Alignment of Molecular Fragments
An analysis of the user-provided molecular structure of the fragment
(e.g., the photosensitizer molecule shown in Figure
2)4 starts from the alignment procedure.
The software allows a user to specify a desired fragmentation scheme.
For example, in Figure 2, the fragmentation scheme is as follows:
methyl-tetrzine (m Tz) is the EET acceptor, while halogentated
bioorthogonal boron dipyrromethene (BODIPY-2I) is the EET donor. Those
are selected as separate fragments, accordingly.4
PyFREC utilizes the Procrustes analysis to perform
alignment.1, 2, 20 In this procedure, a translation
vector a and a rotation matrix R are found so that the
root-mean-square deviation between transformed Cartesian coordinates of
the molecular fragments \(\mathbf{F}^{\mathbf{{}^{\prime}}}\) (Figure 2B,C) and
corresponding coordinates of the molecular system S
(Figure 2A) is minimized:
\(\left\|\mathbf{(FR}+\mathbf{a)}-\mathbf{S}\right\|\longrightarrow min\)(1)
where F are initial coordinates of the molecular
fragment. Then the singular value decomposition of the matrix\(\frac{\mathbf{S}^{\mathbf{t}}\mathbf{F}}{\left\|\mathbf{S}\right\|\left\|\mathbf{F}\right\|}\)of normalized coordinates of the fragment and corresponding atoms of the
molecular system is used to find the rotation matrix R :
\(\frac{\mathbf{S}^{\mathbf{t}}\mathbf{F}}{\left\|\mathbf{S}\right\|\left\|\mathbf{F}\right\|}=\mathbf{\text{Ub}}\mathbf{V}^{\mathbf{t}}\)(2)
where U and V are unitary matrices,
b is a diagonal matrix. Then the rotation matrix is:
\(\mathbf{R}=\mathbf{V}\mathbf{U}^{\mathbf{t}}\) (3)
The translation vector is obtained as the difference between the
centroids of the molecular system and rotated fragment:
\(\mathbf{a}=\left|\mathbf{S}\right|-\left|\mathbf{F}\right|\mathbf{R}\)(4)
The PyFREC software automatically computes metrics for analysis and
quality assessment of the molecular fragment alignment. Two sections of
output are provided. The “transformation section” provides the scaling
factor, translation vector, rotation matrix, and axis–angle
representation of the rotation.
PyFREC provides the sum and root-mean-square deviation between the
transformed atoms of the molecular fragment and those of the molecular
system in the output.