Boundary homogenization for trapping patchy particles

Many systems in chemical and biological physics involve diffusing particles being trapped by absorbing patches on otherwise reflecting surfaces. Such systems are commonly studied by boundary homogenization, in which the heterogenous boundary condition on the patchy surface is replaced by a uniform boundary condition involving a single parameter which encapsulates the effective trapping properties of the surface. In prior works on boundary homogenization, the surface is patchy and the diffusing particles are homogeneous. In this paper, we consider the opposite scenario in which a homogenous surface traps patchy particles, which could model proteins with localized binding sites, cells with membrane receptors, or patchy colloids or nanoparticles. We derive an explicit formula for the effective trapping rate which reveals a fundamental interplay between the translational and rotational diffusivities of the patchy particle, a phenomenon not typically seen in boundary homogenization. Motivated by receptors on the cell membrane, our analysis also allows for the possibility that the patches diffuse on the surface of the particle. We formulate the system in terms of a high-dimensional, time-dependent, anisotropic diffusion equation and employ matched asymptotic analysis to derive the effective trapping rate. We confirm our results by detailed numerical simulations.

Figure 1

(a) A homogeneous particle and a patchy surface. The particle binds to the surface if it touches one of the red patches on the surface. (b) A patchy particle and a homogeneous surface. The particle binds to the surface if one of its patches touches the surface. In both (a) and (b), the particle has translational diffusivity $D_{\mathrm{tr}}$. In (b), the particle's rotational diffusivity $D_{\mathrm{rot}}$ plays a crucial role in binding. Further, the patches in (b) diffuse on the surface of the particle with diffusivity $D_{\mathrm{pat}}$.

Figure 2

(a) Particle diffusing in a one-dimensional slab. The particle orientation is governed by $D_{\mathrm{rot}}$ and its horizontal motion is governed by $D_{\mathrm{tr}}$ (motion in other directions is irrelevant). The blue ball depicts the closest point on the left wall of the slab. (b) Centering the reference frame on the particle and letting the reference frame rotate with the particle, it follows that the blue ball diffuses in an annulus about the particle. The angular diffusion of the blue ball is governed by $D_{\mathrm{rot}}$ and its distance from the particle is governed by $D_{\mathrm{tr}}$. In both (a) and (b), the patches diffuse on the particle with diffusivity $D_{\mathrm{pat}}$.

Figure 3

Curves are the survival probability $\overline{S}$ using $\kappa$ in (29), and the markers are the empirical survival probability $S_{\text{emp}}$ computed from Monte Carlo simulations of the full stochastic process in (7). Here $\varepsilon ={10}^{-2}, N=50, D_{\mathrm{rot}}=0$, and $M=4\times {10}^3$ trials for each value of $D_{\mathrm{pat}}\in \{0.5,1,2\}$.

Figure 4

The error in distribution $\mathcal{E}$ in (30) as a function of $\varepsilon$ for different values of the rotational diffusivity $D_{\mathrm{rot}}$. Here $N=50, D_{\mathrm{pat}}=0$, and $M={10}^4$ trials for each value of $\varepsilon \in \{{10}^{-2},{10}^{-1.5},0.5,0.1,0.2\}$ and $D_{\mathrm{rot}}\in \{0.5,1,2\}$.

Figure 5

Comparing the theoretical trapping rate in (29) and a numerically computed optimal trapping rate. (a) The black curve gives the error $\mathcal{E}$ in (30) between $S_{\text{emp}}$ and $\overline{S}$ for trapping rates $\kappa$ ranging from 0.05 to 0.3. The dashed red curve is the error $\mathcal{E}$ when the theoretical trapping rate in (29) is used. We take $\varepsilon ={10}^{-2}, N=50, D_{\mathrm{rot}}=1, D_{\mathrm{pat}}=0$, and $M={10}^4$ trials. (b) The error in $\kappa$ in (31) as a function of $\varepsilon$ for different values of $D_{\mathrm{rot}}$. The parameters are as in Fig. 4.

Figure 6

Error in distribution $\mathcal{E}$ in (30) as a function of $\kappa$ in (29) for different values of $\varepsilon$. Here $D_{\mathrm{rot}}=1, D_{\mathrm{pat}}=0, \mathrm{\Delta }{t}_{\text{big}}={10}^{-3}, \mathrm{\Delta }{t}_{\text{small}}={10}^{-6}$, and $M={10}^5$ trials for each value of $\varepsilon \in \{0.05,0.1,0.2\}$ and $N\in \{1,10,50,100,200,500,1000\}$.

Figure 7

The thick black curve depicts a curved absorbing left wall. Here (35) is not satisfied, and the wall curvature affects the patch orientation relative to the wall.