computer modeling in molecular biology.pdf

150 Benoi’t Rouxbulk waters. A stable hydrogen bond is formed with a bulk water at 50 ps and ismaintained for the rest of the trajectory leaving the entrance of the channel free. Itis possible that this type of rare event plays a role in the entrance and exit of watermolecules. Appropriate techniques based on biased sampling must be used to studyrare events accurately. This is the object of the next section.6.3.3 Calculation of the Potential of Mean Force:Free Energy SimulationIn Section 6.2, it was shown that the free energy profile, W(x), plays a fundamentalrole in the phenomenological description of ion transport. From its definition, W(x)is the free energy of the system when the ion is at position x along the channel axis,i.e., [79],(6-15)where U(R,x) is the total potential energy of the system and R represents all thecoordinates of the system other than the x coordinate of the ion (the x, y, z coordinatesof all the channel and water atoms including the y and z coordinates ofthe ion), xo is an arbitrary reference position chosen such that W(xo) = Wo. Thefree energy profile is often called “potential of mean force”, following from the propertythat the average force exerted on the ion when it is located at position x, i.e.,( -aU(R, x)/dx>, is equal to - aW(x)/ax [79]. Because the potential of mean force,based on Eq. (6-15), can be related to the average probability distribution functionof the ion,(6-16)it can, in principle, be extracted directly from (p(x)) through a normal moleculardynamics simulation. However, because it is expected that the movements of the ionalong the channel axis are taking place on a very long time-scale compared torealistic simulation times, a direct approach is not practical. It is necessary to usea biased sampling technique to obtain accurate results on the potential of meanforce. Such a technique is free energy perturbation which is illustrated in the nextsection.