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Magnetic anisotropy can create a restoring torque that aligns a specimen in a magnetic field. For a small linearly responding object with uniaxial susceptibility difference $\Delta \chi$, volume $V$ and axis angle $\theta$ in an approximately uniform field, an orientation-dependent energy is $U=-V\Delta \chi B^2{\cos }^2\theta /(2{\mu }_0)$, up to an angle-independent term.

Positive $\Delta \chi$ favors alignment along the field; negative $\Delta \chi$ favors a perpendicular axis. Thermal angular fluctuations compete with this energy, so a preferred orientation becomes pronounced when the anisotropy scale exceeds $k_{B}T$. This induced-moment mechanism differs from the $-\mathbf{\mu }\cdot \mathbf{B}$ energy of a fixed permanent moment and is relevant to interpreting magnetic tweezers.

A magnetic field gradient describes how a magnetic field varies with position. Because the field is a vector field, its complete local variation is the tensor with components $\partial B_i/\partial x_j$; the gradient of the field magnitude is a different, vector-valued quantity.

A fixed magnetic dipole moment in a nonuniform field experiences a force $\mathbf{F}=\nabla (\mathbf{\mu }\cdot \mathbf{B})$ in the simple localized-dipole model. A uniform field can exert a torque without that translational force. This distinction underlies spatial separation in the Stern-Gerlach experiment and forces in magnetic tweezers. Measuring the field at neighboring positions tests a gradient; one field reading cannot determine it.

RNA polymerase first recognizes and binds a gene promoter to form a closed complex with the local DNA duplex still paired. Local strand separation then produces a transcription bubble in an open promoter complex, making the template available for RNA synthesis. Recognition, binding, and opening are distinct events; observing binding alone does not demonstrate strand opening.

In magnetic tweezers, a DNA tether joins a fixed surface to a magnetic bead, with attachments that prevent free swivelling. The magnets pull upward and can impose turns; the bead height reports tether extension. For a torsionally constrained tether, linking number is partitioned into twist and writhe, $Lk=Tw+Wr$. If opening the transcription bubble removes helical twist while $Lk$ is fixed, compensating twist or writhe appears elsewhere. In a supercoiled regime, changes in writhe change the extension, making an opening event visible as a bead-height step.

The sign provides an important control. On a positively supercoiled tether, local unwinding requires more positive compensating writhe and generally shortens the tether. On a negatively supercoiled tether, the same unwinding removes some negative writhe and generally lengthens it. A bending or compaction contribution can have the same sign in both cases, whereas the torsional contribution reverses. Comparing signs, together with an extension-versus-turns response measured for the tether, distinguishes local unwinding from mere binding-induced bending. These claims assume the tether remains torsionally constrained and lies in the calibrated supercoiled regime, rather than an arbitrary stretched-DNA regime.

At initiation, RNA polymerase recognizes a promoter and forms a closed bound complex; bacterial promoter recognition involves its specificity factor. Local strand opening exposes the template in an open complex, after which initial RNA synthesis can precede promoter escape into elongation. Binding, opening and productive escape are distinct transitions.

For DNA unwinding measured by magnetic tweezers, magnetic tweezers report the mechanics of a torsionally constrained DNA tether through the position of an attached magnetic bead. Axial magnetic force extends the tether, while imposed rotation controls its linking constraint. Video localization measures extension changes. The important physical relation isOpening a small helical region decreases twist, so the rest of the tether acquires compensating positive writhe. On a negatively supercoiled tether this removes some writhe magnitude and raises its extension; on a positively supercoiled tether it increases writhe magnitude and lowers extension. A change with opposite signs under the two torsional conditions therefore distinguishes unwinding from simple compaction, which would shorten both tethers. Calibrating extension against imposed turns converts a mechanical step into a change in helical winding. This is a conceptual readout of local opening, rather than a direct optical image of separated bases.^[ref]

Magnetic tweezers apply force and controlled turns to a single torsionally constrained DNA tether joining a surface and a magnetic bead. A magnetic-field gradient pulls the bead; rotation of the field rotates the bead and changes imposed turns. Imaging measures bead height, hence DNA extension. At sufficient excess twisting and suitable force, DNA supercoiling creates writhed loops that shorten the tether. The extension-versus-turn curve supplies a calibration relating the removal or addition of supercoiling to vertical motion. Multiple attachments at both ends are needed conceptually to constrain rotation; freely swivelling DNA would relax the signal.

