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\begin{document}
\title{Movement behavioral plasticity of benthic diatoms driven by optimal
foraging}
\author[1]{We-Si Hu}%
\author[2]{Mingji Huang}%
\author[2]{He-Peng Zhang}%
\author[3]{Feng Zhang}%
\author[4]{Wim Vyverman}%
\author[5]{Quan-Xing Liu}%
\affil[1]{East China Normal University}%
\affil[2]{Shanghai Jiao Tong University - Minhang Campus}%
\affil[3]{Anhui University}%
\affil[4]{Ghent University}%
\affil[5]{State Key Laboratory of Estuarine and Coastal Research}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
Adaptive locomotion of living organisms contributes to their competitive
abilities and helps maintain their fitness in diverse environments. To
date, however, our understanding of searching behavior and its ultimate
cause remains poorly understood in ecology and biology. Here, we
investigate motion patterns of biofilm-inhabiting marine raphid diatom
Navicula arenaria var. rostellata in two-dimensional space. We report
that individual Navicula cells display a ``circular run-and-reversal''
movement behavior at different concentrations of dissolved silicic acid
(dSi). We show that gliding motions of cells can be predicted accurately
with a universal Langevin model. Our experimental results are consistent
with an optimal foraging strategy and a maximized diffusivity of the
theoretical outcomes in which both circular-run and reversal behaviors
are important ingredients. Our theoretical results suggest that the
evolving movement behaviors of diatoms may be driven by optimization of
searching behavioral strategy, and predicted behavioral parameters
coincide with the experimental observations. These optimized movement
behaviors are an evolutionarily stable strategy to cope with
environmental complexity.%
\end{abstract}%
\sloppy
\textbf{ONE SENTENCE SUMMARY} : Novel experiments and modelling reveal
that raphid diatoms can actively exploit resources in complex
environments by adjusting their movement behavior.
\textbf{INTRODUCTION}
The rich diversity of organisms' movement behavior has long invoked
curiosity. Plants may passively adapt their inclining positions to
alleviate competition for light (1,2), while most animals and many
microorganisms can actively move from one place to another to seek
forage, to mate with partners (3,4), or to escape from predators (5,6).
A comprehensive understanding of the drivers, patterns and mechanisms of
organismal movement is central to elucidating its ecological and
evolutionary significance (7-10). In the extensive body of movement
behavioral ecology, particular interest has been paid to foraging, being
a fundamental activity providing energy throughout an organism's life
cycle (5,8). In spite of diverse modes of foraging movement among life
forms (1,8,9), their intrinsic spatiotemporal patterns may converge to
maximize biological fitness of individual foragers as it is predicted by
the optimal foraging theory (OFT) (11, 12).
So far, perhaps the most convincing evidence supporting this prediction
is provided by theoretical and experimental studies on movement patterns
of a range of microorganisms (e.g. swimming bacteria, microalgae and
multi-cellular planktons) in strictly controlled microcosm environments
(13-16). In these systems with low information availability,
microorganisms having weak resource detection capabilities usually
perform random-like movements during the process of foraging. It has
been repeatedly observed that intermittent locomotion (also known as
stop-and-go movement or pause-travel locomotion, Fig. 1A) is common in
these cases, and is characterized by discontinuous movements interwoven
with significant punctuations and reorientations. A prevalent idea is
that foraging efficiency can be maximized by certain statistical
properties provided by their movement patterns. For example, the
probability distributions of time intervals or spatial displacements
between reorientations have been found to fit Brownian type or L\selectlanguage{ngerman}évy type
walks, which are theoretically regarded to be optimal solutions of the
random search problems under specific conditions (13,17-23).
The suggestion that the optimal foraging principle underpins diverse
movement forms is indeed appealing. However, the universality of this
strikingly simple principle remains controversial. It has been argued
that there are exceptions in real-world ecosystems that are more complex
deviating theoretical hypothesis (14,24,25), for instance, some
individuals switching between Lévy and Brownian movement patterns as
they traverse different habitat types (26,27). So far, optimal foraging
is typically referred to as a hypothesis because it has not been
established that the assumptions underlying these theories indeed hold.
Furthermore, it remains elusive to which extent the optimal foraging
principle can be verified to a range of newly discovered movement forms
in microorganisms. Apart from the classical `run-and-tumble' movement
pattern (characterized by almost straight runs that are interrupted by
tumbles) that have been extensively studied since the seminal work
on\emph{Escherichia coli} in the 1970s (10,28), recent studies
discovered a variety of different movement modes (14,16,29), such as the
`run-and-stop' and `run-reverse-flick' patterns (Fig. 1A) in the soil
bacteria \emph{Pseudomonas putida} (30), \emph{Myxococcus
xanthus}(31,32), and the marine bacteria \emph{Vibrio alginoticus} (24).
