Supplemental material for “DNA overwinds when stretched”, Gore et al
Mechanical Properties of the toy DNA model
Our model DNA consists of an isotropic rod wrapped helically by a thin but stiff “wire”
(see Fig. 4 in main text and Fig. S1). In keeping with the expected properties of a
polymeric material1, we assume that the inner rod has a Poisson ratio of υ = 0.5,
corresponding to a material that maintains its volume under stress. In the absence of the
helical wire the inner rod would have mechanical properties1,2
Br 
 Rr4Yr
4
4B
S r  2r   Rr2Yr
Rr
Br
2 Br  Rr4Yr


1 
3
6
gr  0
Cr 
where Br is the bending rigidity, Sr is the stretch modulus, Cr is the twist rigidity, and gr is
the twist-stretch coupling. Yr is the Young’s modulus of the material that makes up the
inner rod and Rr is the rod’s radius. Note that the inner rod by itself cannot have any
twist-stretch coupling (gr = 0) because the material is isotropic. The ratio of bending to
twist rigidity for the inner rod is B/C = 1+ υ = 3/2, and is completely specified by the
Poisson ratio of the material. The B/C ratio of the inner rod therefore contrasts with the
experimentally measured B/C ratio of ~1/2 for DNA.
The outer helical “wire” is assumed to be thin, so that its only contribution to the
mechanical properties of the system is the wire’s resistance to stretching and
compression, specified by its stretch modulus Sh. The outer wire is affixed to the inner
rod such that the radius of the outer helix is always equal to the radius of the inner rod.
1
Supplemental material for “DNA overwinds when stretched”, Gore et al
Under moderate tension (F >> (kBT)2/B ~ 0.1 pN, where kBT ≈ 4.1 pNnm is the
thermal energy of the bath), bending fluctuations are negligible and we can write the
energy of the model DNA as a function of the stretch distance x and excess twist θ:
E DNA x,  
1 S r 2 1 Cr 2 1 S h 2
x 
 
xh x,  ,
2 L
2 L
2 Lh
where L and Lh are the lengths of the inner rod and helical wire, respectively, x is the
distance that the DNA is stretched beyond its contour length L,  is the angle through
which the DNA is twisted from its unperturbed equilibrium value, and xh(x, θ) is the
distance that the helical wire has been stretched (or compressed). The stretch of the
helical wire can be calculated by noting that the final helix length can be expressed as a
Pythagorean sum between the total DNA length and the distance that the helix travels
around the rod:
Lh x,    Lh 0,0   x h x,  
 L2 x   Rr x  0   
2

L0  x 
2



x 
 0   
  Rr 01  
L0  



,
2
where L(0) is the contour length of the helical wire at equilibrium and θ0 is the
equilibrium twist (number of times that the helical wire winds around the inner rod). As
the system is stretched the radius of the inner rod decreases so as to maintain the volume
of the rod. Here we have assumed that the stretch distance x is much less than the total

x 
length L, thus allowing us to make the linear approximation that Rr ( x)  Rr 1    .
L0 

The radius of the inner rod is determined by the amount of stretching in the molecule and
the Poisson ratio of the material (which we have assumed to be 0.5). The stretching of the
2
Supplemental material for “DNA overwinds when stretched”, Gore et al
outer helix is caused by both the stretching and the twisting of the molecule. We can
determine the stretching of the helical wire for small perturbations by taking the
differentials with respect to both x and θ and then simplifying the resulting expression:
Lh
L
x h
x

 sin    R0 cos x  R0 cos 
xh 
This differential for the stretching of the helix has been expressed in terms of the helix
angle α (see Fig. S1). The energy of our model DNA as a function of twisting and
stretching is therefore
E DNA x,  

1 S r 2 1 Cr 2 1 S h 2
x 
 
x h  x,  
2 L
2 L
2 Lh
1 S r 2 1 Cr 2 1 S h
sin    Rr cos x  Rr cos  2
x 
 
2 L
2 L
2 Lh
1 S

1 Sh
sin    Rr cos 2  x 2  Rr S h sin    Rr cos cos x
 r 
Lh
 2 L 2 Lh

 1 Cr 1 S h 2



Rr cos 2   2
 2 L 2 Lh

1

S r  S h csc  sin 2    cos 2 
2L
1

C r  S h Rr2 cos 2    2
2L
1 S eff 2
x 1 C eff 2

x g


2 L
L 2 L



 x  R sin
2
2
r
2
   cos 2  S h 
x
L

We see that the presence of the outer helix modifies the effective stretch and twist
rigidities, and also yields a non-zero twist-stretch coupling:

S eff  S r  S h csc  sin 2    cos 2 
C eff  C r  Rr2 S h sin  cos 2 


g  Rr sin 2    cos 2  S h
3

2
 Sr
Supplemental material for “DNA overwinds when stretched”, Gore et al
In the toy model, negative twist-stretch coupling occurs only for shallow helix angles
below a critical angle  c  arctan
    0.62
rad . The geometry of B-form DNA lies
just within the regime of negative coupling (   arctan 3.4 nm 2Rr   0.53 rad   c ).

The model suggests that helical forms with shallower helix angles (such as A-form DNA
or RNA) might be expected to have a more pronounced negative twist-stretch coupling.
For parameter values consistent with the structure and mechanical properties of DNA, the
stretch modulus is not significantly affected by the presence of the outer helical wire, but
the twist rigidity is dominated by the contribution of the outer helix. The high torsional
rigidity of the model has very little dependence on tension, in contrast to some models
put forward to explain our previous single-molecule measurement of C at high
tensions3,4.
How does the outer helix affect the bending rigidity of the molecule? We have
assumed that the outer helix is very thin (Rh << Rr), which means that the contribution of
the helix will manifest itself as a stretching/compression of the wire, rather than as a
bending or twisting. Bending of our DNA model introduces negligible changes in the
total length of the outer helix (calculations not shown). The bending rigidity of the model
DNA is therefore approximately equal to the bending rigidity of the inner rod:
Beff  Br .
In summary, the presence of the outer helical “wire” is sufficient to generate a
coupling between stretching and twisting of the molecule. The outer helix does not
appreciably increase the bending rigidity or the stretch modulus, but it does radically
increase the stiffness of the molecule to twist. These are precisely the characteristics that
are necessary in order to explain the key mechanical properties of DNA, and we are able
4
Supplemental material for “DNA overwinds when stretched”, Gore et al
to obtain quantitative agreement between the expected mechanical properties of our
“model” DNA and the experimentally measured mechanical properties of real DNA.
References
1.
2.
3.
4.
Hogan, M.E. & Austin, R.H. Importance of DNA stiffness in protein-DNA
binding specificity. Nature 329, 263-6 (1987).
Hegner, M., Smith, S.B. & Bustamante, C. Polymerization and mechanical
properties of single RecA-DNA filaments. Proc Natl Acad Sci U S A 96, 1010914 (1999).
Matsumoto, A. & Olson, W. Effects of Sequence, Cyclization, and Superhelical
Stress on the Internal Motions of DNA, (Chapman & Hall/CRC Press, Boca
Raton, FL, 2006).
Bryant, Z. et al. Structural transitions and elasticity from torque measurements on
DNA. Nature 424, 338-41 (2003).
5