Auxiliary Material for Manuscript:
Abrupt Buckling Transition Observed
During Plectoneme Formation in Individual DNA Molecules
Scott Forth, Christopher Deufel, Maxim Y. Sheinin, Bryan Daniels,
James Sethna, and Michelle D Wang
Auxiliary Text: Predictions by the Marko model
Notation:
Variables:
f  force.
Lp  bending persistence length.
C  twist persistence length (straight DNA segment).
P  plectoneme twist persistence length (plectonemic DNA segment).
kBT  thermal energy.
3.57nm  helical pitch of DNA.
Derived relations/expressions:
0  (2 / 3.57)nm 1
c  kBTC0
2
p  kBTP0
2

C kBT 
cs  c1 
 4 Lp Lp f 


k Tf
g f  b
Lp
Predicted extension change per turn after buckling:
To derive the expected change in extension after the buckling transition, we consider the
following, derived from Marko, PRE 021926, 2007. Extension (as a fraction of relaxed
double helix contour length L, Equation 5 of Marko, 2007) can be written as:
z
F
,

L
f
where F is the free energy.
1) Considering the free energy of the molecule before buckling (Equation 9 of Marko,
2007),
c
F  S ( )   g  s  2
2
and, taking the derivative:
2
z
1 kBT 0 C 2  kBT

 1

L
2 LP f
16  LP f
3/ 2

  2

2) By means of the double tangent construction, one can see that the fractions of stretched
and plectonemic states ( x s and x p ) during the phase coexistence depends linearly on the
linking number density. As the plectonemic state has zero length, extension may be
written as (Equation 6 of Mark, 2007):
z ( s )  p   z ( s )
z
 xs

,
L
L
 p  s L
where  s and  p are linking number densities at the beginning and at the end of the
transition, and z ( s ) is the extension at the beginning of the transition.
It is possible to express the previous formula in terms of known parameters (Equation 14
of Marko, 2007). Therefore, the slope of extension (as a fraction of contour length L)
versus linking number is:
 1 k T  2C 2  k T  3 / 2 2 
B
 B   s 
1 
 0
16  LP f 

 ( z / L)
z ( s ) / L  2 LP f



 p s
 p s
2
 1 k T  2C 2  k T  3 / 2  1
 
2
pg
B
0
B
 
1 

 

16  LP f   cs 1  p / cs  
 2 LP f




2 pg
1 1
  
1  p / cs  p cs 
Above quantity is dimensionless. To convert into nm/turn, note that each turn changes the
1
DNA linking number by one, therefore changing the linking number density by
,
Lk 0
L(nm)
where Lk 0 
is the initial linking number. So, slope in nm/turn is given by:
3.57(nm)
 1 k T  2C 2  k T  3 / 2  1
2 pg
B
 B  
3.57nm 1 
 0
16  LP f   cs 1  p / cs
 2 LP f

z 
2 pg  1 1 
  
1  p / cs  p cs 




Similarly, the predicted torque after buckling (Equation 17 of Marko, 2007):
c 
2k BTPg
.
1 P C
2




.
overwinding
30
Torque (pN nm)
25
20
relaxation
15
10
5
0
700
Extension (nm)
600
500
400
300
200
0
5
10
15
Number of turns added to DNA
Auxiliary Figure 1. Reversibility of over-winding DNA. A 2.2 kbp DNA molecule was
held at a constant force of 2 pN and overwound by 20 turns at a rate of 1 turn/s. Winding
direction was immediately reversed at the same rate to return the DNA to its torsion-free
state. The over-winding and its reversal yielded overlapping data traces for both the
extension and torque signals, indicating the measurements were carried out under quasiequilibrium conditions.