World Journal of Mechanics, 2012, 2, 84-89
doi:10.4236/wjm.2012.22010 Published Online April 2012 (http://www.SciRP.org/journal/wjm)
Analysis of Lattice Size, Energy Density and Denaturation
for a One-Dimensional DNA Model
Gabriel Gouvea Slade, Natalia Favaro Ribeiro, Elso Drigo Filho, Jose Roberto Ruggiero
Department of Physics, Sao Paulo State University, São José do Rio Preto, Brazil
Received January 21, 2012; revised February 30, 2012; accepted March 7, 2012
There are several mechanical models to describe the DNA phenomenology. In this work the DNA denaturation is stu-
died under thermodynamical and dynamical point of view using the well known Peyrard-Bishop model. The thermody-
namics analysis using the transfer integral operator method is briefly reviewed. In particular, the lattice size is discussed
and a conjecture about the minimum energy to denaturation is proposed. In terms of the dynamical aspects of the model,
the equations of motion for the system are integrated and the results determine the energy density where the denatura-
tion occurs. The behavior of the lattice near the phase transition is analyzed. The relation between the thermodynamical
and dynamical results is discussed.
Keywords: DNA Mechanical Model; Peyrard-Bishop Model; Lattice Size; Thermal Denaturation
The DNA molecule contains the genetic information and
it is responsible for the transmission of hereditariness .
Since the discovery of its double helix structure done by
Watson and Crick , researchers of several areas of
science concentrated their attention to understand stru-
ctural and functional aspects of this complex molecule.
In the transcription and replication phenomena of DNA
the ribbons separation in the double helix is an important
effect, because it is necessary to expose the nitrogen
bases to the solution. This means that large amplitudes
and highly located motions are necessary and the dyna-
mic of the molecule should be nonlinear.
Several models have been proposed to describe the
DNA . One of these is the Peyrard-Bishop (PB) model,
proposed at 1989 to study DNA denaturation using sta-
tistical mechanics . The original model consists of a
pair of one dimensional lattice of harmonic oscillators;
the adjacent oscillators of each lattice are bonded by a
nonlinear Morse potential mimicking the hydrogen bond
of the real molecule. The main feature of this model is to
describe the separation of the double strand in terms of
the average stretching of the base pairs.
The PB model and some variations [5-7] have been
used to study the dynamics [8-10] and the thermodyna-
mics [11-13] of DNA. From the dynamical point of view,
there are studies about localized energy modes  that
were identified as precursors of the denaturation and
transcription process. These modes could also prescribe
the dynamic of DNA in room temperature, in which large
amplitude and highly localized motions had been experi-
mentally verified [15,16]. From the thermodynamical
point of view, the original model describes quantitatively
the expected results for the DNA denaturation tempe-
rature. Recently, it has been discussed modifications in
the original model in order to include a more abrupt
phase transition .
In this work, we analyze the thermodynamical and
dynamical aspects of the PB model and relate these two
approaches. In the literature the study of nonlinear lat-
tices is done for a different number of oscillators [8,18,
19]. Besides that, other works discuss the possibility of
phase transition to finite lattices [20,21]. The main ob-
jective of the paper is to suggest a criterion to fix the mi-
nimum size of the lattice. In order to get this result, we
use the transfer integral operator method  and calcu-
late the error committed in the partition function of the
system. Thermodynamic results also lead to conjecture a
minimum energy per base pair for occurrence of the
lattice phase transition. This threshold energy can also be
verified from dynamical results. We propose that the
phase transition can be dynamically observed by follow-
ing the time evolution of the position of the oscillators
for different energies.
2. The Peyrard-Bishop Model
In the Peyrard-Bishop original model each strand of the
macromolecule is represented by a harmonic lattice and
Copyright © 2012 SciRes. WJM
G. G. SLADE ET AL. 85
the interaction between them is described by a nonlinear
potential. The three-dimensional aspects of the helicoi-
dally structure are not considered and only the motions
perpendicular to the strands are analyzed . The nu-
cleotides positions are denoted by
tively, with . For simplicity, we assume
that the chains are homogeneous, i.e., all masses and all
elastics constants are equal.
