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).