Practical Protein Chemistry - A. Darbre 1989

Prediction of Peptide and Protein Conformation
The Arsenal of Modern Theoretical Methods
Statistical Mechanics

The Methods of statistical mechanics, which are also applied in Thermodynamics, allow for the calculation of the average properties of a system that adopts numerous configurations at temperatures above absolute zero. Structure/8.html">Molecular Dynamics methods, the Monte Carlo Method, and direct calculation techniques for various conformation energies can be approximated as generating statistical ensembles. The application of statistical mechanics methods enables the analysis of sampled data. It should be noted, however, that in certain cases—such as the Metropolis algorithm—the method of generating the sample of a physical quantity and the statistical analysis are inseparable from one another.

In fact, averaging via statistical mechanics is identical to standard arithmetic averaging. If each conformation X corresponds to a certain physical property V(X) and a probability of realization P(X), then the expected average is given by

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According to the laws of statistical mechanics, the probability P(X) is proportional to the Boltzmann weighting factor e^-H/kT:

P(X) = Q^-1∙e^-H/kT      (21.9)

where H is the total energy of the system, calculated via the classical Hamiltonian and incorporating kinetic energy as a component:

H = H(p,q) = W(p) + E(q)      (21.10)

Here W is the kinetic energy depending on the nuclear momenta p, and E is the potential energy depending on the nuclear coordinates q and, consequently, on the conformation X. Q is the partition function, the value of which determines all thermodynamic Properties of the system. For example, the Free energy E of the system is related to the partition function as follows:

The coefficient a depends on the number of indistinguishable and therefore interchangeable particles [30], which often leads to confusion. Fortunately, The Effect of coefficient a is negligible when considering the relative free energies of the same molecule.

The main difference between molecular dynamics and the Monte Carlo method is that the latter neglects the kinetic energy component W(p). This omission allows for the Introduction of the configurational partition function Z, which is valid under equilibrium conditions.

V(X) can characterize any physical property that depends on the conformation X of the system under study—for instance, a spin-spin coupling constant in an NMR spectrum, a set of values such as a circular dichroism spectrum over a given wavelength range, or the Flory matrix used in polymer coil theory to determine THE POSITION OF each polymer chain relative to its preceding position. Important thermodynamic parameters include enthalpy (assuming no volume change) V=E and Entropy V=klnP(X), which represents the Amount of Information corresponding to conformation X.