Figures and Tables
Fig. 1: Theoretical hypothesis and experimental setup.(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. (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.
(C ) The schematic of the experimental setup (not to scale).
(D ) An example of the observed movement trajectories.
(E ) Scanning electron microscope image of speciesNavicula arenaria var. rostellata shows an boat-shape
cell, where the two raphes can spray the extracellular polymeric
substances (EPS) to obtain self-propulsion.
Fig. 2: Experimental observations and theoretical predictions of
the circular run-and-reversal behaviors of diatom Navicula
arenaria var. rostellata. (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. (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. (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. (D ) and (E ) Predictions of spatial
trajectory and reversal event obtained from model (1) with parameters
value \(V_{0}\)= 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).
Fig. 3: Comparing the laboratory measurements and simulation
results with theoretical (analytical) predictions on diffusion behaviors
of diatom cells. (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. (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. (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).
(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. (3) and numerical simulations of model (1) respectively.
The dashed line indicates 0 to guide the eye in (B).
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). (B) The distinctive
exponential function,\(\mathbf{n}\left(\mathbf{t}\right)\mathbf{=A}\mathbf{e}^{\mathbf{-\tau t}}\)with the decay rate τ, 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 ν for Navicula arenaria var.rostellata .
Fig. 5: Theoretical and experimental results implicate the
emergence of the foraging efficiency for various behavioral strategies.(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 Navicula
arenaria var. rostellata and the colorscale indicates the scaled
foraging efficiency, \(\tau\) from 0 to 1.0. (B ) Theoretical
prediction of Eq. (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.
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 (x -axis) and mutant
(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\(\nu=0.02\ s^{-1}\), and \(\omega=\pi/36\ rad/s\).
Fig. 7: Reversal behaviors depend on the ambient dSi
concentration. (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. (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.
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.n , number of individuals.