Theory of single-molecule optical spectroscopy

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Single-molecule Forster resonance energy transfer (FRET) measurements on freely diffusing molecules contain information about conformational dynamics because the rate of energy transfer depends on the distance between donor and acceptor labels attached to a molecule. In these experiments, a molecule diffuses through a spot illuminated by a laser, and the donor is excited. The output of these experiments is a sequence of photons of different colors (some emitted by the donor and some by the acceptor) separated by apparently random time intervals. As described in last years report, we have developed a rigorous theory that describes how statistics of photons is influenced by protein conformational dynamics, the diffusion of the protein through the laser spot, shot noise, etc. It was shown that the exact FRET efficiency and photon counting histograms can be obtained by solving an appropriate reaction-diffusion equation. We have obtained a simple analytical yet rigorous result for the width of FRET efficiency distribution and showed that the shape of the distribution depends dramatically on the bin size. This year our continued efforts have lead to four publications in this area. First, we have written (1) a book chapter where we have applied our general theory of photon counting to systems with fluorescence quenching. Second, we introduced a new and simpler to use method of analyzing single-molecule data (2). Third, we applied (3) this new method to interpret fluorescence resonance energy tranfer in polyproline (in collaboration with the W.A. Eaton's group, LCP/NIDDK). Finally,we answered the question when do single-molecule experiments actually measure the properties of one molecule by quantatively determining the influence of fluorophore concentration on the observed photon statistics. An important advance we made (2) was to show how to extract information about conformational dynamics from FRET experiments on diffusing molecules without modeling diffusion. Starting from a rigorous theory that does treat diffusion, we first examined when the single-molecule FRET efficiency distribution can be decomposed into the measured distribution of the total number of photons and the efficiency distribution of an immobilized molecule in the absence of shot noise. If the conformation does not change during the time the molecule spends in the laser spot, this is possible when (I) the efficiency is independent of the location in the laser spot and (II) the total number of photons does not depend on conformation. This decomposition is approximate when the conformation changes during the diffusion time. However, it does provide a simple framework for analyzing data. This is illustrated for a two-state system where the FRET efficiency distribution can be found analytically for all values of the interconversion rates. As an alternative to the analysis of FRET efficiency distributions, we introduced a simpler procedure that allows one to extract the rates of conformational changes by decoding the pattern of colors in the donor/acceptor photon trajectory (2). This can be done in the framework of statistical inference because the likelihood function, which must be optimized with respect to the model rate parameters, depends only on how the conformation changes during the interval between photons with specified colors. The procedure works even when the photon colors appear to be scrambled (i.e., one cannot identify states by visual inspection of a photon trajectory) because the photophysical properties of the conformers are similar and/or conformational dynamics is on a similar time scale as the photon counts. The above work has played an important role in analyzing recent experiments performed by Dr. W. A. Eaton's group (3). In this project, quantitative analysis of the FRET efficiency distribution of polyproline was performed. The analysis resolved the puzzle about the excess width of the distribution and allowed one to draw important conclusions concerning the flexibility of the all-trans form of polyproline and the distribution of cis isomers. More details are provided in Dr. Eaton's report. Finally, the optimal concentrations for single-molecule measurements of freely diffusing molecules (4) was established. It was found that a concentration as low as 0.1 molecule per observation volume may not be small enough for single-molecule FRET efficiency measurements. This result follows from a rigorous theory that takes many molecules into account. We considered the distributions of the number of photons (photon counting histograms) and showed that multiple-molecule effects are pronounced at large photon counts even at low concentrations. FRET efficiency distributions reveal multiple-molecule effects at large threshold values. This might be misinterpreted as multiple conformational states. Multiple-molecule effects strongly depend on the brightness of fluorophores. A simple test was suggested to determine parameters for which the single-molecule description is applicable.