Biopolymer mechanics

Biopolymer mechanics studies bending, stretching, thermal fluctuations, and force-induced deformation of filamentous biological macromolecules.

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.

A homogeneous DNA zipper permitting exactly one contiguous paired block of length $j\ge 1$ has $N-j+1$ block positions and weight $\kappa s^{j-1}$. Consequently its partition function is $q=\kappa {\sum }_{j=1}^N(N-j+1)s^{j-1}=\kappa [s^{N+1}-(N+1)s+N]/(s-1{)}^2$. The removable value at $s=1$ is $\kappa N(N+1)/2$. At large s, $q\sim \kappa s^{N-1}$. Including an unpaired state gives the full partition function $1+q$; the large-s limit is not uniform in arbitrarily small initiation weight.

A homogeneous Zimm-Bragg chain assigns propagation weight $s>0$ to each helical unit and initiation factor $\sigma >0$ to each helical run. With states ordered as coil and helix, one possible transfer matrix for a classical spin chain isIts largest eigenvalue is ${\lambda }_{+}=(1+s+\sqrt{(s-1{)}^2+4s\sigma })/2$. In the long-chain limit the partition function grows as ${\lambda }_{+}^N$, so logarithmic differentiation gives the helical fraction $f_h=s\partial \ln {\lambda }_{+}/\partial s$, equal to the displayed expression. If $Q=(s-1{)}^2+4s\sigma$, thenFor $\sigma =1$, $f_h=s/(1+s)$ and units are noncooperative. A small initiation factor suppresses boundaries and sharpens the crossover around $s=1$, with width of order $\sqrt{\sigma }$. For every fixed $\sigma >0$ the long-chain expression is smooth; a discontinuous step appears only in the singular $\sigma \downarrow 0$ limit. ^[ref]

The drag of a translating polymer depends on conformation and hydrodynamic interactions. A nondraining coil has $\zeta \sim 6\pi \eta R_h$; a long aligned rod has ${\zeta }_{\parallel }\sim 2\pi \eta L/\log (L/a)$ and roughly twice that transverse drag; a freely draining model sums the monomer drag coefficients. Mechanical power is $\zeta U^2$. These models cannot be interchanged without changing the assumed conformation and solvent-mediated coupling.

A polymerization Brownian ratchet advances a filament by incorporating a monomer when thermal motion opens enough space between its tip and a resisting load. The irreversible chemical step rectifies the load's fluctuations. The elementary load advance $a$ depends on tip geometry; two staggered strands can give $a=\delta /2$, whereas two aligned strands do not automatically give this step. An effective addition rate must refer to the same elementary advance.

When load-gap equilibration is fast compared with chemical events, an opposing force gives an exponential gap distribution. The probability of room for an addition is $e^{-Fa/(k_{B}T)}$. With unloaded addition rate $u$ and removal rate $w$, assumed force-independent except for this steric gating, the velocity is the displayed expression. For $u>w$, stall occurs at $F_s=(k_{B}T/a)\log (u/w)$. Concentration is included in $u=k_{\mathrm{on}}c$ if the on-rate constant is bimolecular.

In the instantaneous-addition, negligible-removal ratchet, the gap reflects at zero and a first passage to $a$ triggers addition and resets the gap to zero. With load diffusion coefficient $D$ and opposing drift $DF/(k_{B}T)$, the mean passage time solves $D{T}^{\prime \prime }-DF{T}^{\prime }/(k_{B}T)=-1$, $T^{\prime }(0)=0$, $T(a)=0$. It is $T(0)=a^2(e^z-1-z)/(D{z}^2)$, giving the displayed speed and zero-load limit $2D/a$. The finite-load slowdown is not simply $e^{-z}$ in this diffusion-limited regime.

For independent sliding filaments in a filled bundle, bending energies add and $B\sim NE{r}^4\sim E{r}^2{R}^2$. A coherently bonded one-filament-thick tube has $B\sim Er{R}^3$, while a bonded filled rod has $B\sim E{R}^4$. The same torsional scalings apply with the shear modulus instead of Young's modulus if individual sliding filaments are forced to undergo a common material twist. Completely free filament spin/slip does not define that torsional response: collective helical bending alone begins at fourth order in the twist rate.

