Correcting for time-dependent overall drift.

A major problem in live tissue experiments is uncontrolled global drift due to factors related to sample preparation, including fluid flow, sample movement, and respiration [2, 4, 15]. This background drift introduces experimental artifacts that may lead one to overestimate cell mobility or incorrectly conclude that there is globally directed migration. It has previously been proposed that this drift can be measured by monitoring static structures within the tissue [15]. However, this method cannot account for issues such as sample movement and animal respiration. Instead, drift can be measured by monitoring passive auto-fluorescent particles, which should be nearly immobile. Any motion of these particles should be due to either the background drift or random Brownian-like motion. This phenomenon can be used to quantify drift and correct raw data sets which might have been otherwise unusable.

We calculate a single average trajectory (position as a function of time) from several (typically four, although more could be used) tracked particles in order to minimize the contribution of the random part of their motion to the estimate of global drift. These particles are assumed to move, on average, only as a result of global drift. The total contribution of global drift to cell migration at a given time is thus given by the average passive particle displacement at that time. By subtracting the passive particle displacement from the measured cell displacements, it is possible to obtain drift-corrected cell trajectories and displacements. The actual cell trajectories are more accurately given by these adjusted trajectories. Overall, this correction can lead to a significant shift in the computed trajectories of cells and overall migration statistics. This difference is shown, for example, by the histograms of unadjusted cell displacements (circles) and drift-corrected cell displacements (lines) for the T cells in Fig. 1. Although the drift correction, which is of order μm/min, is small over short observation periods (red in Fig. 1), it noticeably shifts the distributions after several minutes (green and purple). Similarly, other quantities describing cell migration are affected by the drift. All further analyses are performed on these drift-corrected data.

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Figure 1. Correcting for drift.

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.

https://doi.org/10.1371/journal.pcbi.1004058.g001

Testing for global directionality.

Once corrections for global drift have been estimated, we look for directionality in the migration tracks by measuring the directional order parameter, ϕ, which has been used to describe the collective motion of animals [25, 26]. ϕ is calculated by computing the average of the normalized velocity vectors (whose components can take on positive or negative values), $\vec{u}=⟨\vec{v}/\left|\vec{v}\right|⟩$ (where $\vec{v}$ is the velocity vector) and measuring the magnitude of the resulting vector, so that $\varphi =\left|\vec{u}\right|$. If there is a global drift or bias in one direction, there should be a non-zero average velocity and a non-zero value of ϕ.

The order parameter ϕ is complementary to the mean velocity (or displacement) vector, $⟨\vec{v}⟩$ ($⟨\vec{r}⟩=⟨\vec{v}⟩\Delta t$), because ϕ measures only angular direction. In some cases, this may be advantageous since variability in cell speeds contributes an additional component to the error in measuring the velocity vector $⟨\vec{v}⟩$, and may obscure directionality. Furthermore, the magnitudes of cell velocities effectively weight some cells more heavily in directionality measurements using $⟨\vec{v}⟩$; for directionality measurements with ϕ, all cells are weighted equally in the average. In addition, the mean velocity vector may not clearly detect drift along arbitrary combinations of the x, y, and z axes. Nonetheless, the mean velocity vector remains a useful quantity, since it is a speed-weighted average, and could highlight interesting features that the order parameter neglects. Since the utility of $⟨\vec{v}⟩$ has already been demonstrated [5, 11], we present diagnostic results only for the directional order parameter, ϕ, below.

Since the contribution of diffusion may dominate over directed motion until long observation times [5, 27], measurements of ϕ may not be sensitive enough to detect biased motion in cell displacements that occur between just two imaging frames. However, the sensitivity can be amplified by measuring average velocities over a longer time segment rather than “instantaneous” velocity estimated by cellular displacements between adjacent time frames. However, since the duration of the experiment can be broken down into fewer long time segments than short time segments, the statistical error is higher for longer time segments; in addition, data from cells that leave the field of view in less time than the long time segment must be discarded, which can bias data (this issue is described in detail in the section “Analyzing displacement data”). One must therefore choose the length of the time segment to balance these considerations.

