of the
phototactic model reveals that such tuning shortens the time for phototactic
alignment.

Regarding the amount of light necessary for a flagellar photoresponse
appropriate to positive phototaxis, we have converged, through trial and
error, to $\approx$1 µE${\mkern 1.0mu\cdot\mkern 1.0mu}$s${}^{-1}{\mkern
1.0mu\cdot\mkern 1.0mu}$m-2 at a wavelength of $470\,$nm. While this value is
much lower than in other photoresponse experiments [26] where $\approx$60
µE${\mkern 1.0mu\cdot\mkern 1.0mu}$s${}^{-1}{\mkern 1.0mu\cdot\mkern
1.0mu}$m-2 were used at a longer wavelength ($543\,$nm), it is consistent with
the sensitivity profile of channelrhodopsin-2 [57]. More detailed studies of
the wavelength sensitivity of the flagellar photoresponse should be carried
out in order to reveal any possible wavelength dependencies of quantities such
as the time constants $\tau_{\mathrm{r}}$ and $\tau_{\mathrm{a}}$. Our work
has addressed the relationship between the stimulus and the photoresponse of
Chlamydomonas using an adaptive model that has perhaps the minimum number of
parameters appropriate to the problem, each corresponding to a physical
process. Attempts to derive similar relationships between stimulus and
photoresponse [27] used linear system analysis. The result of such a signal-
processing oriented method usually includes a much larger number of parameters
necessary for the description of the system, without necessarily corresponding
to any obvious measurable physical quantities.

The evolutionary perspective that we emphasized in the introduction,
culminating in the results presented in Fig. 2, points to several areas for
future work. Chief among them is an understanding of the biochemical origin of
the response and adaptive timescales of the photoresponse, in light of genomic
information available on the various species. Flagellar and phototaxis mutants
will likely be important in unravelling whether these time scales are
associated with the axoneme directly or arise from coupling to cytoplasmic
components. Additionally, we anticipate that directed evolution experiments
such as those already applied to Chlamydomonas [58] can yield important
information on the dynamics of phototaxis. For example, is it possible to
evolve cells that exhibit faster phototaxis, and if so, which aspect of the
light response changes? For the multicellular green algae, these kinds of
experiments may also impact on the organization of somatic cells within the
extracellular matrix, which has been shown to exhibit significant variability
[59].

Another aspect for future investigation sits within the general are of control
theory; the adaptive phototaxis mechanism that is common to the Volvocine
algae, and to other systems such as Euglena [60], is one in which a
chemomechanical system achieves a fixed point by evolving in time so as to
null out a periodic signal. Two natural questions arise from this observation.
First, what evolutionary pathways may have led to this behavior? Second, are
there lessons for control theory in general and perhaps even for autonomous
vehicles in particular that can be deduced from this navigational strategy?

We close by emphasizing that the flagellar photoresponse – and by extension
phototaxis – is a complex biological process encompassing many variables, and
that in addition to the short-term responses to light stimulation studied here
there are issues of long-term adaptation to darkness or phototactic light that
have only recently have begun to be addressed [61]. Together with the dynamics
of phototaxis in concentrated suspensions [62], these are important issues for
further work.

###### Acknowledgements.

We thank Pierre A. Haas and Eric Lauga for very useful discussions and a
critical reading of the manuscript, Kirsty Y. Wan for sharing code from
previous work on flagellar tracking, David-Page Croft, Colin Hitch, John
Milton, and Paul Mitton or technical support, and Ali Ghareeb for assistance
with the fiber coupling apparatus. This work was supported in part by Wellcome
Grant 207510/Z/17/Z. For the purpose of open access, the authors have applied
a CC BY public copyright license to any Author Accepted Manuscript version
arising from this submission. Additional support was provided by Grant No.
7523 from the Marine Microbiology Initiative of the Gordon and Betty Moore
Foundation, and EPSRC Established Career Fellowship EP/M017982/1.