For a weakly adhering elastic sphere, balancing elastic contact energy $G{a}^5/R^2$ against adhesion $J{a}^2$ gives contact radius $a\sim (J{R}^2/G{)}^{1/3}$. If peeling the rear contact dissipates work of order $J$ per unit area, its rolling moment is $M_r\sim JaR$. Balancing this with a fluid shear moment $\eta \dot{\gamma }{R}^3$ gives the displayed estimate. A perfectly reversible elastic contact has no such net rolling-energy barrier; adhesive hysteresis or bond kinetics must justify a finite threshold, and the corresponding dissipative adhesion energy replaces $J$.

For a long self-avoiding polymer in a tube of diameter $D$, represent each confinement blob by $g$ statistical segments of length $b$, with $D\sim b{g}^{3/5}$. Excluded-volume repulsion orders the blobs along the tube, giving $X\sim (N/g)D=L(b/D{)}^{2/3}$. More generally, a segment excluded-volume parameter $v$ gives $X\sim L[v/(b{D}^2){]}^{1/3}$. This requires many blobs and $D$ smaller than the free coil size; $b\ll D\ll L$ alone does not ensure that. An ideal chain without intersegment repulsion is not a one-dimensional string of nonoverlapping blobs.

In a slit, a confinement blob is three-dimensional but the chain of blobs forms a two-dimensional self-avoiding walk. With $g\sim (D/b{)}^{5/3}$ and in-plane exponent $3/4$, $R_{\parallel }\sim D(N/g{)}^{3/4}$. There is no single preferred axial extension as in a tube. A strictly planar chain confined further to a narrow strip is a different geometry and gives $X\sim L(b/D{)}^{1/3}$ under the two-dimensional blob estimate.

An ideal Gaussian chain is a polymer model whose independent segment vectors have zero mean and isotropic Gaussian distribution. If $⟨|\mathbf{r}{|}^2⟩=b^2$, each Cartesian variance is $b^2/3$, so a displacement across $\ell$ links has density $(3/(2\pi \ell b^2){)}^{3/2}\exp [-3R^2/(2\ell b^2)]$ and characteristic function $\exp [-k^2{b}^2\ell /6]$. The continuous-contour version uses the same characteristic function for any contour separation. It is the large-scale limit of a freely jointed chain, while allowing unbounded Gaussian segment lengths and omitting interactions between distant parts of the chain.

A continuum Gaussian chain has separation characteristic function $e^{-a|s-t|}$, $a=k^2{b}^2/6$. Integrating over a contour of length $N$ gives the polymer scattering functionwhere $R_g^2=N{b}^2/6$ in this continuum convention and $\mathcal{D}(x)=2(e^{-x}-1+x)/x^2$, with continuous value $\mathcal{D}(0)=1$. Its limits are $\mathcal{D}(x)=1-x/3+x^2/12+\cdots$ and $\mathcal{D}(x)\sim 2/x$ for large $x$.

For a discrete chain with $N$ sites, retain the self terms and let $z=e^{-k^2{b}^2/6}$. The exact finite expression is $g_N=1+(2/N){\sum }_{\ell =1}^{N-1}(N-\ell )z^{\ell }=(1+z)/(1-z)-2z(1-z^N)/[N(1-z{)}^2]$, with $g_N(0)=N$. Taking $N\to \infty$ while $x=N{k}^2{b}^2/6$ stays fixed gives $g_N/N\to \mathcal{D}(x)$. The continuum expression describes the coil scaling range, whereas the discrete expression tends to one as $kb\to \infty$.

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.

A freely jointed chain supplemented with axial segment compliance. A simple phenomenological additive law is $z=L\mathcal{L}(fb/(k_{B}T))+Lf/S$, where $L=Nb$, $\mathcal{L}$ is the Langevin function and $S$ is an axial stretch modulus with units of force. This adds energetic compliance to the entropic spring; it is not an exact angular average for arbitrary harmonic-link statistics.

