We perform a detailed statistical analysis of diffusive trajectories of membrane-enclosed vesicles (vacuoles) in the supercrowded cytoplasm of living Acanthamoeba castellanii cells. From the vacuole traces recorded in the center-of-area frame of moving amoebae, we examine the statistics of the time-averaged mean-squared displacements of vacuoles, their generalized diffusion coefficients and anomalous scaling exponents, the ergodicity breaking parameter, the non-Gaussian features of displacement distributions of vacuoles, the displacement autocorrelation function, as well as the distributions of speeds and positions of vacuoles inside the amoeba cells. Our findings deliver novel insights into the internal dynamics of cellular structures in these infectious pathogens.
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This can be an artifact of the tracking procedure and the limited resolution of the image-acquisition setup; see also Sec. V A.
Note that myosin-IC motors are abundant in the actin-rich edge of the cell, while myosin-II motors are present in the entire cytoplasm.
Note that in this setup smaller vacuoles were technically harder to track because our detection algorithm is based on edge detection and subsequent Hough transformation, commonly used to detect circles. This procedure requires a threshold value for the minimal circle radius and for the sensitivity to be preset. So, if the radius is chosen too small, many “circles” that are not vacuoles would be undesirably detected.
Note that the evaluation of the vacuoles’ center-of-mass position20 from their center-of-area coordinate requires an assumption of a uniform cell height. This has certain approximations. Fast-running AC cells appear to have a “fried-egg” geometry13,16 with a varying cell height from the surface. The videos indicate that the cells have thin leading edge in front and rather thick “sack of material” on the rear end, where large vacuoles are often located; see the video files in the supplementary material.
This hampers the detection of small vacuoles for longer times. During amoebae diffusion, larger particles stay in a confident-detection plane for longer times introducing certain bias in the data (see the discussion in Refs. 56, 65, 67, and 68). Specifically, the focus depth still allowing a confident tracking is a couple of μm. Larger vacuoles are, thus, allowed to move larger distances in the vertical direction and still yield a detectable position. By contrast, for smaller vacuoles, the same displacement may lead to its disappearance from the viewfield and to trajectory termination. Thus, a slower subpopulation of smaller vacuoles gets over-represented in the data set.
Note that the discrepancy of the EB parameter from the Brownian behavior may seem inconsistent with a close match of the MSD and mean TAMSD, as seen in Fig. 16 in the Appendix. Theoretically, however, similar discrepancies in the behaviors of the ensemble- and time-averaged displacements versus the EB parameter were found and rationalized previously; see Ref. 76. This is the case, for instance, for diffusive systems where the relaxation time exceeds the measurement time (the length of time series).
The TAMSD exponent varies substantially along the vacuole trajectories in the range of time-shifts probed for the autocorrelation function in Fig. 8. In virtue of a limited length of trajectories, the mean TAMSD does not reveal any extended region of anomalous diffusion with a roughly constant scaling exponent. Therefore, one cannot expect a universal curve for Cδt(t) to emerge when a rescaling of time t/δt is employed; see also the discussion in Ref. 52.
This value, however, has a large standard deviation, again due to the fact that instantaneous speeds of vacuoles take rather discretized values in the current data set. Note here that small vacuoles which are slow can be over-represented in the current data set (generally, smaller tracers are more problematic to track for longer times; Fig. 14 in the Appendix confirms this statement).
We emphasize here, however, that if the mean vacuole radii—rather than the maximum radii—are used for the analysis, the vacuole distributions appear quite different; see Fig. 21 in the Appendix. In this interpretation, for instance, the smallest vacuoles tend to occupy the central regions of the amoebae. The physical interpretation for the mean vacuole radius seems, however, less clear to us than for the maximum radius along a given track.
To cure these “artificial” discreteness-based effects69 in displacements, speeds, and displacement autocorrelations of vacuoles, one can think of smearing out the vacuole positions recorded in these SPT experiments, prior to their statistical analysis. One can use a Gaussian-like smoothening function with width equal to several pixels of the microscopy image [not shown; see the inset of Fig. 7(b)]. This would then make the peaks in the speed distribution of Fig. 9 originating from the discreteness effects less pronounced. The elementary time scale involved in the computation of the average vacuole speed should then also be adjusted correspondingly (instead of setting it to one elementary time step, as in Fig. 9); see Ref. 52. Physically, only those tracer displacements exceeding the position-localization uncertainty52,58,78 should be used in the analysis of physical observables. The effects of varying localization error in these SPT experiments on the behavior of the fundamental quantities such as the TAMSD, the EB parameter, and the autocorrelation function would be interesting to study in the future96 once precision-controlled data are acquired for this motile system.
Possible long-distance correlations in direction and motion speed of diffusing vacuoles—as a function of their separation inside a given amoeba—are an interesting subject to study. They could quantify the “reach” of hydrodynamic and other correlation-inducing interactions being transmitted through the cell cytoplasm. In the current data, however, the mutual distances between vacuoles were not recorded and this question cannot be addressed in principle.