Very recently, one of the most intriguing movement patterns was found in
pennate raphid diatoms, a species-rich and ecologically important group
of microalgae mostly inhabiting benthic habitats in marine and
freshwater environments. They move in a gliding manner, forming
trajectories that highly resemble circular arcs (16). This unique
movement pattern (termed as `circular run-and-reversal', see Fig. 1A,
results and discussion for detailed descriptions) is distinct from those
of previously documented model organisms (28), whose trajectories
typically consist of line segments in contrast with circular arcs. Our
understanding of the statistical properties of these movement patterns
is still rudimentary. In particular, it remains unknown if this type of
movement conforms to optimal foraging behavior.
Here, we performed a systematic study on the novel `circular
run-and-reversal' behavior in the marine biofilm-inhabiting
diatom\emph{Navicula arenaria} var. \emph{rostellata} . By combining
experimental data and theoretical analyses, we demonstrate that the
circular run-and-reversal behavior plays a crucial role in optimizing
searching strategies. Our results suggest that in a silicon-limited
environment, the diatoms can maximize their foraging efficiency by
adapting the key parameters including reversal rate and rotational
diffusivity and they can change the behavior strategy in a silicon-rich
environment (Fig.1B).
\textbf{EXPERIMENTAL SETUP}
In our experimental microcosms (Fig. 1C), enhanced motility of the
diatom \emph{Navicula arenaria} var\emph{. rostellata}
(see\emph{Materials and Methods} for detailed descriptions on its basic
information, Fig. 1E is a picture of electron microscope image of the
studied species) was stimulated by exposing cells to low concentrations
of dissolved silicic acid (dSi, 15 mg/L). Sample is enclosed in a sealed
chamber which consists of a silicone well, and two cover slips. The well
has a diameter of 0.8 cm and a height of 0.1 cm (Fig. 1C). We used a
tracing technique to quantify the movement pattern of individual diatom
cells (Fig. 1D and 2A). We subsequently developed a simple theoretical
model that can well capture the spatial trajectories as well as the
statistical properties of their foraging movement. During the
experiments, cells were placed in a coverslip chamber in dSi depleted
culture medium (about 15 \(\text{cells}/mm^{2}\)). The movement patterns of
the diatom cells were recorded by a Ti-E Nikon phase contrast microscope
with high temporal-spatial resolutions (see \emph{Materials and Methods}
).
\textbf{RESULTS AND DISCUSSION}
\textbf{The `circular run-and-reversal' movement pattern}
We observed that the movement trajectories of the diatom cells are
characterized by two apparently distinguishable components: 1)
continuous spatial displacements following rotation-like (resembling
circular arcs) trajectories (Fig. 2A, Movie S1) in the clockwise (CW) or
counter clockwise (CCW) direction; and 2) reversals of the rotational
direction (Fig. 2B, Movie S1). Here we define this movement pattern as
`circular run-and-reverse' by adapting the term `run-and-tumble' as were
shown in Fig. 1A and 2.
To quantitatively characterize this `circular run-and-reverse' movement
pattern in a comprehensive way, we used \emph{ca.} 30 recorded
continuous individual trajectories to measure a set of key movement
parameters including transitional speed, angular speed, translational
diffusivity and rotational diffusivity (\(D_{\theta}\), indicating
the intensity of random change in particle's orientation, which
resembles the translational diffusion in space) and reversal rate
(\(\nu\), defined as the times of directional reversals per
unit time). Details of the parameters are provided in Table 1.
In our observations, the movement speed as a function of
time\(V\left(t\right)\) was around \(16.2\pm 2.3\ \mu\)m/s (Fig. 2C). The
probability distributions of reversal time intervals of cells are well
characterized by an exponential distribution with mean\(T=\frac{1}{\nu}\)
(\(T\) is the mean interval-reversal time, see Fig. 3A),
and thus the number of reversal events in a fixed interval of time
length conforms to a Poisson distribution. In addition, the statistical
behavior of the rotational diffusivity (\(D_{\theta}\)) satisfies a
Gaussian random variable with log transformation (Table 1 and Fig. S1
for the trajectories with different experimental\(D_{\theta}\)).
Does the circular run-and-reverse pattern satisfy a Gaussianity? The
distribution function of displacements is a fundamental statistic
property for movement behavior, known as the self-part of the van Hove
distribution function is defined as:
\begin{equation}
G_{s}\left(x,t\right)=\frac{1}{N}\sum_{j=1}^{N}\left\langle\delta(x-\left|r_{j}\left(t\right)-r_{j}(0)\right|)\right\rangle\nonumber \\
\end{equation}
where \(N\) is the number of individual cells and
\(\delta\) is the Dirac delta function. They are not Gaussian
behavior at long-term scales (more than 50 sec, Fig. 3B). We find that
this non-Gaussian distribution can be well fitted by a Gumbel law (33):
\(f\left(x\right)=A\left(\lambda\right)\exp\left[-\frac{x}{\lambda}-\exp\left(-\frac{x}{\lambda}\right)\right]\).
Here \(\lambda\) is a length scale, and \(x\) is
the displacement of the cell in the \(x\) direction and
\(A(\lambda)\) is a normalization constant. Therefore, we conclude
that this circular run-and-reversal' movement pattern is a non-Gaussian
process for spatial searching and the rotational diffusivity leads to a
subdiffusive searching behavior at long-time scales (Fig. 3C).
