Eclética Química Print version ISSN 0100-4670On-line version ISSN 1678-4618 Eclet. Quím. vol.26 São Paulo 2001 http://dx.doi.org/10.1590/S0100-46702001000100006 Statistical Model to DNA Melting Paulo Sergio Pilz AUGUSTO*,**,a Elso DRIGO FILHO* Jose Roberto RUGGIERO* ABSTRACT: We explore a DNA statistical model to obtain information about the behavior of the thermodynamics quantities. Special attention is given to the thermal denaturation of this macromolecule. KEYWORDS: DNA, melting denaturation, statistical model for macromolecules. Introduction DNA macromolecule properties have been intensely studied in the last decades. Experimental results and theoretical models has appeared and contributed both to a better understanding of this complex system (see, for example8,13). A special attention has been given to the DNA transcription. This process has some similarities with thermal denaturation. There are evidences that both start from local opening in the double helix. Therefore, the study of thermal denaturation is a preliminary step towards the investigation of the transcription. The thermal denaturation or melting is the separation of the two complementary ribbons by heating the system. A simple model describing the statistical mechanics of the melting and the non-linear motions of the bases was proposed in ref.12. Recently, modifications in this model were introduced1,2 in order to construct a more detailed description of DNA macromolecule. In ref.2, for instance, the discrete case is investigated and the results are compared with the continuum original model. http://www.scielo.br/scielo.php?script=sci_serial&pid=0100-4670&lng=en&nrm=iso http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0100-46702001000100006&lng=en&nrm=iso&tlng=en#notas http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0100-46702001000100006&lng=en&nrm=iso&tlng=en#notas http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0100-46702001000100006&lng=en&nrm=iso&tlng=en#notas http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0100-46702001000100006&lng=en&nrm=iso&tlng=en#notas http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0100-46702001000100006&lng=en&nrm=iso&tlng=en#notas In this work the model before thermal denaturation proposed in ref.12 is adopted and we suggest that after the melting each ribbon is treated in separate. This approach gives a different mathematical description for DNA in two cases: before and after the thermal denaturation. The model, with parameters fixed in previous works1,2,4, gives the melting temperature consistently with experimental data13. The free energy and specific heat are analytically determined. The model The model adopted here for DNA macromolecule is based on Peyrard and Bishop work12. This model is planar and composed of two ribbons jointed by H-bond between adjacent nucleotides. Each ribbon is a spring-mass system where the nucleotides are represented by equal mass (m) and are connected by elastic springs of spring constant k. The H-bonds are simulated by non-harmonic potential, namely the Morse one1,12, although other potentials have been used for this purpose6,15. The Hamiltonian for this model is where un e vn are the transverse displacement for each mass in the ribbons, the indices n represents the n-th nucleotide. V (un – vn) is the potential for H-bond interaction. The longitudinal displacement is not considered here because their amplitudes are typically smaller than the amplitude of the transverse one2. The equation8 can be written in terms of the variables xn = (un + vn)/ and yn = (un - vn)/ as where pn = m n and qn = m n The partition function for N pair of bases is where b = 1/(kBT), kB is the Boltzmanns constant and T is the absolute temperature.� This function may be factored as ZI = ZpZqZxZy, where the kinetic parts are the usual one dimension partition function Zp = Zq = {2pmkBT}N/2. The term Zx after integration of gaussian-like function results in Zx = [2pkBT/k]N/2 . The coupling between the two ribbons involves only the nearest neighbor interaction. Then, Zy can be expressed as where f(yn, yn-1) is the potential energy in y coordinates. The transfer integral operator is used to evaluate exactly the integral (4) in the thermodynamic limit of a large system ( N ® ¥ ) 3,12. The transfer integral operator is written as and the calculations for large system yields where eo is the lowest eingenvalue of the operator. In the continuum limit approximation, the method reduced the problem to solve a