To demonstrate how to use the order parameter, we measure it for a series of numerical simulations of 5000 random walkers (simulated cells). The walkers diffuse with motility coefficient D = 30 μm²/min, similar to T cells in various experiments [1, 5, 28–30] and are biased to move in the +x direction with speed $\left|\vec{v}\right|$. As in our experiments, we track the simulated walkers in a 500 μm × 500 μm × 68 μm imaging field for 10 minutes (thus we track the approximately 1000 simulated walkers that remain in the field for at least two frames). The solid black circles in Fig. 2A show that the order parameter detects global drift as small as fractions of a micron per minute. For large bias, $\left|\vec{v}\right|$, the directional order parameter ϕ is large, indicating that many cellular movements have the same directionality. However, as the drift velocity decreases, the simulated walkers become more like pure Brownian walkers, and thus, ϕ decreases toward zero.

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Figure 2. Testing measures of anisotropy and directionality.

Measures of anisotropy as a function of (A) global drift velocity v_d, (B) ratio, D_∥/D_⊥, of motility coefficients in different directions, and (C) chemotaxis velocity, v_c, towards a central point. In general, when the anisotropy is small (v_d ≈ 0, v_c ≈ 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 v_d ≳ 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, I_n (open red squares) or derived from the average of inertia tensor, $\overline{I}$ (open blue diamonds), and asphericity (solid green triangles, left axes) are non-zero for sufficiently large v_d (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 $v_c\gtrsim 1\mu \text{m/min}$ (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).

https://doi.org/10.1371/journal.pcbi.1004058.g002

We now calculate ϕ for real data for CD8⁺ T cell tracks in the uninflamed lymph node. We find that ϕ for a given imaging experiment is typically more than a standard error of the mean away from zero, but nonetheless lies within a 95% confidence interval of zero for six of eight imaging series. Note that compared to data that has not been adjusted for overall drift, ϕ is decreased, on average, by about 50%. As a means of determining whether the detected bias is significant, we analyzed cell migration data after removing the components of motion directed along $\vec{u}$. We verified that removing the cellular motions along this direction did not significantly alter the measures of migration statistics of the full data set described below. Furthermore, the average magnitude, V = 0.80 ± 0.68 μm/min, of the mean velocity vector is small. Thus, the analysis of ϕ suggests that there is little global directional bias in CD8⁺ T cell tracks in the uninflamed lymph node over the time and volumes imaged.

Calculating the moment of inertia tensor for cell tracks to detect other anisotropies.

While the directional order parameter ϕ can reliably detect preferential motion in a given direction, it cannot detect other types of anisotropy. For instance, ϕ does not indicate differences in the motility coefficients in different directions (e.g., ϕ = 0 for D_x ≠ D_y; solid black circles in Fig. 2B) or directed motion towards a target (solid black circles in ϕ = 0 for a single target at the center of the image; Fig. 2C).

To reliably identify all types of anisotropy, we introduced the moment of inertia tensor, I, which can be used to detect various anisotropies in cellular migration [6, 31, 32]. I is a d × d matrix quantifying geometrical information about the d-dimensional spatial positions visited by the cells (see Methods section for details). In most imaging studies, cellular trajectories are artificially compressed in the z direction due to limited depth of view. Thus, we restrict the analysis to x and y positional information and construct the two-dimensional moment of inertia tensor. Nonetheless, this procedure can easily be extended to analyze the three-dimensional moment of inertia tensor. By calculating the characteristic values (eigenvalues), λ₁ and λ₂, of the matrix I (see Methods section and e.g., refs. [31, 32]), it can be determined whether cells prefer to move along a particular spatial axis (x, y, or some combination). If the characteristic values are equal (λ₁ = λ₂), cell migration is isotropic — every spatial direction is the same. However, if one characteristic value is larger than the other (λ₁ > λ₂; by convention, λ₁ ≥ λ₂), this indicates greater cell migration along a particular axis, which is given by the corresponding eigenvector (see [31, 32]). The ratio, λ₁/λ₂, quantifies the amount of anisotropy. As we describe below, an average moment of inertia tensor, $\overline{I}$, can be calculated for average cell motions, and a set of individual moment of inertia tensors, I_n, can be calculated for the motions of individual cells. These two calculations provide complementary information about migration statistics.

