International Journal of Non-Linear Mechanics 47 (2012) 639–654
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International Journal of Non-Linear Mechanics
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DNA stretching modeled at the base pair level: Overtwisting and shear
instability in elastic linkages
Attila Kocsis a, David Swigon b,n
a
b
Department of Structural Mechanics, Budapest University of Technology and Economics, Hungary
Department of Mathematics, University of Pittsburgh, United States
a r t i c l e i n f o
abstract
Article history:
Received 2 June 2011
Received in revised form
24 October 2011
Accepted 28 October 2011
Available online 12 November 2011
Stretching experiments on single DNA molecules indicate that, counterintuitive to expectations, DNA
overwinds when stretched and, at large forces, undergoes a transition into an overstretched form
indicated by a plateau on the force–displacement diagrams. It is believed that these effects are the
result of non-linearities in the elastic response of DNA. We use a discrete, base pair level model to
simulate the behavior of short DNA molecules, taking into account the sequence dependent physical
properties of DNA alongside with the coupling between the kinematical step parameters, yet retaining
the quadratic form of local elastic energy function. By constructing bifurcation diagrams of equilibrium
configurations and studying the dependence on base pair combinations we show that the quadratic
model is capable of explaining the overtwisting as a result of coupling between modes of deformation
and overstretching as a result of shear instability.
& 2011 Elsevier Ltd. All rights reserved.
Keywords:
DNA mechanics
Overstretching
Discrete elastic model
Simplex algorithm
Bifurcations
1. Introduction
DNA is a double stranded molecule composed of two polynucleotide strands that are bound together by hydrogen bonds
between complementary nucleotide bases. In normal conditions
the strands wind around the DNA axis as two identical righthanded helices with the diameter of 2 nm and pitch of 3.57 nm.
During various important intracellular biological processes, such
as transcription or replication, mechanical forces, exerted by
various proteins, act on DNA and cause its bending, twisting,
stretching, or shearing. Understanding of the role of DNA in such
processes requires us to obtain knowledge about the mechanical
response of DNA to various loading conditions.
Macroscopic properties of DNA can be studied using a range of
techniques, including atomic force microscopy (AFM) observations
[2], electron microscopy visualization, FRET measurements of distances between parts of a molecule, or single molecule manipulation. The latter technique can subject a single DNA molecule to a
mechanical loading, consisting of stretching and twisting, by micromanipulation of objects to which DNA is attached, such as glass
needles [3], glass beads [4–6], or magnetic beads [7–9].
Recent DNA stretching experiments [3,5,6,8,10,11] have
shown that, contrary to natural intuition with twisted ropes,
Abbreviation: BVP, boundary value problem; GRS, general representation space;
SSA, simplex scanning algorithm; PFA, path-following algorithm.
n
Corresponding author.
E-mail address: [email protected] (D. Swigon).
0020-7462/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijnonlinmec.2011.10.008
DNA overwinds under tension until the force reaches a critical
value of 30 pN, above that value it unwinds under tension [8].
The magnitude of the overwinding is small, about 2.5 rotations of
an 8400 base pair segment [8], which corresponds to about 0.3%
of the natural twist of a base pair step. The same effect gives rise
to a lengthening of the molecule as a result of overtwisting [8,12].
It has been suggested [8] that one possible explanation of the
phenomenon can be obtained by treating the DNA as a microstructured material consisting of a soft elastic cylindrical core
surrounded by stiffer helical backbone threads winding on the
surface of the tube. Indeed, when such a structure is stretched,
the diameter of the cylindrical core decreases which leads to the
overtwisting of the backbone threads which corresponds to the
observed phenomena, however the structure maintains the same
behavior at all magnitudes of the stretch and cannot therefore be
the only explanation of the effect.
Its has been also found that when larger forces are applied, DNA
undergoes a transition to an overstretched form. During the transition the force remains almost constant while the DNA lengthens to
1.7 times of its natural B-form contour length [5,6,13]. The threshold
for this transition is 65 pN, if the molecule is torsionally relaxed,
and 110 pN, if it is torsionally constrained. This overstretching
transition has been described by two competing models, both of
which assume that the plateau is a Maxwell line corresponding to a
transition between two phases of the molecule. One model assumes
that the new phase is a double-stranded overstretched form, a
so-called S-DNA [3], in which the base pair separation is increased
but the hydrogen bonding between the bases remains intact.
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A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
Nomenclature
Notations of step parameters according to the ‘‘Cambridge Accord’’
[1]:
yn1
yn2
yn3
hn
rn1
rn2
rn3
qn
tilt in the nth segment
roll in the nth segment
twist in the nth segment
n n n
½y1 , y2 , y3 shift in the nth segment
slide in the nth segment
rise in the nth segment
½rn1 , rn2 , rn3 Intrinsic (stress-free) step parameters of step n:
n
y 1 ¼ yn1 Dyn1 intrinsic tilt
n
y 2 ¼ yn2 Dyn2 intrinsic roll
n
y 3 ¼ yn3 Dyn3 intrinsic twist
Though the exact shape of S-DNA is still unknown, it has been
proposed that in S-DNA form the bases are unwound and unstacked
so that they form a parallel ladder [2,5]. The second model interprets
the overstretched phase as a denatured DNA resulting from strand
separation caused by breaking of hydrogen bonds between complementary bases [14]. Recent studies [15,16] show that these two
states may coexist: GC base pair rich regions appear to prefer to
form an S-DNA, while strand separation is more likely to occur at AT
rich regions [16,17].
Mechanical properties of DNA have been traditionally described
using an ideal elastic rod model [18–20], in which the hydrogen
bonding remains unperturbed, and DNA is assumed to be homogeneous, isotropic, intrinsically straight, unshearable and inextensible. The mechanical response of such a model can be characterized
by a bending modulus and a twisting modulus and the model does
not predict any overtwisting or overstretching transitions. Although
the statistical equivalent of the ideal elastic rod model, the wormlike chain model (WLC) [21], fits stretching experiments well up to
10 pN [6], high-resolution studies on crystal structures [22] and
molecular dynamics simulations [23] show that both the intrinsic
(stress-free) geometry and the elastic properties of DNA are
sequence-dependent. Moreover, relative rotation and displacement
of neighboring base pairs can be strongly coupled [22]. Thus, on a
short length-scale, these sequence-dependent geometrical and physical properties of DNA must be taken into account.
Continuum models of DNA have been be formulated on the basis
of the special Cosserat theory of rods [24], in which intrinsic curvature,
bending anisotropy, shearing and coupling between modes of deformation have (or could be) implemented [25–27]. Such models
assume a relatively smooth deformation of DNA with comparatively
small deviations from the intrinsic helical geometry. Alternatively,
one can construct discrete mechanical models of DNA in which the
molecule is composed of rigid rectangular bodies (representing the
base pairs) with elastic connections between them [28–30]. This
model approximates well the crystal structure properties of DNA
(especially the rigidity of nucleotide bases), while leaving room for
the possibility of elastic deformations that do not disrupt base pairing
and backbone conformation. In a discrete model, the local geometry
of the molecule is defined by six parameters (see below) describing
the angular and translational displacements between base pairs. The
local elastic energy is, in the first approximation, assumed to be a
quadratic function of these parameters, which naturally leads to
h
n
n
n
1
n
2
n
3
n
r ¼r
r ¼r
r ¼r
q
n
n
½y 1 , y 2 , y 3 rn1 intrinsic shift
rn2 intrinsic slide
rn3 intrinsic rise
½r n1 , r n2 , r n3 n
1 D
n
2 D
n
3 D
Notations of basis and basis transformations used in this paper:
n
n
n
Bn ¼ ½d1 ,d2 ,d3 base pair triad of the nth base pair
n n n
n
.B~ ¼ ½d~ 1 , d~ 2 , d~ 3 mid-base triad of the nth segment
n
transformation of vector components from Bn þ 1 into Bn
D
n
~n
D
transformation of vector components from B~ into Bn
n
n
~
~
transformation of vector components from B into Bn
D
Tn
transformation of vector components from Bn into B1
Throughout this paper, small boldface symbol denotes a vector
or its components with respect to a fixed basis E ¼ ½e1 ,e2 ,e3 .
The components of a vector vm with respect to a local basis Bn
are denoted by ½vm n
sequence-dependent intrinsic geometrical and elastic properties,
including coupling between various modes of deformation. The
elastic potentials can be estimated from molecular dynamics
simulation [23,28] or from analysis of high resolution crystal
structure data [22].
In this paper we utilize a discrete mechanical model introduced in [29] to simulate the mechanical response of DNA to
stretching. We study the effects of sequence-dependent elasticity,
shear deformation and couplings between all the step parameters
on equilibrium configurations. We show that shear deformation
plays a crucial role in the mechanical behavior of the stretched
DNA molecule, resulting in configurations in which shear deformation is more significant than elongation. At large tensions the
model undergoes a shearing instability after which any change in
its extension is accommodated locally by an increase in the shear
between the base pairs. Therefore we conjecture that the overstretched S-DNA state should be represented as a stack of sheared
base pairs as opposed to a parallel ladder. We also demonstrate
that due to coupling terms, the discrete model predicts overtwisting of stretched DNA followed by untwisting, just as has
been observed in the experiments.
