Double stranded DNA chain is known to have nontrivial elasticity. We study
the effect of this elasticity on the denaturation profile of DNA oligomer by
constraining one base pair at one end of the oligomer to remain in unstretched
(or intact) state. The effect of this constraint on the denaturation profile of
the oligomer has been calculated using the Peyrard-Bishop Hamiltonian. The
denaturation profile is found to be very different from the free (i.e. without
the constraint) oligomer. We have also examined how this constraint affects the
denaturation profile of the oligomer having a segment of defect sites located
at different parts of the chain.Comment: Appeared in Proceeding of Conference on "Liquid Crystals and other
  Soft materials", held at RRI, Bangalore Indi

Singh, Navin

Singh, Yashwant

English

arXiv

Double stranded DNA chain is known to have non-trivial elasticity. We study the effect of this elasticity on the denaturation profile of DNA oligomer by constraining one base pair at one end of the oligomer to remain in unstretched (or intact) state. The effect of this constraint on the denaturation profile of the oligomer has been calculated using the Peyrard-Bishop Hamiltonian. The denaturation profile is found to be very different from the free (i.e. without the constraint) oligomer. We have also examined how this constraint affects the denaturation profile of the oligomer having a segment of defect sites located at different parts of the chain

Publications of the IAS Fellows

PRAMANA c Indian Academy of Sciences Vol. 61, No. 2
— journal of August 2003
physics pp. 345–352
Statistical mechanics of thermal denaturation of DNA
oligomers
NAVIN SINGH and YASHWANT SINGH
Department of Physics, Banaras Hindu University, Varanasi 221 005, India
Email: ysingh@bhu.ac.in
Abstract. Double stranded DNA chain is known to have non-trivial elasticity. We study the effect
of this elasticity on the denaturation profile of DNA oligomer by constraining one base pair at one
end of the oligomer to remain in unstretched (or intact) state. The effect of this constraint on the
denaturation profile of the oligomer has been calculated using the Peyrard–Bishop Hamiltonian. The
denaturation profile is found to be very different from the free (i.e. without the constraint) oligomer.
We have also examined how this constraint affects the denaturation profile of the oligomer having a
segment of defect sites located at different parts of the chain.
Keywords. DNA; elasticity; defect; denaturation.
PACS Nos 87.10.+e; 63.70.+h; 64.70.-p
1. Introduction
DNA is one of the most complex and important biomolecules as it is central to all living
beings. It contains all the information needed for birth, development, living and proba-
bly sets the average life. Structurally it is a giant double stranded linear molecule with
length ranging from 2 µm for simple viruses to 3:5 10 7 µm for more complex organ-
isms [1]. How such a molecule came into being during the evolution of life and how it
acquired the ability to store and transmit the genetic information are still a mystery. The
recent progress in genome mapping and availability of experimental techniques to study
the physical properties of a single molecule [2,3] has, however, made the field very active
from both biological and physical points of view.
A DNA molecule is not just a static object but a dynamical system having a rather
complex nature of internal motions [4]. The structural elements such as individual atoms,
groups of atoms (bases, sugar rings, phosphates), fragments of double chain including sev-
eral base pairs, are in constant movement and this movement plays an important role in the
functioning of the molecule. The solvent in which the molecule is immersed acts as a ther-
mal bath and provides energy for different motion. In addition, collision with the molecules
of the solution which surrounds DNA, local interactions with proteins, drugs or with some
other ligands also lead to internal motion. These motions are distinguished by activation
energies, amplitudes and characteristic times. The motions which are of our interest here
are opening of base pairs, formation of bubbles along the chain and unwinding of helix (de-
naturation). The energy involved for these motions is of the order of 5–20 kcal/mol. These
345
Navin Singh and Yashwant Singh
motions are activated by increasing temperature, increasing pH of the solvent, action of
denaturation agents etc. The time scale of these motions are of the order of microseconds
and is therefore generally unobservable in atomistic simulations as these simulations are
restricted to time scale of nanoseconds because of computational cost.
The nature of thermal denaturation leading to separation of two strands has been in-
vestigated for several decades [5,6]. Experimentally a sample containing molecules of a
specific length and sequence is prepared. Then the fraction of bound base pairs as a func-
tion of temperature, referred to as the melting curve, is measured through light absorption,
typically at about 260 nm. For heterogeneous DNA, where the sequence contains both
AT and GC pairs, the melting curve exhibits a multistep behaviour consisting of plateaus
with different sizes separated by sharp jumps. These jumps have been attributed to the
unwinding of domains characterized by different frequencies of AT and GC pairs. The
sharpness of the jump in long DNA molecules suggests that the transition from bound to
unbound is first-order. The understanding of this remarkable one-dimensional cooperative
phenomenon in terms of standard statistical mechanics, i.e., a Hamiltonian model with
temperature independent parameters is a subject of current interest [7].
2. Model Hamiltonian and its properties
Since the internal motion that is basically responsible for denaturation is the stretching
of bases from their equilibrium positions along the direction of the hydrogen bonds that
connect the two bases, a DNA molecule can be considered as a quasi-one-dimensional
lattice composed of N base pair units. The forces which stabilize the structure are the hy-
drogen bonds between complementary bases on opposite strands and stacking interactions
between nearest-neighbour bases on opposite strands. Each base pair is in one of the two
states: either open (non-hydrogen bonded) or intact (hydrogen bonded). A Hamiltonian
model that has been found appropriate to include these interactions and to describe the dis-
placement of bases leading to denaturation is the Peyrard–Bishop model (PB model) [8].
The PB model is written as
H =
N
∑
i=1
H(yi;yi+1)
H(yi;yi+1) =
p2i
2m
+W(yi;yi+1)+V(yi); (2.1)
where m is the reduced mass of the base pair, yi denotes the stretching of the hydrogen
bonds connecting the two bases of the ith pair and
pi = m

