# Short-term memory effects in the phototactic behavior of microalgae

Taha Laroussi  Laboratoire d’Hydrodynamique (LadHyX), CNRS, Ecole
Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
Mojtaba Jarrahi  Université Paris-Saclay, CNRS, FAST, 91405 Orsay, France
Gabriel Amselem  Laboratoire d’Hydrodynamique (LadHyX), CNRS, Ecole
Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
<EMAIL_ADDRESS>
###### Abstract

Phototaxis, the directed motion in response to a light stimulus, is crucial
for motile microorganisms that rely on photosynthesis, such as the unicellular
microalga Chlamydomonas reinhardtii. It is well known that microalgae adapt to
ambient light stimuli. On time scales of several dozen minutes, when
stimulated long enough, the response of the microalga evolves as if the light
intensity were decreasing [Mayer, Nature (1968)]. Here, we show experimentally
that microalgae also have a short-term memory, on the time scale of a couple
of minutes, which is the opposite of adaptation. At these short time scales,
when stimulated consecutively, the response of C. reinhardtii evolves as if
the light intensity were increasing. Our experimental results are rationalized
by the introduction of a simplified model of phototaxis. Memory comes from the
interplay between an internal biochemical time scale and the time scale of the
stimulus; as such, these memory effects are likely to be widespread in
phototactic microorganisms.

##  I Introduction

Phototaxis, the directed motion of organisms in response to a light stimulus,
is widespread both in prokaryotes and single-cell eukaryotes [1]. One of the
cellular models for eukaryotic phototaxis is the microalga Chlamydomonas
reinhardtii, which responds to blue-green light [2, 3]. When light hits the
eyespot of the microalga, it induces photocurrents, whose amplitude depend on
the light intensity. These photocurrents then induce flagellar currents which
change the beating pattern of the flagella, leading to reorientation and
eventually phototaxis [3, 4].

At low light intensities, wild-type C. reinhardtii cells swim towards the
light, while they swim away from the light at high light intensities. This
corresponds to positive and negative phototaxis, respectively [5]. What sets
the change in phototactic behavior? The answer to this question is not fully
settled [3].

Several biochemical parameters were found to affect the sign of phototaxis of
C. reinhardtii, such as the amount of calcium ions in the surrounding medium
[6, 7, 8], photosynthetic activity of the microalgae [9], the amount of
intracellular reactive oxygen species [10], or the phosphorylation of
channelrhodopsin-1, one of the photoreceptors of C. reinhardtii [11]. It is
also known that the history of the alga plays a role in its phototactic
response: like many unicellular organisms [12, 13, 14], C. reinhardtii adapt
to their environment. A naive cell population, kept in the dark, exhibits
negative phototaxis in response to an intense light stimulus; the same cell
population, exposed to the same intense light stimulus, undergoes positive
phototaxis when it has been previously exposed to light for a couple dozens
minutes [15]. This change in phototactic behavior of a population depending on
the history of irradiation is consistent with results obtained at the single-
cell level by Rüffer and Nultsch [4], who monitored the change in beating of
the two flagella of C. reinhardtii in response to increasing and decreasing
light-stimuli. Such a history-dependent change in the phototactic response
occurs on long time scales, of the order of a couple dozen minutes.

Here, we show experimentally that the change in phototactic behavior also
depends on the recent history of the cell, where the time scale is of the
order of a couple minutes. At these short time scales, the algae exhibit a
behavior that is the exact opposite of the long-term adaptation: they
integrate consecutive signals over time. When subjected to two consecutive,
closely spaced identical stimuli, an alga essentially adds up the second
stimulus to the first one, and acts as if the second stimulus were of higher
intensity than the first one. Such a signal integration has, to the best of
our knowledge, never been observed in the phototactic response of microalgae.
Our results are rationalized by the introduction of a simplified model of
phototaxis. The memory emerges in the model from the interplay between two
time scales, the time between successive stimuli and the relaxation time of an
inner biochemical process. Since the model is generic, similar short-term
memory effects are likely to be widespread in other organisms that experience
phototaxis.

##  II Methods

###  II.1 Culture preparation

C. reinhardtii strain CC-125 (Chlamydomonas Resource Center, University of
Minnesota, MN, USA) were cultured on a solid medium prepared with Tris-
Acetate-Phosphate (Gibco™ TAP, ThermoFischer Scientific, France) and agar
every 4 weeks, to keep the strain motile, responsive to light and prevent as
much as possible the cells from sticking to the walls of the devices. For
experiments, colonies of C. reinhardtii were picked from solid cultures and
recultured in liquid TAP. The cultures were placed in an agitating incubator
at $176\;\rm rpm$, under a day/light cycle of 14h/10h with an illuminance of
$60\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$ and at a fixed temperature of
$22^{\circ}\rm C$. The cells reached maximum motile cell density within 3 days
[16], after which a solution of swimming C. reinhardtii was obtained.

Before being used for experiments, the cultures underwent a series of
centrifugation steps. This enabled to concentrate the algal solution, get rid
of low-motility and dead algae, and of cellular debris. First, 45 mL of the
liquid culture were centrifuged at 1057$g$ for 10 minutes. Then, 39 mL of the
supernatant was removed, and the bottom 6 mL were centrifuged at 73$g$ for 2
minutes. The supernatant, containing the motile cells, was kept and
centrifuged again at 285$g$ for 5 minutes. The amount of supernatant was
adjusted to obtain the desired algae final concentration.

###  II.2 Experimental setup

An array of shallow cylindrical wells was made in polydimethylsiloxane (PDMS,
Dow-Corning Sylgard 184), using standard soft lithography techniques [17]. The
wells had a depth of $32\;\mu\rm m$ and diameters ranging from $200\;\mu\rm m$
to $500\;\mu\rm m$. The PDMS was rendered hydrophilic by plasma cleaning.
Then, a drop of $\approx 10\;\rm\mu L$ of the algal solution was deposited on
the PDMS. The device was closed with a plasma-activated glass slide by
applying gentle pressure.

The trapped algae were observed under a Nikon TI microscope, using a 4x
objective. Images were recorded at 10 fps with a CMOS camera (Hamamatsu ORCA-
Flash4.0 LT, Hamamatsu Photonics, France). A long-pass red filter with a
$645\;\rm nm$ cut-on wavelength (Newport RG645) was placed between the
microscope’s light source and the microwells. A 3 mm blue LED with a
wavelength $\lambda=470\;\rm nm$ and a light intensity of $7000\;\rm mcd$
(Planète Leds, France) was placed on the microscope’s plate, approximately
$2\;\rm cm$ away from the microfluidic chip, on the same plane. The LED
operating voltage range was between $2.4\;\rm V$ and $3.2\;\rm V$ and its
operating current was $20\;\rm mA$. The light intensity was tuned by varying
the applied voltage. An illustration of the setup is shown in Fig. 1.

Figure 1: Experimental setup. (a) Schematic drawing of a PDMS chip with
shallow cylindrical wells of varying diameters. The algal solution is
deposited in the wells, which are then sealed with a glass slide. The chip is
exposed to blue light from the side. (b) Bright-field microscopy image of an
array of 28 cylindrical wells of different diameters and algae concentrations.
Cell bodies are visible as black dots inside the wells.

The algae concentration $\phi$ was determined in each well from images
captured at the beginning of the experiments, before exposition to a light
stimulus. Initially, the algae swam randomly and were distributed
homogeneously in each well. They were identified by thresholding the images,
and the total area they occupied in each well was determined. The
concentration in algae was then defined as the fraction of surface area
occupied by the algae in the well.

###  II.3 Measuring the local light intensity

We estimated the local light intensity inside a well using the number of
photons reaching the sensor of the camera. First, we determined the camera
offset value by blocking off all the light to the camera and taking a 16-bit
image. The spatial average intensity in grey value of all the pixels in the
image was the camera’s offset. Then, a 16-bit setup image of the sample was
taken at the current experimental condition, with the light of the microscope
turned off and the blue LED light on. The camera offset value was then
subtracted from each pixel in the setup image. The grey values were first
converted to number of electrons by dividing each pixel in the image by the
conversion gain of the camera. The electrons were converted to photons by
dividing the number of electrons from each pixel by the quantum efficiency
(QE) of the sensor at $\lambda=470\;\rm nm$. Last, the local light intensity
inside a well was computed as the average number of photons within the well.

##  III Results

###  III.1 First experimental results

Figure 2: Different phototactic behaviors of C. reinhardtii enclosed in a
well. The blue light stimulus comes from the upper side of the wells and is
turned on at $t=30\;\rm s$. (a) When exposed to a low intensity
$I=0.005\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$, the algae show positive
phototaxis. (b) At intermediate intensities $I=0.089\;\mu\rm mol\cdot
m^{-2}\cdot s^{-1}$, the algae show first positive phototaxis, then negative
phototaxis. (c) At a high intensity $I=1.8\;\mu\rm mol\cdot m^{-2}\cdot
s^{-1}$, the algae show negative phototaxis. The center of mass of the
population is represented with a red arrow pointing from the center of the
well.

The experiments consist in exposing a population of C. reinhardtii to a
directional source of light. When a population of microalgae is exposed to a
light intensity lower than $I_{\rm low}\approx 0.065\;\mu\rm mol\cdot
m^{-2}\cdot s^{-1}$, the algae systematically exhibit positive phototaxis and
swim towards the light source. Within $5$ to $8$ seconds of stimulation, most
of the algae have accumulated at the boundary facing the light, see snapshots
in Fig. 2a. The algae remain accumulated at the boundary during at least one
minute, time after which the imaging was stopped. When the light intensity is
higher than $I_{\rm high}\approx 0.43\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$,
the algae always exhibit negative phototaxis and accumulate at the boundary
opposing the light source, see snapshots in Fig. 2c. These results are well
known in the literature although the precise value of the threshold
intensities varies from experiment to experiment, see e.g. [18, 19].
In between these two light intensities, there is a transition regime, where
the phototactic behavior is hard to reproduce, despite keeping all
experimental parameters identical: the microalgae sometimes exhibit a
transient positive phototaxis followed by a negative phototaxis (see Fig. 2b),
but can also display purely positive phototaxis, or purely negative phototaxis
(see later in text). The aim of our work is to better understand the
parameters responsible for this variety of algal behaviors in the transition
regime. We investigate the effects of well diameter, algae concentration,
light intensity, and finally history of the algae, on the phototactic
behavior.

###  III.2 Quantification of the phototactic behavior

To quantify the phototactic behavior of a microalgae population constrained in
its well, we binarize the experimental images using a simple Otsu threshold on
pixel intensities. This leads to images where the algae are white, on a black
background (see Supp. Fig. 1). In each image, we then calculate the center of
mass $z_{\rm cm}$ of the white pixels, corresponding to the center of mass of
the population. The position of the center of mass is tracked over time, and
eventually normalized to the radius of the well. A value of the center of mass
$z_{\rm cm}=1$ (resp. $z_{\rm cm}=-1$) corresponds to all algae accumulating
at the boundary of the well facing (resp. opposite from) the light.