Schrödinger –like equation; with so = (1/2b)ln(bk/2p). This equation is formally identical to the Schrödinger equation and the potencial V(y) represents the H-bond interaction. As usual11,12 we adopt the Morse potential to simulate H-bond interaction in our analysis. Then, the equation (7) can be rewritten as where V(y) = D(e-ay – 1)2 is the Morse potential. The determination of the parameters D and a for this potential and the spring constant k to stacking interaction is not a trivial matter7,14. We adopt here the parameters indicated in ref.1,2,4, which have been used to study several dynamics and statistical proprieties of DNA macromolecule. The mean stretching áyñ gives a measure of the extent of denaturation of the DNA macromolecule. Using the transfer integral method3,12 áyñ can be calculated as: remembering that, for large systems, the result is again dominated by the lowest energy eigenvalue with the eigenfunction fo(y). The equation (8) is similar to Schrödinger equation for a particle in one-dimensional Morse potential. This equation has exactly analytical solution5,10 and we get a discrete or a continuum spectrum. In this case, different from the usual Schrödinger equation, the solutions are temperature dependent ones. There is a specific temperature Td that can be used to classify the spectrum. For temperatures below Td the spectrum is discrete and corresponds to bond states. On the other hand, for T ñ Td, the spectrum is continuum and corresponds to delocalized states. Then, it is natural to consider Td as the denaturation temperature2,4. The melting temperature can also be determined by analyzing a plot of áyñ vs temperatures. The divergence in the curve indicates the denaturation. The normalized ground state eingenfunction of Schrödinger-like equation (8) is known12 and can be written as where G(2d-1) is a usual gamma function and d = (ba)(2kD)1/2 must be greater than ½ in order to obtain a " bound" state. Using this condition we obtain the value of Td: Then, for T< Td the energies have discrete values and the two ribbons are jointed by the H-bonds. The ground state energy in this case is After thermal desnaturation, T >Td, the two ribbons are not interacting. Then, the H-bond potential becomes zero. Thus, we can rewrite equation (1) considering V(un-vv) equal to zero: where Pn = m 2 n and Qn = m 2 n. This Hamiltonian corresponds to two independents spring-mass systems. The partition function is The integration on the variables reduces to gaussian-like ones and they can be analytically evaluated. For the momenta integrations we get And, for the coordinates, we obtain It is important to note that we have obtained analytical expressions for the partitions functions. This approach permits to determine, in a simple way, the melting temperature and to describe the thermodynamics properties of the system before and after the thermal denaturation. Results The parameters used here are D = 0.04eV, that corresponds to dissociation energy; a = 4.45Å-1, that is the spatial scale of the Morse potential, and the coupling constant is k = 0.06eVÅ-1 1,2,4. The denaturation temperature, using these parameters and the relation (11), is Td = 361K. This value can be confirmed plotting the mean stretching value á y ñ against temperature, as in figure (1). The curve obtained shows a clear divergence near 360K. The free energy and the specific heat can be determined before and after the melting from partition functions (3) and (14), respectively. For T < Td, ZI is computed from (3) as where eo is given in (12). The free energy (ÁI = -kBT ln ZI) in this condition is and the specific heat (CV =¶2ÁI/¶T2) using eo given in equation (12) is For T > Td, ZII is evaluated from (14), using (15) and (16). We get In this case, the free energy is http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0100-46702001000100006&lng=en&nrm=iso&tlng=en#fig01 and the specific heat has the simple form In figure 2 it is shown the specific heat as function of the temperature. We observe that Cv increases up to the melting temperature and, as a consequence of the approach adopted; it drastically decreases and became constant. The curve presented in fig.2 is similar to that one obtained in ref.2,4 by numerical integration. Conclusions We explicitly divided the model in two temperature regions, T < Td and T > Td. This approach gives a simplified mathematical approach to describe the statistical mechanics of thermal DNA denaturation. Other works2,4 get only the dependent part in y (T