Modeling Leukocyte-Leukocyte Non-Contact Interactions in a Lymph Node

<div><p>The interaction among leukocytes is at the basis of the innate and adaptive immune-response and it is largely ascribed to direct cell-cell contacts. However, the exchange of a number of chemical stimuli (chemokines) allows also non-contact interaction during the immunological response. We want here to evaluate the extent of the effect of the non-contact interactions on the observed leukocyte-leukocyte kinematics and their interaction duration. To this aim we adopt a simplified mean field description inspired by the Keller-Segel chemotaxis model, of which we report an analytical solution suited for slowly varying sources of chemokines. Since our focus is on the non-contact interactions, leukocyte-leukocyte contact interactions are simulated only by means of a space dependent friction coefficient of the cells. The analytical solution of the Keller-Segel model is then taken as the basis of numerical simulations of interactions between leukocytes and their duration. The mean field interaction force that we derive has a time-space separable form and depends on the chemotaxis sensitivity parameter as well as on the chemokines diffusion coefficient and their degradation rate. All these parameters affect the distribution of the interaction durations. We draw a successful qualitative comparison between simulated data and sets of experimental data for DC-NK cells interaction duration and other kinematic parameters. Remarkably, the predicted percentage of the leukocyte-leukocyte interactions falls in the experimental range and depends (≅25% increase) upon the chemotactic parameter indicating a non-negligible direct effect of the non-contact interaction on the leukocyte interactions.</p></div

Distribution of the duration times of the interaction between NK and dendritic cells.

<p>Distribution of the duration times for the case of no chemotaxis (χ = 0, panel A) and of high chemotactic effect of the chemokines on the NK cells (, panel B). The results are obtained from the analysis of the interactions of 10 independent simulation runs (100 NK cells per simulation). The error bars are uncertainties obtained by 10 different simulations.</p

Flowchart of the mixed Monte Carlo-Brownian Dynamics algorithm for the simulation of the leukocyte-leukocyte non contact interactions.

<p>Flowchart of the mixed Monte Carlo-Brownian Dynamics algorithm for the simulation of the leukocyte-leukocyte non contact interactions.</p

Sketch of the NK cell motion.

<p>The NK cell is sketched as a red-orange circle: the brightness of the red color codes for increasing simulation time. Panel A: is the deviation angle sampled to implement for the Worm Like Chain model for the diffusion of the NK cell in the tissue. Monte Carlo algorithm for the cell-cell interactions: definition of the displacement parameter, Δd, used for the Metropolis test for the interaction onset. The symbols in panels B and C are: , and . Panel B reports an example of the <i>interaction_start</i> test. The root mean square displacement is . Depicted are two cases corresponding to and . In the first case <b>Eq. 17</b> is satisfied even if , in the second case <b>Eq. 17</b> is satisfied only if . Panel C reports an example of the <i>interaction_stop</i> test. In this case when <b>Eq.18</b> is satisfied even if while when <b>Eq.18</b> is satisfied only if .</p

Sketch representing the main features of the simulation algorithm.

<p>Panel A: implementation of the boundary conditions. When a NK cell (red dots) exits the simulation volume, another NK cell enters it from a random location. The dendritic cells are represented as fixed green circles and the solid arrow indicate the chemotactic force . Panel B sketches action at a distance acted by a dendritic cell that is simulated as a source of chemokines whose spread is represented by dashed circles and dotted arrows. The red lines represent the actual movement of NK cells that is largely determined by the Brownian component. Panel C represent the space dependent NK cell diffusion coefficient D_NK (<b>r</b>) normalized to the intrinsic NK cell diffusion coefficient,.</p

Result of the analysis of the simulated and experimental trajectories of NK cells.

<p>The simulated trajectories were obtained by assuming and were rebinned to reproduce the experimental time acquisition step (25′), as detailed in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0076756#pone.0076756.s001" target="_blank">File S1</a> (paragraph S4:“<i>Analysis of dendritic and NK cell interactions</i>”). The NK cell confinement ratio (C_R), the 3D speed (v_3D) and the trajectory length (L_traj) computed on more than 200 trajectories (NK cells) are reported in Panels A, B (confinement ratio), C, D (speed) and E, F (trajectory length). In each panel the labels “stimulated” and “un-stimulated” indicate the cases under stimulus (lipopolysaccharide injected i.v. in the mouse for the experimental data and for the simulated data) and under no stimulus (no lipopolysaccharide injected i.v. in the mouse for the experimental data and for the simulated data). Interacting and non-interacting NK cells were determined in the experimental and simulated data by the instantaneous NK cell velocity and the NK – dendritic cell distance, as described in the text. The horizontal bars indicate on each data set the average value.</p

Parameters used in the derivation of the NK-DC simulations.

<p>List of the parameters used in the derivation of the simulations of the NK and dendritic cells with an indication of the value and the equation in which the parameter was defined and/or used. When no number is given for the equation, the parameter was used for a direct estimate in the text.</p

Dependence of the duration times of the interaction between NK and dendritic cells on the degradation time.

<p>Distribution of the duration time of the NK – dendritic cell interaction, as a function of the chemokines' degradation time, k: k = 600 s (open black bars, black continuous best fit line); k = 900 s (densely hatched red bars and red open squares best fit line) and k = 1200 s (sparsely hatched green bars and green filled triangles best fit line). The chemotactic parameter was set to . The lines are best fit to the trial function. Best fit parameters are: A (600s)  = 803±28, t₁ (600s)  = 70±6s, B (600s)  = 78±2, t₀ (600s)  = 450±6 s, σ (600s) = 250±14 s for k = 600s; A (900s)  = 859±37, t₁ (900s)  = 70±6s, B (900s)  = 38±2.5, t₀ (900s)  = 570±30s, σ (900s)  = 380±40s for k = 900s and A (1200s)  = 980±70, t₁ (1200s)  = 66±6s, B (1200s)  = 38±3, t₀ (1200s)  = 630±200s, σ (1200s) = 550±90s.</p

Autocorrelation functions for the NK cells under constant chemotactic force.

<p>Panel A: square displacement . Panel B: the single step deviation angle . Panel C: the velocity autocorrelation function . The results are averaged over 100 NK cells. The simulation parameters are reported in the text (<b>Eq.5</b> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0076756#pone-0076756-t001" target="_blank"><b>Table 1</b></a>). The increasing values of the chemotactic parameter are χ = 0.1, 0.2, ….. 0.9, 1.0 .</p