Quantum Conformational Transition in Biological System

The conformational change of biological macromolecule is investigated from the point of quantum transition. A quantum theory on protein folding is proposed. Compared with other dynamical variables such as mobile electrons, chemical bonds and stretching-bending vibrations the molecular torsion has the lowest energy and can be looked as the slow variable of the system. Simultaneously, from the multi-minima property of torsion potential the local conformational states are well defined. Following the idea that the slow variables slave the fast ones and using the nonadiabaticity operator method we deduce the Hamiltonian describing conformational change. It is proved that the influence of fast variables on the macromolecule can fully be taken into account through a phase transformation of slow variable wave function. Starting from the conformationtransition Hamiltonian the nonradiative matrix element is calculated in two important cases: A, only electrons are fast variables and the electronic state does not change in the transition process; B, fast variables are not limited to electrons but the perturbation approximation can be used. Then, the general formulas for protein folding rate are deduced. The analytical form of the formula is utilized to study the temperature dependence of protein folding rate and the curious non-Arrhenius temperature relation is interpreted. The decoherence time of quantum torsion state is estimated and the quantum coherence degree of torsional angles in the protein folding is studied by using temperature dependence data. The proposed folding rate formula gives a unifying approach for the study of a large class problems of biological conformational change. 1 Molecular torsion as slow variable There are huge numbers of variables in a biological system. What are the fundamental variables in the life processes at the molecular level? Since the classical works of B. Pullman and A. Pullman on nucleic acids [1], it is generally accepted that the mobile π electrons play an important role in the biological activities of macromolecules. However,the traditional quantum biochemistry cannot treat a large class problems relating to the conformational variation of biological macromolecules such as protein folding, signal transduction and gene expression regulation, etc. In fact, for a macromolecule consisting of n atoms there are 3n coordinates if each atom is looked as a point. Apart from 6 translational and rotational degrees of freedom there are 3n-6 coordinates describing molecular shape. The molecular shape is the main variables responsible for conformational change. It has been proved that the bond lengths, bond angles and torsion (dihedral) angles form a complete set to describe the molecular shape. For a complex system consisting of many dynamical variables the separation of slow/fast