Past exam of the biological physics course of the University of Oxford / 2015 / 4

In single-molecule force spectroscopy of titin, one end of the protein is attached to a surface and the other to an atomic force microscope cantilever. Moving the support stretches the molecule; a calibrated cantilever deflection measures the force through $f=k_c\delta$. The molecular extension is the imposed separation minus cantilever deflection and other instrumental offsets, rather than the uncorrected support displacement.

At first, increasing extension removes conformational slack from unstructured segments. Their conformational entropy decreases and the resisting force rises. A folded protein domain then unfolds when a thermally activated transition becomes likely under load. Its residues join the stretched chain, increasing the available polymer contour length. At almost the same imposed separation, the newly released segment has slack: the force drops. Further pulling straightens this longer chain until another domain unfolds. Repetition produces a sawtooth sequence; each rising branch mainly measures stretching between unfolding events, and a peak marks the end of such a branch.

The peak height measures the resistance to unfolding under the particular loading history, not a direct equilibrium free energy or an invariant breaking force. Random unfolding times, loading rate, domain identity, attachments, and cantilever compliance affect the record. Releasing tension can allow protein folding, but refolding has its own kinetics and need not retrace the loading curve. Distinct unfolding events explain how a long protein can absorb extension in steps rather than failing at a single threshold.

Source illustration TODO: Figures A and B are absent in the held source and archive parent. This explanation uses the surviving sawtooth description; it supplies no recovered peak heights, spacings, or domain counts.

Fit each pre-unfolding branch with its own polymer contour length $L$ and a persistence length $L_p$, allowing for the measured extension zero. The worm-like chain force-extension interpolation has the limitsThus the low-extension slope and the high-extension curvature constrain the flexible segment's length and bending stiffness. The persistence length is related to the bending modulus by $\kappa =k_{B}T{L}_p$. Successive increases in fitted $L$ measure contour released by unfolding, after subtracting the original folded domain's projected length. With an independently known contour increment per residue, this also estimates how many residues became available. A common fitted $L_p$ across branches supports a common type of unfolded chain; a systematic change can signal altered interactions or a poor model.

For a thermal worm-like chain, $⟨\mathbf{t}(s)\cdot \mathbf{t}(0)⟩=e^{-s/L_p}$. Consequently $s=L_p$ gives $⟨\cos \theta ⟩=1/e$, and $\arccos (1/e)=68.4^{\circ }$. The latter is an angle associated with the mean cosine, not generally the mean angle itself. For example, angles $0$ and $90^{\circ }$ with equal probabilities have mean angle $45^{\circ }$, but the inverse cosine of their mean cosine is $60^{\circ }$. This distinction makes the source's informal angular description precise.

Without Figure B, no numerical fit parameters can be read. The requested physical interpretation of those parameters follows without its numerical data.

An inextensible worm-like chain describes entropic elasticity and diverges as $x\to L$. Real protein backbones and folded protein domains also deform energetically under large forces. For an axial stretch modulus $K_s$, a simple extensible-chain correction isThe last term supplies extension through bond and structural deformation that the inextensible model excludes. It becomes especially noticeable on the high-force ends of rising branches.^[ref] An unfolding transition itself is also outside a single equilibrium fixed-$L$ branch. These mechanisms explain why the source explicitly distinguishes energetic and entropic elasticity.

Source-dependent TODO: the arrows and actual deviations at the first four peaks in Figure B are unavailable. Energetic stretching is the natural explanation of a high-force excess-extension deviation, but its direction, onset, and uniqueness cannot be checked against the missing curves. No particular arrow-marked feature or numerical stretch modulus is invented. This source-dependent identification remains incomplete; the recoverable model limitation is explained above.