The Hamiltonian of the model for a homogeneous
chain is expressed as:
The last term in Equation (1) correspond to the Morse
potential used to describe the hydrogen bonds, the pa-
rameters D and a are related, respectively, with the depth
and width of the potential well, k is the elastic constant of
the harmonic potential used to simulate the stacking in-
teraction and m is the mass. The Hamiltonian (1) can be
uncoupled by introducing new variables
Hamiltonian can be
written as the sum of two terms, one of them depending
only on x variable representing a chain of harmonic os-
cillators and it is not important for our analysis here. On
the other hand, the Hamiltonian dependent on y variable
can be written as
This Hamiltonian represents a chain of harmonic os-
cillators with the additional on site Morse potential that
carries the nonlinearity of the model.
3. Thermodynamical Analysis
The thermodynamic analysis of the system is described
by the partition function Z expressed in terms of the
variable representing the average stretching of the base
pairs (), i.e.,
where N is the number of base pairs in the chain,
k is the Boltzmann constant, T is the tem-
is the uncoupled Hamiltonian
of the model, given by Equation (2).
The transfer integral operator method  is used to
determine the thermodynamical properties from the par-
tition function. This method allows relating the partition
function with eigenfunctions n
and eigenvalues n
given by the following pseudo-Schrödinger equation:
Vy is the Morse potential. From the solutions
of Equation (4), the partition function can be obtained by
. However, in the thermo-
dynamical limit in which the number of particles is very
large (), the partition function is dominated by
the ground state. For this reason, the analysis of the
problem is usually limited to determine the energy eigen-
function and eigenvalue for the ground state.
The characterization of the phase transition for DNA
molecule is done following the dependence of an order
parameter with the temperature of the system. Usually,
the order parameter is the average stretching of base pairs
The eigenfunctions that are solution of the pseudo-
Schrödinger Equation (4) can be characterized as a pro-
bability density. In this way, the average stretching of the
base pairs can be obtained from the equality:
There are several methods that can be used to deter-
mine the solutions of Equation (4), for example, the La-
place transforms  and by using the factorization me-
thod . Then, the eigenenergy and eigenfunction for
the ground state of Morse potential are known (see, for
example, references [4,11]):
Copyright © 2012 SciRes. WJM
G. G. SLADE ET AL.
where the condition
must be ob-
served in order to exist bond states.
From the ground state eigenfunction given by (6), the
average stretching of base pairs can be found by using
Equation (5). To this end it is necessary to fix the pa-
rameters which characterize the model. Following the
reference , the better parameters for Morse potential
to adjust to DNA are D = 0.03 eV and a = 2.81 Å–1 and
the elastic constant to the harmonic potential is k = 0.06
eV·Å–2. Using these parameters, the behavior of <y> with
temperature has an increase in the average stretching of
the base pairs, indicating the thermal denaturation of the
DNA molecule in the temperature range from 300 to 350
Other approach to determine the melting temperature
can be made by noting that the eigenfunction obtained
from Equation (4) must be quadratically integrable. Then,
it is possible to determine from the eigenfunction (6) a
critical temperature of the phase transition (C). The
eigenfunction (6) is quadratically integrable only if d is
higher than 0.5. For d lower or equal 0.5 the system does
not have a discrete bond states and the square of the
wave function integrated in all space diverges. Then, the
critical temperature is obtained from and
it is given by
Substituting the parameters D, a, k and the Boltzmann
, in Equation (8) the
value of the critical temperature is, approximately, 350 K.
The reference  indicates that the denaturation tempe-
rature of the DNA molecule varies in the range 318 to
372 K, depending on the nucleotides composition of the
chain. In this way, the obtained critical temperature is
consistent with experimental results.
