Space exploration, defined

The free energy differences obtained from our constrained simulations refer to strictly specified states, defined by single points in the 14-dimensional dihedral space. Standard concepts of a molecular conformation include some region, or volume in that space, explored by thermal fluctuations around a transient equilibrium structure. To obtain the free energy differences between conformers of the unconstrained peptide, a correction for the thermodynamic state is needed. The volume of explored conformational space may be estimated from the covariance matrix of the coordinates of interest, = ((Ci [13, lOj. For each of the four selected conform-... 

This brief review places emphasis on (1) the use of stereoselective multicomponent reactions to access diverse and complex pure analogues in a powerful combinatorial way and (2) to review recent synthetic pathways and concepts devoted to exploring chemical space (both defined or expanded). 

In a systematic search there is a defined endpoint to the procedure, which is reached whe all possible combinations of bond rotations have been considered. In a random search, ther is no natural endpoint one can never be absolutely sure that all of the minimum energ conformations have been found. The usual strategy is to generate conformations until n new structures can be obtained. This usually requires each structure to be generate many times and so the random methods inevitably explore each region of the conformc tional space a large number of times. 

Eigenvectors and PCA. This is the factor space we are about to explore. We will be working with a factor space defined by the eigenvectors of our data simply because a coordinate system of eigenvectors has certain properties that are convenient and valuable to us. 

Usually the space over which the objective function is minimized is not defined as the p-dimensional space of p continuously variable parameters. Instead it is a discrete configuration space of very high dimensionality. In general the number of elements in the configuration space is exceptionally large so that they cannot be fully explored with a reasonable computation time. 

More generally, MILPs are solved with branch and bound algorithms, similar to the spatial branch and bound method of the previous section, that explore the search space. As seen in Fig. 3-61, binary variables are used to define the search tree, and a number of bounding properties can be noted from the structure of (3-110). 

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