\textbf{Mathematical model}
We developed a basic mechanistic model to capture the movement pattern
of the self-propelled diatom cells at an individual level in a
two-dimensional space. Adapted from the motion behavior of
self-propelled non-living micro-rods (34), our discrete time model uses
the abovementioned 5 movement parameters, assuming white noises with
intensity \(D_{r}\) and \(D_{\theta}\) in translational
diffusivity and rotational diffusivity respectively. The stochastic
difference equations for the movement of a single diatom cell are given
by
\(x\left(t+\Delta t\right)-x\left(t\right)=\kappa(t)V_{0}\cos{\theta(t)}\Delta t+\selectlanguage{greek}\sqrt{2D_{r}\text{Δt}}\selectlanguage{english}\xi_{1},\)(1a)
\(y\left(t+\Delta t\right)-y(t)={\kappa(t)V}_{0}\sin{\theta(t)}\Delta t+\selectlanguage{greek}\sqrt{2D_{r}\text{Δt}}\selectlanguage{english}\xi_{2}\)\emph{,}(1b)
\(\theta\left(t+\Delta t\right)-\theta(t)=\kappa(t)\omega\Delta t+\selectlanguage{greek}\sqrt{2D_{\theta}\text{Δt}}\selectlanguage{english}\xi_{3},\)(1c)
\(\kappa\left(t+\Delta t\right)-\kappa\left(t\right)=-2\kappa\left(t\right)B\left(\nu\right)\)\emph{,}(1d)
where\selectlanguage{ngerman} \selectlanguage{greek}θ \selectlanguage{english}is the direction of the
movement,\selectlanguage{ngerman} x and y indicate
the spatial coordinate and \(t\) is the time.
\(\kappa\) is the rotational direction (\(\kappa=1\ \)for
CCW and -1 for CW), and \(\omega\) is the angular speed.
Reversal events are represented by the telegraph process, characterized
by \(B\left(\nu\right)\) which is a Bernoulli random variable with success
probability \selectlanguage{greek}νΔ\selectlanguage{english}t (\(\nu\) is reversal rate).
The noise terms \(\xi_{1}\), \(\xi_{2}\) and
\(\xi_{3}\) follow a standard normal distribution. The default
values of the parameters were derived from the experimental data (Table
1).
The mathematical derivation of the effective diffusion
coefficient\emph{D} (as a measure of foraging efficiency) can be
obtained by calculating the probability distribution
functions\(\Psi_{\pm}(\mathbf{r},\theta,t)\) (34,35),
\(\frac{\partial\Psi_{\pm}}{\partial t}+\nabla\bullet(\dot{\mathbf{r}}\Psi_{\pm})+\frac{\partial}{\partial\theta}\left(\dot{\theta}\Psi_{\pm}\right)=\nu(\Psi_{\mp}-\Psi_{\pm})\), (2)
with\(\dot{\mathbf{r}}={\pm V}_{0}\mathbf{n(}\theta\mathbf{)}-\ D_{r}\nabla\log\Psi_{\pm}\),\(\dot{\theta}=\pm\omega-D_{\theta}\frac{\partial}{\partial\theta}\log\Psi_{\pm}\). Here, the probability
density function of cells' spatial position\(\mathbf{r}=(x,y)\) changes
following the Fokker-Planck equation associated with the Langevin
equations (Eqs. 1).
We can obtain the time-dependent expected change in orientation of the
diatom cells by multiplying Eq. (2) by \(\cos{\theta}\) and
separately by \(\sin{\theta}\) respectively, and then integrating
both equations over \(\theta\) and \(\mathbf{r}\). By
solving a linear system of ordinary differential equations
for\(\left\langle\operatorname{\kappa\ cos}{\theta}\right\rangle(t)\) and\(\left\langle\kappa\sin{\theta}\right\rangle(t)\) (see Text 1 for details),
the analytical prediction of temporal correlation of orientation
\(\left\langle\cos{\theta}\right\rangle\) is given by
\(\left\langle\cos{\theta}\right\rangle\left(t\right)=e^{-\left(D_{\theta}+\nu\right)t}(\selectlanguage{greek}\cos{\sqrt{\lambda}\text{νt}}\selectlanguage{english}-\frac{1}{\sqrt{\lambda}}\selectlanguage{greek}\sin{\sqrt{\lambda}\text{νt}}\selectlanguage{english})\), (3)
where\(\left\langle\bullet\right\rangle=\int{d\mathbf{r}}\selectlanguage{greek}\int_{0}^{2\pi}{\text{dθ}\left(\Psi_{+}+\Psi_{-}\right)}\selectlanguage{english}\)and \(\lambda=\left(\frac{\omega}{\nu}\right)^{2}-1\).
