computer modeling in molecular biology.pdf

152 Benoit RouxOne advantage of this formulation is that the mean force can be decomposed linearlyinto a sum of contributions, e.g.,(6-22)and the linearity, preserved by the integral of the mean force along the reaction coordinate,allows to determine the contribution Wa(x) of any interaction term to thepotential of mean force,(6-23)with(6-24)This method can be very useful in finding the contribution of the solvent moleculesor specific residues to the free energy profile. The total potential of mean force canbe computed equivalently with the free energy simulation technique, via Eq. (6-17).However, the integrated mean force decomposition can only be obtained usingEqs. (6-22), (6-23) and (6-24).The free energy simulation technique is computationally intensive, i. e., to reachconvergence and obtain accurate results it is necessary to generate long trajectorieswith the ion constrained at various positions along the x-axis. For this reason it isof interest to use a system with as small a number of atoms as possible. To avoidthe large number of water molecules necessary to solvate the mouth of the channel,the potential of mean force was calculated for Na’ ion along the axis of a periodic(L, D) poly-alanine P-helix. Such a periodic helix is appropriate for investigating thesingle file translocation of the ion and its neighboring waters in the interior of thechannel where end effects and sidechains are expected to play a secondary role.Moreover, this choice provides a test of the methodology since any deviation fromperiodicity of the average properties indicates a lack of convergence in virtue of thehelix symmetry.The periodic helix structure was first refined in the absence of any ion and solventmolecules. The helical parameters, the rise and the rotation angle per (LAla, D-Ala)unit, were optimized by energy minimization with the ABNR algorithm [65]. It wasfound that the optimum rise per unit, hL, is 0.155 nm and the optimum turn perunit, 0, is 114.4 degrees, yielding 6.29 residues per turn; this value is similar to thatof the Urry helix model (6.3) [30]. The total number of particles is 229. The fundamentalunit, treated with periodic boundary conditions to avoid end effects (imagesof the system are repeated along the helix axis), includes 34 alanine residues,