analogous to a solenoid conformation in a twisted elastic ring. Under these conditions, it has been shown that although the solenoid conformation is energetically favorable over the straight conformation, plectonemic or supercoiled structures are yet lower in energy [12].
Furthermore, the post-buckling behavior of a twisted elastic filament under tension is shown to exhibit localization and loop formation [11] thus seeding the formation of plectonemes.
In the next section, we consider the post-buckling behavior of the twisted strand, focusing on the formation of loops that lead to plectonemic supercoils.
to conditions where the end effects are negligible orB 1 (previous section). Furthermore, this simple analysis neglects the effect of self-interaction of the chain, which is required to address the formation of the full plectoneme structure. Nonetheless, the initial stage of plectoneme formation is accessible without self-interaction, provided the loop structure is large relative to the diameter of the filament [10]. The looping process studied in Ref. [10]
provides the foundation for our analysis of plectoneme formation; in this section, we review and adapt the results of Ref. [10] to our system.
To begin, we define an energy that contains contributions from bending and twisting deformation along with a constraint that fixes the chain ends at a specified distance apart.
The constrained energy UT is given by
UT = Z L/2
−L/2
dsuT, (6.6)
where
uT = A 2
ω12+ω22+ B
2ω23−T1−zˆ·d~3. (6.7) The Lagrange multiplier T in Eq. 6.7 enforces the length of the chain retraction along the z-axis, thus it is the conjugate variable to ∆ = R−L/2L/2 ds1−zˆ·d~3
. The energy density defined by Eq. 6.7 is not to be confused with the actual energy density of the filament u, which is extracted from Eq. 6.1 to be
u= A 2
ω12+ω22+B
2ω32+f1−zˆ·d~3
(6.8)
after neglecting the self-interaction energy.
Thus far, we have described the chain conformation using cartesian coordinates; how-
ever, general shapes of elastica are better described using Euler angles (φ, θ, ψ) [20], as their use does not require complicated kinematic relationships to constrain the length of the chain. The equations that govern equilibrium shapes for a twisted chain that is clamped and held under tension are given by [1]
uT = A 2
ω21+ω22+B
2ω32−T(1−cosθ) = constant, (6.9) m = −Aω1sinθcosψ+Aω2sinθsinψ+Bω3cosθ= constant, (6.10) ω3 = ∂ψ
∂s +∂φ
∂s cosθ= constant. (6.11)
Eqs. 6.9 through 6.11 are found using the symmetries of the equilibrium conformation, namely invariance with respect to “time” (or path length), rotation about the z-axis, and rotation about the tangent vector. The conformation is found from the Euler angles using the differential relationships: ˆx·∂s~r = sinθcosφ, ˆy·∂s~r= sinθsinφ, and ˆz·∂s~r = cosθ. The other curvature terms are related to the Euler angles throughω1 =∂sθsinψ−∂sφsinθcosψ and ω2 =∂sθcosψ+∂sφsinθsinψ.
Since Eqs. 6.9 through 6.11 are constant for any point along the chain, the constants are evaluated by dictating the values at a single point. Taking into account the end conditions (θ=∂sθ= 0 ats=±L/2), Eqs. 6.9 and 6.10 can be written as
uT = Bω23
2 (6.12)
m = Bω3. (6.13)
From this, we arrive at the governing equations for the chain conformation
∂θ
∂S
= 2 tan θ
2 s
cos2 θ
2
−C2, (6.14)
∂φ
∂S = C
cos2θ2
, (6.15)
whereC=Bω3/(2√
AT). All lengths are non-dimensionalized bypA/T, andS =spT /A.
For L1, Eq. 6.14 has the approximate solution
sin θ
2
=
√ 1−C2 coshS√
1−C2
, (6.16)
and φis found from the solution forθ to be
φ=CS+ arctan
√1−C2tanhS√
1−C2 C
. (6.17)
These equations give the conformation of a growing loop structure from a straight, twisted chain for sufficiently long chains (√
1−C2L/2 1). As the conformation proceeds through the loop formation, the twist density ω3 changes from the initial twist density ω0 due to the writhe of the looped structure. The Lagrange tensionT also responds to the transition. The straight conformation corresponds to a value of C = 1 orT =B2ω20/(4A), which agrees with the conditions for instability found in the previous section.