A freely rotating chain has fixed bond length and fixed angle between successive forward bond vectors, with independently uniform torsion angles. It neglects interactions between distant chain segments. The fixed angle introduces short-range orientational correlations even though the torsional rotations are unrestricted. Its characteristic ratio of a freely rotating chain converts those correlations into an end-to-end size; its equivalent long-chain link is the Kuhn length of a freely rotating chain.

With independently uniform torsion, consecutive forward tangent correlations multiply by $\cos \vartheta$. After $n$ bonds the correlation is $(\cos \vartheta {)}^n$. Matching it to $e^{-nb/{\ell }_p}$ gives the displayed decay length for positive correlation. This discrete orientational definition need not equal half the equivalent Kuhn length exactly.

For $N$ bonds of length $a$, fixed forward angle $\theta$ and uniform independent torsions, conditional averaging gives $⟨{\mathbf{a}}_i\cdot {\mathbf{a}}_j⟩=a^2{\alpha }^{|j-i|}$. Consequently $⟨R^2⟩=N{a}^2+2a^2{\sum }_{k=1}^{N-1}(N-k){\alpha }^k$. Dividing by the independent-step result $N{a}^2$ and summing the geometric series gives the displayed $C_N$, for $\alpha \ne 1$. For fixed $|\alpha |<1$, $C_N\to C_{\infty }=(1+\alpha )/(1-\alpha )$.

The angle is measured between forward vectors; a chemical internal bond angle is supplementary. At $\alpha =1$, the finite chain is a rigid rod with $C_N=N$, so taking the infinite-chain limit before the rigid limit would be invalid. The Kuhn length of a freely rotating chain is asymptotically $a{C}_{\infty }$.

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.

Entropic elasticity is a restoring response arising from the reduction of accessible molecular configurations under deformation. For an ideal freely jointed chain, weak extension has free energy $\Delta \mathcal{F}=3k_{B}T{R}^2/(2N{b}^2)$ and spring constant $k=3k_{B}T/(N{b}^2)$. The proportionality to temperature assumes fixed segment geometry; temperature-dependent Kuhn length modifies it.

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.

An effective step length describing a specified chain-count convention: an ideal polymer with $N$ counted units has mean-squared end-to-end distance of a polymer chain $⟨R^2⟩=N{b}^2$. If $N$ counts freely jointed coarse-grained units, $b$ is their Kuhn length. If $N$ counts chemical repeat units, the effective $b$ incorporates local conformational correlations and is not automatically a physical monomer diameter. A repeat-unit volume $v_0$ is independent information; setting $v_0\sim b^3$ is a scaling approximation.

For $N$ independent links pointing left or right with length $a$, fixed signed extension $L$ gives $N_R=(N+L/a)/2$ and $N_L=(N-L/a)/2$. The multiplicity is $\left(\genfrac{}{}{0ex}{}{N}{N_L}\right)$ when these counts are nonnegative integers. Leading Stirling's approximation gives $\mathcal{F}=(k_{B}T/(2a))\ln (N_R/N_L)$ as the externally applied tension conjugate to $L$. Its small-extension limit is $k_{B}TL/(N{a}^2)$. At fixed tension, the exact one-link partition function is $2\cosh (\beta \mathcal{F}a)$ and the mean extension is $Na\tanh (\beta \mathcal{F}a)$, confirming the same inverse relation without using Stirling at the full-extension endpoint.

The exact count is $\Omega =\left(\genfrac{}{}{0ex}{}{N}{q}\right)=N^q{\prod }_{j=0}^{q-1}(1-j/N)/q!$. For fixed $q$ and large $N$, this is asymptotic to $N^q/q!$. Uniform relative accuracy requires $q^2/N\ll 1$, since the logarithm of the product starts at $-q(q-1)/(2N)$. The weaker assumption $q/N\ll 1$ gives the leading sparse-link entropy but need not give a small relative multiplicity error. In the continuum range $1\ll q\ll N$, the entropic force behaves as $(k_{B}T/(2a))\ln (N/q)$. The fixed-tension mean extension approaches $Na$ only as the tension diverges; at the discrete endpoint $q=0$ the exact entropy is zero and a continuous fixed-extension derivative is not defined.