Theoretically, we can further obtain the analytical predictions of the
time-dependent mean-squared displacements (MSD) and effective diffusion
coefficient from the Fokker-Planck equation (Eq. 2). Using mathematical
derivation, we can obtain the analytical expression of
MSD\(\left\langle\mathbf{r}^{2}\right\rangle\left(t\right)\ \)as
\(\left\langle\mathbf{r}^{2}\right\rangle\left(t\right)=4Dt+\ \frac{2V_{0}^{2}}{\nu^{2}{(\lambda+\alpha^{2})}^{2}}\left[\left(\lambda+2\alpha-\alpha^{2}\right)\left(1-e^{-\alpha\nu t}\selectlanguage{greek}\cos{\sqrt{\lambda}\text{νt}}\selectlanguage{english}\right)+\left(\lambda-2\alpha\lambda-\alpha^{2}\right)\frac{1}{\sqrt{\lambda}}e^{-\alpha\nu t}\selectlanguage{greek}\sin{\sqrt{\lambda}\text{νt}}\selectlanguage{english}\right],\)(4)
where\({D=D}_{r}+\frac{V_{0}^{2}\left(\alpha-1\right)}{2\nu\left(\lambda+\alpha^{2}\right)}\), (5)
is the effective diffusion coefficient (or diffusivity for simplicity
hereafter) with \(\lambda=\frac{\omega^{2}}{\nu^{2}}-1\) and\(\ \alpha=\frac{D_{\theta}}{\nu}+1\). Note that
for noncircular motion, i.e. \(\omega=0,\nu=0\), our model (Eq. 5)
defines a system of persistent random walks characterized by a
diffusivity\(D=D_{r}+\frac{V_{0}^{2}}{(2D_{\theta})}\) (28, 36).
\textbf{Model validation}
Our model can indeed capture the spatial patterns and dynamics of the
diatom movement, as reflected by the agreement between the model
prediction and experiment data (Fig. 2D and E, Fig. 3C and D, movie S2).
A visual check, albeit in a non-quantitative way, suggests that the
circular run-and-reverse mode can be well reproduced by our model (Fig.
2D, movie S2). A rigorous validation usually requires scrutinizing
essential behavioral parameters, including MSD and temporal correlation
of orientation \(\left\langle\cos{\theta}\right\rangle\) (34). With respect to MSD, we do find
that our model (from both analytic and simulated results) is well in
line with the experimental data, in the sense that they both present a
highly consistent two-regime pattern of MSD as a function of time (the
curves of the model results and experiment data are almost completely
overlapping in Fig. 3C). At short time scales (\(tt_{C}\)), MSD changes to sub-diffusive behavior indicated by
a scaling exponent less than 1.0. A decreasing rotational diffusivity
\(D_{\theta}\ \)leads to a decline of the scaling exponent, indicating
a weakened diffusive ability when the cell motion is getting closer to
the circular motion (Fig. S2). In addition, our model predicts that the
diffusion behavior would converge to normal diffusion (with scaling
exponent close to 1.0) after a relative long plateau even for low
rotational diffusivities (see Fig. S3), which is often a general
property across many diffusion behaviors. A further comparison between
the model results and experiment data reveals a consistent pattern of
temporal correlation of orientation\(\ \left\langle\cos{\theta}\right\rangle\) as a function of
time. Specifically, at short time scales, positive correlation
coefficients are consistently found in the model and data, suggesting a
positive feedback in directional persistence (Fig. 3D). The negative
correlation coefficients are consistently present due to cells running a
half arc associated with weak stochastic direction fluctuation. At
longer time scales, correlation of orientation is dominated by noise,
and the coefficients approach \(0\) over time
(\(t<100\ s\)).
\textbf{Movement behavior driven by optimal foraging}
If the prediction of the optimal foraging theory holds for the studied
diatoms, one intuitive corollary is that the observed ``circular
run-and-reversal'' movement mode can provide a statistical property that
can maximize foraging efficiency. Indeed, our model analyses together
with experimental data lend support to this speculation.
In the model setting with homogeneously distributed forage targets (Fig.
4A), our simulation analyses (see \emph{Materials and Methods} ) show
that the amount of resource remaining in the environment decays in an
exponential manner over time (Fig. 4B). We thus use the exponent of the
exponential decay as a straightforward indicator of foraging efficiency,
estimated by the regression slope of unconsumed resource on logarithmic
scale over time. A larger exponent (\(\tau\)) means that more
resource targets can be found per unit time, hence indicating a higher
foraging efficiency. However, this indicator cannot not be feasibly
derived from the analytic model. Instead, we used effective diffusivity
as a measure of foraging efficiency. In both analytic models and
simulations, we consistently found optima of foraging efficiency at
rotational diffusivity \(D_{\theta}\) around 0.1 (Fig. 4C, E). A
striking finding is that this optimal point is very close to the
experimentally observed values of \(D_{\theta}\). This is especially
true for the simulations (there is only 9\% deviation between the model
and experiments, Fig. 4C) with a more realistic measure of foraging
efficiency. The larger discrepancy between the analytical model and
experimental data might be explained by the fact that the theoretically
derived diffusion coefficient, although being strongly correlated, is
insufficiently reflective of actual foraging efficiency under specific
conditions (37). In our model, foraging efficiency does not show any
peak with changing reversal rate (\(\nu\)) (Fig. 4D, F).