The average moment of inertia tensor, $\overline{I}$, over all cellular motions quantifies the geometry of the average of all cell trajectories. By calculating the eigenvalues of this quantity, we can determine whether there is a particular axis along which all cells, on average, prefer to move. Similar to the directional order parameter ϕ, the ratio, ${\overline{\lambda }}_1/{\overline{\lambda }}_2$, of eigenvalues of $\overline{I}$ can be used to detect global drift in numerical simulations (open blue diamonds in Fig. 2A), albeit with slightly lower sensitivity. However, unlike ϕ, the ratio ${\overline{\lambda }}_1/{\overline{\lambda }}_2$ can be used to detect differences in motility coefficients of diffusive motions in different directions. To show this, we simulated walkers where diffusion in one direction occurs with motility coefficient D_∥, which is varied (e.g., we choose the x direction). Diffusion in directions perpendicular to the first (the y and z directions) occurs with a fixed motility coefficient D_⊥ = 10 μm²/min. As shown by the open blue diamonds in Fig. 2B, the ratio ${\overline{\lambda }}_1/{\overline{\lambda }}_2$ grows large for D_∥ noticeably different from D_⊥ (D_∥/D_⊥ ≠ 1).

Neither the average inertia tensor $\overline{I}$ nor the directional order parameter ϕ can be used to identify motion toward a central target (open blue diamonds and solid black circles, respectively, in Fig. 2C). However, by calculating the moment of inertia tensors for individual tracks, I_n, and averaging their eigenvalues to find 〈λ₁〉 and 〈λ₂〉, we can identify motion toward a central target. The open red squares in Fig. 2C show that 〈λ₁〉/〈λ₂〉 can be used to detect chemotaxis toward a central target at velocities of order μm/min or larger. Although Fig. 2C demonstrates the efficacy of using I_n to detect motion towards a known central target, the set of individual track moment of inertia tensors can be used to detect chemotactic motion toward a target at an arbitrary location. Moreover, we emphasize that I_n (as well as the other quantities discussed in this section) can detect anisotropic motion without prior knowledge of the target location or even the existence of a special direction. As shown in Fig. 2A-B, this ratio can also be used to detect other types of anisotropy.

Thus, in conjunction with the order parameter ϕ, the moment of inertia tensor analysis can detect and distinguish between three kinds of anisotropy — global drift, direction-dependent diffusion, and motion toward a single point. These quantities can be used to compare different migration data sets, and the distributions of track eigenvalues, λ₁ and λ₂, provide a constraint for model selection. Table 1 lists the ability of each of these quantities to identify the three types of anisotropy discussed.

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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.

https://doi.org/10.1371/journal.pcbi.1004058.t001

From the moment of inertia tensor and eigenvalues, we calculate the mean asphericity, A, of the tracks [33]. This quantity, defined as $A=\frac{\langle (1/2){({\lambda }_1+{\lambda }_2)}^2-2{\lambda }_1{\lambda }_2)\rangle }{\langle (1/2){({\lambda }_1+{\lambda }_2)}^2\rangle }$ in two dimensions, measures the deviation of the tracks from spherical symmetry; asphericity ranges from A = 0 if tracks are perfectly spherical (circular in two dimensions) to A = 1 for totally elongated tracks. While asphericity, A, does not provide detailed information about the different principal axes and is less precise than the other described measures (e.g., notice the larger error bars in Fig. 2), it does allow the comparison of the measured asphericity to the theoretical value of A = 4/7 ( ≈ 0.57) for a two-dimensional random walk [33].

To determine whether CD8⁺ T cells in lymph nodes in the absence of inflammation exhibit directional motion, we apply this analysis to the experimental data. The asphericity, A = 0.73 ± 0.28, and the ratio of average individual track eigenvalues, 〈λ₁〉/〈λ₂〉 = 28.6 ± 7.1, are both larger than expected for an isotropic random walk. Nonetheless, 〈λ₁〉/〈λ₂〉 is not very large (compare to values at small anisotropy in Fig. 2A-C) and due to the large error in A, asphericity alone does not provide definitive information. The eigenvalues, ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$, of the total inertia tensor, $\overline{I}$, for each movie are typically the same order of magnitude, but differ due to statistical noise in the cell tracks.

