Table 1.

Energy expressions for DNA under different conditions of tension and torsion

Common symbols: Ls, stretched DNA contour length; Ceff, effective twist persistence length (twist modulus/kBT); ΔLk, linking number change in stretched B-form DNA; f, tension; τ, torque; ω0, intrinsic twist of DNA, ω0 = $\frac{2\pi }{3.57}$ nm−1; G, twist-bend coupling parameter (G, 30–40 nm (56,57)); g, twist-stretch coupling parameter (unitless) (g, −22 to −17 (54, 55, 56)); A, bend persistence length; C, twist persistence length.
Energy type	Equations	Annotations		Reference	Equation number
Energy of intertwined, strand-separated DNA	EMelted = EtwM + EbpM EtwM, energy of intertwined, melted region: EtwM = 1/2nκmelt${\overline{\theta }}^2=\frac{k_{B}T{C}_{melt}}{2L_{melt}}$(2πΔLkmelted)2 EbpM, energy cost for denatured basepairs in a melted region: EbpM = εini,melt + $\sum _{i=1}^n$εmelt,i	n, number of unpaired basepairs $\overline{\theta }$, average twist angle of each disrupted basepair Length of melted region: Lmelt ≈ n × 0.54 nm/bp (48), 2πΔLkmelted = $n\overline{\theta }$ κmelt, twist modulus of two intertwined strands, ∼2.3 kBT Twist persistence length, Cmelt = $\frac{L_{melt}\times {\kappa }_{melt}}{n{k}_{B}T}$ ≈ 1.2 nm εini,melt: initial energy for melting, 9–11 kBT. $\sum _{i=1}^n$εmelt,i: sum of energy cost for each individual basepair melting, where εmelt,i depends on each basepair and its neighbors; particular values can be found in (50).	Change of linking number that melted region contributes to flanking DNA: ΔLkB − melt = $\frac{n}{10.5}-\frac{n\overline{\theta }}{2\pi }$	(48,50,57, 58, 59, 60, 61, 62)	(1)
Energy of Z-form DNA (fixed two boundaries)	EZ = EtwZ + EbpZ EtwZ, energy of twisting Z-DNA: EtwZ = $\frac{k_{B}T{C}_Z}{2L_Z}$ (2πΔLkZ)2 EbpM, energy cost for B-Z transition: EbpM = 2εwall + $\sum _{i=1}^m$εB-Z,i	LZ, length of Z-form DNA ΔLkZ, twist change in Z-form DNA CZ: twist persistence length, ∼7 nm for d(pCpG)n repeats εwall: domain wall energy, ∼8.4 kBT. m: number of basepairs that undergo B-Z transition, LZ = m × 0.37 nm/bp (65) $\sum _{i=1}^m$εB-Z,i: sum of energy cost for disrupting each additional basepair, following the nucleation of Z-DNA, where εB-Z,i ≈ 1.1 kBT for d(pCpG) repeats and εB-Z,i ≈ 2.4 kBT for d(pCpA) repeats (63,64).	Change of linking number that B-Z transition contributes to flanking DNA: ΔLkB-Z = −am − 2b, a = 2(1/10.5 + 1/12), b = 0.4.	(49,61,63, 64, 65)	(2)
Twist energy of stretched B-form DNA	EtwB = $\frac{k_{B}T{C}_{eff}}{2L_s}$(2πΔLkB)2	1/Ceff = 1/C∗ + 1/(4A∗)$\sqrt{\frac{k_{B}T}{f{A}^{\ast }}}$	TWLC model (G = 0): A∗ = A (≈50 nm), C∗ = C (≈100 nm)	(66)	(3)
			TWLC (G$\ne$0 and very large G) nonperturbation theory: A∗ = κb = $A\frac{1-\frac{{\epsilon }^2}{A^2}-\frac{G^2}{AC}\left(1+\frac{\epsilon }{A}\right)}{1-\frac{G^2}{2AC}};$C∗ = κt = $C\frac{1-\frac{\epsilon }{A}-\frac{G^2}{AC}}{1-\frac{\epsilon }{A}}$; ε, the bending anisotropy.	(52)	(4)
