Non-affine displacements encode collective conformational fluctuations in proteins

Identifying subtle conformational fluctuations underlying the dynamics of bio macro-molecules is crucial for resolving their free energy landscape. We show that a collective variable, originally proposed for crystalline solids, is able to filter out essential macro-molecular motions more efficiently than other approaches. While homogenous or affine deformations of the biopolymer are trivial, biopolymer conformations are complicated by the occurrence of in-homogenous or non-affine displacements of atoms relative to their positions in the native structure. We show that these displacements encode functionally relevant conformations of macromolecule and, in combination with a formalism based upon time-structured independent component analysis, quantitatively resolve the free energy landscape of a number of macromolecules of hierarchical complexity. The kinetics of conformational transitions among the basins can now be mapped within the framework of a Markov state model. The non-affine modes, obtained by projecting out homogenous fluctuations from the local displacements, are found to be responsible for local structural changes required for transitioning between pairs of macro states.