Collaborative Research: Advances in Nonlocal Dielectric Modeling and Free Energy Calculation for Protein in Ionic Solvent

非局部介电方法考虑了水分子间的极化关系，对经典泊松介电模型有显著的增强作用。然而，目前对该方法的研究大多局限于水溶剂，因为在离子溶剂的情况下会出现建模和算法的复杂性。目前存在的离子模型没有考虑关键的非局部介电效应。最近的发展也表明，由于蛋白质-配体结合引起的熵变化对于理解结合亲和力至关重要。然而，熵的计算仍然是一项非常困难的任务。在这些挑战的激励下，本项目旨在开发新的非局部连续静电模型和新的数值正交模型，用于直接计算离子溶剂中蛋白质的熵和自由能。在约束泛函优化协议下，将非局部介电方法与硬球混合流体的基本测量理论相结合，构建新的非局部模型。与经典泊松-玻尔兹曼方程相比，它们有望显著提高静电势计算的准确性，因为它们既反映了离子大小效应，也反映了水分子之间的极化相关性。新的数值正交将通过使用一个特殊的棱柱元插值来开发，该插值由靠近势能最小点的有界状态区域的棱柱网格构造而成。采用一种新的非局部模型来计算涉及静电势，将进一步降低计算复杂度。最后，将开发新的快速数值算法和程序包来求解新的介电模型和实现新的数值正交。离子溶剂中蛋白质的静电势能、熵和自由能的计算是生物分子模拟的基本任务。新的非局部介电模型、计算熵和自由能的数值正交，以及由此项目产生的高效数值算法和程序包，将对数学生物学、计算生物化学、计算数学和计算机科学领域做出重大贡献。它们将在离子通道研究、合理药物设计和其他生物工程应用中发挥重要作用，并将提高我们对关键生理过程、细胞能量学、蛋白质-配体结合的亲和力以及健康和疾病的定量理解。该项目的研究结果预计将对数学、计算机科学、生物化学和生物工程的发展产生重大影响。由于最近人们对解决高维问题的算法产生了浓厚的兴趣，因此在该项目中取得的任何进展都将对与高维积分和指数衰减积分相关的各种领域产生广泛的潜在影响。

The nonlocal dielectric approach can significantly enhance the classic Poisson dielectric model by considering the polarization correlations among water molecules. However, current studies on the approach are mostly restricted to the water solvent, due to modeling and algorithmic complications that arise in the case of ionic solvents. The current ionic models that do exist fail to incorporate crucial nonlocal dielectric effects. Recent developments also indicate that entropic changes due to protein-ligand association are critical to understanding binding affinity. The computation of entropy, however, remains a very difficult task. Motivated by these challenges, this project aims to develop new nonlocal continuum electrostatic models and new numerical quadratures for the direct calculation of entropy and free energy for protein in ionic solvent. The new nonlocal models will be constructed from a novel combination of the nonlocal dielectric approach with the fundamental measure theory of hard-sphere mixture fluids under the constrained functional optimization protocol. They are expected to significantly improve the accuracy of electrostatic potential calculations in comparison to the classic Poisson-Boltzmann equation, since they reflect both ionic size effects and polarization correlations among water molecules. The new numerical quadratures will be developed by using a special prismatic element interpolation constructed from a prismatic mesh of a bounded state region near a potential energy minimum point. The computing complexity will be further reduced through using a new nonlocal model for computing involved electrostatic potential. Lastly, new fast numerical algorithms and program packages will be developed for solving the new dielectric models and for implementing the new numerical quadratures. Calculation of electrostatic potential energy, entropy, and free energy for protein in ionic solvent is a fundamental task in biomolecular simulations. The new nonlocal dielectric models, numerical quadratures for computing entropy and free energy, and the accompanying efficient numerical algorithms and program packages produced from this project will be a considerable contribution to the fields of mathematical biology, computational biochemistry, computational mathematics, and computer science. They will play important roles in ion channel studies, rational drug design, and other bioengineering applications, and will improve our quantitative understanding of critical physiological processes, cellular energetics, and affinity in protein-ligand binding, and of health and disease in general. The findings from this project are expected to have a significant impact on the development of mathematics, computer science, biochemistry, and bioengineering. Because there has been substantial interest recently in algorithms for solving high-dimensional problems, any advances made in this project will have broad potential impact in a variety of areas that are related to high-dimensional integrals with integrands that decay exponentially.