Reconstruction of Protein Structures from Single-Molecule Time Series

Single-molecule experimental techniques can record a number of observables describing dynamics of molecules, but not an all atom conformational dynamics. Takens’ Delay Embedding Theorem asserts that, under certain conditions, time-delay vectors of scalar observations contain sufficient information to reconstruct full molecular configurations up to an a priori unknown transformation. Applying Takens’ Theorem and tools from statistical mechanics, manifold learning, neural networks, and graph theory, we establish an approach Single-molecule TAkens Reconstruction (STAR) to learn this Jacobian and reconstruct all atom trajectories from experimentally-measurable scalar observables. We apply STAR to molecular dynamics simulations of a C₂₄H₅₀ polymer chain and the mini-protein Chignolin. Trained models reconstruct molecular configurations from synthetic time series data of head-to-tail molecular distances with atomistic root mean squared deviation accuracies better than 0.2 nm. This work exhibits potential to reconstruct protein structures from time series of experimentally-measurable observables and establishes theoretical and algorithmic foundations to do so with real experimental data.