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A biopolymer is a polymer produced by a living system, including proteins, nucleic acids and polysaccharides. Local chemical structure, sequence and interactions can give it stiffness or preferred conformations beyond those of an ideal chain. Biopolymer mechanics uses models such as the worm-like chain and the helix-coil transition model to represent those features.

For an ideal chain inside a blob, $\nu =1/2$ and $\xi \sim b\sqrt{g}$. Together with $\varphi \sim g{v}_0/{\xi }^3$, this gives the displayed concentration-blob size. Under the site-volume approximation $v_0\sim b^3$, $\xi \sim b/\varphi$ and $g\sim {\varphi }^{-2}$. The estimate describes local overlap rather than the full chain size and requires $1\ll g\ll N$ for a well-developed semidilute regime.

An ideal chain with endpoint charges $+q$ and $-q$ has coupling energy $-q\mathbf{E}\cdot \mathbf{R}$ in a uniform electric field, when $\mathbf{R}$ points from negative to positive charge. Completing the square in its Gaussian end-to-end distribution gives the displayed mean extension. The force conjugate to end separation is $qE$, even though the two endpoints experience opposite forces; adding their magnitudes would double-count the work. The zero-field rms length is $a\sqrt{N}$, and the rms of a single Cartesian component is $a\sqrt{N/3}$. Direct endpoint interactions and charge-induced changes of solvent quality are excluded from this model.

The end-to-end vector joins the two ends of a polymer chain. In an isotropic unforced polymer solution, its vector expected value is zero; a useful measure of chain size is its root-mean-square length. For an ideal chain with $N$ independent isotropic steps of length $b$, the cross terms in the square of the step sum have zero expected value, so $R_{\mathrm{rms}}^2=N{b}^2$. For a long Gaussian chain, its radius of gyration obeys $R_g^2=R_{\mathrm{rms}}^2/6$. If $N$ counts chemical repeat units instead of independent Kuhn length segments, the statistical length used in $R_{\mathrm{rms}}^2=N{b}^2$ must be calibrated to that choice of $N$; it is not automatically the physical bond length or the Kuhn length itself.

For a free three-dimensional worm-like chain with contour length $L$ and persistence length ${\ell }_p$, the unit-tangent correlation is $⟨\mathbf{t}(s)\cdot \mathbf{t}(s^{\prime })⟩=e^{-|s-s^{\prime }|/{\ell }_p}$. Since $\mathbf{R}={\int }_0^L\mathbf{t}(s)ds$, integrating this correlation givesFor $L\gg {\ell }_p$, the leading term is $2{\ell }_{p}L$, the ideal chain law with Kuhn length $2{\ell }_p$. For $L\ll {\ell }_p$, a Taylor series gives $⟨R^2⟩=L^2-L^3/(3{\ell }_p)+O(L^4/{\ell }_p^2)$, approaching a rigid rod. This ideal semiflexible-chain model omits long-range excluded volume; sufficiently long dilute chains in a good solvent can cross over to swollen-chain statistics.

An entropic force is the part of a generalized force arising from the dependence of entropy on a coordinate. At fixed temperature, the system's force is $-\partial A/\partial x=-\partial U/\partial x+T\partial S/\partial x$, where $A=U-TS$ is the Helmholtz free energy. If the internal energy is independent of the coordinate, the force is entirely entropic. For an ideal chain whose configurations become less numerous on extension, it acts toward shorter extension. The external tension holding the chain extended has the opposite sign. Thus the force depends on the statistical number of available configurations, even when the energy of each configuration is unchanged.

For an ideal chain of $N$ independent statistical segments of length $a$, the end-to-end probability density is proportional to $\exp [-3R^2/(2N{a}^2)]$. Its constrained Helmholtz free energy has extension-dependent part $3k_{B}T{R}^2/(2N{a}^2)$, so the restoring force is $-3k_{B}T\mathbf{R}/(N{a}^2)$. This is an entropic spring because extension removes available chain configurations, and its stiffness grows with temperature. The Gaussian law applies at extensions small compared with the contour length $Na$; near full extension, finite extensibility matters.

The Flory approximation balances the entropic cost of expanding an ideal chain against a mean-field excluded volume interaction. For $N$ segments of length $b$ in $d$ spatial dimensions, minimizing $R^2/(N{b}^2)+v{N}^2/R^d$ gives $R^{d+2}\sim v{N}^3{b}^2$. This yields the scaling exponents $3/5$ in three dimensions and $3/4$ in two dimensions. It is a scaling approximation, not an exact numerical free energy or proof of critical exponents; confinement replaces $R^d$ by the accessible coil volume.

A good solvent favors contact with polymer segments strongly enough that the effective excluded volume is positive. A long dilute flexible chain swells relative to an ideal chain; the three-dimensional size exponent is approximately $0.588$, often replaced by $3/5$ in a scaling estimate. This designation depends on the polymer, solvent and temperature, rather than on the solvent alone.

