AMC-SS: Theory and Modeling of Rare Events

AMC-SS：罕见事件的理论和建模

• 批准号: 0708140
• 负责人: Eric Vanden-Eijnden
• 金额: $ 36.67万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0708140&HistoricalAwards=false
• 关键词: AMC; SS; Theory; Modeling; Rare

The present proposal focuses on the study of rare events as they arise from real applications in computational chemistry, material sciences or molecular biology (for details on the latter, see the second paragraph below). Examples of such rare events include phenomena as diverse as nucleation events during phase transitions, chemical reactions, conformation changes of biomolecules, or bistable behaviors in genetic switches. The study of these rare events requires going beyond Freidlin-Wentzell theory of large deviations, which in mathematics has been the traditional tool to analyze rare events. It also requires integrating the theory within a computational perspective, which is necessary since traditional numerical tools such as Monte Carlo or direct simulation of SDEs are highly ineffective for rare events. Here the PI proposes to do so by (i) generalizing the theoretical framework of Transition Path Theory (TPT), which allows one to describe the statistical mechanics of rare events in situations when large deviation theory does not apply, and (ii) developing associated numerical algorithms such as the String Method for the effective computation of the various objects in TPT (like the probability density of reactive trajectories, their probability current and flux, and the rate of the reaction).The excitement of using molecular dynamics as a tool to do biology stems for the fact that it would enable us to understand the behavior of biomolecules such as protein, enzymes, ion channels, etc. from the "jiggling and wiggling" of the atoms they are made of (to quote Richard Feynman). This would allow for a much deeper understanding of their function and one that goes beyond what is currently achievable via experiments (in which it is hard or impossible to resolve the actual dynamics of the atoms). But this objective comes with tremendous challenges. While biomolecules are tiny objects from our perspective, they are also huge in that they are typically made of thousands of atoms (hundreds of thousands if one accounts for the molecules of solvent surrounding them). Since these atoms move very fast, the equations of motion governing their evolution must be integrated on the computer using a very small time step--typically of the order of one femtosec (1 femtosec = 1e-15 sec = 0.000000000000001 sec). Every such time step takes some time because it involves updating the position of so many atoms. As a result, it is only possible to simulate directly the motion of a typical biomolecule such as Hemoglobin over a few nanoseconds (1 nanosec = 1e-9 sec = 0.000000001 sec). Such a calculation already takes several days on a large computer. This, however, is a problem because the motion of these large molecules which governs their actual function only arise on a much slower time scale, typically of the order of the microseconds or even more (1 microsec = 1e-6 sec = 0.000001 sec). This is because such motion typically involves reactive events, e.g. large-scale reorganization of the shape of the molecule, which are very rare on the time scale of the molecule (though obviously, they are not on our own daily time scale). The direct simulation of any such reactive event would typically require years of computations, which is neither affordable nor practical. On the other hand, various techniques have been designed recently to bypass this difficulty and describe statistically (rather than on a one to one basis) the reactive events. This proposal is about developing these techniques, first at a theoretical level then by exploiting the theory to design efficient numerical algorithms for the computation of the reactive events which are so important in molecular biology.

本提案的重点是研究罕见事件，这些事件是由计算化学，材料科学或分子生物学中的实际应用引起的（有关后者的详细信息，请参见下面的第二段）。这种罕见事件的例子包括在相变，化学反应，生物分子的构象变化或遗传开关中可行为的构象变化中的各种现象。对这些罕见事件的研究需要超越弗雷德林·韦兹尔的大偏差理论，这在数学中一直是分析稀有事件的传统工具。它还需要在计算角度将理论整合在一起，这是必要的，因为传统的数值工具（例如Monte Carlo或直接模拟SDE）对于罕见事件而言非常无效。在这里，PI建议通过（i）概括过渡路径理论（TPT）的理论框架，该框架允许人们描述大偏差理论不适用罕见事件的统计机制，并且（ii）开发相关的相关性数值算法，例如用于有效计算TPT中各种对象的字符串方法（例如反应性轨迹的概率密度，它们的概率电流和磁通量以及反应速率）。使用分子动力学作为一种工具来兴奋生物学是因为它可以使我们能够理解诸如蛋白质，酶，离子通道等生物分子的行为。从它​​们制成的原子的“跳动和摇摆”（引用Richard Feynman）。 这将使人们对其功能有更深入的了解，并且超出了通过实验目前可以实现的功能（在这种实验中很难解决原子的实际动态）。 但是这个目标带来了巨大的挑战。尽管从我们的角度来看，生物分子是微小的物体，但它们也很大，因为它们通常是由数千原子制成的（如果有成千上万的原子是围绕它们的溶剂分子）。由于这些原子的移动非常快，因此管理其演变的运动方程必须使用非常小的时间步长将其集成在计算机上，这是一个the的顺序（1 fomtosec = 1e-15 sec = 0.00000000000000001 sec）。每这样的时间步骤都需要一些时间，因为它涉及更新许多原子的位置。结果，只能直接模拟典型的生物分子（例如血红蛋白）在几纳秒内的运动（1 nanosec = 1e-9 sec = 0.000000001 sec）。 这样的计算已经在一台大型计算机上需要几天。 但是，这是一个问题，因为这些大分子的运动控制它们的实际功能仅在时间尺度上出现，通常是微秒或更多的时间范围的时间（1 microSec = 1e-6 sec = 0.000001 sec） 。这是因为这种运动通常涉及反应性事件，例如分子形状的大规模重组，在分子的时间尺度上非常罕见（尽管显然，它们并不是我们自己的日常时间尺度）。对任何此类反应事件的直接模拟通常需要数年的计算，这既不是可负担也不是实际的。另一方面，最近已经设计了各种技术来绕过这一难度，并在统计上（而不是一对一地描述）反应性事件。该建议是关于开发这些技术的，首先是在理论层面上利用理论来设计有效的数值算法来计算反应性事件，这在分子生物学中非常重要。