Single-molecule Forster resonance energy transfer (FRET) between fluorescent donor and acceptor labels attached to a protein or nucleic acid can be used to probe a molecules structure, dynamics and function. In these experiments, a molecule is either immobilized on a surface or diffuses through a spot illuminated by a laser, and the donor is excited. The donor can emit a photon or transfer the excitation to an acceptor which then can emit a photon of a different color. The rate of transfer depends on (interdye distance)-6 and this is why there is information about conformational dynamics (FRET is the optical analog of the NOE in NMR that is used in structure determination). The output of these experiments is a photon trajectory (the color of the photons emitted by the donor differ from those emitted by the acceptor). The observed sequence of photons can be binned, and a histogram of the FRET efficiencies for each bin, defined as the fraction of the photons emitted from the acceptor, can be constructed. The shape of the histogram depends on the conformational states of the molecule and their interconversion rates. Although this technique provides unique information that cannot be obtained at the ensemble level, the possibility of studying fast molecular dynamics is limited by the number of photons detected per unit time (photon count rate), which is proportional to the illumination intensity. To improve the dynamic range of single-molecule fluorescence spectroscopy at a given photon count rate, we consider each and every photon and use a maximum likelihood method to get the information about fast conformational dynamics. For a photon trajectory with recorded photon colors and inter-photon times, the parameters of a model describing molecular dynamics are obtained by maximizing the appropriate likelihood function. In collaboration with H. S. Chung, we discussed the theory and experimental applications of the maximum likelihood method (reference 1). We considered various likelihood functions, their applicability, and various experiments on fast two-state protein folding to extract folding rates and to measure transition path times. The review contains several original results that have not been published before, such as the study of the accuracy of the extracted parameters and the influence of blinking. We also presented an experimental demonstration of utilizing other information such as fluorescence lifetimes. This was discussed in the framework of the theory for two-dimensional FRET efficiency-lifetime histograms, as a continuation of our previous work . Repeated modification of a multisite protein plays a key role in cell signaling. After an enzyme catalyzes a reaction at one site, it can modify another site before diffusing away. This physical process, which is amplified in the crowded environment of the cell, has been recently shown by stochastic simulations of hundreds of particles to dramatically change the response of a biochemical network. The simplest way to describe the influence of the relative diffusion of the reactants on the time course of bimolecular reactions is to modify or renormalize the phenomenological rate constants that enter into the rate equations of conventional chemical kinetics. However, for macromolecules with multiple inequivalent reactive sites, this is no longer sufficient, even in the low concentration limit. The physical reason is that an enzyme (or a ligand) that has just modified (or dissociated from) one site can bind to a neighboring site rather than diffuse away. This process is not described by the conventional chemical kinetics, which is only valid in the limit that diffusion is fast compared with reaction. Using an exactly solvable many-particle reaction-diffusion model, we showed (reference 2) that the influence of diffusion on the kinetics of multisite binding and catalysis can be accounted for by not only scaling the rates, but also by introducing new connections into the kinetic scheme. The rate constants that describe these new transitions or reaction channels turn out to have a transparent physical interpretation: The chemical rates are scaled by the appropriate probabilities that a pair of reactants, which are initially in contact, bind rather than diffuse apart. The theory is illustrated by application to phosphorylation of a multisite substrate. This is work is an example of a new line of research which we plan to pursue in the future. Our paper (reference 3) on diffusion along the splitting/commitment probability reaction coordinate and the relation to the protein folding was described in last year report but actually appeared in print during the present reporting period.