heteroduplex DNA is a duplex whose strands originate from different parental molecules. Related but nonidentical sequences can introduce mismatches; heteroduplex formation does not require that every position be mismatched.

Twisting of the axis of double-stranded DNA associated with torsional constraint. Supercoiling can influence the energetic cost of local strand opening.

A-DNA is a right-handed duplex family with predominantly C3′-endo sugar puckering, tilted and displaced base pairs, and a deep narrow major groove. Double-stranded RNA and many DNA-RNA hybrids have related A-form geometry.

B-DNA is the familiar hydrated right-handed duplex family, with predominantly C2′-endo sugars and accessible major groove base edges. Its local geometry varies with sequence and bound molecules; approximately 10.5 base pairs per turn is an average rather than an exact universal repeat.

Z-DNA is a left-handed duplex family with a zigzag backbone and alternating nucleotide conformations. Alternating purine–pyrimidine sequences can favour it; local torsional stress and binding partners influence its occurrence. Left-handed nucleosome wrapping is a different structural level.

A DNA double helix contains two antiparallel, base-paired DNA strands wound around a common axis. Handedness and groove geometry can change between conformational families without changing the basic fact of duplex pairing.

The major groove of an ordinary B-DNA duplex exposes base-pair edges whose donor, acceptor and hydrophobic patterns distinguish oriented base pairs. Groove accessibility changes with helical conformation, and protein recognition can combine these patterns with backbone and shape contacts.

The minor groove of duplex DNA exposes a different set of base edges from the major groove. Similar chemical patterns for some oriented base-pair alternatives limit simple direct readout, but groove shape, hydration and electrostatics can still contribute to specificity.

Negative supercoiling describes torsionally constrained DNA with linking number below its relaxed reference value. Under appropriate conditions this stress can favour local strand opening or a transition to a different helical conformation, including Z-DNA.

DNA looping brings separated sequence positions into spatial proximity through bending and folding of the DNA. In transcriptional regulation it can permit a distant regulatory protein to influence promoter-associated machinery, although proximity alone does not prove productive activation.

DNA recovered from historical or prehistoric biological material. Authentication must distinguish endogenous sequence from contamination and account for short fragments and damage.

A chemical or structural alteration of DNA that can disrupt its normal function or copying. Damage, activation of a damage response and successful repair are separate measurements.

DNA hybridization is association of nucleic-acid strands through complementary base pairing. It can form DNA-DNA or DNA-RNA duplexes, depending on the partners.

Sequence complementarity, temperature, ionic conditions, and strand concentration affect the equilibrium and rate of association. Mismatches can weaken binding, which makes hybridization useful for sequence-selective detection without making it an absolute all-or-nothing identity test.

An assay asking whether nucleic acid remains inaccessible to a nuclease under specified conditions. Protection can reflect protein binding or compartmental enclosure; appropriate disruption controls distinguish them.

Consider two identical quantum harmonic oscillators with a stable coordinate coupling,The normal mode coordinates $x_{\pm }=(x_1\pm x_2)/\sqrt{2}$ have positive frequencies ${\omega }_{\pm }=\sqrt{{\omega }_0^2\pm \kappa }$. In their joint ground state, the first oscillator has zero means andThese are exactly the second moments of a thermal density operator for an effective oscillator withThe partial trace of the Gaussian quantum state remains a Gaussian quantum state, so matching the second moments identifies its full density operator. In the effective number state basis it isHere $q=e^{-{\beta }_{\mathrm{eff}}\hslash \nu }$; zero coupling gives $q=0$ and a pure ground state. The effective frequency generally differs from the bare ${\omega }_0$. Thus thermality in this effective oscillator basis does not imply a diagonal thermal mixture in the original bare-frequency basis; in those bare coordinates it is generally a squeezed thermal state.

The one-body density matrixuses a creation operator and an annihilation operator to describe first-order spatial quantum coherence. Its diagonal is particle-number density and its trace is the expectation value of the particle number operator, $\mathrm{Tr}{G}^{(1)}=⟨N⟩$. Dividing by $⟨N⟩$ instead gives a unit-trace convention, so the normalization matters when interpreting its eigenvalues.

For bosons, an eigenvalue that remains proportional to particle number in the thermodynamic limit identifies Bose-Einstein condensation. This criterion extends occupation of a single-particle mode to interacting systems.^[ref]

In a homogeneous system of volume $V$, an occupied plane-wave mode contributesA constant long-distance limit therefore corresponds to a zero-momentum condensate; a moving condensate can retain a spatial phase. Finite local density or short-distance quantum coherence alone does not establish off-diagonal long-range order.

Neglecting commutators between kinetic and potential energy permits a leading product approximation to the imaginary-time evolution operator. The kinetic factor is a Gaussian Fourier integral; the potential factor multiplies it in position space. The unsymmetrized product places the potential at one endpoint and need not be exactly Hermitian at the discarded order. Splitting the potential equally between both endpoints gives a Hermitian approximation with exponent $-\beta [V(x)+V(x^{\prime })]/2$ and the same leading diagonal normalization.

The reduced density matrix obtained by taking the partial trace of a larger entangled state. It predicts the local measurement statistics, but its mixed form does not imply that the total state has collapsed into one of the displayed components. Different ensembles can represent the same reduced density matrix.

The mean mass density of a body is its total mass divided by its volume. A nonuniform density distribution can have the same mean as a uniform one; orbital information that determines only $M/V$ does not determine the internal density profile.

A degron whose recognition depends on phosphorylation. Phosphorylation can thereby couple protein turnover to a regulatory signal; not every phosphorylated motif is a degradation signal.