The fact that the observed ratio, 〈λ₁〉/〈λ₂〉, and asphericity, A, are larger than expected for Brownian random walks might be perceived as indicative of anisotropic walks. But, it is important to note that (1) the observed walks are necessarily rather short due to cells leaving the field of view through the shallow z direction and (2) the walks may be isotropic and random but not Brownian. Both of these factors will give rise to apparent anisotropy at short time scales. Moreover, nearly all migrating cells move unidirectionally on short enough time scales [15, 16], and some move persistently for several minutes or longer [5, 6, 19, 21].

To demonstrate this principle, we simulate 5000 persistent random walkers with representative values of motility coefficient, D = 29 μm/min, and persistence time, t_p = 81 s [1, 5, 19, 21, 28–30] in a finite volume (note that the values of D and t_p were calculated in a simulated infinite field of view). Although the model is isotropic so that cells do not have a preferred direction, many individual cell tracks are nearly unidirectional on the experimental time scale. This is reflected by the measurements of 〈λ₁〉/〈λ₂〉 and A, which are large for short observation periods, but decreases to smaller values for simulated experiments lasting more than 15 minutes (Fig. 2D). This example highlights the difficulty in interpreting the described quantities even if they indicate that anisotropic motion is present. Before concluding that cell migrate directionally, one must rule out alternative sources of apparent directionality, such as the limited duration of the experiment. For the case of CD8⁺ T cells, our simulations suggest that the observed values of 〈λ₁〉/〈λ₂〉 and A are not the result of directional migration. Rather, as described below, they are expected from our non-directional migration model with a limited experimental observation time.

Individually, each of these approaches have the ability to detect different types of anisotropic migration, but they must be used together in order to properly classify anisotropy. Note that once anisotropy is detected, experimental artifacts must be ruled out. This is most easily done within the context of a random walk model that fits the displacement data, as discussed below. One can then calculate the ratio 〈λ₁〉/〈λ₂〉 within the model for tracks of the time duration of the experiment, and compare the expected anisotropy from the model to the observed anisotropy from the experiment to see whether there is a significant difference. In the case of T cells in the lymph node in the absence of inflammation, we find that the distributions of λ₁ and λ₂ agree reasonably well with expectations for the isotropic non-Brownian random walk model presented below (Fig. 3A). Furthermore, the eigenvalues, ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$, of the total inertia tensor, $\overline{I}$, for each imaging series are typically the same order of magnitude, but differ due to statistical noise in the cell tracks. Altogether, these measurements suggest that CD8⁺ T cell migration in lymph nodes during the steady state is globally isotropic on the time and length scales of the experiment performed.

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Figure 3. Analysis of T cell migration.

(A) The experimentally observed distributions of eigenvalues, λ₁ and λ₂, for individual track moment of inertia tensors, I_n (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, $\tilde{P}(\rho )$, 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 $\sqrt{t}$, 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 $\vec{r}(t_1,t_1+\tau )$ and $\vec{r}(t_2,t_2+\tau )$ (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, $\propto e^{-v/v_0}$, 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.

https://doi.org/10.1371/journal.pcbi.1004058.g003

Summary of directionality analysis.

Traditionally, directionality is identified qualitatively by visualization of the cell tracks and quantitatively by measuring the average drift velocity, mean-squared displacement exponent, and comparing meandering indices across data sets. However, these methods are insensitive to some of the types of directionality described above. Moreover, quantities such as the MSD are biased by the finite imaging field, while others, such as the meandering index, are not robust across different experiments. In contrast, the methods we have described are more general and less affected by these limitations and they do not require prior knowledge of the target location or the axis of directional motion. When used together, the quantities described in this section can be used to identify several types of anisotropic motion; these results are summarized in Table 1. Moreover, the methods described directly measure the shape of cell tracks instead of a particular aspect of directional motion. Thus, these quantities, especially the individual track moment of inertia tensors, I_n, can also describe non-directional motion and provide constraints for migration model selection. Using these measurements, we found that migration of CD8⁺ T cells in lymph nodes is globally isotropic on the time and length scales of this experiment.

Displacement probability distributions.