In Section 2 we describe the applied model and the boundary
conditions used in the study. The geometrical, equilibrium and
constitutive equations of the model with the boundary conditions
lead us to formulate the non-linear Boundary Value Problem (BVP)
of N base pair steps. Solutions of the BVP and a thorough study of
bifurcation diagrams are given in Section 3, finally conclusions are
drawn in Section 4.
2. Discrete mechanical model of DNA
We utilize a discrete model of DNA introduced in [29]. In this
model the base pairs are represented by rigid rectangular plates
and the connections between the neighboring base pairs are
taken to be linearly elastic.
2.1. Geometry of the model
The position of the nth base pair is specified by the location xn
of the center of the base pair and a right-handed orthonormal
n n n
triad Bn ¼ ½d1 ,d2 ,d3 embedded in the rectangle. The triad Bn is
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
n
n
n
respect to the d~ 2 axis. Here b ¼
defined so that d3 is perpendicular to the rectangle, pointing
n
toward the 50 -30 direction of Strand I, d1 is perpendicular to the
n
minor/major groove sides and d2 points from Strand II to Strand I,
n n n
n
as detailed in Fig. 1. The triad B~ ¼ ½d~ , d~ , d~ is called the mid1
2
3
tude of bending and the ‘‘hinge’’ is the axis of bending.
n
base triad [31] of the nth base step. The mid-base triad B~ is a sort
As the starting point, assume that the base pair triads of the
neighboring base pairs n and n þ1 and the mid-base triad
n
nþ1
n
coincide, i.e., d
¼ d ¼ d~ . Define the ‘‘hinge’’ as an axis in
i
i
i
n
n
n
n
n
the d~ 1 d~ 2 plane so that it is inclined at f ¼ arcsinðy1 =b Þ with
n+1
d1
n+1
d3
n+1
d2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
n
ðy1 Þ2 þ ðy2 Þ2 is the magni-
Apply twist in such a way that both the (n þ1)th and the nth
n
of average of the triads Bn and Bn þ 1 .
The position of the (n þ1)th base pair relative to the nth one
is specified by three parameters describing the relative rotation
n
n
n
(the tilt y1 , the roll y2 and the twist y3 ) and by another three
parameters describing the translation (the shift rn1 , the slide rn2
and the rise rn3 ) [1], as it is visualized in Fig. 2. We here employ
the parametrization of ElHassan and Calladine [31], in which
the relative positions of two neighboring base pairs are obtained
n n n
from hn ¼ ½y1 , y2 , y3 and qn ¼ ½rn1 , rn2 , rn3 as a sequence of the
following steps:
641
n
base pairs are rotated by angles y3 =2 and y3 =2, respectively,
n
n
nþ1
about the axis d~ 3 (which still coincides with d3 and d3 Þ.
Next, rotate both the (n þ1)th and the nth base pairs about the
n
n
‘‘hinge’’ by angles b =2 and b =2, respectively.
Finally, translate both base pairs by 12 ðrn1 , rn2 , rn3 Þ and
n
n
n
1 ðrn , rn , rn Þ respectively, along d~ , d~ and d~ .
2
1
2
1
3
2
3
Any vector v can be represented by the components ½vn with
respect to the base pair triad Bn , the components ½vn þ 1 with
n
respect to Bn þ 1 , or the components ½vn~ with respect to B~ :
n
v ¼ Bn ½vn ¼ Bn þ 1 ½vn þ 1 ¼ B~ ½vn~ :
The relative rotation of B
nþ1
ð1Þ
n
with respect to B is given, in the
n
nþ1
units of Bn , by a 3 3 matrix Dn ¼ ðBn ÞT Bn þ 1 (i.e. Dnij ¼ di dj
).
n
Similarly, the relative rotation of B~ with respect to Bn is given by
n
n
n
n
n
n
T
~ ¼ d d~ ). Both Dn and D
~ n are orthogonal and
~ ¼ ðB Þ B~ (i.e. D
D
i
ij
j
can be parameterized by hn (see [29] for details). The translational
step parameters qn are the components of r n ¼ xn þ 1 xn with
n
~ n qn .
respect to the mid-base triad B~ , therefore ½r n n ¼ D
(n+1)th base pair
The transformation
Tn ¼
5’
n
1
Y
Dm ¼ ðB1 ÞT Bn
ð2Þ
m¼1
3’
n
d3
transforms the components of a vector with respect to basis Bn
into components of the same vector with respect to basis B1 .
n
d1
nth base step
2.2. Equilibrium
The stress-free state of the nth base pair step is given by the
I
nd I
nd I
Stra
Stra
n
d2
n
d2
5’
n
d3
n
n
nth base pair
Minor groove side
n
n
n
intrinsic base step parameters h ¼ ½y 1 , y 2 , y 3 and q n ¼ ½r n1 , r n2 , r n3 .
We use the so-called dinucleotide model in which the local elastic
energy of the nth base pair step depends on the composition of
that step (and no other base pairs). This elastic energy is taken to
d1
3’
Fig. 1. One base pair step and the corresponding base pair triads. The asymmetric
ends of DNA strands are called the 50 and 30 ends, with the 50 end having a terminal
phosphate group and the 30 end a terminal hydroxyl group. The strands are closer
together on one side of the helix (on the minor groove side) than on the other (that
is the major groove side). Bases are encountered along either one of these strands
(Strand I or Strand II) on the 50 -30 direction. The complementary bases are situated
on the other strand and also encountered in the 50 -30 direction. For example,
bases A, T, C, C have the complementary bases G, G, A, T on the other strand,
backward encountered.
be a quadratic function of the deformations Dhn ¼ hn h
Dqn ¼ qn q n
" n
#"
#
Gn
Dhn
F
1
cn ¼ ½ðDhn ÞT ðDqn ÞT :
ðGn ÞT H n
Dqn
2
n
and
ð3Þ
The total elastic energy is the sum of the elastic energies of the
P
n
n
n
n
base pair steps: c ¼ N
n ¼ 1 c . The elements of F , G and H are
the elastic (or force) constants of the nth step; F n and H n are
Tilt θ1
Roll θ2
Twist θ3
Shift ρ1
Slide ρ2
Rise ρ3
Fig. 2. The rotational (tilt, roll and twist) and the translational (shift, slide and rise) base step parameters describing the relative positions of two neighboring base pairs.
642
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
fn
m
(n+1)th base pair
n
nth base step
rn
nth base pair
−mn−1
−fn−1
n1
Fig. 3. Forces and moments balancing the nth base pair. The opposite of f
and
mn1 are exerted by the ðn1Þ th base pair, acting at the center of the nth base
n
pair. f and mn are exerted by the (nþ 1)th base pair and are excentric: they act at
the center of the (n þ1)th base pair.
focus on configurations of long DNA that can be represented as a
periodic repetition of a relatively short segment of a few base
pairs. In other words, we shall look for configurations of short
DNA segments formed by N þ1 base pairs that can be periodically
extended. This condition restricts the configurations and also the
base pair composition of DNA molecules that we study, as the
(N þ1)th base pair must be identical in composition to the 1st
base pair of the segment.
Let Q be the orthogonal map that takes the triad of the first
base pair, B1 , into the triad BN þ 1 , i.e., Q ¼ BN þ 1 ðB1 ÞT . An equilibrium configuration obeys the periodic boundary conditions (and
hence can be periodically extended to an equilibrium configura0
tion of a larger segment) if the components of f and m0 with
N
1
respect to the triad B are identical to the components of f and
0
N
mN with respect to the triad BN þ 1 , i.e., ½f 1 ¼ ½f N þ 1 , ½m0 1 ¼
½mN N þ 1 , which implies
0
N
Qf ¼ f ,
ð8Þ
n
symmetric matrices. The off-diagonal elements of F are the
coupling terms between the angular step parameters, the offdiagonal elements of H n are the coupling terms between the
translational step parameters, while the elements of Gn are the
coupling terms between the angular and the translational step
parameters. For instance, Gn33 is the negative twist-stretch coupling term discussed in [8], while F n33 is the twist stiffness.
Estimates of the values of elastic constants and intrinsic base
step parameters can be derived from statistical analysis of base
pair configurations recorded in X-ray observation of crystal
structures [22].
The DNA segments with Nþ 1 base pairs studied here are free
from external forces and moments. In [29] it is shown that each
base pair step in an equilibrium configuration satisfies the
variational (equilibrium) equations:
n
f f
n
n1
m m
¼ 0,
n1
ð4Þ
n
n
¼f r ,
n
n ¼ 2, . . . ,N:
ð5Þ
n
Here f and m are the force and moment that the (n þ1)th base
pair exerts on the nth base pair. The nth base pair is balanced by
n
f and mn , which are exerted by the (n þ1)th base pair, and by the
n1
opposite of f
and mn1 exerted by the ðn1Þth base pair. Note
n
that f and mn are excentric and act in the center of the (n þ1)th
base pair (see Fig. 3).
n
n
The components ½f n , ½mn n of f and mn with respect to the
n
local basis B are given [29] as functions of the step parameters qn
and hn , force constants F n , Gn , H n and stress-free initial values of
n
the step parameters h and q n as
n
n
n
~ ðGn ÞT Dhn þ D
~ H n Dqn ,
½f n ¼ D
½mni n ¼
3
X
j¼1
þ
Gnij
3
X
ð6Þ
n
ðF njk Dyk þ Gnjk Drnk Þ
k¼1
3 X
3
X
l¼1k¼1
n n
j Lkl rl
3
X
!
r¼1
ð7Þ
Here both Gnij and j Lnkl depend on hn and are given in details in
[29].
