Figure 1.
Probability distributions, P(r(t)), of displacements, r(t), at various times, t, are distorted by global drift (lines). To remove drift, we calculate adjusted probability distributions (circles) by removing the average motion of passive autofluorescent particles. The adjusted and unadjusted probability distributions differ significantly, especially at late times.
Figure 2.
Testing measures of anisotropy and directionality.
Measures of anisotropy as a function of (A) global drift velocity vd, (B) ratio, D∥/D⊥, of motility coefficients in different directions, and (C) chemotaxis velocity, vc, towards a central point. In general, when the anisotropy is small (vd ≈ 0, vc ≈ 0, and D∥/D⊥ ≈ 1), the anisotropy measures are small too. The directional order parameter, ϕ, measured with non-overlapping time intervals of 5 minutes, (solid black circles, left axes), is non-zero for vd ≳ 1 μm/min (panel (A)) but is zero or very small otherwise (panels (B) and (C)). The ratios of eigenvalues, λ1/λ2 (open symbols, right axes), averaged over all individual track inertia tensors, In (open red squares) or derived from the average of inertia tensor, (open blue diamonds), and asphericity (solid green triangles, left axes) are non-zero for sufficiently large vd (panel (A)) and D∥/D⊥ that deviates sufficiently from 1 (panel (B)). Only the average eigenvalue ratio over individual track inertia tensors and asphericity are non-zero for (panel (C)). (D) The ratio, 〈λ1〉/〈λ2〉, of average eigenvalues of individual track moment of inertia tensors (open red squares, right axes) and asphericity, A (solid green triangles, left axes), for an isotropic persistent random walk is large at early times, but decays as a function of time. The red dashed line is a guide to the eye. Error bars show standard error of the mean (SEM).
Table 1.
Summary of measures of directionality.
The four quantities described in section “Identifying directional motion” can be used to identify several types of anisotropy. Together, they can be used to identify each of the three types of anisotropy discussed.
Figure 3.
(A) The experimentally observed distributions of eigenvalues, λ1 and λ2, for individual track moment of inertia tensors, In (black and red solid lines, respectively). The distributions of track eigenvalues for the two populations walk model agree well (dashed lines) with experiment. (B) Drift-corrected probability distributions, P(r(t)), of T cell displacements, r(t), at various times, t (circles) and probability distributions of simulated cell displacements (solid lines). Inset: The scaled distributions, , of ρ = r/ζ, of T cell displacements fall on one curve (circles) at small ρ. This is clearly not a single Gaussian distribution (dashed line). Colored dotted lines in the inset are a guide for the eye to more clearly see systematic deviations from perfect scaling collapse at large ρ. (C) The scaling factor, ζ(t) (black circles), and RMSD (red squares) as a function of time. Initially, these grow faster than , but at intermediate to late times, they approach the scalings expected for long-time diffusive behavior. The slopes are different on the log-log plot, indicating that ζ(t) and the RMSD have different time dependences and are not proportional to each other. Inset: The MSD of walkers in the model (line) agrees well with the experimentally observed MSD (circles). (D) Normalized correlations, C(t1, t2;τ), between the displacements and (which occur over the time intervals t1 < t < t1+τ and t2 < t < t2+τ, respectively), plotted as a function of t2 – t1. C(t1, t2;τ) for T cells (circles) initially drops sharply, and then decays as a straight line on the semi-logarithmic plot, indicating exponential decay. At long time intervals, t2 – t1, there are deviations from exponential decay. The model (line) agrees qualitatively. Although the magnitude of normalized correlations is greater, the correlations have the same functional dependence and decay with the same persistence time. Error bars in (C) and (D) show SEM. (E) Experimentally observed instantaneous speed distribution, P(v) (circles). This is estimated by the distance traveled over 20 s, the time between imaging frames. Dashed line is an exponential decay, , where v0 = 5.63 μm/min. (F) Simulated trajectories for walkers in our model projected into 2D. (G) Experimentally observed trajectories of T cells in uninflamed lymph nodes.