dyi
dt

:
The on-site potential V (yi) describes the interaction of the two bases of the ith pair. The
Morse potential
V (yi) = Di(e
 ayi
 1)2; (2.2)
which is taken to represent the on-site interaction, represents not only the hydrogen bonds
connecting two bases belonging to opposite strands, but also the repulsive interactions
346 Pramana – J. Phys., Vol. 61, No. 2, August 2003
Thermal denaturation of DNA oligomers
of the phosphates and the surrounding solvent effects. The flat part at large values of the
displacement of this potential emulates the tendency of the pair ‘melt’ at high temperatures
as thermal phonons drive the molecules outside the well and towards the flat portion of the
potential.
The stacking interactions are contributed by dipole–dipole interactions, pi-electron sys-
tems, London dispersion forces and in water solution, the hydrophobic interactions. These
forces result in a complex interaction pattern between overlapping base pairs, with min-
imum energy distance close to 3.4 A˚ in the normal DNA double helix. The following
anharmonic potential model mimics these features of the stacking energy:
W (yi;yi+1) =
1
2
k[1+ρ exp( α(yi + yi+1))](yi  yi+1)2; (2.3)
where k is the coupling constant and the second term in the bracket represents the an-
harmonic term. When the hydrogen bonds connecting the bases break due to stretching,
the electronic distribution on the bases is modified causing the stacking interactions to de-
crease. This is taken into account by the exponential term in eq. (2.3). It may be noted
that the effective coupling constant decreases from k(1+ ρ) to k when either of the two
interacting base pairs is stretched. This decrease in coupling provides a large entropy in
the denaturation. The parameter α in eq. (2.3) defines the ‘anharmonic range’.
The model described above can be viewed as a model of one-dimensional monoatomic
lattice (see figure 1) with each atom having mass m and nearest-neighbour interaction given
by eq. (2.3). Furthermore, each atom is subjected to ‘external’ potential given by eq. (2.1)
whose effect is to confine the chain in the potential well. The melting takes place because
of the competition between the thermal energy which leads to displacement and ‘external’
field and the nearest-neighbour interactions which lead to the confinement. The model,
therefore, represents a one-dimensional system that differs from the usual one-dimensional
systems that do not show phase transitions.
The model Hamiltonian of eq. (2.1) has extensively been used to study the melting pro-
file of a very long (N ! ∞) homogeneous DNA chain using both statistical mechanical
calculations and constrained temperature molecular dynamics [9,10]. Analytical investi-
gation of non-linear dynamics of the model suggests that intrinsic energy localization can
initiate the denaturation [11]. The model for a long homogeneous chain exhibits a pecu-
liar type of first-order transition with finite melting entropy, a discontinuity in the fraction
of bound pairs and divergent correlation lengths. However, as the value of the stacking
parameter α increases and the range of the ‘entropy barrier’ becomes shorter than or com-
parable to the range of the Morse potential, the transition changes to second order. The
cross-over is seen at α=a = 0:5 [7]. Though the PB model seems capable of explain-
ing the multi-step melting in a sequence-specific disorder [12], how this disorder affects
the nature of transition has yet to be understood. In other work the PB model has been
used to understand the melting profile of short chains [13] and the effect of defects on this
profile [14].
3. Denaturation profile
In a given system of DNA chains, the average fraction θ of bonded base pairs can be
written as θ = θextθint. θext represents the average fraction of the strands forming duplexes(double strands), while θint is the average fraction of unbroken bonds in the duplexes. The
Pramana – J. Phys., Vol. 61, No. 2, August 2003 347
Navin Singh and Yashwant Singh
  
