Contact time periods in immunological synapse

This paper resolves the long standing debate as to the proper time scale $\langle \tau \rangle$ of the onset of the immunological synapse bond, the noncovalent chemical bond defining the immune pathways involving T cells and antigen presenting cells. Results from our model calculations show $\langle \tau \rangle$ to be of the order of seconds instead of minutes. Close to the linearly stable regime, we show that in between the two critical spatial thresholds defined by the integrin:ligand pair (${\Delta }_2\sim$ 40–45 nm) and the T-cell receptor TCR:peptide-major-histocompatibility-complex pMHC bond (${\Delta }_1\sim$ 14–15 nm), $\langle \tau \rangle$ grows monotonically with increasing coreceptor bond length separation $\delta$ (= ${\Delta }_2-{\Delta }_1\sim$ 26–30 nm) while $\langle \tau \rangle$ decays with ${\Delta }_1$ for fixed ${\Delta }_2$. The nonuniversal $\delta$-dependent power-law structure of the probability density function further explains why only the TCR:pMHC bond is a likely candidate to form a stable synapse.

Figure 1

$t_{\Delta }^{+}$ and $t_{\Delta }^{-}$ regions: The time evolution of the separation distance for a point in the membrane interface, $\varphi (\mathbf{x},t)$. The $t_{{\Delta }_i}^{+}$ regions are periods of time where close contact exists between the membrane surfaces at distances ${\Delta }_i$ ($i=1,2$) and during the $t_{{\Delta }_i}^{-}$ period the membrane separation distance is not favorable for ligand-receptor bond formation. The shaded regions indicate the time persistence [20] between two thresholds, ${\Delta }_1$ and ${\Delta }_2$.

Figure 2

Time persistence for different thresholds, $\langle {\tau }^{+}\rangle$ vs ${\Delta }_i$: The ensemble average for the time persistence above the ${\Delta }_i$ threshold. The dots show the result obtained by numerically solving Eq. (1) based on the scheme detailed in Eq. (11) while the solid line indicates the analytical result obtained from a solution of Eq. (10). The results have been linearly scaled for comparison.

Figure 3

Time persistence, $\langle {\tau }_{\delta }\rangle$ vs $\delta$: The ensemble average time persistence between two thresholds, where the distance between the thresholds is given by $\delta$.

Figure 4

Variation of the probability density function $p\left(\tau \right)$ with $\tau$ for $\delta =15$ nm between two thresholds for the ${\tau }_{11}$, ${\tau }_{12}$, ${\tau }_{21}$, and ${\tau }_{22}$ cases.

Figure 5

Variation of the probability density function $p\left({\tau }_{12}\right)$ with time ${\tau }_{12}$ between thresholds ${\Delta }_1$ and ${\Delta }_2$ for a range of $\delta$ values. As expected, the smallest $\delta$ (=15 nm) turns up the highest peak (represented by crosses), followed by $\delta =20$ nm (squares), with $\delta =25$ nm showing the lowest peak with the maximum spread (solid squares).