Past exam of the biological physics course of the University of Oxford / 2019 / 5

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.

An inextensible freely jointed chain of $N$ links of Kuhn length $b$ has contour length $L_c=Nb$ and cannot be extended beyond it at any finite force. There is an explicit numerical contradiction in the printed parameters:The discrepancy is present in the original page, not introduced by digitization. To obtain $26\mu \mathrm{m}$ with $186$ links requires $b\simeq 140\mathrm{nm}$; to obtain it with $b=7\mathrm{nm}$ requires $N\simeq 3714$. No single inextensible FJC can satisfy all three stated numbers. The graph's much larger extension is an immediate counterexample to treating the literal $N,b$ as an accurate global fit.

Separately, accepting the graph's nominal $26\mu \mathrm{m}$ contour as an effective low-force model, its continuation to $32.7\mu \mathrm{m}$ is about $26\%$ beyond that contour. An inextensible ideal chain contains orientational entropy only; it excludes stretching of segments, energetic compliance of bonds, changes in helix structure or base pairing, and heterogeneous attachments. An extensible freely jointed chain or a semiflexible-chain model with appropriate energetic terms can describe some departures. Calibration or contour-length assignment can also be wrong. The graph alone does not identify which mechanism contributes, and its displayed moderate forces do not justify assigning a specific high-force DNA structural transition. Arithmetic inconsistency in $Nb$ is not explained away by these physical mechanisms.

Let the applied force be $\mathbf{f}=f\hat{\mathbf{z}}$. A segment with polar angle $\theta$ has energy $-fb\cos \theta$. Independent 3D orientations give the one-link partition functionTaking the force derivative of $k_{B}T\ln Z_N$ yields the mean end-to-end projection,For $u\ll 1$, $\coth u-1/u=u/3+O(u^3)$, soIsotropy supplies the vector relation. It is the mean vector or its projection, not the mean magnitude of a fluctuating chain; at zero force $⟨\mathbf{R}⟩=0$ while $⟨R^2⟩=N{b}^2$ remains positive. The last panel of the model figure in solution shows the exact orientational response and its weak-force tangent.

The weak-force spring constant isUsing the literal printed $N=186$, $b=7\mathrm{nm}$ at $25^{\circ }\mathrm{C}=298.15\mathrm{K}$ givesThis is a calculation with the quoted parameters, not a self-consistent fit to the plotted $26\mu \mathrm{m}$ contour. If instead $N=186$ and $L_c=26\mu \mathrm{m}$ are retained, then $b=L_c/N$ and $k=3.40\times 10^{-9}\mathrm{N}{\mathrm{m}}^{-1}$. Retaining $b=7\mathrm{nm}$ and $L_c=26\mu \mathrm{m}$ gives $k=6.79\times 10^{-8}\mathrm{N}{\mathrm{m}}^{-1}$. These distinct alternatives quantify the unresolved parameter inconsistency rather than silently choosing a correction.

The restoring force is entropic. Stretching selects a small fraction of the many disordered orientations, reducing conformational entropy. The Gaussian chain free-energy cost is $\Delta \mathcal{F}\simeq 3k_{B}T{R}^2/(2N{b}^2)$, whose gradient gives the Hooke-law coefficient above. In an ideal inextensible chain, individual bonds need not change their length to generate this force.

At fixed $N$ and $b$, the entropic spring constant is proportional to absolute temperature, because the free-energy penalty for a given loss of conformational entropy is $-T\Delta S$. ThereforeFor the literal printed chain this is $\boxed{1.287\times 10^{-6}\mathrm{N}{\mathrm{m}}^{-1}}$. The same factor multiplies either alternative contour-consistent spring constant from part (d), provided its Kuhn length is held fixed. Real DNA's persistence length, base pairing and solvent interactions can themselves vary with temperature and salt; then $b(T)$ contributes through $k\propto T/b(T{)}^2$ at fixed segment count, or $k\propto T/[L_{c}b(T)]$ if the physical contour is instead held fixed. The simple linear temperature result belongs to the specified ideal-chain model.

After the load is removed, a reversibly stretched DNA molecule relaxes towards a thermally fluctuating coil, releasing stored free energy as heat through the surrounding fluid. There are many more disordered chain conformations than strongly aligned ones, so entropy favours retraction; viscous drag sets how rapidly it occurs. At long times an unconstrained ideal chain has zero mean end-to-end vector but a nonzero typical size, $R_{\mathrm{rms}}=b\sqrt{N}$; it does not contract to a point.

At finite temperature the chain continues fluctuating, and an anchored or otherwise constrained molecule need not regain an isotropic ensemble. If stretching broke attachments, denatured the DNA or caused irreversible structural changes, recovery can be incomplete or hysteretic. Those outcomes cannot be decided from the ideal FJC alone.