  
  



  
  
  



  
  


  
  


  
  


V(y)
y
D
n
Spring representing stacking energy
particle of mass "m"
"external" potential acting
on each particle
Figure 1. Schematic of quasi-one-dimensional model of DNA.
equilibrium dissociation of the duplexes C2 to single strand C1 may be represented by the
relation C2
 2C1. The dissociation equilibrium can be neglected in the case of long chains
as θext is practically 1 while θint and therefore θ goes to zero. This is because in the case of
long DNA chains when θ goes practically from 1 to zero near the denaturation transition,
while most bonds are disrupted and the DNA has denatured, the few bonds still remaining
prevent the two strands from getting apart from each other. There will be real separation
only at T  Tm (Tm being the melting temperature at which half of the bonds are broken).
Therefore, at the transition the double strand is always a single molecule and in calculation
based on PB model one has to calculate only θ int( θ ). On the contrary, in the case of short
chains the processes of single bond disruption and strand dissociation tend to happen in the
same temperature range, therefore, the computation of θ ext in addition to θint is essential.
Unfortunately, at present, we do not have any reliable method for calculating θ ext. The
method which has been used is based on the partition function of rigid molecules and
adjustable parameters [5,13] to be determined from experimental data. To avoid this short-
coming of the theory we in the present article discuss the denaturation profile of oligonu-
cleotides of a given sequence with a base pair at one end of the chain held in such a way
that it remains at its equilibrium seperation (i.e. there is no stretching) at all temperatures.
This can be done by creating a deep potential well for this base pair or attaching one end
of both strands to a substrate. This will be referred to as chain with constraint in order to
348 Pramana – J. Phys., Vol. 61, No. 2, August 2003
Thermal denaturation of DNA oligomers
distinguish it from the ‘free chain’. One of the advantage of having a constraint of this type
is obvious; the problem of divergence of the partition function for the PB model for short
chains no more exists.
The DNA molecule is known to have non-trivial elastic properties: When the two strands
of the DNA molecule are pulled apart by applying a force at one end of the chain, a novel
phase transition is found (in case of infinitely long chain) to take place at which the two
strands are pulled completely apart [15]. The phase digram plotted in the plane of temper-
ature and force reveals the elastic properties of the DNA chain. Our study reported in this
paper differs from the situation just described in two ways: (i) a short chain of 21 given
base sequence is considered and (ii) instead of pulling the chain apart, the end base pair is
constrained to be in unstreched or intact position.
The oligonucleotide which we consider has the following sequence given by:
50A C G C T A T A C T C A C G T T A A C A G30
30T G C G A T A T G A G T G C A A T T G T C
50
; (3.1)
The denaturation profile of this oligonucleotide has been studied by Campa and Giansanti
[13] and by us [14]. We take the same parameters as in the previous study. Thus, D AT =
0:05 eV, DGC = 0:075 eV, aAT = 4:2 A˚ 1, aGC = 6:9 A˚ 1, k = 0:025 eV A˚ 2, ρ = 2 and
α = 0:35 A˚ 1. When one end of the chain is held fixed at y = 0 distance, the fraction of
intact bonds is found using the relation
θ = 1
N
N
∑
i=1
hϑ (y0  yi)i; (3.2)
where ϑ (y) is Heaviside step function, N the total number of base pairs and the canonical
average hi is evaluated by considering all the configuration of the chain with one end fixed.
The ith bond is considered bound if the value ofy i is smaller than a chosen threshold y0. For
y0 we have taken a value of 2 A˚. Since the model Hamiltonian of eq. (2.1) couples only the
nearest neighbours the calculation of canonical average reduces to multiplication of finite
matrices. The discretization of the coordinate variables and introduction of a proper cut-off
on the matrices and the number of base pairs in the chain fixes the number of matrices to
be multiplied.
In figure 2 we compare the value of θ as a function of temperature for the free and the
constrained chain. The influence of the constraint on the denaturation profile is enormous.
While in case of free chain the breaking of bonds takes place in a narrow temperature range,
in the constrained chain it is over a wide temperature range. The value of temperature at
which dθ=dT is maximum shifts by about 45ÆC and the peak is much wider and smaller
compared to that of free chain.
Next we calculated the denaturation profile for the following two nucleotides having a
segment of chain with defect sites:
(a) 5
0
A C G C T A T A C T C A C G T T A A C A G30
30T C G C T A T A C T C T G C A A T T G T C
50 (3.3)
(b) 50A C G C T A T A C T C A C G T T A A C A G30
30T G C G A T T A C T C A C G T T T T G T C
50
: (3.4)
Pramana – J. Phys., Vol. 61, No. 2, August 2003 349
Navin Singh and Yashwant Singh
200 250 300 350 400 450 500
T(K)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
θ
200 250 300 350 400 450 500
T(K)
0
0.02
0.04
0.06
0.08
dθ/dT
(a) (b)
Figure 2. Variation of θ and dθ=dT as a function of temperature for free (solid line)
and constrained chain (dashed line).
200 250 300 350 400 450 500
T(K)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
θ
200 250 300 350 400 450 500
T(K)
0
0.002
0.004
0.006
0.008
0.01
dθ/dT
(a) (b)
Figure 3. (a) Variation of θ as a function of temperature for the constrained chain
without defects (solid line) with 10 defects at one end of the chain (dashed line) and 10
defects in the middle (dot–dashed line). (b) Plot of dθ=dT for the same.
While both oligonucleotides have 10 defect sites, their locations differ. In (a) the defects
are on the left end from site 2 to 11 while in (b) it is in the middle from site 6 to 16. The
position of the base pair is counted from left. The purpose is to see how these defects
affect the denaturation profile and the formation of loop and stem as often seen in a single
strand DNA or RNA. In figure 3 we plot the variation of θ and dθ=dT as functions of
temperature. While the denaturation takes place at lower temperatures the shift is more
when the defects are in the middle.
The other quantities of interest are the mean value of displacement of nth base pairs
defined as
hyni=
1
Z
Z
 
