T-cell motility in the early stages of the immune response modeled as a random walk amongst targets

The transport process by which a T cell makes high-frequency encounters with antigen-presenting cells following infection is an important element of adaptive immunity. Recent experimental work has allowed in vivo cell motility to be characterized in detail. On the basis of experimental data we develop a quantitative model for encounters between T cells and antigen-presenting cells. We model this as a transport-limited chemical reaction with the dynamics dependent on physical contact between randomly moving reactants. We use asymptotic methods to calculate a time distribution which characterizes the delay before a T cell is activated and use Monte Carlo simulations to verify the analysis. We find that the density of antigen-primed dendritic cells within the lymph node paracortex must be greater than $35\mathrm{cells}∕{\mathrm{mm}}^3$ for a T cell to have a more than $50\%$ chance of encountering a dendritic cell within $24\mathrm{h}$. This density is much larger than existing estimates based on calculations which neglect the transport process. We also use simulations to compare a T cell which re-orients isotropically with a T cell which turns according to an experimentally observed distribution and find that the effects of anisotropy on the solution are small.

Figure 1

Diagram showing the histological regions of a typical lymph node. Antigen enters through the afferent lymphatic vessels and is sequestered by DCs. T cells enter via the blood and survey antigen-bearing DCs in the paracortex.

Figure 2

The model geometry.

Figure 3

Plots of the survival distribution (37) for a variety of target radii and initial positions. The solution (37) is plotted to leading order (dashed line), to $O\left(\epsilon \right)$ (dotted line), and to $O\left({\epsilon }^2\right)$ (solid line). Shown are plots for (a) $\epsilon =1∕3$, $b=1∕3$, $r=\mid {\mathbf{x}}_0\mid =4∕3$, (b) $\epsilon =1∕3$, $b=2∕3$, $r=5∕3$, (c) $\epsilon =1∕3$, $b=1$, $r=2$, and (d) $\epsilon =1∕10$, $b=3$, $r=4$. The crosses in each graph show the survival distribution calculated from $7\times {10}^4$ Monte Carlo trials. The analytical solution becomes increasingly inaccurate as the ratio of the reaction radius to the step length, $a\lambda ∕s_t=b∕\epsilon$, decreases.

Figure 4

The survival probability for a single cell within a lymph node; with reaction radius $a=25\mathrm{\mu }\mathrm{m}$, and the DC concentration calculated supposing there are (i) $200$, (ii) $1000$, and (iii) $3000$ DCs uniformly distributed within a spherical paracortex of radius $2\mathrm{mm}$. The solid lines are calculated from $2\times {10}^4$ Monte Carlo trials, assuming the T cell re-orients isotropically; the dotted lines show the analytical solution from equations (7, 42, 43, 44). The dashed line is the Monte Carlo solution (for $1000$ targets, from $2\times {10}^4$ trials) for a walker which re-orients according to the empirical turning distribution measured by Mempel et al. [9] (shown as dots in Fig. 5).

Figure 5

A graph of data taken from Mempel et al. [9] showing the turning distributions of T cells within the paracortex of a lymph node. The two data sets correspond to the T cells turning in the presence (dots) and absence (crosses) of peptide-bearing antigen-presenting cells. The two solid lines are fifth-order polynomial fits, and the dotted line is the PDF which characterizes the distribution of the turning angle $\Theta$ for an isotropic random walk, $\mathrm{Prob}\{\theta <\Theta <\theta +d\theta \}=\frac{1}{2}\sin \theta d\theta$.

Figure 6

The mean (dash) and standard deviation (dot-dash) of displacements for a random walker which turn isotropically, compared with the mean (solid) and standard deviation (dot) of displacements for a random walker which turns according to the empirical distribution from Ref. 9 (shown as dots in Fig. 5). Each line is calculated from $2\times {10}^4$ trials, using parameter values appropriate for a T cell: $s_t=11\mathrm{\mu }\mathrm{m}{\min }^{-1}$, $\lambda =1∕2{\min }^{-1}$. The mean and standard deviation are larger for the persistent walk than for the isotropic walk.