Persistence-Speed Coupling Enhances the Search Efficiency of Migrating Immune Cells

Migration of immune cells within the human body allows them to fulfill their main function of detecting pathogens. We present experimental evidence showing the optimality of the search strategy of these cells, which is of crucial importance to achieve an efficient immune response. We find that the speed and directional persistence of migrating dendritic cells in our in vitro experiments are highly correlated, which enables them to reduce their search time. We introduce theoretically a new class of random search optimization problems by minimizing the mean first-passage time (MFPT) with respect to the strength of the coupling between influential parameters. We derive an analytical expression for the MFPT in a confined geometry and verify that the correlated motion enhances the search efficiency if the mean persistence length is sufficiently shorter than the confinement size. Our correlated search optimization approach provides an efficient searching recipe and predictive power in a broad range of correlated stochastic processes.

Figure 1

(a) Sample cell trajectory, color coded with respect to instantaneous migration speed $v$. (b) Mean persistence length over speed bins, $⟨{\ell }_p{⟩}_v$, versus mean speed in each bin. Red (gray) data represent speed binning intervals of $1.0(0.1)\mu \mathrm{m}/\min$. The dashed line shows the fit via Eq. (1). Inset: Log-lin plot of $⟨{\ell }_p{⟩}_v$ vs $⟨v⟩$. (c) Comparison between the conditional MFPT ${\tau }^{*}$ of two categories of cells with low and high $p\text{−}v$ correlations but similar mean persistence lengths.

Figure 2

(a) Sketch of the correlated persistent random search on a square lattice. (b) MFPT $\tau$, obtained from numerical simulations of a constant speed ($⟨v⟩=1$) and uncorrelated $p\text{−}v$ process, vs ${\ell }_p$ normalized by $L$ ($L=200$). $\tau$ is scaled by ${\tau }_o$ of an ordinary random walk; the ratio of their asymptotic diffusion coefficient is $D/D_o=[(1+p)/(1-p)]$. (c) MFPT of a correlated $p\text{−}v$ process, scaled by the MFPT of an uncorrelated search (i.e., ${\kappa }_{pv}=0$), vs the coupling strength ${\kappa }_{pv}$. Each curve belongs to a different regime of $⟨{\ell }_p⟩$ shown in (b). The symbols are simulation results, and the dashed lines represent analytical predictions via Eq. (5) using a persistence extracted from Eq. (6). Insets: Schematics of trajectories with the same $⟨{\ell }_p⟩$ but different ${\kappa }_{pv}$. (d) MFPT scaled by the optimal search time of the uncorrelated process vs ${\kappa }_{pv}$ and scaled $⟨{\ell }_p⟩$. Insets: $\tau$ vs ${\kappa }_{pv}$ in the extreme regime $⟨{\ell }_p⟩\to L$ (top, linear scales) and $⟨{\ell }_p⟩\to 0$ (bottom, logarithmic scales).

Figure 3

(a) Example of correlated $(v,{\ell }_p)$ pairs generated in simulations with the coupling strength ${\kappa }_{pv}=0.8$, drawn from a distribution with $⟨{\ell }_p⟩=30$ (dashed line) and width $\mathrm{\Delta }=1$. The solid line represents the $p\text{−}v$ coupling according to Eq. (6). $L=200$. (b) Influence of the distribution width $\mathrm{\Delta }$ on the MFPT as a function of the coupling strength ${\kappa }_{pv}$.