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 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).