However, the experimentally observed values of \(\nu\) are
consistently located within the ranges presenting maximum foraging
efficiency. The consistency between the experimental observations and
model results remains robust if we plot the data and modeled foraging
efficiency in two-dimensional parameter space (\(D_{\theta},\nu\)),
clearly showing that the experimental points fall within the optimal
strategy regions (yellow regions in Fig. 5A, B).
Unlike the rotational diffusivity, the lack of optimum in foraging
efficiency as a function of reversal rate \(\nu\) in our
random-environment model suggests that a relatively wide range
of\(\nu\) can have maximized foraging efficiency. This seems
somewhat counterintuitive, and contrasts with our experimental
observation presenting a rather narrow range of \(\nu\). We
infer there should be other benefit to obtain from the reversal
behavior, such as self-organized biofilm formation, whereas a
substantial experimental evidence is still lack.
Taken together, the experimentally observed movement parameters
(both\(\nu\) and \(D_{\theta}\)) are consistently found
in the vicinity of theoretically predicted optimal foraging efficiency.
This suggests the plausibility that the movement pattern of the diatom
is in line with the optimal foraging theory.
\textbf{Evolutionary invasion analysis of behavioral plasticity}
If resource (forage) availability can indeed have a significant effect
on the movement behavior, then the question is whether the movement
strategy corresponds with an evolutionary stable strategy (ESS). We
therefore generated a pairwise invasibility plot (PIP, Fig. 6), by
performing an evolutionary invasibility analysis (see \emph{Materials
and Methods} ) to determine whether the optimal value is a long-term
outcome of competition selection, or just be exploited by free-riding
strategies (38). The PIP reveals that diatom movement strategy of
diatoms observed in our experiments is not only an evolutionarily stable
strategy but also convergence stable. Here, we did not observe a
branching to occur when the parameters of the model are changed in the
evolutionary dynamics. In addition, integrating the effect of multiple
attractors into evolutionary strategies remain a fascinating topic for
future research. Our model provides a universal way to understand the
ecologically relevant functions of movement behavior from the
perspective of foraging theory.
\textbf{IMPLICATIONS}
Our results have several useful implications. The biomechanical
mechanism underlying the `circular run-and-reversal' movement behavior
of the diatom cells remains puzzling. A reasonable speculation is that
the physical constraints of boat-shaped cells with apically located
sensory receptors gliding in fluids might lead to this type of movement
trajectories (39), but this is beyond the scope of this paper. Despite
that, our work provides a clear demonstration that the statistical
properties of this unique behavior can be `optimized' towards enhanced
foraging efficiency. Both theoretically and experimentally, moving
beyond the statistical descriptions of movement behaviors in previous
literature (13,16), our minimal model may thus serve as a useful
framework for follow-up studies unravelling the ecological and
evolutionary consequences of this movement behavioral plasticity in a
broader context.
One fundamental question is how diatoms would adapt their movements, at
individual and collective levels, in response to different foraging
conditions. Indeed, our observations show that the key movement
parameters revealed in our study, including reversal rate and rotational
diffusivity, are sensitive to changing resource availability (see Fig.
7). The diatom cells move with low reversal rate and high effective
diffusivity \(D\) at intermediate dSi concentrations (from
10 to 50 mg/L), whereas low and high dSi will lead to a decreased
efficiency diffusivity to cells (Fig. 7A). We attribute this to the
hypothesis that when silicon becomes the limiting factor, diatom cells
increase searching activity to meet dSi demand for survival with a
higher effective diffusivity to explore larger areas to take up dSi. It
is surprising that the peak of effective diffusivity coincides with
typical dSi concentrations of many coastal scenarios (Fig. 7A). The
effective diffusivity shows a monotonic decline with increased reversal
rates (Fig. 7B). This adaptive response suggests that diatom cells are
able to sense the local dSi concentration and adjust their reversal rate
to adapt to their physical surroundings. The searching efficiency within
a low nutrient environment is thus strongly dependent on cell movement
behaviors. Extending our results beyond dSi scavenging, there may be
other attractors server as the same role to impact motion behaviors of
diatoms. For instance, \emph{in silico} comparison of experimental data
led to the suggestion that diatoms have a more efficient behavioral
adaptation to pheromone gradients as opposed to dSi (40). Our
observations thus pave the roads for follow-up work to look further into
why different movement behaviors have evolved with changing of cell body
shape among diatom species, depending on cell size and shape and in
response to different environmental stimuli\emph{.}
Insights into the movement behavioral plasticity of microorganisms in
aquatic environments have been generated from disciplines such as
biophysics (41-43), but the focus of these studies has largely been on
the statistical physical causes of behavior and not on the ultimate
cause. Cases of reversal behavior were reported independently in
different species of marine bacteria (24, 44, 45), and it has been
suggested that it can contribute to increase foraging efficiency (24,
43) and group social effects (41), but similar evidence is still lacking
for motile microalgae. This study underscores the need to study the
significance of these questions in other microorganisms.