Figure 3: The diameter of the well $d_{c}$ does not influence the phototactic
behavior of the algae. The light stimulus is turned on at $t=30\;\rm s$ and
the position of the center of mass $z_{\rm cm}$ is tracked over time. (a) When
exposed to an intensity $I=0.019\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$, the
algae show positive phototaxis. (b) At intermediate intensities
$I=0.12\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$, the algae show first positive
phototaxis, then negative phototaxis. (c) At a high intensity $I=1.4\;\mu\rm
mol\cdot m^{-2}\cdot s^{-1}$, the algae show negative phototaxis. Each curve
is an average over 4 to 7 experiments.

In one experimental run, the phototactic response of 15 – 30 wells containing
microalgae is quantified. The field of view contains wells with at least 3
different diameters, and there are at least 4 wells of each diameter, see Fig.
1b. We start with an experiment where the algae are at the same concentration
in all wells, and do not find any impact of the well diameter on the
phototactic behavior of populations of C. reinhardtii: for all diameters, in
one given experiment, the algae exhibit the same behavior, as shown by the
evolution of the center of mass of the population, see Fig. 3. Note that the
center of mass never reaches $\pm 1$. This is mainly due to a fraction of the
algae not responding to light, see Supp. Fig. 2. A second-order effect is that
algae take space, and so the center of mass of the population can never reach
$\pm 1$.

The time scales for pure positive and pure negative phototaxis are different:
positive phototaxis leads to an accumulation on the side of the light source
within $t_{\rm pos}\approx 5.3\pm 0.9\;\rm s$ (mean $\pm$ std. deviation)
after the stimulus is turned on. The time scale for negative phototaxis is
slightly longer: $t_{\rm neg}\approx 9.1\pm 1.1\;\rm s$. In the case where
there is a back-and-forth between positive and negative phototaxis, the first
accumulation occurs within $t_{\rm pos}\approx 5.1\pm 0.3\;\rm s$ while the
second accumulation is within $t_{\rm neg}\approx 34\pm 10\;\rm s$. Negative
phototaxis always occurs on a time scale longer than positive phototaxis,
irrespective of the behavior (back-and-forth or not) of the population.

###  III.3 Effect of cell concentration

Figure 4: The concentration $\phi$ of algae in the well does not influence the
phototactic behavior. The light stimulus is turned on at $t=30\;\rm s$ and the
position of the center of mass $z_{\rm cm}$ is tracked over time. (a) When
exposed to an intensity $I=0.059\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$, the
algae show positive phototaxis. (b) At intermediate intensities
$I=0.069\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$, the algae show first positive
phototaxis, then negative phototaxis. (c) At a high intensity $I=0.88\;\mu\rm
mol\cdot m^{-2}\cdot s^{-1}$, the algae show negative phototaxis.

One possibility for the alternating phototactic behavior would be to invoke
screening of the light by the algae. We would then expect that at high algae
concentration, the algae closest to the light source screen the light
intensity, so that the algae further away from the light source see a dimmer
light. This could result in a seemingly biphasic behavior, with, at high light
intensities, algae close to the light source showing negative phototaxis, and
algae further away from the light source exhibiting positive phototaxis.

To test this hypothesis, we enclosed microalgae at concentrations $\phi=0.008$
to $\phi=0.6$ in the microwells, and monitored the evolution of their center
of mass. The concentration $\phi$ is defined as the surface fraction of algae
in a microwell, see Methods. For $\phi>0.5$, the well is essentially packed
with algae, which impedes their motion and so leads to the center of mass not
moving even during the light stimulus, see Supp. Fig. 3. For $\phi\leq 0.5$,
we do not see any effect of the concentration on the sign of phototaxis: the
phototactic behavior depends solely on the light intensity, see Fig. 4. There
is however an effect of concentration on the position of the center of mass
$z_{\rm cm}$: when the concentration is higher, the algae take more space and
the center of mass of the population gets closer to the center of the well,
see Fig. 4a and c.

The experiments were reproduced over a range of light intensities and algae
concentrations, in more than 300 wells. In each well, the behavior was
quantified as positive phototaxis, negative phototaxis, or back-and-forth. The
resulting phase diagram of phototactic behavior as a function of light
intensity and algae concentration shows no clear influence of the algae
concentration on the phototactic behavior, see Fig. 5.

Figure 5: Phase diagram for the phototactic behavior of 367 populations of
algae, as a function of light intensity and algae concentration.

###  III.4 Effect of history

We finally investigated the effect of history on the behavior of the algae. To
do so, a population of algae was kept in the dark for one hour and then
exposed to a given light stimulus during 90 seconds. The light was then turned
off during 90 seconds, after which the exact same stimulus was re-applied. At
low light intensities, the qualitative behavior of algae does not change
between the two stimuli: the algae always exhibit positive phototaxis, see the
first two graphs in Fig. 6a. At high light intensities, the algae always
exhibit negative phototaxis, see the first two graphs in Fig. 6c. At
intermediate light intensities however, the behavior of the algae changes
between the two stimuli: in the first stimulus, the algae exhibit a back-and-
forth motion, first towards the light and then away from it. During the second
stimulus, the algae show negative phototaxis, see the first two graphs in Fig.
6b.

To understand whether the change in behavior was permanent, the algae were
allowed to rest in the dark for 30 minutes after the end of the second
stimulus. Then, the same protocol was followed: a third stimulus, of the same
intensity as the two previous ones, was applied. After 90 seconds in the dark,
the algae were restimulated with a fourth, identical stimulus. In the case of
low and high light intensity, the qualitative response of the algae was the
same for the third and fourth stimuli as for the two first ones, see the last
two graphs in Fig. 6a,c. At intermediate light intensities, the algae showed
positive phototaxis during the third stimulus, and then qualitatively changed
their behavior during the fourth stimulus to exhibit slightly negative
phototaxis, see the last two graphs in Fig. 6b. This last response to light,
averaged over 31 wells, is quite subtle. Indeed, after several stimuli, most
of the algae stick to the wells, a likely effect of light stimulation [20].
These stuck algae move very slowly by gliding [21], see Supp. Fig. 4. The
center of mass of the population then largely reflects the position of the
stuck algae. The motile algae however do exhibit negative phototaxis, see
Supp. Fig. 4.

Note that the quantitative response of the algae, as measured by the position
of the center of mass $z_{\rm cm}$, differs between the four stimuli – even in
the case of low and high light intensities. For example, at low light
intensities (positive phototaxis), the shift in the center of mass is
identical for stimuli 1 and 2. The shift is less pronounced for stimuli 3 and
4, see Fig. 6a. In contrast, in the case of negative phototaxis, the shift of
the center of mass is identical for stimuli 1 and 3, as well as for stimuli 2
and 4, see Fig. 6c: there, resting in the dark for 30 minutes seems to reset
the phototactic behavior.

What is important in our case however is the qualitative response: the algae
can change phototactic behavior at intermediate light intensities due to the
stimuli they previously experienced. The change in behavior is not an adaptive
response, which would make the algae show first negative phototaxis, then
positive phototaxis in response to the same stimulus. It rather corresponds to
an integration of the signal over time. This effect explains the variety of
behaviors observed in the transition regime between positive and negative
phototaxis: one cannot predict the phototactic response of an alga without
knowing its history.

Figure 6: Memory effects on the phototactic response of algae. The light
stimulus is turned on at $t=30\;\rm s$ and the position of the center of mass
$z_{\rm cm}$ is tracked over time. Four consecutive experiments are performed
with varying rest times, where the light is turned off in between. There is a
90 s pause between Exp. 1 and Exp. 2, then a 30 min pause between Exp. 2 and
Exp. 3 and finally a 90 s pause between Exp. 3 and Exp. 4. (a) When exposed to
a low intensity $I=0.003\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$, the algae show
positive phototaxis in all experiments. The center of mass is averaged over 24
experiments. (b) At intermediate intensities $I=0.093\;\mu\rm mol\cdot
m^{-2}\cdot s^{-1}$, the algae show different phototactic behaviors in each
experiment. The center of mass is averaged over 31 experiments. (c) At a high
intensity $I=1.4\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$, the algae show negative
phototaxis in all experiments. The center of mass is averaged over 18
experiments. Each orange line represents the center of mass $z_{\rm cm}$ of
one experiment. The dark red solid line represents the average $z_{\rm cm}$
for the set of experiments, and the shaded area in red gives the standard
deviation around the mean.

##  IV Model

We now propose a simplified model to describe our experimental observations.
We call $I_{0}$ the light intensity within a well, and $S$ an intracellular
biochemical species responsible for the sign of phototaxis. The total
concentration of $S$ is conserved and is called $s_{\rm tot}$. This species
exists in two states: active (with concentration $s^{\star}$) and inactive
(with concentration $s$), such that $s^{\star}+s=s_{\rm tot}$.

We assume the transition from inactive to active state depends on the light
intensity $I_{0}$, while the reverse transition occurs at a constant rate
$\tau^{-1}$:

$\frac{ds^{\star}}{dt}=\gamma I_{0}(s_{\rm
tot}-s^{\star})-\tau^{-1}s^{\star},$ (1)

where $\gamma$ is the reaction rate at which $s$ is converted into
$s^{\star}$. When the concentration of active molecules $s^{\star}$ is large
(resp. small), the algae undergo negative (resp. positive) phototaxis, which
can be summarized as:

$\frac{dz}{dt}=-{\rm sign}(s^{\star}-s_{T})v_{0},$ (2)

where $z$ is the position of an alga, $s_{T}$ is the threshold concentration
at which cells transition from positive to negative phototaxis, and $v_{0}$ is
the characteristic speed of the alga. Equations (3) and (5) constitute a
minimal model of phototaxis.

Figure 7: Simulation results for three different light intensities $I_{0}$. We
recover positive phototaxis at $I_{0}=0.01\;\mu\rm mol\cdot m^{-2}\cdot
s^{-1}$, back-and-forth motion at $I_{0}=0.1\;\mu\rm mol\cdot m^{-2}\cdot
s^{-1}$, and negative phototaxis at $I_{0}=1\;\mu\rm mol\cdot m^{-2}\cdot
s^{-1}$. Other simulation parameters: $s_{T}=0.035$, $s_{\rm tot}=1$,
$v_{0}=100\;\mu\rm m\cdot\rm s^{-1}$, $\tau=600\;\rm s$ and $\gamma=1/60\;\rm
m^{2}\cdot\mu mol^{-1}$.

In the remaining part of this study, all simulations parameters are fixed
except for the light intensity $I_{0}$. We normalize the concentrations of $S$
so that $s_{\rm tot}=1$, and place the algae in a well of radius $500\;\rm\mu
m$. The position of the alga $z$ is then renormalized by this radius to obtain
a number between -1 and 1. The light intensities are expressed, as in the
experiment, as a flux of photons in $\mu\rm mol\cdot m^{-2}\cdot s^{-1}$. We
choose $\tau=600\;\rm s$, $\gamma=1/60\;\rm m^{2}\cdot\mu mol^{-1}$,
$s_{T}=0.035$, $s_{\rm tot}=1$ and $v_{0}=100\;\mu\rm m\cdot\rm s^{-1}$.
Initially, the algae are naive ($s^{\star}=0$) and $z=0$. A detailed
description of the choice of the model parameters can be found in the
Supplementary Material.

