Theoretical Studies On The Dynamic Aspects Of Macromolecular Function

蛋白质或核酸上的荧光供体和受体标记之间的单分子Forster共振能量转移(FRET)可以用来探测分子结构、动力学和功能。在这些实验中，分子要么被固定在表面上，要么通过激光照射的光斑扩散，供体被激发。施主可以发射一个光子，或者将激发传递给接受者，然后接受者可以发射不同颜色的光子。转移速度取决于(染料间距离)-6，这就是为什么有关于构象动力学的信息(FRET是用于结构确定的核磁共振中NOE的光学模拟)。这些实验的输出是一个光子轨迹(施主发射的光子的颜色与受主发射的光子的颜色不同)。可以对观察到的光子序列进行入库，并且可以构建每个入库的FRET效率的直方图，该直方图被定义为从接受者发射的光子的分数。直方图的形状取决于分子的构象状态和它们的相互转化率。或者，可以使用基于似然性的方法在不入库的情况下分析光子序列。 我们为这种单分子FRET实验的分析建立了一个严格的理论框架。该理论描述了光子统计如何受到构象动力学、分子在激光光斑中的扩散、散粒噪声、染料光物理等的影响。该理论的各个方面已在全面综述中进行了总结(见参考文献1)。该理论已被用于在W.A.Eaton博士(LCP，NIDDK)的实验室进行的单分子实验的分析。 当单个分子被一串激光脉冲激发时，不仅可以检测到发射的光子的颜色和到达时间，而且可以检测到激光脉冲和光子之间的时间间隔。这一所谓的延迟时间与荧光团的荧光寿命有关。荧光寿命取决于能量转移的速率，因此随着给体和受体的靠近而减小。在2中，我们推广了我们以前关于FRET效率直方图的工作，以包括延迟时间。我们的主要理论贡献是推导出了考虑构象动力学在所有时间尺度上的影响的介子中施主和受主光子数和施主寿命的联合分布的精确表达式。也许最有趣的发现是，通过简单的肉眼检查实验联合分布在效率-寿命平面上的投影，就可以直接确定基本构象状态的连接性。 在一种互补的方法中，通过确定模型对整个光子轨迹的描述程度来分析整个光子轨迹。这是通过关于构象动力学模型的参数最大化适当的似然函数来实现的。在3中，我们以前基于可能性的分析被扩展到只记录连续时间间隔中的光子数量的一类实验。这允许改进传统上在隐马尔可夫模型中使用的似然函数。 具有单一催化位点的酶将底物转化为产物的速度以双曲线方式依赖于底物浓度，正如著名的Michaelis-Menten方程所描述的那样。在经典的酶动力学中，偏离这种双曲线行为被认为是协作性的标志(1.E.该酶有几个相互作用的结合部位)。在4中，我们证明了即使由于基态结合的扩散性质而存在单一结合位置，也可能偏离Michaelis-Menten方程的预测。 数学理论背后的基本思想是，在高亚态浓度下，扩散并不重要，因为一些衬底总是靠近结合位置。在较低的基态浓度下，它们相距很远，底物必须分散一个显著的密度才能达到结合。因此，两种情况下的结合率不同，导致周转率对底物浓度的依赖关系更加复杂。我们的理论预测 证实了这一点。

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. Alternatively, photon sequences can be analyzed without binning using likelihood-based methods. We have developed a rigorous theoretical framework for the analysis of such single-molecule FRET experiments. The theory describes how statistics of photons are influenced by conformational dynamics, diffusion of the molecule through the laser spot, shot noise, dye photophysics, etc. Various aspects of the theory have been summarized in a comprehensive review (see reference 1). The theory has been used in the analysis of single-molecule experiments performed in the laboratory of Dr. W. A. Eaton (LCP, NIDDK). When a single molecule is excited by a train of laser pulses, it is not only possible to detect the colors and arrival times of the emitted photons, but also the time interval between the laser pulse and the photon. This so-called delay time is related to the fluorescence lifetime of the fluorophore. The fluorescence lifetime depends on the rate of energy transfer and hence decreases as the donor and acceptor come closer together. In 2, we have generalized our previous work on FRET efficiency histograms to include delay times. Our main theoretical contribution was to derive an exact expression for the joint distribution of the numbers of donor and acceptor photons and donor lifetimes in a bin that treats the influence of conformational dynamics on all time scales. Perhaps the most interesting finding is that the connectivity of the underlying conformational states can be determined directly by simple visual inspection of the projection of the experimental joint distribution on the efficiency-lifetime plane. In a complimentary approach, the whole photon trajectory is analyzed by determining how well a model describes it. This is done by maximizing the appropriate likelihood function with respect to the parameters of the model of conformational dynamics. In 3, our previous likelihood-based analysis was extended to a class of experiments in which only the number of photons in consecutive time intervals is recorded. This allows one to improve the likelihood function that is traditionally used in Hidden Markov Models. The rate with which an enzyme with a single catalytic site converts a substrate into product, depends on the substrate concentration in a hyperbolic way, as described by the well-known Michaelis- Menten equation. In classical enzyme kinetics, deviations from such hyperbolic behavior is taken to be a hallmark of cooperativity (1.e. the enzyme has several interacting binding sites). In 4, we show that deviations from the predictions of the Michaelis-Menten equation can occur even if there is there is a single binding site as a result of the diffusive nature of substate binding. The basic idea behind out mathematical theory, is that at high substate concentrations, diffusion is unimportant because some substrate is always close to the binding site. At low substate concentration, they are far apart and the substrate has to difuusive a significant dististance to reach the binding. Thus the binding rates are different different in the two cases, resulting in a more complex dependence of the turnover rate on substrate concentration. Our theoretical predictions avaits experimental confirmation.