			TWLC (G$\ne$0 and G is small) perturbation theory (f ∈ (0.1 pN, 10 pN): 1/A∗ = 1/A$\left(1+\frac{G^2}{2AC}\right)$, 1/C∗ = 1/C$\left(1+\frac{G^2}{AC}\right)$.	(53)	(5)
Stretched DNA, TWLC model (G = 0)	E(f, ΔLk) = [−f +$\sqrt{\frac{f{k}_{B}T}{A}}+\frac{C_{eff}{k}_{B}T}{2}{\left(\frac{2\pi \text{Δ}Lk}{L_s}\right)}^2$]Ls	For definitions of f, A, Ceff, ΔLk, and Ls, see above.		(66)	(6)
Stretched-twisted DNA, (perturbation theory) (G$\ne$0)	E(f, τ) ≈ $\left(-f+\sqrt{\frac{f{k}_{B}T}{A^{\ast }}}+\text{Γ}\tau -\frac{{\tau }^2}{2k_{B}T{C}_{eff}}\right)$Ls	Torque, $\tau \approx \frac{2\pi k_{B}T{C}_{eff}}{L_s}\Delta Lk$	proportionality constant Γ, Γ = $\frac{G^2}{8A^2{C}^2{\omega }_0}$	(53)	(7)
Stretched-twisted DNA, (CTWLC model)	$\frac{E}{k_{B}T{L}_s}=\frac{1}{A}\sqrt{\frac{Af}{k_{B}T}-\frac{1}{4}{\left(C_{eff}{\omega }_0\sigma +\frac{gf}{K_0}\right)}^2}+\frac{C_{eff}}{2}{\omega }_0^2{\sigma }^2-\frac{f}{k_{B}T}-\frac{k_{B}T}{2K_0}{\left(\frac{f}{k_{B}T}-g{\omega }_0\sigma \right)}^2$	σ: supercoiling levelσ = ΔLk/Lk0	K0, the stretch modulusK0 ≈ 1200 pN	(55,67)	(8)
			For f, A, Ceff, ΔLk, Ls, g, and ω0, see above.
Energy barrier of DNA from unbuckled to soliton state	$E_s=\frac{8k_{B}TA}{l}\text{tanh}(\frac{L}{2l})-2\pi W{r}_s{\left(\tau +\frac{\pi k_{B}T{C}_{eff}W{r}_s}{L}\right)}^2$	L, DNA length in nm,l, soliton length scale: ${\left[{\left(\frac{A}{f}\right)}^{-1}-{\left(\frac{2C_{eff}}{f}\right)}^{-2}\right]}^{-\frac{1}{2}}$	Wrs: writhe in soliton: $\begin{array}{l}\frac{2}{\pi }ta{n}^{-1}\left[\frac{2A}{\tau l}\text{tanh}\left(\frac{L}{2l}\right)\right]\\ \end{array}$	(68,69)	(9)
Curled DNA	Ec = $\left(8-\frac{3.14f{\left(0.8+2.2{\kappa }_D^{-1}\right)}^2}{A{k}_{B}T}\right)\sqrt{k_{B}TAf}$	${\kappa }_D^{-1}$, Debye length; f and A, see above.		(26)	(10)
Plectonemic DNA	Ep = $\frac{2{\pi }^{2}C{k}_{B}T}{L_p+q\text{Γ}}{\left(\text{Δ}L{k}_p-W{r}_p\right)}^2$ + Lp$\left[\frac{A{k}_{B}T{\left(sin\alpha \right)}^4}{2r^2}+U\left(r,\alpha \right)\right]$ + qεp$\sqrt{Af{k}_{B}T}$	q, number of plectonemes along DNA; Lp, plectoneme length; ΔLkp, linking number change of plectoneme, Wrp, total writhe of the plectonemic regions Wrp =$\frac{L_{p}sin2\alpha }{4\pi r}$+ qωp (ωp ≈ 1); U(r, α), electrostatic repulsion and entropic confinement free energy, Γ, length of the end loop and tail region of a plectoneme (Fig. 1); for more details, please refer to (26). The top part of left expression represents the twist energy of plectoneme. The middle term in the sum represents bending energy as well as the electrostatic repulsion and entropic confinement free energy (whole plectonemes except end loop and tail region). The bottom part contains the energy contribution of end loop and tail region.		(26)	(11)

Some variable names have been changed and values/units have been converted from those used in the original reports for uniformity.