The probability distribution P_Δt(r) is the fundamental quantity underlying the migration statistics, and in principle, all of the statistical properties of migration, including directional biases, mean-squared displacement, and correlations, can be obtained from it. However, in order to determine the best way to construct P_Δt(r), it is useful to have an idea in advance of whether directional bias is present. For example, if there is overall motion in the x-direction, it is useful to focus on the x-displacements and to average over the y- and z-displacements, while if there is overall motion towards a central target, it is useful to focus on the distribution of radial displacements.

One can construct P_Δt(r) by plotting the histogram of cell displacements (or components of displacements), r, that occur over a time interval, Δt, and normalizing by the total number, N, of displacements measured. As shown by animal migration studies, low resolution and linear or logarithmic binning methods can bias the data, leading to spurious results [37–39]. In order to avoid these artifacts and capture the true shape of P_Δt(r), we collect a large amount of data (several hundred tracks), place a fixed number of displacements in each bin, and set the bin widths according to the composition of each bin [6]. Specifically, we put m displacements in each bin, so that there will be N/m bins. The i^th bin is centered at the average over all m displacements in that bin and the bin is assigned a weight of $W_i=1/(r_{\text{max}}^{(i)}-r_{\text{min}}^{(i)})$, where $r^{{(i)}_{\mathrm{\text{max}}}}$ and $r^{{(i)}_{\text{min}}}$ are, respectively, the maximum and minimum displacements in the i^th bin. The weighting procedure normalizes P_Δt(r), and more importantly, accounts for the variable width of bins, ensuring that the constructed P_Δt(r) accurately reflects the probability density function.

The effect of statistical errors can be evaluated by varying the number of points per bin and the number of bins in the histogram for both the complete set and subsets of the displacement data [6]. For a sufficiently large data set (typically ≳ 1000 displacements) errors in the positions and weights of the data points in the middle of P_Δt(r) are on the order of a few percent. Errors in bin locations and bin weights for, respectively, the smallest and largest displacement bins are significantly larger, but these errors do not affect the observed shape of P_Δt(r) and have a negligible effect on the ensuing analysis [6].

Statistical accuracy of the distribution can be improved by utilizing symmetries within the data. Since CD8⁺ T cell migration is isotropic, we binned the x-, y-, and z-components of displacements together for the displacement distributions in Fig. 3B (circles). For cases where T cell migration is not isotropic, x, y, and z displacements should be binned separately and displacements in the positive direction (+x, +y, or +z) should be distinguished from displacements in the negative (−x, −y, or −z) direction.

For example, consider N = 10,065 cell displacements occurring over all Δt = 2.7 minute time intervals in the experiment. To construct the distribution of cell displacements over this time interval (green circles in Fig. 3B), we sorted the displacements in ascending order and constructed histograms with m = 700 displacements the first 14 bins and put the remaining 265 displacements in the final bin at large displacements, r. Each bin is centered at the average displacement for the bin, and its probability P, is m/N divided by the width of the bin (maximum minus minimum displacement in the bin). This process was repeated for Δt = 0.33, 1.0, 2.7, 4.3, and 9.0 minutes using m = 850,800,700,600, and 330 counts per bin, respectively (Fig. 3B). These data serve as the foundation for the analysis of CD8⁺ T cell migration.

By constructing P_Δt(r) for multiple time intervals, Δt, migration statistics can be characterized over the entire time course of the experiment. In Fig. 3B, histograms of cell motions over Δt = 0.33,1.0,2.7,4.3, and 9.0 minutes are shown. As expected, the distributions broaden with increasing Δt, because cells can move further over larger Δt. However, the overall shape of P_Δt(r) at all observed times is broader than Gaussian.

To investigate the possibility of non-Gaussian scaling behavior, we scaled the probability distributions by a time-dependent scaling factor, ζ(Δt), and plotted probability distributions, ${\tilde{P}}_{\Delta t}(\rho )=\zeta (\Delta t)P_{\Delta t}(r)$, of scaled displacements, ρ = r/ζ(Δt) (inset to Fig. 3B). This factor is chosen for each time, Δt, so that the rescaled probability distribution, ${\tilde{P}}_{\Delta t}(\rho )$, for that time, will lie on top of the other rescaled distributions. Often, this factor can be chosen by numerically fitting P_Δt(r) to a distribution function with a known scaling factor. However, for the data in Fig. 3B, we chose ζ(Δt) so that the average ${\tilde{P}}_{\Delta t}(\rho )$ for the three smallest ρ in the histogram is equal to one. The qualitative behavior observed in the inset to Fig. 3B is not sensitive to the precise details of the procedure for choosing ζ(Δt). In general, it may not be possible to perfectly collapse the probability distributions in this manner; thus, the scalability of the distributions provides useful information about the underlying migration statistics.

