Reconstruction of protein structures from single-molecule time series.

Single-molecule experimental techniques track the real-time dynamics of molecules by recording a small number of experimental observables. Following these observables provides a coarse-grained, low-dimensional representation of the conformational dynamics but does not furnish an atomistic representation of the instantaneous molecular structure. Takens's delay embedding theorem asserts that, under quite general conditions, these low-dimensional time series can contain sufficient information to reconstruct the full molecular configuration of the system up to an a priori unknown transformation. By combining Takens's theorem with tools from statistical thermodynamics, manifold learning, artificial neural networks, and rigid graph theory, we establish an approach, Single-molecule TAkens Reconstruction, to learn this transformation and reconstruct molecular configurations from time series in experimentally measurable observables such as intramolecular distances accessible to single molecule Förster resonance energy transfer. We demonstrate the approach in applications to molecular dynamics simulations of a C