At low light intensities ($I_{0}=0.01\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$),
the amount of activated chemical species $s^{\star}$ is lower than the
threshold $s_{T}$, and the resulting behavior is purely positive phototaxis.
At intermediate light intensities ($I_{0}=0.1\;\mu\rm mol\cdot m^{-2}\cdot
s^{-1}$), the usage rate $\gamma I_{0}$ and relaxation rate $\tau^{-1}$ are of
similar orders of magnitude, and the behavior can alternate between positive
and negative phototaxis, resulting in a back-and-forth motion of the algae.
Finally, at high light intensities ($I_{0}=1\;\mu\rm mol\cdot m^{-2}\cdot
s^{-1}$), all molecules are instantly converted to an active state, and cannot
relax back, resulting in negative phototaxis, see Fig. 7. Note that, at high
light intensities, the simulations show an initial small positive phototactic
effect; this is similar to what is observed in experiments, see Fig. 6c.

Apart from the qualitative agreement between the different phototactic
behaviours, the model also reproduces well the different experimental time
scales. Positive phototaxis leads to accumulation within $t_{\rm pos}\approx
5\;\rm s$, which depends solely on the swimming speed of the alga.
Accumulation due to negative phototaxis occurs on a longer time scale, $t_{\rm
neg}\approx 9\;\rm s$, due to the initial small positive phototactic effect.
The back-and-forth motion leads to accumulation within $\approx 32\;\rm s$, a
consequence of the addition of three time scales: the time to swim to $z=1$,
the time to produce enough activated species to overcome the threshold
$s_{T}$, and the time to eventually swim to $z=-1$.

We now simulate the application of successive stimuli of $90\;\rm s$, as in
the experiments. Between two stimuli, the alga position $z$ is reset to 0.
Indeed, in the experiments, the algae repopulate homogeneously the well after
one minute without light stimulus, and so the center of mass of the population
goes back to 0 before the next experiment. At low ($I_{0}=0.01\;\mu\rm
mol\cdot m^{-2}\cdot s^{-1}$) and high ($I_{0}=1\;\mu\rm mol\cdot m^{-2}\cdot
s^{-1}$) light intensities, applying the same stimulus multiple times does not
change the phototactic behavior, independently of the time between stimuli,
see Fig. 8a,c.

At intermediate light intensity ($I_{0}=0.1\;\mu\rm mol\cdot m^{-2}\cdot
s^{-1}$), we find that the phototactic behavior depends on the number of
stimuli applied, their duration, and on the timing between stimuli. At the
beginning of the first simulation, algae are in their naive state. When the
stimulus is applied, the number of active molecules increases but is initially
lower than the threshold $s_{T}$, which leads to positive phototaxis. When the
amount of active molecules increases above the threshold value $s_{T}$, the
phototactic behavior reverts during the experiment, and leads to the overall
back-and-forth behavior. The light is then turned off for a duration
$t=90\;\rm s$. After this rest time, the amount of active molecules has
decayed but is still above $s_{T}$, so that when the algae are restimulated,
they display purely negative phototaxis. We then remove the light stimulus for
$30\;\rm min$, allowing almost all active molecules to revert back to their
inactive state. When the light is switched back on during the third
experiment, the algae exhibit the back-and-forth motion: positive phototaxis
at the beginning of the experiment, negative phototaxis once the threshold
amount of active molecules is reached. Finally, the light is turned off once
more for a duration $t=90\;\rm s$. After this rest time, the amount of active
molecules is still above $s_{T}$, and the algae display purely negative
phototaxis in the last simulation. Simulations results can be seen in Fig. 8b,
where the pause times are illustrated with gray-shaded areas.

A prediction of the model is that algae can switch behavior successively from
positive phototaxis, to back-and-forth, to negative phototaxis before
reverting back to positive phototaxis. Such is the case for three consecutive
stimuli of $I_{0}=0.02\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$ spaced by $90\;\rm
s$ and a pause of $30\;\rm min$ before the fourth stimulus, see Fig. 8d. This
switch through all phototactic behaviors requires a precise tuning of both the
light intensity and the waiting times, which is experimentally tedious. We
were however able to experimentally observe the switch from positive to back-
and-forth behavior and from back-and-forth to negative behavior, see Supp.
Fig. 5.

Figure 8: Simulation results for four consecutive experiments performed at a
constant light intensity $I_{0}$, with different rest times in between. Each
experiment lasts for $90$ s. Rest times are shown in shaded gray areas.
($t_{\rm{pause,1}}=90$ s, $t_{\rm{pause,2}}=30$ min and $t_{\rm{pause,3}}=90$
s for (a), (b) and (c)). (a) When exposed to a low light intensity
$I_{0}=0.01\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$, the alga always displays
positive phototaxis. (b) At intermediate intensities $I_{0}=0.1\;\mu\rm
mol\cdot m^{-2}\cdot s^{-1}$, the alga varies its phototactic behavior between
each experiment. In the first experiment, the alga shows a back-and-forth
motion. In the second, the alga shows only negative phototaxis. In the third
experiment, the alga displays a back-and-forth motion again. Finally, in the
last experiment, the behavior is negative phototaxis. (c) At a high intensity
$I_{0}=1\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$, the alga always shows negative
phototaxis. (d) At low to intermediate intensities $I_{0}=0.02\;\mu\rm
mol\cdot m^{-2}\cdot s^{-1}$ and with adjusted pause times
($t_{\rm{pause,1}}=90$ s, $t_{\rm{pause,2}}=90$ s and $t_{\rm{pause,3}}=30$
min) it is possible to successively go through positive phototaxis in the
first experiment, back-and-forth motion in the second, negative phototaxis in
the third and then revert back to positive phototaxis in the last experiment.
Other simulation parameters: $s_{T}=0.035$, $s_{\rm tot}=1$,
$v_{0}=100\;\mu\rm m\cdot\rm s^{-1}$, $\tau=600\;\rm s$ and $\gamma=1/60\;\rm
m^{2}\cdot\mu mol^{-1}$.

##  V Discussion

We have shown experimentally that the change in phototactic behavior of C.
reinhardtii can be due to its recent history of illumination. When exposed to
two closely spaced, identical stimuli, the algae act as if they were time-
integrating the stimuli. This is visible when the stimulus is close to that
delimiting positive and negative phototaxis. Then, the algae show two
different behaviors in response to the two consecutive stimuli: first a back-
and-forth response, then a negative phototaxis. When the stimuli are
sufficiently spaced apart, the algae have had time to relax to a basal state,
and show the same qualitative response to the two stimuli, see Fig. 6. In our
experiments, we find this time of complete relaxation to be approximately
equal to 30 minutes.

A simplified model of phototaxis explains this memory effect by the
introduction of a time-scale of relaxation in the inner biochemistry of the
cell. When the time-scale of relaxation is larger than the time between two
consecutive stimuli, the alga integrates the signal and can display different
behaviors in response to the two identical stimuli, see Fig. 8. The time
needed to return to the basal state should be of order $3\tau$, which
underlies our choice of $\tau=10\;\rm min$ for the relaxation time constant.
Our experimental data are consistent with an inner relaxation time $\tau$ in
the range $\approx 5$ to 20 minutes. What could be the biochemical species
corresponding to the intracellular species $S$ introduced in the model? It is
very likely that our simplified model compounds the dynamics of multiple
biochemical species, activated sequentially in a signalling cascade. Still, it
is known that the photoreceptors for phototaxis in C. reinhardtii are
channelrhodopsins, that become phosphorylated within milliseconds after the
application of a light stimulus. A high level of phosphorylated
channelrhodopsin-1 (ChR1) in the cell correlates with a change from positive
to negative phototaxis [11], much like a high level of activated $S$
corresponds to negative phototaxis in our model. Moreover, the time scale of
dephosphorylation of ChR1 in C. reinhardtii has been shown to be of the order
of $\approx 5$ to 15 minutes [11], which is compatible with our value of
$\tau=10\;\rm min$. A possibility is therefore to identify $S$ with ChR1.

A more physical model of phototaxis was recently introduced by Leptos et al.
to describe the response of C. reinhardtii to light [22]. In their model,
Leptos et al. take into account the rotation of the microalga around its axis
while swimming, and the resulting oscillating light intensity seen by the
eyespot. The consecutive step-up and step-down stimuli sensed by the eyespot
during one body rotation lead to a realignment of the trajectory towards the
light source. Our model averages out the body rotation of the microalgae and
describes the behavior of the microorganism at a more coarse-grained time-
scale. Another difference is that the model of Leptos et al. does not take
into account the fact that phototaxis changes sign when the light intensity
increases. This change of phototactic behavior requires the introduction of a
threshold value in the model, below (resp. above) which phototaxis is positive
(resp. negative). Such a threshold is incorporated in our simplified model.

Note that the threshold in the model, introduced as a simple means to obtain a
change in behavior between positive and negative phototaxis at different light
intensities, leads to the back-and-forth behavior at intermediate light
intensities. The existence of this threshold also enables to explain why the
time scale of negative phototaxis is longer than that of positive phototaxis.
Negative phototaxis occurs when the amount of active species $s^{\star}$
overcomes a threshold $s_{T}$; this takes time, during which the alga exhibits
positive phototaxis. Only when $s^{\star}>s_{T}$ do the algae show negative
phototaxis, and accumulate at one boundary of the well. The small initial
positive phototactic response predicted by the model is in agreement with
experimental data. The model also predicts that it is possible to switch the
phototactic behavior of algae from positive phototaxis to back-and-forth to
negative phototaxis, using three consecutive identical stimuli, by adjusting
the waiting times between the stimuli and choosing carefully the stimulus
intensity, see Fig. 8d. While we were not able to find the experimental
conditions corresponding to this theoretical prediction, we were able to
experimentally observe the switch from back-and-forth to negative phototaxis,
and from positive phototaxis to back-and-forth in a couple experiments, see
Supp. Fig. 5.

While the model successfully reproduces the phototactic behavior and the time
scales at play, it fails to reproduce two experimental observations. First, in
a couple experiments, multiple consecutive back-and-forth of the center of
mass were observed at intermediate intensities, see Supp. Fig. 6. Second, in
experiments with consecutive stimuli at intermediate light intensities, algae
in the third experiment (after a waiting time of 30 minutes) display positive
phototaxis instead of the expected back-and-forth behavior observed during the
first experiment, see Fig. 6b and Supp. Fig. 5. Both behaviors can be obtained
in the model by supposing that the threshold $s_{T}$ increases with the time
spent under a light stimulus, and so depends on the history of the
microorganism. Such an increasing threshold would also enable to recover the
traditional adaptative behavior at longer time scales [15]. This implies
having a third equation describing the time-evolution of $s_{T}$, thus
introducing another time constant into the problem.