\textbf{MATERIALS AND METHODS}
\textbf{Diatom cell culture and image acquisition}
The \emph{Navicula arenaria} var\emph{. rostellata} strain 0488 (size
ranges from 30\textasciitilde{}50 \(\mu\)m in length and
5\textasciitilde{}15 \(\mu\)m in width) is maintained in the
BCCM/DCG diatom culture collection at Ghent University,
http://bccm.belspo.be/about-us/bccm-dcg. It was isolated in January 2013
from high-nitrate intertidal flats of Paulina Schor, The Netherlands
(51\selectlanguage{ngerman}°21'N, 3°43'E). The isolate has since been maintained in unialgal
culture in artificial seawater medium Aquil (f/2+Si). Like other
naviculoid diatoms, \emph{N. arenaria} is boat-shaped with on each valve
a raphe, a specialized slit in their silica cell wall, running along its
longitudinal axis. Although the precise mechanism remains unknown,
diatom gliding involves an actin/myosin motility system and the
secretion of adhesive EPS strands through the raphe (46).
Diatom culture were maintained using a standard protocol. One months
before the experiment, cells were acclimated to 2000 Lux light intensity
with a light dark cycle of 12:12 hours (INFORS HT Multitron pro,
Switzerland). A 100 ml flask suspension was grown on a shaker at 20°C
rotating with 100 rpm. For motility experiments, diatom cells at period
of exponential phase were diluted with filtered autoclaved seawater and
introduced into the test chamber for observations. The densities of
individuals (about 15 cells/mm\textsuperscript{2}) were used in order to
minimize effects of cell-cell interference.
\textbf{Numerical simulations}
The parameters of the simulation correspond to Fig. 2D and E, Fig. 3C.
For each parameters \(D_{\theta}\) and \(\nu\), 1000
trajectories of 600 sec were simulated using a time-step
\(\delta t=0.1\ \)sec with the parameter values \(V_{0}=17\ \mu m/s\),
\(D_{r}=0\ \mu m^{2}/s\),\(\omega=\pi/36\) rad/s. In Fig. 4C,
\(\nu=0.02\ s^{-1}\), in Fig. 4D, \(D_{\theta}=0.0054\ rad/s\).
We start by analyzing an active cell with a sensing radius
\(r_{c}\), blindly searching for 4000 nutrient resource
(targets) in an environment with a homogeneous topography (47). As a
diatom cruises throughout the searching space, it continuously captures
nutrients that come within a capture radius \(r_{c}\) from the
cells center. At each step, the `nutrients' that come within a capture
radius \(r_{c}\) from the cell center will be removed. We
evaluate the individual search efficiency by calculating the leftover
nutrients \(n\) in the searching space. The amount of
leftover nutrients \(n\) in each run shows a monotonous
decline as a function of the area swept by the active cell. Here, we
assume that all cells use the same strategy of reversal and rotational
diffusivity for the simulations. Fig. 4B plots the average amount of
leftover nutrients \(n\), obtained from 1,000 simulated
trajectories as a function of various rotational diffusivity, so that
the decay rate\(\tau\) of the exponential fitted is defined
as the foraging efficiency.
For the evolutionarily stable strategy analysis, up to 1000 cells are
simulated with various prescribed rotational
diffusivities\(D_{\theta}\). Fitness is given by the product of
survival probability and division rate. We assumed that survival
probability is proportional to the foraging efficiency in the ESS
analysis. A mutant strategy with a relative fitness value larger than
the resident population will invade and potentially take over them. For
any combination of resident and mutant movement strategy, the relative
fitness of the mutants is depicted in a pairwise invasibility plot (Fig.
6).
\textbf{Calculation of time-dependent orientation correlation and MSDs
of moving cells}
We computed the average temporal correlation as
follows:\(\left\langle\cos{\theta}\right\rangle\left(t\right)\equiv\left\langle\hat{\mathbf{v}}\left(t+t\right)\bullet\hat{\mathbf{v}}\left(t\right)\right\rangle\), where \(\hat{\mathbf{v}}\) is the unit vector
of velocity and calculated the mean square displacement
via\(\text{MSD}(t)=\left\langle\left|\mathbf{r}\left(t+t\right)-\mathbf{r}(t)\right|^{2}\right\rangle\).
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\textbf{Acknowledgments}
We thank Chi Xu fruitful discussion that greatly improved the
manuscript. \textbf{Funding:} This research was a product of the project
``Coping with deltas in transition'' within the Programme of Strategic
Scientific Alliances between China and The Netherlands (PSA) financed by
the Chinese Ministry of Science and Technology (2016YFE0133700), and the
National Natural Science Foundation of China (41676084). \textbf{Author
contributions:} Q.-X. L. designed research; W.-S. H. and M. H. performed
research; W.-S. H. and H. P. Z. contributed new analyzed data; W.-S. H.
and F. Z. contributed computer code; W.V. contributed the experimental
diatom strains and experimental details; All authors contributed
substantially to discussion of the content, wrote the article and
reviewed and edited the manuscript before submission. \textbf{Competing
interests:} The authors declare that they have no competing
interests.\textbf{Data and materials availability:} all data needed to
evaluate the conclusions in the paper are present in the paper and/or
the supplementary Materials. The code to reproduce the results of this
study is available on the publicly repository Dryad, DOI:
10.5061/dryad.547d7wm41.
