Unified Model of Dynamic Forced Barrier Crossing in Single Molecules

Thermally activated barrier crossing in the presence of an increasing load can reveal kinetic rate constants and energy barrier parameters when repeated over a range of loading rates. Here we derive a model of the mean escape force for all relevant loading rates—the complete force spectrum. Two well-known approximations emerge as limiting cases, one of which confirms predictions that single-barrier spectra should converge to a phenomenological description in the slow loading limit.

Figure 1

Schematic of the linear-cubic model energy landscape along the reaction coordinate $x$. When under an increasing force field, $f=\dot{f}t$ (dashed lines), the microscopic details of the barrier follow a time dependence (arrow) which modulates the rate of diffusive passage over the barrier.

Figure 2

Nonlinear least-squares fit of Eq. (5) (solid curve passing behind points), with $b=3/2$, against Brownian dynamics simulations of forced escape (open circles) from a Morse potential (drawn in the lower inset). Fit results give $U_0=9.49\pm 0.04k_{B}T$, $x^{‡}=3.14\pm 0.03\AA$ (arrow), and $k_0=36.45\pm 1.41{\mathrm{s}}^{-1}$. From simulations at zero force the intrinsic rate is found to be $k_0=37.06{\mathrm{s}}^{-1}$. Equations (7) (dashed curve, converging at low $\dot{f}$) and (8) (dotted curve, converging at high $\dot{f}$) are shown for comparison. The upper inset displays the variance in the simulated escape forces (diamonds) compared to Eq. (6) (solid curve), and the high-force approximation (dashed curve, see text), calculated at the fitted parameter values.

Figure 3

(a) Fit of Eq. (5) (solid curve, $b=3/2$) to biotin-avidin bond rupture data (△) reproduced with permission from 23, copyright 1999 by MacMillan Publishers, Ltd.: Nature. (b) Mean switching field $⟨H⟩$ versus sweep-rate ($\dot{H}\equiv dH/dt$) for a single-molecule magnet at temperatures from 0.4 K (upper most spectrum, ○) to 2.6 K (bottom most spectrum, ×), in 0.2 K increments; reproduced with permission from [1], copyright 2005 by the American Physical Society. Globally fitting to these spectra yields $v_0=1.21\pm 0.31\times {10}^8{\mathrm{s}}^{-1}$, ${\mu }_0{H}_c=6.15\pm 0.06\mathrm{T}$, and $U_0/k_B=50.65\pm 0.68\mathrm{K}$. Using the fitted parameters, the right panel in (b) shows scaling of the data onto a single master curve (see text) for temperatures above 0.8 K.