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 (\(t<t_{C}=25\ s\)), MSD as a scaling function of time
shows ballistic dynamics with a scaling exponent of \(2.0\), whereas at
long-time scales (\(t>t_{C}\)), 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 (\(t<100\ s\)).