computer modeling in molecular biology.pdf

88 Shoshana .L Wodak, Daniel van Belle, and Martine Prhost(4-15)(g)Ajwhere the canonical average is equal to ( Aprovided the hybrid poten-tial energy function V(r”, Ii) is linear in I (as in Eq. (4-12)). Ensemble averages ofA V are computed at each A value. Like in the EF procedure, these averages are obtainedfrom a series of simulations performed at successive A values. The integrationover A, which is also an exact expression, approximated here by a summation overa finite number of AI “windows”, then yields the requested free energy value.The accuracy of the results obtained with either the EF or TI procedures will dependon the quality of the empirical potential function V(rN, Ai), on whether or notit is appropriate to use the linear form for the hybrid potential (in particular for theTI procedure), and on the method employed to sample the configuration ensemble.Differences due to changes in the number of particles may arise due to the kineticenergy contribution to A G. These differences cancel out however when identicalalchemical transformations are considered in two different states. Here they are thenative and unfolded states described below.4.3.2 Computing Free Energy Differences :Practical Aspects4.3.2.1 Implementation of the Perturbation MethodPrevious studies have shown that the linear dependence of the hybrid potential onthe coupling parameter A, is adequate for treating charged to non-polar (e.g.Asp + Ala [ll]), and charged to charged (e. g. Arg -+ His [94]) mutations. However,in cases where van der Waals interactions dominate the transformation, non linearforms of the hybrid potential were shown to lead to faster convergence [85]. In theapplication illustrated here which involves a non-polar to non-polar mutation(Ile + Ala) dominance of van der Waals interactions was also expected. To ensureconvergence of the calculations a variant of the classical procedure described in Section4.3.1 was therefore used. Several intermediate states were defined along apathway from A (wild type) to B (mutant) as illustrated in Figure 4-16. These stateswere generated by gradually modifying van der Waals parameters and bond lengthsof the relevant sidechains in the direction of the transformation. The free energy differencesA Gi+i+l between successive states along the transformation pathway werethen computed with Eq. (4-13) or Eq. (4-15), using three values of I (A = 1/6, 3/6,