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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.

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.

The coefficient in the linear force law isStretching suppresses the many disordered orientations available to the chain. The Gaussian chain free energy cost $\Delta \mathcal{F}\simeq 3k_{B}T{R}^2/(2N{b}^2)$ has gradient equal to this restoring force. This is entropic elasticity; individual covalent bonds need not stretch to produce the weak-force spring.

At moderate force, even an ideal FJC becomes nonlinear as its orientations align, so failure of a linear fit alone need not imply nonideal chemistry. But failure of the full FJC curve can reflect backbone spring constant and correlated segment angles, nonuniform effective links, remaining folded contacts, force-induced conformational changes, energetic bond compliance, or tether/calibration effects. An extensible FJC or an appropriate semiflexible model includes some of these effects; their parameters require independent justification. The stated inextensible $14\mathrm{nm}$ polymer contour length already excludes a literal $15\mathrm{nm}$ extension.

The first panel of the original model figure below shows the full orientational response and weak-force tangents without tracing the experimental data.

In a tensile atomic force microscopy measurement, a DNA tether connects the cantilever tip and a surface. Moving the support stretches the molecule, and calibrated cantilever deflection supplies the tensile force; extension must be corrected for cantilever deflection and any linkers. At low force, extension mainly straightens an initially fluctuating coil, giving approximately linear entropic elasticity. At higher force the available conformations become increasingly aligned, so additional extension costs progressively greater force. The readable source curve then continues beyond the nominal inextensible-model contour length, which signals a failure of that simple model or of the stated parameter interpretation, rather than negative elasticity.