Analysis of the Contribution of Internal Vibrations to the Statistical Weights of Equilibrium Conformations of Macromolecules

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Abstract
The statistical weights of equilibrium conformations of macromolecules contain contributions from internal vibrations. An analysis of such vibrations, in the absence and presence of solvent, is presented from a quantum‐statistical‐mechanical point of view. Several classical approximations to the quantum‐mechanically correct expression, with different degrees of accuracy, are derived. In all of these approximations, all internal degrees of freedom of the polymer are divided into two classes: hard (bond stretching and bond angle bending) and soft (torsional rotations around single bonds, i.e., variation of dihedral angles). Since the hard variables oscillate many times before the soft variables change in value by an appreciable amount, the hard variables can be treated effectively as parameters (i.e., not as independent variables), which are (i) functions of the instantaneous values of the soft variables, or (ii) simply constants, the latter treatment being less accurate. In treatment (i) the molecule is regarded as flexible, whereas in treatment (ii) bond lengths and bond angles are assumed to be rigidly fixed, while the dihedral angles can change. In both treatments, the sum of the zero‐point energies corresponding to the hard degrees of freedom must be added to the soft‐mode part of the energy unless the errors in the calculation of the latter are ≥0.1 kcal/unit, which is the usual magnitude for the change in zero‐point energies for various conformations. The soft degrees of freedom are treated classically, i.e., the statistical weight is given by integration of the Boltzmann factor over phase space. The integration over the momentum space of the soft variables yields a conformation‐dependent term, proportional to ln detG, which may be called the kinetic entropy (G being a coefficient matrix for the kinetic energy of a polymer in the canonical expression for the Hamiltonian). A practical method is given for the calculation of detG, and it is applied to a simple example. The result shows that the dependence of ln detG on the coordinates is usually not negligible. For states with small conformational fluctuations (helical polymer structures and globular proteins), the result of treatment (ii) can be used as a perturbational step to proceed to treatment (i). Further approximations, necessary for treating random‐coil states (ones with large conformational fluctuations), are discussed. The effect of solvent on the statistical weights is also discussed.