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Statistical Mechanics of thermal denaturation of DNA oligomers

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.


I. INTRODUCTION
DNA is one of the most complex and important biomolecule as it is central to all living beings. It contains all the informations needed for birth, development, living and probably sets the average life. Structurally it is a giant double stranded linear molecule with length ranging from 2µ m for simple viruses to 3.5 × 10 7 µm for more complex organism [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 informations 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 point of views.
A DNA molecule is not just a static object but a dynamical system having rather complex nature of internal motions [4]. The nature of thermal denaturation leading to separation of two strands has been investigated 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 function 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 phenomena in terms of standard statistical mechanics, i.e., a Hamiltonian model with temperature independent parameters is a subject of current interest [7].

II. MODEL HAMILTONIAN AND ITS PROPERTIES
Since the internal motion that is basically responsible for denaturation is the stretching of bases from their equilibrium position along the direction of the hydrogen bonds that connect the two bases, a DNA molecule can be considered as a quasi one dimensional lattice either open(non hydrogen bonded) or intact (hydrogen bonded). A Hamiltonian model that has been found appropriate to include these interactions and describe the displacement of bases leading to denaturation is the Peyrard-Bishop model (PB model) [8]. The PB model is written as where m is the reduced mass of the base pair, y i denotes the stretching of the hydrogen bonds connecting the two bases of the i th pair and The on-site potential V (y i ) describes the interaction of the two bases of the i th pair. The Morse potential 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 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, π-electron systems, London dispersion forces and in water solution, the hydrophobic interactions. These forces result in a complex interaction pattern between overlapping base pairs, with minimum energy distance close to 3.4Å in the normal DNA double helix. The following anharmonic potential model mimic these features of the stacking energy: where k is the coupling constant and the second term in the bracket represents the anharmonic term. When due to stretching the hydrogen bonds connecting the bases break, the electronic distribution on the bases is modified causing the stacking interactions to decrease.
This is taken into account by the exponential term in Eq.  which effect is to confine the chain in the potential well. The melting takes place because of 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 profile of a very long (N → ∞) homogeneous DNA chain using both statistical mechanical calculations and constrained temperature molecular dynamics [9,10]. Analytical investigation of nonlinear dynamics of the model suggests that intrinsic energy localization can initiate the denaturation [11]. The model for a long homogeneous chain exhibits a peculiar 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 comparable to the range of the Morse potential the transition changes to second order. The crossover is seen at α/a = 0.5 [7]. Though the PB model seems capable of explaining the multistep melting in a sequence-specific disorder [12], how this disorder affect 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 effect of defects on this profile [14].

III. DENATURATION PROFILE
In a given system of DNA chains the average fraction θ of bonded base pairs can be written as θ = θ ext θ int . θ ext represents the average fraction of the strands forming duplexes (double strands), while θ int is the average fraction of unbroken bonds in the duplexes. The equilibrium dissociation of the duplexes C 2 to single strand C 1 may be represented by the relation The dissociation equilibrium can be neglected in the case of long chains as θ ext is practically 1 while θ int 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, few bonds still remaining prevent the two strands getting apart from each other. It is only at T >> T m (T m being the melting temperature at which half of the bonds are broken) there will be real separation. 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 θ int 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 shortcoming of the theory we in the present article discuss the denaturation profile of oligonucleotides of 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 the one end of both strand to a substrate. This will be referred to as chain with constraint in order to distinguish it form 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 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 temperature 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: 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, D GC = 0.075 eV, a AT = 4.2Å −1 , a GC = 6.9Å −1 , k = 0.025 eVÅ −2 , ρ = 2 and α = 0.35Å −1 . When one end of chain is held fixed at y = 0 distance, the fraction of intact bonds is found using the relation where ϑ(y) is Heaviside step function, N the total number of base pairs and the canonical average · is evaluated by considering all configuration of the chain with one end fixed. The In Fig.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 about 45 0 C 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: as and its fluctuations is the partition function of the chain. Because of the constraint that the first base pair of the chain is in unstreched condition the value of y n as well as |δy n | 2 depend on the site n. In Fig. 4 we plot the value of y n as a function of n at several temperatures for all three constrained chain as described above. We find in all cases the opening of the chain starts from the open end of the chain.
In Fig. 5 we plot |δy n | 2 as a function of temperature for several base pairs. The quantity |δy n | 2 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 |δy n | 2 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 very significant effect on the denaturation profile.