Collaborative Research: Laplacian-Centered Poisson Solvers and Multilevel Summation Algorithms

溶剂相互作用在决定生物分子体系的结构和功能中起着至关重要的作用。然而，由于长程关联和大量的溶剂原子和离子，精确的建模是一项具有挑战性的任务。对于非常大的系统，减少这一费用的一种方法是应用隐式溶剂模型，它用光电连续统来代替显式的原子和离子。一种较精确的隐式溶剂模型可用广义泊松方程或带点电荷源的泊松-玻尔兹曼方程来描述。虽然广义Poisson方法在许多情况下被认为是足够精确的，但大多数现有的近似其解的方法对于非常大的系统和需要快速重复计算的应用来说被认为是非常昂贵的。本项目的核心是创建和分析逼近广义Poisson方程(GPE)及其扩展的解的非常有效的算法。该方法涉及到求解GPE解的拉普拉斯所给出的“有效电荷分布”。相应的静电势可以用一个快速的N体求解器恢复。该项目的更广泛影响将通过在模拟软件中纳入新算法来实现。科学家手中掌握的更好的方法和软件将能够以更高的精度对更大的系统进行生物分子模拟，这将为社会做出贡献。此外，这个项目为毕业生提供了一个从事计算机科学、物理科学和数学研究的机会。

Solvent interactions play a critical role in determining the structure andfunction of biomolecular systems. However, accurate modeling is a challengingtask due to long-range correlations and vast numbers of solvent atoms and ions.For very large systems, one method for reducing this expense is the applicationof an implicit solvent model which replaces the explicit atoms and ions with adielectric continuum. One of the more accurate implicit solvent models isdescribed by a generalized Poisson equation or Poisson-Boltzmann equation withpoint charge source terms. Although the generalized Poisson approach isconsidered in very many cases to be sufficiently accurate, most current methodsfor approximating its solution have been deemed prohibitively expensive forvery large systems and for applications requiring rapid repetitive evaluation.This project centers on the creation and analysis of algorithms of unsurpassedeffectiveness for approximating the solution to the generalized Poissonequation (GPE) and its extensions. The approach involves solving for an"effective charge distribution" given by the Laplacian of the solutionto the GPE. The corresponding electrostatic potential can be recovered usinga fast N-body solver. Broader impact of this project will be realized throughthe incorporation of novel algorithms in simulation software. Better methodsand software in the hands of scientists will enable biomolecular simulation ofmuch larger systems with improved accuracy which will result in contributionsto society. In addition, this project provides an opportunity for a graduatestudent to engage in research that spans computer science, physical science,and mathematics.