Theoretical Studies On The Dynamic Aspects Of Macromolecular Function

自由扩散分子的单分子福斯特共振能量转移（FRET）测量包含构象动力学信息，因为能量转移速率取决于分子上附着的供体和受体标签之间的距离。在这些实验中，一个分子通过激光照射的光斑扩散，供体被激发。这些实验的输出是一系列不同颜色的光子（一些由供体发射，一些由受体发射），这些光子被明显随机的时间间隔隔开。正如在之前的报告中所描述的，我们已经开发了一个严格的形式化理论，描述了光子统计是如何受到蛋白质构象动力学、蛋白质通过激光光斑的扩散、射击噪声等的影响的。结果表明，通过求解相应的反应扩散方程，可以得到准确的FRET效率和光子计数直方图。我们对FRET效率分布的宽度得到了一个简单而严谨的分析结果，并表明分布的形状与料仓尺寸有很大的关系。

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 previous reports, we have developed a rigorous formal 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. Last year ( see reference 1), we introduced a novel procedure to extract the rates of conformational changes by decoding the pattern of colors in the donor/acceptor photon trajectory. For a photon trajectory with measured arrival times, the pattern of colors is analyzed by maximizing the likelihood of observing such a trajectory within the framework of a microscopic model. The likelihood function, which must be optimized with respect to the model rate parameters, depends only on how the structure of the molecule changed during the time interval between photons with specified colors. This approach can be applied to bursts of photons emitted by diffusing molecules as well as to long photon trajectories generated by immobilized molecules. This procedure has been illustrated using simulated photon trajectories obtained for systems with two and three different conformational states. Our method works even when the photon colors appear to be random because of high background noise, the photophysical properties of the conformers are similar and/or the photon count rates are similar to the rates of conformational transitions. This year we developed a complimentary approach based on a simple analytic expression for the FRET efficiency histograms ( see references 2 and 3). The histograms constructed from photon trajectories generated by a molecule that has multiple conformational states are approximated by a sum of Gaussians each with a weight, mean and variance that are not adjustable but rather explicit functions of the rates of the transitions between the states. The basic idea of our approximation is to describe the contribution of time bins that contain no transitions and those that involve transitions between just two states by Gaussians with the appropriate mean and variance. The contribution of all other bins is described by a single Gaussian with parameters chosen to ensure that the entire FRET efficiency histogram is normalized and has the exact mean and variance for all bin times. The theory has been tested against simulated data for two, three and four conformational states. It accurately describes how the peaks in the histograms collapse as the transition rates or the bin time increases. The Gaussian approximation can be readily used to analyze experimental data and extract the rates of conformational changes and the FRET efficiencies of the conformers. Both approaches, the maximum liklehood method and the Gaussian approximation to FRET efficiency histograms, have been used to analyze single molecule folding experiments performed in the group of Dr. W.A. Eaton (LCP/NIDDK)(see reference 4). In these experiments, folding and unfolding rate coefficients were extracted from single-molecule Forster resonance energy transfer (FRET) data for proteins with fast kinetics. Two types of experiments and two different analyses were performed. In one experiment, bursts of photons were collected from donor and acceptor fluorophores attached to a protein freely diffusing through the illuminated volume of a confocal microscope. In the second, the protein was immobilized by attaching it to a surface, and photons were collected until one of the fluorophores bleached. Folding and unfolding rate coefficients and mean FRET efficiencies were determined using the maximum likelihood method described above. The ability of the method to describe the data in terms of a two-state model was checked by recoloring the photon trajectories with the extracted parameters and comparing the calculated FRET efficiency histograms with the measured histograms. The sum of the rate coefficients for the two-state model agreed well with the relaxation rate obtained from the decay of the donor-acceptor intensity correlation function, confirming the high accuracy of the method. The rate coefficients and mean FRET efficiencies were also obtained by fitting the FRET efficiency histograms, calculated by binning the donor and acceptor photons, with a sum of three-Gaussians. The kinetics are manifested in these histograms by the appearance of a FRET efficiency peak in between the folded and unfolded peaks as the bin size increases, a phenomenon similar to NMR exchange broadening. When the populations of folded and unfolded molecules are comparable, the extracted rate coefficients are in very good agreement with those obtained with the maximum likelihood method. We expect that these complementary procedures will become a standard tool for analyzing single molecule fluorescence experiments to learn about both structure and dynamics.