##  Appendix A Details of the iterated map from solution of the initial value
problem

Here we provide details of the derivation of the iterated map for phototurns
based on explicit solution of the initial value problem for the adaptive
response. For conciseness we fix the eyespot position at $\kappa=0$ and set
the time delay $\tau_{d}=0$. We start from the dynamics (50), rewritten for
each full turn $n\geq 0$ as

$\alpha H^{(n)}_{T}+H^{(n)}=\begin{cases}P^{*}\cos\phi\sin{T};&n\pi\leqslant
T<(n+1)\pi,\,n\,{\rm even}\\\ 0;&n\pi\leqslant T<(n+1)\pi,\,n\,{\rm
odd}\end{cases}$ (61)

and

$\beta
P^{(n)}_{T}+P^{(n)}=\begin{cases}P^{*}\cos\phi\sin{T}-H^{(n)};&n\pi\leqslant
T<(n+1)\pi,\,n\,{\rm even}\\\ -H^{(n)};&n\pi\leqslant T<(n+1)\pi,\,n\,{\rm
odd}.\end{cases}$ (62)

Solving this in a piecewise fashion we obtain

$H^{(n)}=P^{*}\cos\phi\frac{\alpha}{1+\alpha^{2}}\times\begin{cases}C_{n}e^{-(T-n\pi)/\alpha}+\frac{1}{\alpha}\sin{T}-\cos{T};&n\pi\leqslant
T<(n+1)\pi,\,n\,{\rm even}\\\ C_{n}e^{-(T-n\pi)/\alpha};&n\pi\leqslant
T<(n+1)\pi,n\,{\rm odd}.\end{cases}$ (63)

where $C_{n}=(1-r^{n+1})/(1-r)$ and $r=e^{-\pi/\alpha}$. Continuity of $H$ at
the end of each light interval can be verified by noting that
$H^{(n)}(T=(n+1)\pi)\propto rC_{n}+1$ for even $n$, while
$H^{(n)}(T=n\pi)\propto C_{n+1}$ for the subsequent odd $n$, and observing
that $1+rC_{n}=C_{n+1}$.

The solution for the photoresponse variable can be expressed as
$P^{(n)}=P^{*}\cos\phi\,\tilde{P}^{(n)}$, where

$\tilde{P}^{(n)}=\begin{cases}\Lambda_{1}D_{n}e^{-(T-n\pi)/\beta}-\Lambda_{2}C_{n}e^{-(T-n\pi)/\alpha}+\Lambda_{3}\sin{T}+\Lambda_{4}\cos{T};&n\pi\leqslant
T<(n+1)\pi,n\,{\rm even}\\\
\Lambda_{1}D_{n}e^{-(T-n\pi)/\beta}-\Lambda_{2}C_{n}e^{-(T-n\pi)/\alpha};&n\pi\leqslant
T<(n+1)\pi,n\,{\rm odd}\end{cases}$ (64)

with $D_{n}=(1-q^{n+1})/(1-q)$, $q=e^{-\pi/\beta}$, and

$\Lambda_{1}=\frac{\alpha\beta}{(1+\beta^{2})(\alpha-\beta)},\ \
\Lambda_{2}=\frac{\alpha^{2}}{(1+\alpha^{2})(\alpha-\beta)},\ \
\Lambda_{3}=\frac{\alpha(\alpha+\beta)}{(1+\beta^{2})(1+\alpha^{2})},\ \
\Lambda_{4}=\frac{\alpha(1-\alpha\beta)}{(1+\beta^{2})(1+\alpha^{2})}.$ (65)

Since $n$ represents the number of half-turns, with even(odd) values for the
illuminated(shaded) periods, we integrate for each value of $n\geqslant 0$ to
obtain

$\phi_{n+1}=\phi_{n}-P^{*}\cos\phi\int_{n\pi}^{(n+1)\pi}\tilde{P}^{(n)}\sin{T}dT.$
(66)