2.3. Periodic boundary conditions
We wish to study configurations of long DNA segments.
However, the method we use for finding all equilibrium configurations of a given mechanical system works efficiently only if the
number of variables of the system is small. Therefore, we shall
ð9Þ
n
Equilibrium equation (4) implies that f is independent of n, i.e.
n
0
f ¼f ,
n ¼ 1, . . . ,N:
ð10Þ
The orthogonal matrix Q is a proper rotation (detðQ Þ ¼ 1) in three
dimensions, and hence it has at least one real unit eigenvalue
l ¼ 1 with a corresponding real eigenvector v. In view of (8) and
n
(10) the vectors f ðn ¼ 1, . . . ,N) are all parallel to v. By adding up
the equilibrium equations (5) from n ¼1 to N, one also obtains
mN m0 ¼
N
X
n
f rn :
ð11Þ
n¼1
Scalar product of (11) with v reduces to
v mN ¼ v m0 ,
ð12Þ
and hence, in equilibrium, the projections of the moments m0 and
mN onto v are the same.1
Boundary condition (9) implies that the projections of m0 onto
an orthonormal triad are the same as the projections of mN onto
the same orthonormal triad transformed by Q . It gives us three
equations, but only two of them are linearly independent from
the equilibrium equations. In order to prove it, let us use the
orthonormal triad ½v,w,u, where v is the real eigenvector of the
matrix Q with eigenvalue l ¼ 1. Boundary condition (9) implies
that the projections of m0 onto the triad ½v,u,w are equal to the
projections of mN onto the orthonormal triad ½Qv,Qu,Qw. Since
Qv ¼ 1 v and m0v ¼ mN
v is guaranteed by the linear combination
(12) of equilibrium equations, boundary condition (9) does
provide us with only two independent scalar equations.
The additional constraint to be prescribed can be related to the
torque of the configuration about v. We shall focus on the
configurations of segments that are torsionally relaxed and hence
obey the condition
m0 v ¼ 0:
n
ðF nkr Dyr þ Gnrk Drnr Þ ,
i ¼ 1; 2,3:
Qm0 ¼ mN :
ð13Þ
Schematic diagram of the conditions (8), (9) and (13) is shown in
Fig. 4.
Once an equilibrium configuration of an N þ1 base pair long
segment with periodic boundary conditions is found, it can be
extended into an equilibrium configuration of a kN þ1 base pair
long DNA (with integer k) by taking copies of the original segment
and attaching them end-to-end in such a way that the first base
pair of one segment coincides with the (Nþ 1)th base pair of the
1
In fact, every moment fmn gN
n ¼ 0 has the same component with respect to the
eigenvector v.
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
Another two equations come from reformulating the periodic
boundary condition (9) as
fN
(N+1)th
643
base pair
u m0 ðQuÞ mN ¼ 0,
0
1
1
w m0 ðQwÞ mN ¼ 0,
ð15Þ
1
with m ¼ m f r . The last equation is (13).
mN
2.5. Numerical solution
Q
α
Qf0=fN
Q m 0 = mN
v
f0
1st base pair
m0
Fig. 4. Periodic boundary conditions.
next segment. Eventually, a helical structure is obtained in
this way with helical axis identical to the eigenvector v of the
matrix Q . Thus the molecule is stretched along its helical axis and
it is torque-free about it.
It follows from periodicity that the same dinucleotide steps are
situated at the nth and the ðn þ kNÞ th base steps (k¼1,2,y) and
that the corresponding step parameters of the extended DNA
are the same: hn þ kN ¼ hn , qn þ kN ¼ qn . And, in general, vectors
n þ kN
n
n
n þ kN
f
¼ ðQ ÞkN f ¼ f , mn þ kN ¼ ðQ ÞkN mn , r n þ kN ¼ ðQ ÞkN r n , di
¼
n
ðQ ÞkN di .
2.4. Formulation of the periodic BVP for N base pair steps
The equilibrium equations and the periodic boundary conditions can be conveniently computed as a set of 3N non-linear
functions of the variables fhn gN
n ¼ 1 , such that the zeros of those
functions define an equilibrium configuration. We shall use the
0
magnitude of the force F ¼ 9f 9 as a bifurcation parameter.
In view of (2), Q ¼ BN þ 1 ðB1 ÞT ¼ B1 T N þ 1 ðB1 ÞT , and hence Q is a
non-linear function of fhn gN
n ¼ 1 . Now the real eigenvector v of the
matrix Q is computed, and an orthonormal triad ½v,u,w is
n
formulated. The force vectors are f ¼ F v (n ¼1,y,N), therefore
fDqn gN
can
be
obtained
by
solving
the linear system (6).
n¼1
Of the 3N non-linear equations to be solved, 3(N 1) come from (5)
n
½mn n ½mn1 n ½f n ½r n n ¼ 0,
n ¼ 2, . . . ,N:
ð14Þ
Here
n
½f n ¼ F ðT n ÞT ½v1 ,
n
n
n
n
~ ðq n þ Dqn Þ ¼ D
~ ðq n þðH n Þ1 ðD
~ ÞT ½f n ðH n Þ1 ðGn ÞT Dhn Þ
½r n n ¼ D
and the components of moments f½mn n gN
n ¼ 1 are given by Eq. (7) as
non-linear functions of fhn gN
n ¼ 1 and F.
Numerical solutions of the BVP can be computed with the
Simplex Algorithm that allows one to determine the bifurcation
diagram for the system within a specified range of the configuration parameters [32,33]. There are d 1¼3N non-linear equations
to solve, written in the form gi ¼0 (i¼1,y,d 1). The solutions are
searched for in a finite d dimensional space, which is called the
General Representation Space (GRS), spanned by d 1 independent
variables and a parameter (F in our case). The solution set can be
embedded uniquely into this GRS. Since the BVP contains one
parameter, the solutions are typically one-dimensional sets
locally. These sets, the solutions of the BVP, are formed up by
equilibrium states, therefore we call them equilibrium paths (or
branches). The Simplex Algorithm is capable of either following a
solution set (equilibrium path) from a known initial point in the
GRS (this version is called the Path-Following Algorithm, PFA) [33],
or to scan a part of the GRS for all approximate solutions of the
BVP (that is called Simplex Scanning Algorithm, SSA) [32].
Scanning the GRS with SSA is accomplished in the following
steps. First the d-dimensional GRS is discretized over a given
domain of the variables and the parameter by setting up an
orthogonal (cubic) grid. Then each of these cubes is divided into d!
non-overlapping simplices with an appropriate method [32]. The
d 1 non-linear functions gi are evaluated at the mesh points of
the symplectic grid and the solution of the BVP over each simplex
is approximated with a piece-wise linear interpolation [34],
resulting in a line. If that line crosses the simplex, its segment
inside the simplex can be considered as a local approximation
of the equilibrium path. The endpoints of that segment are
recorded and then shown on the bifurcation diagrams in the
Section 3. The algorithm is robust, iteration-free and capable of
finding isolated equilibrium branches. The drawback of SSA is that
the number of the computational steps is proportional to
ðd1Þ3 d!ad , where d is the dimension of the GRS and a is the
resolution of the discretization grid of the GRS. In order to achieve
a good approximation, a fine grid must be used in the GRS. Thus
the application of the scanning algorithm is limited by the
dimension d of the GRS of the problem. For further details of
the algorithm, see e.g. [32]. Note that because of discretization
and linearization, bifurcation points typically appear imperfect
(branches split around the bifurcation point). Therefore, results of
the SSA can be used as inputs for a non-linear equation solver
(such as Newton–Raphson iteration) in order to reduce the error of
the solutions and to clarify whether a perfect bifurcation points
exist in the solution set.
Following an equilibrium path with the recursive procedure
PFA [33] is possible if a point on the path (an equilibrium state) is
known. The known solution point is placed on the centroid of a
face (entering face) of a simplex in the GRS. The (d 1) equations
are solved using piece-wise linearization [34] inside the simplex.
The resulting line enters the simplex at the known initial point
(entering point), and exits through another face (exit face) at an
exit point. The next simplex is defined by reflecting the vertex of
the simplex that does not lie on the exit face to the hyperplane of
the exit face. In this way the exit face of the previous simplex
becomes the entering face of the new one, while the previous exit
point becomes the new entering point, and the procedure starts
over. For further details see [33].
644
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
In our case, SSA is applied to search for periodic configurations
with periodicity up to N ¼2 base steps (ie. up to d¼7-dimensional
GRS). The domain of the GRS in which solutions are searched for
1
is: 451 r Dyi r451, 0 rF r 3 nN, with 75 grid points on each
axis in case of configurations with periodicity N ¼1 and 301 r
Dy1i r 301, 301 r Dy2i r301, 0 r F r2 nN, with 17 grid points on
each axis in case of configurations with periodicity N¼2. For longer
chains, PFA is used to trace the intrinsic, stress-free equilibrium
configurations.
2.6. Sequence dependent parameters of DNA
We employ sequence-dependent elastic constants and intrinsic step parameters for DNA that were obtained by analysis of
the statistical distribution of configurational parameters in the
Nucleic Acid Database [22]. In that work, the elastic constants
were obtained by inversion of the covariance matrix of step
parameters, computed separately for each dinucleotide step type,
while the intrinsic parameter values were equated with the
average values of the step parameters, and the data is rescaled
to fit the experimentally measured value of DNA bending stiffness
in solution (based on the WLC model) such that ð1=F 11 þ1=F 22 Þ ¼
2=A, where A¼0.0427 kT deg 2 [35]. Besides real dinucleotide
steps, we also study the mechanical properties of DNA with
average elastic constants and intrinsic values, labeled as MN. Its
elastic constants and intrinsic step parameters are obtained from
the covariance matrix and average parameter values computed
for the sample of all base pair combinations [22]. The most recent
values of the force constants and intrinsic values of the step
parameters can be obtained from [36].
