Simulated Annealing SA

As described in Section 9.4, the determination and refinement of molecular conformations comprehends three main methods DG, MD and SA. Other techniques like Monte Carlo calculations have only a limited applicability in the field of structure elucidation. In principle, it is possible to exclusively make use of DG, MD or SA, but normally it is strongly suggested to combine these methods in order to obtain robust and reliable structural models. Only when the results of different methods match a 3D structure should be presented. There are various ways of combining the described techniques and the procedural methods may differ depending on what kind of molecules are investigated. However, with the flowchart in Fig. 9.13 we give an instruction on how to obtain a reliable structural model. 

The starting point of each structure elucidation is the collection of experimental data like NOE/ROE-derived distances, angular restraints from J-couphngs or CCRs and RDCs as orientational restraints (1). 

Exoerimsntal HMR Data NOEs/RDEs, (J-Couplings), (RDCs) 

At this point it is essential to compare the calculated structure with both the experimental data and the results of the rMD run (6). On the one hand, the interatomic distances of the final model must match the NMR restraints additionally, the fMD-averaged structure should correspond with the refined conformation obtained by the rMD. Only if the rMD and the fMD simulations result in the same conformational model and no experimental restraints are severely violated the calculated structure can be presented as a 3D image (7). 

If the final structure either deviates from the refined model or does not match the NMR restraints (8) one has to revise the experimental data and the parameters used in the DG and MD computations (9). In many cases, mistakes are made when preparing and performing the computational processes (10) or even experimental errors might be present (11). Those errors include a wrong NMR peak assignment, no precise calibration of the NOE/ROE signals, an incorrect conversion of the experimental data to constraints, and a nonfactual parameterization of the rMD and fMD trajectories. In such cases either new calculations or new experiments must be performed. 

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This criterion resumes all the a priori knowledge that we are able to convey concerning the physical aspect of the flawed region. Unfortunately, neither the weak membrane model (U2 (f)) nor the Beta law Ui (f)) energies are convex functions. Consequently, we need to implement a global optimization technique to reach the solution. Simulated annealing (SA) cannot be used here because it leads to a prohibitive cost for calculations [9]. We have adopted a continuation method like the GNC [2]. 

An approach to overcome the multi minima problem of proteins is simulated annealing (SA) run. Besides global molecular properties such as structural and thermal motions, functional properties of fast biological reactions can also be studied by MD. 

Restrained MD (rMD) is followed by the use of MD in explicit solvent, i.e. the conformation as determined above is taken into a box containing many solvent molecules around the molecule. Subsequently, simulated annealing (SA) and energy minimizahon steps are performed to draw the molecule into the global energy minimum. An MD run (the so-called trajectory) over at least 150ps to Ins is followed and a mean structure is calculated from such a trajectory. The con-formahon must be stable under this condihon even when the experimental constraints are removed. 

Refinement of Conformations by Computational Methods 243 9.4.2.6 Simulated Annealing (SA)... 

The second problem can be overcome by using a stochastic search. Thus Corana s simulating annealing SA [89], a sophisticated global search method arising from the Metropolis algorithm [90] was employed. A synthetic seven species problem was constructed and the elements in T-jx were correctly determined using SA. The recovered spectra are essentially identical to the synthetic 7 pure component spectra. [91]... 

These first applications were simply too inefficient, and the cluster examples chosen were too small, for them to be serious competition for established methods like simulated annealing (SA) (see Sect. 4) therefore, they went largely unnoticed, and the following year, 1994, did not see much activity in this field at all. 

In the next section the ideas behind several methods, including the GA and a simulated annealing (SA) approach [19,20], then their implementation used to generate ionic crystal structures are reviewed. This will contain an introduction to the types of move class operators and the various types of cost functions used to modify the current trial structure(s) and to assess the quality of the trial structures, respectively. In the third section recent applications of the GA and SA approaches to closest-packed ionic systems and then to open-framework crystal structures are reviewed. 

Fig. 13 Cost function as a function of the logarithm of temperature during the annealing of NbF4. Inset structural arrangement of NbF4 at the beginning of the simulation (right) and as obtained by simulated annealing (SA) (left) [58]. Reproduced with the kind permission of the Nature Publishing Group (http //www.nature.com/)...

The first two examples, which imply that similarity between compounds is related to similar activity profiles, use two popular selection methods genetic algorithms (GA 96-100) and simulated annealing (SA 101, 102). 

Separability theorem, 309 SHAKE algorithm, 385 SHAPES force field, 40 Simulated Annealing (SA), global optimization, 342 Simulation methods, 373 Supidfiidiil, iulcs, 3j6 Susceptibility, 237 Symbolic variables, for optimizations, 416 Symmetrical orthogonalization of basis sets, 314 Symmetry adapted functions, 75 Symmetry breaking, of wave functions, 76 ... 

Linear Models. Variable selection approaches can be applied in combination with both linear and nonlinear optimization algorithms. Exhaustive analysis of all possible combinations of descriptor subsets to find a specific subset of variables that affords the best correlation with the target property is practically impossible because of the combinatorial nature of this problem. Thus, stochastic sampling approaches such as genetic or evolutionary algorithms (GA or EA) or simulated annealing (SA) are employed. To illustrate one such application we shall consider the GA-PLS method, which was implemented as follows (136). 

Some researchers have combined various optimization algorithms to improve the search efficiency and computational effort, including evolutionary algorithms (EA), simulated annealing (SA), particle swarm optimization (PSO), ant colony optimization (AGO), hybrid PSO-SQP, hybrid GA-ACO. Nevertheless, the combination of the GA and SQP algorithms is reported only in a few works [1,2]. 

Besides the measure of the dispersion of the one-dimensional projection i.e. the projective index, another distinction of PP PCA from the classical PCA is the procedure of computation. Since the projective index is the quadratic form of X as stated above, the extremal problem of Eqn. 1 can be turned into the problem of finding the eigenvalues and eigenvectors of the sample covariance matrix for which a lot of algorithms such as SVD, QR are available. Because of the adoption of the robust projective index in PP PCA, some nonlinear optimization approaches should be used. In order to guarantee the global optimum. Simulated annealing (SA) is adopted which is the main topic of this book. 

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