Numerical Methods for Fluctuating Motion of Interface

University Of California-San Diego, La Jolla CA

Investigators

Abstract

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