Finally, we examine the mechanics of segments with modified
elastic properties derived from MN. The segment MNst refers to a
modified MN segment with helically twisted but straight initial
n
n
configuration, i.e., with y 1 ¼ y 2 ¼ r n1 ¼ r n2 ¼ 0. The segment MNdg
refers to a modified MN segment with coupling constants set to
zero. Thus the base step MNdg is free from coupling between step
parameters, but has the same intrinsic shape as MN, while MNst
has straight initial configuration, but coupling is taken into account.
The segment MNdg
st incorporates both idealizations, i.e., it is without
coupling terms and with straight (but helically twisted) initial
geometry.
2.7. Stability analysis of equilibrium configurations
Stability of an equilibrium configuration of the discrete model is
generally investigated by computing the Hessian of the total potential
energy function with respect to all possible perturbations. If all
eigenvalues of this matrix are positive then the configuration is
considered stable. In the present case there is a major complication
associated with the application of this method. By joining together
configurations obeying periodic boundary conditions one can, in
principle, construct an equilibrium configuration of an arbitrarily
long DNA molecule. The stability of this molecule will depend on its
length, which makes it impractical to analyze. Therefore we restrict
our analysis to stability with respect to a selected subset of perturbations of the molecule.
n
In particular, the elastic energy c of the nth segment is given
as a quadratic function of xn þ 1 , xn and of the two right handed
n
n
n
nþ1
orthonormal triads Bn ¼ ½d1 ,d2 ,d3 , Bn þ 1 ¼ ½d1
nþ1
,d2
nþ1
,d3
n
: c ¼
cn ðBn ,xn ,Bn þ 1 ,xn þ 1 Þ. The total elastic energy c is the sum
P
c ¼ Nn ¼ 1 cn . We use periodic perturbation with wave length
M¼kN, where N is the periodicity of the equilibrium configuration and k is an integer. The terminal base pairs of the M base
steps are unperturbed, while the middle base pairs are perturbed.
2
A necessary condition for stability is that d c, the second variation of the elastic energy is strictly positive for all admissible
2
variations of the periodic perturbation. The second variation d c
is expressed as a quadratic form of the components of dðxn Þ and
dðwn Þ. The former one describes the variation in the displacement
of base pairs, while the latter one describes the variation in the
n
orientation of base pairs, the variation in the triads di through the
n
n
n
relation dðdi Þ ¼ w di [29].
We compute the Hessian H from the second variations obtained
by central difference approximation for k¼ 4. If the smallest eigenvalue of H is not positive, then the configuration is unstable. The
unstable equilibrium configurations are plotted with thin symbols in
all the corresponding figures of this paper. If all the eigenvalues of H
are positive, then the equilibrium configuration is declared potentially stable and is represented by a thick symbol in the graphs.
Clearly, if the configuration minimizes the total energy over all
configurations with the same boundary conditions, then it must be
globally stable. Such is the case for configurations on the primary
branch if the branch has no bifurcations.
3. Results
Equilibrium configurations of N r 2 base step periodic DNA
were computed with SSA and the primary equilibrium paths of
configurations with periodicity N ¼3 were traced with PFA. The
results are obtained as equilibrium paths, i.e., curves in the 3N þ1
dimensional GRS spanned by fhn gN
n ¼ 1 and F. The projections of
these paths onto subspaces of the GRS are visualized as bifurcation diagrams with the relative extension x=x0 of the segment as
the bifurcation parameter. Here the distance x between the center
of the terminal rectangles is measured along the eigenvector v:
x¼v
N
X
rn :
n¼1
Thus it is the extension along the axis of the helix of the periodic
configuration. The value of x in the stress-free state is denoted by
x0. Only configurations with relative extension 1 rx=x0 r 2 are
shown.
We shall monitor the magnitude of average excess of local
shear s, rise r and bending b for each configuration. These average
deformations are defined for a segment of periodicity N as
N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 X
s¼
ðDrn1 Þ2 þ ðDrn2 Þ2 ,
ð16Þ
Nn¼1
r¼
N
1 X
Drn3 ,
Nn¼1
ð17Þ
b¼
N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 X
n
n
ðDy1 Þ2 þ ðDy2 Þ2 :
Nn¼1
ð18Þ
The twisting of a segment can be monitored by measuring the
angle a between the terminal base pairs. The angle a can be
determined from the eigenvalues l, x þ Zi, xZi of the orthogonal
transformation Q as
Z
a ¼ arctan :
x
Thus a describes the rotation about the axis of the helix of the
periodic configuration. The stress-free value of a is denoted by a0 .
The quantity ðaa0 Þ=N is the over/undertwisting density.
A periodic configuration is referred to by its repeating unit of
base pairs on one strand. For instance, A refers to a poly(A) DNA,
which is formed by bases A taking position on one strand (thus
their complementary T bases are laced up on the other strand),
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
while ACT refers to poly(ACT) DNA with bases ACTACTACTACTy
following on one strand (therefore the complementary bases
on the opposite strand recorded in the opposite direction are
yAGTAGTAGTAGT). The periodicity of the segment is N ¼1 in the
first case and N ¼3 in the second one. Note that poly(MN) has
periodicity N ¼1, nonetheless it is marked by two letters by
convention.
3.1. Configurations with periodicity N ¼1
Configurations with periodicity N ¼1 correspond to DNA segments that are homogeneous in their elastic response, i.e., are
composed of identical dinucleotide steps. There are two real types
of such segments, namely poly(A) and poly(G), labeled for short as
F [nN]
2
F [nN]
A and G. We also consider four idealized types labeled as MN,
MNst, MNdg and MNdg
st .
The first type of non-linear response we wish to explore is the
transition to an overstretched configuration. As we shall see, such
a transition is observed for segments with periodicity N ¼1 and it
can be best described as shear instability.
Fig. 5 shows the diagram of the force F vs. relative extension
x=x0 . Starting from the stress-free state (x=x0 ¼ 1, F¼ 0), the
primary equilibrium paths of each segment type are monotonically increasing and linear near F¼0. For segment MNdg
st the
primary branch shows a linear dependence of F on x=x0 throughout the entire range studied (see Fig. 5(b)). It loses stability at the
first bifurcation point at x=x0 1:264 where it branches off into a
pair of secondary branches (overlapping in this view) through a
A
G
MN
2.5
1.5
2.5
MNst
2
MNdg
MNdg
st
1.5
1
1
0.5
0.5
0
0
1
0.6
1.4
x/x0
1.6
1.8
1
0.4
s [nm]
s [nm]
1.2
A
G
MN
0.5
0.3
0.6
MNst
0.5
MNdg
MNdg
st
0.4
0.2
0.1
0.1
0
1.2
1.4
x/x0
1.6
1.8
1
0.3
1.8
1.2
1.4
x/x0
1.6
1.8
1.4
x/x0
1.6
1.8
MNst
0.25
0.2
Δ ρ3 [nm]
0.15
1
1
Δ ρ3 [nm]
1.6
0.3
A
G
MN
0.25
0.1
0.05
MNdg
dg
MNst
0.2
0.15
0.1
0.05
0
0
1
1.2
1.4
x/x0
1.6
1.8
1
A
G
MN
b [deg]
b [deg]
1.4
x/x0
0
1
20
1.2
0.3
0.2
25
645
15
10
5
25
MNst
20
MNdg
MNdg
st
1.2
15
10
5
0
0
1
1.2
1.4
x/x0
1.6
1.8
1
1.2
1.4
x/x0
1.6
1.8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fig. 5. (a) and (b) Force F vs. relative extension x=x0 along the eigenvector v. (c) and (d) Shear deformation s ¼ ðDr11 Þ2 þ ðDr12 Þ2 vs. relative extension x=x0 . (e) and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
1
(f) Excess rise r ¼ Dr13 vs. relative extension x=x0 . (g) and (h) Bending deformation b ¼ ðDy1 Þ2 þ ðDy2 Þ2 vs. relative extension x=x0 . Thin points represent unstable
equilibrium configurations and thick points represent potentially stable configurations. The spatial periodicity is N ¼ 1.
646
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
pitchfork bifurcation. There is an additional pitchfork bifurcation
on the primary branch, at x=x0 1:366, where a pair of tertiary
branches (overlapping in this view) appear. The secondary branches
are potentially stable, while the tertiary branches are unstable. For
the segments A, G, MN and MNst there are no pitchfork bifurcation
points, but separate branches appear. The primary branch plateaus at
values of the relative extension x=x0 4 1:2. There are two separated
branches for A and G, and one secondary branch for MN and MNst.
Poly(A) and poly(G) have potentially stable equilibrium configurations on the separated branches, while MN and MNst do not. For
segment MNdg there seem to be bifurcations on the primary branch
in Fig. 5(b). In spite of this, the plot of shear deformation s vs. x=x0 in
Fig. 5(d) reveals that there are no bifurcations on the primary branch,
but there is a pitchfork bifurcation on the secondary branch. (A more
thorough discussion is given later in Section 3.4.) The primary branch
and a part of the secondary branch are potentially stable.
