Theoretical Studies On The Dynamic Aspects Of Macromolecular Function

自由扩散分子上的单分子福斯特共振能量转移（FRET）测量包含关于构象动力学的信息，因为能量转移的速率取决于附着于分子的供体和受体标记之间的距离。在这些实验中，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（一些由施主发射，一些由受主发射），这些光子被明显随机的时间间隔分开。如在以前的报告中所描述的，我们已经开发了一个严格的正式理论，描述了如何统计的光子是由蛋白质构象动力学，扩散的蛋白质通过激光光斑，散粒噪声等的影响。它表明，精确的FRET效率和光子计数直方图可以通过求解适当的反应扩散方程。我们已经得到了一个简单的分析，但严格的结果FRET效率分布的宽度，并表明分布的形状取决于显着的bin大小。 去年（见参考文献1），我们介绍了一种新的程序，通过解码供体/受体光子轨迹中的颜色模式来提取构象变化率。 对于具有测量的到达时间的光子轨迹，通过在微观模型的框架内最大化观察这种轨迹的可能性来分析颜色的图案。 似然函数必须根据模型速率参数进行优化，它仅取决于在具有指定颜色的光子之间的时间间隔期间分子结构如何变化。 这种方法可以应用于扩散分子发出的光子的爆发，以及固定分子产生的长光子轨迹。这一过程已经说明了使用模拟光子轨迹获得的系统具有两个和三个不同的构象状态。我们的方法工作，即使当光子的颜色似乎是随机的，因为高的背景噪声，构象的物理性质是相似的和/或光子计数率是相似的构象转变的速率。 今年，我们开发了一种基于FRET效率直方图的简单解析表达式的补充方法（参见参考文献2和3）。由具有多个构象状态的分子产生的光子轨迹构建的直方图由高斯的总和近似，每个高斯具有不可调节的权重、均值和方差，而是状态之间的转换速率的显式函数。我们的近似的基本思想是描述的贡献的时间仓，不包含过渡和那些涉及过渡之间的高斯与适当的均值和方差的两个状态。所有其他箱的贡献由单个高斯描述，选择参数以确保整个FRET效率直方图被标准化，并且具有所有箱时间的精确均值和方差。该理论已被测试对模拟数据的两个，三个和四个构象状态。它准确地描述了直方图中的峰值如何随着转换速率或仓时间的增加而塌陷。高斯近似可以很容易地用来分析实验数据，并提取构象变化的速率和FRET效率的构象。 这两种方法，最大似然法和高斯近似FRET效率直方图，已被用来分析单分子折叠实验中进行的博士W.A.伊顿（LCP/NIDDK）（参见参考文献4）。在这些实验中，折叠和展开速率系数提取的单分子福斯特共振能量转移（FRET）的数据与快速动力学的蛋白质。进行了两种类型的实验和两种不同的分析。在一个实验中，从附着于蛋白质的供体和受体荧光团收集光子的爆发，所述蛋白质自由扩散通过共聚焦显微镜的照明体积。在第二个实验中，蛋白质通过附着在表面上而被固定，光子被收集，直到其中一个荧光团漂白。使用上述最大似然法确定折叠和解折叠速率系数和平均FRET效率。通过将光子轨迹与提取的参数进行比较，并将计算的FRET效率直方图与测量的直方图进行比较，来检查该方法在双态模型方面描述数据的能力。两态模型的速率系数之和与从施主-受主强度相关函数的衰减获得的弛豫速率一致，证实了该方法的高精度。速率系数和平均FRET效率也通过拟合FRET效率直方图获得，通过将供体和受体光子合并计算，具有三高斯的总和。动力学在这些直方图中表现为随着箱大小增加，在折叠峰和未折叠峰之间出现FRET效率峰，这是一种类似于NMR交换加宽的现象。当折叠和未折叠分子的群体是可比的，提取的速率系数是在非常好的协议与最大似然法获得的。我们希望这些互补的程序将成为分析单分子荧光实验的标准工具，以了解结构和动力学。

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.