\textbf{Figures and Tables}
\textbf{Fig. 1: Theoretical hypothesis and experimental
setup.}(\textbf{A} ) Three typical patterns of movement behaviors of
microorganisms, showing the `run-and-stop', `run-reverse-flick',
`run-and-tumble', and the `circular run-and-reverse' pattern of marine
diatoms. (\textbf{B} ) The characteristics of an optimization model by
adjusting movement behavioral plasticity. The dash line shows peaks
predicted fitness and therefore what would be expected in nature. When
the environment changes, its optimal value would change accordingly.
(\textbf{C} ) The schematic of the experimental setup (not to scale).
(\textbf{D} ) An example of the observed movement trajectories.
(\textbf{E} ) Scanning electron microscope image of
species\emph{Navicula arenaria} var. \emph{rostellata} shows an
boat-shape cell, where the two raphes can spray the extracellular
polymeric substances (EPS) to obtain self-propulsion.
\textbf{Fig. 2: Experimental observations and theoretical predictions of
the circular run-and-reversal behaviors of diatom \emph{Navicula
arenaria} var. \emph{rostellata}.} (\textbf{A} ) A typical cell
trajectory containing circular run and reversal behaviors captured with
a microscopy at 4 frames per second (see Movie S1 for more trajectories)
for 5 min. (\textbf{B} ) Cropping of the partial trajectory depicts a
reversal behavior with zoom in on the panel (A), where the running from
CCW switches to CW through a reversal behavior, and vice versa. The
arrows indicate the moving direction of the diatom cells. (\textbf{C} )
Experimental data showing the movement velocity before and after a
reversal occurrence; for clarity, not all speeds of the time series are
shown here. (\textbf{D} ) and (\textbf{E} ) Predictions of spatial
trajectory and reversal event obtained from model (1) with parameters
value \(V_{0}\)\emph{=} 17 \(\mu m/s\),
\(D_{\theta}=0.0054\)\(\text{rad}^{2}\)/s, \(\nu=0.02\ s^{-1}\), and
\(\omega=\pi/3\)6 rad/s. Colorbars in panel (A, D) depict the time
(see Movie S2 for theoretical simulations).
\textbf{Fig. 3: Comparing the laboratory measurements and simulation
results with theoretical (analytical) predictions on diffusion behaviors
of diatom cells.} (\textbf{A} ) Statistical distribution of 1704
reversal interval time \(t\) from the 29 experimental
individuals trajectories, which can be well fitted by an exponential
distribution\(f\propto\ e^{-0.016t}\) with the slope of -0.016. (\textbf{B} )
The measured probability density functions of cells' displacements as a
function of displacement normalized by its standard deviation
(\(\sigma=\sqrt{\left\langle{x}^{2}\right\rangle}\)) along the\(\ x\)-axis direction for
different times. A fit to the data with Gumbel law (solid black lines)
and Gaussian model (dashed green lines) are shown for two different time
scales, where the Gumbel law of the distribution imply slower diffusion
at a long-time scale. (\textbf{C} ) Mean squared displacement (MSD) for
three different values of the rotational diffusion coefficient
\(D_{\theta}\) obtained by performing the numerical simulations of
model (1) and comparison with the experiments (circles symbols),
respectively. By decreasing the strength of rotational diffusion in the
model, the scaling behaviors of the MSD vs. time becomes consistent with
confined diffusivity from ballistic behaviors similarly to cage-effect
emergence after the characteristic times (\(\sim 25\ s\)).
Parameters are \(\omega=\pi/36\) rad/s,\(\nu=0.02\ s^{-1}\),
\(D_{\theta}=0.0054\) \(\text{rad}^{2}\)/s and \(V_{0}=\)17
\(\mu\)m/s. The dashed lines are a guide to the eye to mark
the change of the scaling law with 2.0 and 1.0 respectively, the solid
line corresponds to the trend predicted by theory Eq. (4). (\textbf{D} )
Correlation of measured and predicted changes in the direction of cells
moving. Experimental data ( symbols) have error bars representing lower
and upper SD. Corresponding analytical predictions (solid line and
dashed line with triangle symbols) are given by theory Eq. (\textbf{3)}
and numerical simulations of model (1) respectively. The dashed line
indicates 0 to guide the eye in (B).
