Potential of Mean Force

The molecular dynamics simulation described above has as a principal goal the calculation of the PMF $\Delta G(Rc)$ of the reaction. In our case the PMF calculation has been performed using the umbrella sampling technique[196,15].

In the umbrella sampling method, outlined in section 1.5.2, at every simulation window the reaction coordinate is restrained within a limited range by imposing a harmonic umbrella potential.

The simulations are carried out by overlapping regions (windows) until covering the entire reaction coordinate. For every window we run 15 ps for equilibration and 50 ps for sampling. The reaction coordinate is divided in finite bins of length 0.01 Å. During the sampling the probability distribution P(Rc) is built counting the configurations that fall in the range Rc $\pm$ 0.005 Å of every bin.

The WHAM technique has been used to join the different windows of simulations. The WHAM results have been compared with those obtained by directly matching every two overlapping windows and few differences have been encountered^5.1. For a detailed description of umbrella sampling and WHAM technique see the introductory section 1.5.2. Unless indicated explicitly the free energy profile using WHAM will be shown.

In general, the simulation techniques need many heuristic parameters. Magnitudes such as the length of a dynamics for equilibration or sampling, the number of waters to solvate, the cutoff for non-bonded interactions ...etc. Most of these parameters and procedures are not tested here, they are taken from the literature or by personal communication from other experienced researchers.

Multi-dimensional PMF:
There has been several publications paving the methodology needed for multi-dimensional PMF calculations [201,199,200]. A multi-dimensional PMF would be useful for many reasons, mainly for the usage of several independent parameters to fully represent the reaction path. A two-dimensional free energy surface would permit to observe how two reaction coordinates participate in the reaction mechanism. The decision in our study of preferring a one-dimensional PMF was mainly due to the computational cost.

An approximated prediction of the computational requirements for a 2D PMF applied to our system is the following. It took about 30 hours of CPU for every 65 ps window (15ps equilibration + 50ps sampling). In order to scan from reactants to products the geometrical coordinate (see next section) R$_{NHC}$ goes from -0.7 to 1.9 Å, R$_{CHN}$ from -1.9 to 0.6 Å. If the coordinate increment for two adjacent windows is 0.2 Å, we need 169 windows, or 5070 hours of computation. Unless an automated and parallelized process is designed it means 211 days of uninterrupted computation time.