This has the form of (55), but with an $n$-dependent $\xi_{n}$,

$\xi_{n}=-P^{*}\Xi_{n}(\omega_{3})$ (67)

where

$\Xi_{n}(\omega_{3})=\begin{cases}\displaystyle\beta^{2}\Lambda_{1}D_{n}(q+1)\\\
\displaystyle\qquad-\alpha^{2}\Lambda_{2}C_{n}(r+1)+\frac{\pi}{2}\Lambda_{3},&n\,{\rm
even}\\\
\displaystyle-\beta^{2}\Lambda_{1}D_{n}(q+1)+\alpha^{2}\Lambda_{2}C_{n}(r+1),&n\,{\rm
odd}.\end{cases}$ (68)

From the general structure of the iterated map, it is clear that the larger is
$\xi_{n}$ the larger the angular change within a given half-turn. It is of
interest then to consider the average
$\bar{\Xi}=\left(\Xi_{0}+\Xi_{1}\right)/2$ over the first two half turns,
which gives the average coefficient

$\bar{\xi}=-P^{*}\bar{\Xi}(\omega_{3}).$ (69)

The quantity $\bar{\Xi}$ can be interpreted as the initial photoresponse
function analogous to the steady-state response embodied in the amplitude
$G(\omega_{3})$ and phase $\chi_{0}$ in (30). The functions
$G(\omega_{3})\cos\chi_{0}(\omega_{3})$ and $\bar{\Xi}$ are compared in Fig.
19, where we see that the transient response function $\bar{\Xi}$ is about
$10\%$ higher at its peak, a feature that can be attributed to the fact that
the hidden variable $H$ has not yet built up to its steady value. But the two
functions are otherwise remarkably similar, indicating the accuracy of the
steady-state approximation.

More generally, the coefficients $\Xi_{n}$ exhibit an oscillating decay with
$n$, converging as $n\to\infty$ to

$\displaystyle\Xi_{\infty}(\omega_{3})=\begin{cases}\displaystyle\beta^{2}\Lambda_{1}\frac{1+q}{1-q}-\alpha^{2}\Lambda_{2}\frac{1+r}{1-r}+\frac{\pi}{2}\Lambda_{3},&n\,{\rm
even}\\\
\displaystyle-\beta^{2}\Lambda_{1}\frac{1+q}{1-q}+\alpha^{2}\Lambda_{2}\frac{1+r}{1-r},&n\,{\rm
odd}.\end{cases}$

The connection to the steady-state approximation is obtained by considering
the average $\bar{\Xi}_{\infty}(\omega_{3})$ over the light and dark cycles
(hence over even and odd values of $n$), where one finds
$\bar{\Xi}_{\infty}(\omega_{3})=(\pi/4)\Lambda_{3}$, or

$\bar{\Xi}_{\infty}(\omega_{3})=\frac{\pi}{4}G(\omega_{3})\cos(\chi_{0}),$
(70)

completely consistent with the steady-state analysis (56).

In the animation shown in Supplementary Video 2 [43] of the cell reorientation
dynamics we evolve the iterated map using the $\left\\{\xi_{n}\right\\}$ and
linearly interpolate between the values $\phi_{n}$ to obtain a smooth function
of time.

Figure 19: Response functions. For $\kappa=\tau_{d}=0$, the graph compares the
steady-state response function $(\pi/4)G(\omega_{3})\cos{\chi_{0}}$ in (30)
(dashed blue), the initial average response $\bar{\Xi}(\omega_{3})$ in (69)
(green) and the $n\to\infty$ limit of the transient response
$\bar{\Xi}_{\infty}(\omega_{3})$ (dashed red), as functions of
$f_{r}=\omega_{3}/2\pi$.  Figure 20: The functions $Q^{(n)}(T)$. Approximating
$Q^{(n)}(T)$ as a straight line $Q_{\mathrm{approx}}(T)=\bar{\mathcal{Q}}T$.
Both $Q^{(n)}(t)$ and $\bar{\mathcal{Q}}$ were computed with the
experimentally derived $\tau_{r}=0.009$ s and $\tau_{a}=0.520$ s. The value of
$f_{r}$ was taken to be 1.67 Hz. The value of
$\bar{\mathcal{Q}}(\omega_{3},\tau_{r},\tau_{a})$ was calculated to be 0.265.

