In order to explain the denaturation and melting in replication and transcription of deoxyribonucleic acid (DNA) at physiological temperature, we propose a dynamical model on the basis of structure and motion of DNA under the actions of energy released in hydrolysis of adenosine triphosphate (ATP) and enzymes. The model admits three degrees of freedom per base pair: two displacement variables associated with the vibrations of hydrogen atom in the hydrogen bonds and base (nucleotide), respectively, and an angular variable describing the rotation of base. The important role of hydrogen atom in the hydrogen bonds is stressed in this model. The Hamiltonian of the intact double helix system is given first, subsequently the equations of motion and solutions of a melting system are developed. The solutions represent the excited states formed by the displacements of hydrogen atoms and bases and the rotations of bases, arising from the energy absorbed by DNA, respectively. The results obtained show that the displacements of hydrogen atoms increase with time and along the helix. Once the displacements of hydrogen atoms and rotations of bases reach to certain limits, the hydrogen bonds are disrupted, and the two double helical strands of DNA are separated, a melting state in the replication or transcription process occurs. The properties of thermal motion of hydrogen atoms in this state at physiological temperature is further studied by using a transfer integral method and the thermodynamic properties such as free energy and entropy of the thermally excited state are obtained in the process. Values of characteristic parameters and critical temperature of melting in transcription process are derived. Finally the validity of the theory was demonstrated through comparison between experimental and theoretical data of specific heat and force of phase transition. This investigation is helpful to understand mechanism and essence of the replication or transcription process and promote quantitative description of these processes.