= Solution
Use a <force-tilted activation barrier> for a generic one-coordinate escape model. A tensile <force> tilts a <free-energy landscape> to $G_F(x)=G_0(x)-Fx$. If $x_b$ is the initial <minimum> and $x_s$ the saddle, define $\Delta x=x_s-x_b>0$ and the zero-force barrier $\Delta G_0^\ddagger=G_0(x_s)-G_0(x_b)$. The relevant activation barrier is measured relative to the initial <minimum>, not an arbitrary absolute transition-state <energy>. To first order in <force>,
$$
\Delta G^\ddagger(F)=\Delta G_0^\ddagger-F\Delta x,
\qquad
\boxed{k(F)=k_0e^{F/F_0},\quad F_0=\frac{k_BT}{\Delta x},\quad
k_0=\nu e^{-\Delta G_0^\ddagger/(k_BT)}.}
$$
The attempt prefactor $\nu$ is assumed <force> independent, and the saddle displacement is approximated as constant. This single-barrier approximation need not survive large landscape deformations or a change of escape pathway. TODO: \b[user-deferred biomolecular implementation]; the protein-specific shear-force arrangement and attachment chemistry are not solved.
For a generic force-ramp measurement, a calibrated <atomic force microscope> <cantilever> pulls an attachment while a stage changes separation. <cantilever> deflection $d$ gives <force> $k_cd$, and the <cantilever compliance correction> subtracts $d$ from imposed separation to give tether extension, with further corrections for other apparatus compliance. A prescribed stage speed does not automatically produce constant <force> loading rate when tether stiffness varies; the assumed $F(t)=ft$ must be measured or controlled. The original schematic below illustrates a generic nonbiological bond/tether geometry. Protein-specific construction is deferred.
The no-rebinding approximation is a physical escape assumption: after the link breaks, separation removes the partners from the capture region before appreciable reassociation. It requires the integrated return/association probability during that separation to be small. If the separated state stays nearby or the <force> is held without separation, setting $k_+=0$ can fail. Under the stipulated approximation, with the link initially intact,
$$
\frac{dp}{dF}=-\frac{k_0}{f}e^{F/F_0}p,
\qquad
\boxed{p(F)=\exp\left[-\frac{k_0F_0}{f}(e^{F/F_0}-1)\right],\quad F\geq0.}
$$
This <survival under a linear force ramp> differs from the <probability density> for rupture. It starts at one, decreases to zero and shifts to larger <force> as the loading rate increases. For small $F/F_0$, $p\simeq e^{-k_0F/f}$; a slow ramp allows much escape before substantial <force> is reached.
The <median rupture force under a linear ramp> satisfies $p=1/2$, giving
$$
\boxed{F_{1/2}=F_0\ln\left(1+\frac{f\ln2}{k_0F_0}\right).}
$$
At high loading rate, the slope of $F_{1/2}$ against $\ln f$ tends to $F_0$; it is not exactly $F_0$ for a slow ramp. Fit the full expression across loading rates to recover $F_0$ and $k_0$. The rupture-force <probability density> is $\rho(F)=k(F)p(F)/f$, whose <most probable rupture force under a linear ramp> is $F_*=F_0\ln[f/(k_0F_0)]$ if $f>k_0F_0$, and zero otherwise. <median> and <mode> are different observables and must not be interchanged when fitting.
At known <temperature>, $F_0$ yields the activation distance $\Delta x=k_BT/F_0$; $1/k_0$ gives the zero-force mean lifetime in this exponential model. Absolute barrier height additionally requires the attempt prefactor, and neither equilibrium binding <free energy> nor an association rate follows from dissociation-only data. A loading-rate-dependent rupture <force> is a kinetic readout, not an intrinsic equilibrium strength. TODO: \b[user-deferred biological conclusions]. The protein-specific stability interpretation and cell-adhesion strategy are left unsolved.
\Image[/past-exam-of-the-soft-matter-and-biological-physics-course-of-the-university-of-cambridge/2013/paper-1-generic-force-ramp.png]
{title=Original generic pulling arrangement, tilted energy landscape and calculated survival under a force ramp}
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