N
∏
i=1
dyi
!
yn exp
"
 β
N
∑
i=1
H(yi;yi+1)
#
(3.5)
and its fluctuations
350 Pramana – J. Phys., Vol. 61, No. 2, August 2003
Thermal denaturation of DNA oligomers
1 3 5 7 9 11 13 15 17 19
0
1
2
3
4
5
6
1 3 5 7 9 11 13 15 17 19
0
1
2
3
4
5
6
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
6
1 3 5 7 9 11 13 15 17 19
0
1
2
3
4
5
6
360 K 370 K
380 K 390 K
Figure 4. Plot of hyni vs. n (site position) at four different temperatures. Æ denotes the
constrained chain without defect,2 denotes the chain with 10 defects at one end while
 denotes the chain with 10 defects in the middle.
200 250 300 350 400 450 500
T(K)
0
2
4
6
8
10
12
14
<δy
n
2
>
n = 2
n = 5
n = 10
n = 15
n = 20
Figure 5. Plot showing the variation of transverse correlation length with temperature
for different values of n.
hjδynj2i=
1
Z
Z
 
N
∏
i=1
dyi
!
(yn hyni)2 exp
"
 β
N
∑
i=1
H(yi;yi+1)
#
: (3.6)
Here Z =
R
 
∏Ni=1 dyi

exp[ β ∑Ni=1 H(yi;yi+1)] is the partition function of the chain.
Pramana – J. Phys., Vol. 61, No. 2, August 2003 351
Navin Singh and Yashwant Singh
Because of the constraint that the first base pair of the chain is in unstretched condition,
the value of hyni as well as hjδynj2i depend on the site n. In figure 4 we plot the value of
hyni as a function of n at several temperatures for all three constrained chain as described
above. We find in all cases that the opening of the chain starts from the open end of the
chain. In figure 5 we plot hjδynj2i as a function of temperature for several base pairs.
The quantity hjδynj2i measures the transverse correlation length for the base pair n. This
correlation length remains almost zero for all n when the oligonucleotide is in the native
state but at the denaturation its value increases. At a given temperature the value of hjδy nj2i
depends on n and it increases with n. For a long chain we expect it to diverge for large value
of n.
In conclusion we wish to emphasize that the effect of constraining a base pair at one end
of a given oligonucleotide has a very significant effect on the denaturation profile.
Acknowledgements
This work has been supported through research grants by Department of Science and Tech-
nology and Council of Scientific and Industrial Research, New Delhi, Government of India.
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352 Pramana – J. Phys., Vol. 61, No. 2, August 2003


Statistical Mechanics of thermal denaturation of DNA oligomers

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