Quantifying the degree of persistence in random amoeboid motion based on the Hurst exponent of fractional Brownian motion

Amoebae explore their environment in a random way, unless external cues like, e.g., nutrients, bias their motion. Even in the absence of cues, however, experimental cell tracks show some degree of persistence. In this paper, we analyzed individual cell tracks in the framework of a linear mixed effects model, where each track is modeled by a fractional Brownian motion, i.e., a Gaussian process exhibiting a long-term correlation structure superposed on a linear trend. The degree of persistence was quantified by the Hurst exponent of fractional Brownian motion. Our analysis of experimental cell tracks of the amoeba Dictyostelium discoideum showed a persistent movement for the majority of tracks. Employing a sliding window approach, we estimated the variations of the Hurst exponent over time, which allowed us to identify points in time, where the correlation structure was distorted (“outliers”). Coarse graining of track data via down-sampling allowed us to identify the dependence of persistence on the spatial scale. While one would expect the (mode of the) Hurst exponent to be constant on different temporal scales due to the self-similarity property of fractional Brownian motion, we observed a trend towards stronger persistence for the down-sampled cell tracks indicating stronger persistence on larger time scales.

Figure 1

Estimated Hurst exponents along the $X$ coordinate (top) and along the $Y$ coordinate (bottom). The white dashed line corresponds to a Hurst exponent of $H=0.5$, separating the persistent area with $H>0.5$ from the antipersistent area with $H<0.5$. The posterior density of $H$ is color coded. Dark colors for low density values to increasingly brighter colors for higher values. The tracks are sorted with increasing most probable Hurst exponent $\widehat{H}$. Therefore it is clearly visible that for larger tracks most of the Hurst exponents are in the persistent region $H>0$.

Figure 2

(left) Illustrative original track (No. 683) and (right) reconstructed track data of the movement of a Dictyostelium cell in the absence of external cues.

Figure 3

Projection of Bayesian estimation of the Hurst exponent for the $X$ coordinate of some Dictyostelium cell as a function of time for some illustrative track (No. 683). In red color we depicted posterior distributions before deletion of the outliers, whereas in green they are shown after deletion. Shift to persistent area is clearly observed.

Figure 4

Posterior density of $X$ (solid line) and Y coordinate (dashed line) of an illustrative track (No. 683); based on original track data (left) and after outliers were detected and deleted (right).

Figure 5

Posterior density of the Hurst exponent obtained by the Bayesian approach applied to fBm with Hurst exponent $H=0.595$ (based on 1000 realizations).

Figure 6

Posterior density of the Hurst exponent obtained by the Bayesian approach applied to the illustrative track (No. 683); see Fig. 2 (left).