The stretch modulus S is the slope of the linear range of Fig. 5(a).
It is 4299, 2612 and 2621 pN for poly(A), poly(G) and poly(MN),
respectively. Measured values of the stretch modulus from stretching
experiments [6] are 1000 pN. The force at which the transition
starts is approximately 1100, 480 and 670 pN for poly(A),
poly(G) and poly(MN), respectively. According to the experiments
[5,6,13] this transition occurs at 65 pN. We will discuss the
possible reason for these differences, in Section 4.
Fig. 5(c) and (d) shows the plots of shear deformation s vs.
relative extension x=x0 . These plots confirm that the major change
occurring at the primary bifurcation point or the point at which
the force plateaus is in local shear. Plots for segments A, G and MN
are shown in Fig. 5(c). The beginning parts of the primary branches
of these segments are fairly horizontal, i.e. shear deformation is not
significant. However, at around the point at which force levels off,
the primary branches show a steep increase of shear deformation
with x=x0 , and the growth of the magnitude of shear deformation
eventually becomes linear in x=x0 . The potentially stable secondary
equilibrium branches show even larger shear deformation than the
primary ones. In Fig. 5(d) plots for idealized segments are shown.
Note, that there is no shear deformation along the primary branch of
idealized segment MNdg
st .
Fig. 5(e) and (f) shows the dependence of excess rise r (increase
in separation between neighboring base pairs) on relative extension
x=x0 . Not surprisingly, the increase in magnitude of shear deformation corresponds to no change, or even a decrease in the magnitude
of excess rise. Potentially stable secondary branches also show a
decrease in rise. Note, that these branches show an increase in shear
deformation while rise decreases, hence deformations are shear
dominated during increasing extension.
Fig. 5(g) and (h) shows bending deformation b vs. relative
extension x=x0 . The graphs show that bending accompanies shearing
during the bifurcation, mainly due to the intrinsic helicity of DNA.
Primary branches of the segments A and MN (shown in Fig. 5(g)) are
initially similar (a gradual incline is followed by a steeper one), while
α−α0 [deg]
10
A
G
MN
α−α0 [deg]
15
the primary branch of G has a constant incline for smaller extension
(x=x0 o1:3). At larger values of x=x0 , the increase in bending
deformation levels off on the primary branches and secondary
branches appear. Fig. 5(h) shows the results for idealized segments.
Note that bending deformation for the primary branch of idealized
segment MNdg
st is zero.
Although large bending deformation occurs during stretching,
it does not contribute to elongation, only shear deformation and
rise do. Since the shear deformation dominates at around the
plateau on force–extension diagram while rise remains fairly constant, the phase transition is a consequence of shear instability.
The second unusual type of behavior of long DNA is overtwisting followed by untwisting as a result of stretching. As we
show, this result is observed already for DNA segments with
periodicity N ¼ 1. Fig. 6(a) and (b) shows rotation aa0 about the
eigenvector v vs. relative extension x=x0 . For segments A, G, MN
and MNst, the primary branch shows that the molecule overtwists
(i.e., a is positive) up to a certain value of the extension, then it
untwists. This feature is related to coupling terms in the elastic
coefficient matrix, since the primary branch of the idealized
segments MNdg and MNdg
st remains horizontal, see Fig. 6(b).
(While their secondary and tertiary branches show untwisting.)
The secondary branches of A, G, MN and MNst also show overtwisting, but only A and G have potentially stable overtwisted
secondary equilibrium states. As it was pointed out, along the
primary branch of polyðMNdg
st Þ, there is no bending, twisting nor
shear deformation; only the separation of base pairs (the excess
rise) increases linearly with increasing F. The reason for this is the
intrinsically straight (but helically twisted) geometry and the lack
of coupling between the modes of deformation: the straight
geometry implies that the eigenvector v and the force F are
perpendicular to the base pairs, and the zero coupling terms imply
that this force causes only separation. There are, however, bifurcations on the primary branch that correspond to shear instability
(similar to the phenomenon observed with straight, continuous,
stretched rods described in [37]), which lead to new branches of
equilibrium configurations. On these secondary branches twist,
bending and shear deformations are non-zero.
According to the experiments, the magnitude of maximal
overtwisting is 2:5 rotations/8400 base pairs [8]. Our model
shows higher overtwisting, approximately 4.95, 6.12 and 11.40
rotations/1000 base pairs for poly(A), poly(G) and poly(MN),
respectively. A discussion about this difference is given in
Section 4.
In Fig. 7 equilibrium branches of poly(G) DNA are plotted and
configurations are visualized as segments of 15 base pair steps at
selected values of the relative extension x=x0 ¼ 1,1:2,1:4 and 1.6.
It is apparent how the shear deformation dominates with increasing
extension. It is important to observe that none of the deformed
shapes resembles a parallel ladder. Configurations labeled a, b, c,
and d lie on the potentially stable primary branch of the bifurcation
5
0
−5
15
MNst
10
MNdg
MNdg
st
5
0
−5
−10
−10
1
1.2
1.4
x/x0
1.6
1.8
1
1.2
1.4
x/x0
1.6
1.8
Fig. 6. Rotation aa0 about the eigenvector v vs. relative extension x=x0 . Here a0 refers to the rotation corresponding to the stress-free state. The spatial periodicity is
N ¼1.
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
647
2.5
j
F [nN]
2
i
h
f
d
1.5
1
0.5
0 a
1
b
1.2
g
e
c
1.4
x/x0
1.6
e
1.8
19.67 kT
g
20.32 kT
a
f
c
b
34.23 kT
h
40.62 kT
j
65.49 kT
i
63.21 kT
d
0 kT
3.89 kT
11.71 kT
20.78 kT
x/x 0=1
x/x 0=1.2
x/x 0=1.4
x/x 0=1.6
Fig. 7. Bifurcation diagram of poly(G) DNA with spatial periodicity N ¼ 1 shown in the subspace (F,x=x0 ) and periodic equilibrium configurations of 15 base pair steps at
x=x0 ¼ 1,1:2,1:4 and 1.6. The corresponding elastic energies (in kT) per a base pair step are also presented. The possible transitions through energy barriers are indicated by
arrows.
diagram and have the lowest elastic energy of all configurations at a
fixed extension. These configurations would be observed when the
segment was stretched in the absence of thermal fluctuations.
Configurations e and f lie on the secondary, potentially stable
branch and are local minimizers of the elastic energy. In the
presence of fluctuations, at the extension x=x0 ¼ 1:4 the DNA can
potentially exist in c or e state. The transition path between these
states travels through the mountain pass defined by the configuration g. Likewise, at x=x0 ¼ 1:6 the DNA can exist in d or f state. There
are two transition paths between these states, one which passes
through h and the other one through i. Configuration j has two
unstable perturbation dimensions and hence is not at an energy
barrier.
3.2. Global bifurcation diagrams of configurations with periodicity
N ¼2
For configurations with periodicity N ¼2, the dependence of
deformational characteristics on x=x0 is generally similar to those
for N ¼1. Both shear instability and overtwisting followed by
untwisting as a result of stretching are present. The main new
features are the presence of secondary bifurcations for a few real
segments and the large variability of the magnitude of effects,
depending on the combination of base pairs.
The dependence of force F on relative extension x=x0 is linear
at the beginning of the primary branch for all the base pair
compositions (Fig. 8(a)). At larger extension it plateaus and
secondary branches appear. There are potentially stable configurations along parts of the secondary branches. For each composition, the primary branch shows overtwisting up to a certain
value of extension x=x0 (which depends on the sequence),
followed by a plateau, then it begins to untwist (see Fig. 8(b)).
On the primary branch of GC and on the secondary branch of AT
there are bifurcations seen in these projections. We shall comment more on this feature in Section 3.4, where we show that
these are true bifurcations in the 3(N þ1) dimensional GRS.
Shear deformation is more and more pronounced again at
larger extensions (Fig. 8(c)). The slope of the primary branch
increases monotonically up to at least x=x0 1:3 and potentially
stable parts of the secondary equilibrium branches show even
larger shear deformation. Fig. 8(d) shows excess rise r vs. relative
extension x=x0 . The first part of each primary branch is linearly
increasing, then it levels off. Potentially stable secondary
branches show a decrease in excess rise. Note that it is correlated
with an increase in shear deformation on those branches in
Fig. 8(c), which confirms that it is not the rise which is important
during large stretching, but the shear deformation.
As the projections of equilibrium branches onto various subspaces show significantly different characteristics depending on
the base pair composition of the periodic DNA, it is evident that
the mechanical response of the molecule on stretching is strongly
sequence-dependent. In the next section we shall focus on the
primary equilibrium path and discuss the sequence dependence
of the evolution of shear and bending deformations, excess rise,
648
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
1.4
F [nN]
1.2
AC
AG
AT
GC
10
(α−α0)/2 [deg]
1.6
1
0.8
0.6
0.4
0.2
5
AC
AG
AT
GC
0
−5
−10
0
1
s [nm]
0.5
0.4
1.4
x/x0
1.6
1
1.8
AC
AG
AT
GC
r [nm]
0.6
1.2
0.3
0.2
0.1
0
1
1.2
1.4
x/x0
1.6
1.8
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1.2
1.4
x/x0
1.6
1.8
AC
AG
AT
GC
1
1.2
1.4
x/x0
1.6
1.8
Fig. 8. (a) Force F, (b) rotation ðaa0 Þ=2, (c) shear deformation s and (d) excess rise r vs. relative extension x=x0 . The spatial periodicity is N ¼2.
twist and force along the extension for all the possible combinations of base pairs with periodicity N ¼1,2 and N ¼3.
