Weak Ergodicity Breaking of Receptor Motion in Living Cells Stemming from Random Diffusivity

Molecular transport in living systems regulates numerous processes underlying biological function. Although many cellular components exhibit anomalous diffusion, only recently has the subdiffusive motion been associated with nonergodic behavior. These findings have stimulated new questions for their implications in statistical mechanics and cell biology. Is nonergodicity a common strategy shared by living systems? Which physical mechanisms generate it? What are its implications for biological function? Here, we use single-particle tracking to demonstrate that the motion of dendritic cell-specific intercellular adhesion molecule 3-grabbing nonintegrin (DC-SIGN), a receptor with unique pathogen-recognition capabilities, reveals nonergodic subdiffusion on living-cell membranes In contrast to previous studies, this behavior is incompatible with transient immobilization, and, therefore, it cannot be interpreted according to continuous-time random-walk theory. We show that the receptor undergoes changes of diffusivity, consistent with the current view of the cell membrane as a highly dynamic and diverse environment. Simulations based on a model of an ordinary random walk in complex media quantitatively reproduce all our observations, pointing toward diffusion heterogeneity as the cause of DC-SIGN behavior. By studying different receptor mutants, we further correlate receptor motion to its molecular structure, thus establishing a strong link between nonergodicity and biological function. These results underscore the role of disorder in cell membranes and its connection with function regulation. Because of its generality, our approach offers a framework to interpret anomalous transport in other complex media where dynamic heterogeneity might play a major role, such as those found, e.g., in soft condensed matter, geology, and ecology.

Popular Summary

The transport of proteins and organelles in cells is relevant for facilitating biomolecular reactions and regulating cellular function. Recently, it has been shown that some cellular components display nonergodic motion; i.e., the average properties of one molecule observed for a long time differ from those of a large ensemble of molecules. This behavior has been attributed to the occurrence of very long trapping events, but its biological implications are not yet clear.

We employ single-particle tracking to measure receptor diffusion, a motion characterized by random displacements, in living-cell membranes. We show that a membrane receptor involved in the capture of pathogens displays nonergodic motion not caused by trapping but consistent with a model of spatiotemporal disorder of the cell membrane. Our observations, based on video microscopy of over 600 trajectories of quantum-dot-labeled receptors, are complemented by numerical simulations of Brownian diffusion in a heterogeneous medium. Besides the typical form of the receptor, we furthermore investigate three mutated versions; we find distinct dynamics corresponding to impaired functioning of receptors.

Our results confirm that the plasma membrane is a complex environment, and they provide an interpretative framework applicable to a range of diffusive systems, spanning from the life sciences to geology.

Figure 1

DC-SIGN diffusion shows weak ergodicity breaking and aging. (a) Representative video frames of a quantum-dot-labeled WT DC-SIGN molecule diffusing on the dorsal membrane of a CHO cell. The centroid position of the bright spot ($+$), corresponding to a single quantum dot, is tracked and reconnected to build up the DC-SIGN trajectory, shown by the cyan line. (b) Representative trajectories for the same recording time (3.2s). (c) Log-log plot of the time-averaged MSD for individual trajectories (blue lines). The dashed lines scale linearly in time, showing that tMSD is compatible with pure Brownian motion ($\beta =1$). The symbols (circle) correspond to the average tMSD. A linear fit to the log-log transformed data (black line) provides $\beta =0.95\pm 0.05$. (d) Distribution of short-time diffusion coefficients as obtained from linear fitting of the time-averaged MSD for all the trajectories. Inset: Cumulative distribution function (CDF) of the exponent $\beta$ obtained from nonlinear fitting of the tMSD of all the trajectories. (e) Log-log plot of the ensemble-averaged MSD. A power-law fit of the data (dashed line) provides an exponent $\beta =0.84$, showing subdiffusion. (f) Log-log plot of the time-ensemble-averaged diffusion coefficient as a function of the observation time $T$. The diffusion coefficients are obtained by linear fitting of the time-ensemble-averaged MSD. A power-law fit (dashed line) provides an exponent $\beta -1=-0.17$, revealing aging and in good agreement with the value of $\beta$ found in (e).

Figure 2

DC-SIGN receptor dynamics is inconsistent with the CTRW model. (a) A trajectory of WT DC-SIGN on living-cell membranes showing a short-lived transient immobilization event, highlighted by the yellow circular area. (b) Plot of the $x$ (blue) and $y$ (red) displacements as a function of time. The occurrence of transient immobilization (yellow region) corresponds to a reduction in the trajectory displacement. (c) Time-averaged MSD for the entire trajectory (filled square) and for the immobilization region only (open circle). (d) Survival function of the duration of immobilization events for WT DC-SIGN trajectories (black line). Red and blue lines correspond to fits to exponential and power-law distribution functions, respectively. The power-law fit provides $\beta =0.83\pm 0.05$, in agreement with the exponent obtained for the eMSD. (e) Schematic representation of the calculation of the escape-time probability from circular areas of different radius $R_{\mathrm{TH}}$. (f) CDF of the trajectory escape time for different radii $R_{\mathrm{TH}}=20$ (open circle), 50 (open inverted triangle), 100 (open square), 200 (open triangle), 300 (filled circle), 500 (filled inverted triangle), and 1000 (filled square) nm. Dashed lines are guides to the eye. The gray-shaded region represents times shorter than the acquisition frame rate.

