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方面起着至关重要的作用。然而，由于长期的相关性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，减少这种费用的一种方法是应用隐式溶剂模型，用介电连续体代替显式原子和离子。一个更精确的隐式溶剂模型是由广义泊松方程或泊松-玻尔兹曼方程与点电荷源项描述。尽管广义泊松方法在很多情况下被认为是足够精确的，但是对于非常大的系统和需要快速重复评估的应用来说，目前用于近似其解的大多数方法都被认为是过于昂贵的。这个项目的中心是创建和分析算法的无与伦比的有效性近似解的广义泊松方程（GPE）及其扩展。该方法涉及求解由GPE解的拉普拉斯算子给出的“有效电荷分布”。使用快速n体求解器可以恢复相应的静电势。通过在仿真软件中加入新颖的算法，将实现该项目更广泛的影响。科学家手中更好的方法和软件将使更大系统的生物分子模拟具有更高的准确性，从而为社会做出贡献。此外，这个项目为研究生提供了一个从事跨计算机科学、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.