C. Principal Component Analysis

are preferentially spread along one dimension, as reflected by the broad distribution along one of the new axes and a narrow distribution along the other in Figure 7b. The above procedure is what is done by PCA. Starting from a multidimensional distribution of data points (in our case, of molecular conformations), PCA performs a similarity transformation on the original axes to find a new set of axes that best fits the data. The first new axis is selected such that the variance of the distribution along it is the largest possible. The second axis is placed orthogonal to the first in the direction of the second largest variance of the distribution, and so forth. In this new axes set it is usually possible to identify a low-dimensional subspace that captures most of the relative distances between individual conformations.

In general, two related techniques may be used: principal component analysis (PCA) and principal coordinate analysis (PCoorA). Both methods start from the n m data matrix M, which holds the m coordinates defining n conformations in an m-dimensional space. That is, each matrix element Mij is equal to qij, the jth coordinate of the ith conformation. From this starting point PCA and PCoorA follow different routes.

Principal component analysis (PCA) takes the m-coordinate vectors q associated with the conformation sample and calculates the square m m MTM matrix, reflecting the relationships between the coordinates. This matrix, also known as the covariance matrix C,

where the averaging is over the conformation sample (in Cartesian space m 3N for an N-atom molecule). The covariance matrix C is diagonalized to obtain the eigenvectors that capture most of the variation in atomic position fluctuations.

Principal coordinate analysis (PCoorA) [37], on the other hand, operates on the square n n MMT matrix, reflecting the relationships between the conformations. The elements of this matrix, also known as the distance matrix , are distances dij between two conformations i and j [such as those defined in Eqs. (12) and (13)]. Since the distances dij can also be obtained from the n n matrix A of latent roots (eigenvectors), one can use this matrix for the projection, defining Aij 1/2d2ij and Aii 0 (for i, j 1, 2, . . . , n). To guarantee that the matrix A has a zero root (and thus guarantee that it corresponds to a real configuration) it is ‘‘centered,’’ so that the sum of every row and the sum of every column of A is zero. This centering, which does not alter the distances dij, is defined as

where k is the mean over all specific indices k i, j, ij. The centered matrix A* is diagonalized using standard matrix algebra to obtain the latent eigenvectors and the diagonal matrix of eigenvalues. The resulting eigenvalues (normalized) give the percentage of the projection of the original distribution on the new coordinate set, and the eigenvectors (scaled by their corresponding eigenvalues) give the new coordinates of the original points in the new axes set. For a more detailed description of this method, see Refs. 29 and 37.

It should be stressed that PCA and PCoorA are dual methods that give the same analytical results. Using one or the other is simply a matter of convenience, whether one prefers to work with the covariance matrix C or with the distance matrix ∆.

As stated earlier, the main motivation for using either PCA or PCA is to construct a low-dimensional representation of the original high-dimensional data. The notion behind this approach is that the effective (or essential, as some call it [33]) dimensionality of a molecular conformational space is significantly smaller than its full dimensionality (3N- 6 degrees of freedom for an N-atom molecule). Following the PCA procedure, each new

Figure 8 A joint principal coordinate projection of the occupied regions in the conformational spaces of linear (Ala)6 (triangles) and its conformational constraint counterpart, cyclic-(Ala)6 (squares), onto the optimal 3D principal axes. The symbols indicate the projected conformations, and the ellipsoids engulf the volume occupied by the projected points. This projection shows that the conformational volume accessible to the cyclic analog is only a small subset of the conformational volume accessible to the linear peptide, (Adapted from Ref. 41.)

axis k is associated with a normalized eigenvalue λk that reflects the relative weight of that axis in reproducing the original data. An axis with a high λk value is significant for the projection, whereas axes with small λk values are insignificant. By sorting the new axes according to their λk weight it is possible to select a small subset of effective coordinates that capture most of the conformational relationships of the original high-dimen- sional space. The quality of such a projection can be estimated by the average difference between conformation distances reconstructed in the low s-dimensional subspace d(ijs) and the original distance dij. The reconstructed distances are defined as

k 1

where Qik is the coordinate of the ith conformation along the kth new (principal) axis. It can be shown that the average deviation of the distances in s dimensions from the exact distances is given by the sum of the first s eigenvalues,

k 1

Thus by summing the normalized eigenvalues of the first s dimensions one can judge the quality of a projection onto that subspace.

Fortunately, it was found that in polypeptide systems the effective dimensionality of conformational spaces is significantly smaller than the dimensionality of the full space, with only a few principal axes contributing to the projection [38–41]. In fact, in many cases a projection quality of 70–90% can be achieved in as few as three dimensions [42], opening the way for real 3D visualization of molecular conformational space. Figure 8

shows a 3D visualization of the conformational spaces of two hexapeptides, (Ala)6 and cyclic-(Ala)6, jointly projected on the same principal coordinate set. The comparison shows that the conformation volume occupied by the linear peptide is about 10 times larger than the conformation volume occupied by its conformationally constrained counterpart. Quantifying relative flexibility of analogous peptides through joint principal projections was shown to be useful, for example, in predicting their relative bioactivity [41].

V.CONCLUSION

In this chapter we surveyed a variety of computational methods that contribute to the ‘‘conformational analysis’’ of complex molecules. These include methods for searching and sampling the molecular conformation space, methods for local optimization of the sampled conformations, and basic analytical techniques. In practice, many variations of the basic methodologies are reported as researchers continuously try to improve and enhance these procedures. The different methods are often used in conjunction to form a complete conformational analysis study of a bimolecular system of interest. However, each of the different procedures can also be used separately as part of computational studies with other goals. The need for these analytical techniques is to a large extent brought about by the continuous increase is simulation times, which generates more data than ever before, requiring systematic ways to interpret and represent them.

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