Mapping of spatiotemporal heterogeneous particle dynamics in living cells

Colloidal particles embedded in the cytoplasm of living mammalian cells have been found to display remarkable heterogeneity in their amplitude of motion. However, consensus on the significance and origin of this phenomenon is still lacking. We conducted experiments on Hmec-1 cells loaded with about 100 particles to reveal the intracellular particle dynamics as a function of both location and time. Central quantity in our analysis is the amplitude $\left(A\right)$ of the individual mean-squared displacement (iMSD), averaged over a short time. Histograms of $A$ were measured, (1) over all particles present at the same time and (2) for individual particles as a function of time. Both distributions showed significant broadening compared to particles in Newtonian liquid, indicating that the particle dynamics varies with both location and time. However, no systematic dependence of $A$ on intracellular location was found. Both the (strong) spatial and (weak) temporal variations were further analyzed by correlation functions of $A$. Spatial cross correlations were rather weak down to interparticle distances of $1\mu \text{m}$, suggesting that the precise intracellular probe distribution is not crucial for observing a dynamic behavior that is representative for the whole cell. Temporal correlations of $A$ decayed at $\sim 10\text{s}$, possibly suggesting an intracellular reorganization at this time scale. These findings imply (1) that both individual particle dynamics and the ensemble averaged behavior in a given cell can be measured if there are enough particles per cell and (2) that the amplitude and power-law exponent of iMSDs can be used to reveal local dynamics. We illustrate this by showing how superdiffusive and subdiffusive behaviors may be hidden under an apparently diffusive global dynamics.

Figure 1

(Color online) Individual mean-squared displacement functions obtained by simulation for a purely diffusive system. The yellow central line indicates the total MSD obtained by averaging over all particles. Inset: normalized distribution of the iMSD amplitude as a function of the number of contributions to the average: 20 (red dashes), 50 (blue dots), and 200 (black solid line).

Figure 2

(Color online) Illustration of two exemplary behaviors of the iMSD amplitude $A_p$ of particle $p$ as a function of real time. Dotted lines show the average over the 100 observations. The inset shows the corresponding histograms of $A_p$ values.

Figure 3

Schematic illustration of our analysis of iMSD amplitudes. $P$ labels individual particles (and their positions), while $t$ is the index of a time segment in a chronological series. Filled circles indicate for which matrix elements $\left(p,t\right)$ a contribution to $A$ exists. Analysis of the rows allows studying fluctuations of $A$ over time via autocorrelation functions $g\left(\tau \right)$ or via variances. Analysis of the columns allows quantifying spatial variations in $A$ via cross correlations $f\left(d\right)$ or via variances.

Figure 4

(Color online) (a) Microscopy image of a Hmec-1 cell containing endogenous granules, visible as dark objects. (b) Reconstruction of (a) in which each particle has been assigned an MSD amplitude, represented via a color scale. (c) Individual and whole cell MSDs as found in a typical experiment. (d) Amplitude histogram for EGs, averaged over all experiments. For comparison, also a histogram for Brownian particles is included. Note the amplitude normalization on the abscissa.

Figure 5

(Color online) Histogram of trajectory lengths obtained from 41 movies of endogenous granules in a confluent Hmec-1 cell.

Figure 6

(Color online) Spatial correlation function [cf. Eq. (18)] for the iMSD amplitude of EGs in Hmec-1 cells. Inset: reference case of polystyrene latex particles in glycerol. Vertical bars indicate standard deviations.

Figure 7

(Color online) Time autocorrelation function [cf. Eq. (13)] of the iMSD of EGs in Hmec-1 cells. Red diamonds and green squares correspond to segmentation of trajectories into blocks of 50 and 20 units, respectively, of 60 ms. $+$: latex spheres in glycerol. △: computer simulation for Brownian spheres.

Figure 8

(Color online) Relative standard deviations in the iMSD amplitude of EGs in Hmec-1 cells, calculated in two different manners. Spatial: cf. Eq. (24). Temporal: cf. Eq. (21). Dotted lines indicate averages over the 41 movies. For Brownian particles in Newtonian liquids, both averages coincide at 0.20.

Figure 9

(Color online) Representative set of 100 iMSD functions [based on trajectory segments cf. Eq. (5)] for BIPs (left) and EGs (right) in Hmec-1 cells. Average MSDs [cf. Eq. (2)] are shown in yellow. Solid magenta lines indicate a power-law exponent of 1.

Figure 10

(Color online) Probability distributions for the logarithm of amplitude $\left(\text{ln}A\right)$ and power-law exponent $\left(\alpha \right)$ measured at $\tau \approx 0.1\text{s}$ for two probe particles (EGs and BIPs) in individual Hmec-1 cells. For comparison, also the case of monodisperse particles in a Newtonian liquid is included. Distributions for $\text{ln}A$ have been centered on zero. For BIPs the distribution of $\alpha$ is also shown for segmentation into blocks of 200 steps (dotted line).