A three-dimensional ideal chain of $N$ independent segments of length $b$ has $⟨R^2⟩=N{b}^2=Lb$. The isotropic small-force susceptibility gives $⟨z⟩=F⟨R^2⟩/(3k_{B}T)$, so $k_s=3k_{B}T/(Lb)$. For a long worm-like chain, the statistical segment is $b=2l_p$. This stiffness describes entropic end-to-end extension, not axial stretching of the material backbone, whose spring constant is $EA/L$.

For an isotropic fixed-length step $b$, angular integration gives the characteristic function $\hat{\psi }(\mathbf{k})=\sin (kb)/(kb)$. Independence makes the characteristic function of the sum of $N$ steps its $N$th power. Since $\ln [\sin (kb)/(kb)]=-k^2{b}^2/6+O(k^4{b}^4)$, the central scaling $k=O((b\sqrt{N}{)}^{-1})$ gives $\hat{\Phi }\to \exp [-N{b}^2{k}^2/6]$. The inverse Fourier transform isThis is a density with respect to three-dimensional vector volume, not a radial density; the radial density is $4\pi R^2\Phi$. It has $⟨R^2⟩=N{b}^2$. The approximation holds on the central coil scale $R=O(b\sqrt{N})$, not near the fully stretched finite-length boundary $R=Nb$.

If each link is restricted to a plane, $Z_1={\int }_0^{2\pi }{e}^{\xi \cos \theta }d\theta =2\pi I_0(\xi )$, with $\xi =Fb/(k_{B}T)$. Differentiating gives $L/(Nb)=I_1(\xi )/I_0(\xi )$, where $I_n$ are Modified Bessel functions of the first kind. The high-force angular Gaussian has $⟨{\theta }^2⟩\sim 1/\xi$, so $Nb-L\sim N{k}_{B}T/(2F)$, half the three-dimensional coefficient.

Independent link orientations make the canonical partition function $Z_1^N$, with the one-link integral defining the Langevin function. Differentiating its logarithm with respect to force gives $L=Nb\mathcal{L}(Fb/(k_{B}T))$. At high extension, $Nb-L\sim N{k}_{B}T/F$, so the force diverges inversely with remaining length deficit. This requires rigid links and excludes axial bond stretching.

The Langevin function is the mean projection of a freely orientable three-dimensional unit vector weighted by $e^{\xi \cos \theta }$. Integrating over solid angle gives $Z_1=4\pi \sinh \xi /\xi$, and differentiation gives $⟨\cos \theta ⟩={\partial }_{\xi }\log Z_1=\coth \xi -1/\xi$. It behaves as $\xi /3$ at small argument and $1-1/\xi$ at large positive argument.

An inextensible filament keeps the length of every material segment fixed. With material arc length $s$, its centerline obeys $|{\mathbf{r}}_s|=1$ and has unit tangent vector $\mathbf{t}={\mathbf{r}}_s$. A filament tension acts as a Lagrange multiplier for this constraint.

For a periodic inextensible filament with arclength period $\Lambda$ and projected axial period $\lambda$, material points slide tangentially through the wave frame at speed $Q$. An axial pattern speed $V$ relative to mean filament motion gives $Q\lambda /\Lambda =V$. Confusing the two speeds changes the swimming-to-wave-speed ratio by $\lambda /\Lambda$.

Write a small planar displacement in arclength coordinates as $X=s+X_2$ and $Y=y$. The inextensibility relation $X_s^2+Y_s^2=1$ gives $X_{2,s}=-y_s^2/2$ to quadratic order. Thus longitudinal material velocity enters one order later than the transverse velocity. For a periodic material stroke with no mean axial drift, its mean velocity vanishes, although its instantaneous contribution is needed for exact local force balance.

