Expectation-Maximization of the Potential of Mean
Force and Diffusion Coefficient in Langevin Dynamics from Single Molecule
FRET Data Photon by Photon
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Abstract
The dynamics of a protein along a
well-defined coordinate can be
formally projected onto the form of an overdamped Lagevin equation.
Here, we present a comprehensive statistical-learning framework for
simultaneously quantifying the deterministic force (the potential
of mean force, PMF) and the stochastic force (characterized by the
diffusion coefficient, <i>D</i>) from single-molecule Förster-type
resonance energy transfer (smFRET) experiments. The likelihood functional
of the Langevin parameters, PMF and <i>D</i>, is expressed
by a path integral of the latent smFRET distance that follows Langevin
dynamics and realized by the donor and the acceptor photon emissions.
The solution is made possible by an eigen decomposition of the time-symmetrized
form of the corresponding Fokker–Planck equation coupled with
photon statistics. To extract the Langevin parameters from photon
arrival time data, we advance the expectation-maximization algorithm
in statistical learning, originally developed for and mostly used
in discrete-state systems, to a general form in the continuous space
that allows for a variational calculus on the continuous PMF function.
We also introduce the regularization of the solution space in this
Bayesian inference based on a maximum trajectory-entropy principle.
We use a highly nontrivial example with realistically simulated smFRET
data to illustrate the application of this new method