##  Appendix B Details of continuous model

Figure 21: Contour plot maps of $\bar{\mathcal{Q}}$ (a) and $\mathcal{Y}$ (b)
with ($\alpha$,$\beta$) pairs acquired from micropipette experiments shown
with red markers (step-up as “x” and frequency as “o”). (c-e) Plots of
$Q^{(n)}(T)$ (blue line) and linear approximation
$Q_{\mathrm{approx}}(T)=\bar{\mathcal{Q}}T$ (red line) for three
($\alpha$,$\beta$) value pairs shown in (a) and (b) as circle, square and
triangle respectively. (c) is based on experimental data (open red circle),
(d) corresponds to $\mathcal{Y}\approx 0.7$ (solid purple square) and (e)
corresponds to $\mathcal{Y}\approx 0$ (solid purple triangle).

Using the solution of the initial value problem we can compute the continuous
approximation to the evolution equation for $\phi$ by integrating over the
fast photoresponse variables within a turn. From the governing equation
$\phi_{T}=-P^{*}\cos\phi\,\tilde{P}^{(n)}(T)\sin{T}$, we obtain

$\Phi(T)=2\tan^{-1}\left(e^{-P^{*}Q^{(n)}(T)}\right),$ (71)

where

$Q^{(n)}(T)=\int_{0}^{T}\tilde{P}^{(n)}(T^{\prime})\sin{T^{\prime}}{\rm
d}T^{\prime}.$ (72)

From (64) we find

$Q^{(n)}(T)=\begin{cases}\displaystyle-\beta\Lambda_{1}D_{n}e^{-\frac{T-n\pi}{\beta}}(\sin{T}+\beta\cos{T})+\alpha\Lambda_{2}C_{n}e^{-\frac{T-n\pi}{\alpha}}(\sin{T}+\alpha\cos{T})\\\
\displaystyle\qquad+\Lambda_{3}\left(\frac{T}{2}-\frac{\sin{2T}}{4}\right)+\Lambda_{4}\frac{\sin^{2}{T}}{2}-\Lambda_{3}\frac{n\pi}{4}+\beta^{2}\Lambda_{1}-\alpha^{2}\Lambda_{2}&n\pi\leqslant
T<(n+1)\pi,n\,{\rm even}\\\
\displaystyle-\beta\Lambda_{1}D_{n}e^{-\frac{T-n\pi}{\beta}}(\sin{T}+\beta\cos{T})+\alpha\Lambda_{2}C_{n}e^{-\frac{T-n\pi}{\alpha}}(\sin{T}+\alpha\cos{T})\\\
\displaystyle\qquad+\Lambda_{3}\frac{(n+1)\pi}{4}&n\pi\leqslant
T<(n+1)\pi,n\,{\rm odd}.\end{cases}$ (73)

As shown in Figure 20, the function $Q^{(n)}(T)$ typically increases
monotonically with $T$, exhibiting small oscillations around an interpolant
that grows nearly linearly with time. These magnitudes of these oscillations
vary between the light and dark halves of each turn. To quantify this
asymmetry we compute the values $Q^{(n)}(n\pi)$ at the start of each half
turn,

$Q^{(n)}(n\pi)=\begin{cases}\beta^{2}\Lambda_{1}(1-D_{n})-\alpha^{2}\Lambda_{2}(1-C_{n})+\Lambda_{3}\frac{n\pi}{4},&\text{\qquad$n$\,even
}\\\
\displaystyle\beta^{2}\Lambda_{1}D_{n}-\alpha^{2}\Lambda_{2}C_{n}+\Lambda_{3}\frac{(n+1)\pi}{4}&\text{\qquad$n$\,odd
}\end{cases}$ (74)