In the lab, the memory effects reported here imply that it is not possible to
repeat the same phototaxis experiments on a given batch of cells and expect
similar outcomes. Such an effect was already known in the case of long-term
adaptation [15]. The data reported here show that even in the case of short-
term experiments, the behavior is not necessarily reproducible. To study
phototaxis quantitatively, experimental repeats should be carried out with
different batches of naive cells. Having identified this short time-scale, our
experiments pave the way for the study of phototaxis in the presence of time-
varying light stimuli, much as the algae are subject to in their natural
environment. There, light is constantly fluctuating. The short-term memory
could then be a beneficial way to integrate a light signal randomly
interrupted by the shadow of other motile organisms or objects.

## Acknowledgments

We thank David Gonzalez-Rodriguez for insightful comments on an early version
of the model, Maxime Simon for preliminary experiments, Salome Guttierez-Ramos
for help with microfabrication, and Caroline Frot for precious technical
support. This work was supported by “Investissements d’Avenir” LabEx PALM
(ANR-10-LABX-0039-PALM) and by PEPS “Mécanique du futur”.

## References

  * Jékely [2009] G. Jékely, Philosophical Transactions of the Royal Society B: Biological Sciences 364, 2795 (2009).
  * Foster and Smyth [1980] K. W. Foster and R. D. Smyth, Microbiological reviews 44, 572 (1980).
  * Hegemann and Berthold [2009] P. Hegemann and P. Berthold, in  _The Chlamydomonas Sourcebook_ (Elsevier, 2009) pp. 395–429.
  * Rüffer and Nultsch [1991] U. Rüffer and W. Nultsch, Cell motility and the cytoskeleton 18, 269 (1991).
  * Feinleib and Curry [1971] M. E. H. Feinleib and G. M. Curry, Physiologia Plantarum 25, 346 (1971).
  * Nultsch  _et al._ [1986] W. Nultsch, J. Pfau, and R. Dolle, Archives of microbiology 144, 393 (1986).
  * Morel-Laurens [1987] N. Morel-Laurens, Photochemistry and Photobiology 45, 119 (1987).
  * Dolle  _et al._ [1987] R. Dolle, J. Pfau, and W. Nultsch, Journal of plant physiology 126, 467 (1987).
  * Takahashi and Watanabe [1993] T. Takahashi and M. Watanabe, FEBS Letters 336, 516 (1993).
  * Wakabayashi  _et al._ [2011] K.-i. Wakabayashi, Y. Misawa, S. Mochiji, and R. Kamiya, Proceedings of the National Academy of Sciences 108, 11280 (2011).
  * Böhm  _et al._ [2019] M. Böhm, D. Boness, E. Fantisch, H. Erhard, J. Frauenholz, Z. Kowalzyk, N. Marcinkowski, S. Kateriya, P. Hegemann, and G. Kreimer, The Plant Cell 31, 886 (2019).
  * Macnab and Koshland Jr [1972] R. M. Macnab and D. E. Koshland Jr, Proceedings of the National Academy of Sciences 69, 2509 (1972).
  * Berg and Tedesco [1975] H. C. Berg and P. Tedesco, Proceedings of the National Academy of Sciences 72, 3235 (1975).
  * Van Haastert [1983] P. Van Haastert, The Journal of cell biology 96, 1559 (1983).
  * Mayer [1968] A. Mayer, Nature 217, 875 (1968).
  * Harris  _et al._ [2009] E. Harris, D. Stern, and G. Witman, The chlamydomonas sourcebook, vol. 1 san diego (2009).
  * Xia and Whitesides [1998] Y. Xia and G. M. Whitesides, Annual review of materials science 28, 153 (1998).
  * Ueki  _et al._ [2016] N. Ueki, T. Ide, S. Mochiji, Y. Kobayashi, R. Tokutsu, N. Ohnishi, K. Yamaguchi, S. Shigenobu, K. Tanaka, J. Minagawa, _et al._ , Proceedings of the National Academy of Sciences 113, 5299 (2016).
  * Ramamonjy  _et al._ [2022] A. Ramamonjy, J. Dervaux, and P. Brunet, Physical Review Letters 128, 258101 (2022).
  * Kreis  _et al._ [2018] C. T. Kreis, M. Le Blay, C. Linne, M. M. Makowski, and O. Bäumchen, Nature Physics 14, 45 (2018).
  * Till  _et al._ [2022] S. Till, F. Ebmeier, A. A. Fragkopoulos, M. G. Mazza, and O. Bäumchen, Physical Review Research 4, L042046 (2022).
  * Leptos  _et al._ [2023] K. C. Leptos, M. Chioccioli, S. Furlan, A. I. Pesci, and R. E. Goldstein, Physical Review E 107 (2023).

## Supplementary Material

###  .1 Determining the parameters of the simplified phototactic model

The dynamics of the inner biochemical species $S$ in our simplified phototaxis
model evolve according to:

$\frac{ds^{\star}}{dt}=\gamma I_{0}(s_{\rm
tot}-s^{\star})-\tau^{-1}s^{\star},$ (3)

where $\gamma$ is the reaction rate at which the inactive species of
concentration $s$ is converted into the active species of concentration
$s^{\star}$. The total concentration is conserved and is called $s_{\rm
tot}=s+s^{\star}$. The transition from inactive to active state depends on the
light intensity $I_{0}$, while the reverse transition occurs at a constant
rate $\tau^{-1}$.

The solution to this equation writes

$s^{\star}(t)=\frac{\gamma I_{0}\tau}{\gamma I_{0}\tau+1}s_{\rm
tot}\left(1-{\rm exp}\left\\{-\left[\frac{\gamma
I_{0}\tau+1}{\tau}\right]t\right\\}\right).$ (4)

We use $s_{\rm tot}=1$ for simplicity. To obtain time scales close to the
experimental time scales, we choose $\tau=10\;\rm min$. Indeed, we know
experimentally that there is a memory effect when stimulating a second time
after a pause of a couple minutes, so $\tau>2\;\rm\min$. We also know that the
phototactic behavior is reset after 30 minutes, so $\tau\lesssim 15\rm min$.

Then, we assume the position $z$ of the alga evolves according to:

$\frac{dz}{dt}=-{\rm sign}(s^{\star}-s_{T})v_{0},$ (5)

where $s_{T}$ is the threshold concentration at which cells transition from
positive to negative phototaxis, and $v_{0}$ is the characteristic speed of
the alga. We know experimentally that $v_{0}\approx 100\;\rm\mu m.s^{-1}$.

Two parameters now need to be determined, for which there is no direct
experimental input: $\gamma$ and $s_{T}$. To estimate them, we use the fact
that, at $I_{0}=0.1\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$, the algae exhibit a
back-and-forth behavior, with negative phototaxis occuring at $t\approx
20\;\rm s$. We then know that at this intensity, $s^{\star}(t=20)=s_{T}$, and
so:

$\frac{60\gamma}{60\gamma+1}\left(1-{\rm
exp}\left\\{\left[-\frac{60\gamma+1}{600}\right]20\right\\}\right)=s_{T}.$ (6)

For each value of $\gamma$ there is therefore a corresponding value of $s_{T}$
that enables to obtain the back-and-forth behavior at $I_{0}=0.1\;\mu\rm
mol\cdot m^{-2}\cdot s^{-1}$. We choose arbitrarily $\gamma=1/60\;\rm
m^{2}\cdot\mu mol^{-1}$, so that $\gamma I_{0}\tau=1$ for $I_{0}=0.1\;\mu\rm
mol\cdot m^{-2}\cdot s^{-1}$. This determines $s_{T}=0.035$.

###  .2 Supplementary figures

Supp. Fig. 1: Binarization of the experimental images using a threshold on
pixel intensity. (a) An experimental image of a well with algae in it. The
algae are darker than the background. (b) Thresholding the experimental image
leads to a binarized image where algae appear in white in a dark background.
Supp. Fig. 2: The light stimulus is turned on at $t=30\;\rm s$ and the
fraction $f$ of algae accumulated at the wall is tracked over time. (a)
$I=0.019\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$, the algae show positive
phototaxis. 75% of the algae react to the light stimulus and accumulate at the
wall. (b) $I=0.12\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$, the algae show first
positive phototaxis, then negative phototaxis. At most, 65% of the algae react
to the light stimulus and accumulate at the wall during the transient positive
regime and 50% during negative phototaxis. (c) $I=1.4\;\mu\rm mol\cdot
m^{-2}\cdot s^{-1}$, the algae show negative phototaxis. 89% of the algae
react to the light stimulus and accumulate at the wall. Each curve is the
fraction of accumulated algae averaged over 5 experiments. (Same experiments
as in Fig. 3) Supp. Fig. 3: C. reinhardtii enclosed in a well. The blue light
stimulus is turned on at $t=30\;\rm s$. The stimulus comes from the upper side
of the wells. The algal population fills the entire well, preventing the algae
from migrating towards or away from the light. Supp. Fig. 4: C. reinhardtii
enclosed in a well. The blue light stimulus is turned on at $t=30\;\rm s$. The
stimulus comes from the upper side of the wells. (a) In the first experiment
performed at intermediate intensity $I=0.089\;\mu\rm mol\cdot m^{-2}\cdot
s^{-1}$, the algae show first positive phototaxis, then negative phototaxis.
The algae are motile. (b) At the beginning of the fourth experiment performed
at intermediate intensity $I=0.089\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$ many
algae remain accumulated at the walls from earlier experiments. We can see
aggregates forming at the center of the well and at the walls during the
course of the experiment. (c) The sixth experiment of a series performed at
intensity $I=0.33\;\mu\rm mol\cdot m^{-2}\cdot s^{-1}$ is displayed. The algae
have formed aggregates at the wall and at the center and do not react to the
light stimulus anymore. Supp. Fig. 5: Memory effects on the phototactic
response of algae. The light stimulus is turned on at $t=30\;\rm s$ and the
position of the center of mass $z_{\rm cm}$ is tracked over time. Four
consecutive experiments are performed with varying rest times, where the light
is turned off in between. There is a 90 s pause between Exp. 1 and Exp. 2,
then a 30 min pause between Exp. 2 and Exp. 3 and finally a 90 s pause between
Exp. 3 and Exp. 4. At intermediate intensities $I=0.023\;\mu\rm mol\cdot
m^{-2}\cdot s^{-1}$, the algae show different phototactic behaviors in each
experiment. The behavior switches from positive phototaxis to back-and-forth
behavior between the third and fourth experiments. The center of mass is
averaged over 6 experiments. Supp. Fig. 6: C. reinhardtii enclosed in a well.
A blue light stimulus of intensity $I=0.102\;\mu\rm mol\cdot m^{-2}\cdot
s^{-1}$ is turned on at $t=30\;\rm s$. The stimulus comes from the upper side
of the wells. In response to the stimulus, the algae go back-and-forth twice
in the well.