3.3. Sequence-dependent mechanics of segments with periodicity
1 rN r 3
We have seen that equilibrium states on primary branches are
potentially stable as long as the branch has no bifurcation points.
The one exception is poly(GC) which has a bifurcation point on its
primary path; for this segment we show just the potentially
stable states, namely the primary branch up to the bifurcation
point and then the secondary branches (overlapping in the
projections shown here). The projections of these paths onto
various subspaces for all the possible base pair compositions with
periodicity N ¼1,2 and 3 are discussed in this section.
In general, the force–extension diagram is linear up to an
overstretching not larger than 20%, then primary branches start
plateauing at different force–extension values depending on the
sequence composition (see Fig. 9(a) and (b)). The position of the
plateau (i.e., the force causing the transition) depends primarily
on the GC base pair content of the molecule according to our
results. Indeed, higher GC percentage results in a molecule that
is softer for shearing (see the values of H11 and H22 in [36] for
comparison) and (consequently) shearing instability occurs at
smaller stretching force. The approximate value P of the force F
where the transition occurs is summarized in Table 1 for each
base pair composition studied.
The average shear deformation s and excess rise r seem to be
correlated in that base pair combinations responding by small shear
deformation to stretching undergo large rise at large extension and
vice versa (see Fig. 9(c)–(f)). Average shear deformation s does not
show much difference among different compositions and monotonously increases with the extension. The largest deviation in shear
deformation is smaller than 0.1 nm even for large extension.
The excess rise r, however, shows significant deviations at large
extension among different compositions, and the peak of the curve is
reached at x=x0 o 2 in most of the cases. Despite the large deviation
in H33, the slope of the projection of the primary branch on the
subspace (r,x=x0 ) for small extensions is essentially independent of
the composition (see Fig. 9(e) and (f)). The dependence of average
bending deformation b on relative extension x=x0 is surprisingly
similar for all the base pair combinations (Fig. 9(g) and (h)). The
largest deviation in bending is smaller than 51 even for large
extension. Segment MN with average physical properties undergoes
the smallest bending for relatively small (x=x0 o 1:3) and for relatively large (x=x0 4 1:75) extension, thus again showing an extreme
behavior.
Using the modulus H33 given by [36] for all types of base steps,
an averaged normal stiffness Sn can be computed for any combination of base pairs. This stiffness Sn is given in Table 1 for each
composition studied here in addition to the stretch modulus S
obtained from the slope of the linear range of Fig. 9 (a) and (b).
One can conclude that for the chains studied here Sn (stiffness
against the separation of base pairs) is 20–70% bigger than the
(effective) stretch modulus S obtained from the force–extension
curves. The molecule is more stretchable than what the modulus
H33 would suggest because of intrinsic curvature and coupling
between the modes of deformation. The only exception is the
idealized segment MNdg
st which lacks such couplings and is
intrinsically straight. (Note in Table 1 that MNst is also softer.)
The most rigid sequence composition for stretching (i.e. having
the biggest stretching modulus S, see Table 1) is poly(A), then
poly(ATT) and poly(AT) follows, and the least stiff ones are
poly(AG), poly(AGC) and poly(AGG). For large extension the
stretch modulus becomes smaller because of the shear instability.
As one can see from Table 1, there is a large variation in both S
(from 2341 pN to 5059 pN) and P (from 450 pN to 1100 pN)
depending on the base pair composition.
All sequence compositions show overtwisting at low x=x0 ,
followed by untwisting at higher x=x0 (Fig. 10). Poly(AGT) overtwists by far the most and poly(A) overtwists the least. The
composition which untwists the least is poly(GC). The composition poly(MN) untwists the most—it is surprising that the configuration formed up by segments with average mechanical (and
geometrical) properties has an extreme mechanical response in
this respect. In addition, it is remarkable that configurations
having all purine (A or G bases) on one strand (like A, AG, AGA,
AGG) untwist significantly at large extension.
The twist-stretch coupling term G33 given by [36] for each
base step is the smallest in case of step AA and the biggest in
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
1.2
F [nN]
1
0.8
0.6
1.4
A
G
MN
AC
AG
AT
GC
1.2
1
F [nN]
1.4
0.8
0.6
0.4
0.4
0.2
0.2
0
ACA ATT
ACC GCG
ACG
AGA
AGC
AGG
AGT
ATC
0
1
1.2
1.4
1.6
1.8
2
1
1.2
1.4
x/x0
s [nm]
0.5
0.4
0.3
0.6
A
G
MN
AC
AG
AT
GC
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
1.2
1.4
1.6
1.8
2
1
1.2
1.4
0.06
0.12
A
G
MN
AC
AG
AT
GC
0.1
0.08
0.06
0.04
0.04
0.02
0.02
0
1.2
1.4
1.6
1.8
2
1
1.2
1.4
1.6
1.8
2
x/x0
A
G
MN
AC
AG
AT
GC
25
20
b [deg]
b [deg]
15
2
ACA ATT
ACC GCG
ACG
AGA
AGC
AGG
AGT
ATC
x/x0
20
1.8
0
1
25
1.6
x/x0
r [nm]
r [nm]
0.08
2
ACA
ACC
ACG
AGA
AGC
AGG
AGT
ATC
ATT
GCG
x/x0
0.1
1.8
0
1
0.12
1.6
x/x0
s [nm]
0.6
649
10
5
15
10
5
0
ACA
ACC
ACG
AGA
AGC
AGG
AGT
ATC
ATT
GCG
0
1
1.2
1.4
1.6
1.8
2
x/x0
1
1.2
1.4
1.6
1.8
2
x/x0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
P
n
Fig. 9. (a) and (b) Force F, (c) and (d) shear deformation s ¼ ð1=NÞ N
ðDrn1 Þ2 þ ðDrn2 Þ2 , (e) and (f) excess rise r ¼ ð1=NÞ N
n¼1
n ¼ 1 Dr3 and (g) and (h) bending deformation
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PN
n
n
b ¼ ð1=NÞ n ¼ 1 ðDy1 Þ2 þ ðDy2 Þ2 vs. relative extension x=x0 for configurations with periodicity N ¼ 1,2 and N¼ 3. Only the primary equilibrium branches are shown.
case of steps AG, then GC, GG and AC follows. That could explain
why poly(A) overtwists the least, but cannot explain itself why
poly(AGT) overtwists by far the most. It is another proof of
that coupling terms effect significantly the mechanical response
of the molecule. Our model again shows a higher overtwisting
than the experiments, with a large variation depending on the
base pair composition (from 4.95 to 21.41 rotations/1000 base
pairs). This is summarized in Table 1, and further discussed in
Section 4.
3.4. Special cases: sequence-symmetric steps
It was noted in [31] that when the numbering of the base pairs
in a segment of even number M of base pairs is inverted, i.e.,
650
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
n-Mn þ1,
n
n
n
n
n
and
the
Mn þ 1
base
Mn þ 1
pair
triads
are
Mn þ 1
changed
as
n
½d1 ,d2 ,d3 -½d1
,d2
,d3
, so that d3 again points
n
from the new nth base pair to the new (n þ1)th base pair and d1
points away from the minor groove side, then the values of roll,
twist, slide and rise do not change (apart from re-indexing), i.e.,
Mn
½y2 , y3 , rn2 , rn3 -½y2
n
½y1 ,
Mn
, Mn
, Mn
, while tilt and shift change
2
3
Mn
n
Mn
, 1 . The inversion in numbering
1 -½y1
, y3
r
r
r
in signs, i.e.,
r
also leads to the change of the DNA sequence of the segment to a
complementary sequence in which every base is replaced by its
complementary base and the bases are read in the opposite order,
for example ACTG is changed into CAGT. A palindromic sequence,
such as AGCT, is identical to its inversion and hence for a
palindromic sequence one can expect that the inversion of an
equilibrium configuration corresponding to symmetric loading
conditions is again an equilibrium configuration for which
n
n
n
Mn
½y1 , y2 , y3 , rn1 , rn2 , rn3 new ¼ ½y1
Mn
, y2
Mn
, y3
,rMn
, rMn
, rMn
old ,
1
2
3
ð19Þ
where ‘‘old’’ refers to the equilibrium configuration before inversion and ‘‘new’’ refers to the configuration after inversion.
A base pair step is sequence-symmetric if the bases on one
strand in the 50 -30 direction are identical to the bases on the
other strand (backward) in the 50 -30 direction. There are four
Table 1
Force P at the transition for various compositions of base pairs, maximum of the
magnitude of overtwisting (given as the number nmax
rot of rotations scaled to 1000
base pair steps), effective stretch modulus S, and pure stretch modulus Sn
computed from normal stiffnesses Hn33 of the compositions. (Extreme values are
emphasized in bold.)
