Theoretical Studies On The Dynamic Aspects Of Macromolecular Function

单分子Forster共振能量转移(FRET)测量可以用来测量亚群的结构和动力学性质，以及亚群之间的跃迁动力学。最常见的单分子FRET实验是收集附着在分子上的给体和受体荧光团发出的光子，它们通过共聚焦显微镜的照明体积自由扩散。光子被入库，并且构建每个库的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。在1中，我们发展了一个简单的解析理论来描述FRET效率直方图，该直方图由具有多个构象状态的分子产生的光子轨迹构成。直方图由高斯和近似，其参数由状态的FRET效率和状态之间的转换速率明确确定。该理论已经针对两个构象状态的精确直方图和三个和四个构象状态的模拟数据进行了测试，准确地描述了直方图中的峰如何随着结合时间或跃迁速率的增加而坍塌。在2中，我们使用这种方法和我们之前开发的一种互补的最大似然方法来从单分子FRET数据中提取折叠和去折叠速率系数，这些数据的动力学太快，无法测量等待时间分布。 激光光钳或原子力显微镜施加在单个分子上的机械力可以诱导结构转变，如蛋白质或核酸的展开或复合体的解离。在这些实验中，分子系统可以被驱使远离平衡，从而允许探索亚稳态和罕见的分子过程。。2001年，我们展示了如何通过将Jarzynski方程从依赖于实验可控参数的系统自由能关系扩展到依赖于波动坐标的分子自由能表面来重构分子自由能表面。在3中，我们证明了作为分子伸展函数的无扰自由能分布可以从这样的实验中严格地获得，而不需要使用我们前面介绍的功加权位置直方图。利用魏尔斯特拉斯逆变换将系统的自由能直接与分子自由能表面联系起来。用最陡下降法计算反变换，得到了自由能面的精确近似。将这种形式应用于从RNA折叠动力学模型获得的模拟数据，在该模型中，动力学由连接子主导的折叠和未折叠的自由能表面之间的跳跃组成。 酶中的底物结合部位常常被掩埋。在许多情况下，它们可以通过一条狭窄的隧道从高分子表面到达。为了到达活性部位，配基必须扩散到通往活性部位的缝隙中，然后通过缝隙进入。乙酰胆碱酯酶就是一个典型的例子。跨膜通道还可以具有由可以共价结合特定离子或分子的孔衬残留物产生的结合部位。在4中，我们考虑了扩散影响的结合到掩埋的结合部位，该结合部位通过一个狭窄的隧道连接到表面。在隧道截面上配体平衡分布的单一假设下，我们将含时速率系数的计算简化为具有适当边界条件的一维扩散方程的求解。我们得到了一个简单的稳态速率的解析表达式，它取决于管内平均作用力的势和与隧道入口处结合的扩散控制速率。我们的理论的潜在应用包括底物结合到酶的埋藏的活性部位，以及真正的离子结合到跨膜通道的内部部位。 当存在时间尺度分离时，通过绝热消除快变量，可以得到对慢变量动力学的有效描述。例如，对于二维各向异性的朗之万动力学，传统的方法得到了慢坐标下的朗之万方程，该方程涉及平均力的势。沿此坐标的摩擦常数保持不变。在文5中，我们证明了用与位置有关的摩擦力可以得到更准确但仍然是马尔可夫式的慢动力学描述，它与由柯克伍德式公式表示的实际力与平均力之差的自相关函数的时间积分有关。结果被推广到许多维，其中慢速或反应坐标是笛卡尔坐标的任意函数。当快速变量是有效的一维时，沿慢速坐标的附加摩擦力可以表示为任意势的闭合形式。对于具有弯曲中心线的变截面圆柱对称流道，我们的解析表达式立即给出了位置相关扩散系数的多维版本的Zwanzig-Bradley公式。

Single-molecule Forster resonance energy transfer (FRET) measurements on single biological molecules can be used to measure the structural and dynamical properties of subpopulations, as well as the kinetics of transitions between subpopulations. The most frequent single molecule FRET experiment has been to collect photons emitted by donor and acceptor fluorophores attached to the molecule freely diffusing through the illuminated volume of a confocal microscope. The photons are binned, and a histogram of the FRET efficiencies for each bin, defined as the fraction of the photons emitted from the acceptor, is constructed. The shape of the histogram depends on the conformational states of the molecule and their interconversion rates. In 1 we developed a simple analytic theory to describe FRET efficiency histograms constructed from a photon trajectory generated by a molecule with multiple conformational states. The histograms are approximated by a sum of Gaussians with the parameters explicitly determined by the FRET efficiencies of the states and the rates of the transitions between the states. The theory, which has been tested against exact histograms for two conformational states and simulated data for three and four conformational states, accurately describes how the peaks in the histograms collapse as the bin time or the transition rates increase. In 2 this approach and a complimentary maximum likelihood approach we developed earlier were used to extract folding and unfolding rate coefficients from single-molecule FRET data for proteins with kinetics too fast to measure waiting time distributions. Mechanical forces exerted on single molecules by laser optical tweezers or atomic force microscopes can induce structural transitions such as the unfolding of a protein or nucleic acid or the dissociation of a complex. In these experiments, the molecular system can be driven far from equilibrium, permitting the exploration of metastable states and rare