0.4 Molecular distance measures  (Page 4/6)

Where $V$ and $W^T$ are the matrices of left and right eigenvectors, respectively, of the covariance matrix $C=X{Y}^T$ . This formula ensures that $U$ is orthonormal (the reader should carry out the high-level matrix multiplication and verify this fact).

The only remaining detail to take care of is to make sure that $U$ is a proper rotation, as discussed before. It could indeed happen that $\mathrm{det(U)=-1}$ if its rows/columns do not make up a right-handed system. When this happens, we need to compromise between two goals: maximizing $\mathrm{Tr(}$ $Y^T\mathrm{X\text{'}}$ $)$ and respecting the constraint that $\mathrm{det(U)=+1}$ . Therefore, we need to settle for the second largest value of $\mathrm{Tr(}$ $Y^T\mathrm{X\text{'}}$ $)$ . It is easy to see what the second largest value is; since:

then the second largest value occurs when $T_{11}=T_{22}=\mathrm{+1}$ and $T_{33}=-1$ . Now, we have that $T$ cannot be the identity matrix as before, but instead it has the lower-right corner set to -1. Now we finally have a unified way to represent the solution. If $\mathrm{det(C)>0}$ , $T$ is the identity; otherwise, it has a -1 as its last element. Finally, these facts can be expressed in a single formula for the optimal rotation $U$ by stating:

where $d=\mathrm{sign(det(C))}$ . In the light of the preceding derivation, all the facts that have been presented as a proof can be succinctly put as an algorithm for computing the optimal rotation to align two data sets $x$ and $y$ :

Optimal rotation

• Build the 3xN matrices $X$ and $Y$ containing, for the sets $x$ and $y$ respectively, the coordinates for each of the N atoms after centering the atoms by subtracting the centroids.
• Compute the covariance matrix $C=X{Y}^T$
• Compute the SVD (Singular Value Decomposition) of $C=VS{W}^T$
• Compute $d=\mathrm{sign(det(C))}$
• Compute the optimal rotation $U$ as

Optimal alignment for lrmsd using quaternions

Another way of solving the optimal rotation for the purposes of computing the lRMSD between two conformations is to use quaternions . These provide a very compact way of representing rotations (only 4 numbers as compared to 9 or 16 for a rotation matrix) and are extremely easy to normalize after performing operations on them. Next, a general introduction to quaternions is given, and then they will be used to compute the optimal rotation between two point sets.

Introduction to quaternions

Quaternions are an extension of complex numbers. Recall that complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the canonical imaginary number, equal to the square root of -1. Quaternions add two more imaginary numbers, j and k. These numbers are related by the set of equalities in the following figure:

Given this definition of i, j, and k, we can now define a quaternion.