For CD8⁺ T cell migration statistics, the scaling factor, ζ(t), is not proportional to the root-MSD (RMSD), as would be expected for purely Gaussian behavior (Fig. 3C). Instead, ζ(t) (black circles in Fig. 3C) and the RMSD (red squares) have different time dependences. In addition, the distributions, ${\tilde{P}}_{\Delta t}(\rho )$, are not Gaussian (dashed line in the inset to Fig. 3B), indicating that migration is not Brownian-like over these time scales. However, although the collapsed distributions approximately fall on a single curve for small scaled displacements (ρ ≲ 1), deviations from the collapse at large values of ρ are systematic, so migration is also not Lévy-like on these time scales [6]. Intriguingly, the deviations from perfect collapse also indicate behavior that increasingly deviates from a single Gaussian distribution over time. From the central limit theorem, one would expect behavior to converge towards a Gaussian distribution at long times for a random walk whose step size is bounded, so the opposing trend apparent in the experimental data suggests that longer steps occur with greater frequency than expected.

Correlations between displacements.

Four-point time correlation functions quantify persistence and memory in cell migration and provide an additional constraint for model selection [6, 40]. In particular, it is useful to measure the statistical relation between a quantity, such as cellular displacement, $\Delta \vec{r}(t_1,t_1+\tau )$, between images taken at t₁ and t₁+τ, and another quantity, such as a later cellular displacement, $\Delta \vec{r}(t_2,t_2+\tau )$ between times, t₂ and t₂+τ. Thus, in contrast to the meandering index [15], such correlations can precisely measure the distance and duration of straight runs before turns or stalls [40]. Moreover, unlike the meandering index, the overall shape of the correlation function does not depend on the duration of the experiment or the time interval, τ, between imaging frames (provided τ < τ_p, where τ_p is the persistence time).

Correlation functions may also provide additional evidence or quantification of anisotropic or directional motion. This can clarify the interpretation of anisotropies identified by the measures in the section “Identifying directional motion.” For instance, the moment of inertia tensor might identify several distinct principal axes of motion. If this is due to different persistence times or lengths along these axes, then correlations between motions along these axes should differ.

Finally, it is in principle possible to investigate the response to different events, such as the introduction of a chemotactic signal, by investigating differences between correlation functions of cells at different times or locations.

To describe CD8⁺ T cell migration in the lymph node, we use the correlation function: (2) where $\Delta \vec{r}(t,t+\tau )$ is the vector change in position over the time interval starting at time t and ending at time t+τ. C(t₁, t₂;τ) measures the correlations between cell motions at different times, normalized by the average motion. The semi-logarithmic plot of the correlation function, C(t₁, t₂;τ), in Fig. 3D (circles), shows the decay of correlations over time. The correlation function decays primarily in a straight line on this scale, indicating that correlations typically drop off exponentially; thus, there is a well-defined apparent persistence time for C(t₁, t₂;τ), t_p ≈ 2 min (alternatively, a persistence length can be defined by ${\ell }_p\equiv \langle v\rangle t_p\approx 13\mu \text{m}$.

There are two features of the correlation function that differ from a pure exponential decay. First, there is a sharp decrease in correlations over the smallest time interval, indicating that there is a short-time decoherence in cellular motions. This could result from various factors such as small fluctuations in cell centers due to cell shape changes, short-time diffusive-like motions of cells, and experimental noise. Second, the correlation function C(t₁, t₂;τ) is nearly constant at late times, possibly indicating that a small percentage of cells move persistently for very long times. Any model designed to fully describe the behavior of T cells should take these factors into consideration.

Modeling cell migration.

