Optimization and universality of Brownian search in a basic model of quenched heterogeneous media

The kinetics of a variety of transport-controlled processes can be reduced to the problem of determining the mean time needed to arrive at a given location for the first time, the so-called mean first-passage time (MFPT) problem. The occurrence of occasional large jumps or intermittent patterns combining various types of motion are known to outperform the standard random walk with respect to the MFPT, by reducing oversampling of space. Here we show that a regular but spatially heterogeneous random walk can significantly and universally enhance the search in any spatial dimension. In a generic minimal model we consider a spherically symmetric system comprising two concentric regions with piecewise constant diffusivity. The MFPT is analyzed under the constraint of conserved average dynamics, that is, the spatially averaged diffusivity is kept constant. Our analytical calculations and extensive numerical simulations demonstrate the existence of an optimal heterogeneity minimizing the MFPT to the target. We prove that the MFPT for a random walk is completely dominated by what we term direct trajectories towards the target and reveal a remarkable universality of the spatially heterogeneous search with respect to target size and system dimensionality. In contrast to intermittent strategies, which are most profitable in low spatial dimensions, the spatially inhomogeneous search performs best in higher dimensions. Discussing our results alongside recent experiments on single-particle tracking in living cells, we argue that the observed spatial heterogeneity may be beneficial for cellular signaling processes.

Figure 1

Schematic of the model system: (a) A spherical target with radius $a$ is placed in the center of a spherical domain of radius $R$. The free space between the radii $a$ and $R$ is divided into two regions denoted by subscripts. The inner region is bounded by a shell at radius $r_{\mathrm{I}}$. The outer region ranges from $r_{\mathrm{I}}$ to the reflective boundary at $R$. Initially, the particle's starting position is uniformly distributed over the surface of the sphere with radius $r_0$. (b) Microscopic picture of the problem starting from a discrete random walk between spherical shells. The hopping rates are assumed to obey detailed balance and the interface position is chosen to be placed symmetrically between two concentric spherical surfaces.

Figure 2

Schematic of the equivalence of MFPTs in inhomogeneous and homogeneous systems in the case of (a) a searcher starting in the inner region and (b) a searcher starting in the outer region. The radius $x_1$ of the initial particle position is shown by the thin dashed circle. We show direct trajectories as solid lines. Each panel also contains an indirect trajectory (dashed line) that leads the particle into the outer region of the system. As our analysis shows direct trajectories dominate the MFPT.

Figure 3

Ratio $\theta \left(x_0\right)=\mathbf{T}\left(x_0\right)/{\mathbf{T}}^0\left(x_0\right)$ as a function of $\phi =D_1/D_2$ in dimensions (a) $d=1$, (b) $d=2$, and (c) $d=3$ for initial radii $x_0=0.3$ (dashed lines) and $x_0=0.8$ (solid lines), respectively. In (b) and (c) we plot the results for a target size $x_a=0.1$. The solid black lines denote the $\simeq 1/\phi$ and $\simeq \phi$ scalings, respectively, and the dashed black line corresponds to $\simeq \chi \left(x_{\mathrm{I}}\right)$.

Figure 4

Optimal heterogeneity as a function of the starting position $x_0$ for three interface positions $x_{\mathrm{I}}=0.25$ (blue), $x_{\mathrm{I}}=0.5$ (red), and $x_{\mathrm{I}}=0.75$ (orange) in dimensions (a) $d=1$, (b) $d=2$, and (c) $d=3$. The target sizes in (b) and (c) are $x_a=0.1$ (solid lines) and $x_a=0.2$ (dashed lines). Also shown is the ratio $\theta \left(x_0\right)=\mathbf{T}\left(x_0\right)/{\mathbf{T}}^0\left(x_0\right)$ for the optimal heterogeneity ${\phi }^{*}$ in dimensions (d) $d=1$, (e) $d=2$, and (f) $d=3$. In (e) and (f) the target size is $x_a=0.1$.

Figure 5

Ratio $\overline{\theta }=\overline{\mathbf{T}}/{\overline{\mathbf{T}}}^0$ as a function of $\phi =D_1/D_2$ for different dimensions and interface positions $x_{\mathrm{I}}=0.4$ (blue) and $x_{\mathrm{I}}=0.6$ (orange). The target sizes in (b) and (c) are $x_a=0.1$ (solid lines) and $x_a=0.2$ (dashed lines). (d)–(f) Optimal heterogeneity for different dimensions as a function of the interface position $x_{\mathrm{I}}$. In (e) and (f) the target sizes are $x_a=0.05$ (blue), $x_a=0.1$ (red), and $x_a=0.2$ (orange). The insets in (d)–(f) show $\overline{\theta }$ for the optimal heterogeneity ${\overline{\phi }}^{*}$.

Figure 6

Ratio $\{\theta \}=\{\mathbf{T}\}/\mathbf{T}\left(x_0\right)$ as a function of $\phi =D_1/D_2$ for different dimensions and $x_0=0.4$ (blue) and $x_0=0.6$ (orange). The target sizes in (b) and (c) are $x_a=0.1$ (solid lines) and $x_a=0.2$ (dashed lines). (d)–(f) Optimal heterogeneity for different dimensions as a function of the starting position $x_0$. In (e) and (f) the target sizes are $x_a=0.05$ (blue), $x_a=0.1$ (red), and $x_a=0.2$ (orange). The insets in (d)–(f) show $\{\theta \}$ for the optimal heterogeneity $\{{\phi }^{*}\}$.

Figure 7

(a) Ratio $\{\overline{\theta }\}=\{\overline{\mathbf{T}}\}/\overline{\mathbf{T}}$ as a function of $\phi =D_1/D_2$ for different dimensions and target sizes $x_a=0.1$ (solid lines) and $x_a=0.01$ (dashed lines). (b) Optimal heterogeneity as a function of the target size. (c) $\{\overline{\theta }\}$ for the optimal strategy as a function of the target size.

Figure 8

Ratio $\theta =\mathbf{T}\left(x_0\right)/{\mathbf{T}}^0\left(x_0\right)$ as a function of $x_{\mathrm{I}}$ for $x_0=0.6$ and $x_a=0.1$ in different dimensions. The colors depict results for various heterogeneities: $\phi =0.6$ (blue), 2 (red), 10 (green), and 150 (orange). The symbols are results of simulations and the solid lines correspond to the analytical results in Eqs. (12).

Figure 9

Ratio $\overline{\theta }=\overline{\mathbf{T}}/{\overline{\mathbf{T}}}^0$ as a function of $x_{\mathrm{I}}$ for $x_a=0.1$ in different dimensions. The colors depict results for various heterogeneities: $\phi =0.5$ (blue), 2 (red), 10 (green), and 120 (orange). The symbols are results of simulations and the solid lines correspond to the analytical results in Eqs. (25).

Figure 10

Ratio $\{\theta \}=\{\mathbf{T}\}/\mathbf{T}\left(x_0\right)$ as a function of $x_0$ for $x_a=0.1$ in different dimensions. The colors depict results for various heterogeneities: $\phi =0.5$ (blue), 2 (red), 10 (green), and 120 (orange). The symbols are results of simulations and the solid lines correspond to the analytical results in Eqs. (20).

Figure 11

(a) Ratio $\{\overline{\theta }\}=\{\overline{\mathbf{T}}\}/\overline{\mathbf{T}}$ as a function of $\phi$ for different dimensions and target sizes. In dimensions (b) 2 and (c) 3 the target sizes correspond to 0.05 (blue), 0.1 (red), and 0.2 (green).