Group Scaling and Molecular Scaling

We have seen in section 3.3 that the center of mass or molecular pressure is equivalent to the atomic pressure. The atomic pressure is the natural quantity that enters in the virial theorem [12] irrespective of the form of the interaction potential among the particles. So, in principle it is safer to adopt atomic scaling in the extended system constant pressure simulation. For systems in confined regions, the equivalence between atomic or true pressure and molecular pressure (see sec. 3.3) holds for any definition of the molecular subsystem irrespective of the interaction potentials. In other words we could have defined virtual molecules made up of atoms selected on different real molecules. We may expect that, as long as the system, no matter how its unities or particles are defined, contains a sufficiently large number of particles, generates a distribution function identical to that generated by using the ``correct'' atomic scaling. From a computational standpoint molecular scaling is superior to atomic scaling. The fast varying Liouvillean in Eq. (3.70) for the atomic scaling contains the two terms $iL_{z}, iL_{u}$. These terms are slowly varying when molecular scaling is adopted and are assigned to the slow part of the Liouvillean in Eq. (3.69). The inner part of the time propagation is therefore expected to be more expensive for the multiple time step integration with atomic scaling rather than with molecular scaling. Generally speaking, given the equivalence between the molecular and atomic pressure, molecular scaling should be the preferred choice for maximum efficiency in the multiple time step integration.^3.10

For large size molecules, such as proteins, molecular scaling might be inappropriate. The size of the molecule clearly restricts the number of particles in the MD simulation box, thereby reducing the statistics on the instantaneous calculated molecular pressure which may show nonphysical large fluctuations. Group scaling [27] is particularly convenient for handling the simulation of macromolecules. A statistically significant number of groups can be selected in order to avoid all problems related to the poor statistics on molecular pressure calculation for samples containing a small number of large size particles. Notwithstanding, for solvated biomolecules and provided that enough solvent molecules are included, molecular scaling again yields reliable results. In Ref. [27] Marchi and Procacci showed that the scaling method in the $N{\bf P}T$ ensemble does not affect neither the equilibrium structural and dynamical properties nor the kinetic of non equilibrium MD. For group-based and molecular-based scaling methods in a system of one single molecule of BPTI embedded in a box of about a 1000 water molecules, they obtained identical results for the system volume, the Voronoi volumes of the proteins and for the mean square displacement of both solvent and protein atoms under normal and high pressure.