A helix-coil transition model assigns local states of a biopolymer to an ordered helical conformation or a disordered coil conformation. Their competing free energies determine the tendency to form a helix, while the cost of boundaries between regions gives cooperativity: forming a new helical run can be less favorable than extending an existing one. A coil state is a class of disordered conformations, not a claim that the whole molecule is an ideal chain. The Zimm-Bragg model is a simple nearest-neighbour statistical model of this competition.

An ideal chain, also called a freely jointed chain, consists of fixed-length links with independent orientations and no bending energy between successive links. It neglects segment interactions and excluded volume. A one-dimensional freely jointed chain restricts each link to two orientations. At fixed force its canonical partition function factors into one-link angular integrals. The resulting extension is entropic and depends on temperature and on whether the orientations are allowed in one, two, or three dimensions.

The Kuhn length is the link length of an equivalent long ideal chain with the same contour length and mean squared end-to-end distance. A three-dimensional worm-like chain satisfies $⟨R^2⟩\sim 2l_{p}L$ for $L\gg l_p$, hence $b_K=2l_p$.

For a long ideal chain of $N$ independent statistical steps of length $b$, $R_g^2=N{b}^2/6$. With fixed chain chemistry and $b$, the molar mass $M$ is proportional to $N$. Thus multiplying molar mass by $s$ multiplies the radius of gyration by $\sqrt{s}$; solvent quality can change both the statistical segment length and the size exponent.

This is the one-dimensional version of a freely jointed chain, an ideal chain. A chain of $n$ links of length $a$, each pointing independently right or left, has extensionand multiplicity $n!/(n_{+}!n_{-}!)$.

On releasing the tensile load, the extended chain first retracts towards less aligned conformations because they have much greater configurational multiplicity. Viscous friction and internal relaxation determine the time course; stored free energy is dissipated as heat. An unconstrained ideal chain does not shrink to a point: its mean vector is zero but its root mean square end-to-end length is $b\sqrt{N}\simeq 2.37\mathrm{nm}$ for the supplied ideal-chain parameters.

A real polypeptide also has sequence-dependent interactions. If the original native structure is the stable state in the prevailing solvent and the molecule remains intact and capable of folding, it can subsequently refold, gaining favourable intramolecular and solvent free energy. The folded state is not selected by chain entropy alone, and formation of native contacts need not be instantaneous. Kinetic traps, proline isomerization, protein aggregation, misfolding, surface attachments or chemical damage may prevent complete recovery or produce hysteresis. Thus reversible refolding is a physically motivated possibility, not guaranteed simply by removal of force. This distinction connects the recoil of a stretched polymer to the thermodynamics discussed in Question 6(c).

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.

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.

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 $⟨z⟩\ll Nb$.

Let $b\simeq 2l_p$ be the Kuhn length, $N=L/b$ the number of statistical segments, and $v$ their effective excluded volume. We first assume a good-solvent, strongly confined self-avoiding walk, rather than an ideal chain. A confinement blob contains $g$ links and has diameter $D\sim b{g}^{3/5}$ in the simple $v\sim b^3$ scaling model. Therefore $g\sim (D/b{)}^{5/3}$. Distant blobs cannot freely pass through one another in a narrow tube, so the $N/g$ blobs form an axial string. The de Gennes confinement scaling is consequentlyThe free energy is of order $k_{B}T$ per blob, not the energy of stretching a straight elastic rod.

The role of solvent and chain thickness can be made explicit by a Flory approximation: an axial extension $X$ costs an entropic term $k_{B}T{X}^2/(N{b}^2)$, while the confined number density gives an interaction term $k_{B}Tv{N}^2/(X{D}^2)$. Minimizing their sum gives $X^3\sim v{N}^3{b}^2/D^2$, henceFor a thin statistical segment of effective diameter $d$, $v\sim b^{2}d$ instead gives $X\sim L(bd/D^2{)}^{1/3}$. Thus the simpler answer implicitly fixes the excluded volume; the DNA thickness and ionic environment can matter quantitatively.

Both blob arguments require $b\ll D\ll R_{\mathrm{free}}$ and $N/g\gg 1$, where $R_{\mathrm{free}}\sim b{N}^{3/5}$ in the simple good-solvent model. The printed comparison with $L$ alone does not ensure confinement relative to the unconfined coil. If $D\gg R_{\mathrm{free}}$, the tube barely perturbs the coil. An ideal chain without excluded volume has axial spread of order $\sqrt{Lb}$ and does not acquire the extensive extension of a nonoverlapping blob string. These are distinct physical regimes, not changes to a numerical prefactor.

At the theta condition, effective excluded volume approximately cancel and a freely jointed ideal chain has $⟨R^2⟩=N{b}^2$. With the stipulated segment length $b=2a$,In a good solvent repulsive excluded volume swells the coil, giving $R\sim b{N}^{\nu }$ with $\nu \simeq 0.588$ in three dimensions, or the Flory approximation $\nu =3/5$. In a poor solvent effective segment attraction favours a compact globule, whose linear size scales as $b{N}^{1/3}$ at roughly fixed internal density. The latter are scaling laws with model-dependent prefactors. A real chemical monomer need not be a statistical Kuhn length; using $2a$ as the independent segment length is part of this question's idealized model.