and the gradients
$\displaystyle\mathcal{Q}_{n}=[Q^{(n+1)}((n+1)\pi)-Q^{(n)}(n\pi)]/\pi$ of line
segments connecting the node. One can easily show that
$\mathcal{Q}_{n}=\Xi_{n}/\pi$. The light-dark variation of these slopes serves
as a measure of the smoothness of the reorientation dynamics, and from the
first two values $\mathcal{Q}_{0}$ and $\mathcal{Q}_{1}$ we define two
relevant quantities: the strength of the initial response as measured by the
average of the slopes of the first two line segments
$\bar{\mathcal{Q}}=\left(\mathcal{Q}_{0}+\mathcal{Q}_{1}\right)/2=\bar{\Xi}/\pi$,
and its smoothness, as measured by the ratio
$\mathcal{Y}=\mathcal{Q}_{1}/\mathcal{Q}_{0}$.

If, as in Fig. 20, we approximate $Q^{(n)}(T)$ by the line
$\bar{\mathcal{Q}}T$, then the reorientation dynamics (72) takes the simple
form

$\Phi(T)=2\tan^{-1}\left(e^{-P^{*}\bar{\mathcal{Q}}T}\right),$ (75)

from which we identify the characteristic relaxation time $\tau_{x}$ (in
physical units) analogous to (60),

$\tau_{x}=\frac{1}{|\omega_{1}^{*}|\bar{\mathcal{Q}}}=\frac{\tilde{\zeta_{r}}}{|\mathcal{T}_{\mathrm{p}}^{*}|\bar{\mathcal{Q}}}.$
(76)

Finally, we explore the space of reorientation dynamics by probing
$Q^{(n)}(T)$ through its dependency on parameters $\alpha$ and $\beta$. Our
strategy is to observe how the quantities $\bar{\mathcal{Q}}(\alpha,\beta)$
(Fig. 21(a)) and $\mathcal{Y}(\alpha,\beta)$ (Fig. 21(b)), which are also
functions of $\alpha$ and $\beta$, and essentially describe the curve’s shape,
vary. Firstly, we make the observation that the ($\alpha$,$\beta$) pairs
acquired from micropipette experiments (step-up and frequency response; Fig.
10(b)) lie in the high-slope area ($\bar{\mathcal{Q}}=0.27$) of the
$\bar{\mathcal{Q}}(\alpha,\beta)$ function (Fig. 21(a) and Fig. 21(c)). We
also observe that the same data lie in an area of relatively moderate symmetry
($\mathcal{Y}\approx 0.34$) of the $\mathcal{Y}(\alpha,\beta)$ function (red
markers in Fig. 21(b)), as opposed to the extreme cases of highest symmetry
i.e. $\mathcal{Y}\approx 0.7$ (solid purple square in Fig. 21(b) and Fig.
21(d)) and lowest symmetry i.e. $\mathcal{Y}\approx 0$ (solid purple triangle
in Fig. 21(b) and Fig. 21(e)).