In Fig. 6.2, we show the formation of a loop via a quasi-equilibrium transition using Eqs. 6.16 and 6.17. Each of the conformations in Fig. 6.2 represents a stable structure for the tensionT found in C, which marks the progress through the transition. The beginning of the transition occurs through a helical formation, much like the structures analyzed in the previous section. However, further progress shows a localization of the helix into a loop
Figure 6.2: Equilibrium conformations for the looping transition determined from Eqs. 6.16 and 6.17. These snapshots represent, from left to right, the stable conformations for C = 0.99, 0.89, 0.79, 0.69, 0.59, 0.49, 0.39, 0.29, 0.19, and 0.09. The length scale in this figure is scaled by pA/T.
structure similar to that found in Ref. [11]. This loop formation marks the initial stage of plectoneme formation.
We now turn to the energy of the growing loop. The energetic contributions are found using Eqs. 6.16 and 6.17 along with the total energy (Eq. 6.1). This calculation makes use of the relationship between the retraction distance ∆ and the tension T found in Ref. [10], which is given by
T = B2 4A
ω0−L4 arcsin
∆ 4
qT A
2
1−
∆ 4
qT A
2 , (6.18)
which is a transcendental equation forT in terms of ∆ and the material parameters of the chain. Equation 6.18 exhibits a double value for T at a given retraction distance ∆; this double value is the source of an instability associated with looping and popping-out [10].
The total energy of the looped chain is given by an integral of the energy density, which is found from Eqs. 6.7 and 6.8 to be
u=T(1−cosθ) +B
2ω23+f(1−cosθ). (6.19)
The first term in Eq. 6.19 is the bending deformation energy, the second term is the twisting deformation energy, and the third term is the work against the external load f. The total energy is given by
U = BL 2
ω0− 4 Larcsin
∆ 4
s T A
2
+ (f +T)∆, (6.20)
where the Lagrange tension T is found using Eq. 6.18. The total energy represents a non- equilibrium energy function relating the energy to the distance of retraction ∆, and the equilibrium value of ∆ is determined by the minimization of Eq. 6.20.
0 2 4 6 -4
-3 -2 -1 0 1 2 3 4 5 6 7 8
DU (k BT)
D (nm)
Figure 6.3: The total energy ∆U =U −U(∆ = 0) (in kBT) versus the end retraction ∆ (in nm) for a chain with A =B = 50kBTnm, L= 1000 nm, and 100 turns of twist (ω0 = 0.6283 nm−1) subjected to end tension f = 20.2327 pN (dotted curve), f = 22.5561 pN (dashed-dotted curve), f = 24.8796 pN (dashed curve), and f = 27.2031 pN (solid curve).
In Fig. 6.3, we plot the total energy ∆U = U −U(∆ = 0) (Eq. 6.20) measured in units of the thermal energy kBT versus the end retraction ∆ (in nm) for a chain forming a looped structure from a straight conformation. The properties of the chain in Fig. 6.3 are A = B = 50kBTnm, L = 1000 nm, and 100 turns of twist (ω0 = 0.6283 nm−1), and the end tension f is f = 20.2327 pN (dotted curve), f = 22.5561 pN (dashed-dotted curve), f = 24.8796 pN (dashed curve), and f = 27.2031 pN (solid curve). We note that for f = 20.2327 pN (dotted curve) the conditions are at the point of instability governed by the stability diagram (Fig. 6.1), thus the other curves with larger tension f show the non-equilibrium looping energy for conditions in the stable region of Fig. 6.1. These curves exhibit an initial increase in the energy with the end retraction; however, these energy plots eventually turn and decrease with further end retraction. The chain properties in Fig. 6.3 correspond to the material properties of a 3000 basepair strand of DNA subjected to end tension. The predictions of Fig. 6.3 demonstrate an energy barrier to the formation of a
loop for tensions above the stability curve; although the straight conformation is stable, loop growth eventually leads to an energy decrease and the subsequent growth of a plectoneme.
In this analysis, we neglect thermal fluctuations. The non-equilibrium energy displayed in Fig. 6.3 suggests that loop formation requires an energetic nudge in order to overcome a barrier for sufficient end tension (conditions above the stability diagram in Fig. 6.1). A molecular strand is subject to thermal fluctuations with characteristic energy kBT, thus energies comparable to kBT are attainable by a fluctuating strand. For the conditions in Fig. 6.3, thermal energy can easily overcome the boundary to loop formation, thus an AFM or tweezer experiment conducted under these conditions is predicted to exhibit transient loop formations leading to plectonemic supercoils, even under stable conditions.
In the following section, we numerically solve the equations of motion found in Sec. 6.2 in order to show the progression from straight, twisted chain through the instability and the subsequent nonlinear dynamics. In particular, we compare and contrast the predictions of this section to assess the validity of the pseudo-equilibrium approximation.