Numerical Methods for Fluctuating Motion of Interface

界面波动在许多物理和生物系统中是常见的。理解这种波动背后的原理具有深远的科学和技术后果。例如，在生长中的纳米级半导体材料的所谓分子束外延中，对界面波动的控制可以极大地提高此类材料的质量和功能，这些材料广泛用于高科技电子和军事传感器设备。一些致命疾病的有效治疗关键依赖于我们对水-蛋白质界面异常结构的了解，这种结构是由波动和生物突变引起的，是此类疾病的特征。该项目开发了一个最先进的计算机程序来研究波动如何影响界面的结构和长期动态，并特别应用于药物分子与目标蛋白质的结合，这是计算机辅助药物设计中的关键步骤。因此，该项目的成功可能有助于降低实验室实验经常需要的高昂成本，并加快药物发现的进程。此外，这种高度跨学科的研究为不同水平的学生带来了独特的机会，特别是那些来自代表性不足群体的学生，可以接受数学和生物科学相结合的培训。这样的培训对于在竞争激烈的国际环境中保持我国的科研实力至关重要。在计算上跟踪波动界面的运动通常是相当具有挑战性的，因为这种运动涉及多个但相关的时空尺度、一个稳定的界面结构与另一个稳定界面结构之间的高能垒以及界面和整体过程的耦合。PI构建了两种方法来克服其中的一些困难。一种是随机水平集方法，它通过求解随机微分方程来描述起伏界面。方程中的噪声在空间上位于界面上或界面附近。严格的随机分析被用来重新表述这样的方程，以便进行准确和高效的计算。另一种是随机晶格相方法，它同时处理界面涨落和体涨落的耦合。这种方法通过在每个离散位置上分配一个二进制值来描述界面几何，并使用蒙特卡罗模拟方法最小化所有可能的离散二进制场的哈密顿量。这种哈密顿量模拟了具有空间梯度平方项和双势垒势的连续统哈密顿量。利用Gamma收敛的概念进行的数学分析揭示了数值网格尺寸和界面宽度之间的相互作用，并直接指导快速算法的设计。PI还开发了与GPU实现的并行计算算法，以加快它们的计算速度。他们将他们的新技术与分子溶剂化理论相结合，研究分子识别，特别是小药物分子与目标蛋白质的结合。在该项目中开发的计算模型、数值算法和计算机代码可以被合并到现有的软件中，这些软件每天被用于研究生物分子相互作用，特别是用于计算机辅助药物设计

Interfacial fluctuations are common in many physical and biological systems. Understanding the principles that underlie such fluctuations has far-reaching scientific and technological consequences. For instance, the manipulation of interfacial fluctuations in the so-called molecular beam epitaxy of growing nanometer-scale semiconductor materials can largely improve the quality and functionality of such materials that are widely used for high-technology electronic and military sensor devices. Effective treatment of some fatal diseases relies critically on our knowledge of anomalous water-protein interfacial structures that result from fluctuations and biological mutations and that characterize such diseases. This project develops a state-of-the-art computer program to investigate how the fluctuation affects the structures and long-time dynamics of interfaces, with a particular application to the binding of a drug molecule to a target protein that is a crucial step in the computer-aided drug design. The success of this project can therefore potentially help reduce the high cost often needed for laboratory experiments and speed up the process of drug discovery. In addition, this highly interdisciplinary research brings unique opportunities for students at different levels, particularly those from under represented groups, to receive training at the interface of mathematical and biological sciences. Such training is critical to keeping our nation's strength in scientific research in a highly competitive international environment. Computationally tracking the motion of fluctuating interface is in general rather challenging, as such motion involves multiple but correlated spatial and temporal scales, high energy barriers between one stable interfacial structure to another, and the coupling of interface and bulk processes. The PIs construct two methods to overcome some of these difficulties. One is the stochastic level-set method that describes the fluctuating interface by solving a stochastic differential equation. The noise in the equation is spatially localized on or near the interface. Rigorous stochastic analysis is carried out to reformulate such an equation for accurate and efficient computations. The other is a stochastic lattice-phase method that treats the coupling of both interfacial and bulk fluctuations. This method describes the interface geometry by assigning a binary value on each of the discrete sites, and minimizes a Hamiltonian of all possible discrete binary fields using a Monte Carlo simulation method. This Hamiltonian mimics the continuum one with spatial gradient-square term and a double-well potential. The mathematical analysis using the notion of Gamma-convergence reveals the interplay between the numerical grid size and the interfacial width, and directly guides the design of fast algorithms. The PIs also develop a parallel computational algorithm with the GPU implementation to speed up their computations. They combine their new techniques with a molecular solvation theory to study molecular recognition, particularly the binding of a small drug molecule to a target protein. The computational models, numerical algorithms, and computer codes developed in the project can be incorporated into existing software that are used on a daily basis to study biomolecular interactions, and in particular, for computer-aided drug design