## References

  * [1] S.W. Bendix, Phototaxis, Bot. Rev. 26, 145–208 (1960).
  * [2] G. Thorson, Light as an ecological factor in the dispersal and settlement of larvae of marine bottom invertebrates, Ophelia 1, 167–208 (1964.
  * [3] P. Hegemann, Vision in microalgae, Planta 203, 265–274 (1997).
  * [4] G. Jékely, J. Colombelli, H. Hausen, and K. Guy and E. Stelzer and F. Nédélec and D. Arendt, Mechanism of phototaxis in marine zooplankton, Nature 456, 395–399 (2008).
  * [5] P. Hegemann, Algal Sensory Photoreceptors, Annu. Rev. Plant Biol. 59, 167–189 (2008).
  * [6] K.W. Foster and R. D. Smyth, Light antennas in phototactic algae, Microbiol. Rev. 44, 572–630 (1980).
  * [7] A. Giometto, F. Altermatt, A. Maritan, R. Stocker and A. Rinaldo, Generalized receptor law governs phototaxis in the phytoplankton Euglena gracilis, Proc. Natl. Acad. Sci. USA 112, 7045–7050 (2015).
  * [8] G. Jékely, Evolution of phototaxis, Phil. Trans. R. Soc. B 364, 2795–2808 (2009).
  * [9] N. Ueki, T. Ide, S. Mochiji, Y. Kobayashi, R. Tokutsu, N. Ohnishi, K. Yamaguchi, S. Shigenobu, K. Tanaka, J. Minagawa, T. Hisabori, M. Hirono, and K. Wakabayashi, Eyespot-dependent determination of the phototactic sign in Chlamydomonas reinhardtii Proc. Natl. Acad. Sci. USA 113, 5299–5304 (2016).
  * [10] J.O. Kessler, A.M. Nedelcu, C.A. Solari, and D.E. Shelton, Cells Acting as lenses: Possible Role for Light in the Evolution of Morphological Asymmetry in Multicellular Volvocine Algae, in Evolutionary Transitions to Multicellular Life, Advances in Marine Genomics, vol. 2, I. Ruiz-Trillo and A. Nedelcu, editors (Springer, Dordrecht, 2015).
  * [11] H.S. Jennings, On the significance of the spiral swimming of organisms, Am. Nat. 35, 369–378 (1901).
  * [12] E.E. Wildman, Why do ciliated animals rotate counter-clockwise while swimming? Science 63, 385–386 (1926).
  * [13] H.S. Jennings, Studies on Reactions to Stimuli in Unicellular Organisms. II.—The Mechanism of the Motor Reactions of Paramecium, Am. J. Phys. 2, 311–341 (1899).
  * [14] H.S. Jennings, Studies on Reactions to Stimuli in Unicellular Organisms. V.—On the Movements and Motor Reflexes of the Flagellata and Ciliata, Am. J. Phys. 3, 229–260 (1900).
  * [15] R. Kamiya and G.B. Witman, Submicromolar Levels of Calcium Control the Balance of Beating between the Two Flagella in Demembranated Models of Chlamydomonas, J. Cell Bio. 98, 97–107 (1984).
  * [16] U. Rüffer and W. Nultsch, High-speed cinematographic analysis of the movement of Chlamydomonas Cell Motility 5, 251–263 (1985).
  * [17] K. Schaller, R. David, and R. Uhl, How Chlamydomonas keeps track of the light once it has reached the right phototactic orientation, Biophys. J. 73, 1562–1572 (1997).
  * [18] S.K. Choudhary, A. Baskaran, and P. Sharma, Reentrant efficiency of phototaxis in Chlamydomonas reinhardtii cells, Biophys. J. 117, 1508–1513 (2019).
  * [19] D. Cortese and K.Y. Wan, Control of Helical Navigation by Three-Dimensional Flagellar Beating, Phys. Rev. Lett. 126, 088003 (2021).
  * [20] U. Rüffer and W. Nultsch, Flagellar photoresponses of Chlamydomonas cells held on micropipettes: I. Change in flagellar beat frequency, Cell Mot. Cytoskeleton 15, 162–167 (1990).
  * [21] U. Rüffer and W. Nultsch, Flagellar photoresponses of Chlamydomonas cells held on micropipettes: II. Change in flagellar beat pattern, Cell Mot. Cytoskeleton 18, 269–278 (1991).
  * [22] U. Rüffer and W. Nultsch, Flagellar Photoresponses of ptx1, a Nonphototactic Mutant of Chlamydomonas Cell. Mot. Cytoskeleton 37, 111–119 (1997).
