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For the generic instrument, magnetic tweezers combine a magnetically responsive bead, a controllable magnetic field gradient, a tether attached to a fixed support, and optical tracking of bead coordinates. For a fixed magnetic dipole moment, the force is $\nabla (\mathbf{m}\cdot \mathbf{B})$; an induced linear isotropic moment instead gives $\mathbf{F}=V\Delta \chi \nabla B^2/(2{\mu }_0)$. A uniform magnetic field can orient an anisotropic bead, but a gradient is needed for translation. Magnet height sets the load, while field rotation can impose torque if bead orientation is coupled to the field. Tracking uses lateral images and a calibrated axial image/defocus response; a fixed reference removes stage drift. A sufficiently slowly varying gradient approximates a force clamp over the bead's displacement. TODO: user-deferred biomolecular implementation; DNA-specific attachment, preparation and response are not treated.

For transverse-fluctuation calibration of a tensioned tether, use an ideal tether of vertical extension $h$ whose endpoint is displaced by small $x,y$. Keeping its length fixed to quadratic order lowers its vertical coordinate by $(x^2+y^2)/(2h)$, increasing load potential by $F_z(x^2+y^2)/(2h)$. The equipartition theorem therefore givesHere fluctuations are about the mean. The height is the effective pendulum length of the measured coordinate, not automatically the polymer contour length. The displayed $h$ formula uses the point-endpoint or fixed-bead-orientation idealization; freely rotating large beads can require a bead-radius correction to that lever arm. The finite-exposure correction to fluctuation force calibration restores variance lost through exposure averaging; ignoring it would overestimate force; localization noise increases it and would underestimate force. Drift, wall coupling, finite observation time and magnetic lateral stiffness must be checked. Vertical fluctuations measure longitudinal stiffness and do not obey the same geometrical formula.

For the stretch and twist coefficients of an elastic filament, take $s$ as reference contour length and $ϵ$ as local axial strain. The three positive quadratic costs represent bending of the tangent, stretching of the material and excess twist about its relaxed orientation. With the paper's coefficient convention,$K_b,K_t$ are bending and torsional rigidities with units of energy times length; $S$ is an axial force modulus. Thus $A,C$ have units of length and $B$ inverse length. Positive quadratic forms describe a stable relaxed filament; odd terms vanish when expansion is about its zero-stress, zero-torque reference configuration.

At weak and intermediate tension, changing thermal bending conformations can extend the filament without appreciably stretching the backbone. Axial strain is $ϵ\simeq F_z/S$, so material stretching must be included when this is no longer negligible. For a long strongly aligned but nearly inextensible three-dimensional worm-like chain, the thermal extension deficit is approximately $\frac{1}{2}\sqrt{k_{B}T/(A{F}_z)}$; stretch adds approximately $F_z/S$ to fractional extension. Twist matters whenever torque or imposed end rotation is present, at any tension. A freely swivelling, torque-free tether need not store excess twist. There is no universal force-only threshold for switching on torsional energy.

The circular bending energy of an elastic filament follows because curvature is $1/R$ over contour length $2\pi R$. Integrating the bending term givesThis is bending energy alone: imposed tension, stretching and twist would add costs. The dimensionless bending action is $(A/2)\int |{\partial }_s\mathbf{t}{|}^{2}ds$. Its tangent propagator in three dimensions obeys rotational diffusion on the unit sphere, ${\partial }_{s}P=(2A{)}^{-1}{\nabla }_{S^2}^{2}P$. The vector spherical harmonic has angular eigenvalue $-2$, givingHence $A$ is the persistence length in this three-dimensional convention. A strictly planar filament instead has tangent-correlation length $2A$; one cannot infer the factor merely by setting a loop's energy equal to $k_{B}T$.

TODO: user-deferred biomolecular applications. Both enzyme-driven unwinding experiments, their experimental sketches, the requested numerical force and plectoneme radius, energy loss per turn and enzyme-specific dissipation mechanisms remain unsolved. None of their biomolecular parameters has been used in a calculation.