Maximizing dynamical systems information embedded in experimental observables of molecules through statistical learning enabled Takens reconstruction

Single molecule Forster Resonance Energy Transfer (smFRET) allows experimentalists to track changes in molecular conformations over time by tagging the molecule with two or more fluorescent probes whose emissivities correspond to inter-probe distances. While experiments of molecular dynamics are restricted to scalar time series, dye placement choice allows for the tuning of the intramolecular distance observable recorded. Takens' Delay Embedding Theorem guarantees that there exists a bijective Jacobian between manifolds embedding the full dimensional dynamics of a system and a time delayed embedding of scalar observables of that system under some minor technical conditions. The theorem is silent on the question of observable choice leading to the question of which dye placement choices maximally embed system dynamics. We combine Takens' Theorem, manifold learning, nonlinear mapping techniques and statistical mechanics to obtain mappings from experimental observation space to reconstructions of atomic coordinates using molecular dynamics training data. In previous work, we have reconstructed the mini-proteins Chignolin and Villin to accuracies better than 0.4 nm from synthetic noisy and time averaged trajectories generated from molecular dynamics simulations using single variate head to tail molecular distances. In this work, we expand this Single-molecule TAkens Reconstruction (STAR) technique to use synthetic multivariate data streams corresponding to a 3 dye multichannel smFRET system generated from molecular dynamics simulations of the mini-protein Villin to learn optimal dye placement to maximize information embedded in experiments. This work not only shows how dye placement can optimize reconstructed trajectories via STAR but in principle also suggests correct dye placement to enhance performance of standard smFRET analysis protocols such as Hidden Markov Models for identifying conformational states.