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Many phenomena in molecular biophysics happen over time and length scales that are inaccessible by fully atomistic computer simulations. Therefore, coarse-graining has become a common strategy for bridging the gap in time and length scale between the atomistic simulation and biological processes. Furthermore, in many cases the system dynamics is better represented in terms of collective coordinates. This study is concerned with a rigorous coarse-graining method for dynamics of linear systems using collective coordinates of the resiudes rather than coordinates of individual atoms. In this method an invertible linear time-independent map is considered to relate the original displacements to the collective coordinates. Then, the conformational space of the transformed system is divided into master and slave degrees of freedom. Under the assumption that the masters are slower than the slaves and by expanding the masters’ displacements in Taylor series with respect to time variable, the method results in effective stiffness, friction and mass for the coarse-grained system in terms of collective coordinates of the residues. Center of mass and hydrodynamic center of reaction coordinates of the residues are considered as collective coordinates. Application of the method to finding the relaxation dynamics of various proteins shows that using center of mass coordinates significantly improves the results.

Keywords: coarse-graining, collective coordinates, center of mass, relaxation, protein dynamics, hydrodynamic center of reaction

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