Base pair
composition
P (pN)
nmax
rot (rot/kb)
S (pN)
Sn (pN)
A
G
MN
MNst
MNdg
1100
480
670
650
850
850
4.95
6.12
11.40
9.23
0
0
4299
2612
2621
2723
2971
3195
5059
4520
3195
3195
3195
3195
540
570
720
450
610
670
540
690
570
540
720
600
780
470
8.57
10.27
15.04
10.82
7.40
11.61
7.38
9.59
13.69
9.20
21.41
8.29
11.40
10.40
2598
2341
3582
2508
2970
2694
2540
2751
2369
2427
2912
2623
3845
2482
3622
4000
5132
3664
4096
3924
3603
4353
3919
4177
4722
3766
5107
3952
actual sequence-symmetric steps: AT, TA, GC and CG. Sequencesymmetric base pair steps have the property that their sequence
is invariant under the inversion and as a result [22,29], sequencesymmetric steps have energy functions that are invariant under
n n n
n n n
the following transformation: ½y1 , y2 , y3 , rn1 , rn2 , rn3 -½y1 , y2 , y3 ,
n n n
r1 , r2 , r3 . Therefore, the intrinsic (stress-free) values of tilt and
shift of a sequence-symmetric step are zero
n
y 1 ¼ r n1 ¼ 0:
ð20Þ
Furthermore, in the case of quadratic energy function (3), the
force constants of a sequence-symmetric step obey (see [29] for
further details)
F n12 ¼ F n13 ¼ 0,
Hn12 ¼ Hn13 ¼ 0,
Gn12 ¼ Gn13 ¼ Gn21 ¼ Gn31 ¼ 0:
Namely, the coupling terms between tilt and roll, tilt and twist,
shift and slide, shift and rise, tilt and slide, tilt and rise, shift and
n
roll, shift and twist are all zero. In other words, tilt y1 and shift rn1
are decoupled from the other modes of deformation.
For an equilibrium configuration of length of period N, obeying
PBCs, it holds that hn ¼ hn þ N and qn ¼ qn þ N . Besides, if N ¼2 then
hn ¼ hN þ 2n and qn ¼ qN þ 2n also hold because of the symmetry
of the configuration. Taking a slice of the equilibrium configuration with even number M¼N þ2 of base pairs an equilibrium
configuration is obtained which also obeys PBCs (except for the
terminal base pairs are not identical). If this slice has a palindromic
sequence and N¼2, then it follows from (19) and the periodicity of
the configuration that
n
N þ 2n
½y1 , rn1 new ¼ ½y1
and, in view of (20)
n
Therefore, as a special case, a chain composed of identical
sequence-symmetric steps with length of periodicity N ¼ 2 has a
bifurcation diagram which is reflection symmetric with a change
n
in sign of all Dy1 and Drn1 . This symmetry may lead to a pitchfork
bifurcation on one of the equilibrium branches. We note here that
if a chain composed of the same identical sequence-symmetric
steps but with higher periodicity N is investigated, this symmetry
of the bifurcation diagram may not hold for every branch, but
since the equilibrium states obtained for N ¼2 are also solutions
for higher (even) N values, the reflection symmetric solutions of
case N ¼2 also form reflection symmetric branches in the bifurcation diagram of higher (even) N values with a change in sign of all
Dyn1 and Drn1 .
In addition to the true sequence-symmetric steps, there are
also virtual sequence-symmetric steps: the average step MN and
the derived steps MNdg, MNst and MNdg
st . Global solutions of
10
10
5
5
0
−5
A
G
MN
AC
n
½Dy1 , Drn1 new ¼ ½Dy1 ,Drn1 old :
(α−α0)/N [deg]
(α−α0)/N [deg]
MNdg
st
AC
AG
AT
GC
ACA
ACC
ACG
AGA
AGC
AGG
AGT
ATC
ATT
GCG
n
þ 2n old
,rN
¼ ½y1 ,rn1 old
1
AG
AT
GC
−10
0
ACA
ACC
ACG
AGA
AGC
−5
AGG
AGT
ATC
ATT
GCG
−10
1
1.2
1.4
1.6
x/x0
1.8
2
1
1.2
1.4
1.6
1.8
2
x/x0
Fig. 10. Rotation ðaa0 Þ=N vs. relative extension x=x0 for configurations with periodicity (a) N ¼ 1,2 and (b) N ¼ 3. Only the primary equilibrium branches are shown.
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
chains composed of these virtual sequence-symmetric steps with
periodicity N ¼1 and N ¼2 were identical in the studied regime of
the GRS. For base pair steps MNst and MNdg
st the intrinsic values of
n
roll and slide are zero: y 2 ¼ r n2 ¼ 0. As for the force constants, only
MNdg
st obey
F n12 ¼ F n23 ¼ 0,
Hn12 ¼ Hn23 ¼ 0,
651
induced pitchfork bifurcations arise from two independent bending
and shearing modes originating from anisotropy, straight initial
geometry, and material symmetry.
We note here that if segment MNdg
st is further idealized by
making it unshearable, only a primary branch with linear riseextension and linear force–extension diagram exists, with no
bending, twisting or shear deformation. This confirms that shear
deformation is crucial in the mechanical behavior of DNA.
Gn12 ¼ Gn23 ¼ Gn21 ¼ Gn32 ¼ 0:
From these facts, we can conclude, similarly as above, that if a
chain composed of MNdg
st steps with length of periodicity N ¼2
then
n
3.4.2. PolyðMNdg ) and poly(MN)
The step MNdg differs from MNdg
st by having non-zero values of
1
the intrinsic roll y 2 and slide r 12 . Comparing the projections of the
bifurcation diagram of MNdg
st (Fig. 11(a)) to the projections of the
n
½Dy2 , Drn2 new ¼ ½Dy2 ,Drn2 old ,
namely it has a bifurcation diagram which is reflection symmetric
n
not only with a change in sign of all Dy1 and Drn1 but with a
n
change in sign of all Dy2 and Drn2 , too.
3.4.1. PolyðMNdg
st Þ
As expected, there are two pitchfork bifurcation points on the
primary branch which can be observed in the projections of equili1
1
brium branches onto the 3d subspace (Dy1 , Dy2 , x=x0 ) shown in
Fig. 11(a). The paths shown in the figure reveal reflection symmetries
1
1
with respect to the planes ðDy1 ,x=x0 Þ, ðDy2 ,x=x0 Þ, ðDr11 ,x=x0 Þ,
2
2
1
2
ðDr2 ,x=x0 Þ, ðDy1 ,x=x0 Þ, ðDy2 ,x=x0 Þ, ðDr1 ,x=x0 Þ and ðDr22 ,x=x0 Þ. The
solution sets for the two base pair steps are identical.
In this case the intrinsic shape is straight, i.e. there is no
intrinsic curvature and shear deformation, but there is intrinsic
helical twist. The primary branch consists of configurations with
no bending, twisting or shear deformation, but with considerable
excess rise: the molecule remains straight, only the separation of
base pairs increases. At the first bifurcation point at x=x0 1:264,
the equilibrium configuration on the primary branch loses stability and two potentially stable secondary branches arise, with
configurations that obey non-zero roll and slide, coupled with
untwisting and a decrease in rise (see also Fig. 6(b) and 5(f)). At
the second bifurcation point at x=x0 1:366 a tertiary unstable
equilibrium branch originates with configurations that show nonzero tilt and shift, while untwists, and its rise decreases more
intensely than that of the secondary branch. These shear instability
bifurcation diagram of MNdg (shown in Fig. 11(b)), one can
conclude that in the latter case there are no bifurcation points
on the trivial path but two disconnected branches appear (the
primary and the secondary), while a bifurcation point on the
secondary branch exists. This bifurcation point is at x=x0 1:377,
Dy11 ¼ y11 ¼ 0, Dy12 2:351. The equilibrium paths show reflection
1
symmetries with respect to the planes ðDy1 ,x=x0 Þ, ðDr11 ,x=x0 Þ,
2
ðDy1 ,x=x0 Þ and ðDr21 ,x=x0 Þ.
All the configurations related to the primary branch and configurations corresponding to a part of the secondary branch are
potentially stable, as indicated in the figures. Equilibrium configurations on the primary and secondary branches have zero tilt and
shift. Configurations obey zero or negative twist (see Fig. 6(b)).
The poly(MN) sequence has coupling between base pair step
parameters which results in a bifurcation diagram that is different
from the previous two cases in that no bifurcations occur in the
range of parameters studied (not shown). There are only primary
and secondary branches within the scanned domain. The primary
branch is formed by potentially stable equilibrium states, while
the secondary branch is formed by unstable configurations.
1
2
Values of tilt Dy1 , Dy1 and shift Dr11 , Dr21 are zero on the primary
and secondary equilibrium branches.
3.4.3. Poly(AT)
The bifurcation diagram for poly(AT) is topologically the same
as for polyðMNdg Þ: there is no bifurcation on the primary path but
1.6
x/x0
x/x0
1.6
1.4
1.2
1.2
−10
1
−10
0
Δ θ21 [deg]
10
10
−10
1
0
]
1 [deg
Δ θ1
−10
0
0
Δ θ21 [deg
10
10
]
1.4
1
Δ θ1
g]
[de
1.5
x/x0
x/x0
1.4
1.2
−5
1
−10
0
−5
Δ
0
θ21 [deg]
5
5
10
1
Δ θ1
[d
]
eg
1.25
1
−10 −5
0
5
Δ θ12 [deg]
5
10
10
1
1
−10
−5
]
eg
1 [d
θ1
Δ
0
dg
Fig. 11. Global bifurcation diagram of base pair steps (a) MNdg
shown as a projection onto the subspace (Dy1 , Dy2 , x=x0 ). Global bifurcation diagram of base
st and (b) MN
1
1
1
2
pair steps ATA shown as a projection onto (c) the subspace (Dy1 , Dy2 , x=x0 ) and (d) the subspace (Dy1 , Dy1 , x=x0 ). The projection of the equilibrium paths onto the
parameter plane is shown in gray.