\textbf{Fig. 4: Theoretical prediction of optimal foraging strategies
with spatially randomized nutrient targets.} (A) Schematic
representation (not to scale) of diatom cells blindly searching for
randomly distributed nutrient resources (dots). The cells placed in a
two-dimensional space move with constant speed\(\mathbf{V}_{\mathbf{0}}\) and
variable orientation described in model (1). The capture radius
\(\mathbf{r}_{\mathbf{c}}\) is about 20\(\mathbf{\mu}\)m size (dashed circle
area). \textbf{} (B) The distinctive exponential
function,\(\mathbf{n}\left(\mathbf{t}\right)\mathbf{=A}\mathbf{e}^{\mathbf{-\tau t}}\)with the decay rate \selectlanguage{greek}τ, \selectlanguage{english}was used to describes
the foraging efficiency of diatom movement strategy with respect to
various value of\(\mathbf{D}_{\mathbf{\theta}}\) and \(\mathbf{\nu}\). (C, D) The
efficiency of captured nutrients as a function of\(\mathbf{D}_{\mathbf{\theta}}\) and
\(\mathbf{\nu}\), respectively. Foraging efficiency is calculated by
averaging over 1000 trajectories with various \(\mathbf{\omega}\), where
the plot is scaled to the maximum value at \(\mathbf{D}_{\mathbf{\theta}}\mathbf{=0.3}\)
and\(\mathbf{\nu=0.0001\ }\mathbf{s}^{\mathbf{-1}}\) respectively (see fig. S6 for without scaling).
(E, F) The analytical prediction of effective diffusivity from theory
Eq. (5), coinciding with directly numerical simulations of model (1).
The dashed lines and gray shaded area represent mean
\(\mathbf{\pm}\mathbf{2\ }\)SD from the experimentally measured values of
\(\mathbf{D}_{\mathbf{\theta}}\)and\selectlanguage{ngerman} \selectlanguage{greek}ν \selectlanguage{english}for \emph{Navicula arenaria}
var.\emph{rostellata} .
\textbf{Fig. 5: Theoretical and experimental results implicate the
emergence of the foraging efficiency for various behavioral
strategies.}(\textbf{A} ) Heatmap of foraging efficiency (colorbar) with
respect to (\(D_{\theta},\nu\))-parameter space obtained from randomly
distributed nutrient targets and constant movement speed for
\(\omega=\pi/36\)rad/s and \(V_{0}=17\) \(\mu\)m/s.
Optimal foraging occurs over a window of behavioral parameters of
\(\nu\ \)and \(D_{\theta}\), and is indicated by the
yellow areas. The boundaries of the optimal regions change sharply with
increasing reversal rate (white dashed lines with intervals
\(\tau=0.1\)). In the low reversal rate limit, there are
nonlinear effects of the rotational diffusion on diatom foraging. The
colored-solid dots correspond to the experimentally measured rotational
diffusion coefficients versus reversal rate on diatom \emph{Navicula
arenaria} var\emph{. rostellata} and the colorscale indicates the scaled
foraging efficiency, \(\tau\) from 0 to 1.0. (\textbf{B} )
Theoretical prediction of Eq. (\textbf{5)} on the effective diffusivity
as functions of the rotational diffusivity and reversal rate. It shows a
similar spatial profile comparison with directly numerical simulations.
\textbf{Fig. 6: Pairwise invasibility plot (PIP) of behavioral strategy}
. The PIP indicates that the movement behavioral strategy of rotational
diffusivity evolves toward a stable point 0.2 (vertical dashed line).
For a range of resident (\emph{x} -axis) and mutant (\emph{y} -axis)
movement strategies, the PIP describes whether a mutant has a higher
(green) or a lower (blue) fitness than the resident. Plus and minus
symbols indicate combinations resulting in positive and negative
invasion fitness, respectively. Here, the PIP shows that the rotational
diffusivity with 0.2 is the sole evolutionarily stable strategy (ESS).
Simulation parameters with\(\nu=0.02\ s^{-1}\)\emph{,} and
\(\omega=\pi/36\ rad/s\)\emph{.}
\textbf{Fig. 7: Reversal behaviors depend on the ambient dSi
concentration.} (\textbf{A} ) The diffusivity of diatom cells maximizes
at an ambient dSi concentration of about 30 mg/L and declines at low and
high dSi concentrations. The reversal events show a sharply increase
when dSi goes beyond 60 mg/L, but it maintains a plateau at low dSi. The
grayscale rectangle indicates typical dSi concentrations in coastal
ecosystems. (\textbf{B} ) Efficiency diffusion coefficient, showing a
monotonic decline with increased reversal events, which have a maximized
dispersal coefficient about \(\nu=0.02\ s^{-1}\) coincident with model
predictions.
\textbf{Table 1. Statistical properties of measured experimental
parameters on diatom movement behaviors.} Experimental statistics of
behavioral parameters on diatom cells at dSi concentrations of 15
mg/L.\emph{n} , number of individuals.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}
\end{center}
\end{figure}
SUPPLEMENTARY MATERIALS
Text S1. Theoretical derivation for the expected value of MSD and
diffusivity.
Fig. S1. Distribution and trajectories of experimental
\(D_{\theta}\).
Fig. S2. Trajectories at three different values of \(D_{\theta}\).
Fig. S3. Behaviors of mean squared displacement (MSD) at different
values of \(D_{\theta}\).
Fig. S4. The efficiency without scaling to the maximal foraging
efficiency.
Movie S1. Typical trajectories of swimming diatoms in real experiments
on diatom movement behaviors.
Movie S2. Typical trajectories of swimming diatoms in theoretical
simulations of Eqs. (1).
\par\null\par\null
\selectlanguage{english}
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