molecular processes. . In 2001 we showed how to reconstruct molecular free energy surfaces by extending the Jarzynski equality from a relation for the system free energy that depends on an experimentally controllable parameter to a relation for a molecular free energy surface that depends on a fluctuating coordinate. In 3 we show that unperturbed free energy profiles as a function of molecular extension can be obtained rigorously from such experiments without using work-weighted position histograms we introduced previously.. An inverse Weierstrass transform is used to relate the system free energy directly to the underlying molecular free energy surface. An accurate approximation for the free energy surface is obtained by using the method of steepest descent to evaluate the inverse transform. The formalism is applied to simulated data obtained from a kinetic model of RNA folding, in which the dynamics consists of jumping between linker-dominated folded and unfolded free energy surfaces. Substrate binding sites in enzymes are often buried. In many cases they are accessible from the surface of the macromolecule by a narrow tunnel. In order to reach the active site, a ligand must diffuse into and then through the crevice leading to the site. A classic example is acetylcholinesterase . Transmembrane channels also can have binding sites that are produced by a pore-lining residue that can covalently bind a permeant ion or molecule. In 4 we consider diffusion-influenced binding to a buried binding site that is connected to the surface by a narrow tunnel. Under the single assumption of an equilibrium distribution of ligands over the tunnel cross section, we reduce the calculation of the time-dependent rate coefficient to the solution of a one-dimensional diffusion equation with appropriate boundary conditions. We obtain a simple analytical expression for the steady-state rate that depends on the potential of mean force in the tuunnel and the diffusion-controlled rate of binding to the tunnel entrance. Potential applications of our theory include substrate binding to a buried active site of an enzyme and permeant ion binding to an internal site in a transmembrane channel. When there is a separation of time scales, an effective description of the dynamics of the slow variables can be obtained by adiabatic elimination of fast ones. For example, for anisotropic Langevin dynamics in two dimensions, the conventional procedure leads to a Langevin equation for the slow coordinate that involves the potential of the mean force. The friction constant along this coordinate remains unchanged. In 5, we show that a more accurate, but still Markovian, description of the slow dynamics can be obtained by using position-dependent friction that is related to the time integral of the autocorrelation function of the difference between the actual force and the mean force by a Kirkwood-like formula. The result is generalized to many dimensions, where the slow or reaction coordinate is an arbitrary function of the Cartesian coordinates. When the fast variables are effectively one-dimensional, the additional friction along the slow coordinate can be expressed in closed form for an arbitrary potential. For a cylindrically symmetric channel of varying cross section with winding centerline, our analytical expression immediately yields the multidimensional version of the Zwanzig-Bradley formula for the position-dependent diffusion coefficient.