FIGURE 5.

DDM analysis reveals ballistic-like transport of particles entrained in active composites at mesoscopic spatiotemporal scales. (A) Sample image structure functions $D(q,\mathrm{\Delta }t)$ for composites with actin fractions ${\varphi }_A$ indicated in the legend. All curves shown are evaluated at $q=3.92\mu m^{-1}$, and solid black lines are fits to the data to determine corresponding $q$-dependent decay times $\tau (q)$ and stretching exponents $\gamma$, as described in Methods. (B) Decay times $\tau (q)$ for each composite shown in (A). Dashed and dotted black lines show scaling $\tau (q)~q^{-\beta }$ for ballistic $(\beta =1)$ and diffusive $(\beta =2)$ transport. (C) $\tau (q)\times q^2$, normalized by $\tau \left(q_{max}\right)\times {\left(q_{max}\right)}^2$, for the data shown in (B). Horizontal dotted line and unity-sloped dashed line correspond to scaling indicative of normal diffusion $(\alpha =2/\beta =1)$ and ballistic motion $(\alpha =2/\beta =2)$. Color-coded solid lines correspond to power-law fits, with the corresponding exponents $\beta$ and $\alpha$ shown in (E) and (F). For ${\varphi }_A>0$ composites, the fitting range is truncated to $q>1.5\mu {\mathrm{m}}^{-1}$ where a single power-law is observed. (D) Scatter plot of $1/\tau (q)$, a measure of the transport rate, for all measured $q$ values for each ${\varphi }_A$. Color coding and gradient indicate ${\varphi }_A$ and $q$, respectively, with light to dark shades of each color indicating increasing $q$ values. (E) DDM scaling exponents $\beta$ determined from fits shown in (C). (F) Anomalous scaling exponents ${\alpha }_2$ determined from $\tau (q)$ fits (filled circles, ${\alpha }_{DDM}=2/\beta$), as well as the large-$\mathrm{\Delta }t$ regime fits of the $MSD\mathrm{s}$ (open triangles, ${\alpha }_{MSD,2}$) and van Hove distributions (open triangles, ${\alpha }_{vH,2}$) measured via SPT (see Figures 2, 3). Error bars indicate 95% confidence intervals of fits. (G) Stretching exponent $\gamma (q)$, averaged over all $q$ values, for each composite ${\varphi }_A$, with error bars indicating standard error.