Theory of single-molecule optical spectroscopy

自由扩散分子的单分子Forster共振能量转移(FRET)测量包含构象动力学的信息，因为能量转移的速度取决于分子上标记的供体和受体之间的距离。在这些实验中，分子通过激光照射的光斑扩散，施主被激发。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，另一些由受主发射)，由明显随机的时间间隔隔开。正如去年的报告所描述的，我们发展了一种严格的理论，描述了蛋白质构象动力学、蛋白质在激光光斑中的扩散、散粒噪声等因素对光子统计的影响。结果表明，通过求解合适的反应扩散方程，可以获得准确的FRET效率和光子计数直方图。我们得到了FRET效率分布宽度的一个简单的解析而严格的结果，并且表明分布的形状显著地依赖于箱的大小。 今年，我们继续努力，在这一领域出版了四份出版物。首先，我们写了(1)书中的一章，在这一章中，我们将光子计数的一般理论应用于具有荧光猝灭的系统。其次，我们介绍了一种新的、简单易用的单分子数据分析方法(2)。第三，我们应用这种新方法(与W.A.Eaton的小组LCP/NIDDK合作)解释了多聚脯氨酸中的荧光共振能量转移。最后，我们回答了单分子实验通过定量确定荧光团浓度对观察到的光子统计的影响来实际测量一个分子的性质的问题。 我们取得的一个重要进展(2)是展示了如何从扩散分子的FRET实验中提取关于构象动力学的信息，而不需要对扩散进行建模。从处理扩散的严格理论出发，我们首先研究了在没有散粒噪声的情况下，单分子FRET效率分布何时可以分解为光子总数的测量分布和固定分子的效率分布。如果分子在激光光斑中停留的时间内构象没有变化，当(I)效率与激光光斑中的位置无关，以及(Ii)光子总数不取决于构象时，这是可能的。当构象在扩散过程中发生变化时，这种分解是近似的。然而，它确实为分析数据提供了一个简单的框架。这是以两态系统为例说明的，其中FRET效率分布可以解析地求出所有相互转换率的值。 作为FRET效率分布分析的另一种选择，我们引入了一种更简单的程序，允许人们通过解码给体/受体光子轨迹中的颜色图案来提取构象变化的速率(2)。这可以在统计推断的框架内完成，因为必须相对于模型速率参数优化的似然函数仅取决于在具有指定颜色的光子之间的间隔期间构象如何变化。即使当光子颜色看起来被扰乱时(即，人们不能通过对光子轨迹的目视检查来识别状态)，该过程也起作用，因为构象的光物理性质相似和/或构象动力学处于与光子计数类似的时间尺度上。 上述工作在分析W.A.Eaton博士团队最近进行的实验中发挥了重要作用(3)。在本项目中，我们对多聚脯氨酸的FRET效率分布进行了定量分析。这一分析解决了关于分布的额外宽度的难题，并使人们能够就多聚脯氨酸的全反式的灵活性和顺式异构体的分布得出重要的结论。伊顿博士的报告提供了更多细节。 最后，建立了自由扩散分子(4)单分子测量的最佳浓度。研究发现，低至每观察体积0.1个分子的浓度可能不足以用于单分子FRET效率的测量。这一结果源于一种考虑了许多分子的严格理论。我们考虑了光子数(光子计数直方图)的分布，结果表明，即使在低浓度下，在大光子计数时，多分子效应也是显著的。FRET效率分布在大阈值下表现出多分子效应。这可能会被误解为多个构象状态。多分子效应很大程度上依赖于荧光团的亮度。建议用一种简单的试验来确定适用于单分子描述的参数。

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.