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 (\(t<100\ s\)).