The thermodynamical treatment of the Peyrard-Bishop
model to DNA was realized assuming a very large num-
ber of oscillators in the chain. In this limit, the excited
states to the Morse potential were disregard. If the num-
ber of oscillators is not large, it is necessary to estimate
the numerical error in this approach. The partition func-
tion (3), written in terms of the eigenvalues of Equation
(4), can be rearranged in the following form:
Then, the first order error in the partition function is
. It is important to note that the se-
ries (9) must be convergent otherwise the thermodynamic
limit could not be valid to none value of the particles
number N. So the higher terms in (9) are always lower
than the first one. In order to estimate the value of N, we
should know the difference between the energy eigen-
. This estimative can be done re-
membering that the Morse potential permits bond states
to the values of 1
. Using the parameters given above and
the temperature equal to 300 K, the value obtained to
is approximately 0.08. This means that there is
only one bond state inside the potential well at this tem-
perature, the ground state. Increasing the temperature
above 350 K even the ground state will be out of the po-
tential well. On the other hand, a discrete first excited
state appears only for temperature lower than 150 K.
Thus, for the temperature range of interest, the first
excited state is in the continuum part of the spectrum. In
this case, the difference between the ground state and the
first energy level of the continuum spectrum can be esti-
mated as the value of the depth of the well, D, minus the
value of the ground state energy, 0
. Consequently, the error by disre-
garding the first state above the ground state is
. From the used parameters and consider-
ing the size of the chain equal to 21 oscillators, the error
in the partition function by disregarding the second term
in the expansion (9) at 300 K is lower than 10–5. This
result indicates that a chain with 21 oscillators can be
considered as being sufficiently large to disregard the
states above the ground state. This procedure can be ado-
pted to control the error in terms of the lattice size. It is
important to observe that for the system close the de-
naturation temperature, the eigenvalue 0
is close the
top of the potential well, i.e.,
becomes minor. Then,
the continuum levels of the spectrum become more and
more important to the description of the system.
Considering the above arguments, the characterization
of the phase transition of the DNA molecule by the
thermodynamical formalism can be done admitting that
all the base pairs have sufficient energy to break the
Morse potential, i.e., when the oscillators have energy
above the potential well. In this way, when the energy of
each base pair is equal to or higher than D, the wave
function is no more quadratically integrable and the
phase transition occurs. An alternative way to analyze
this relation between the melting process and the energy
of each base pair is to note that the energy density of the
lattice can be related with the melting temperature by the
NkT. As the critical temperature obtained
Copyright © 2012 SciRes. WJM
G. G. SLADE ET AL. 87
is 350 K, we can predict that the energy density neces-
sary to denaturation is 0.03
observes that this value is the same that was used to the
depth of the potential well D which reinforce the previ-
ous arguments. Summarizing, from this analysis, the de-
naturation occurs when the energy per oscillator is at
least equal to the depth of the Morse potential well.
4. Dynamical Study of the Model
In this section the dynamical results for the Peyrard-
Bishop model are discussed. The numerical calculation
can be simplified if we introduce dimensionless variables:
. The Hamiltonian (2)
can be rewritten as a function of a single parameter:
DH. So the
equations of motion are given by:
We integrate the equations of motion using the tenth
order Runge-Kutta Nystrom method . The initial
condition is such that only the central oscillator of the
chain is excited and all its energy is kinetic. Then, at time
all oscillators are in their equilibrium position and
at rest with the exception of the central oscillator that has
an initial velocity. The periodic boundary conditions is
used, i.e., 0
. Following the previ-
ous discussion (Section 3), the number of oscillators used
to perform the simulation is 21.
