Decision-making at a T-junction by gradient-sensing microscopic agents

Active navigation relies on effectively extracting information from the surrounding environment, and often features the tracking of gradients of a relevant signal—such as the concentration of molecules. Microfluidic networks of closed pathways pose the challenge of determining the shortest exit pathway, which involves the proper local decision-making at each bifurcating junction. Here, we focus on the basic decision faced at a T-junction by a microscopic particle, which orients among possible paths via its sensing of a diffusible substance's concentration. We study experimentally the navigation of colloidal particles following concentration gradients by diffusiophoresis. We treat the situation as a mean first passage time (MFPT) problem that unveils the important role of a separatrix in the concentration field to determine the statistics of path taking. Further, we use numerical experiments to study different strategies, including biomimetic ones such as run and tumble or Markovian chemotactic migration. The discontinuity in the MFPT at the junction makes it remarkably difficult for microscopic agents to follow the shortest path, irrespective of adopted navigation strategy. In contrast, increasing the size of the sensing agents improves the efficiency of short-path taking by harvesting information on a larger scale. It inspires the development of a run-and-whirl dynamics that takes advantage of the mathematical properties of harmonic functions to emulate particles beyond their own size.

Figure 1

(a) Sketch of the experimental setup. Continuous lateral flows set the boundary conditions and allow the development of a steady concentration map in the microfluidic network. Colloids (white particles) added to the lateral flow enter the network by diffusiophoresis, migrating towards the high concentration of salt. Engineered “hairy” colloids do not exhibit diffusiophoretic migration and allow one to ensure the absence of transversal flow as controlled by a pressure controller. The color map represents the steady concentration map of the Laplace equation $\mathrm{\Delta }c=0$. (b),(c) Curvilinear displacement of colloidal particles by diffusiophoresis. (b) In the absence of background salt, the time dependence of the curvilinear position is independent of the path (color for the paths is defined in the inset). The experimental results are in good agreement with the logarithmic sensing, $\mathbf{v}=D_{DP}\mathbf{\nabla }lnc$. (c) With the presence of background salt $\overline{c}$, the time dependence of the curvilinear position depends on the path (the color for the paths is defined in the inset). The speed on each segment is constant, consistent with the linear sensing, $\mathbf{v}=\frac{D_{DP}}{\overline{c}}\mathbf{\nabla }c$.

Figure 2

(a) Microfluidic network. Three different paths are coded with different colors, with red the shortest and blue the longest. (b) Histogram of exit time for particles traveling through the microfluidic network, where each path has the same color scheme as in (a). The histogram agrees well with the stochastic modeling (solid lines). (c) Numerical solution of the mean first passage time (MFPT) in the geometry of (a). The discontinuity in the MFPT is highlighted in the inset. (d) Discontinuity of MFPT is prescribed by a separatrix in the concentration field, whose position is set by the geometry and the relative gradients in paths exiting the T-junction.

Figure 3

(a) The efficiency of the path taking for particles with radius $R$ is improved due to the exclusion zone (shaded area) near the wall. A “run-and-whirl” navigation scheme (red trajectories) based on the mathematical properties of the analytic is proposed, which allows point particles to emulate the performance of particles with larger radii. (b) Path selectivity $P$ for different navigation strategies (different symbols). $f_2=1/c$ represents the phoretic strategy and $f_3={\mathbf{1}}_{c>\overline{c}}$ represents a strategy that has a nonsmooth dependence on $c$. Chemotactic migration is obtained by evolving a three-state system, run-and-tumble strategy and pure diffusion give roughly 50% selectivity at the junction. All strategies but the run and tumble and pure diffusion compare similarly and agree with our analytic prediction for particles of finite size (dashed line). The selectivity for point particles at the considered gradient is $P\left(0\right)=d_1/d=0.65$, with the critical particle size $R_c=d_2=0.35d$, beyond which the selectivity is 1.