The filament bending modulus multiplies squared curvature of a space curve in the bending energy, $E_b=(B/2)\int {\kappa }^{2}ds$. It has dimensions force times length squared. In the small-slope Monge representation, the leading elastic force per unit length is $-B{y}_{xxxx}$.

Filament tension is the axial force enforcing local inextensibility. Its force density is ${\partial }_s(T{\mathbf{r}}_s)$. A pre-existing constant tension contributes $T{y}_{xx}$ to small-slope transverse dynamics; in an unloaded, initially straight filament, tension induced by transverse motion begins at second order in amplitude.

The Monge representation describes a nearly flat curve or surface as a single-valued height over a reference line or plane. For a planar curve $(x,h(x))$ with small slope, arclength is $ds=dx+O(h_x^2)$ and signed curvature is $\kappa =h_{xx}+O(h_x^2{h}_{xx})$.

In local Stokesian filament dynamics, viscous drag balances the variational elastic force without inertia. A small-slope filament of bending modulus $A$ and transverse drag $\zeta$ obeys $\zeta h_t=-A{h}_{xxxx}$.

The overdamped filament bending equation diagonalizes in the complete free-free bending spectrum. Positive modes decay at rates $A{k}_n^4/\zeta$, while translation and tilt remain fixed. The long-time shape is the affine least-squares projection of the initial height. Free-end boundary values conserve the total height integral and its first spatial moment. For square-integrable initial data the expansion gives the smoothing semigroup solution.

For a straight, unloaded inextensible filament with small transverse displacement, resistive-force theory gives viscous force density $-c_{\perp }{y}_t$. Variation of the bending energy gives $-B{y}_{xxxx}$, while the induced filament tension is higher order. Their instantaneous balance yields $c_{\perp }{y}_t=-B{y}_{xxxx}$. It is an overdamped fourth-order diffusion equation: short-wavelength bends relax much faster than long ones. Wiggins and Goldstein's flexive-propulsion paper develops the elastic-wave mechanism for low-Reynolds-number propulsion.

Let positive transverse displacement point along a uniform load per unit length $w$. The filament bending modulus $B$ gives energy ${\int }_0^L[B(h_{xx}{)}^2/2-wh]dx$. The variational derivative is $B{h}_{xxxx}-w$, so resistive-force theory gives ${\zeta }_{\perp }{h}_t=-B{h}_{xxxx}+w$. Clamping imposes $h(0)=h_x(0)=0$, while a free tip imposes $h_{xx}(L)=h_{xxx}(L)=0$. The steady solution isIts tip displacement is $w{L}^4/(8B)$. For a submerged cylindrical filament under Newtonian gravity, buoyancy gives $w=({\rho }_f-\rho )\pi a^{2}g$. This linear approximation requires small slope and negligible inertia.

For a uniform-load bending of a clamped filament, total load $F=wL$ and tip displacement $\delta =w{L}^4/(8B)$ obey Hooke's law in the form $F=k_{\mathrm{eff}}\delta$, with $k_{\mathrm{eff}}=8B/L^3$. This stiffness depends on the specified distributed loading. A concentrated tip force instead gives $3B/L^3$, so equal total forces with different distributions are not interchangeable.

The anisotropic-drag contribution to axial force from a small-slope transverse motion is $F=(c_{\perp }-c_{\parallel })\int y_x{y}_{t}dx$. Using the small-slope elastohydrodynamic filament equation and integration by parts givesThus this propulsive contribution is determined by endpoint slope, bending moment, and shear. A separately imposed axial translation adds ordinary longitudinal drag. Exact second-order longitudinal motion required by inextensibility also contributes to instantaneous total drag, but its time-derivative contribution averages to zero over a deformation cycle.