  * [23] N. Okita, N. Isogai, M. Hirono, R. Kamiya, and K. Hoshimura, Phototactic activity in Chlamydomonas ‘non-phototactic’ mutants deficient in Ca2+-dependent control of flagellar dominance or in inner-arm dynein, J. Cell. Sci. 118, 529–537 (2005).
  * [24] M. Bessen, R.B. Fay, and G.B. Witman, Calcium Control of Waveform in Isolated Flagellar Axonemes of Chlamydomonas J. Cell. Biol. 86, 446–455 (1980).
  * [25] K. Yoshimura and R. Kamiya, The sensitivity of Chlamydomonas photoreceptor is optimized for the frequency of cell body rotation, Plant Cell Physiol. 42, 665–672 (2001).
  * [26] K. Josef, J. Saranak, and K.W. Foster, Ciliary behavior of a negatively phototactic Chlamydomonas reinhardtii, Cell Mot. Cytoskel. 61, 97–111 (2005).
  * [27] K. Josef, J. Saranak, and K.W. Foster, Linear systems analysis of the ciliary steering behavior associated with negative-phototaxis in Chlamydomonas reinhardtii, Cell Mot. Cytoskel. 63, 758–777 (2006).
  * [28] K. Drescher, R.E. Goldstein, and I. Tuval, Fidelity of adaptive phototaxis, Proc. Natl. Acad. Sci. USA 107, 11171–11176 (2010).
  * [29] P.A. Spiro, J.S. Parkinson, and H.G. Othmer, A model of excitation and adaptation in bacterial chemotaxis, Proc. Natl. Acad. Sci. USA 94, 7263–-7268 (1997).
  * [30] B. Friedrich, and F. Jülicher, Chemotaxis of sperm cells Proc. Natl. Acad. Sci. USA 104, 13256–-13261 (2007).
  * [31] M.J. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small reynolds numbers, Comm. Pure Appl. Math. 5, 109–118 (1952).
  * [32] M.B. Short, C.A. Solari, S. Ganguly, T.R. Powers, J.O. Kessler, and R.E. Goldstein, Flows driven by flagella of multicellular organisms enhance long-range molecular transport, Proc. Natl. Acad. Sci. USA 103, 8315–8319 (2006).
  * [33] H.A. Stone and A. Samuel, Propulsion of microorganisms by surface distortions, Phys. Rev. Lett. 77, 4102–-4104 (1996).
  * [34] H. de Maleprade, F. Moisy, T. Ishikawa, and R.E. Goldstein, Motility and phototaxis in Gonium, the simplest differentiated colonial alga Phys. Rev. E 101, 022416 (2020).
  * [35] The present paper is a major revision to an earlier preprint: K.C. Leptos, M. Chioccioli, S. Furlan, A.I. Pesci, and R.E. Goldstein, An adaptive flagellar photoresponse determines the dynamics of accurate phototactic steering in Chlamydomonas, bioRxiv doi: 10.1101/254714.
  * [36] R.R. Bennett and R. Golestanian, A steering mechanism for phototaxis in Chlamydomonas, J. R. Soc. Interface 12, 20141164 (2015).
  * [37] D.L. Kirk, A twelve-step program for evolving multicellularity and a division of labor, BioEssays 27, 299–310 (2005).
  * [38] E.H. Harris, The Chlamydomonas Sourcebook, Vol. 1-3 (Academic Press, Oxford, 2009).
  * [39] J.-D. Rochaix, S. Mayfield, M. Goldschmidt-Clermont, and J.M. Erickson, Molecular biology of Chlamydomonas, in Plant Molecular Biology: a Practical Approach, ed. C.H. Schaw (IRL Press, Oxford, 1988), pp. 253–275.
  * [40] M. Polin, I. Tuval, K. Drescher, J.P. Gollub, and R.E. Goldstein, Chlamydomonas swims with two “gears” in a eukaryotic version of run-and-tumble locomotion Science 325, 487–490 (2009).
  * [41] K.C. Leptos, K.Y. Wan, M. Polin, I. Tuval, A.I. Pesci, and R.E. Goldstein, Antiphase synchronization in a flagellar-dominance mutant of Chlamydomonas Phys. Rev. Lett. 111, 158101 (2013).
  * [42] K. Drescher, K.C. Leptos, and R.E. Goldstein, How to track protists in three dimensions, Rev. Sci. Instrum. 80, 014301 (2009).
