Second generation Mining-Minima algorithm (M2) -- A New Free Energy Calculation Method

The M2 method computes the standard free energy of binding as the difference between the standard chemical potentials of the receptor-ligand complex, the free receptor, and the free ligand:

Each chemical potential, u∘x, where X = R; L; or RL, is computed via a sum of contributions from the low-energy conformations of the molecule or complex. For example, in a ligand L, the energy profile shown on the left has three low energy conformations. The chemical potential is computed by summing from three configuration integrals, 1 to 3 in the right figure.

where C∘ is the standard concentration (here 1 mole/liter), R is the gas constant, and T is absolute tempperature.

To compute the chemical potential (or Free energy), we need two very important steps. First is identifying the low-energy conformations, and we apply Tork conformational search method to find low energy minima. The second one is computing the configuration integral of each energy minimum found. The configuration integral in each unique energy well is computed with an augmented form of the harmonic approximation, the Harmonic Approximation/Mode Scanning (HA/MS) method, as detailed in the following section. Note that in order to avoid double-counting, duplicate conformations are eliminated by a method that accounts for molecular symmetry.

We have developed  "HA/MS", a novel algorithm for the direct calculation of configuration integrals in all degree of freedom. The method uses the harmonic approximation (HA) with finite integration ranges, along with Mode Scanning (MS), a fast numerical integration based upon internal bond-angle-torsion coordinates. A major concern in the use of harmonic approximations is that sometimes the modes don't behave harmonically [1,2]. Thus, we introduced  "Mode Scanning" to account for local anharmonicity without the need for expensive Monte Carlo integration. The method is shown to be accurate via comparisons with analytic or highly detailed Monte Carlo integration, and is efficient for cyclic, acyclic, and macrocyclic molecules and also host-guest complexes. We also showed that the use of a Cartesian coordinate basis set may produce significant errors in free energy calculations, because the linear combination of x, y, z can't properly express a bond rotation.