Worm-like chain

The worm-like chain models an inextensible filament with bending energy $A\int {\kappa }^{2}ds/2$ and thermal tangent fluctuations.

An inextensible worm-like chain gains extension by reducing configurational fluctuations. At high force its fractional entropic extension approaches $1-\frac{1}{2}\sqrt{k_{B}T/(f{L}_p)}$. A finite axial stretch modulus $K_s$ adds the leading energetic extension $f/K_s$. This approximation distinguishes suppressed bending fluctuations from actual stretching of the chain backbone; it does not describe an unfolding event that changes contour length.

The interpolation links chain extension to entropic tensile force, with contour length $L$ and persistence length $L_p$. Small extension gives spring constant $3k_{B}T/(2L_{p}L)$; near $x=L$ the force stiffens as $(1-x/L{)}^{-2}$. The divergence assumes an inextensible backbone and is not a physical infinite tensile force. A branch fit estimates available contour and flexibility, not an unfolding activation barrier by itself.

The curvature-dependent elastic energy of a semiflexible DNA polymer in the isotropic worm-like chain model. Here $l_p$ is the persistence length, $\kappa$ the curvature and $\ell$ the polymer contour length. A complete circular loop of radius $R$ costs $\pi k_{B}T{l}_p/R$, obtained by integrating constant curvature $1/R$ over length $2\pi R$.

A circular loop of radius $R$ has constant curvature $1/R$ and contour length $2\pi R$. Integrating the quadratic bending energy density $K_b/(2R^2)$ gives the displayed energy. With $K_b=k_{B}TA$ this is $\pi k_{B}TA/R$. Applied-force work, axial strain, twist and contact energies are separate terms; the loop formula alone does not determine an actual loaded loop's radius.

For an elastic filament with reference arclength $s$, curvature $c$, axial strain $ϵ$ and excess twist rate $\Omega$, a harmonic energy density is $(K_b{c}^2+S{ϵ}^2+K_t{\Omega }^2)/2$. Bending and torsional rigidities have units of energy times length; the stretch modulus has units of force. Factoring out $k_{B}T$ therefore makes $A,C$ lengths but $B$ an inverse length. Axial strain under tension is approximately $F/S$. Twist is controlled by torque and rotational boundary conditions, not force alone. A three-dimensional thermally fluctuating worm-like chain has persistence length $K_b/(k_{B}T)$.

A stiff chain's transverse wandering over contour length $s$ scales as $s^3/{\ell }_p$. Equating this to the squared confining radius gives the displayed deflection length when $R\ll {\ell }_p$. A free-energy cost of order $k_{B}T$ per deflection yields confinement free energy per inserted contour length of order $k_{B}T/\lambda$. Numerical prefactors depend on geometry and boundary constraints.

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.

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 fixed bond length $b$, fixed forward bond angle $\alpha$, and independently uniform torsion angles, the bond correlations are $⟨{\mathbf{b}}_i\cdot {\mathbf{b}}_{i+s}⟩=b^2{c}^s$. Thus for $N$ bondsFor fixed $|c|<1$ and a sufficiently long chain, divide by contour length $Nb$ and sum the geometric series to obtain the title formula. Positive forward persistence raises the Kuhn length, and negative correlation lowers it relative to $b$. A chemical internal angle is supplementary to the forward-vector angle. The long-chain formula cannot be used uniformly as $c\to 1$ at fixed $N$: the exact chain obeys $R\le Nb$ and hence $l_K\le Nb$, with equality in the rigid-rod limit.

A strongly stretched inextensible worm-like chain has two small transverse tangent components. Its quadratic energy is $\frac{1}{2}\int [A|{\partial }_s{\mathbf{t}}_{\perp }{|}^2+f|{\mathbf{t}}_{\perp }{|}^2]ds-fL$. For a long bulk chain, the equipartition theorem assigns each Fourier mode covariance $k_{B}T/(A{q}^2+f)$. Integrating over $q$ and summing both components gives $⟨|{\mathbf{t}}_{\perp }{|}^2⟩=k_{B}T/\sqrt{Af}$, hence the extension deficit is half that value. With persistence length $L_p=A/(k_{B}T)$, the deficit is $\sqrt{k_{B}T/(4L_{p}f)}$. It requires small slopes and negligible stretching of the backbone.

The bulk tangent covariance is twice the inverse Fourier transform of $k_{B}T/(A{q}^2+f)$, one term for each transverse direction. Equivalently the inverse of $-A{\partial }_r^2+f$ is $e^{-|r|/\xi }/(2\sqrt{Af})$. This gives the displayed correlation and identifies a force-dependent correlation length, different from the zero-force persistence length. Finite chains require endpoint conditions or a periodic convention.

Periodizing the bulk transverse tangent correlation of a stretched worm-like chain gives $C_L(r)={\sum }_{j\in \mathbb{Z}}C(r+jL)$, where $d_L(r)$ is the distance to zero on the periodic interval. Summing the geometric series yields the displayed result. At zero separation its finite-length factor is $\coth (L/(2\xi ))$; for $L\gg \xi$ the bulk result is recovered. Periodic tangent fluctuations are a mathematical boundary convention, not a claim that a pulled physical chain forms a closed loop.

The persistence length is the decay length of tangent correlations. A common three-dimensional convention is $L_p=A/(k_{B}T)$; a strictly planar chain has tangent-correlation length $2A/(k_{B}T)$.