During transcription initiation, RNA polymerase binds a promoter in a closed complex and then opens a local DNA bubble, exposing template for RNA synthesis. In bacterial initiation a sigma factor helps recognize the promoter. Opening changes local helical twist. Because the total linking number is constrained,Unwinding the bubble means $\Delta T{w}_{\mathrm{bubble}}<0$, so the remainder acquires a positive turn increment. A negatively supercoiled tether therefore loses some negative writhe and becomes longer; a positively supercoiled tether gains positive writhe and becomes shorter. Ordinary DNA compaction instead shortens both tethers. Comparing both signs separates local unwinding from compaction rather than assigning every height change to opening.

If $q$ is the number of turns unwound, $s$ the magnitude of the extension change per imposed turn, and $c$ the compaction, the symmetric-slope approximation predicts $\Delta z_{-}=sq-c$ and $\Delta z_{+}=-sq-c$. HenceMeasured unequal slopes can be retained in the two separate calibration equations. Multiplying $q$ by the helical repeat converts turns to an estimated number of unwound base pairs, subject to the mechanical model and the distinction between twist and writhe.

Repeated extension transitions reveal open-complex lifetimes, closed-complex lifetimes and their dependence on mechanical torque. Initial short RNA synthesis can occur while RNA polymerase remains associated with the promoter, including abortive initiation and DNA scrunching. Addition of nucleotides allows progression to promoter escape; an unwound bubble can then persist with elongation, so loss of the bubble is not the definition of escape. Successive cumulative unwinding events can identify renewed access to the same promoter after clearance. These are the measurement principles demonstrated by the paired-sign experiments.^[ref]

The experiment observes mechanics rather than directly imaging individual opened base pairs. Drift, nonspecific binding, tether integrity, force-dependent buckling and limited time resolution can mimic or obscure steps. A control lacking productive promoter recognition and agreement of the opposite-sign responses help identify the origin of a signal.

The bacterial RNA polymerase core associates with a sigma factor to form a holoenzyme. The sigma factor recognizes promoter features and helps position the DNA and stabilize its opening. Initial binding gives a closed promoter complex; local DNA unwinding creates an open promoter complex with the template accessible. Initial RNA synthesis can produce short abortive products while the RNA polymerase remains attached to the promoter, taking in downstream DNA by scrunching. Growth of the RNA eventually permits promoter escape and productive elongation. The sigma factor changes its interactions during this transition; immediate complete dissociation is not obligatory.

In magnetic tweezers, torsionally constrained DNA joins a surface to a magnetic bead. A field gradient applies upward force, rotation of the magnets controls the imposed DNA supercoiling, and optical tracking measures bead height. At suitable modest force, DNA supercoiling forms writhed loops, shortening the tether. Opening roughly one helical turn at fixed linking number makes the remaining DNA more positively twisted: $\Delta T{w}_{\mathrm{rest}}+\Delta Wr=-\Delta T{w}_{\mathrm{bubble}}$. Thus opening lengthens a negatively supercoiled tether by removing some negative writhe, but shortens a positively supercoiled tether by adding positive writhe. This opposite sign distinguishes unwinding from ordinary compaction, which shortens both tethers. Height-versus-turn calibration converts the two extension changes into an unwinding angle and a compaction contribution. Repeated height transitions give opening and closing dwell times.^[ref]

The sketch is a schematic prediction, not a copy of an experimental trace. Opening must be distinguished from promoter escape: an unwound DNA region can remain associated with an elongating RNA polymerase.

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.

The generic instrument uses a magnetically responsive bead, a controllable external field and optical tracking of bead position or orientation. A field gradient supplies translational force; rotating an appropriate anisotropic field or bead moment can supply torque. For a permanent dipole,For an isotropic unsaturated induced moment with effective susceptibility contrast $\Delta \chi$ and bead volume $V$, the induced magnetic bead force instead follows fromThe factor one-half accounts for inducing the moment. A spatially uniform field alone supplies no translational force; an isotropic induced moment parallel to the field supplies no orienting torque either. Saturation or magnetic anisotropy requires the corresponding magnetization model.

Position tracking needs a length calibration, and force calibration needs measured field gradients, a known reference load, or calibrated thermal fluctuations. For a generic tensioned tether of extension $\ell$, small transverse displacement $x$ costs energy $F{x}^2/(2\ell )$, so $F=k_{B}T\ell /⟨x^2⟩$ under equilibrium, small-angle and single-mode assumptions. Attachment geometry, bead rotation, wall drag and finite acquisition bandwidth can affect this calibration. These equations describe generic magnetic tweezers mechanics, without assigning a biological response.

TODO: biological single-molecule applications are user-deferred; no biological systems or protocols have been researched or solved.