Past exam of the biological physics course of the University of Oxford / 2016 / 4

A rising force branch means that an already exposed polypeptide segment is being stretched while the remaining structure resists further extraction. At a force peak a structural barrier is crossed; protein denaturation or release of additional chain increases the available polymer contour length, so the force falls at almost the same imposed separation. Stretching then loads the next remaining barrier. The sawtooth pattern therefore reports a sequence of mechanically resistant intermediates, not a regular spring oscillation. The cantilever deflection gives force, while its own deformation must be removed from the imposed separation to infer molecular extension.

A membrane protein also requires work to disrupt membrane anchoring, helix packing and contacts with lipids or neighboring molecules. The retinal-containing interior contributes a bulky, specifically packed scaffold; exposing chain must overcome these steric hindrance as well as hydrogen bond and electrostatic interactions. A force peak need not correspond one-to-one to one complete helix, and its height depends on loading rate and pathway, not just an equilibrium folding Gibbs free energy.

Salt changes electrostatic screening and specific ion binding. Screening can reduce either repulsion or attraction, so “more salt always weakens the protein” is false. The primary study underlying these kinds of traces identifies relatively salt-insensitive steric hindrance around the retinal and additional salt-sensitive barriers. Its interpretation associates some added barriers with screening of destabilizing electrostatic repulsion. The supplied traces likewise show a changed set of intermediate peaks, rather than a uniform rescaling of one curve.^[ref]

Exact assignment of every arrow to a residue requires structural and statistical information beyond one representative trace. No such assignment is inferred here. The printed reference to titin in the following paragraph is inconsistent with the stated specimen and the membrane-protein traces; the physical interpretation here concerns bacteriorhodopsin, while the worm-like-chain expression is a generic polymer approximation.

On each branch the worm-like chain force-extension interpolation relates force to the extension of the already released chain. Fitting a sufficiently long branch estimates its polymer contour length $L$ and, if the data constrain it independently, its persistence length $L_p$. In practice $L_p$ may be fixed across branches so that the changes in $L$ are better identified. Consecutive fitted polymer contour lengths reveal how much additional chain became available at an protein denaturation event; using a justified polymer contour length per residue can then estimate residue counts. A projected folded length and the pulling geometry must be included before assigning a structural boundary.

Writing $u=x/L$, expansion at small extension givesNear full extension, $f\sim k_{B}T/[4L_p(1-x/L{)}^2]$. Larger $L$ shifts a branch toward larger extension at fixed force. The effective flexural rigidity is $\kappa =k_{B}T{L}_p$, with units of force times length squared. This is entropic straightening of a worm-like chain; bond stretching and barrier rupture themselves are not included in the formula.

The persistence-length wording requires care. The three-dimensional model gives $⟨\mathbf{t}(s)\cdot \mathbf{t}(0)⟩=e^{-s/L_p}$. At $s=L_p$, $\arccos ⟨\cos \theta ⟩=\arccos (e^{-1})=68.4^{\circ }$. This is not a statement that $⟨\theta ⟩=68.4^{\circ }$. For example, angles zero and $90^{\circ }$ with equal probabilities have mean angle $45^{\circ }$, but $\arccos ⟨\cos \theta ⟩=60^{\circ }$. The distinction prevents a false literal interpretation of the printed definition.

The earliest branch is close to the membrane and the point of tip contact. Little flexible chain has yet been exposed, so a long homogeneous continuum chain with purely entropic elasticity is a poor description. Tip–surface interactions, membrane deformation, steric obstruction, and extraction of tightly anchored structure can contribute strongly to the measured force. The retinal-associated structure and local electrostatic contacts can still be mechanically active at these small separations.

The worm-like-chain formula assumes that the measured extension chiefly stretches an already available flexible segment with an identifiable polymer contour length. It does not model detachment of the tip from a surface or the energetic rupture of a short structured segment. That explains why fitting these first peaks is unreliable even if later branches admit reasonable fits. A single trace cannot distinguish uniquely among these short-range effects; claiming that one particular arrow proves a specific retinal bond rupture would exceed the available evidence.