652
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
the secondary path bifurcates. The bifurcation point found on the
1
1
2
secondary branch is at x=x0 1:33, Dy1 ¼ 0, Dy2 9:571, Dy1 ¼ 0,
2
Dy2 9:971 (see Fig. 11(c) and 11(d)). The bifurcation diagram,
again, shows reflection symmetries with respect to the planes
1
2
ðDy1 ,x=x0 Þ, ðDr11 ,x=x0 Þ, ðDy1 ,x=x0 Þ and ðDr21 ,x=x0 Þ. The secondary
and tertiary equilibrium branches contain only unstable equilibrium states. Equilibrium configurations related to the primary
and secondary branches obey zero tilt and shift, while configurations related to the tertiary branches have non-zero values of tilt
and shift. The solution sets for the two different base pair steps AT
and TA are different: see the projection of the secondary equili1
2
brium paths onto (Dy1 , Dy1 ), for example.
3.4.4. Poly(GC)
In Fig. 12 equilibrium paths of poly(GC) shown as a projection
1
1
onto the subspace (y1 , y2 , x=x0 ) and some equilibrium configurations are visualized. Although the stress-free configuration is not
straight, there are non-trivial equilibrium states possessing a sort
of spatial symmetry. This type of symmetry appearing both in the
bifurcation diagrams and in space is a unique feature of sequencesymmetric steps.
A pitchfork bifurcation point is found on the primary branch
1
1
1
2
2
2
at x=x0 1:185, Dy1 ¼ y1 ¼ 0, Dy2 5:621, Dy1 ¼ y1 ¼ 0, Dy2 1:721 (see Fig. 12). On the primary branch, tilt and shift are zero.
At the bifurcation point, the primary branch loses stability and
potentially stable secondary branches arise, where tilt and shift have
non-zero values. The equilibrium paths show reflection symmetries
d
c
x/x0
1.5
b
1.25
1
−10
a
−5
Δθ 1
0
2 [d
eg
]
c
5
10
15
5 10
0
−5
1 [deg]
−15−10
Δ θ1
b
d
a
Fig. 12. (Top) Global bifurcation diagram of base pair step GC shown as a
1
1
projection onto the subspace (Dy1 , Dy2 , x=x0 ). The projection of the equilibrium
paths onto the parameter plane is shown in gray. (Bottom) Examples of periodic
equilibrium configurations of a poly(GC) segment of length 12 with periodicity
N ¼2.
1
2
with respect to the planes ðDy1 ,x=x0 Þ, ðDr11 ,x=x0 Þ, ðDy1 ,x=x0 Þ and
ðDr21 ,x=x0 Þ.
4. Conclusions
We have applied the sequence-dependent base pair level
elastic model of DNA to simulate DNA stretching and found that
experimentally observed properties of overtwisting followed by
untwisting and the overstretching transition can be reproduced
by this model. The explored behavior is tied to the non-linear
geometry of the model and to coupling between the modes of
deformation, and does not require one to make any correction to
the linear elastic constitutive equations.
The overtwisting during stretching is primarily due to coupling
terms in the elastic coefficient matrix that relate stretching and
twisting, obtained by statistical analysis of DNA fluctuations
observed in crystal structures. However, for large extensions,
the twist–extension curve shows a peak followed by untwisting
in every case. This is due to the non-linear geometry and to
coupling between twist and other modes of deformation. The
magnitudes of overtwisting and of extension at the peak are
strongly dependent on the base pair composition and vary widely.
Our model predicts at least 16 times larger rotation than the
experimental observations (from 4.95 to 21.41 compared to 0:3
per 1000 base pairs). The results are difficult to compare directly,
because in the experiments the base pair compositions are
different from ours and the segments are much longer. The
experimental result of overtwisting could be a combination of
larger overtwisting in some parts of the molecule and untwisting
in others: as the peak on the twist–extension curve is reached
during the stretching process, some of the segments may be
already unwinding while other ones may be still overwinding and
the resultant rotation between the terminal base pairs can be
lower than if all segments were in synchrony.
The peak in the computed force–extension diagrams does not
coincide with the peak in the corresponding twist–extension diagrams. This is in agreement with experimental studies [8].
The overstretching transition observed in the model is a consequence of shear instability, in which the global extension of the
molecule is accommodated by local changes in shear as opposed to
increased separation between base pairs. Similar phenomenon was
found in the behavior of stretched, shearable, bendable and extensionable elastic rods [24,37] and in continuous [38] and discrete [39]
modeling of compressed bar with infinite bending and finite shear
stiffness. The bifurcation can be seen only in special cases, such as
the idealized MNdg
st composition. For other compositions the bifurcation point disappears and separate branches appear which allow
for bistability at a constant force between two states with different
extensions. For long segments this bistability would result in phase
separation along DNA and an observation of a plateau in the stress–
strain curve, i.e., a Maxwell line.
The magnitude of the force at which the transition occurs in
the model is 7:2216:9 times larger (it is 450 pN in case of
poly(GC)) than the actual value ( 65 pN) observed in stretching
experiments. The stretch modulus obtained with the model is
2:324:3 times larger than it is measured ( 1000 pN) in
experiments [6] (see Table 1 for details). There are three likely
reasons for this discrepancy. First, the elastic modulus used here
were renormalized by comparison with bending stiffness derived
from the WLC model, but not with stretching experiments on
DNA. WLC model in fact describes the behavior of a long (longer
than 150 base pairs) DNA with relatively small deformations well,
but may fail to explain short length-scale properties, especially
for large deformations. We mention here that recent cyclization
studies [40,41] and sharp bending experiments [10,42] show DNA
A. Kocsis, D. Swigon / International Journal of Non-Linear Mechanics 47 (2012) 639–654
much (3–5 times) softer than it is predicted by the WLC models. It
suggests softening of the elastic coefficients with large displacement, which might be due to strain induced melting, major–
minor groove bending discrepancies [10], or base-pair-level
flipout events [42]. This softening effect could be implemented
in the model by using non-linear elastic behavior, caused by the
non-linear nature of the hydrophobic stacking force between the
base pairs, the ionic effects, and the non-linear characteristics of
the force–extension diagram of the sugar–phosphate backbones.
Alternatively, renormalizing the elastic constants such that the
stretch modulus of the average segment MN fits the experimental
data, for example, may bring us closer to the reality. We note that
renormalization of the elastic constants would only affect the
scale of the force and not the bifurcation diagrams.
The second and more likely reason is that our model only
captures part of the physical behavior of DNA under stretching.
Specifically, we do not account for any disruptions in the secondary
structure of DNA, such as base stacking or base pair denaturation. If
the energy threshold for unstacking is smaller than the energy of our
structures then transition of DNA into an overstretched form is the
natural outcome. Theories of DNA denaturation have been developed
[43,44] which could be used in combination with our model of DNA
deformation to produce a combine model of DNA deformation and
denaturation that could provide better fit to experimental results.
Our numerical results show that the plateaus on the force–extension
diagrams are much higher for AT rich segments. AT base pairs are,
however, easier to separate because they are bounded only by two
hydrogen bonds. This also suggests that AT rich segments are more
likely to separate and GC rich segments prefer to form an S-DNA
shape, as it is observed in experiments [15,16].
The third reason is that the experimental observations are
influenced by the presence of thermal fluctuations which are not
included in our model. The result of such fluctuations is a
smoothing out and averaging of any mechanical features we have
found using the model.
This study demonstrates the power of using the Simplex
Scanning Algorithm to obtain bifurcation diagrams. Many of the
diagrams consist of disconnected branches that would be difficult
to locate using standard parameter continuation methods. It also
demonstrates the difficulty in describing mechanical behavior of
DNA as some results, especially the overtwisting, show large
sensitivity on the sequence. We have constrained ourselves to
periodic segments of period three and less. It is possible that there
are sequences which show intriguing mechanical behavior but it
is currently not possible to find them because of combinatorial
complexity of exploring large number of possible sequences (4N
where N is the length of the segment). Overcoming the combinatorial explosion and characterizing mechanical behavior of all
DNA segments of particular length, within the realm of sequencedependent elasticity, remains a currently open problem. Besides,
studies on global equilibrium configurations of longer molecules
possibly yield to spatial chaos [45] explored in less complicated
mechanical models [45,46].
We have demonstrated here that a model with linear constitutive equations can give rise to non-linear behavior if its geometry is
non-linear. This idea is not new and it has been applied to DNA
research before [47]. An important point to make is that the nonlinear behavior of the molecule is preserved even though the high
twist density of the molecule would suggest that the material
properties should be homogeneous and isotropic [48].
Acknowledgment
Financial support from Human Frontiers in Science Program
no. RGP0051, OTKA-NKTH no. 68415 and OTKA no. PD 100786 is
653
greatly acknowledged. We thank that Institute for Mathematics and
its Applications helped starting our common work. AK was supported by the Hungarian Eötvös Scholarship. We wish to thank
György Károlyi and Róbert Németh for valuable discussions. Many
numerical computations were obtained on PittGrid [49].
Appendix A. Supplementary data
Supplementary data associated with this article can be found
in the online version at doi:10.1016/j.ijnonlinmec.2011.10.008.
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