11institutetext: Department of Astronomy and Atmospheric Sciences, Kyungpook
National University, Daegu 41566, Korea
11email<EMAIL_ADDRESS>22institutetext: Kavli Institute for Astronomy
and Astrophysics, Peking University, Beijing 100871, China 33institutetext:
Department of Astronomy, School of Physics, Peking University, Beijing 100871,
China

# The Size-luminosity Relation of the AGN Torus Determined from the Comparison
between Optical and Mid-infrared Variability

Minjin Kim 11 Suyeon Son 11 Luis C. Ho 2233

(Received)

We investigate the optical variability of low-redshift ($0.15<z\leq 0.4$)
active galactic nuclei using the multi-epoch data from the Zwicky Transient
Facility. We find that a damped random walk model well describes the ensemble
structure function in the $g$ band. Consistent with previous studies, more
luminous active galactic nuclei tend to have a steeper structure function at a
timescale less than the break timescale and smaller variability amplitude. By
comparing the structure functions in the optical with the mid-infrared
obtained from the Wide-field Infrared Survey Explorer, we derive the size of
the dusty torus using a toy model for the geometry of the torus. The size of
the torus positively correlates with the luminosity of the active nucleus,
following a relation that agrees well with previous studies based on
reverberation mapping. This result demonstrates that the structure function
method can be used as a powerful and highly efficient tool to examine the size
of the torus.

###### Key Words.:

galaxies: active — galaxies: photometry — quasars: general

##  1 Introduction

The ultraviolet-optical continuum from unobscured active galactic nuclei
(AGNs) originates from the thermal emission from the accretion disk. It is
known to vary over a wide range of timescales from days to years (e.g., Ulrich
et al. 1997). While the physical origin of the ultraviolet-optical variability
is still under debate, recent observational studies argue that it could be
attributed to thermal instability, in which the variability timescale scales
with the orbital timescale or black hole mass (e.g., Kelly et al. 2009; Sun et
al. 2020; Burke et al. 2021; Arévalo et al. 2023; Tang et al. 2023). The
intrinsic variability is often modeled with a damped random walk, in which the
power spectral density (PSD) decreases with increasing frequency as a power
law (i.e., PSD $\propto\nu^{-2}$) at high frequency above the break frequency
($\nu\gg\nu_{\rm break}$) and flattens at low frequency ($\nu\leq\nu_{\rm
break}$; Kelly et al. 2011). At the same time, AGNs exhibit a strong mid-
infrared (MIR) continuum emitted from dust heated in a donut-shaped torus
around the supermassive black hole and accretion disk. MIR emission also
varies in response to the ultraviolet-optical continuum (e.g., Kozlowski et
al. 2016; Son et al. 2022). The characteristics of MIR variability are mostly
determined by the combination of the intrinsic variability in the accretion
disk and the geometry of the dusty torus, indicating that it can be used to
probe the structure of the torus (e.g., Li & Shen 2023; Son et al. 2023b).

Variability has been widely used to investigate the physical properties of the
central structures in AGNs. In particular, the reverberation mapping (RM;
Blandford & McKee 1982) technique allows us to derive the size of central
components robustly. This method has been applied primarily to estimate the
size of the broad-line region (BLR) by estimating the time lag between the
ultraviolet-optical continuum from the accretion disk and the fluxes of
various broad emission lines emitted from the dense material in the BLR, which
is ionized by the high-energy photons from the accretion disk (e.g., Peterson
et al. 2004; Bentz et al. 2009; Du et al. 2015; Shen et al. 2023; Woo et al.
2024). Recently, the size of the accretion disk has been evaluated for a
handful of nearby AGNs (e.g., Mrk 142, NGC 4593, NGC 5548) using X-ray and
ultraviolet-optical continuum reverberation mapping (e.g., Fausnaugh et al.
2016; Cackett et al. 2018, 2020).

The time lag between the ultraviolet-optical continuum and IR continuum has
been used to constrain the size of the torus (Suganuma et al. 2006; Koshida et
al. 2014 for $K$-band, and Lyu et al. 2019; Yang et al. 2020; Mandal et al.
2024 for $3.4-4.6\ \mu$m), with the aid of multi-epoch near-infrared (NIR)
data from ground-based telescopes and MIR light curves from the Wide-field
Infrared Survey Explorer (WISE; Wright et al. 2010). These studies reveal that
the size of the torus is positively correlated with the AGN luminosity. Prior
or parallel to these RM works, observational studies with the MIR
interferometry also demonstrated similar results (e.g. Kishimoto et al. 2011;
Burtscher et al. 2013; GRAVITY Collaboration et al. 2020). This size-
luminosity relation is also found in the BLR reverberation mapping studies. In
addition, the size of the torus increases with increasing rest-frame
wavelength, indicating the inner part of the torus is hotter than the outer
part, consistent with the torus model (e.g., Lyu et al. 2019). However,
because the reverberation mapping method requires intensive observations of
high cadence and long duration for individual objects, it can only be applied
to a limited number of targets.

Alternatively, Li & Shen (2023) demonstrated that the comparison between the
ultraviolet-optical and MIR variability can provide statistically valuable
constraints on the size of the dusty torus. Although this method cannot be
applied to an individual object, it permits the investigation of the average
geometry of the torus in a sample of AGNs sharing similar properties.
Moreover, unlike the reverberation mapping method, it does not require
simultaneous, multi-epoch, multiwavelength data with a regular cadence for a
single target. Therefore, this method is suitable for existing time-series
surveys.

AGN variability is known to follow a damped random walk, which is well
described by a broken power-law PSD (e.g., Kelly et al. 2009; Kozlowski et al.
2010; but see Mushotzky et al. 2011; Kasliwal et al. 2015). However,
characterization of the PSD requires multi-epoch data with regular and dense
sampling, as a consequence of which the PSD of ultraviolet-optical light
curves has been estimated robustly for relatively small samples (Smith et al.
2018). An alternative, effective strategy to study the characteristics of AGN
variability utilizes the structure function (SF), which is defined as the
root-mean-square of the magnitudes at a given time difference $\Delta t$. The
SF is relatively free from bias due to sparse and irregular sampling, and it
reasonably approximates a damped random walk (e.g., Hughes et al. 1992;
MacLeod et al. 2010; Kozlowski 2016). On the other hand, unless the multi-
epoch data are well sampled with a dense cadence and sufficiently long
baseline, it is hard to fully constrain the SF of a single object (Kozlowski
2016). To overcome this limitation, one can utilize an ensemble SF computed by
averaging the SFs from multiple objects at a given timescale. In particular,
ensemble SFs have been widely used to examine how AGN variability depends on
the physical properties of AGNs (e.g., Trevese et al. 1994; Vanden Berk et al.
2004; Son et al. 2023b).

Li & Shen (2023) adopted a simple toy model for the torus, one solely
determined by the torus geometry without considering radiative transfer, in
order to compute the torus transfer function. From a direct comparison between
the ensemble SFs of the optical and MIR variability, in conjunction with the
torus transfer function, Li & Shen (2023) investigated the size of the torus
for a sample of relatively luminous [$\log\leavevmode\nobreak\ (L_{\rm
bol}/{\rm erg\leavevmode\nobreak\ s^{-1}})\geq 45.3$] and distant ($0.5\leq
z\leq 1.2$) quasars selected from the multi-epoch optical photometric data
from Stripe 82 of the Sloan Digital Sky Survey (SDSS; Abazajian et al. 2009;
Annis et al. 2014) and MIR photometry from WISE. In good agreement with
previous studies conducted with the reverberation mapping method, this
experiment showed that the size of the torus positively correlates with the
AGN luminosity (Li & Shen 2023).

In this study, we apply the methodology of Li & Shen (2023) to examine whether
the size-luminosity relation applies to quasars of lower redshift ($0.15<z\leq
0.4$) and lower luminosity [$44.4\leq\log\leavevmode\nobreak\ (L_{\rm
bol}/{\rm erg\leavevmode\nobreak\ s^{-1}})\leq 45.9$]. In particular, the
narrow range of redshift is essential to minimize the dependence of the torus
size on rest-frame wavelength. Section 2 describes the sample selection and
data. Section 3 presents the detailed method to construct the ensemble SFs.
The comparison between the optical and MIR SFs is shown in Section 4, and
size-luminosity relation from our method is given in Section 5. Section 6
summarizes the overall results. We adopt the cosmological parameters
$H_{0}=100h=67.4$ km ${\rm s}^{-1}$ ${\rm Mpc}^{-1}$, $\Omega_{m}=0.315$, and
$\Omega_{\Lambda}=0.685$ (Planck Collaboration et al. 2020).

Figure 1:  Distribution of SFs divided by $\sqrt{2}$ and the original
photometric uncertainty ($\sigma_{o}$) of the $g$-band light curves of the
non-variable sources. The blue solid line denote the median value ($0.90$),
and the dashed lines give its median absolute deviation ($0.06$). If the
photometric noise is estimated robustly, the median value should be equal to
1; the median value of 0.90 indicates that the photometric uncertainty is
underestimated.

##  2 Sample and data

For a direct comparison between the optical and MIR SFs, we adopt the same
sample of type 1 AGNs from Son et al. (2023b), for which the MIR SF was
computed using the light curves from WISE. The parent sample was initially
drawn from the Data Release 14 quasar catalog of the SDSS (Pâris et al. 2018).
To minimize the effect of cosmic evolution on the dust properties and
photometric uncertainties introduced by the extended features of the host
galaxy, Son et al. (2023b) imposed a redshift cut of $0.15<z\leq 0.4$. To
secure NIR and MIR data, Son et al. (2023b) selected sources with 2MASS
(Skrutskie et al. 2006) and WISE counterparts with a matching radius of 2′′,
to arrive at a sample of 4,295 type 1 AGNs. Son et al. (2023b) demonstrated
that the NIR data are crucial to constrain properly the flux contribution from
the host galaxy (Son et al. 2022, 2023b).

To obtain the optical light curves in the $g$ band, we crossmatch the initial
sample from Son et al. (2023b) with the Zwicky Transient Facility (ZTF) Data
Release 19 using a matching radius of 2′′ (Bellm et al. 2019). We find that
all the sources have counterparts in ZTF. The average baseline and cadence of
the optical light curves are $\sim 1785$ days and $\sim 21.7$ days,
respectively. To remove any suspicious measurements, we only use the
photometric data with flag=0, separation angle
$<0\aas@@fstack{\prime\prime}4$, and limiting $g$-band magnitude $<20.2$.
These criteria were carefully determined from visual inspection of the light
curves to discard outliers in the photometric measurements (e.g., timescale of
spikes and dips less than a few days). To reconstruct the SF robustly, we
choose sources that have ZTF photometric data from at least 20 epochs.
Finally, to only consider variable sources, we adopt the criterion $P_{\rm
var}\geq 0.95$, which is the probability that the target is genuinely
variable, calculated from $\chi^{2}$ statistics (Sánchez-Sáez et al. 2018; Son
et al. 2023b). $P_{\rm var}$ is estimated after the host galaxy contribution
is subtracted. Note that the selection criterion of 0.95 is suitable for
selecting weakly or moderately variable sources: $\sim 87$% of the entire
sample meets this criterion. Without this selection, SFs with large
photometric uncertainties can introduce systematic bias in the ensemble
estimation. Radio-loud AGNs may exhibit enhanced variability due to the
relativistic jet. We compute the radio-loudness of the sample using the 1.4GHz
luminosity ($L_{\rm 1.4GHz}$) from the FIRST survey and $i$-band luminosity
($L_{i}$), in which a $K$-correction is applied by assuming a spectral index
of $-0.5$ Richards et al. (2006); Kimball & Ivezić (2008). The radio-loud AGNs
are classified with a criterion of $R>1$, where $R$ is defined as $L_{\rm
1.4GHz}/L_{i}$ (Baloković et al. 2012), finally excluded from further
analysis. This reduces the final sample to 3,578 objects.

