Path optimization

To compute the above expression, short molecular dynamics runs (with a small time step) are calculated and serve as exact trajectories. Using the exact trajectory as an initial guess for path optimization (with a large time step) we optimize a discrete Onsager-Machlup path. The variation of the action with respect to the optimal trajectory is computed and used in the above formula. 

The expander turbine is designed to minimize the erosive effect of the catalyst particles stiU remaining in the flue gas. The design ensures a uniform distribution of the catalyst particles around the 360° aimulus of the flow path, optimizes the gas flow through both the stationary and rotary blades, and uses modem plasma and flame-spray coatings of the rotor and starter blades for further erosion protection (67). 

As explained above, the QM/MM-FE method requires the calculation of the MEP. The MEP for a potential energy surface is the steepest descent path that connects a first order saddle point (transition state) with two minima (reactant and product). Several methods have been recently adapted by our lab to calculate MEPs in enzymes. These methods include coordinate driving (CD) [13,19], nudged elastic band (NEB) [20-25], a second order parallel path optimizer method [25, 26], a procedure that combines these last two methods in order to improve computational efficiency [27],... 

The organization of the paper is as follows. In the Methods Section we present a brief review of the procedures employed to determine the MEP, namely QM/MM geometry optimizations and the CD method (Section 3.2.1). NEB and the second order parallel path optimizer methods are reviewed in Section 3.2.2. In Section 3.2.3 we present a modified NEB method which allows the addition of extra images to a converged path. Finally we describe the FEP method developed in our lab in Section 3.2.4. In the Computational Details Section we present a description of the procedures employed for the determination of all four paths. Subsequently, we analyze the results obtained from all four paths and conclude with closing remarks. 

The procedure and methods for the MEP determination by the NEB and parallel path optimizer methods have been explained in detail elsewhere [25, 27], Briefly, these methods are types of chain of states methods [20, 21, 25, 26, 30, 31]. In these methods the path is represented by a discrete number of images which are optimized to the MEP simultaneously. This parallel optimization is possible since any point on the MEP is a minimum in all directions except for the reaction coordinate, and thus the energy gradient for any point is parallel to the local tangent of the reaction path. 

The path optimizations are carried out by an iterative optimization procedure [25]. In the case of enzyme systems, because of the large number of degrees of freedom, we partition them into a core set and an environmental set. The core set is small and contains all the degrees of freedom that are involved with the chemical steps of the reaction, while all the remaining degrees of freedom are included in the environmental set. In all the QM/MM calculations presented below, the core set is defined by the QM subsystem and the environmental set by the MM subsystem. 

Once the path has been optimized with NEB, it is used as the initial guess for the parallel path optimizer method. In this second step the path is again iteratively optimized with the parallel path optimizer method for the core set, followed by the optimization of the environment set. In this part of the calculation no restraints are imposed on the environment set during the optimization. The iterations are continued until all the convergence criteria are met and the final optimized MEP is obtained. 

Initially, we have applied the modified NEB method to the calculation of both steps of the 40T catalyzed reaction. The free energy profiles and relative free energies obtained with this method were compared to our previously determined profiles [33], As we had previously shown, the calculated MEPs for Ref. [33] determined with the reaction coordinate driving method, and the MEPs for Ref. [25] calculated with the parallel path optimizer method, agree in the overall shape and relative potential energies. This provides a good starting point for our comparison. 

Figures 3-4 and 3-5 show the optimized paths with the added images and the original combined method [27] and parallel path optimizer method [25] calculated paths for the first and second steps of the reaction respectively. In both cases, the addition of extra images on the converged path, and subsequent optimization of these extra images produces a smoother path since the additional images allows for a better mapping of the potential energy surfaces (PESs).

Reinhardt, W. P. Hunter III, J. E., Variational path optimization and upper and lower bounds for the free energy via finite time minimization of the external work, J. Chem. Phys. 1992, 97, 1599-1601... 

By using open-equation formats and infeasible path optimization algorithms, the type of difficulty described above can be avoided. All the equations in the NLP problem can be solved simultaneously, driving the residuals to zero. The open-equation format for the heat exchanger is... 

Biegler, L. T. Improved Infeasible Path Optimization for Sequential Modular Simulators—I The Interface. Comput Chem Eng 9 245-256 (1985). 

A feasible path optimization approach can be very expensive because an iterative calculation is required to solve the undetermined model. A more efficient way is to use an unfeasible path approach to solve the NLP problem however, many of these large-scale NLP methods are only efficient in solving problems with few degrees of freedom. A decoupled SQP method was proposed by Tjoa and Biegler (1991) that is based on a globally convergent SQP method. 

Biegler. L. T., and Cuthrell, 1. E., Improved infeasible path optimization for sequential modular simulators — II The optimization algorithm, Comp, and Chem. Eng. 9(3), 257-267 (1985). 

Our judgment is that feasible path methods in which the solution of the model equations over time is carried out by conventional integration software, which has been extensively developed and refined, are at present more reliable than infeasible path methods. Feasible path optimization methods are also easier to implement as the size of the optimization problem is much smaller. For these reasons, we have pursued feasible path methods despite evidence that infeasible path methods are more efficient on some problems. 

A number of good reviews on TS optimization have appeared in recent years [9,11, 12,21,23-25,128]. In this section, we provide an overview of the three general classes of TS optimization methods—local schemes (Section 10.4.1), climbing, bracketing, and interpolation algorithms (Section 10.4.2), and path optimization approaches (Section 10.4.3). In Section 10.4.4 we discuss practical considerations related to TS optimization and offer suggestions for difficult cases. 

For local methods, it can be difficult to build a guess at the TS structure. Using a guess at the TS is also useful for interpolating, bracketing, and path optimization methods. However, generating an initial structure of a TS is usually non-trivial. Unlike minima, there are no direct experimental observations of TS geometries. Instead, the best tools available to computational chemists for this purpose are chemical intuition and the theoretical literature. Over the past two decades, thousands of optimized TSs have been... 

Table 10.6 Comparison of the number of gradient evaluations required to complete TS optimization using a QN with RFO, three point STQN methods, and the path optimization ...

Reaction QN with RFO Three point STQN Path optimization... 

W. P. Reinhardt and J. E. Hunter III, /. Chem. Phys., 97,1599 (1992). Variational Path Optimization and Upper and Lower Bounds of Free Energy Changes via Finite Time Minimization of External Work. 

Robinson, S. M. (1996), Analysis of Sample-Path Optimization, Mathematics of Operations Research, Vol. 21, pp. 513—528. 

As an alternative to RSM, simulation responses can be used directly to explore the sample space of control variables. To do so, a lot of combinatorial optimization approaches were adapted for simulation optimization. In general, there are four main classes of methods that have shown a particular applicability in (multi-objective) simulation optimization Meta-heuristics, gradient-based procedures, random search, and sample path optimization. Of particular interest are meta-heuristics as they have shown a good performance for a wide range of combinatorial optimization approaches. Therefore, commercial simulation software primarily uses these techniques to incorporate simulation optimization routines. Among meta-heuristics, tabu search, scatter search, and genetic algorithms are most widely used. Table 4.13 provides an overview on aU aforementioned techniques. 

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