Theoretical Studies On The Dynamic Aspects Of Macromolecular Function

对单个生物分子的单分子福斯特共振能量转移（FRET）测量可用于测量亚群的结构和动力学性质，以及亚群之间的转变动力学。最常见的单分子FRET实验是收集由附着在分子上的供体和受体荧光团发射的光子，这些荧光团通过共聚焦显微镜的照射体积自由扩散。光子被分箱，并构建每个箱的FRET效率的直方图，定义为从受体发射的光子的分数。直方图的形状取决于分子的构象状态及其相互转化率。在1中，我们开发了一个简单的分析理论来描述FRET效率直方图构建从光子轨迹产生的分子与多个构象状态。直方图近似的高斯与明确确定的状态和状态之间的转换率的FRET效率的参数的总和。该理论，已被测试对两个构象状态和模拟数据的三个和四个构象状态的精确直方图，准确地描述了如何在直方图中的峰崩溃的bin时间或过渡速率增加。 在2这种方法和一个免费的最大似然方法，我们开发的早期被用来提取折叠和展开速率系数从单分子FRET数据的蛋白质动力学太快，以测量等待时间分布。 通过激光光镊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，例如蛋白质或核酸的解折叠或复合物的解离。在这些实验中，分子系统可以远离平衡，允许探索亚稳态和罕见的分子过程。. 在2001年，我们展示了如何重建分子的自由能表面延长Jarzynski等式从一个关系的系统自由能依赖于一个实验可控的参数的关系的分子自由能表面依赖于一个波动的坐标。 在3中，我们表明，未扰动的自由能分布作为分子延伸的函数，可以严格地从这样的实验中获得，而不使用我们以前介绍的功加权位置直方图。一个逆Weierstrass变换是用来直接相关的系统自由能的基本分子自由能表面。利用最速下降法求逆变换，得到了自由能曲面的精确近似。的形式主义被施加到模拟的数据从RNA折叠的动力学模型，其中的动力学由连接器为主的折叠和展开的自由能表面之间的跳跃。 酶中的底物结合位点常常被掩埋。在许多情况下，它们可以通过狭窄的通道从大分子表面进入。为了到达活性位点，配体必须扩散进入并穿过通向该位点的缝隙。一个典型的例子是乙酰胆碱酯酶。跨膜通道也可以具有由可以共价结合渗透离子或分子的孔衬残基产生的结合位点。 在4中，我们考虑了扩散影响的结合到一个掩埋的结合位点，该结合位点通过一个狭窄的隧道连接到表面。在单一的假设下的平衡分布的配位体的隧道横截面，我们减少了计算的时间依赖性的速率系数的一维扩散方程的解决方案与适当的边界条件。我们得到一个简单的解析表达式的稳态速率，取决于潜在的隧道中的平均力和扩散控制的隧道入口结合率。我们的理论的潜在应用包括底物结合到一个掩埋的活性位点的酶和渗透离子结合到一个跨膜通道的内部网站。 当时间尺度分离时，通过绝热消去快变量，可以得到慢变量动力学的有效描述。例如，对于二维的各向异性朗之万动力学，传统的程序导致了一个朗之万方程的慢坐标，涉及潜在的平均力。沿该坐标的摩擦常数沿着保持不变。在5中，我们表明，一个更准确的，但仍然马尔可夫，描述的缓慢的动力学可以通过使用位置相关的摩擦，这是相关的时间积分的自相关函数之间的差异，实际的力量和平均力的柯克伍德公式。该结果被推广到许多维度，其中慢或反应坐标是笛卡尔坐标的任意函数。当快变量是有效的一维时，沿慢坐标沿着的附加摩擦力可以用任意势的封闭形式表示。对于具有缠绕中心线的变化横截面的圆柱对称通道，我们的分析表达式立即产生了位置相关扩散系数的Zwanzig-Bradley公式的多维版本。

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.