AMC-SS: Theory and Modeling of Rare Events

本提案侧重于研究计算化学、材料科学或分子生物学中的真实的应用所产生的罕见事件（关于后者的详细情况，见下文第二段）。这种罕见事件的例子包括相变过程中的成核事件、化学反应、生物分子的构象变化或遗传开关中的突变行为等多种现象。对这些罕见事件的研究需要超越Freidlin-Wentzell大偏差理论，这在数学中一直是分析罕见事件的传统工具。它还需要在计算的角度整合理论，这是必要的，因为传统的数值工具，如蒙特卡罗或直接模拟的SDES是非常无效的罕见事件。在此，PI建议通过（i）推广过渡路径理论（TPT）的理论框架来实现这一目标，该理论框架允许人们在大偏差理论不适用的情况下描述罕见事件的统计力学，以及（ii）开发相关的数值算法，如用于有效计算TPT中各种对象的String方法（如反应轨迹的概率密度，它们的概率电流和通量，以及反应的速率）。使用分子动力学作为生物学工具的兴奋源于这样一个事实，即它将使我们能够理解生物分子的行为，如蛋白质，酶，离子通道等，从“抖动和摆动”的原子，他们是由（引用理查德费曼）。 这将使人们能够更深入地了解它们的功能，并且超越目前通过实验所能实现的功能（在实验中很难或不可能解决原子的实际动力学）。 但这一目标也带来了巨大的挑战。虽然从我们的角度来看，生物分子是微小的物体，但它们也是巨大的，因为它们通常由数千个原子组成（如果考虑到它们周围的溶剂分子，则有数十万个原子）。由于这些原子的运动速度非常快，控制它们演化的运动方程必须在计算机上使用非常小的时间步长进行积分-通常为1飞秒（1飞秒= 1 e-15秒= 0.0000000000001秒）。每一个这样的时间步都需要一些时间，因为它涉及更新如此多的原子的位置。因此，只能直接模拟典型生物分子（如血红蛋白）在几纳秒内的运动（1纳秒= 1 e-9秒= 0.000000001秒）。 这样的计算在大型计算机上已经需要几天时间。 然而，这是一个问题，因为这些大分子的运动控制着它们的实际功能，只出现在一个慢得多的时间尺度上，通常是微秒或甚至更长的量级（1微秒= 1 e-6秒= 0.000001秒）。这是因为这种运动通常涉及反应事件，例如分子形状的大规模重组，这在分子的时间尺度上非常罕见（尽管很明显，它们不在我们自己的日常时间尺度上）。任何此类反应事件的直接模拟通常需要数年的计算，这既不经济也不实用。另一方面，最近已经设计了各种技术来绕过这一困难，并统计地描述（而不是一对一的基础上）的反应事件。这项建议是关于发展这些技术，首先在理论层面上，然后利用理论来设计有效的数值算法，用于计算在分子生物学中非常重要的反应事件。

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.