Systematic determination of order parameters for chain dynamics using diffusion maps

We employ the diffusion map approach as a nonlinear dimensionality reduction technique to extract a dynamically relevant, low-dimensional description of n-alkane chains in the ideal-gas phase and in aqueous solution. In the case of C-8 we find the dynamics to be governed by torsional motions. For C-16 and C-24 we extract three global order parameters with which we characterize the fundamental dynamics, and determine that the low free-energy pathway of globular collapse proceeds by a kink and slide mechanism, whereby a bend near the end of the linear chain migrates toward the middle to form a hairpin and, ultimately, a coiled helix. The low-dimensional representation is subtly perturbed in the solvated phase relative to the ideal gas, and its geometric structure is conserved between C-16 and C-24. The methodology is directly extensible to biomolecular self-assembly processes, such as protein folding.