Non-Gaussian Displacements in Active Transport on a Carpet of Motile Cells

We study the dynamics of micron-sized particles on a layer of motile cells. This cell carpet acts as an active bath that propels passive tracer particles via direct mechanical contact. The resulting nonequilibrium transport shows a crossover from superdiffusive to normal-diffusive dynamics. The particle displacement distribution is distinctly non-Gaussian even at macroscopic timescales exceeding the measurement time. We obtain the distribution of diffusion coefficients from the experimental data and introduce a model for the displacement distribution that matches the experimentally observed non-Gaussian statistics. We argue why similar transport properties are expected for many composite active matter systems.

Figure 1

Characteristics of cargo particle transport on cellular monolayers. (a) Illustration of an experimental bright-field microscopy recording with particle trajectories displayed as colored overlays (see the Supplemental Material [40] for a video). Bright spots with a black halo are particles ($46\mathrm{\mu }\mathrm{m}$ diameter). Cells with an extension of about $10\mathrm{\mu }\mathrm{m}$ appear translucent in the background. (b) MSD as function of lag time $\tau$: TAMSDs $\overline{{\delta }_j^2(\tau )}$, Eq. (1), of individual particles (light gray lines), their ensemble average $⟨\overline{{\delta }_j^2(\tau )}⟩$ (blue), and the ensemble-averaged MSD $m_2(\tau )$ (black). Error bars indicate $1\sigma$-confidence intervals. These MSDs reveal two distinct regimes: superdiffusion at short timescales ($\tau \lesssim 2\min$) and normal diffusion at long times ($\tau \gtrsim 2\min$). Inset: scaling exponents $\gamma$ obtained by fitting $\overline{{\delta }_j^2(\tau )}\sim {\tau }^{\gamma }$ to the first three data points of TAMSDs (time interval: 45 s). (c)–(f) The non-Gaussian displacement PDFs along the $x$ axis for different lag times. Red line in (c): stretched exponential distribution, proportional to $\exp (-a|\mathrm{\Delta }x{|}^{\delta })$ with an exponent $\delta \approx 0.77$; red lines in (d)–(f): predicted displacement distributions, based on heterogeneous Brownian motion. The respective parameters inferred from the distributions shown in Figs. 3 were used to plot the PDFs in (d)–(f) [for details, see Eqs. (5) and main text].

Figure 2

(a) Ensemble-averaged DACFs for different lag times $\tau$ as listed in panel (b) (all times in minutes). Gray lines show single-particle DACFs ${\tilde{C}}_{\tau }^{(j)}(\mathrm{\Delta })$ for lag time $\tau =2\min$ (inset); the ensemble average $⟨{\tilde{C}}_{\tau }(\mathrm{\Delta })⟩$ is depicted in blue—error bars indicate $3\sigma$-confidence intervals. (b) The same data as in (a) as function of the rescaled time shift $\mathrm{\Delta }/\tau$. For lag times $\tau \ge 2\min$, the DACF collapses onto a triangular correlation function (black solid line), in line with Brownian motion [40].

Figure 3

Histograms of diffusion coefficients for different lag times $\tau$. Solid lines represent fits with a gamma distribution [Eq. (3)], inferred by maximum likelihood estimation: (a) $\alpha =2.55$, $D_{\beta }=3.23\mathrm{\mu }{\mathrm{m}}^2/\min$, (b) $\alpha =2.26$, $D_{\beta }=3.40\mathrm{\mu }{\mathrm{m}}^2/\min$, and (c) $\alpha =1.91$, $D_{\beta }=4.24\mathrm{\mu }{\mathrm{m}}^2/\min$. Sample means of diffusion coefficients: (a) $⟨D⟩=(8.2\pm 0.5)\mathrm{\mu }{\mathrm{m}}^2/\min$ for $\tau =2\min$, (b) $⟨D⟩=(7.7\pm 0.4)\mathrm{\mu }{\mathrm{m}}^2/\min$ for $\tau =8\min$, and (c) $⟨D⟩=(8.0\pm 0.5)\mathrm{\mu }{\mathrm{m}}^2/\min$ for $\tau =32\min$. (b) The fit from (a) is shown as an overlay; (c) contains all three fits. Differences are not statistically significant [40].