In units of elastohydrodynamic penetration length and inverse forcing frequency, $y_t=-y_{xxxx}$. Prescribing $y(0,t)=y_0\cos t$, zero bending moment $y_{xx}(0,t)=0$, and decay at infinity givesThe two components have phase velocities $-1/S$ and $1/C$. Their unequal spatial attenuation produces a nonreciprocal shape cycle even though the endpoint actuation is reciprocal.

A microtubule is a stiff hollow cytoskeletal filament assembled from tubulin. Its millimetre-scale persistence length makes bending fluctuations measurable over micrometre lengths.

A clamped filament of length $L$ and bending modulus $A$ has equilibrium tip variance $k_{B}T{L}^3/(3A)$. Under local transverse viscous drag, each bending mode relaxes exponentially at a rate proportional to its fourth power of wavenumber.

The bending modes of a filament clamped at one end and force-free and torque-free at the other obey $\cos {\alpha }_n\cosh {\alpha }_n=-1$. The slowest mode has ${\alpha }_1\simeq 1.875$.

For the positive roots $q_n$ of $\cos q\cosh q=-1$, ${\sum }_n{q}_n^{-4}=1/12$. The tip-force compliance of a cantilever is $L^3/(3A)$ directly, whereas expansion in normalized clamped--free bending modes and $W_n(L{)}^2=4L^{-1}\int W_n^{2}dx$ gives $4L^3{\sum }_n{q}_n^{-4}/A$. Equating the two responses proves the sum and evaluates the full thermal endpoint variance.

The elastic energy $U=A\int (W^{\prime \prime }{)}^{2}dx/2$ minimized without forcing gives $W^{\prime \prime \prime \prime }=0$. With fixed $\int W^{2}dx$, stationarity of $U-A{k}^4\int W^{2}dx/2$ gives $W^{\prime \prime \prime \prime }=k^{4}W$: a bending eigenmode. A general admissible shape expands in these normal modes. The natural boundary conditions for a free endpoint are $W^{\prime \prime }(L)=W^{\prime \prime \prime }(L)=0$, while the clamp imposes $W(0)=W^{\prime }(0)=0$.

For ${\zeta }_{\perp }{h}_t=-B{h}_{xxxx}$, separated normal modes decay at rates $B{\alpha }_n^4/({\zeta }_{\perp }{L}^4)$. Clamping at zero and vanishing moment and shear at $L$ give $\cos {\alpha }_n\cosh {\alpha }_n=-1$. A mode isThe fourth-order operator is self-adjoint: two integration by parts operations give $\int f{g}^{\prime \prime \prime \prime }=\int f^{\prime \prime }{g}^{\prime \prime }$ under these boundary conditions. Its zero eigenspace is trivial and its eigenfunctions form an orthogonal complete basis. Thus a nonzero projection on the first mode gives long-time decay proportional to ${\varphi }_1(x/L)\exp [-B{\alpha }_1^{4}t/({\zeta }_{\perp }{L}^4)]$, where ${\alpha }_1=1.8751040687\dots$. If this projection vanishes, the lowest excited mode replaces it.

A filament free of moment and shear at both ends has $h_{xx}=h_{xxx}=0$. Its positive biharmonic eigenvalues $q_n^4$ obey $\cos (q_{n}L)\cosh (q_{n}L)=1$, in addition to the two rigid zero modes representing translation and rotation.

A follower force acts along the instantaneous tangent at its point of application. Its direction changes with the filament, making its generalized force nonconservative and allowing oscillatory instabilities in overdamped dynamics.

A two-link follower-force model replaces a filament by two rigid links with torsional springs and point drags. The nonsymmetric linearized stiffness can produce a Hopf bifurcation even without inertia.

The dimensionless follower load compares the moment scale $\Gamma \ell$ of a tangential tip force with a torsional spring constant $k$. In the equal-link two-degree-of-freedom model, straight motion loses stability through a Hopf bifurcation at $\Sigma =3$.

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 $2A/(k_{B}T)$.

A pinned filament of length $L$ and bending modulus $A$ first buckles under compressive force $F_c={\pi }^{2}A/L^2$.