  Measuring the F\"{o}rster resonance energy transfer (FRET) efficiency of
freely diffusing single molecules provides information about the sampled
conformational states of the molecules. Under equilibrium conditions, the
distribution of the conformational states is independent of time, whereas it
can vary over time under non-equilibrium conditions. In this work, we consider
the problem of parameter inference on non-equilibrium solution-based
single-molecule FRET data. With a non-equilibrium model for the conformational
dynamics and a model for the conformation-dependent FRET efficiency
distribution, the likelihood function could be constructed. The model
parameters, such as the rate constants of the non-equilibrium conformational
dynamics model and the average FRET efficiencies of the different
conformational states, have been estimated from the data by maximizing the
appropriate likelihood function via the Expectation-Maximization algorithm. We
illustrate the likelihood method for a few simple non-equilibrium models and
validated the method by simulations. The likelihood method could be applied to
study protein folding, macromolecular complex formation, protein conformational
dynamics and other non-equilibrium processes at the single-molecule level and
in solution.