The data presented in Fig. 3 provide strict constraints that a proposed model for CD8⁺ T cell migration must satisfy. Considering these factors together, a variety of multimodal persistent random walks and generalized Lévy walks were tested. Some of these models could generate the observed correlation function and/or the MSD, but none of the models tested could produce the displacement distributions in Fig. 3B (e.g., the run/pause model proposed in [21] can reproduce the MSD, but not the displacement distributions).

The model that describes all aspects of Fig. 3 was realized through the observation that each of the displacement distributions in Fig. 3B can be fit reasonably well using a double Gaussian fitting function, Q_Δt(r): (3) The fit parameters are the motility parameters, D₁ and D₂ and the weights, c₁ and c₂, of each of the Gaussians, where c₁ + c₂ = 1 for normalization of the entire probability distribution. The fit of Eq. 3 to the cell displacement distributions, P_Δt(r), at various times was found using Mathematica (Wolfram Research, Inc., Champaign, IL). All probability distributions, P_Δt(r), in Fig. 3B are well-described by c₁ = 0.3, c₂ = 0.7, D₁ = 0.2 μm²/min, and D₂ = 9 μm²/min.

The fact that the displacement distributions could be fit by a double Gaussian suggests a model with two distinct populations. Therefore we constructed a model in which one population, comprising c₁ = 30% of all walkers, migrates via Brownian walk with a three-dimensional motility coefficient of $D_1^{(3D)}=3D_1=0.6\mu {\text{m}}^{\text{2}}\text{/min}$. The other c₂ = 70% of walkers migrates through a persistent random walk. This second population is required for obtaining the long tails of the probability distributions, P(r(t)), and its persistence is required for the exponential decay of the correlation function, C(t₁, t₂;τ). We made the simplifying assumption that persistent walkers have the same average velocity 〈v〉 = 6.6 μm/min, the value measured by our experiments. The actual persistence time, τ_p = 3.5 min (≠ t_p, the apparent persistence time), for these walkers was determined through the relation $D_2^{(3D)}={\langle v\rangle }^2{\tau }_p/6$. In this model, these walkers move at a velocity, v, drawn from an exponential distribution with mean 〈v〉 for a time, t, drawn from an exponential distribution of mean τ_p. At the end of a run, the walker chooses a new v, t, and migration direction. This additional randomness adds variability to our model to more accurately mimic experimental data. Moreover, drawing speeds, v, from an exponential distribution mimics the observed instantaneous speed distribution, P(v), which decays exponentially for large speeds (Fig. 3E). Finally, in order to qualitatively capture the short-time decorrelation of cell motions, the second population of walkers also experiences a small amount of Brownian noise with diffusion coefficient D_noise = 0.6 μm²/min.

In these simulations, walkers were modeled in a finite imaging volume with dimensions 500 μm × 500 μm × 68 μm. Walkers could both enter and leave the field of view. As in the actual experiments, a walker only contributes to the migration statistics during the time interval in which it is in the imaging volume, which is typically a few minutes. Consequently, this model should reflect many of the limitations inherent in the actual experiment.

Indeed, the model reproduces all of the statistical measures shown in Fig. 3 reasonably well and simulated trajectories (Fig. 3F) are qualitatively similar to experimentally observed cell motions (Fig. 3G). The track moment of inertia eigenvalue distribution (dashed lines in Fig. 3A), walker displacement distributions (solid lines in Fig. 3B), and MSD (line in inset to 3C) closely match the experimentally observed statistical measures. The agreement between the model and data explains why the scaled distributions $\tilde{P}(\rho )$ (inset to Fig. 3B) do not collapse; clearly, a double Gaussian distribution cannot be fully characterized by a single time-dependent length scale, ζ(t). In addition, while the model does not precisely reproduce the correlation function, it captures the main qualitative features (line in Fig. 3D): marked decorrelation over short times and decay with an apparent persistence time of t_p ≈ 2 min. Other models for T cell migration neither quantitatively nor qualitatively reproduce these correlations (S1 Fig.). It is likely that a more complicated model could more accurately reproduce the correlation function. Such a model might include, for instance, features such as finely-tuned short-time decorrelation, a small third population of extremely persistent walkers, and cells that switch populations. However, our model has the advantage of describing the data while retaining relative simplicity.