The bolometric luminosity is estimated from the continuum luminosity at 5100 Å
with a bolometric correction of $L_{\rm bol}=9.26\,L_{5100}$ derived from the
mean spectral energy distribution of type 1 quasars (Richards et al. 2006),
where $L_{5100}$ is computed from spectral fitting of the SDSS spectra
(Rakshit et al. 2020). We adopt the bolometric luminosities based on
$L_{5100}$ instead of alternatives such those derived from the strength of [O
iii]$\lambda 5007$ (Son et al. 2023b), which is sensitive to the covering
factor of the narrow-line region, the ionization parameter, and the shape of
the ionizing continuum (e.g., Heckman et al. 2004; Shen et al. 2011; Kong & Ho
2018; Netzer 2019). While the bolometric correction is known to be sensitive
to the AGN luminosity, its uncertainty is typically smaller than 0.1 dex in
the range of the AGN luminosity in our sample (e.g., Hopkins et al. 2007;
Netzer 2019). The final sample has a median $z=0.296$, $M_{g}=-22.03$ mag,
$\log\leavevmode\nobreak\ (M_{\rm BH}/M_{\odot})=8.19$, $\log\ (L_{\rm
bol}/{\rm erg\leavevmode\nobreak\ s^{-1}})=45.2$, and
$\log\leavevmode\nobreak\ (L_{\rm bol}/L_{\rm Edd})=-1.05$. We follow Son et
al. (2023b) and calculate the black hole mass $M_{\rm BH}$ using the virial
method and spectral measurements of Rakshit et al. (2020).

Figure 2:  Ensemble SFs derived from the mock light curves. The 10,000 light
curves were generated to mimic the real multi-epoch data from ZTF. The dashed
line denotes the input SF. The red squares and blue circles represent the
square root of the mean ${\rm SF}^{2}$ and the mean of SF, respectively.

##  3 Structure function

We quantify the optical variability with the SF, defined as the mean
difference of the magnitude in the light curve as a function of the time lag
($\Delta t$):

$\begin{split}{\rm SF}^{2}(\Delta t)=\frac{1}{N_{\Delta t,{\rm
pair}}}\sum_{i=1}^{N_{\Delta t,{\rm pair}}}(m(t)-m(t+\Delta t))^{2}\\\
-\sigma_{\rm e}^{2}(t)-\sigma_{\rm e}^{2}(t+\Delta t),\end{split}$ (1)

where $N_{\Delta t,{\rm pair}}$ denotes the number of pairs associated with
$\Delta t$, $m$ is the magnitude, and $\sigma_{\rm e}$ represents the
uncertainty of the magnitude in each epoch. In general, AGN light curves can
be described by a damped random walk in which the SF is proportional to the
time lag at $\Delta t<\tau$ (${\rm SF}\propto\Delta t^{1}$) and flattens at
$\Delta t\geq\tau$ (${\rm SF}\propto\Delta t^{0}$), where $tau$ is the break
timescale. This can be expressed more generally as

${\rm SF}=[{\rm SF^{2}_{\infty}}(1-e^{(-\Delta t/\tau)^{\gamma}})]^{0.5},$ (2)

where ${\rm SF_{\infty}}$ is the SF at $\Delta t\gg\tau$ and $\gamma$ is the
power-law slope for $\Delta t<\tau$ (i.e., ${\rm SF}\propto\Delta
t^{\gamma}$). The damped random walk model can be expressed with $\gamma=1$.
The uncertainty of the SF for an individual target is computed from the square
root of the standard deviation of $\Delta{\rm mag}\equiv(m(t)-m(t+\Delta
t))^{2}-\sigma_{\rm e}^{2}(t)-\sigma_{\rm e}^{2}(t+\Delta t)$ at a given
$\Delta t$ (see Equation 1).

###  3.1 Host galaxy contribution

Although the ZTF survey provides photometry in both $g$ and $r$, we use the
$g$-band data because it is less contaminated by the flux from the host
galaxy. Nevertheless, we estimate the flux contribution from the host in the
$g$ band by fitting the spectral energy distribution that spans from the
optical to the MIR, constructed from measurements from SDSS, 2MASS, and WISE.
Following the detailed fitting method described in Son et al. (2023a), we
adopt three spectral energy distribution templates for AGNs (hot dust-
deficient, warm dust-deficient, and normal AGNs) from Lyu et al. (2017) and
seven templates for the host galaxy from Polletta et al. (2007), which
comprise an old (7 Gyr) stellar population and galaxies of Hubble type E, S0,
Sa, Sb, Sc, and Sd. The derived host flux from the individual target is
subtracted from the original light curve. The average host fraction in the $g$
band is only $\sim 0.12$, which has a minor effect on the SF measurements.

###  3.2 Photometric uncertainty

The SF is known to be highly sensitive to photometric error, and its effect is
severe when estimating the slope on short timescales (e.g. Kozlowski 2016). To
check whether the photometric uncertainties ($\sigma_{\rm o}$) provided by the
ZTF survey are reliable, we use the ZTF light curves of non-variable sources
drawn from the inactive galaxies from the Max Planck Institute for
Astrophysics and the Johns Hopkins University (MPA–JHU) catalog (Brinchmann et
al. 2004). From the SF calculated from the individual object without removing
the contribution from the photometric noise, ${\rm SF}^{2}(\Delta
t)=\frac{1}{N_{\Delta t,{\rm pair}}}\sum_{i=1}^{N_{\Delta t,{\rm
pair}}}(m(t)-m(t+\Delta t))^{2}$. We apply the same criteria as the science
data for the AGN sample to select reliable photometric measurements. If the
original photometric error ($\sigma_{\rm o}$) is estimated robustly, the SFs
are expected to equal $\sqrt{2}\sigma_{\rm o}$. Instead, we find ${\rm
SF}/\sqrt{2}\approx 0.9\sigma_{\rm o}$ (Figure 1), which indicates that the
photometric noise is underestimated. To account for this, we adopt a final
photometric error of $\sigma_{\rm e}=0.9\sigma_{\rm o}$.

Table 1: Structure function parameters

Subsample | ${\rm SF_{\infty}}$ | $\gamma$ |  $\tau$ (days) |  | ${\rm SF_{\infty}}$ |  $\gamma$ (fixed) |  $\tau$ (days)
---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) |  | (6) | (7) | (8)
All | $0.37\pm 0.02$ | $0.99\pm 0.05$ | $326\pm 54$ |  | $0.37\pm 0.02$ | $1$ | $314\pm 50$
$\log L_{\rm bol}\leq 44.67$ | $0.40\pm 0.02$ | $0.81\pm 0.06$ | $281\pm 83$ |  | $0.36\pm 0.02$ | $1$ | $155\pm 36$
$44.67<\log L_{\rm bol}\leq 45.17$ | $0.38\pm 0.01$ | $0.98\pm 0.04$ | $278\pm 42$ |  | $0.38\pm 0.01$ | $1$ | $265\pm 38$
$45.17<\log L_{\rm bol}\leq 45.67$ | $0.36\pm 0.02$ | $1.14\pm 0.05$ | $293\pm 44$ |  | $0.40\pm 0.05$ | $1$ | $482\pm 190$
$45.67<\log L_{\rm bol}$ | $0.27\pm 0.01$ | $1.26\pm 0.03$ | $277\pm 22$ |  | $0.35\pm 0.11$ | $1$ | $751\pm 658$

111Col. (1): Subsample, where $L_{\rm bol}$ is the AGN bolometric luminosity
in units of erg s-1. Col. (2): SF amplitude at $\Delta t\gg\tau$. Col. (3):
Power-law index at $\Delta t<\Delta\tau$. Col. (4): Break in the power law.
Col. (5): SF amplitude at $\Delta t\gg\tau$ with a fixed $\gamma=1$. Col. (6):
Power-law index at $\Delta t<\Delta\tau$ fixed as 1. Col. (7): Break in the
power law with a fixed $\gamma=1$.

Figure 3:  (a) Ensemble SFs of the entire sample. Open circles represent the
SF for the optical data from ZTF, with the solid line denoting their fit to a
broken power law. Filled circles give the SF for the MIR data from Son et al.
(2023b). (b) Ensemble SFs of the subsamples binned by the continuum luminosity
at 5100 Å. Open symbols and solid lines are the SFs and fitting results for
the optical data, while the dashed lines show the corresponding SFs for the
MIR data.

###  3.3 Ensemble structure function

As in the work of Son et al. (2023b) on the MIR SF of AGNs, this study
utilizes the ensemble SF, which is the average value of the SFs for a number
of objects at a given $\Delta t$. The ensemble SFs are useful to examine
statistically the characteristics of the variability as a function of various
AGN properties, using light curves with sparse and irregular cadences (e.g.,
Vanden Berk et al. 2004; Bauer et al. 2009).

The reliability of the SF also depends on the cadence and baseline of the
light curves. To test this, we made 10,000 mock light curves modeled with a
damped random walk, for which the sampling of the mock light curve is randomly
taken from the real observations of ZTF for our sample. To mimic the ZTF light
curves of our sample, we generate the input light curves assuming that ${\rm
SF_{\infty}}=0.38$, $\tau=350$ days, $\gamma=1$, and that the photometric
uncertainty is 0.068 mag. We then compute the ensemble SF using two different
estimators: the square root of the mean ${\rm SF}^{2}$ and the mean of SF. Our
simulations show that the ensemble SF is slightly overestimated at $\Delta
t<15\leavevmode\nobreak\ {\rm days}$, as is also seen in the observed data
(Figure 2). If the photometric error is larger than the intrinsic SF, the mean
observed ${\rm SF}^{2}$ occasionally could be smaller than zero, which should
be excluded in the calculation of the ensemble SF. This can boost the SF at
small $\Delta t$, where the intrinsic SF is smaller than the photometric
noise. Therefore, this study uses the ensemble SF at $\Delta t\geq
15\leavevmode\nobreak\ {\rm days}$. Son et al. (2023b) demonstrated, and the
simulation in Figure 2 confirms, that the square root of the mean ${\rm
SF}^{2}$ is a better tracer of the real ensemble SF than the mean of SF. We
also compute the ensemble SFs in the same manner. Finally, the uncertainties
in the ensemble SF are estimated from bootstrap resampling of the SFs of the
individual target within the uncertainty of the individual SF. We perform 1000
realizations of resampling, and the final uncertainty is estimated from half
of the difference between the 16th and 84th percentiles.

