Local approximation, dynamic optimization

We present below some easily implementable methods for improving the robustness and efficiency of feasible path dynamic optimization codes which have proved useful in our work. Here, we cover methods for preventing simulation error from disrupting optimization, representation of path constraints, and handling poor local approximations during the optimization. 

Local approximations (linear or quadratic) are often particularly poor in dynamic optimization problems. For instance, this situation is found to occur when taking the full step predicted from the local approximation, 6, causes a path constraint to become active or the system to become unstable. 

Perhaps the most accurate calculations performed to date are the MP2, LMP2, and LCCSD(TO) calculations on chorismate mutase (CM) and para hydroxy-benzoate-hydroxylase (PHBH) (the L in the acronyms indicates that local approximations were used, and TO is an approximate triples correction).41,42 These are coupled-cluster calculations that account for the effects of conformational fluctuations through an averaging over multiple pathways (16 for CM and 10 for PHBH). Initial structures were sampled from semiempirical QM/MM dynamics, using B3LYP/MM optimized reaction pathways. 

Figure 5 Optimization of the objective function in Modeller. Optimization of the objective function (curve) starts with a random or distorted model structure. The iteration number is indicated below each sample structure. The first approximately 2000 iterations coiTespond to the variable target function method [82] relying on the conjugate gradients technique. This approach first satisfies sequentially local restraints, then slowly introduces longer range restraints until the complete objective function IS optimized. In the remaining 4750 iterations, molecular dynamics with simulated annealing is used to refine the model [83]. CPU time needed to generate one model is about 2 mm for a 250 residue protein on a medium-sized workstation.

The incorporation of non-Gaussian effects in the Rouse theory can only be accomplished in an approximate way. For instance, the optimized Rouse-Zimm local dynamics approach has been applied by Guenza et al. [55] for linear and star chains. They were able to obtain correlation times and results related to dynamic light scattering experiments as the dynamic structure factor and its first cumulant [88]. A similar approach has also been applied by Ganazzoli et al. [87] for viscosity calculations. They obtained the generalized ZK results for ratio g already discussed. 

An alternative way to treat the surrounding protein is by including it in a MM description. All QM/MM calculations in this review are made with the program Qsite [34, 35, 36]. In this program the QM/MM interface is built by frozen localized molecular orbitals. Only predefined cuts between MM and QM parts are allowed. The QM parts of the QM/MM models therefore include the entire residue side chains. The MM part in a Qsite approach is treated with the OPLS-AA force field [37]. A dimer of the yeast enzyme consists of more than 8,000 atoms, which is the maximum number of atoms that Qsite can handle. To reduce the size of the MM part below this Hmit, parts of the second chain (B) that are more than 10 A away from the substrate are removed as well as the entire C and D chains. To keep the overall structure of the enzyme, approximately 40 atoms located close to the surface are kept frozen during the optimizations. The QM/MM treatment does not include dynamics and minima are located by optimizations starting from the X-ray structure. 

If the ions are treated explicitly, the ion-electron interaction is described by pseudopotentials. This allows us to eliminate all core electrons from the actual calculation and to deal only with the valence electrons. There exists a wide variety of pseudopotentials. The most elaborate of them are nonlocal operators because they project out of the occupied core electron states, see e.g. [18]. Separable approximations to projection can simplify the handling [19]. The simplest to use are, of course, local pseudopotentials, as e.g. the old empty-core potential of [20] or the more recent and extensive adjustment of [21]. It is a welcome feature that most simple metals, except for Li, can be treated fairly well with local pseudopotentials [22]. It is to be remarked that all these choices have been optimized with respect to structural properties. The performance concerning dynamical features has not yet been explored systematically. In fact, nSost of the available pseudopotentials tend to produce a blue-shifted plasmon position. An optimization of pseudopotentials for simple metals with simultaneous adjustment of static and dynamic properties is presently under way [23]. For noble metals, it is known that pseudopotentials alone (even when nonlocal) cannot reproduce the proper plasmon position. One needs to explicitly take into account the considerable polarizability of the ionic core, in particular of the rather soft last d shell [24]. 

---