Expectation-Maximization of the Potential of MeanForce and Diffusion Coefficient in Langevin Dynamics from Single MoleculeFRET Data Photon by Photon.

The dynamics of a protein along awell-defined coordinate can beformally projected onto the form of an overdamped Lagevin equation.Here, we present a comprehensive statistical-learning framework forsimultaneously quantifying the deterministic force (the potentialof mean force, PMF) and the stochastic force (characterized by thediffusion coefficient, D) from single-molecule Förster-typeresonance energy transfer (smFRET) experiments. The likelihood functionalof the Langevin parameters, PMF and D, is expressedby a path integral of the latent smFRET distance that follows Langevindynamics and realized by the donor and the acceptor photon emissions.The solution is made possible by an eigen decomposition of the time-symmetrizedform of the corresponding Fokker–Planck equation coupled withphoton statistics. To extract the Langevin parameters from photonarrival time data, we advance the expectation-maximization algorithmin statistical learning, originally developed for and mostly usedin discrete-state systems, to a general form in the continuous spacethat allows for a variational calculus on the continuous PMF function.We also introduce the regularization of the solution space in thisBayesian inference based on a maximum trajectory-entropy principle.We use a highly nontrivial example with realistically simulated smFRETdata to illustrate the application of this new method. [ABSTRACT FROM AUTHOR]