Machine learning of protein folding funnels from experimentally measurable observables.

The stable conformations and dynamical motions of proteins are governed by the underlying single-molecule free energy surface. By integrating ideas from dynamical systems theory with nonlinear machine learning, we establish a technique to recover single-molecule free energy surfaces from univariate time series in a single esperimentally-accessible observable. Using Takens’ Delay Embedding Theorem, we expand the univariate time series into a high dimensional space in which the dynamics are equivalent to those of the molecular motions inreal space. We then apply the diffusion map nonlinear manifold learning algorithm to extract a low-dimensional representation of the free energy surface that is diffeomorphic to the one from the all atom system. We validate our approach in molecular dynamics simulations of a C₂₄H₅₀ n-alkane chain and the Trp-cage mini protein to demonstrate that the low-dimensional free energy surfaces extracted from the atomistic simulation trajectory are geometrically and topologically equivalent to that recovered from a knowledge of only the head-to-tail extent of the molecule. Our approach lays the foundations to extract empirical single-molecule free energy surfaces directly from experimental measurements.