Collaborative Research: A Regularized Poisson Boltzmann Model for Fast Computation of the Ensemble Average Polar Solvation Energy

该合作研究项目将开发数学模型和模拟工具，用于研究大型生物分子（例如蛋白质）与周围水分子之间的相互作用，并模拟水环境。表征这种相互作用的能量计算是复杂的，因为生物分子的结构不是完全固定或刚性的，而且周围的水分子也在不断运动。因此，为了提供与实验可测量能量相当的量，必须在相应的数学描述中考虑这些构象变化。该项目将结合适当的生物物理学考虑和数学进步，制定一个新的理论模型，允许模拟大分子和水原子构象变化的影响。提出的数学发展将有利于分子生物科学和生物物理学的研究人员。此外，所提出的模型和算法将在DelPhi中实现，该DelPhi免费分发给学术用户，以确保数学，化学，物理和生物领域的从业者广泛使用。此外，该项目将为本科生和研究生在生物建模、计算和数学分析方面提供跨学科的研究和培训机会。实验观测到的溶剂化能是系综平均。然而，这种能量的直接泊松-玻尔兹曼（PB）计算需要根据数千个快照生成具有代表性的结构系综，这在计算上是非常昂贵的。通过模拟非均质介电分布对大分子构象变化的影响，采用单一结构计算总体平均溶剂化能，可大大节省计算时间。在本项目中，将建立一个新的超高斯PB模型，其中包含三个关键创新：(1)在连续统静电偏微分方程（PDE）中纳入大分子的环境相关原子特征；(2)通过严格的数学分析开发了新的椭圆偏微分方程，开发了一种新的正则化公式来处理奇异电荷：部分电荷和水分子（在腔内，与蛋白质结合，以及在体溶剂中）将不再像目前的高斯模型那样被建模为均匀的空间区域，而是将反映整个溶质-溶剂系统的灵活性；(3)消除了需要确定一个尖锐的分子表面，对于大分子在真空和水环境。将进行模型基准测试和生物应用测试以进行验证，并将结果工具纳入广泛使用的DelPhi程序包中，用于计算整体平均溶剂化能。本项目由数学科学部数学生物学项目、化学部化学理论、模型和计算方法项目以及促进竞争性研究的既定计划（EPSCoR）共同支持。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This collaborative research project will develop mathematical models and simulation tools for studying the interactions between large biological molecules, for example proteins, and surrounding water molecules modeling an aqueous environment. The energy computation for characterizing such interactions is complex because the structures of biomolecules are not completely fixed or rigid, and the surrounding water molecules are also in constant motion. Thus, to deliver quantities that are comparable with experimentally-measurable energies, one must account for these conformational changes in the corresponding mathematical description. A new theoretical model will be formulated in this project by combining appropriate biophysical considerations with mathematical advances, allowing simulations to mimic the effect of conformational changes in both macromolecule and water atoms. The proposed mathematical development will benefit researchers in molecular biosciences and biophysics. Moreover, the proposed models and algorithms will be implemented in DelPhi, which is distributed free of charge to academic users, to ensure extensive usage by practitioners from mathematics, chemistry, physics, and biology. In addition, this project will provide interdisciplinary research and training opportunities for undergraduate and graduate students in biological modeling, computation and mathematical analysis.Experimentally-observable solvation energies are ensemble averaged. However, direct Poisson-Boltzmann (PB) calculations of such energies require the generation of a representative ensemble of structures in terms of thousands of snapshots, which is computationally very expensive. Tremendous savings in computational time can be achieved if one can calculate the ensemble average solvation energy by employing a single structure by mimicking the effect of conformation changes of macromolecules via heterogeneous dielectric distributions. In this project, a novel super-Gaussian PB model will be formulated embodying three key innovations: (1) incorporation of environment-dependent atomic characteristics of macromolecules within the continuum electrostatic partial differential equation (PDE); (2) development of a novel, regularized formulation to treat singular charges, with new elliptic PDEs developed through rigorous mathematical analysis: partial charges and water molecules (inside cavities, bonded to the protein, and in bulk solvent) will no longer be modeled as homogeneous spatial regions as in the current Gaussian model, but will reflect the flexibility of the entire solute-solvent system; and (3) elimination of the need to determine a sharp molecular surface, for macromolecules in both vacuum and water environments. Model benchmarking and biological applications tests will be carried out for validation, and the resulting tool will be incorporated into the widely-used DelPhi program package for computing ensemble-average solvation energies. This project is supported jointly by the Division of Mathematical Sciences Mathematical Biology Program, the Division of Chemistry Chemical Theory, Models and Computational Methods Program, and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.