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

非局域介电方法通过考虑水分子之间的极化关联，可以显著地改进经典的泊松介电模型。然而，目前对该方法的研究大多局限于水溶剂，这是由于在离子溶剂的情况下出现的建模和算法复杂性。目前确实存在的离子模型未能考虑关键的非局域介电效应。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是至关重要的。然而，熵的计算仍然是一项非常困难的任务。在这些挑战的推动下，本项目旨在开发新的非局部连续介质静电模型和新的数值积分，用于直接计算蛋白质在离子溶剂中的熵和自由能。新的非局域模型将由非局域介电方法和硬球混合流体的基本测量理论在约束函数优化协议下的新组合来构造。与经典的泊松-玻尔兹曼方程相比，它们将显著提高静电势计算的精度，因为它们既反映了离子尺寸效应，也反映了水分子之间的极化关联。新的数值积分将通过使用由势能极小点附近的有界态区域的棱柱面网格构造的特殊棱柱体单元来展开。通过使用一种新的非局部模型来计算所涉及的静电势，进一步降低了计算的复杂性。最后，将开发新的快速数值算法和程序包，用于求解新的介质模型和实现新的数值积分。计算蛋白质在离子溶剂中的静电势能、熵和自由能是生物分子模拟中的一项基本任务。由该项目产生的新的非局部介电模型、用于计算熵和自由能的数值求积法以及随之而来的高效的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、合理的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、细胞能量学和蛋白质与配体结合的亲和力以及一般健康和疾病的定量理解。这个项目的发现预计将对数学、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有很大的兴趣，所以在这个项目中取得的任何进展都将在许多领域产生广泛的潜在影响，这些领域与具有指数衰减的被积分项的高维积分有关。

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.