Theory of DNA melting based on the Peyrard-Bishop model

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Abstract
The DNA melting based on the Peyrard-Bishop (PB) model is systematically investigated. Our study on the eigenvalues and eigenvectors of the transfer integral equation for the original PB model points out that the eigenvectors are composed of two kinds: discrete bound states that constitute the internal states of a DNA molecule and the continuous unbound states that represent its dissociated states. Another process controlling the melting of DNA—the dissociation equilibrium between duplex and single-stranded DNAs—is introduced, which leads to an extended model applicable for a realistic DNA chain with a finite number of base pairs. Based on the expansion of kernels, the calculations of the thermodynamic quantities of the system are reduced to multiplication of matrix series. Calculations on model block DNAs show the method is much more efficient than molecular dynamic simulation and has enough high precision to handle the melting of a natural DNA with arbitrary sequence. The discreteness effect and nonlinear effect of the model are discussed based on the Gaussian model. Rigorous melting curves for periodic DNA with two and three base pairs in a unit cell and boundary effects are presented by the transfer integral approach.