Practical Protein Chemistry - A. Darbre 1989

Prediction of Peptide and Protein Conformation
The Arsenal of Modern Theoretical Methods
Rigid Geometry

Monte Carlo calculations and energy Minimization can be performed in Cartesian coordinates, though this approach is typically reserved for special cases, such as the energetic refinement of molecular Conformations. More commonly, the studied compound is assumed to have a rigid geometry, allowing only the torsion angles around single bonds to vary, while all other parameters (Bond Lengths, valence angles, etc.) are held fixed. Naturally, such an assumption saves significant computing time, as it greatly simplifies the search across the potential energy surface. Moreover, reducing the number of variables yields additional practical benefits.

A key feature of this approach is the lack of a straightforward relationship between the new internal coordinate system and the actual Cartesian coordinates of the molecule. This means that searching within the space of torsion angles (either randomly or using fixed increments) is not equivalent to searching the real conformational space defined by the Cartesian coordinates of all atoms. The mathematical link between partial derivatives with respect to Cartesian variables and internal coordinates is provided by the elements of a special determinant (the Jacobian), which can be viewed as a correction factor during calculations. In practice, the Jacobian is almost always neglected, except in cases where its computation is relatively simple—for instance, when accounting for Water molecule rotation during Monte Carlo simulations of solution behavior. It is also safe to assume that neglecting the Jacobian is entirely justified for molecules with deep potential energy minima, particularly when only the minimum value of the function is of interest (e.g., in minimization Procedures).

The rigid-geometry approximation is generally not used in Structure/139.html">Molecular Dynamics Calculations, where the Cartesian coordinates of the atoms serve as variables. Such an assumption is fundamentally incompatible with the foundations of the method, although imposing constraints on bond lengths and certain other parameters can substantially save computer time. While applying constraints is feasible in practice, implementing them is not straightforward. Naturally, such constraints slightly distort the realistic Description of the system's dynamic behavior; however, the resulting errors are seemingly outweighed by the overall uncertainty introduced by various other factors.