Figure 3

DC-SIGN motion experiences changes in diffusivity. (a) Representative WT DC-SIGN trajectory displaying changes of diffusivity. Change-point analysis evidences five different regions represented with different colors. (b) Plot of the $x$ (blue) and $y$ (red) displacements for the trajectory in (a) as a function of time. The shaded areas indicate the regions of different diffusivity. The lower panel displays the corresponding short-time diffusion coefficient as obtained from a linear fit of the time-averaged MSD for the five different regions. Gray areas correspond to the 95% confidence level. (c) Plot of time-averaged MSD versus time lag for the first three regions of the trajectory in (a). (d) Histogram of the number of changes of diffusion per trajectory. Most of the trajectories (63%) display at least one dynamical change, with an average of 2.2 changes per trajectory.

Figure 4

Annealed model of heterogeneous diffusion quantitatively reproduces DC-SIGN motion. (a) A simulated trajectory composed by five time intervals with different transit times ${\tau }_i$ and diffusivity $D_i$. (b) Contour plot of the probability distribution of the simulated diffusion coefficient $D$ and transit time $\tau$ for the parameters reproducing the dynamics of WT DC-SIGN ($\sigma =1.16$, $\gamma =1.38$, $b=0.12\mu {\mathrm{m}}^2/\mathrm{s}$, $k=0.10\mu {\mathrm{m}}^{2\gamma }{\mathrm{s}}^{\gamma +1}$). The white line represents the power-law dependence between diffusivity and average transit time with exponent $-\gamma$. (c) Log-log plot of the time-averaged MSD for simulated trajectories (black lines). The symbols (circle) correspond to the average tMSD. The linear fit to the log-log transformed data provides $\beta =0.98\pm 0.03$. (d) Log-log plot of the ensemble-averaged MSD for the simulated trajectories. The dashed line represents a power law with the theoretical exponent $\beta =\sigma /\gamma =0.84$. (e) Log-log plot of the time-ensemble-averaged diffusion coefficient as a function of the observation time $T$. The dashed line represents a power law with the theoretical exponent $\beta -1=-0.16$. (f) Simulated trajectories for the same recording time (3.2s). (g) Distribution of short-time diffusion coefficients as obtained from linear fitting of the time-averaged MSD for all the simulated trajectories. Inset: CDF of the exponent $\beta$ obtained from nonlinear fitting of the tMSD of all the trajectories. (h) CDF of trajectory escape time for different radii. Curves from left to right correspond to radii $R_{\mathrm{TH}}=20$, 50, 100, 200, 300, 500, and 1000 nm.

Figure 5

Effect of mutations on the dynamics of DC-SIGN. (a) Schematic representation of WT DC-SIGN and its mutated forms. (b) Contour plot of the probability distribution of the simulated diffusion coefficient ($D$) and transit time ($\tau$) for the parameters reproducing the dynamics of N80A ($\sigma =0.58$, $\gamma =0.68$, $b=0.09\mu {\mathrm{m}}^2/\mathrm{s}$, $k=0.74\mu {\mathrm{m}}^{2\gamma }{\mathrm{s}}^{\gamma +1}$), $\mathrm{\Delta }35$ ($\sigma =1.04$, $\gamma =1.23$, $b=0.08\mu {\mathrm{m}}^2/\mathrm{s}$, $k=0.07\mu {\mathrm{m}}^{2\gamma }{\mathrm{s}}^{\gamma +1}$), and $\mathrm{\Delta }\text{Rep}$ ($\sigma =2.11$, $\gamma =1.91$, $b=0.07\mu {\mathrm{m}}^2/\mathrm{s}$, $k=0.07\mu {\mathrm{m}}^{2\gamma }{\mathrm{s}}^{\gamma +1}$). The white line represents the power-law dependence between diffusivity and average transit time with exponent $-\gamma$. (c) Log-log plot of the ensemble-averaged MSD for N80A trajectories (filled circle) and simulated data ($+$). (d) Log-log plot of the time-ensemble-averaged diffusion coefficient for N80 A trajectories (filled circle) and simulated data ($+$) as a function of the observation time $T$. (e) Distribution of short-time diffusion coefficients as obtained from linear fitting of the time-averaged MSD for the N80A (filled bars) and the simulated trajectories (empty bars). (f) CDF of the escape time for N80A (symbols) and simulated trajectories (lines) for different radii. The meaning of the symbols is the same as in Fig. 2. (g)–(l) Dynamical behavior of the $\mathrm{\Delta }35$ mutant. (m)–(p) Dynamical behavior of the $\mathrm{\Delta }\text{Rep}$ mutant.