Principal component analysis (PCA)

In order to explore impacts of candidate molecules on dynamical and structural properties of M^pro homodimer, principal components of the systems, which reflected the collective dynamics of trajectories, were calculated. To do so, the Cα atom of each residue was picked up and the systems were aligned with respect to the Cα atoms of the crystal structure. Subsequently, the eigenvalues and eigenvectors of the systems were calculated by diagonalizing the co-variance matrix of each system using the following equation.

where covariance matrix is denoted by C_mn as well as M_mn Δr_mΔr_n corresponds to a change in position from time-averaged structure for each coordinate of m and n atoms.

where set of eigenvalues and eigenvectors of the diagonalized co-variance matrix are shown by δ², and v, respectively.

Consequently, for calculating 2D PCAs, trajectories of each system were aligned with respect to the first two eigenvalues of holo in 2D space. To this aim, the ’gmx covar’ and ’gmx anaeig’ modules of the GROMACS were utilized to generate the diagonalized co-variance matrix pertaining to each system and calculate corresponding eigenvalues and eigenvectors (Abraham et al., 2015). In addition, each system was projected in 2D space with respect to the first eigenvalue to unveil the collective change in each Cα atom throughout the trajectory. To do so, an open-source Python package, namely ProDy, was used (Bakan et al., 2011, 2014).