Figure 4:  Fitting results of the MIR SFs for (a) the entire sample, (b) the
least luminous sources, and (c) the most luminous sources. The solid line is
the optical SF estimated from the ZTF survey. Red points represent the MIR SF
from the WISE survey, with the blue dashed line denoting the best fit.

##  4 Results

###  4.1 The $g$-band structure function

We first estimate the ensemble $g$-band SF of the entire sample (Figure 3;
Table 1). The break timescale $\tau$ is known to be correlated with black hole
mass and AGN luminosity (e.g., Kelly et al. 2009; MacLeod et al. 2010; Burke
et al. 2021; Tang et al. 2023). However, the relatively short time baseline of
the ZTF survey makes it difficult to estimate $\tau$ robustly with this
dataset. Although we only use Equation (2) to investigate the qualitative form
of the SF, without attaching any specific physical meaning to it at this
stage, we nevertheless derive a power-law slope of $\gamma=0.99\pm 0.05$,
which agrees well with the prediction from the damp random walk model.

Dividing the sample into four luminosity bins reveals that $\gamma$ is
proportional to luminosity, such that more luminous AGNs exhibit a steeper SF
over short timescales than fainter sources. The parameter ${\rm SF_{\infty}}$
is inversely correlated with the AGN luminosity (Table 1). These findings are
consistent with the trends reported in previous studies, not only in terms of
MIR variability (Son et al. 2023b) but also optical variability (e.g., Vanden
Berk et al. 2004; MacLeod et al. 2010; Tang et al. 2023).

###  4.2 Fitting with torus model

The MIR light curves can be expressed as the combination of the intrinsic
light curve from the accretion disk and the transfer function of the dusty
torus,

$L_{\rm MIR}(t)=\int{\Psi(\tau)L_{\rm opt}(t-\tau)d\tau},$ (3)

where $L_{\rm MIR}$ is the MIR light curve, $\Psi(\tau)$ is the transfer
function of the torus, and $L_{\rm opt}$ is the ultraviolet-optical continuum
light curve. We adopt the torus model of Li & Shen (2023), for which the
transfer function of the torus is determined solely by its geometry, which is
described by an inner radius ($R_{\rm in}$), a half-opening angle ($\theta$)
representing the angle between the equatorial plane and the edge of the torus,
an outer to inner radius ratio ($Y$), and an inclination angle ($i$). A total
of 50,000 clouds are generated following a power-law density distribution
($p=-1$ for $\rho\propto r^{p}$) in a radial direction.

We additionally consider a damping factor ($d\equiv{\rm SF_{\infty,MIR}}/{\rm
SF_{\infty,opt}}$), which is the fraction of the variability amplitude between
the optical and the MIR. This term is necessary because the variability
amplitude of the MIR data is significantly smaller than that of the optical
data (i.e., $d<1$; Neugebauer et al. 1989; Kozlowski et al. 2010, 2016). We
adopt the Python codes from Li & Shen
(2023)222https://github.com/bwv1194/geometric-torus-variability but modify
them by adding the damping factor in order to fit the MIR SFs more robustly.
The grid of predicted MIR SFs, generated with the input optical light curve
and the torus transfer function computed from the torus model, spans the
following range of parameters: $-1.85\leq\log(R_{\rm in}/{\rm pc})\leq-0.1$,
$0^{\circ}<i<45^{\circ}$, $1.3\leq Y\leq 1.5$, $\sigma=30^{\circ}$, and
$0.45\leq d\leq 0.8$. While the range of $R_{\rm in}$ is determined based on
the results from the previous NIR and MIR RM projects (e.g., Koshida et al.
2014; Lyu et al. 2019), that of $Y$ is inferred from the morphology of hot
dust derived from the NIR interferometric observations (e.g., Kishimoto et al.
2009). We use the grid of modeled SFs to fit the observed MIR SFs from Son et
al. (2023b). Because the optical SFs only securely cover $\Delta t<1000$ days,
we use the same range of $\Delta t$ to fit the MIR SFs.

The transfer function and, therefore, the shape of the MIR SFs are mostly
determined by the combination of $R_{\rm in}$, $Y$, and a power-law index for
the density distribution ($p$; e.g., Almeyda et al. 2020). To keep the
consistency of the method with Li & Shen (2023), we first adopt the small $Y$
and a fixed $p$. The best fit for the entire sample yields $R_{\rm in}=0.18\pm
0.01$ pc (Figure 4a), which is in good agreement with the torus sizes derived
from the reverberation mapping method and interferometric observations of
nearby Seyferts [$42.5\lesssim\log\leavevmode\nobreak\ (L_{\rm bol}/{\rm
erg\leavevmode\nobreak\ s^{-1}})\lesssim 46.5$] at $\sim
2.2\leavevmode\nobreak\ \mu{\rm m}$ (e.g., Kishimoto et al. 2011; Koshida et
al. 2014).

Table 2: Radius of the torus

Subsample | $\log(L_{\rm bol}/{\rm erg\leavevmode\nobreak\ s^{-1}})$ | $\log(R_{\rm in}/{\rm pc})$ | $\log(R_{\rm eff}/{\rm pc})$ | $\lambda_{\rm rest}$
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
All | $45.21\pm 0.20$ | $-0.75\pm 0.01$ | $-0.69\pm 0.01$ | $2.63\pm 0.11$
$\log L_{5100}\leq 44.67$ | $44.51\pm 0.11$ | $-1.01\pm 0.06$ | $-0.92\pm 0.06$ | $2.75\pm 0.10$
$44.67<\log L_{5100}\leq 45.17$ | $45.01\pm 0.10$ | $-0.76\pm 0.01$ | $-0.70\pm 0.01$ | $2.70\pm 0.11$
$44.67<\log L_{5100}\leq 45.17$ | $45.34\pm 0.10$ | $-0.70\pm 0.01$ | $-0.64\pm 0.01$ | $2.58\pm 0.09$
$45.67<\log L_{5100}$ | $45.80\pm 0.09$ | $-0.42\pm 0.02$ | $-0.36\pm 0.02$ | $2.52\pm 0.06$
333 Col. (1): Subsample, where $L_{\rm bol}$ is the AGN bolometric luminosity
in units of erg s-1. Col. (2): Median $L_{\rm bol}$ and its median absolute
deviation. Col. (3): Inner radius of the torus. Col. (4): Effective radius of
the torus, measured from the effective $\tau$. Col. (5): Effective wavelength
of the W1 filter in the rest-frame.

Figure 5:  The size-luminosity relation of the torus of type 1 AGNs. Gray
squares denote the inner radius of the torus derived from fitting the MIR SFs
with the toy model for the torus. Red circles represent the effective size of
the torus estimated from the best-fit transfer function, with the red solid
line giving their ordinary distance regression fit to the size-luminosity
relation. The uncertainties of $R_{\rm in}$ are the same as those of $R_{\rm
eff}$. The grey dashed line shows the size-luminosity relation of low-redshift
($z<0.5$) PG quasars estimated using the reverberation mapping method (Lyu et
al. 2019). The grey shaded area represents the $1\,\sigma$ scatter ($\sim
0.17$ dex) estimated from a sample of relatively high-redshift ($0.3<z<2$)
AGNs (Yang et al. 2020). The blue dotted line is the relation of intermediate-
redshift ($0.5\leq z\leq 1.2$) quasars in Stripe 82, based on the direct
comparison between the optical SFs and MIR (W1) SFs, similar to this study.
The cyan dashed-dotted line denotes the relation of the local Seyferts in the
$K$ band inferred from the reverberation mapping method. Blue stars represent
the size of the torus estimated from the $K$-band interferometry measurements
(GRAVITY Collaboration et al. 2020)

.

##  5 The size-luminosity relation of the torus

We apply the fit for the ensemble SFs to the subsamples divided by AGN
luminosity. Fitting the faintest (Figure 4b) and brightest (Figure 4c) AGNs
clearly shows that, as expected, $R_{\rm in}$ is correlated with the AGN
luminosity (Table 2). We compare our results with those obtained from
reverberation mapping studies, which investigated the size-luminosity relation
of the torus using the time lag between the optical light curves obtained from
ground-based telescopes and MIR light curves derived from WISE (e.g., Koshida
et al. 2014; Lyu et al. 2019; Yang et al. 2020). The MIR light curves of low-
redshift ($z<0.5$) Palomar-Green (PG; Schmidt & Green 1983) quasars studied by
Lyu et al. (2019) are comparable to those of our sample. They demonstrated a
strong correlation between the size of the torus determined in the W1 band and
the AGN luminosity, in the form $R\propto L_{\rm AGN}^{0.47}$. While the
intrinsic scatter in the size-luminosity relation was not reported in Lyu et
al. (2019), using more distant quasars at $0.3<z<2$ Yang et al. (2020) found a
size-luminosity relation similar to that of Lyu et al. (2019); this is
illustrated with the grey area in Figure 5, which has an intrinsic scatter of
0.17 dex.

For a direct comparison with previous reverberation mapping studies, we
compute the effective radius ($R_{\rm eff}$) of the torus determined from the
best-fit transfer function. This is equivalent to the effective time delay
($\tau_{\rm eff}$) directly measured from the reverberation mapping method,
which is expressed as

$\tau_{\rm eff}=\frac{\int{\tau\Psi(\tau)d\tau}}{\int{\Psi({\tau})d\tau}}.$
(4)

The effective time delay depends on the inner radius and the geometry of the
torus (e.g., the inclination angle and the ratio of the outer and inner
radius). Figure 5 shows that the size-luminosity relation from our study is in
good agreement with previous studies within the intrinsic scatter. An ordinary
distance regression for our measurements yields

$\begin{split}\log\leavevmode\nobreak\ (R_{\rm eff}/{\rm pc})=(0.42\pm
0.06)\times\log\leavevmode\nobreak\ (L_{\rm bol}/{\rm erg\leavevmode\nobreak\
s^{-1}})\\\ -(19.81\pm 2.59).\end{split}$ (5)

This result demonstrates that the ensemble SF can be as effective as the
reverberation mapping method in statistically estimating the size of the
torus. Interestingly, the zero point of the size-luminosity relation from this
study is slightly larger than that from Li & Shen (2023). The offset in the
torus size ranges from 0.1 to 0.2 dex at $\log\ (L_{5100}/{\rm erg\
s^{-1}})=44.5-46.$ This may be attributed to the dependence of the torus size
on the rest-frame wavelength (e.g., Lyu et al. 2019). The rest-frame
wavelength ($\lambda_{\rm rest}\approx 1.4-2.8\,\mu{\rm m}$) of the MIR data
in the sample of Li & Shen (2023) is significantly shorter than that of our
sample ($\lambda_{\rm rest}\approx 2.5-2.8\,\mu{\rm m}$). The shorter
wavelengths likely probe the inner regions of the torus. In support of this
interpretation, we note that the $K$-band time lag from Koshida et al. (2014)
($\lambda_{\rm rest}\approx 2\,\mu{\rm m}$) is marginally shorter than that
inferred from the W1-based studies. Interestingly, our estimation agrees well
with the results from the $K$-band interferometry measurements (GRAVITY
Collaboration et al. 2020). As our results depend on the adopted model for the
torus geometry, the discrepancy in the size-luminosity relation among the
different studies may originate from the methodology.