  * [43] See Supplemental Material at http://link.aps.org/supplemental/10.1103/XXX for further experimental videos.
  * [44] R.G. Cox, The motion of long slender bodies in a viscous fluid. Part 1: General theory, J. Fluid Mech. 44, 791–810 (1970).
  * [45] E. Lauga, The Fluid Dynamics of Cell Motility, (Cambridge University Press, Cambridge, 2020).
  * [46] R. Sager and G.E. Palade, Structure and Development of the Chrloroplast in Chlamydomonas. I. The Normal Green Cell. J. Biophys. Biochem. Cytol. 3, 463–488 (1957).
  * [47] R.P. McCord, J.N. Yukich, and K.K. Bernd, Analysis of force generation during flagellar assembly through optical trapping of free-swimming Chlamydomonas reinhardtii, Cell Mot. Cyto. 61, 137-144 (2005).
  * [48] D.R. Brumley, K.Y. Wan, M. Polin and R.E. Goldstein, Flagellar synchronization through direct hydrodynamic interactions, eLife 3, e02750 (2014).
  * [49] J.S. Guasto, K.A. Johnson, and J.P. Gollub, Oscillatory Flows Induced by Microorganisms Swimming in Two Dimensions, Phys. Rev. Lett. 105, 168102 (2010).
  * [50] G.S. Klindt and B.M. Friedrich, Flagellar swimmers oscillate between pusher- and puller-type swimming, Phys. Rev. E 92, 063019 (2015).
  * [51] T.J. Böddeker, S. Karpitschka, C.T. Kreis, Q. Magdelain and O. Bäumchen, Dynamics force measurements on swimming Chlamydomonas cells using micropipette force sensors, J. R. Soc. Interface 17, 20190580 (2020).
  * [52] K.C. Leptos, J.S. Guasto, J.P. Gollub, A.I. Pesci, and R.E. Goldstein, Dynamics of Enhanced Tracer Diffusion in Suspensions of Swimming Eukaryotic Microorganisms, Phys. Rev. Lett. 103, 198103 (2009).
  * [53] H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Boston, MA, 1980).
  * [54] H.C. Crenshaw, Orientation by helical motion - I. Kinematics of the helical motion of organisms with up to six degrees of freedom, Bull. Math. Biol. 55, 197–212 (1993).
  * [55] H.C. Crenshaw and L. Edelstein-Keshet, Orientation by helical motion - II. Changing the direction of the axis of motion, Bull. Math. Biol. 55, 213–230 (1993).
  * [56] H.C. Crenshaw, Orientation by helical motion - III. Microorganism can orient to stimuli by changing direction of their rotational velocity, Bull. Math. Biol. 55, 197–212 (1993).
  * [57] O.A. Sineshchekov and K.-H. Jung and J.L. Spudich, Two rhodopsins mediate phototaxis to low- and high-intensity light in Chlamydomonas reinhardtii, Proc. Natl. Acad. Sci. USA 99, 8689–8694 (2002).
  * [58] W.C. Ratcliff, M.D. Herron, K. Howell, J.T. Pentz, F. Rosenzweig, and M. Travisano, Experimental evolution of an alternating uni- and multicellular life cycle in Chlamydomonas reinhardtii, Nat. Commun. 4, 2742 (2013).
  * [59] T.C. Day, S.S. Höhn, S.A. Zamani-Dehaj, D. Yanni, A. Burnetti, J. Pentz, A.R. Honerkamp-Smith, H. Wioland, H.R. Sleath, W.C. Ratcliff, R.E. Goldstein, and P.J. Yunker, Cellular organization in lab-evolved and extant multicellular species obeys a maximum entropy law, eLife 11, e72707 (2022).
  * [60] A.C.H. Tsang, A.T. Lam, and I.H. Riedel-Kruse, Polygonal motion and adaptable phototaxis via flagellar beat switching in the microswimmer Euglena gracilis, Nature Phys. 14, 1216–1222 (2018).
  * [61] J. Arrieta, A. Barreira, M. Chioccioli, M. Polin, and I. Tuval, Phototaxis beyond turning: persistent accumulation and response acclimation of the microalga Chlamydomonas reinhardtii, Sci. Rep. 7, 3447 (2017).
  * [62] A. Ramamonjy, J. Dervaux, and P. Brunet, Nonlinear phototaxis and instabilities in suspensions of light-seeking algae, Phys. Rev. Lett. 128, 258101 (2022).