In Section 3, we conjecture that the denaturation hap-
pens if each oscillator shares an energy corresponding to
the depth of the Morse potential well, D. By using the
dimensionless variables, as writing in (10), this depth is
tracked back to the value 0.5. In order to characterize the
denaturation, we follow the position value (y) of each
oscillator and its Morse potential energy as a function of
the time. We consider that the chain reaches the denatu-
ration when all the oscillators get the Morse potential
equal to or larger than 0.5. In this case all the hydrogen
bonds should be broken and the double strand should be
separated. Figure 1 shows the position y of all the oscil-
lators superposed as a function of the time. In this simu-
lation it is used and the energy density is
0.7857. Thus, in Figure 1 there are 21 curves superposed,
there is one curve for each site of the lattice. When all the
oscillators acquire energy of the Morse potential equiva-
Figure 1. (a) Position of each of the oscillators superposed
as a function of the time; (b) A zoon in the region where the
onsite potential is broken down. N = 21 and 0.7857
lent to the depth of the well, the positions of all oscilla-
tors begin to rise constantly, so the chain starts a uniform
translation motion. Once y represents the separation of a
base pair, this behavior characterizes qualitatively the
broken of the hydrogen bonds and consequently the
Another result obtained from the dynamics is related
with the behavior of the chain before the phase transition
of the lattice. Following the time evolution of the energy
of Morse for each oscillator we observe the behavior of
the hydrogen bond in the denaturation process. A co-
operative effect in this process is observed. Specifically,
it is noted that when one hydrogen bond is broken it in-
duces a broken of some neighbour hydrogen bonds form-
ing “bubbles” in the lattice. This occurs in different parts
of the chain and these regions grow up until the complete
separation of the lattices, i.e., the denaturation occurs.
This behavior is shown in the Figure 2 where the tem-
poral evolution of the hydrogen bond break of each os-
cillator can be observed. In this figure the site bonded is
Copyright © 2012 SciRes. WJM
G. G. SLADE ET AL.
Figure 2. Hydrogen bond of each oscillator in function of
the time. Black color represents the formed bonds, while
white color represents the broken bond. N = 21 and
represented by black color and the broken hydrogen bond
is indicated by the white color.
Finally we analyze the energy density in the model.
Then we simulate different energy densities and verify
the necessary time to denaturation. This result is pre-
sented in Figure 3. We note that small values of the en-
ergy density correspond to high values of time to obtain
the denaturation. The results suggest also that an asymp-
totic behavior could be achieved for energy density near
0.5, which is the well depth, in accordance with the
thermodynamic results. Others initial conditions were
tested and the results are qualitatively in agreement with
In this work we studied the thermodynamic and dynamic
properties of the one dimensional model to study the DNA
macromolecule. The thermodynamical analysis inferred
that a chain of 21 oscillators should be sufficient large to
analyze the behavior of the system for room temperatures
(300 K), once that the error in the partition function is
about 10–5 when despising the excited states of the
Schrödinger-type Equation (4). It is important to observe
that, if the system is near the denaturation temperature,
the thermodynamic results indicate that it is necessary a
large chain to describe its properties, otherwise the error
in the partition function becomes significant.
It is suggested a dynamical approach to analyze the
thermal denaturation by studying the stretching of the
base pairs (y) for each oscillator on the lattice. This ana-
lysis permits to observe that the formation of bubbles is
Figure 3. Energy density of the system as a function of the
necessary time to occur denaturation. N = 21.
part of the melting process. The minimum energy density
needed to the denaturation is related with the energy of
the potential well, i.e., it is necessary that all oscillators
of the chain get energy higher than the parameter D of
the Morse potential to get the melting. The dynamical
calculations of the model, in a lattice of 21 oscillators,
show that the denaturation occurs when the energy densi-
ties tends to the value of D. However, for the obtained
results the energy is always higher than that indicated
from thermodynamics analysis. Two causes can be iden-
tified to this behavior, first the energy in the dynamic si-
mulation are not localized only in the Morse potential,
part of it can be localized in the harmonic potentials.
Other cause is related with the fact that the thermody-
namics results indicate that near the denaturation tem-
perature it is necessary a larger chain to describe the sys-
tem. Nevertheless the behavior of the chain is in agree-
ment with the DNA phenomenology and both formalisms
converge to the same result.
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