The studies with the MIR interferometry showed that the MIR emitting region
($8.5-13\ \mu$m ) is more extended than the NIR emitting region (e.g.,
Kishimoto et al. 2011). To account for this effect, we examine the dependence
of $R_{\rm eff}$ on the outer-to-inner radius ratio ($Y$) and the radial
density distribution ($p$). The SF fits with a model from $1.5\leq Y\leq 10$
and $-0.5\leq p\leq-1.0$ yield worse results compared to the best-fits with a
smaller $Y$ ($1.3\leq Y\leq 1.5$) and a fixed $p=-1$. In addition, the
inferred $R_{\rm eff}$ becomes remarkably smaller than that from the best-fit
results with the smaller $Y$ by up to $\sim 0.5$ dex. This result is not
physically meaningful as the inner radius is substantially smaller than the
sublimation radius. It reveals that the hot dust emitting the MIR continuum at
2.5-2.8 $\mu$m is likely to locate the innermost region of the torus and may
not be extended along the radius, which is consistent with the previous
studies (e.g., Kishimoto et al. 2009, 2011).

##  6 Conclusions

We derive the optical ensemble structure function of the low-redshift and
moderately luminous AGNs selected from SDSS using the $g$-band multi-epoch
data obtained from the ZTF survey. We find that a broken power law, similar to
the damped random walk, can well describe the ensemble SF. However, more
luminous AGNs tend to have steeper slopes in the ensemble SF at short
timescale than less luminous AGNs, which may be due to the geometric effect of
the accretion disk. Consistent with previous studies, the variability
amplitude also is anti-correlated with the AGN luminosity.

Using the toy model of the torus geometry from (Li & Shen 2023), we make a
comparison between the optical and MIR SFs to estimate the effective size of
the torus. As with reverberation mapping experiments, we find that the
effective size of the torus positively correlates with AGN luminosity. The
slope of the size-luminosity relation agrees well with that of previous
studies, although the zero point of the relation is marginally higher than
that derived from the analysis of $K-$band data for local AGNs and W1-band
data for high-redshift AGNs, which implies that the size of the torus
increases with increasing rest-frame wavelength. These results demonstrate
that comparison between the optical and MIR SFs for a large sample can be a
powerful tool for estimating the size of the torus.

###### Acknowledgements.

We thank the anonymous referee for the constructive comments that
substantially improved the manuscript. This work was supported by the National
Key R&D Program of China (2022YFF0503401), the National Science Foundation of
China (11991052, 12233001), the China Manned Space Project (CMS-CSST-2021-A04,
CMS-CSST-2021-A06), and the National Research Foundation of Korea (NRF),
through grants funded by the Korean government (MSIT) (Nos. 2022R1A4A3031306,
2023R1A2C1006261, and RS-2024-00347548).

## References

  * Abazajian et al. (2009) Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., et al. 2009, ApJS, 182, 543
  * Almeyda et al. (2020) Almeyda, T., Robinson, A., Richmond, M., Nikutta, R., & McDonough, B. 2020, ApJ, 891, 26
  * Annis et al. (2014) Annis, J., Soares-Santos, M., Strauss, M. A., et al. 2014, ApJ, 794, 120
  * Arévalo et al. (2023) Arévalo, P., Lira, P., Sánchez-Sáez, P., et al. 2023, MNRAS, 526, 6078
  * Baloković et al. (2012) Baloković, M., Smolčić, V., Ivezić, Ž., et al. 2012, ApJ, 759, 30
  * Bauer et al. (2009) Bauer, A., Baltay, C., Coppi, P., et al. 2009, ApJ, 696, 1241
  * Bellm et al. (2019) Bellm, E. C., Kulkarni, S. R., Graham, M. J., et al. 2019, PASP, 131, 018002
  * Bentz et al. (2009) Bentz, M. C., Walsh, J. L., Barth, A. J., et al. 2009, ApJ, 705, 199
  * Blandford & McKee (1982) Blandford, R. D., & McKee, C. F. 1982, ApJ, 255, 419
  * Brinchmann et al. (2004) Brinchmann, J., Charlot, S., White, S. D. M., et al. 2004, MNRAS, 351, 1151
  * Burke et al. (2021) Burke, C. J., Shen, Y., Blaes, O., et al. 2021, Science, 373, 789
  * Burtscher et al. (2013) Burtscher, L., Meisenheimer, K., Tristram, K. R. W., et al. 2013, A&A, 558, A149
  * Cackett et al. (2018) Cackett, E. M., Chiang, C.-Y., McHardy, I., et al. 2018, ApJ, 857, 53
  * Cackett et al. (2020) Cackett, E. M., Gelbord, J., Li, Y.-R., et al. 2020, ApJ, 896, 1
  * Du et al. (2015) Du, P., Hu, C., Lu, K.-X., et al. 2015, ApJ, 806, 22
  * Fausnaugh et al. (2016) Fausnaugh, M. M., Denney, K. D., Barth, A. J., et al. 2016, ApJ, 821, 56
  * GRAVITY Collaboration et al. (2020) GRAVITY Collaboration, Dexter, J., Shangguan, J., et al. 2020, A&A, 635, A92
  * Heckman et al. (2004) Heckman, T. M., Kauffmann, G., Brinchmann, J., et al. 2004, ApJ, 613, 109
  * Hopkins et al. (2007) Hopkins, P. F., Richards, G. T., & Hernquist, L. 2007, ApJ, 654, 731
  * Hughes et al. (1992) Hughes, P. A., Aller, H. D., & Aller, M. F. 1992, ApJ, 396, 469
  * Kasliwal et al. (2015) Kasliwal, V. P., Vogeley, M. S., & Richards, G. T. 2015, MNRAS, 451, 4328
  * Kelly et al. (2009) Kelly, B. C., Bechtold, J., & Siemiginowska, A. 2009, ApJ, 698, 895
  * Kelly et al. (2011) Kelly, B. C., Sobolewska, M., & Siemiginowska, A. 2011, ApJ, 730, 52
  * Kimball & Ivezić (2008) Kimball, A. E., & Ivezić, Ž. 2008, AJ, 136, 684
  * Kishimoto et al. (2009) Kishimoto, M., Hönig, S. F., Antonucci, R., et al. 2009, A&A, 507, L57
  * Kishimoto et al. (2011) —. 2011, A&A, 536, A78
  * Kong & Ho (2018) Kong, M., & Ho, L. C. 2018, ApJ, 859, 116
  * Koshida et al. (2014) Koshida, S., Minezaki, T., Yoshii, Y., et al. 2014, ApJ, 788, 159
  * Kozlowski (2016) Kozlowski, S. 2016, ApJ, 826, 118
  * Kozlowski et al. (2016) Kozlowski, S., Kochanek, C. S., Ashby, M. L. N., et al. 2016, ApJ, 817, 119
  * Kozlowski et al. (2010) Kozlowski, S., Kochanek, C. S., Stern, D., et al. 2010, ApJ, 716, 530
  * Li & Shen (2023) Li, J., & Shen, Y. 2023, ApJ, 950, 122
  * Lyu et al. (2017) Lyu, J., Rieke, G. H., & Shi, Y. 2017, ApJ, 835, 257
  * Lyu et al. (2019) Lyu, J., Rieke, G. H., & Smith, P. S. 2019, ApJ, 886, 33
  * MacLeod et al. (2010) MacLeod, C. L., Ivezić, Ž., Kochanek, C. S., et al. 2010, ApJ, 721, 1014
  * Mandal et al. (2024) Mandal, A. K., Woo, J.-H., Wang, S., et al. 2024, arXiv e-prints, arXiv:2403.01885
  * Mushotzky et al. (2011) Mushotzky, R. F., Edelson, R., Baumgartner, W., & Gandhi, P. 2011, ApJ, 743, L12
  * Netzer (2019) Netzer, H. 2019, MNRAS, 488, 5185
  * Neugebauer et al. (1989) Neugebauer, G., Soifer, B. T., Matthews, K., & Elias, J. H. 1989, AJ, 97, 957
  * Pâris et al. (2018) Pâris, I., Petitjean, P., Aubourg, É., et al. 2018, A&A, 613, A51
  * Peterson et al. (2004) Peterson, B. M., Ferrarese, L., Gilbert, K. M., et al. 2004, ApJ, 613, 682
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6
  * Polletta et al. (2007) Polletta, M., Tajer, M., Maraschi, L., et al. 2007, ApJ, 663, 81
  * Rakshit et al. (2020) Rakshit, S., Stalin, C. S., & Kotilainen, J. 2020, ApJS, 249, 17
  * Richards et al. (2006) Richards, G. T., Lacy, M., Storrie-Lombardi, L. J., et al. 2006, ApJS, 166, 470
  * Sánchez-Sáez et al. (2018) Sánchez-Sáez, P., Lira, P., Mejía-Restrepo, J., et al. 2018, ApJ, 864, 87
  * Shen et al. (2011) Shen, Y., Richards, G. T., Strauss, M. A., et al. 2011, ApJS, 194, 45
  * Shen et al. (2023) Shen, Y., Grier, C. J., Horne, K., et al. 2023, arXiv e-prints, arXiv:2305.01014
  * Skrutskie et al. (2006) Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163
  * Smith et al. (2018) Smith, K. L., Mushotzky, R. F., Boyd, P. T., et al. 2018, ApJ, 857, 141
  * Son et al. (2022) Son, S., Kim, M., & Ho, L. C. 2022, ApJ, 927, 107
  * Son et al. (2023a) —. 2023a, ApJ, 953, 175
  * Son et al. (2023b) —. 2023b, ApJ, 958, 135
  * Suganuma et al. (2006) Suganuma, M., Yoshii, Y., Kobayashi, Y., et al. 2006, ApJ, 639, 46
  * Sun et al. (2020) Sun, M., Xue, Y., Brandt, W. N., et al. 2020, ApJ, 891, 178
  * Tang et al. (2023) Tang, J.-J., Wolf, C., & Tonry, J. 2023, Nature Astronomy, 7, 473
  * Trevese et al. (1994) Trevese, D., Kron, R. G., Majewski, S. R., Bershady, M. A., & Koo, D. C. 1994, ApJ, 433, 494
  * Ulrich et al. (1997) Ulrich, M.-H., Maraschi, L., & Urry, C. M. 1997, ARA&A, 35, 445
  * Vanden Berk et al. (2004) Vanden Berk, D. E., Wilhite, B. C., Kron, R. G., et al. 2004, ApJ, 601, 692
  * Woo et al. (2024) Woo, J.-h., Wang, S., Rakshit, S., et al. 2024, ApJ, 962, 67
  * Wright et al. (2010) Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868
  * Yang et al. (2020) Yang, Q., Shen, Y., Liu, X., et al. 2020, ApJ, 900, 58