Umbrella sampling chemical potential

Tx and Tx are the kinetic energies of the atomic coordinates and X variables, respectively. The As are treated as volumeless particles with mass mx. Since the X variables are associated with the chemical reaction coordinates , the A-dynamics method can utilize the power of specific biasing potentials in the umbrella sampling method to overcome sampling problems that require conventional FEP calculations to be performed in multiple steps. 

The basic formalism of the X-dynamics method has taken various forms in its application to problems of interest. In an early prototype calculation to assess umbrella sampling in chemical coordinates, the X-dynamics method was used to evaluate the relative free energy of hydration for a set of small molecules which included both nonpolar (C2H6,) and polar (CH3OH, CH3SH, and CH3CN) solutes.1 By assigning a separate X variable to the Lennard-Jones and Coulomb interactions, a linear partition of the potential part of the hybrid Hamiltonian was constructed... 

Using umbrella sampling, Tieleman and Marrink [18] determined a PMF for transferring a DPPC lipid from water to the center of a DPPC bilayer (Figure 3B). The DPPC PMF has a deep minimum at its equilibrium position and a steep slope in free energy as it moved into bulk water. The free energy of desorption (AGdesorb) was 80 kj/mol, and is directly related to its excess chemical potential in the bilayer compared to water. 

K. S. Shing and K. E. Gubbins, Mol. Phys., 43, 717 (1981). The Chemical Potential from Computer Simulation. Test Particle Method with Umbrella Sampling. 

AB, = B, % = 1) - B, (A, =0) and Pum is the probability function of the hybrid potential with the umbrella potential. If (AB/ = ABj), the effect of the umbrella potential will be canceled completely. An iterative procedure is sometimes required to produce complete sampling of important configurations along the chemical coordinates. In such cases, WHAM,... 

---