Past exam of the biological physics course of the University of Oxford / 2022 / 5

Write the end-to-end random vector as $\mathbf{r}={\sum }_{j=1}^{N}b{\mathbf{u}}_j$, with independent isotropically distributed unit vectors. At zero applied force, $⟨{\mathbf{u}}_j⟩=0$, ${\mathbf{u}}_j\cdot {\mathbf{u}}_j=1$, and for $i\ne j$ independence gives $⟨{\mathbf{u}}_i\cdot {\mathbf{u}}_j⟩=0$. ThereforeHere “size” is the root mean square end-to-end distance of a polymer chain. It is not the magnitude of the mean random vector, which is zero, nor the mean distance $⟨r⟩$. In the long-chain Gaussian limit of the end-to-end distribution of a freely jointed chain, $⟨r⟩\simeq b\sqrt{8N/(3\pi )}$, illustrating why these definitions cannot be interchanged.

Each Cartesian component of one link has zero mean and variance $b^2/3$, with zero cross-component covariance. By the multivariate central limit theorem, for large $N$ on the coil scale the vector probability density isIt is normalized with respect to $d^{3}r$; the radial probability density is $4\pi r^{2}p(\mathbf{r})$. The Gaussian limit of the end-to-end distribution of a freely jointed chain is an approximation, not the exact finite-$N$ law near $r=Nb$.

At prescribed vector separation the configurational entropy is $S(\mathbf{r})=k_B\ln p(\mathbf{r})+\text{constant}$. The ideal chain has no extension-dependent internal energy, so its Helmholtz free energy isThe externally required force is ${\nabla }_{\mathbf{r}}A$, while the chain's restoring force has the opposite sign. ThusThe last equation also follows by multiplying $p(\mathbf{r})$ by the Boltzmann factor $e^{fz/(k_{B}T)}$ and completing the square. This entropic spring resists extension because fewer link configurations realize a stretched chain. At the same small applied force, increasing temperature reduces extension; at the same extension it increases the required force. Do not differentiate $-k_{B}T\ln [4\pi r^{2}p(\mathbf{r})]$ to obtain the fixed-vector Hooke law, since that radial measure includes an additional geometric multiplicity. The low-force condition is $fb/(k_{B}T)\ll 1$, equivalently $⟨z⟩\ll Nb$.

In atomic force microscopy, one end of the polyacrylic acid chain is attached to the substrate and the other to a compliant cantilever tip. Controlled separation displaces the cantilever; calibrated cantilever stiffness converts its deflection into force. The molecular extension is the tip–surface separation after correcting for cantilever deflection and instrumental offsets. A quasi-static interpretation requires relaxation to be fast compared with the pulling motion.

At low force a coiled polymer has many available configurations and extends mainly by changing segment orientations. This is the entropic spring regime. As the available contour is aligned, the loss of configurational entropy becomes steep. For an inextensible three-dimensional freely jointed chain,where $\mathcal{L}$ is the Langevin function. The limiting extension is the polymer contour length $Nb$; the ideal model approaches it only as $f\to \infty$. Real polymer backbone stretching permits extension beyond the fitted rigid-link contour, explaining why the experimental curve lies on the more extensible side at high force.

The original trace has a sharp force drop after its peak. Finite contour length explains the steep preceding rise, but does not itself cause that drop: release of an attachment, desorption or mechanical failure ends tensile loading. The graph alone does not distinguish these possibilities. If the intact chain is released from tension, thermal motion restores a coil with characteristic end-to-end distance of a polymer chain of order $b\sqrt{N}$, because this macrostate has many more configurations. Recoil is reversible for an intact equilibrating chain; irreversible chain scission or attachment loss can prevent repetition of the same pulling curve.

Identifying the fitted limiting extension with the polymer contour length givesThese are statistical links of the freely jointed chain, not necessarily chemical polymer repeat units. A measured maximum just before detachment need not equal the true contour length exactly, so the identification is itself an estimate.

The stated temperature request mixes two distinct quantities. At unchanged $N$ and $b$, the true freely jointed chain maximum is geometric:In contrast, the low-force relation $⟨z⟩=N{b}^{2}f/(3k_{B}T)$ predicts a ratio at the same small force, not a new maximum:Formally multiplying $500\mathrm{nm}$ by this ratio gives about $400\mathrm{nm}$, but $500\mathrm{nm}=Nb$ is outside the small-extension domain, so that is not a justified maximum-extension prediction. A concrete internally consistent counterexample uses the same chain at $f=0.10\mathrm{pN}$: the low-force extensions are about $2.59\mathrm{nm}$ and $2.07\mathrm{nm}$, while both contour lengths remain $500\mathrm{nm}$. Any temperature dependence of the segment length, solvent interactions, attachment strength or rupture force needs information absent from the model.

The rigid-link freely jointed chain stores extension entirely in orientation. It predicts $⟨z⟩\to L=Nb$ and a divergent force as that limit is approached. Real chemical bonds and bond angles can deform, adding energetic compliance. This permits the measured chain to stretch farther at the same large force and removes the rigid-link vertical asymptote over the experimentally accessible range. Other limitations include segment correlations, solvent effects and eventual failure; a fit alone does not isolate their contributions.

Introduce one axial stretch modulus $S$ with units of force. A useful phenomenological extensible freely jointed chain law adds a linear strain contribution:Equivalently, if each aligned segment is a spring of stiffness $k_s$ in force per length, $S=b{k}_s$ and the additional extension is $Nf/k_s$. At high force, $⟨z⟩\simeq L-N{k}_{B}T/f+Lf/S$; the extra term supplies finite axial compliance. At small force, the compliances add: $d⟨z⟩/df=N{b}^2/(3k_{B}T)+L/S$.

This additive law is a simple requested one-term modification, rather than an exact angular average for harmonic links. Another common approximation is $L\mathcal{L}(fb/(k_{B}T))(1+f/S)$, which couples stretching to orientation. Neither should be extrapolated through bond rupture or claimed to predict a universal finite maximum for an extensible chain. A positive fitted $S$ could improve the high-force extension while leaving the entropic part largely intact.