= Solution
<single-molecule force spectroscopy> measures <force> and extension of an individual tether while controlling position or load. In an atomic-force arrangement, a synthetic chain connects a cantilever probe to a translated substrate. For the <cantilever compliance correction>, a deflection sensor gives $d$, a calibrated spring constant gives \b[$F=k_cd$], and substrate displacement $Z$ gives chain extension $x=Z-d$ after geometric offsets are removed. Additional linkers and instrument compliance must be included if appreciable. Thermal deflection calibration uses $k_c\langle d^2\rangle=k_BT$ only after isolating the cantilever's calibrated thermal contribution.
An optical-tweezers implementation detects a tethered bead's displacement from a calibrated trap centre, with $F=k_{\rm trap}\Delta x$ in the harmonic regime. Detector calibration, drag and trap linearity determine the usable <force> and frequency range. These instruments measure individual <force>–extension curves and distributions; averaging many curves can hide multiple tether populations.
For a synthetic <freely jointed chain> with <Kuhn length> $b$,
$$
\frac{x}{L}=\coth\frac{Fb}{k_BT}-\frac{k_BT}{Fb}.
$$
A strongly stretched <worm-like chain> instead has $1-x/L\simeq\sqrt{k_BT/(4F\ell_p)}$ before backbone stretching becomes important. Fits can estimate contour and <persistence lengths> only in their appropriate regimes. Hysteresis and pulling-rate dependence test whether an apparent mechanical transition is quasistatic or activated.
Control single versus multiple attachments, subtract nonspecific adhesion, vary pulling <speed>, calibrate <temperature> and stiffness, and correct drift. A rupture peak is not a universal bond strength independent of loading history. A primary nonbiological example measured individual <polydimethylsiloxane> chains by atomic-force methods; its flexibility interpretation combines those measurements with molecular calculations.<Understanding the Extraordinary Flexibility of Polydimethylsiloxane through Single-Molecule Mechanics by Song Lu and others>[abstract, synthetic PDMS single-chain mechanics]{ref}
\Image[/past-exam-of-the-soft-matter-and-biological-physics-course-of-the-university-of-cambridge/2014/paper-1-force-spectroscopy.png]
{title=Original single-tether force-measurement schematic and the instrument-compliance relation}
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