First-contact time to a patch in a multidimensional potential well

The escape of a diffusing particle from a potential well is an important aspect of many dynamic processes in chemistry, physics, and biology, and such an escape process often involves finding a restricted region or patch in a multidimensional potential well. We study an idealized model of this process via simulation and analytic theory. By combining results from special cases having either high symmetry or zero potential, we obtain a simple formula for the first-contact time for a particle moving to a boundary patch in an arbitrary number of dimensions. We apply this formula in two, three, and six dimensions. The predicted dependences of the first-contact time on the well depth and patch size are compared to results from simulations, and close agreement is found. We extend the theory to calculate the first-contact time between two particles in separate harmonic potential wells. As an application of this extended theory, we calculate the first-contact time for two parallel semiflexible biopolymer filaments and compare these results to previous simulations.

Figure 1

Schematic of the first-contact time for a particle to a patch in three-dimensional space. The patch size is ${\theta }_p$.

Figure 2

Calculations and simulations of the first-contact time for a particle in a potential well to the entire boundary (filled circles and accompanying lines) and patch (open circles and accompanying lines) as functions of potential in (a) two-, (b) three-, and (c) six-dimensional space. The patch size is fixed, ${\theta }_p=0.35$.

Figure 3

Calculations and simulations of the first-contact time for a particle in a potential well (open circles and accompanying lines) and free particle (filled circles and accompanying lines) moving to a patch, as a function of patch size in (a) two-, (b) three-, and (c) six-dimensional space. The potential is fixed, $E_0=11k_{B}T$. Solid lines: calculations using Eqs. (20, 22). Dashed lines: calculations using Eq. (24).

Figure 4

Contour plot of analytic theory (solid lines) and simulations (dashed lines) of the first contact time for a particle in a potential well to a patch as a function of potential and patch size in (a) two-, (b) three-, and (c) six-dimensional space. Numbers on contour lines give first-contact time in seconds. Solid lines are calculated using Eq. (23).

Figure 5

Contour plot of the corrected calculations (solid lines) and simulations (dashed lines) of the first-contact time for a particle in a potential well to a patch as a function of potential and patch size in (a) two-, (b) three-, and (c) six-dimensional space. Numbers on contour lines give first-contact time in seconds. The solid lines are calculated using Eq. (24).

Figure 6

Contour plot of the distribution of contact position between two particles in two-dimensional space. Each particle is in its own harmonic potential well. Circles represent the simulation results. Lines in the one-dimensional projections are Gaussian distributions fitted to the simulations. Distance between two potential minima is $0.2375\mu \text{m}$. The vertical line is $x=1/2\times 0.2375\mu \text{m}$, halfway between the minima.

Figure 7

Simulations (open circles) and calculations (line) of the first-contact time between two actin filaments with length $L=0.6\mu \text{m}$ as a function of distance $R$ between their bases. The two filaments are fixed and parallel at one end, and free at the other.